Subgroups — C₂₃×#60824 #19208450

Order 5888.

2058 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3511	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3225	C₂	1	8	yes	maximal, complemented
4	C₂₃×#3210	C₂	1	2	yes	maximal, complemented
5	C₂₃×#3209	C₂	1	2	yes	maximal, complemented
6	C₂₃×#3207	C₂	1	2	yes	maximal, complemented
7	C₂₃×#3206	C₂	1	2	yes	maximal, complemented
8	C₂₃×#3152	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#3147	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#3141	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#3046	C₂	1	4	yes	maximal, complemented
13	C₂₃×#3013	C₂	1	4	yes	maximal, complemented
14	C₂₃×#2922	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₂₃×2₋¹⁺⁴	C₂²	1	2	yes	—
16	C₂×C₂₃×#137	C₂²	1	4	yes	—
17	C₂×C₂₃×#135	C₂²	1	4	yes	—
18	C₂₃×#556	—	2	1	no	—
19	C₂₃×#555	—	2	1	no	—
20	C₂₃×#554	—	2	1	no	—
21	C₂₃×#553	—	2	2	no	—
22	D₄×Q₈×C₂₃	—	2	1	no	—
23	D₄×Q₈×C₂₃	—	2	1	no	—
24	C₂₃×#548	—	2	2	no	—
25	C₂₃×#547	—	2	1	no	—
26	C₂₃×#547	—	2	2	no	—
27	C₂₃×#547	—	2	2	no	—
28	C₂₃×#547	—	2	1	no	—
29	C₂₃×#547	—	2	1	no	—
30	C₂₃×#542	C₂²	1	4	yes	—
31	C₂₃×#537	C₂²	1	2	yes	—
32	C₂₃×#536	C₂²	1	4	yes	—
33	C₂₃×#536	C₂²	1	2	yes	—
34	C₂₃×#533	C₂²	1	2	yes	—
35	C₂₃×#532	C₂²	1	2	yes	—
36	C₂₃×#532	C₂²	1	1	yes	characteristic
37	C₂₃×#532	C₂²	1	1	yes	characteristic
38	C₂₃×#532	C₂²	1	2	yes	—
39	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
40	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
41	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
42	C₂₃×#520	C₂²	1	2	yes	—
43	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
44	C₂₃×#497	C₂²	1	4	yes	—
45	C₂₃×#495	C₂²	1	4	yes	—
46	C₂₃×#494	C₂²	1	1	yes	characteristic
47	C₂₃×#494	C₂²	1	2	yes	—
48	C₂₃×#493	C₂²	1	1	yes	characteristic
49	C₂₃×#492	C₂²	1	1	yes	characteristic
50	C₂₃×#492	C₂²	1	2	yes	—
51	C₂₃×#492	C₂²	1	1	yes	characteristic
52	C₂₃×#489	C₂²	1	4	yes	—
53	C₂₃×#487	C₂²	1	2	yes	—
54	C₂₃×#487	C₂²	1	2	yes	—
55	C₂₃×#487	C₂²	1	2	yes	—
56	C₂₃×#487	C₂²	1	4	yes	—
57	C₂₃×#486	C₂²	1	2	yes	—
58	C₂₃×#464	C₂²	1	8	yes	—
59	C₂₃×#463	C₂²	1	8	yes	—
60	C₂₃×#462	C₂²	1	8	yes	—
61	C₂₃×#461	C₂²	1	8	yes	—
62	C₂₃×#452	C₂²	1	8	yes	—
63	C₂₃×#451	C₂²	1	8	yes	—
64	C₂₃×#449	C₂²	1	8	yes	—
65	C₂₃×#448	C₂²	1	8	yes	—
66	C₂₃×#433	C₂²	1	4	yes	—
67	C₂₃×#431	C₂²	1	2	yes	—
68	C₂₃×#431	C₂²	1	2	yes	—
69	C₂₃×#424	C₂²	1	2	yes	—
70	C₂₃×#417	C₂²	1	4	yes	—
71	C₂×C₂₃×#103	C₂²	1	4	yes	—
72	C₂₃×#408	C₂²	1	2	yes	—
73	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
74	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
75	C₂₃×2₋¹⁺⁴	—	4	8	no	—
76	C₂×#41×C₂₃	—	2	2	no	—
77	C₂×#41×C₂₃	—	2	4	no	—
78	C₂×#41×C₂₃	—	2	2	no	—
79	C₂×#41×C₂₃	C₂³	1	2	yes	—
80	C₂×#41×C₂₃	C₂³	1	2	yes	—
81	C₂×#41×C₂₃	C₂³	1	4	yes	—
82	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
83	C₂²×Q₈×C₂₃	—	2	2	no	—
84	C₂²×Q₈×C₂₃	—	2	1	no	—
85	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
86	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
87	C₂₃×#137	—	4	8	no	—
88	C₂₃×#135	—	4	8	no	—
89	C₂×Q₁₆×C₂₃	C₂³	1	4	yes	—
90	C₂×Q₁₆×C₂₃	C₂³	1	8	yes	—
91	C₂×SD₁₆×C₂₃	C₂³	1	4	yes	—
92	C₂×SD₁₆×C₂₃	C₂³	1	4	yes	—
93	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
