Subgroups — C₂₃×#19752 #19167378

Order 5888.

706 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3185	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3182	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3182	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#3165	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#3141	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#3139	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1723	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1720	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1719	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1718	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1718	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1714	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1710	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1708	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1455	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#533	C₂²	1	1	yes	characteristic
17	C₂₃×#533	C₂²	1	1	yes	characteristic
18	C₂₃×#532	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂₃×#501	—	2	2	no	—
21	C₂₃×#500	C₂²	1	1	yes	characteristic
22	C₂₃×#500	C₂²	1	1	yes	characteristic
23	C₂₃×#500	C₂²	1	1	yes	characteristic
24	C₂₃×#500	C₂²	1	1	yes	characteristic
25	C₂₃×#498	C₂²	1	1	yes	characteristic
26	C₂₃×#498	C₂²	1	1	yes	characteristic
27	C₂₃×#498	C₂²	1	1	yes	characteristic
28	C₂₃×#494	C₂²	1	1	yes	characteristic
29	C₂₃×#492	C₂²	1	1	yes	characteristic
30	C₂₃×#492	C₂²	1	1	yes	characteristic
31	C₂₃×#491	—	2	2	no	—
32	C₂₃×#489	—	2	2	no	—
33	C₂₃×#487	C₂²	1	1	yes	characteristic
34	C₂₃×#487	C₂²	1	1	yes	characteristic
35	C₂₃×#487	C₂²	1	1	yes	characteristic
36	C₂₃×#487	C₂²	1	1	yes	characteristic
37	C₂₃×#487	C₂²	1	1	yes	characteristic
38	C₂₃×#487	C₂²	1	1	yes	characteristic
39	C₂₃×#487	C₂²	1	1	yes	characteristic
40	C₂₃×#487	C₂²	1	1	yes	characteristic
41	C₂₃×#487	C₂²	1	1	yes	characteristic
42	C₂₃×#487	C₂²	1	1	yes	characteristic
43	C₂₃×#486	C₂²	1	1	yes	characteristic
44	C₂₃×#486	C₂²	1	1	yes	characteristic
45	C₂₃×#486	C₂²	1	1	yes	characteristic
46	C₂₃×#486	C₂²	1	1	yes	characteristic
47	C₂₃×#470	—	2	2	no	—
48	C₂₃×#468	—	2	2	no	—
49	C₂₃×#468	—	2	2	no	—
50	C₂₃×#441	—	2	2	no	—
51	C₂₃×#440	—	2	2	no	—
52	C₂₃×#440	—	2	2	no	—
53	C₂₃×#428	—	2	2	no	—
54	C₂₃×#419	—	2	2	no	—
55	C₂₃×#417	—	2	2	no	—
56	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
57	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
58	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
59	C₂₃×#322	C₂²	1	1	yes	characteristic
60	C₂₃×#322	C₂²	1	1	yes	characteristic
61	C₂₃×#322	C₂²	1	1	yes	characteristic
62	C₈²×C₂₃	C₂²	1	1	yes	characteristic
63	C₂×#41×C₂₃	—	2	1	no	—
64	C₂²×Q₈×C₂₃	—	2	1	no	—
65	C₂×Q₁₆×C₂₃	—	2	2	no	—
66	C₂×SD₁₆×C₂₃	—	2	2	no	—
67	C₂×SD₁₆×C₂₃	—	2	2	no	—
68	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
69	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
70	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
71	C₂₃×C₄⋊Q₈	—	2	1	no	—
72	C₂₃×C₄⋊Q₈	—	2	1	no	—
73	C₂₃×C₄⋊Q₈	—	2	1	no	—
74	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
75	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
76	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
77	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
