Subgroups — C₂₃×#61053 #19208679

Order 5888.

2666 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#578	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3486	C₂	1	2	yes	maximal, complemented
4	D₄×Q₁₆×C₂₃	C₂	1	4	yes	maximal, complemented
5	C₂₃×#3285	C₂	1	4	yes	maximal, complemented
6	C₂₃×#3270	C₂	1	8	yes	maximal, complemented
7	C₂₃×#3156	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#3059	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#3050	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#3044	C₂	1	2	yes	maximal, complemented
11	C₂₃×#3011	C₂	1	2	yes	maximal, complemented
12	C₂₃×#3004	C₂	1	2	yes	maximal, complemented
13	C₂₃×#2950	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#2929	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₂₃×2₋¹⁺⁴	C₂²	1	2	yes	—
16	C₂³×Q₈×C₂₃	C₂²	1	2	yes	—
17	C₂₃×#578	—	2	8	no	—
18	C₂×C₂₃×#137	C₂²	1	2	yes	—
19	C₂×C₂₃×#137	C₂²	1	2	yes	—
20	C₂×C₂₃×#137	C₂²	1	4	yes	complemented
21	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#135	C₂²	1	2	yes	complemented
23	C₂²×Q₁₆×C₂₃	—	2	8	no	—
24	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	—
25	C₂²×Q₁₆×C₂₃	C₂²	1	4	yes	—
26	C₂²×Q₁₆×C₂₃	C₂²	1	4	yes	—
27	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
28	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
29	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	complemented
30	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
31	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
32	D₄×Q₈×C₂₃	—	2	4	no	—
33	D₄×Q₈×C₂₃	C₂²	1	8	yes	—
34	C₂₃×#548	—	2	4	no	—
35	C₂₃×#548	C₂²	1	8	yes	—
36	C₂₃×#536	—	2	4	no	—
37	C₂₃×#536	C₂²	1	2	yes	—
38	C₂₃×#536	C₂²	1	2	yes	—
39	C₂₃×#525	—	2	4	no	—
40	C₂₃×#525	C₂²	1	2	yes	—
41	C₂₃×#525	C₂²	1	1	yes	characteristic
42	C₂₃×#519	C₂²	1	2	yes	—
43	C₂₃×#497	C₂²	1	4	yes	complemented
44	C₂₃×#495	C₂²	1	2	yes	complemented
45	C₂₃×#494	C₂²	1	2	yes	complemented
46	C₂₃×#470	C₂²	1	4	yes	complemented
47	C₂₃×#467	C₂²	1	2	yes	—
48	C₂₃×#467	C₂²	1	4	yes	complemented
49	C₂₃×#466	C₂²	1	2	yes	—
50	C₂₃×#465	C₂²	1	4	yes	—
51	C₂₃×#464	C₂²	1	8	yes	complemented
52	C₂₃×#462	C₂²	1	8	yes	complemented
53	C₂₃×#452	C₂²	1	8	yes	complemented
54	C₂₃×#451	C₂²	1	8	yes	—
55	C₂₃×#451	C₂²	1	8	yes	complemented
56	C₂₃×#448	C₂²	1	8	yes	—
57	C₂₃×#441	C₂²	1	4	yes	complemented
58	C₄×Q₁₆×C₂₃	C₂²	1	4	yes	complemented
59	C₂₃×#435	C₂²	1	4	yes	—
60	C₈×D₄×C₂₃	C₂²	1	4	yes	—
61	C₂₃×#427	C₂²	1	1	yes	characteristic
62	C₂₃×#424	C₂²	1	1	yes	characteristic
63	C₂₃×#419	C₂²	1	2	yes	—
64	C₂₃×#419	C₂²	1	2	yes	—
65	C₂₃×#416	C₂²	1	2	yes	—
66	C₂₃×#408	C₂²	1	1	yes	characteristic
67	C₂₃×#407	C₂²	1	1	yes	characteristic
68	C₂₃×#405	C₂²	1	1	yes	characteristic
69	C₂₃×2₋¹⁺⁴	—	2	8	no	—
70	C₂₃×2₋¹⁺⁴	—	4	4	no	—
71	C₂×#41×C₂₃	—	2	2	no	—
72	C₂×#41×C₂₃	—	2	4	no	—
73	C₂×#41×C₂₃	C₂³	1	2	yes	—
74	C₂×#41×C₂₃	C₂³	1	4	yes	—
75	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
76	C₂²×Q₈×C₂₃	—	2	2	no	—
77	C₂²×Q₈×C₂₃	—	2	4	no	—
78	C₂²×Q₈×C₂₃	—	2	2	no	—
79	C₂²×Q₈×C₂₃	—	2	8	no	—
80	C₂²×Q₈×C₂₃	—	4	4	no	—
81	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
82	C₂²×Q₈×C₂₃	C₂³	1	8	yes	—
