Subgroups — C₂₃×#61157 #19208783

Order 5888.

2088 conjugacy classes of subgroups.

The subgroup lattice has 918 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3475	C₂	1	1	yes	maximal, characteristic
2	C₂×D₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3329	C₂	1	2	yes	maximal, complemented
4	C₂₃×#3328	C₂	1	2	yes	maximal, complemented
5	C₂₃×#3320	C₂	1	4	yes	maximal, complemented
6	C₂₃×#3312	C₂	1	4	yes	maximal, complemented
7	C₂₃×#3299	C₂	1	2	yes	maximal, complemented
8	C₂₃×#3298	C₂	1	2	yes	maximal, complemented
9	C₂₃×#3139	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#3095	C₂	1	2	yes	maximal, complemented
11	C₂₃×#3078	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#3056	C₂	1	2	yes	maximal, complemented
13	C₂₃×#3039	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#3004	C₂	1	2	yes	maximal, complemented
15	C₄×C₂₃×#137	C₂	1	1	yes	maximal, characteristic, complemented
16	D₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
17	C₂×C₂₃×#419	C₂	1	2	yes	maximal, complemented
18	C₂³×Q₈×C₂₃	C₂²	1	2	yes	—
19	C₂×C₂₃×#137	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#137	C₂²	1	2	yes	—
21	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
22	C₂₃×#556	C₂²	1	1	yes	characteristic
23	C₂₃×#555	C₂²	1	1	yes	characteristic
24	C₂₃×#554	C₂²	1	1	yes	characteristic
25	C₂₃×#553	C₂²	1	2	yes	—
26	C₂₃×#552	—	2	1	no	—
27	C₂₃×#552	C₂²	1	2	yes	—
28	C₂₃×#551	—	2	2	no	—
29	D₄×Q₈×C₂₃	—	2	1	no	—
30	D₄×Q₈×C₂₃	—	2	2	no	—
31	D₄×Q₈×C₂₃	—	2	1	no	—
32	D₄×Q₈×C₂₃	C₂²	1	2	yes	—
33	D₄×Q₈×C₂₃	C₂²	1	4	yes	—
34	D₄×Q₈×C₂₃	C₂²	1	2	yes	—
35	C₂₃×#547	C₂²	1	1	yes	characteristic
36	C₂₃×#546	—	2	1	no	—
37	C₂₃×#533	C₂²	1	1	yes	characteristic
38	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
39	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
40	C₂₃×#529	—	2	2	no	—
41	C₂₃×#529	—	2	1	no	—
42	C₂₃×#525	C₂²	1	2	yes	—
43	C₂×C₂₃×C₂²⋊Q₈	—	2	2	no	—
44	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
45	C₄×#41×C₂₃	—	2	1	no	—
46	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
47	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
48	C₂×C₄×D₄×C₂₃	—	2	1	no	—
49	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
50	C₂×C₂₃×#117	C₂²	1	2	yes	—
51	C₂₃×#491	C₂²	1	4	yes	complemented
52	C₂₃×#487	C₂²	1	2	yes	complemented
53	C₂₃×#487	C₂²	1	2	yes	complemented
54	C₂₃×#485	C₂²	1	4	yes	complemented
55	C₂₃×#485	C₂²	1	4	yes	complemented
56	C₂₃×#484	C₂²	1	4	yes	complemented
57	C₂₃×#483	C₂²	1	4	yes	complemented
58	C₂₃×#479	C₂²	1	4	yes	complemented
59	C₂₃×#477	C₂²	1	2	yes	complemented
60	C₂₃×#475	C₂²	1	2	yes	complemented
61	C₂₃×#470	C₂²	1	4	yes	complemented
62	C₂₃×#468	C₂²	1	4	yes	complemented
63	C₂₃×#467	C₂²	1	4	yes	complemented
64	C₂₃×#465	C₂²	1	4	yes	complemented
65	C₂₃×#462	C₂²	1	2	yes	complemented
66	C₂₃×#462	C₂²	1	2	yes	complemented
67	C₂₃×#461	C₂²	1	2	yes	complemented
68	C₂₃×#461	C₂²	1	2	yes	complemented
69	C₂₃×#451	C₂²	1	4	yes	complemented
70	C₂₃×#451	C₂²	1	4	yes	complemented
71	C₂₃×#448	C₂²	1	4	yes	complemented
72	C₂₃×#448	C₂²	1	4	yes	complemented
73	C₂₃×#441	C₂²	1	2	yes	complemented
74	C₂₃×#440	C₂²	1	2	yes	complemented
75	C₄×Q₁₆×C₂₃	C₂²	1	2	yes	complemented
76	C₄×SD₁₆×C₂₃	C₂²	1	2	yes	complemented
77	C₂₃×#436	C₂²	1	2	yes	—
78	C₂₃×#435	C₂²	1	4	yes	—
79	C₈×D₄×C₂₃	C₂²	1	2	yes	—
80	C₂₃×#428	C₂²	1	1	yes	characteristic
81	C₂₃×#428	C₂²	1	2	yes	—
82	C₂₃×#423	C₂²	1	1	yes	characteristic
83	C₂₃×#419	C₂²	1	1	yes	characteristic
84	C₂₃×#419	C₂²	1	2	yes	—
85	C₂₃×#419	C₂²	1	2	yes	—
86	C₂₃×#419	C₂²	1	2	yes	—
87	C₂₃×#419	C₂²	1	2	yes	—
88	C₂₃×#419	C₂²	1	4	yes	complemented
89	C₂₃×#419	C₂²	1	4	yes	complemented
90	C₂₃×#417	C₂²	1	1	yes	characteristic
91	C₂×C₂₃×#103	C₂²	1	4	yes	complemented
92	C₂×C₂₃×#103	C₂²	1	4	yes	complemented
93	C₂₃×#407	C₂²	1	2	yes	—
94	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
95	C₂×#41×C₂₃	—	2	1	no	—
96	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
97	C₂²×Q₈×C₂₃	—	2	2	no	—
98	C₂²×Q₈×C₂₃	—	2	2	no	—
99	C₂²×Q₈×C₂₃	—	2	2	no	—
100	C₂²×Q₈×C₂₃	—	2	8	no	—
101	C₂²×Q₈×C₂₃	—	4	4	no	—
102	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
103	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
104	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
105	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
106	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