94	C₂×D₈×C₂₃	C₂³	1	4	yes	—
95	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
96	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
97	C₂₃×C₄⋊Q₈	—	2	2	no	—
98	C₂₃×C₄⋊Q₈	—	2	1	no	—
99	C₂₃×C₄⋊Q₈	—	2	1	no	—
100	C₂₃×C₄⋊Q₈	—	2	2	no	—
101	C₂₃×C₄⋊Q₈	—	2	1	no	—
102	C₂₃×C₄⋊Q₈	—	2	1	no	—
103	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
104	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
105	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
106	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
107	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
108	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
109	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
110	C₂₃×#127	C₂³	1	1	yes	characteristic
111	C₂₃×#127	C₂³	1	1	yes	characteristic
112	C₂₃×#126	—	2	4	no	—
113	C₂₃×#125	—	2	2	no	—
114	C₂₃×#124	—	2	4	no	—
115	C₂₃×#124	—	2	2	no	—
116	C₂₃×#124	C₂³	1	4	yes	—
117	C₂₃×#124	C₂³	1	2	yes	—
118	C₂₃×#124	C₂³	1	2	yes	—
119	C₂₃×#123	—	2	4	no	—
120	C₂₃×#123	—	2	4	no	—
121	C₂₃×#123	—	2	4	no	—
122	C₂₃×#123	—	2	4	no	—
123	C₂₃×C₂²⋊Q₈	—	2	2	no	—
124	C₂₃×C₂²⋊Q₈	—	2	2	no	—
125	C₂₃×C₂²⋊Q₈	—	2	2	no	—
126	C₂₃×C₂²⋊Q₈	—	2	2	no	—
127	C₂₃×C₂²⋊Q₈	—	2	4	no	—
128	C₂₃×C₂²⋊Q₈	—	2	4	no	—
129	C₂₃×C₂²⋊Q₈	—	2	4	no	—
130	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
131	C₂₃×#121	—	2	2	no	—
132	C₂₃×#121	—	2	4	no	—
133	C₂₃×#121	—	2	1	no	—
134	C₂₃×#121	—	2	2	no	—
135	C₂₃×#121	—	2	1	no	—
136	C₂₃×#121	—	2	2	no	—
137	C₂₃×#121	—	2	1	no	—
138	C₂₃×#121	—	2	1	no	—
139	C₂₃×#121	—	2	2	no	—
140	C₂₃×#121	C₂³	1	4	yes	—
141	C₄×Q₈×C₂₃	—	2	2	no	—
142	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
143	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
144	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
145	C₄×D₄×C₂₃	—	2	4	no	—
146	C₄×D₄×C₂₃	—	2	2	no	—
147	C₄×D₄×C₂₃	—	2	1	no	—
148	C₄×D₄×C₂₃	—	2	1	no	—
149	C₄×D₄×C₂₃	—	2	2	no	—
150	C₄×D₄×C₂₃	—	2	1	no	—
151	C₄×D₄×C₂₃	—	2	2	no	—
152	C₄×D₄×C₂₃	—	2	1	no	—
153	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
154	C₄×D₄×C₂₃	D₄	1	2	yes	—
155	C₄×D₄×C₂₃	D₄	1	2	yes	—
156	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
157	C₄×D₄×C₂₃	C₂³	1	4	yes	—
158	C₂₃×#117	—	2	2	no	—
159	C₂₃×#117	—	2	2	no	—
160	C₂₃×#117	C₂³	1	1	yes	characteristic
161	C₂₃×#117	C₂³	1	2	yes	—
162	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
163	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
164	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
165	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
166	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
167	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
168	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
169	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
170	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
171	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
172	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
173	C₂₃×#103	C₂³	1	8	yes	—
174	C₂₃×#103	C₂³	1	8	yes	—
175	C₂₃×#103	C₂³	1	8	yes	—
176	C₂₃×#102	C₂³	1	8	yes	—
177	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
178	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
179	C₂₃×#97	C₂³	1	2	yes	—