83	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
84	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
85	C₂³×C₄×C₂₃	—	2	2	no	—
86	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
87	C₂₃×#137	—	2	8	no	—
88	C₂₃×#137	—	2	16	no	—
89	C₂₃×#137	—	4	4	no	—
90	C₂₃×#135	—	2	8	no	—
91	C₂₃×#135	—	2	4	no	—
92	C₂×Q₁₆×C₂₃	—	2	8	no	—
93	C₂×Q₁₆×C₂₃	—	2	16	no	—
94	C₂×Q₁₆×C₂₃	—	2	16	no	—
95	C₂×Q₁₆×C₂₃	—	2	8	no	—
96	C₂×Q₁₆×C₂₃	—	2	4	no	—
97	C₂×Q₁₆×C₂₃	D₄	1	8	yes	complemented
98	C₂×Q₁₆×C₂₃	C₂³	1	8	yes	—
99	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
100	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
101	C₂×Q₁₆×C₂₃	C₂³	1	1	yes	characteristic
102	C₂×Q₁₆×C₂₃	C₂³	1	4	yes	—
103	C₂×Q₁₆×C₂₃	C₂³	1	4	yes	—
104	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
105	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
106	C₂×SD₁₆×C₂₃	C₂³	1	2	yes	—
107	C₂×SD₁₆×C₂₃	C₂³	1	4	yes	—
108	C₂×SD₁₆×C₂₃	C₂³	1	4	yes	—
109	C₂×D₈×C₂₃	C₂³	1	1	yes	characteristic
110	C₂₃×#131	—	2	4	no	—
111	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
112	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
113	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
114	C₂²×C₈×C₂₃	—	2	4	no	—
115	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
116	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
117	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
118	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
119	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
120	C₂₃×C₄⋊Q₈	—	2	2	no	—
121	C₂₃×C₄⋊Q₈	—	2	8	no	—
122	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
123	C₂₃×#124	—	2	2	no	—
124	C₂₃×#124	—	2	8	no	—
125	C₂₃×#124	C₂³	1	4	yes	—
126	C₂₃×#123	—	2	4	no	—
127	C₂₃×#123	—	2	2	no	—
128	C₂₃×#123	—	2	4	no	—
129	C₂₃×#123	D₄	1	2	yes	—
130	C₂₃×C₂²⋊Q₈	—	2	4	no	—
131	C₂₃×C₂²⋊Q₈	—	2	8	no	—
132	C₂₃×C₂²⋊Q₈	—	2	8	no	—
133	C₂₃×C₂²⋊Q₈	—	2	8	no	—
134	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
135	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
136	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
137	C₂₃×#121	—	2	8	no	—
138	C₂₃×#121	D₄	1	2	yes	—
139	C₂₃×#121	C₂³	1	4	yes	—
140	C₂₃×#120	—	2	2	no	—
141	C₂₃×#120	D₄	1	2	yes	—
142	C₄×Q₈×C₂₃	—	2	4	no	—
143	C₄×Q₈×C₂₃	C₂³	1	8	yes	—
144	C₄×D₄×C₂₃	—	2	8	no	—
145	C₄×D₄×C₂₃	—	2	4	no	—
146	C₄×D₄×C₂₃	—	2	8	no	—
147	C₄×D₄×C₂₃	C₂³	1	4	yes	—
148	C₂₃×#117	—	2	4	no	—
149	C₂₃×#117	C₂³	1	2	yes	—
150	C₂₃×#117	C₂³	1	1	yes	characteristic
151	C₂₃×#107	C₂³	1	2	yes	—
152	C₂₃×#106	C₂³	1	2	yes	—
153	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
154	C₂₃×#103	C₂³	1	4	yes	—
155	C₂₃×#103	C₂³	1	8	yes	—
156	C₂₃×#103	C₂³	1	8	yes	—
157	C₂₃×#102	C₂³	1	4	yes	—
158	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
159	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
160	C₂₃×#97	C₂³	1	2	yes	—
161	C₄×C₈×C₂₃	C₂³	1	2	yes	—
162	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
163	#41×C₂₃	—	2	2	no	—
164	#41×C₂₃	—	2	2	no	—
165	#41×C₂₃	—	2	8	no	—
166	#41×C₂₃	—	2	4	no	—
167	#41×C₂₃	—	2	4	no	—
168	#41×C₂₃	—	4	4	no	—
169	#41×C₂₃	—	4	16	no	—