107	C₂³×C₄×C₂₃	—	2	2	no	—
108	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
109	C₂₃×#137	—	2	4	no	—
110	C₂₃×#137	—	4	4	no	—
111	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
112	C₂×Q₁₆×C₂₃	C₂³	1	4	yes	—
113	C₂×SD₁₆×C₂₃	C₂³	1	2	yes	—
114	C₂×SD₁₆×C₂₃	C₂³	1	4	yes	—
115	C₂×M₁₆×C₂₃	—	4	4	no	—
116	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
117	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
118	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
119	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
120	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
121	C₂₃×C₄⋊Q₈	—	2	2	no	—
122	C₂₃×C₄⋊Q₈	—	2	2	no	—
123	C₂₃×C₄⋊Q₈	—	2	4	no	—
124	C₂₃×C₄⋊Q₈	—	2	1	no	—
125	C₂₃×C₄⋊Q₈	—	2	1	no	—
126	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
127	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
128	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
129	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
130	C₂₃×#126	—	2	2	no	—
131	C₂₃×#126	—	2	2	no	—
132	C₂₃×#126	—	2	2	no	—
133	C₂₃×#125	—	2	2	no	—
134	C₂₃×#125	—	2	1	no	—
135	C₂₃×#125	C₂³	1	2	yes	—
136	C₂₃×#124	—	2	1	no	—
137	C₂₃×#124	—	2	2	no	—
138	C₂₃×#123	—	2	2	no	—
139	C₂₃×#123	—	2	2	no	—
140	C₂₃×#123	—	2	2	no	—
141	C₂₃×#123	—	2	2	no	—
142	C₂₃×C₂²⋊Q₈	—	2	2	no	—
143	C₂₃×C₂²⋊Q₈	—	2	2	no	—
144	C₂₃×C₂²⋊Q₈	—	2	1	no	—
145	C₂₃×C₂²⋊Q₈	—	2	1	no	—
146	C₂₃×C₂²⋊Q₈	—	2	1	no	—
147	C₂₃×C₂²⋊Q₈	—	2	4	no	—
148	C₂₃×C₂²⋊Q₈	—	2	4	no	—
149	C₂₃×C₂²⋊Q₈	—	2	4	no	—
150	C₂₃×C₂²⋊Q₈	—	2	4	no	—
151	C₂₃×C₂²⋊Q₈	—	2	2	no	—
152	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
153	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
154	C₂₃×C₂²⋊Q₈	C₂³	1	2	yes	—
155	C₂₃×C₂²⋊Q₈	C₂³	1	2	yes	—
156	C₂₃×#121	—	2	2	no	—
157	C₂₃×#121	—	2	1	no	—
158	C₂₃×#121	C₂³	1	2	yes	—
159	C₂₃×#121	C₂³	1	2	yes	—
160	C₂₃×#120	—	2	2	no	—
161	C₄×Q₈×C₂₃	—	2	1	no	—
162	C₄×Q₈×C₂₃	—	2	1	no	—
163	C₄×Q₈×C₂₃	—	2	1	no	—
164	C₄×Q₈×C₂₃	—	2	1	no	—
165	C₄×Q₈×C₂₃	—	2	1	no	—
166	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
167	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
168	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
169	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
170	C₄×D₄×C₂₃	—	2	4	no	—
171	C₄×D₄×C₂₃	—	2	2	no	—
172	C₄×D₄×C₂₃	—	2	2	no	—
173	C₄×D₄×C₂₃	—	2	1	no	—
174	C₄×D₄×C₂₃	—	2	1	no	—
175	C₄×D₄×C₂₃	—	2	1	no	—
176	C₄×D₄×C₂₃	—	2	2	no	—
177	C₄×D₄×C₂₃	—	2	2	no	—
178	C₄×D₄×C₂₃	—	2	2	no	—
179	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
180	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
181	C₄×D₄×C₂₃	D₄	1	2	yes	—
182	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
183	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
184	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
185	C₄×D₄×C₂₃	C₂³	1	2	yes	—
186	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
187	C₂₃×#117	—	2	2	no	—
188	C₂₃×#117	—	2	2	no	—
189	C₂₃×#117	—	2	1	no	—
190	C₂₃×#117	—	2	2	no	—
191	C₂₃×#117	—	2	1	no	—
192	C₂₃×#117	C₂³	1	2	yes	—
193	C₂₃×#117	C₂³	1	2	yes	—
194	C₂₃×#117	C₂³	1	1	yes	characteristic
195	C₂₃×#117	C₂³	1	2	yes	—
196	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
197	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
198	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
199	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
200	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
201	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
202	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
203	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
204	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
205	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
206	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
207	C₂×C₄²×C₂₃	—	2	2	no	—
208	C₂×C₄²×C₂₃	—	2	1	no	—
209	C₂×C₄²×C₂₃	—	2	1	no	—
210	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
211	C₂₃×#107	C₂³	1	2	yes	—
212	C₂₃×#107	C₂³	1	4	yes	—
213	C₂₃×#106	C₂³	1	2	yes	—
214	C₂₃×#106	C₂³	1	4	yes	—
215	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
216	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
217	C₂₃×#103	C₂³	1	2	yes	—
218	C₂₃×#103	C₂³	1	2	yes	—
219	C₂₃×#103	C₂³	1	4	yes	—
220	C₂₃×#103	C₂³	1	4	yes	—
221	C₂₃×#103	C₂³	1	4	yes	—
222	C₂₃×#103	C₂³	1	4	yes	—
223	C₂₃×#103	C₂³	1	4	yes	—
224	C₂₃×#103	C₂³	1	4	yes	—