180	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
181	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
182	C₄×C₈×C₂₃	C₂³	1	2	yes	—
183	C₄×C₈×C₂₃	C₂³	1	2	yes	—
184	#41×C₂₃	—	2	4	no	—
185	#41×C₂₃	—	2	8	no	—
186	#41×C₂₃	—	2	8	no	—
187	#41×C₂₃	—	4	8	no	—
188	#41×C₂₃	—	4	16	no	—
189	#41×C₂₃	—	4	8	no	—
190	#41×C₂₃	—	8	4	no	—
191	C₂×Q₈×C₂₃	—	2	2	no	—
192	C₂×Q₈×C₂₃	—	2	4	no	—
193	C₂×Q₈×C₂₃	—	2	2	no	—
194	C₂×Q₈×C₂₃	—	2	2	no	—
195	C₂×Q₈×C₂₃	—	2	8	no	—
196	C₂×Q₈×C₂₃	—	4	8	no	—
197	C₂×Q₈×C₂₃	—	4	8	no	—
198	C₂×Q₈×C₂₃	—	8	2	no	—
199	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
200	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
201	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
202	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
203	C₂×D₄×C₂₃	—	2	2	no	—
204	C₂×D₄×C₂₃	—	2	4	no	—
205	C₂×D₄×C₂₃	—	2	2	no	—
206	C₂×D₄×C₂₃	—	2	4	no	—
207	C₂×D₄×C₂₃	—	2	1	no	—
208	C₂×D₄×C₂₃	—	2	1	no	—
209	C₂×D₄×C₂₃	—	2	1	no	—
210	C₂×D₄×C₂₃	—	2	1	no	—
211	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
212	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
213	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
214	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
215	C₂²×C₄×C₂₃	—	2	4	no	—
216	C₂²×C₄×C₂₃	—	2	2	no	—
217	C₂²×C₄×C₂₃	—	2	2	no	—
218	C₂²×C₄×C₂₃	—	2	2	no	—
219	C₂²×C₄×C₂₃	—	2	4	no	—
220	C₂²×C₄×C₂₃	—	2	2	no	—
221	C₂²×C₄×C₂₃	—	2	4	no	—
222	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
223	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
224	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
225	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
226	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
227	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
228	Q₁₆×C₂₃	—	4	4	no	—
229	Q₁₆×C₂₃	—	4	8	no	—
230	SD₁₆×C₂₃	—	4	4	no	—
231	SD₁₆×C₂₃	—	4	4	no	—
232	SD₁₆×C₂₃	—	4	8	no	—
233	D₈×C₂₃	—	4	4	no	—
234	M₁₆×C₂₃	—	2	4	no	—
235	C₂×C₈×C₂₃	—	2	4	no	—
236	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
237	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
238	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
239	C₄⋊C₄×C₂₃	—	2	2	no	—
240	C₄⋊C₄×C₂₃	—	2	1	no	—
241	C₄⋊C₄×C₂₃	—	2	2	no	—
242	C₄⋊C₄×C₂₃	—	2	1	no	—
243	C₄⋊C₄×C₂₃	—	2	2	no	—
244	C₄⋊C₄×C₂₃	—	2	2	no	—
245	C₄⋊C₄×C₂₃	—	2	4	no	—
246	C₄⋊C₄×C₂₃	—	2	4	no	—
247	C₄⋊C₄×C₂₃	—	2	2	no	—
248	C₄⋊C₄×C₂₃	—	2	2	no	—
249	C₄⋊C₄×C₂₃	—	2	4	no	—
250	C₄⋊C₄×C₂₃	—	2	2	no	—
251	C₄⋊C₄×C₂₃	—	2	1	no	—
252	C₄⋊C₄×C₂₃	—	2	1	no	—
253	C₄⋊C₄×C₂₃	—	2	4	no	—
254	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
255	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
256	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
257	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
258	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
259	C₂²⋊C₄×C₂₃	—	2	4	no	—
260	C₂²⋊C₄×C₂₃	—	2	4	no	—
261	C₂²⋊C₄×C₂₃	—	2	4	no	—
262	C₂²⋊C₄×C₂₃	—	2	4	no	—
263	C₂²⋊C₄×C₂₃	—	2	2	no	—
264	C₂²⋊C₄×C₂₃	—	2	2	no	—
265	C₂²⋊C₄×C₂₃	—	2	4	no	—
266	C₂²⋊C₄×C₂₃	—	2	2	no	—
267	C₂²⋊C₄×C₂₃	—	2	2	no	—
268	C₂²⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
269	C₂²⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