170	#41×C₂₃	—	4	8	no	—
171	C₂×Q₈×C₂₃	—	2	2	no	—
172	C₂×Q₈×C₂₃	—	2	8	no	—
173	C₂×Q₈×C₂₃	—	2	2	no	—
174	C₂×Q₈×C₂₃	—	2	4	no	—
175	C₂×Q₈×C₂₃	—	2	4	no	—
176	C₂×Q₈×C₂₃	—	2	16	no	—
177	C₂×Q₈×C₂₃	—	2	8	no	—
178	C₂×Q₈×C₂₃	—	2	4	no	—
179	C₂×Q₈×C₂₃	—	2	4	no	—
180	C₂×Q₈×C₂₃	—	4	8	no	—
181	C₂×Q₈×C₂₃	—	4	8	no	—
182	C₂×Q₈×C₂₃	—	4	4	no	—
183	C₂×Q₈×C₂₃	—	4	16	no	—
184	C₂×Q₈×C₂₃	—	4	8	no	—
185	C₂×Q₈×C₂₃	C₂×D₄	1	8	yes	—
186	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
187	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
188	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
189	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
190	C₂×D₄×C₂₃	—	2	4	no	—
191	C₂×D₄×C₂₃	—	2	2	no	—
192	C₂×D₄×C₂₃	—	2	4	no	—
193	C₂×D₄×C₂₃	C₂×D₄	1	4	yes	—
194	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
195	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
196	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
197	C₂²×C₄×C₂₃	—	2	2	no	—
198	C₂²×C₄×C₂₃	—	2	2	no	—
199	C₂²×C₄×C₂₃	—	2	2	no	—
200	C₂²×C₄×C₂₃	—	2	8	no	—
201	C₂²×C₄×C₂₃	—	2	4	no	—
202	C₂²×C₄×C₂₃	—	2	2	no	—
203	C₂²×C₄×C₂₃	—	2	4	no	—
204	C₂²×C₄×C₂₃	—	2	2	no	—
205	C₂²×C₄×C₂₃	—	4	8	no	—
206	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
207	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
208	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
209	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
210	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
211	Q₁₆×C₂₃	—	2	16	no	—
212	Q₁₆×C₂₃	—	2	8	no	—
213	Q₁₆×C₂₃	—	2	4	no	—
214	Q₁₆×C₂₃	—	2	4	no	—
215	Q₁₆×C₂₃	—	2	2	no	—
216	Q₁₆×C₂₃	—	4	4	no	—
217	Q₁₆×C₂₃	C₂×D₄	1	8	yes	—
218	Q₁₆×C₂₃	C₂×D₄	1	8	yes	—
219	SD₁₆×C₂₃	—	2	8	no	—
220	SD₁₆×C₂₃	—	2	4	no	—
221	SD₁₆×C₂₃	—	4	4	no	—
222	D₈×C₂₃	—	2	2	no	—
223	M₁₆×C₂₃	—	2	2	no	—
224	M₁₆×C₂₃	—	2	2	no	—
225	M₁₆×C₂₃	—	2	4	no	—
226	C₂×C₈×C₂₃	—	2	2	no	—
227	C₂×C₈×C₂₃	—	2	4	no	—
228	C₂×C₈×C₂₃	—	2	4	no	—
229	C₂×C₈×C₂₃	—	2	4	no	—
230	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
231	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
232	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
233	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
234	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
235	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
236	C₄⋊C₄×C₂₃	—	2	8	no	—
237	C₄⋊C₄×C₂₃	—	2	8	no	—
238	C₄⋊C₄×C₂₃	—	2	8	no	—
239	C₄⋊C₄×C₂₃	—	2	4	no	—
240	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
241	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
242	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
243	C₂²⋊C₄×C₂₃	—	2	2	no	—
244	C₂²⋊C₄×C₂₃	—	2	4	no	—
245	C₂²⋊C₄×C₂₃	—	2	8	no	—
246	C₂²⋊C₄×C₂₃	—	2	8	no	—
247	C₂²⋊C₄×C₂₃	—	2	4	no	—
248	C₂²⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
249	C₂²⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
250	C₂²⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
251	C₄²×C₂₃	—	2	8	no	—
252	C₄²×C₂₃	—	2	2	no	—
253	C₄²×C₂₃	C₂⁴	1	2	yes	—
254	#61053	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
255	C₂³×C₂₃	—	2	2	no	—
256	C₂³×C₂₃	—	2	4	no	—
257	C₂³×C₂₃	—	2	2	no	—