225	C₂₃×#103	C₂³	1	2	yes	—
226	C₂₃×#103	C₂³	1	4	yes	—
227	C₂₃×#103	C₂³	1	4	yes	—
228	C₂₃×#102	C₂³	1	2	yes	—
229	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
230	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
231	C₂₃×#97	C₂³	1	2	yes	—
232	C₄×C₈×C₂₃	C₂³	1	2	yes	—
233	C₂⁴×C₂₃	C₂×D₄	1	2	yes	—
234	#41×C₂₃	—	2	2	no	—
235	#41×C₂₃	—	2	2	no	—
236	#41×C₂₃	—	4	4	no	—
237	C₂×Q₈×C₂₃	—	2	1	no	—
238	C₂×Q₈×C₂₃	—	2	1	no	—
239	C₂×Q₈×C₂₃	—	2	4	no	—
240	C₂×Q₈×C₂₃	—	2	4	no	—
241	C₂×Q₈×C₂₃	—	2	2	no	—
242	C₂×Q₈×C₂₃	—	2	1	no	—
243	C₂×Q₈×C₂₃	—	2	2	no	—
244	C₂×Q₈×C₂₃	—	2	8	no	—
245	C₂×Q₈×C₂₃	—	2	8	no	—
246	C₂×Q₈×C₂₃	—	2	4	no	—
247	C₂×Q₈×C₂₃	—	2	4	no	—
248	C₂×Q₈×C₂₃	—	4	8	no	—
249	C₂×Q₈×C₂₃	—	4	4	no	—
250	C₂×Q₈×C₂₃	—	4	4	no	—
251	C₂×Q₈×C₂₃	—	4	8	no	—
252	C₂×Q₈×C₂₃	—	4	8	no	—
253	C₂×Q₈×C₂₃	#41	1	4	yes	complemented
254	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
255	C₂×Q₈×C₂₃	C₂⁴	1	1	yes	characteristic
256	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
257	C₂×D₄×C₂₃	—	2	4	no	—
258	C₂×D₄×C₂₃	—	2	1	no	—
259	C₂×D₄×C₂₃	—	2	2	no	—
260	C₂×D₄×C₂₃	—	2	2	no	—
261	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
262	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
263	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
264	C₂×D₄×C₂₃	C₂⁴	1	1	yes	characteristic
265	C₂²×C₄×C₂₃	—	2	1	no	—
266	C₂²×C₄×C₂₃	—	2	4	no	—
267	C₂²×C₄×C₂₃	—	2	4	no	—
268	C₂²×C₄×C₂₃	—	2	1	no	—
269	C₂²×C₄×C₂₃	—	2	2	no	—
270	C₂²×C₄×C₂₃	—	2	2	no	—
271	C₂²×C₄×C₂₃	—	2	4	no	—
272	C₂²×C₄×C₂₃	—	2	4	no	—
273	C₂²×C₄×C₂₃	—	2	2	no	—
274	C₂²×C₄×C₂₃	—	2	1	no	—
275	C₂²×C₄×C₂₃	—	2	1	no	—
276	C₂²×C₄×C₂₃	—	2	1	no	—
277	C₂²×C₄×C₂₃	—	4	8	no	—
278	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
279	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
280	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
281	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
282	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
283	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
284	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
285	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
286	Q₁₆×C₂₃	—	2	4	no	—
287	Q₁₆×C₂₃	—	4	4	no	—
288	SD₁₆×C₂₃	—	2	4	no	—
289	SD₁₆×C₂₃	—	4	4	no	—
290	M₁₆×C₂₃	—	2	2	no	—
291	M₁₆×C₂₃	—	2	2	no	—
292	M₁₆×C₂₃	—	2	4	no	—
293	M₁₆×C₂₃	—	4	4	no	—
294	C₂×C₈×C₂₃	—	4	4	no	—
295	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
296	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
297	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
298	C₄⋊C₄×C₂₃	—	2	2	no	—
299	C₄⋊C₄×C₂₃	—	2	2	no	—
300	C₄⋊C₄×C₂₃	—	2	2	no	—
301	C₄⋊C₄×C₂₃	—	2	2	no	—
302	C₄⋊C₄×C₂₃	—	2	2	no	—
303	C₄⋊C₄×C₂₃	—	2	2	no	—
304	C₄⋊C₄×C₂₃	—	2	4	no	—
305	C₄⋊C₄×C₂₃	—	2	4	no	—
306	C₄⋊C₄×C₂₃	—	2	2	no	—
307	C₄⋊C₄×C₂₃	—	2	2	no	—
308	C₄⋊C₄×C₂₃	—	2	1	no	—
309	C₄⋊C₄×C₂₃	—	2	1	no	—
310	C₄⋊C₄×C₂₃	—	2	2	no	—
311	C₄⋊C₄×C₂₃	—	2	2	no	—
312	C₄⋊C₄×C₂₃	—	2	1	no	—
313	C₄⋊C₄×C₂₃	—	2	1	no	—
314	C₄⋊C₄×C₂₃	—	2	2	no	—
315	C₄⋊C₄×C₂₃	—	2	2	no	—
316	C₄⋊C₄×C₂₃	—	2	1	no	—
317	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
318	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
319	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
320	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
321	C₄⋊C₄×C₂₃	C₂⁴	1	1	yes	characteristic
322	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
323	C₄⋊C₄×C₂₃	C₂⁴	1	1	yes	characteristic
324	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
325	C₂²⋊C₄×C₂₃	—	2	1	no	—
326	C₂²⋊C₄×C₂₃	—	2	2	no	—
327	C₂²⋊C₄×C₂₃	—	2	2	no	—
328	C₂²⋊C₄×C₂₃	—	2	2	no	—
329	C₂²⋊C₄×C₂₃	—	2	4	no	—
330	C₂²⋊C₄×C₂₃	—	2	4	no	—
331	C₂²⋊C₄×C₂₃	—	2	2	no	—
332	C₂²⋊C₄×C₂₃	—	2	2	no	—
333	C₂²⋊C₄×C₂₃	—	2	2	no	—
334	C₂²⋊C₄×C₂₃	—	2	1	no	—
335	C₂²⋊C₄×C₂₃	—	2	1	no	—
336	C₂²⋊C₄×C₂₃	—	2	2	no	—
337	C₂²⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
338	C₂²⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
339	C₂²⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
340	C₂²⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
341	C₄²×C₂₃	—	2	1	no	—
342	C₄²×C₂₃	—	2	2	no	—
343	C₄²×C₂₃	—	2	1	no	—
344	C₄²×C₂₃	—	2	2	no	—
345	C₄²×C₂₃	—	2	2	no	—
346	C₄²×C₂₃	—	2	1	no	—
347	C₄²×C₂₃	—	2	2	no	—
348	C₄²×C₂₃	—	2	2	no	—
349	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
350	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