270	C₂²⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
271	C₄²×C₂₃	—	2	4	no	—
272	C₄²×C₂₃	C₂×D₄	1	2	yes	—
273	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
274	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
275	C₄²×C₂₃	C₂×D₄	1	2	yes	—
276	C₄²×C₂₃	C₂⁴	1	2	yes	—
277	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
278	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
279	#60824	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
280	C₂³×C₂₃	—	2	2	no	—
281	C₂³×C₂₃	—	2	2	no	—
282	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
283	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
284	Q₈×C₂₃	—	2	2	no	—
285	Q₈×C₂₃	—	2	4	no	—
286	Q₈×C₂₃	—	2	8	no	—
287	Q₈×C₂₃	—	2	4	no	—
288	Q₈×C₂₃	—	4	8	no	—
289	Q₈×C₂₃	—	4	4	no	—
290	Q₈×C₂₃	—	4	4	no	—
291	Q₈×C₂₃	—	8	2	no	—
292	D₄×C₂₃	—	2	4	no	—
293	D₄×C₂₃	—	2	2	no	—
294	D₄×C₂₃	—	2	4	no	—
295	D₄×C₂₃	—	2	4	no	—
296	D₄×C₂₃	—	4	8	no	—
297	D₄×C₂₃	—	4	4	no	—
298	D₄×C₂₃	—	4	4	no	—
299	D₄×C₂₃	—	4	2	no	—
300	D₄×C₂₃	—	4	2	no	—
301	D₄×C₂₃	—	4	2	no	—
302	D₄×C₂₃	—	4	2	no	—
303	D₄×C₂₃	—	8	4	no	—
304	C₂×C₄×C₂₃	—	2	2	no	—
305	C₂×C₄×C₂₃	—	2	4	no	—
306	C₂×C₄×C₂₃	—	2	2	no	—
307	C₂×C₄×C₂₃	—	2	4	no	—
308	C₂×C₄×C₂₃	—	2	4	no	—
309	C₂×C₄×C₂₃	—	2	2	no	—
310	C₂×C₄×C₂₃	—	2	4	no	—
311	C₂×C₄×C₂₃	—	2	4	no	—
312	C₂×C₄×C₂₃	—	2	2	no	—
313	C₂×C₄×C₂₃	—	4	8	no	—
314	C₂×C₄×C₂₃	—	4	8	no	—
315	C₂×C₄×C₂₃	—	4	4	no	—
316	C₂×C₄×C₂₃	—	4	4	no	—
317	C₂×C₄×C₂₃	—	4	4	no	—
318	C₂×C₄×C₂₃	—	8	4	no	—
319	C₂×C₄×C₂₃	—	8	2	no	—
320	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
321	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
322	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
323	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
324	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
325	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
326	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
327	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
328	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
329	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
330	C₈×C₂₃	—	2	2	no	—
331	C₈×C₂₃	—	2	2	no	—
332	C₈×C₂₃	—	2	4	no	—
333	#3511	C₂×C₂₃	1	1	yes	characteristic, complemented
334	#3499	C₂×C₂₃	1	1	yes	characteristic, complemented
335	#3225	C₂×C₂₃	1	8	yes	complemented
336	#3210	C₂×C₂₃	1	2	yes	complemented
337	#3209	C₂×C₂₃	1	2	yes	complemented
338	#3207	C₂×C₂₃	1	2	yes	complemented
339	#3206	C₂×C₂₃	1	2	yes	complemented
340	#3152	C₂×C₂₃	1	1	yes	characteristic, complemented
341	#3147	C₂×C₂₃	1	1	yes	characteristic, complemented
342	#3141	C₂×C₂₃	1	1	yes	characteristic, complemented
343	C₂×#487	C₂×C₂₃	1	1	yes	characteristic, complemented
344	#3046	C₂×C₂₃	1	4	yes	complemented
345	#3013	C₂×C₂₃	1	4	yes	complemented
346	#2922	C₂×C₂₃	1	1	yes	characteristic, complemented
347	C₂²×C₂₃	—	2	1	no	—
348	C₂²×C₂₃	—	2	1	no	—
349	C₂²×C₂₃	—	2	2	no	—
350	C₂²×C₂₃	—	2	2	no	—
351	C₂²×C₂₃	—	2	2	no	—
352	C₂²×C₂₃	—	4	2	no	—
353	C₂²×C₂₃	—	4	2	no	—
354	C₂²×C₂₃	—	4	2	no	—
355	C₂²×C₂₃	—	4	2	no	—
356	C₂²×C₂₃	—	4	2	no	—
357	C₂²×C₂₃	—	4	2	no	—
358	C₂²×C₂₃	#535	1	2	yes	—
359	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
360	C₄×C₂₃	—	2	2	no	—