258	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
259	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
260	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
261	Q₈×C₂₃	—	2	8	no	—
262	Q₈×C₂₃	—	2	2	no	—
263	Q₈×C₂₃	—	2	4	no	—
264	Q₈×C₂₃	—	2	2	no	—
265	Q₈×C₂₃	—	2	4	no	—
266	Q₈×C₂₃	—	2	4	no	—
267	Q₈×C₂₃	—	2	2	no	—
268	Q₈×C₂₃	—	4	16	no	—
269	Q₈×C₂₃	—	4	8	no	—
270	Q₈×C₂₃	—	4	4	no	—
271	Q₈×C₂₃	C₂²×D₄	1	8	yes	—
272	D₄×C₂₃	—	2	4	no	—
273	D₄×C₂₃	—	2	8	no	—
274	D₄×C₂₃	—	2	2	no	—
275	D₄×C₂₃	—	2	2	no	—
276	D₄×C₂₃	—	2	2	no	—
277	D₄×C₂₃	—	4	8	no	—
278	D₄×C₂₃	—	4	4	no	—
279	D₄×C₂₃	—	4	8	no	—
280	C₂×C₄×C₂₃	—	2	2	no	—
281	C₂×C₄×C₂₃	—	2	8	no	—
282	C₂×C₄×C₂₃	—	2	4	no	—
283	C₂×C₄×C₂₃	—	2	4	no	—
284	C₂×C₄×C₂₃	—	2	2	no	—
285	C₂×C₄×C₂₃	—	2	2	no	—
286	C₂×C₄×C₂₃	—	2	2	no	—
287	C₂×C₄×C₂₃	—	2	2	no	—
288	C₂×C₄×C₂₃	—	2	4	no	—
289	C₂×C₄×C₂₃	—	2	4	no	—
290	C₂×C₄×C₂₃	—	2	2	no	—
291	C₂×C₄×C₂₃	—	4	8	no	—
292	C₂×C₄×C₂₃	—	4	16	no	—
293	C₂×C₄×C₂₃	—	4	8	no	—
294	C₂×C₄×C₂₃	—	4	4	no	—
295	C₂×C₄×C₂₃	—	4	4	no	—
296	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
297	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
298	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
299	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
300	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
301	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
302	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
303	C₈×C₂₃	—	2	2	no	—
304	C₈×C₂₃	—	2	2	no	—
305	C₈×C₂₃	—	2	2	no	—
306	C₈×C₂₃	C₂²×D₄	1	2	yes	—
307	C₈×C₂₃	C₂²×D₄	1	2	yes	—
308	C₂×#578	C₂×C₂₃	1	1	yes	characteristic, complemented
309	C₂³×Q₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
310	#3486	C₂×C₂₃	1	2	yes	complemented
311	D₄×Q₁₆	C₂×C₂₃	1	4	yes	complemented
312	#3285	C₂×C₂₃	1	4	yes	complemented
313	#3270	C₂×C₂₃	1	8	yes	complemented
314	#3156	C₂×C₂₃	1	1	yes	characteristic, complemented
315	#3059	C₂×C₂₃	1	1	yes	characteristic, complemented
316	#3050	C₂×C₂₃	1	1	yes	characteristic, complemented
317	#3044	C₂×C₂₃	1	2	yes	complemented
318	#3011	C₂×C₂₃	1	2	yes	complemented
319	#3004	C₂×C₂₃	1	2	yes	complemented
320	#2950	C₂×C₂₃	1	1	yes	characteristic, complemented
321	#2929	C₂×C₂₃	1	1	yes	characteristic, complemented
322	C₂²×C₂₃	—	2	8	no	—
323	C₂²×C₂₃	—	2	2	no	—
324	C₂²×C₂₃	—	2	2	no	—
325	C₂²×C₂₃	—	2	2	no	—
326	C₂²×C₂₃	—	2	2	no	—
327	C₂²×C₂₃	—	2	4	no	—
328	C₂²×C₂₃	—	4	2	no	—
329	C₂²×C₂₃	—	4	4	no	—
330	C₂²×C₂₃	D₄²	1	2	yes	—
331	C₂²×C₂₃	#578	1	4	yes	complemented
332	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
333	C₄×C₂₃	—	2	2	no	—
334	C₄×C₂₃	—	2	8	no	—
335	C₄×C₂₃	—	2	2	no	—
336	C₄×C₂₃	—	2	4	no	—
337	C₄×C₂₃	—	4	2	no	—
338	C₄×C₂₃	—	4	4	no	—
339	C₄×C₂₃	—	4	4	no	—
340	C₄×C₂₃	D₄²	1	2	yes	—
341	C₄×C₂₃	C₂³×D₄	1	1	yes	characteristic
342	C₄×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
343	C₂×2₋¹⁺⁴	C₂²×C₂₃	1	2	yes	—
344	C₂³×Q₈	C₂²×C₂₃	1	2	yes	—
345	#578	—	2	8	no	—
346	C₂×#137	C₂²×C₂₃	1	2	yes	—
347	C₂×#137	C₂²×C₂₃	1	2	yes	—
348	C₂×#137	C₂²×C₂₃	1	4	yes	complemented
349	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
350	C₂×#135	C₂²×C₂₃	1	2	yes	complemented