351	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
352	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
353	#61157	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
354	C₂³×C₂₃	—	2	4	no	—
355	C₂³×C₂₃	—	2	4	no	—
356	C₂³×C₂₃	—	2	1	no	—
357	C₂³×C₂₃	—	4	2	no	—
358	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
359	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
360	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
361	Q₈×C₂₃	—	2	4	no	—
362	Q₈×C₂₃	—	2	2	no	—
363	Q₈×C₂₃	—	2	1	no	—
364	Q₈×C₂₃	—	2	4	no	—
365	Q₈×C₂₃	—	2	4	no	—
366	Q₈×C₂₃	—	2	1	no	—
367	Q₈×C₂₃	—	4	8	no	—
368	Q₈×C₂₃	—	4	8	no	—
369	Q₈×C₂₃	—	4	4	no	—
370	Q₈×C₂₃	—	4	2	no	—
371	Q₈×C₂₃	—	8	1	no	—
372	Q₈×C₂₃	C₂×#41	1	4	yes	—
373	D₄×C₂₃	—	2	4	no	—
374	D₄×C₂₃	—	2	2	no	—
375	D₄×C₂₃	—	2	1	no	—
376	D₄×C₂₃	—	2	1	no	—
377	D₄×C₂₃	—	4	2	no	—
378	D₄×C₂₃	—	4	4	no	—
379	D₄×C₂₃	—	8	2	no	—
380	D₄×C₂₃	#137	1	4	yes	complemented
381	C₂×C₄×C₂₃	—	2	4	no	—
382	C₂×C₄×C₂₃	—	2	4	no	—
383	C₂×C₄×C₂₃	—	2	2	no	—
384	C₂×C₄×C₂₃	—	2	4	no	—
385	C₂×C₄×C₂₃	—	2	4	no	—
386	C₂×C₄×C₂₃	—	2	2	no	—
387	C₂×C₄×C₂₃	—	2	2	no	—
388	C₂×C₄×C₂₃	—	2	2	no	—
389	C₂×C₄×C₂₃	—	2	2	no	—
390	C₂×C₄×C₂₃	—	2	2	no	—
391	C₂×C₄×C₂₃	—	2	1	no	—
392	C₂×C₄×C₂₃	—	2	1	no	—
393	C₂×C₄×C₂₃	—	2	2	no	—
394	C₂×C₄×C₂₃	—	2	1	no	—
395	C₂×C₄×C₂₃	—	2	2	no	—
396	C₂×C₄×C₂₃	—	2	2	no	—
397	C₂×C₄×C₂₃	—	2	4	no	—
398	C₂×C₄×C₂₃	—	4	1	no	—
399	C₂×C₄×C₂₃	—	4	8	no	—
400	C₂×C₄×C₂₃	—	4	8	no	—
401	C₂×C₄×C₂₃	—	4	2	no	—
402	C₂×C₄×C₂₃	—	4	4	no	—
403	C₂×C₄×C₂₃	—	4	4	no	—
404	C₂×C₄×C₂₃	—	4	2	no	—
405	C₂×C₄×C₂₃	—	4	2	no	—
406	C₂×C₄×C₂₃	—	8	1	no	—
407	C₂×C₄×C₂₃	—	8	2	no	—
408	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
409	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
410	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
411	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
412	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
413	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
414	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
415	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
416	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
417	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
418	C₈×C₂₃	—	2	2	no	—
419	C₈×C₂₃	—	2	2	no	—
420	C₈×C₂₃	—	2	4	no	—
421	#3475	C₂×C₂₃	1	1	yes	characteristic
422	C₂×D₄×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
423	#3329	C₂×C₂₃	1	2	yes	complemented
424	#3328	C₂×C₂₃	1	2	yes	complemented
425	#3320	C₂×C₂₃	1	4	yes	complemented
426	#3312	C₂×C₂₃	1	4	yes	complemented
427	#3299	C₂×C₂₃	1	2	yes	complemented
428	#3298	C₂×C₂₃	1	2	yes	complemented
429	#3139	C₂×C₂₃	1	1	yes	characteristic, complemented
430	#3095	C₂×C₂₃	1	2	yes	complemented
431	#3078	C₂×C₂₃	1	1	yes	characteristic, complemented
432	#3056	C₂×C₂₃	1	2	yes	complemented
433	#3039	C₂×C₂₃	1	1	yes	characteristic, complemented
434	#3004	C₂×C₂₃	1	2	yes	complemented
435	C₄×#137	C₂×C₂₃	1	1	yes	characteristic, complemented
436	D₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
437	C₂×#419	C₂×C₂₃	1	2	yes	complemented
438	C₂²×C₂₃	—	2	1	no	—
439	C₂²×C₂₃	—	2	2	no	—
440	C₂²×C₂₃	—	2	2	no	—
441	C₂²×C₂₃	—	2	1	no	—
442	C₂²×C₂₃	—	2	2	no	—
443	C₂²×C₂₃	—	2	2	no	—
444	C₂²×C₂₃	—	2	2	no	—
445	C₂²×C₂₃	—	4	8	no	—
446	C₂²×C₂₃	—	4	1	no	—
447	C₂²×C₂₃	—	4	1	no	—
448	C₂²×C₂₃	—	4	1	no	—
449	C₂²×C₂₃	#546	1	2	yes	—
450	C₂²×C₂₃	C₂×#137	1	4	yes	—
451	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
452	C₄×C₂₃	—	2	1	no	—
453	C₄×C₂₃	—	2	1	no	—
454	C₄×C₂₃	—	2	2	no	—
455	C₄×C₂₃	—	2	2	no	—
456	C₄×C₂₃	—	2	4	no	—
457	C₄×C₂₃	—	2	1	no	—
458	C₄×C₂₃	—	4	2	no	—
459	C₄×C₂₃	—	4	1	no	—
460	C₄×C₂₃	—	4	2	no	—
461	C₄×C₂₃	—	4	4	no	—
462	C₄×C₂₃	—	4	1	no	—
463	C₄×C₂₃	—	4	1	no	—
464	C₄×C₂₃	—	4	2	no	—
465	C₄×C₂₃	#546	1	2	yes	—
466	C₄×C₂₃	C₂×#137	1	2	yes	—
467	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
468	C₄×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
469	C₂³×Q₈	C₂²×C₂₃	1	2	yes	—
470	C₂×#137	C₂²×C₂₃	1	1	yes	characteristic
471	C₂×#137	C₂²×C₂₃	1	2	yes	—
472	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
473	#556	C₂²×C₂₃	1	1	yes	characteristic
474	#555	C₂²×C₂₃	1	1	yes	characteristic
475	#554	C₂²×C₂₃	1	1	yes	characteristic
476	#553	C₂²×C₂₃	1	2	yes	—
477	#552	—	2	1	no	—
478	#552	C₂²×C₂₃	1	2	yes	—
479	#551	—	2	2	no	—
480	D₄×Q₈	—	2	1	no	—
481	D₄×Q₈	—	2	2	no	—
482	D₄×Q₈	—	2	1	no	—
483	D₄×Q₈	C₂²×C₂₃	1	2	yes	—