361	C₄×C₂₃	—	2	2	no	—
362	C₄×C₂₃	—	2	1	no	—
363	C₄×C₂₃	—	2	2	no	—
364	C₄×C₂₃	—	2	1	no	—
365	C₄×C₂₃	—	4	4	no	—
366	C₄×C₂₃	—	4	2	no	—
367	C₄×C₂₃	—	4	2	no	—
368	C₄×C₂₃	—	4	4	no	—
369	C₄×C₂₃	#535	1	2	yes	—
370	C₄×C₂₃	#577	1	2	yes	—
371	C₄×C₂₃	#578	1	2	yes	—
372	C₄×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
373	C₂×2₋¹⁺⁴	C₂²×C₂₃	1	2	yes	—
374	C₂×#137	C₂²×C₂₃	1	4	yes	—
375	C₂×#135	C₂²×C₂₃	1	4	yes	—
376	#556	—	2	1	no	—
377	#555	—	2	1	no	—
378	#554	—	2	1	no	—
379	#553	—	2	2	no	—
380	D₄×Q₈	—	2	1	no	—
381	D₄×Q₈	—	2	1	no	—
382	#548	—	2	2	no	—
383	#547	—	2	1	no	—
384	#547	—	2	2	no	—
385	#547	—	2	2	no	—
386	#547	—	2	1	no	—
387	#547	—	2	1	no	—
388	#542	C₂²×C₂₃	1	4	yes	—
389	#537	C₂²×C₂₃	1	2	yes	—
390	#536	C₂²×C₂₃	1	4	yes	—
391	#536	C₂²×C₂₃	1	2	yes	—
392	#533	C₂²×C₂₃	1	2	yes	—
393	#532	C₂²×C₂₃	1	1	yes	characteristic
394	#532	C₂²×C₂₃	1	1	yes	characteristic
395	#532	C₂²×C₂₃	1	2	yes	—
396	#532	C₂²×C₂₃	1	2	yes	—
397	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
398	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
399	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
400	#520	C₂²×C₂₃	1	2	yes	—
401	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
402	#497	C₂²×C₂₃	1	4	yes	—
403	#495	C₂²×C₂₃	1	4	yes	—
404	#494	C₂²×C₂₃	1	1	yes	characteristic
405	#494	C₂²×C₂₃	1	2	yes	—
406	#493	C₂²×C₂₃	1	1	yes	characteristic
407	#492	C₂²×C₂₃	1	1	yes	characteristic
408	#492	C₂²×C₂₃	1	1	yes	characteristic
409	#492	C₂²×C₂₃	1	2	yes	—
410	#489	C₂²×C₂₃	1	4	yes	—
411	#487	C₂²×C₂₃	1	2	yes	—
412	#487	C₂²×C₂₃	1	2	yes	—
413	#487	C₂²×C₂₃	1	4	yes	—
414	#487	C₂²×C₂₃	1	2	yes	—
415	#486	C₂²×C₂₃	1	2	yes	—
416	#464	C₂²×C₂₃	1	8	yes	—
417	#463	C₂²×C₂₃	1	8	yes	—
418	#462	C₂²×C₂₃	1	8	yes	—
419	#461	C₂²×C₂₃	1	8	yes	—
420	#452	C₂²×C₂₃	1	8	yes	—
421	#451	C₂²×C₂₃	1	8	yes	—
422	#449	C₂²×C₂₃	1	8	yes	—
423	#448	C₂²×C₂₃	1	8	yes	—
424	#433	C₂²×C₂₃	1	4	yes	—
425	#431	C₂²×C₂₃	1	2	yes	—
426	#431	C₂²×C₂₃	1	2	yes	—
427	#424	C₂²×C₂₃	1	2	yes	—
428	#417	C₂²×C₂₃	1	4	yes	—
429	C₂×#103	C₂²×C₂₃	1	4	yes	—
430	#408	C₂²×C₂₃	1	2	yes	—
431	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
432	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
433	C₂×C₂₃	—	2	2	no	—
434	C₂×C₂₃	—	4	2	no	—
435	C₂×C₂₃	—	8	2	no	—
436	C₂×C₂₃	—	8	2	no	—
437	C₂×C₂₃	C₂×#535	1	1	yes	characteristic
438	C₂×C₂₃	C₂×#577	1	1	yes	characteristic
439	C₂×C₂₃	C₂×#578	1	1	yes	characteristic
440	2₋¹⁺⁴	—	4	8	no	—
441	C₂×#41	—	2	2	no	—
442	C₂×#41	—	2	4	no	—
443	C₂×#41	—	2	2	no	—
444	C₂×#41	C₂³×C₂₃	1	2	yes	—
445	C₂×#41	C₂³×C₂₃	1	2	yes	—
446	C₂×#41	C₂³×C₂₃	1	4	yes	—
447	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
448	C₂²×Q₈	—	2	2	no	—
449	C₂²×Q₈	—	2	1	no	—
450	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
451	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
452	#137	—	4	8	no	—
453	#135	—	4	8	no	—
454	C₂×Q₁₆	C₂³×C₂₃	1	4	yes	—
455	C₂×Q₁₆	C₂³×C₂₃	1	8	yes	—
456	C₂×SD₁₆	C₂³×C₂₃	1	4	yes	—
457	C₂×SD₁₆	C₂³×C₂₃	1	4	yes	—
458	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
459	C₂×D₈	C₂³×C₂₃	1	4	yes	—
460	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
461	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
462	C₄⋊Q₈	—	2	2	no	—
463	C₄⋊Q₈	—	2	1	no	—