351	C₂²×Q₁₆	—	2	8	no	—
352	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
353	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	—
354	C₂²×Q₁₆	C₂²×C₂₃	1	4	yes	—
355	C₂²×Q₁₆	C₂²×C₂₃	1	4	yes	—
356	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
357	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	complemented
358	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
359	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
360	D₄×Q₈	—	2	4	no	—
361	D₄×Q₈	C₂²×C₂₃	1	8	yes	—
362	#548	—	2	4	no	—
363	#548	C₂²×C₂₃	1	8	yes	—
364	#536	—	2	4	no	—
365	#536	C₂²×C₂₃	1	2	yes	—
366	#536	C₂²×C₂₃	1	2	yes	—
367	#525	—	2	4	no	—
368	#525	C₂²×C₂₃	1	2	yes	—
369	#525	C₂²×C₂₃	1	1	yes	characteristic
370	#519	C₂²×C₂₃	1	2	yes	—
371	#497	C₂²×C₂₃	1	4	yes	complemented
372	#495	C₂²×C₂₃	1	2	yes	complemented
373	#494	C₂²×C₂₃	1	2	yes	complemented
374	#470	C₂²×C₂₃	1	4	yes	complemented
375	#467	C₂²×C₂₃	1	2	yes	—
376	#467	C₂²×C₂₃	1	4	yes	complemented
377	#466	C₂²×C₂₃	1	2	yes	—
378	#465	C₂²×C₂₃	1	4	yes	—
379	#464	C₂²×C₂₃	1	8	yes	complemented
380	#462	C₂²×C₂₃	1	8	yes	complemented
381	#452	C₂²×C₂₃	1	8	yes	complemented
382	#451	C₂²×C₂₃	1	8	yes	—
383	#451	C₂²×C₂₃	1	8	yes	complemented
384	#448	C₂²×C₂₃	1	8	yes	—
385	#441	C₂²×C₂₃	1	4	yes	complemented
386	C₄×Q₁₆	C₂²×C₂₃	1	4	yes	complemented
387	#435	C₂²×C₂₃	1	4	yes	—
388	C₈×D₄	C₂²×C₂₃	1	4	yes	—
389	#427	C₂²×C₂₃	1	1	yes	characteristic
390	#424	C₂²×C₂₃	1	1	yes	characteristic
391	#419	C₂²×C₂₃	1	2	yes	—
392	#419	C₂²×C₂₃	1	2	yes	—
393	#416	C₂²×C₂₃	1	2	yes	—
394	#408	C₂²×C₂₃	1	1	yes	characteristic
395	#407	C₂²×C₂₃	1	1	yes	characteristic
396	#405	C₂²×C₂₃	1	1	yes	characteristic
397	C₂×C₂₃	—	2	2	no	—
398	C₂×C₂₃	—	2	4	no	—
399	C₂×C₂₃	—	4	2	no	—
400	C₂×C₂₃	—	8	2	no	—
401	C₂×C₂₃	C₂×D₄²	1	1	yes	characteristic
402	C₂×C₂₃	C₂×#578	1	2	yes	—
403	2₋¹⁺⁴	—	2	8	no	—
404	2₋¹⁺⁴	—	4	4	no	—
405	C₂×#41	—	2	4	no	—
406	C₂×#41	—	2	2	no	—
407	C₂×#41	C₂³×C₂₃	1	2	yes	—
408	C₂×#41	C₂³×C₂₃	1	4	yes	—
409	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
410	C₂²×Q₈	—	2	2	no	—
411	C₂²×Q₈	—	2	2	no	—
412	C₂²×Q₈	—	2	4	no	—
413	C₂²×Q₈	—	2	8	no	—
414	C₂²×Q₈	—	4	4	no	—
415	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
416	C₂²×Q₈	C₂³×C₂₃	1	8	yes	—
417	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
418	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
419	C₂³×C₄	—	2	2	no	—
420	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
421	#137	—	2	8	no	—
422	#137	—	2	16	no	—
423	#137	—	4	4	no	—
424	#135	—	2	8	no	—
425	#135	—	2	4	no	—
426	C₂×Q₁₆	—	2	8	no	—
427	C₂×Q₁₆	—	2	16	no	—
428	C₂×Q₁₆	—	2	16	no	—
429	C₂×Q₁₆	—	2	8	no	—
430	C₂×Q₁₆	—	2	4	no	—
431	C₂×Q₁₆	D₄×C₂₃	1	8	yes	complemented
432	C₂×Q₁₆	C₂³×C₂₃	1	8	yes	—
433	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
434	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
435	C₂×Q₁₆	C₂³×C₂₃	1	1	yes	characteristic
436	C₂×Q₁₆	C₂³×C₂₃	1	4	yes	—
437	C₂×Q₁₆	C₂³×C₂₃	1	4	yes	—
438	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
439	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
440	C₂×SD₁₆	C₂³×C₂₃	1	2	yes	—
441	C₂×SD₁₆	C₂³×C₂₃	1	4	yes	—