484	D₄×Q₈	C₂²×C₂₃	1	2	yes	—
485	D₄×Q₈	C₂²×C₂₃	1	4	yes	—
486	#547	C₂²×C₂₃	1	1	yes	characteristic
487	#546	—	2	1	no	—
488	#533	C₂²×C₂₃	1	1	yes	characteristic
489	C₂×C₄⋊Q₈	—	2	1	no	—
490	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
491	#529	—	2	2	no	—
492	#529	—	2	1	no	—
493	#525	C₂²×C₂₃	1	2	yes	—
494	C₂×C₂²⋊Q₈	—	2	2	no	—
495	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
496	C₄×#41	—	2	1	no	—
497	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
498	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
499	C₂×C₄×D₄	—	2	1	no	—
500	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
501	C₂×#117	C₂²×C₂₃	1	2	yes	—
502	#491	C₂²×C₂₃	1	4	yes	complemented
503	#487	C₂²×C₂₃	1	2	yes	complemented
504	#487	C₂²×C₂₃	1	2	yes	complemented
505	#485	C₂²×C₂₃	1	4	yes	complemented
506	#485	C₂²×C₂₃	1	4	yes	complemented
507	#484	C₂²×C₂₃	1	4	yes	complemented
508	#483	C₂²×C₂₃	1	4	yes	complemented
509	#479	C₂²×C₂₃	1	4	yes	complemented
510	#477	C₂²×C₂₃	1	2	yes	complemented
511	#475	C₂²×C₂₃	1	2	yes	complemented
512	#470	C₂²×C₂₃	1	4	yes	complemented
513	#468	C₂²×C₂₃	1	4	yes	complemented
514	#467	C₂²×C₂₃	1	4	yes	complemented
515	#465	C₂²×C₂₃	1	4	yes	complemented
516	#462	C₂²×C₂₃	1	2	yes	complemented
517	#462	C₂²×C₂₃	1	2	yes	complemented
518	#461	C₂²×C₂₃	1	2	yes	complemented
519	#461	C₂²×C₂₃	1	2	yes	complemented
520	#451	C₂²×C₂₃	1	4	yes	complemented
521	#451	C₂²×C₂₃	1	4	yes	complemented
522	#448	C₂²×C₂₃	1	4	yes	complemented
523	#448	C₂²×C₂₃	1	4	yes	complemented
524	#441	C₂²×C₂₃	1	2	yes	complemented
525	#440	C₂²×C₂₃	1	2	yes	complemented
526	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	complemented
527	C₄×SD₁₆	C₂²×C₂₃	1	2	yes	complemented
528	#436	C₂²×C₂₃	1	2	yes	—
529	#435	C₂²×C₂₃	1	4	yes	—
530	C₈×D₄	C₂²×C₂₃	1	2	yes	—
531	#428	C₂²×C₂₃	1	2	yes	—
532	#428	C₂²×C₂₃	1	1	yes	characteristic
533	#423	C₂²×C₂₃	1	1	yes	characteristic
534	#419	C₂²×C₂₃	1	2	yes	—
535	#419	C₂²×C₂₃	1	2	yes	—
536	#419	C₂²×C₂₃	1	2	yes	—
537	#419	C₂²×C₂₃	1	2	yes	—
538	#419	C₂²×C₂₃	1	1	yes	characteristic
539	#419	C₂²×C₂₃	1	4	yes	complemented
540	#419	C₂²×C₂₃	1	4	yes	complemented
541	#417	C₂²×C₂₃	1	1	yes	characteristic
542	C₂×#103	C₂²×C₂₃	1	4	yes	complemented
543	C₂×#103	C₂²×C₂₃	1	4	yes	complemented
544	#407	C₂²×C₂₃	1	2	yes	—
545	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
546	C₂×C₂₃	—	2	2	no	—
547	C₂×C₂₃	—	2	4	no	—
548	C₂×C₂₃	—	4	2	no	—
549	C₂×C₂₃	—	8	1	no	—
550	C₂×C₂₃	C₂×#546	1	1	yes	characteristic
551	C₂×C₂₃	C₂²×#137	1	1	yes	characteristic
552	C₂×C₂₃	#3586	1	1	yes	characteristic
553	C₂×#41	—	2	1	no	—
554	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
555	C₂²×Q₈	—	2	2	no	—
556	C₂²×Q₈	—	2	2	no	—
557	C₂²×Q₈	—	2	2	no	—
558	C₂²×Q₈	—	2	8	no	—
559	C₂²×Q₈	—	4	4	no	—
560	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
561	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
562	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
563	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
564	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
565	C₂³×C₄	—	2	2	no	—
566	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
567	#137	—	2	4	no	—
568	#137	—	4	4	no	—
569	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
570	C₂×Q₁₆	C₂³×C₂₃	1	4	yes	—
571	C₂×SD₁₆	C₂³×C₂₃	1	2	yes	—
572	C₂×SD₁₆	C₂³×C₂₃	1	4	yes	—
573	C₂×M₁₆	—	4	4	no	—
574	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
575	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
576	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
577	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
578	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
579	C₄⋊Q₈	—	2	2	no	—
580	C₄⋊Q₈	—	2	2	no	—
581	C₄⋊Q₈	—	2	4	no	—
582	C₄⋊Q₈	—	2	1	no	—
583	C₄⋊Q₈	—	2	1	no	—
584	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
585	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
586	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
587	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
588	#126	—	2	2	no	—
589	#126	—	2	2	no	—
590	#126	—	2	2	no	—
591	#125	—	2	2	no	—
592	#125	—	2	1	no	—
593	#125	C₂³×C₂₃	1	2	yes	—
594	#124	—	2	1	no	—
595	#124	—	2	2	no	—
596	#123	—	2	2	no	—
597	#123	—	2	2	no	—
598	#123	—	2	2	no	—
599	#123	—	2	2	no	—
600	C₂²⋊Q₈	—	2	2	no	—
601	C₂²⋊Q₈	—	2	2	no	—
602	C₂²⋊Q₈	—	2	1	no	—
603	C₂²⋊Q₈	—	2	1	no	—