464	C₄⋊Q₈	—	2	1	no	—
465	C₄⋊Q₈	—	2	2	no	—
466	C₄⋊Q₈	—	2	1	no	—
467	C₄⋊Q₈	—	2	1	no	—
468	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
469	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
470	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
471	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
472	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
473	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
474	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
475	#127	C₂³×C₂₃	1	1	yes	characteristic
476	#127	C₂³×C₂₃	1	1	yes	characteristic
477	#126	—	2	4	no	—
478	#125	—	2	2	no	—
479	#124	—	2	4	no	—
480	#124	—	2	2	no	—
481	#124	C₂³×C₂₃	1	4	yes	—
482	#124	C₂³×C₂₃	1	2	yes	—
483	#124	C₂³×C₂₃	1	2	yes	—
484	#123	—	2	4	no	—
485	#123	—	2	4	no	—
486	#123	—	2	4	no	—
487	#123	—	2	4	no	—
488	C₂²⋊Q₈	—	2	2	no	—
489	C₂²⋊Q₈	—	2	2	no	—
490	C₂²⋊Q₈	—	2	2	no	—
491	C₂²⋊Q₈	—	2	2	no	—
492	C₂²⋊Q₈	—	2	4	no	—
493	C₂²⋊Q₈	—	2	4	no	—
494	C₂²⋊Q₈	—	2	4	no	—
495	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
496	#121	—	2	2	no	—
497	#121	—	2	4	no	—
498	#121	—	2	1	no	—
499	#121	—	2	2	no	—
500	#121	—	2	1	no	—
501	#121	—	2	2	no	—
502	#121	—	2	1	no	—
503	#121	—	2	1	no	—
504	#121	—	2	2	no	—
505	#121	C₂³×C₂₃	1	4	yes	—
506	C₄×Q₈	—	2	2	no	—
507	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
508	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
509	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
510	C₄×D₄	—	2	4	no	—
511	C₄×D₄	—	2	2	no	—
512	C₄×D₄	—	2	1	no	—
513	C₄×D₄	—	2	1	no	—
514	C₄×D₄	—	2	2	no	—
515	C₄×D₄	—	2	1	no	—
516	C₄×D₄	—	2	2	no	—
517	C₄×D₄	—	2	1	no	—
518	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
519	C₄×D₄	D₄×C₂₃	1	2	yes	—
520	C₄×D₄	D₄×C₂₃	1	2	yes	—
521	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
522	C₄×D₄	C₂³×C₂₃	1	4	yes	—
523	#117	—	2	2	no	—
524	#117	—	2	2	no	—
525	#117	C₂³×C₂₃	1	1	yes	characteristic
526	#117	C₂³×C₂₃	1	2	yes	—
527	C₂×C₄⋊C₄	—	2	1	no	—
528	C₂×C₄⋊C₄	—	2	2	no	—
529	C₂×C₄⋊C₄	—	2	1	no	—
530	C₂×C₄⋊C₄	—	2	2	no	—
531	C₂×C₄⋊C₄	—	2	1	no	—
532	C₂×C₄⋊C₄	—	2	2	no	—
533	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
534	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
535	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
536	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
537	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
538	#103	C₂³×C₂₃	1	8	yes	—
539	#103	C₂³×C₂₃	1	8	yes	—
540	#103	C₂³×C₂₃	1	8	yes	—
541	#102	C₂³×C₂₃	1	8	yes	—
542	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
543	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
544	#97	C₂³×C₂₃	1	2	yes	—
545	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
546	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
547	C₄×C₈	C₂³×C₂₃	1	2	yes	—
548	C₄×C₈	C₂³×C₂₃	1	2	yes	—
549	C₂₃	#60824	1	1	yes	characteristic, direct product factor, Sylow
550	#41	—	2	4	no	—
551	#41	—	2	8	no	—
552	#41	—	2	8	no	—
553	#41	—	4	8	no	—
554	#41	—	4	16	no	—
555	#41	—	4	8	no	—
556	#41	—	8	4	no	—
557	C₂×Q₈	—	2	2	no	—
558	C₂×Q₈	—	2	4	no	—
559	C₂×Q₈	—	2	2	no	—
560	C₂×Q₈	—	2	2	no	—
561	C₂×Q₈	—	2	8	no	—
562	C₂×Q₈	—	4	8	no	—
563	C₂×Q₈	—	4	8	no	—
564	C₂×Q₈	—	8	2	no	—