442	C₂×SD₁₆	C₂³×C₂₃	1	4	yes	—
443	C₂×D₈	C₂³×C₂₃	1	1	yes	characteristic
444	#131	—	2	4	no	—
445	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
446	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
447	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
448	C₂²×C₈	—	2	4	no	—
449	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
450	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
451	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
452	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
453	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
454	C₄⋊Q₈	—	2	2	no	—
455	C₄⋊Q₈	—	2	8	no	—
456	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
457	#124	—	2	2	no	—
458	#124	—	2	8	no	—
459	#124	C₂³×C₂₃	1	4	yes	—
460	#123	—	2	4	no	—
461	#123	—	2	2	no	—
462	#123	—	2	4	no	—
463	#123	D₄×C₂₃	1	2	yes	—
464	C₂²⋊Q₈	—	2	8	no	—
465	C₂²⋊Q₈	—	2	4	no	—
466	C₂²⋊Q₈	—	2	8	no	—
467	C₂²⋊Q₈	—	2	8	no	—
468	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
469	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
470	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
471	#121	—	2	8	no	—
472	#121	D₄×C₂₃	1	2	yes	—
473	#121	C₂³×C₂₃	1	4	yes	—
474	#120	—	2	2	no	—
475	#120	D₄×C₂₃	1	2	yes	—
476	C₄×Q₈	—	2	4	no	—
477	C₄×Q₈	C₂³×C₂₃	1	8	yes	—
478	C₄×D₄	—	2	8	no	—
479	C₄×D₄	—	2	4	no	—
480	C₄×D₄	—	2	8	no	—
481	C₄×D₄	C₂³×C₂₃	1	4	yes	—
482	#117	—	2	4	no	—
483	#117	C₂³×C₂₃	1	2	yes	—
484	#117	C₂³×C₂₃	1	1	yes	characteristic
485	#107	C₂³×C₂₃	1	2	yes	—
486	#106	C₂³×C₂₃	1	2	yes	—
487	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
488	#103	C₂³×C₂₃	1	4	yes	—
489	#103	C₂³×C₂₃	1	8	yes	—
490	#103	C₂³×C₂₃	1	8	yes	—
491	#102	C₂³×C₂₃	1	4	yes	—
492	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
493	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
494	#97	C₂³×C₂₃	1	2	yes	—
495	C₄×C₈	C₂³×C₂₃	1	2	yes	—
496	C₂₃	#61053	1	1	yes	characteristic, direct product factor, Sylow
497	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
498	#41	—	2	2	no	—
499	#41	—	2	2	no	—
500	#41	—	2	8	no	—
501	#41	—	2	4	no	—
502	#41	—	2	4	no	—
503	#41	—	4	4	no	—
504	#41	—	4	16	no	—
505	#41	—	4	8	no	—
506	C₂×Q₈	—	2	2	no	—
507	C₂×Q₈	—	2	8	no	—
508	C₂×Q₈	—	2	2	no	—
509	C₂×Q₈	—	2	4	no	—
510	C₂×Q₈	—	2	4	no	—
511	C₂×Q₈	—	2	16	no	—
512	C₂×Q₈	—	2	8	no	—
513	C₂×Q₈	—	2	4	no	—
514	C₂×Q₈	—	2	4	no	—
515	C₂×Q₈	—	4	8	no	—
516	C₂×Q₈	—	4	8	no	—
517	C₂×Q₈	—	4	4	no	—
518	C₂×Q₈	—	4	16	no	—
519	C₂×Q₈	—	4	8	no	—
520	C₂×Q₈	C₂×D₄×C₂₃	1	8	yes	—
521	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
522	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
523	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
524	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
525	C₂×D₄	—	2	4	no	—
526	C₂×D₄	—	2	2	no	—
527	C₂×D₄	—	2	4	no	—
528	C₂×D₄	C₂×D₄×C₂₃	1	4	yes	—
529	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
530	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
531	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
532	C₂²×C₄	—	2	2	no	—
533	C₂²×C₄	—	2	2	no	—
534	C₂²×C₄	—	2	2	no	—
535	C₂²×C₄	—	2	8	no	—