604	C₂²⋊Q₈	—	2	1	no	—
605	C₂²⋊Q₈	—	2	4	no	—
606	C₂²⋊Q₈	—	2	4	no	—
607	C₂²⋊Q₈	—	2	4	no	—
608	C₂²⋊Q₈	—	2	4	no	—
609	C₂²⋊Q₈	—	2	2	no	—
610	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
611	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
612	C₂²⋊Q₈	C₂³×C₂₃	1	2	yes	—
613	C₂²⋊Q₈	C₂³×C₂₃	1	2	yes	—
614	#121	—	2	2	no	—
615	#121	—	2	1	no	—
616	#121	C₂³×C₂₃	1	2	yes	—
617	#121	C₂³×C₂₃	1	2	yes	—
618	#120	—	2	2	no	—
619	C₄×Q₈	—	2	1	no	—
620	C₄×Q₈	—	2	1	no	—
621	C₄×Q₈	—	2	1	no	—
622	C₄×Q₈	—	2	1	no	—
623	C₄×Q₈	—	2	1	no	—
624	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
625	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
626	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
627	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
628	C₄×D₄	—	2	4	no	—
629	C₄×D₄	—	2	2	no	—
630	C₄×D₄	—	2	2	no	—
631	C₄×D₄	—	2	1	no	—
632	C₄×D₄	—	2	2	no	—
633	C₄×D₄	—	2	1	no	—
634	C₄×D₄	—	2	1	no	—
635	C₄×D₄	—	2	2	no	—
636	C₄×D₄	—	2	2	no	—
637	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
638	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
639	C₄×D₄	D₄×C₂₃	1	2	yes	—
640	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
641	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
642	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
643	C₄×D₄	C₂³×C₂₃	1	2	yes	—
644	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
645	#117	—	2	2	no	—
646	#117	—	2	2	no	—
647	#117	—	2	1	no	—
648	#117	—	2	2	no	—
649	#117	—	2	1	no	—
650	#117	C₂³×C₂₃	1	1	yes	characteristic
651	#117	C₂³×C₂₃	1	2	yes	—
652	#117	C₂³×C₂₃	1	2	yes	—
653	#117	C₂³×C₂₃	1	2	yes	—
654	C₂×C₄⋊C₄	—	2	1	no	—
655	C₂×C₄⋊C₄	—	2	1	no	—
656	C₂×C₄⋊C₄	—	2	1	no	—
657	C₂×C₄⋊C₄	—	2	2	no	—
658	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
659	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
660	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
661	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
662	C₂×C₂²⋊C₄	—	2	2	no	—
663	C₂×C₂²⋊C₄	—	2	2	no	—
664	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
665	C₂×C₄²	—	2	2	no	—
666	C₂×C₄²	—	2	1	no	—
667	C₂×C₄²	—	2	1	no	—
668	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
669	#107	C₂³×C₂₃	1	2	yes	—
670	#107	C₂³×C₂₃	1	4	yes	—
671	#106	C₂³×C₂₃	1	2	yes	—
672	#106	C₂³×C₂₃	1	4	yes	—
673	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
674	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
675	#103	C₂³×C₂₃	1	2	yes	—
676	#103	C₂³×C₂₃	1	2	yes	—
677	#103	C₂³×C₂₃	1	4	yes	—
678	#103	C₂³×C₂₃	1	4	yes	—
679	#103	C₂³×C₂₃	1	4	yes	—
680	#103	C₂³×C₂₃	1	4	yes	—
681	#103	C₂³×C₂₃	1	4	yes	—
682	#103	C₂³×C₂₃	1	4	yes	—
683	#103	C₂³×C₂₃	1	2	yes	—
684	#103	C₂³×C₂₃	1	4	yes	—
685	#103	C₂³×C₂₃	1	4	yes	—
686	#102	C₂³×C₂₃	1	2	yes	—
687	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
688	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
689	#97	C₂³×C₂₃	1	2	yes	—
690	C₄×C₈	C₂³×C₂₃	1	2	yes	—
691	C₂₃	#61157	1	1	yes	characteristic, direct product factor, Sylow
692	C₂⁴	C₂×D₄×C₂₃	1	2	yes	—
693	#41	—	2	2	no	—
694	#41	—	2	2	no	—
695	#41	—	4	4	no	—
696	C₂×Q₈	—	2	1	no	—
697	C₂×Q₈	—	2	1	no	—
698	C₂×Q₈	—	2	4	no	—
699	C₂×Q₈	—	2	4	no	—
700	C₂×Q₈	—	2	2	no	—
701	C₂×Q₈	—	2	1	no	—
702	C₂×Q₈	—	2	2	no	—
703	C₂×Q₈	—	2	8	no	—
704	C₂×Q₈	—	2	8	no	—
705	C₂×Q₈	—	2	4	no	—
706	C₂×Q₈	—	2	4	no	—
707	C₂×Q₈	—	4	8	no	—
708	C₂×Q₈	—	4	4	no	—
709	C₂×Q₈	—	4	4	no	—
710	C₂×Q₈	—	4	8	no	—
711	C₂×Q₈	—	4	8	no	—
712	C₂×Q₈	#41×C₂₃	1	4	yes	complemented
713	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
714	C₂×Q₈	C₂⁴×C₂₃	1	1	yes	characteristic
715	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
716	C₂×D₄	—	2	4	no	—
717	C₂×D₄	—	2	1	no	—
718	C₂×D₄	—	2	2	no	—
719	C₂×D₄	—	2	2	no	—
720	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
721	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
722	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
723	C₂×D₄	C₂⁴×C₂₃	1	1	yes	characteristic
724	C₂²×C₄	—	2	1	no	—
725	C₂²×C₄	—	2	4	no	—
726	C₂²×C₄	—	2	4	no	—
727	C₂²×C₄	—	2	1	no	—
728	C₂²×C₄	—	2	2	no	—
729	C₂²×C₄	—	2	2	no	—
730	C₂²×C₄	—	2	4	no	—
731	C₂²×C₄	—	2	4	no	—
732	C₂²×C₄	—	2	2	no	—
733	C₂²×C₄	—	2	1	no	—
734	C₂²×C₄	—	2	1	no	—
735	C₂²×C₄	—	2	1	no	—
736	C₂²×C₄	—	4	8	no	—
737	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
738	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