565	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
566	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
567	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
568	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
569	C₂×D₄	—	2	2	no	—
570	C₂×D₄	—	2	4	no	—
571	C₂×D₄	—	2	2	no	—
572	C₂×D₄	—	2	4	no	—
573	C₂×D₄	—	2	1	no	—
574	C₂×D₄	—	2	1	no	—
575	C₂×D₄	—	2	1	no	—
576	C₂×D₄	—	2	1	no	—
577	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
578	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
579	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
580	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
581	C₂²×C₄	—	2	4	no	—
582	C₂²×C₄	—	2	2	no	—
583	C₂²×C₄	—	2	2	no	—
584	C₂²×C₄	—	2	2	no	—
585	C₂²×C₄	—	2	4	no	—
586	C₂²×C₄	—	2	2	no	—
587	C₂²×C₄	—	2	4	no	—
588	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
589	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
590	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
591	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
592	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
593	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
594	Q₁₆	—	4	4	no	—
595	Q₁₆	—	4	8	no	—
596	SD₁₆	—	4	4	no	—
597	SD₁₆	—	4	4	no	—
598	SD₁₆	—	4	8	no	—
599	D₈	—	4	4	no	—
600	M₁₆	—	2	4	no	—
601	C₂×C₈	—	2	4	no	—
602	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
603	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
604	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
605	C₄⋊C₄	—	2	2	no	—
606	C₄⋊C₄	—	2	1	no	—
607	C₄⋊C₄	—	2	2	no	—
608	C₄⋊C₄	—	2	1	no	—
609	C₄⋊C₄	—	2	2	no	—
610	C₄⋊C₄	—	2	2	no	—
611	C₄⋊C₄	—	2	4	no	—
612	C₄⋊C₄	—	2	4	no	—
613	C₄⋊C₄	—	2	2	no	—
614	C₄⋊C₄	—	2	2	no	—
615	C₄⋊C₄	—	2	4	no	—
616	C₄⋊C₄	—	2	2	no	—
617	C₄⋊C₄	—	2	1	no	—
618	C₄⋊C₄	—	2	1	no	—
619	C₄⋊C₄	—	2	4	no	—
620	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
621	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
622	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
623	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
624	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
625	C₂²⋊C₄	—	2	4	no	—
626	C₂²⋊C₄	—	2	4	no	—
627	C₂²⋊C₄	—	2	4	no	—
628	C₂²⋊C₄	—	2	4	no	—
629	C₂²⋊C₄	—	2	2	no	—
630	C₂²⋊C₄	—	2	2	no	—
631	C₂²⋊C₄	—	2	4	no	—
632	C₂²⋊C₄	—	2	2	no	—
633	C₂²⋊C₄	—	2	2	no	—
634	C₂²⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
635	C₂²⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
636	C₂²⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
637	C₄²	—	2	4	no	—
638	C₄²	C₂×D₄×C₂₃	1	2	yes	—
639	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
640	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
641	C₄²	C₂×D₄×C₂₃	1	2	yes	—
642	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
643	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
644	C₄²	C₂⁴×C₂₃	1	2	yes	—
645	C₂³	—	2	2	no	—
646	C₂³	—	2	2	no	—
647	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
648	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
649	Q₈	—	2	2	no	—
650	Q₈	—	2	4	no	—
651	Q₈	—	2	8	no	—
652	Q₈	—	2	4	no	—
653	Q₈	—	4	8	no	—
654	Q₈	—	4	4	no	—
655	Q₈	—	4	4	no	—
656	Q₈	—	8	2	no	—
657	D₄	—	2	4	no	—
658	D₄	—	2	4	no	—