536	C₂²×C₄	—	2	4	no	—
537	C₂²×C₄	—	2	2	no	—
538	C₂²×C₄	—	2	4	no	—
539	C₂²×C₄	—	2	2	no	—
540	C₂²×C₄	—	4	8	no	—
541	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
542	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
543	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
544	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
545	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
546	Q₁₆	—	2	16	no	—
547	Q₁₆	—	2	8	no	—
548	Q₁₆	—	2	4	no	—
549	Q₁₆	—	2	4	no	—
550	Q₁₆	—	2	2	no	—
551	Q₁₆	—	4	4	no	—
552	Q₁₆	C₂×D₄×C₂₃	1	8	yes	—
553	Q₁₆	C₂×D₄×C₂₃	1	8	yes	—
554	SD₁₆	—	2	8	no	—
555	SD₁₆	—	2	4	no	—
556	SD₁₆	—	4	4	no	—
557	D₈	—	2	2	no	—
558	M₁₆	—	2	2	no	—
559	M₁₆	—	2	2	no	—
560	M₁₆	—	2	4	no	—
561	C₂×C₈	—	2	2	no	—
562	C₂×C₈	—	2	4	no	—
563	C₂×C₈	—	2	4	no	—
564	C₂×C₈	—	2	4	no	—
565	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
566	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
567	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
568	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
569	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
570	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
571	C₄⋊C₄	—	2	8	no	—
572	C₄⋊C₄	—	2	8	no	—
573	C₄⋊C₄	—	2	8	no	—
574	C₄⋊C₄	—	2	4	no	—
575	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
576	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
577	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
578	C₂²⋊C₄	—	2	2	no	—
579	C₂²⋊C₄	—	2	4	no	—
580	C₂²⋊C₄	—	2	8	no	—
581	C₂²⋊C₄	—	2	8	no	—
582	C₂²⋊C₄	—	2	4	no	—
583	C₂²⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
584	C₂²⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
585	C₂²⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
586	C₄²	—	2	8	no	—
587	C₄²	—	2	2	no	—
588	C₄²	C₂⁴×C₂₃	1	2	yes	—
589	C₂³	—	2	4	no	—
590	C₂³	—	2	2	no	—
591	C₂³	—	2	2	no	—
592	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
593	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
594	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
595	Q₈	—	2	4	no	—
596	Q₈	—	2	8	no	—
597	Q₈	—	2	2	no	—
598	Q₈	—	2	2	no	—
599	Q₈	—	2	4	no	—
600	Q₈	—	2	4	no	—
601	Q₈	—	2	2	no	—
602	Q₈	—	4	16	no	—
603	Q₈	—	4	8	no	—
604	Q₈	—	4	4	no	—
605	Q₈	C₂²×D₄×C₂₃	1	8	yes	—
606	D₄	—	2	8	no	—
607	D₄	—	2	4	no	—
608	D₄	—	2	2	no	—
609	D₄	—	2	2	no	—
610	D₄	—	2	2	no	—
611	D₄	—	4	8	no	—
612	D₄	—	4	8	no	—
613	D₄	—	4	4	no	—
614	C₂×C₄	—	2	4	no	—
615	C₂×C₄	—	2	2	no	—
616	C₂×C₄	—	2	8	no	—
617	C₂×C₄	—	2	2	no	—
618	C₂×C₄	—	2	2	no	—
619	C₂×C₄	—	2	2	no	—
620	C₂×C₄	—	2	2	no	—
621	C₂×C₄	—	2	4	no	—
622	C₂×C₄	—	2	4	no	—
623	C₂×C₄	—	2	4	no	—
624	C₂×C₄	—	2	2	no	—
625	C₂×C₄	—	4	8	no	—
626	C₂×C₄	—	4	16	no	—
627	C₂×C₄	—	4	8	no	—
628	C₂×C₄	—	4	4	no	—
629	C₂×C₄	—	4	4	no	—
630	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
631	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
632	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
633	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
634	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
635	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