739	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
740	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
741	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
742	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
743	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
744	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
745	Q₁₆	—	2	4	no	—
746	Q₁₆	—	4	4	no	—
747	SD₁₆	—	2	4	no	—
748	SD₁₆	—	4	4	no	—
749	M₁₆	—	2	2	no	—
750	M₁₆	—	2	2	no	—
751	M₁₆	—	2	4	no	—
752	M₁₆	—	4	4	no	—
753	C₂×C₈	—	4	4	no	—
754	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
755	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
756	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
757	C₄⋊C₄	—	2	2	no	—
758	C₄⋊C₄	—	2	2	no	—
759	C₄⋊C₄	—	2	2	no	—
760	C₄⋊C₄	—	2	2	no	—
761	C₄⋊C₄	—	2	2	no	—
762	C₄⋊C₄	—	2	2	no	—
763	C₄⋊C₄	—	2	4	no	—
764	C₄⋊C₄	—	2	4	no	—
765	C₄⋊C₄	—	2	2	no	—
766	C₄⋊C₄	—	2	2	no	—
767	C₄⋊C₄	—	2	1	no	—
768	C₄⋊C₄	—	2	1	no	—
769	C₄⋊C₄	—	2	2	no	—
770	C₄⋊C₄	—	2	2	no	—
771	C₄⋊C₄	—	2	1	no	—
772	C₄⋊C₄	—	2	1	no	—
773	C₄⋊C₄	—	2	2	no	—
774	C₄⋊C₄	—	2	2	no	—
775	C₄⋊C₄	—	2	1	no	—
776	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
777	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
778	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
779	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
780	C₄⋊C₄	C₂⁴×C₂₃	1	1	yes	characteristic
781	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
782	C₄⋊C₄	C₂⁴×C₂₃	1	1	yes	characteristic
783	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
784	C₂²⋊C₄	—	2	1	no	—
785	C₂²⋊C₄	—	2	2	no	—
786	C₂²⋊C₄	—	2	2	no	—
787	C₂²⋊C₄	—	2	2	no	—
788	C₂²⋊C₄	—	2	4	no	—
789	C₂²⋊C₄	—	2	4	no	—
790	C₂²⋊C₄	—	2	2	no	—
791	C₂²⋊C₄	—	2	2	no	—
792	C₂²⋊C₄	—	2	2	no	—
793	C₂²⋊C₄	—	2	1	no	—
794	C₂²⋊C₄	—	2	1	no	—
795	C₂²⋊C₄	—	2	2	no	—
796	C₂²⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
797	C₂²⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
798	C₂²⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
799	C₂²⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
800	C₄²	—	2	1	no	—
801	C₄²	—	2	2	no	—
802	C₄²	—	2	1	no	—
803	C₄²	—	2	2	no	—
804	C₄²	—	2	2	no	—
805	C₄²	—	2	1	no	—
806	C₄²	—	2	2	no	—
807	C₄²	—	2	2	no	—
808	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
809	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
810	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
811	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
812	C₂³	—	2	4	no	—
813	C₂³	—	2	4	no	—
814	C₂³	—	2	1	no	—
815	C₂³	—	4	2	no	—
816	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
817	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
818	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
819	Q₈	—	2	2	no	—
820	Q₈	—	2	4	no	—
821	Q₈	—	2	1	no	—
822	Q₈	—	2	4	no	—
823	Q₈	—	2	4	no	—
824	Q₈	—	2	1	no	—
825	Q₈	—	4	8	no	—
826	Q₈	—	4	8	no	—
827	Q₈	—	4	2	no	—
828	Q₈	—	4	4	no	—
829	Q₈	—	8	1	no	—
830	Q₈	C₂×#41×C₂₃	1	4	yes	—
831	D₄	—	2	4	no	—
832	D₄	—	2	2	no	—
833	D₄	—	2	1	no	—
834	D₄	—	2	1	no	—
835	D₄	—	4	4	no	—
836	D₄	—	4	2	no	—
837	D₄	—	8	2	no	—
838	D₄	C₂₃×#137	1	4	yes	complemented
839	C₂×C₄	—	2	2	no	—
840	C₂×C₄	—	2	4	no	—
841	C₂×C₄	—	2	4	no	—
842	C₂×C₄	—	2	2	no	—
843	C₂×C₄	—	2	2	no	—
844	C₂×C₄	—	2	2	no	—
845	C₂×C₄	—	2	2	no	—
846	C₂×C₄	—	2	2	no	—
847	C₂×C₄	—	2	4	no	—
848	C₂×C₄	—	2	4	no	—
849	C₂×C₄	—	2	1	no	—
850	C₂×C₄	—	2	1	no	—
851	C₂×C₄	—	2	2	no	—
852	C₂×C₄	—	2	1	no	—
853	C₂×C₄	—	2	2	no	—
854	C₂×C₄	—	2	2	no	—
855	C₂×C₄	—	2	4	no	—
856	C₂×C₄	—	4	1	no	—
857	C₂×C₄	—	4	8	no	—
858	C₂×C₄	—	4	8	no	—
859	C₂×C₄	—	4	2	no	—
860	C₂×C₄	—	4	4	no	—
861	C₂×C₄	—	4	4	no	—
862	C₂×C₄	—	4	2	no	—
863	C₂×C₄	—	4	2	no	—
864	C₂×C₄	—	8	1	no	—
865	C₂×C₄	—	8	2	no	—
866	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
867	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
868	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
869	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
870	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
871	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
872	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
873	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
874	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
875	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
876	C₈	—	2	2	no	—
877	C₈	—	2	2	no	—
878	C₈	—	2	4	no	—
879	C₂²	—	2	2	no	—