659	D₄	—	2	4	no	—
660	D₄	—	2	2	no	—
661	D₄	—	4	8	no	—
662	D₄	—	4	4	no	—
663	D₄	—	4	4	no	—
664	D₄	—	4	2	no	—
665	D₄	—	4	2	no	—
666	D₄	—	4	2	no	—
667	D₄	—	4	2	no	—
668	D₄	—	8	4	no	—
669	C₂×C₄	—	2	2	no	—
670	C₂×C₄	—	2	4	no	—
671	C₂×C₄	—	2	2	no	—
672	C₂×C₄	—	2	4	no	—
673	C₂×C₄	—	2	4	no	—
674	C₂×C₄	—	2	2	no	—
675	C₂×C₄	—	2	4	no	—
676	C₂×C₄	—	2	4	no	—
677	C₂×C₄	—	2	2	no	—
678	C₂×C₄	—	4	8	no	—
679	C₂×C₄	—	4	8	no	—
680	C₂×C₄	—	4	4	no	—
681	C₂×C₄	—	4	4	no	—
682	C₂×C₄	—	4	4	no	—
683	C₂×C₄	—	8	4	no	—
684	C₂×C₄	—	8	2	no	—
685	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
686	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
687	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
688	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
689	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
690	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
691	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
692	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
693	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
694	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
695	C₈	—	2	2	no	—
696	C₈	—	2	2	no	—
697	C₈	—	2	4	no	—
698	C₂²	—	2	1	no	—
699	C₂²	—	2	1	no	—
700	C₂²	—	2	2	no	—
701	C₂²	—	2	2	no	—
702	C₂²	—	2	2	no	—
703	C₂²	—	4	2	no	—
704	C₂²	—	4	2	no	—
705	C₂²	—	4	2	no	—
706	C₂²	—	4	2	no	—
707	C₂²	—	4	2	no	—
708	C₂²	—	4	2	no	—
709	C₂²	C₂₃×#535	1	2	yes	—
710	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
711	C₄	—	2	2	no	—
712	C₄	—	2	2	no	—
713	C₄	—	2	1	no	—
714	C₄	—	2	2	no	—
715	C₄	—	2	1	no	—
716	C₄	—	4	4	no	—
717	C₄	—	4	2	no	—
718	C₄	—	4	2	no	—
719	C₄	—	4	4	no	—
720	C₄	C₂₃×#535	1	2	yes	—
721	C₄	C₂₃×#577	1	2	yes	—
722	C₄	C₂₃×#578	1	2	yes	—
723	C₄	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
724	C₂	—	2	2	no	—
725	C₂	—	4	2	no	—
726	C₂	—	8	2	no	—
727	C₂	—	8	2	no	—
728	C₂	C₂×C₂₃×#535	1	1	yes	characteristic
729	C₂	C₂×C₂₃×#577	1	1	yes	characteristic
730	C₂	C₂×C₂₃×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3511	C₂	semidirect
2. C₂₃×#3499	C₂	semidirect
3. C₂₃×#3225	C₂	semidirect
4. C₂₃×#3210	C₂	semidirect
5. C₂₃×#3209	C₂	semidirect
6. C₂₃×#3207	C₂	semidirect
7. C₂₃×#3206	C₂	semidirect
8. C₂₃×#3152	C₂	semidirect
9. C₂₃×#3147	C₂	semidirect
10. C₂₃×#3141	C₂	semidirect
11. C₂×C₂₃×#487	C₂	semidirect
12. C₂₃×#3046	C₂	semidirect
13. C₂₃×#3013	C₂	semidirect
14. C₂₃×#2922	C₂	semidirect
279. #60824	C₂₃	direct (N × H)
333. #3511	C₂×C₂₃	semidirect
334. #3499	C₂×C₂₃	semidirect
335. #3225	C₂×C₂₃	semidirect
336. #3210	C₂×C₂₃	semidirect
337. #3209	C₂×C₂₃	semidirect
338. #3207	C₂×C₂₃	semidirect
339. #3206	C₂×C₂₃	semidirect
340. #3152	C₂×C₂₃	semidirect
341. #3147	C₂×C₂₃	semidirect
342. #3141	C₂×C₂₃	semidirect
343. C₂×#487	C₂×C₂₃	semidirect
344. #3046	C₂×C₂₃	semidirect
345. #3013	C₂×C₂₃	semidirect
346. #2922	C₂×C₂₃	semidirect
549. C₂₃	#60824	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3511	2
C₂₃×#3499	2
C₂₃×#3225	2
C₂₃×#3210	2
C₂₃×#3209	2
C₂₃×#3207	2
C₂₃×#3206	2
C₂₃×#3152	2
C₂₃×#3147	2
C₂₃×#3141	2
C₂×C₂₃×#487	2
C₂₃×#3046	2
C₂₃×#3013	2
C₂₃×#2922	2
#60824	23

Subgroups — C23×#60824 #19208450

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60824 #19208450