636	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
637	C₈	—	2	2	no	—
638	C₈	—	2	2	no	—
639	C₈	—	2	2	no	—
640	C₈	C₂²×D₄×C₂₃	1	2	yes	—
641	C₈	C₂²×D₄×C₂₃	1	2	yes	—
642	C₂²	—	2	8	no	—
643	C₂²	—	2	2	no	—
644	C₂²	—	2	2	no	—
645	C₂²	—	2	2	no	—
646	C₂²	—	2	2	no	—
647	C₂²	—	2	4	no	—
648	C₂²	—	4	2	no	—
649	C₂²	—	4	4	no	—
650	C₂²	D₄²×C₂₃	1	2	yes	—
651	C₂²	C₂₃×#578	1	4	yes	complemented
652	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
653	C₄	—	2	2	no	—
654	C₄	—	2	8	no	—
655	C₄	—	2	2	no	—
656	C₄	—	2	4	no	—
657	C₄	—	4	2	no	—
658	C₄	—	4	4	no	—
659	C₄	—	4	4	no	—
660	C₄	D₄²×C₂₃	1	2	yes	—
661	C₄	C₂³×D₄×C₂₃	1	1	yes	characteristic
662	C₄	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
663	C₂	—	2	4	no	—
664	C₂	—	2	2	no	—
665	C₂	—	4	2	no	—
666	C₂	—	8	2	no	—
667	C₂	C₂×D₄²×C₂₃	1	1	yes	characteristic
668	C₂	C₂×C₂₃×#578	1	2	yes	—

	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 ◁ C₂₃×#525 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#578	C₂	semidirect
2. C₂³×Q₁₆×C₂₃	C₂	semidirect
3. C₂₃×#3486	C₂	semidirect
4. D₄×Q₁₆×C₂₃	C₂	semidirect
5. C₂₃×#3285	C₂	semidirect
6. C₂₃×#3270	C₂	semidirect
7. C₂₃×#3156	C₂	semidirect
8. C₂₃×#3059	C₂	semidirect
9. C₂₃×#3050	C₂	semidirect
10. C₂₃×#3044	C₂	semidirect
11. C₂₃×#3011	C₂	semidirect
12. C₂₃×#3004	C₂	semidirect
13. C₂₃×#2950	C₂	semidirect
14. C₂₃×#2929	C₂	semidirect
20. C₂×C₂₃×#137	C₂²	semidirect
22. C₂×C₂₃×#135	C₂²	semidirect
29. C₂²×Q₁₆×C₂₃	C₂²	semidirect
43. C₂₃×#497	C₂²	semidirect
44. C₂₃×#495	C₂²	semidirect
45. C₂₃×#494	C₂²	semidirect
46. C₂₃×#470	C₂²	semidirect
48. C₂₃×#467	C₂²	semidirect
51. C₂₃×#464	C₂²	semidirect
52. C₂₃×#462	C₂²	semidirect
53. C₂₃×#452	C₂²	semidirect
55. C₂₃×#451	C₂²	semidirect
57. C₂₃×#441	C₂²	semidirect
58. C₄×Q₁₆×C₂₃	C₂²	semidirect
97. C₂×Q₁₆×C₂₃	D₄	semidirect
254. #61053	C₂₃	direct (N × H)
308. C₂×#578	C₂×C₂₃	semidirect
309. C₂³×Q₁₆	C₂×C₂₃	semidirect
310. #3486	C₂×C₂₃	semidirect
311. D₄×Q₁₆	C₂×C₂₃	semidirect
312. #3285	C₂×C₂₃	semidirect
313. #3270	C₂×C₂₃	semidirect
314. #3156	C₂×C₂₃	semidirect
315. #3059	C₂×C₂₃	semidirect
316. #3050	C₂×C₂₃	semidirect
317. #3044	C₂×C₂₃	semidirect
318. #3011	C₂×C₂₃	semidirect
319. #3004	C₂×C₂₃	semidirect
320. #2950	C₂×C₂₃	semidirect
321. #2929	C₂×C₂₃	semidirect
331. C₂²×C₂₃	#578	semidirect
348. C₂×#137	C₂²×C₂₃	semidirect
350. C₂×#135	C₂²×C₂₃	semidirect
357. C₂²×Q₁₆	C₂²×C₂₃	semidirect
371. #497	C₂²×C₂₃	semidirect
372. #495	C₂²×C₂₃	semidirect
373. #494	C₂²×C₂₃	semidirect
374. #470	C₂²×C₂₃	semidirect
376. #467	C₂²×C₂₃	semidirect
379. #464	C₂²×C₂₃	semidirect
380. #462	C₂²×C₂₃	semidirect
381. #452	C₂²×C₂₃	semidirect
383. #451	C₂²×C₂₃	semidirect
385. #441	C₂²×C₂₃	semidirect
386. C₄×Q₁₆	C₂²×C₂₃	semidirect
431. C₂×Q₁₆	D₄×C₂₃	semidirect
496. C₂₃	#61053	direct (N × H)
651. C₂²	C₂₃×#578	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#578	2
C₂³×Q₁₆×C₂₃	2
C₂₃×#3486	2
D₄×Q₁₆×C₂₃	2
C₂₃×#3285	2
C₂₃×#3270	2
C₂₃×#3156	2
C₂₃×#3059	2
C₂₃×#3050	2
C₂₃×#3044	2
C₂₃×#3011	2
C₂₃×#3004	2
C₂₃×#2950	2
C₂₃×#2929	2
#61053	23

Subgroups — C23×#61053 #19208679

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#61053 #19208679