880	C₂²	—	2	1	no	—
881	C₂²	—	2	1	no	—
882	C₂²	—	2	2	no	—
883	C₂²	—	2	2	no	—
884	C₂²	—	2	2	no	—
885	C₂²	—	2	2	no	—
886	C₂²	—	4	8	no	—
887	C₂²	—	4	1	no	—
888	C₂²	—	4	1	no	—
889	C₂²	—	4	1	no	—
890	C₂²	C₂₃×#546	1	2	yes	—
891	C₂²	C₂×C₂₃×#137	1	4	yes	—
892	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
893	C₄	—	2	2	no	—
894	C₄	—	2	2	no	—
895	C₄	—	2	4	no	—
896	C₄	—	2	1	no	—
897	C₄	—	2	1	no	—
898	C₄	—	2	1	no	—
899	C₄	—	4	4	no	—
900	C₄	—	4	1	no	—
901	C₄	—	4	1	no	—
902	C₄	—	4	2	no	—
903	C₄	—	4	2	no	—
904	C₄	—	4	1	no	—
905	C₄	—	4	2	no	—
906	C₄	C₂₃×#546	1	2	yes	—
907	C₄	C₂×C₂₃×#137	1	2	yes	—
908	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
909	C₄	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
910	C₂	—	2	2	no	—
911	C₂	—	2	4	no	—
912	C₂	—	4	2	no	—
913	C₂	—	8	1	no	—
914	C₂	C₂×C₂₃×#546	1	1	yes	characteristic
915	C₂	C₂²×C₂₃×#137	1	1	yes	characteristic
916	C₂	C₂₃×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂₃×#3475	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄×D₄×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×D₄×Q₈×C₂₃	C₂	semidirect
3. C₂₃×#3329	C₂	semidirect
4. C₂₃×#3328	C₂	semidirect
5. C₂₃×#3320	C₂	semidirect
6. C₂₃×#3312	C₂	semidirect
7. C₂₃×#3299	C₂	semidirect
8. C₂₃×#3298	C₂	semidirect
9. C₂₃×#3139	C₂	semidirect
10. C₂₃×#3095	C₂	semidirect
11. C₂₃×#3078	C₂	semidirect
12. C₂₃×#3056	C₂	semidirect
13. C₂₃×#3039	C₂	semidirect
14. C₂₃×#3004	C₂	semidirect
15. C₄×C₂₃×#137	C₂	semidirect
16. D₄×M₁₆×C₂₃	C₂	semidirect
17. C₂×C₂₃×#419	C₂	semidirect
51. C₂₃×#491	C₂²	semidirect
52. C₂₃×#487	C₂²	semidirect
53. C₂₃×#487	C₂²	semidirect
54. C₂₃×#485	C₂²	semidirect
55. C₂₃×#485	C₂²	semidirect
56. C₂₃×#484	C₂²	semidirect
57. C₂₃×#483	C₂²	semidirect
58. C₂₃×#479	C₂²	semidirect
59. C₂₃×#477	C₂²	semidirect
60. C₂₃×#475	C₂²	semidirect
61. C₂₃×#470	C₂²	semidirect
62. C₂₃×#468	C₂²	semidirect
63. C₂₃×#467	C₂²	semidirect
64. C₂₃×#465	C₂²	semidirect
65. C₂₃×#462	C₂²	semidirect
66. C₂₃×#462	C₂²	semidirect
67. C₂₃×#461	C₂²	semidirect
68. C₂₃×#461	C₂²	semidirect
69. C₂₃×#451	C₂²	semidirect
70. C₂₃×#451	C₂²	semidirect
71. C₂₃×#448	C₂²	semidirect
72. C₂₃×#448	C₂²	semidirect
73. C₂₃×#441	C₂²	semidirect
74. C₂₃×#440	C₂²	semidirect
75. C₄×Q₁₆×C₂₃	C₂²	semidirect
76. C₄×SD₁₆×C₂₃	C₂²	semidirect
88. C₂₃×#419	C₂²	semidirect
89. C₂₃×#419	C₂²	semidirect
91. C₂×C₂₃×#103	C₂²	semidirect
92. C₂×C₂₃×#103	C₂²	semidirect
253. C₂×Q₈×C₂₃	#41	semidirect
353. #61157	C₂₃	direct (N × H)
380. D₄×C₂₃	#137	semidirect
422. C₂×D₄×Q₈	C₂×C₂₃	semidirect
423. #3329	C₂×C₂₃	semidirect
424. #3328	C₂×C₂₃	semidirect
425. #3320	C₂×C₂₃	semidirect
426. #3312	C₂×C₂₃	semidirect
427. #3299	C₂×C₂₃	semidirect
428. #3298	C₂×C₂₃	semidirect
429. #3139	C₂×C₂₃	semidirect
430. #3095	C₂×C₂₃	semidirect
431. #3078	C₂×C₂₃	semidirect
432. #3056	C₂×C₂₃	semidirect
433. #3039	C₂×C₂₃	semidirect
434. #3004	C₂×C₂₃	semidirect
435. C₄×#137	C₂×C₂₃	semidirect
436. D₄×M₁₆	C₂×C₂₃	semidirect
437. C₂×#419	C₂×C₂₃	semidirect
502. #491	C₂²×C₂₃	semidirect
503. #487	C₂²×C₂₃	semidirect
504. #487	C₂²×C₂₃	semidirect
505. #485	C₂²×C₂₃	semidirect
506. #485	C₂²×C₂₃	semidirect
507. #484	C₂²×C₂₃	semidirect
508. #483	C₂²×C₂₃	semidirect
509. #479	C₂²×C₂₃	semidirect
510. #477	C₂²×C₂₃	semidirect
511. #475	C₂²×C₂₃	semidirect
512. #470	C₂²×C₂₃	semidirect
513. #468	C₂²×C₂₃	semidirect
514. #467	C₂²×C₂₃	semidirect
515. #465	C₂²×C₂₃	semidirect
516. #462	C₂²×C₂₃	semidirect
517. #462	C₂²×C₂₃	semidirect
518. #461	C₂²×C₂₃	semidirect
519. #461	C₂²×C₂₃	semidirect
520. #451	C₂²×C₂₃	semidirect
521. #451	C₂²×C₂₃	semidirect
522. #448	C₂²×C₂₃	semidirect
523. #448	C₂²×C₂₃	semidirect
524. #441	C₂²×C₂₃	semidirect
525. #440	C₂²×C₂₃	semidirect
526. C₄×Q₁₆	C₂²×C₂₃	semidirect
527. C₄×SD₁₆	C₂²×C₂₃	semidirect
539. #419	C₂²×C₂₃	semidirect
540. #419	C₂²×C₂₃	semidirect
542. C₂×#103	C₂²×C₂₃	semidirect
543. C₂×#103	C₂²×C₂₃	semidirect
691. C₂₃	#61157	direct (N × H)
712. C₂×Q₈	#41×C₂₃	semidirect
838. D₄	C₂₃×#137	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂₃×#3475	2
C₂×D₄×Q₈×C₂₃	2
C₂₃×#3329	2
C₂₃×#3328	2
C₂₃×#3320	2
C₂₃×#3312	2
C₂₃×#3299	2
C₂₃×#3298	2
C₂₃×#3139	2
C₂₃×#3095	2
C₂₃×#3078	2
C₂₃×#3056	2
C₂₃×#3039	2
C₂₃×#3004	2
C₄×C₂₃×#137	2
D₄×M₁₆×C₂₃	2
C₂×C₂₃×#419	2
#61157	23

Subgroups — C23×#61157 #19208783

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#61157 #19208783