78	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
79	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
80	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
81	C₂₃×#127	C₂³	1	1	yes	characteristic
82	C₂₃×#125	—	2	1	no	—
83	C₂₃×#125	—	2	1	no	—
84	C₂₃×#124	—	2	1	no	—
85	C₂₃×C₂²⋊Q₈	—	2	1	no	—
86	C₂₃×C₂²⋊Q₈	—	2	1	no	—
87	C₂₃×C₂²⋊Q₈	—	2	1	no	—
88	C₂₃×C₂²⋊Q₈	—	2	1	no	—
89	C₂₃×#121	—	2	1	no	—
90	C₂₃×#121	—	2	1	no	—
91	C₄×Q₈×C₂₃	—	2	1	no	—
92	C₄×Q₈×C₂₃	—	2	1	no	—
93	C₄×Q₈×C₂₃	—	2	1	no	—
94	C₄×Q₈×C₂₃	—	2	1	no	—
95	C₄×D₄×C₂₃	—	2	1	no	—
96	C₄×D₄×C₂₃	—	2	1	no	—
97	C₂₃×#117	—	2	1	no	—
98	C₂₃×#117	—	2	1	no	—
99	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
100	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
101	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
102	C₂₃×#107	—	2	2	no	—
103	C₂₃×#107	—	2	2	no	—
104	C₂₃×#107	—	2	2	no	—
105	C₂₃×#107	—	2	2	no	—
106	C₂₃×#106	—	2	2	no	—
107	C₂₃×#106	—	2	2	no	—
108	C₂₃×#106	—	2	2	no	—
109	C₂₃×#103	—	2	2	no	—
110	C₂₃×#103	—	2	2	no	—
111	C₂₃×#103	—	2	2	no	—
112	C₂₃×#103	—	2	2	no	—
113	C₂₃×#103	—	2	2	no	—
114	C₂₃×#103	—	2	2	no	—
115	C₂₃×#103	—	2	2	no	—
116	C₂₃×#103	—	2	2	no	—
117	C₂₃×#103	—	2	2	no	—
118	C₂₃×#103	—	2	2	no	—
119	C₂₃×#102	—	2	2	no	—
120	C₂₃×#102	—	2	2	no	—
121	C₂₃×#102	—	2	2	no	—
122	C₂₃×#102	—	2	2	no	—
123	C₂₃×#97	D₄	1	2	yes	—
124	C₂₃×#97	D₄	1	2	yes	—
125	C₂₃×#97	D₄	1	2	yes	—
126	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
127	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
128	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
129	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
130	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
131	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
132	#41×C₂₃	—	4	4	no	—
133	C₂×Q₈×C₂₃	—	2	1	no	—
134	C₂×Q₈×C₂₃	—	2	1	no	—
135	C₂×Q₈×C₂₃	—	2	1	no	—
136	C₂×Q₈×C₂₃	—	2	1	no	—
137	C₂×Q₈×C₂₃	—	2	1	no	—
138	C₂×Q₈×C₂₃	—	2	1	no	—
139	C₂×Q₈×C₂₃	—	2	1	no	—
140	C₂×Q₈×C₂₃	—	4	4	no	—
141	C₂×Q₈×C₂₃	—	4	1	no	—
142	C₂×D₄×C₂₃	—	2	1	no	—
143	C₂×D₄×C₂₃	—	2	1	no	—
144	C₂×D₄×C₂₃	—	2	1	no	—
145	C₂×D₄×C₂₃	—	4	1	no	—
146	C₂²×C₄×C₂₃	—	4	1	no	—
147	C₂²×C₄×C₂₃	—	4	1	no	—
148	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
151	Q₁₆×C₂₃	—	4	4	no	—
152	SD₁₆×C₂₃	—	4	4	no	—
153	SD₁₆×C₂₃	—	4	4	no	—
154	M₁₆×C₂₃	—	2	4	no	—
155	M₁₆×C₂₃	—	4	2	no	—
156	M₁₆×C₂₃	—	4	2	no	—
157	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
158	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
159	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
160	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
161	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
162	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
163	C₄⋊C₄×C₂₃	—	2	1	no	—
164	C₄⋊C₄×C₂₃	—	2	1	no	—
165	C₄⋊C₄×C₂₃	—	2	1	no	—
166	C₄⋊C₄×C₂₃	—	2	1	no	—
167	C₄⋊C₄×C₂₃	—	2	1	no	—
168	C₄⋊C₄×C₂₃	—	2	1	no	—
169	C₄⋊C₄×C₂₃	—	2	1	no	—
170	C₄⋊C₄×C₂₃	—	2	1	no	—
171	C₄⋊C₄×C₂₃	—	2	1	no	—
172	C₄⋊C₄×C₂₃	—	2	1	no	—
173	C₄⋊C₄×C₂₃	—	2	1	no	—
174	C₄⋊C₄×C₂₃	—	2	1	no	—
175	C₄⋊C₄×C₂₃	—	2	1	no	—
176	C₄⋊C₄×C₂₃	—	2	1	no	—
177	C₄⋊C₄×C₂₃	—	4	1	no	—
178	C₄⋊C₄×C₂₃	—	4	1	no	—
179	C₄⋊C₄×C₂₃	—	4	1	no	—
180	C₄⋊C₄×C₂₃	—	4	1	no	—
181	C₄⋊C₄×C₂₃	—	4	1	no	—
182	C₄⋊C₄×C₂₃	—	4	1	no	—
183	C₂²⋊C₄×C₂₃	—	4	1	no	—
184	C₂²⋊C₄×C₂₃	—	4	1	no	—
185	C₂²⋊C₄×C₂₃	—	4	1	no	—
186	C₂²⋊C₄×C₂₃	—	4	1	no	—
187	C₄²×C₂₃	—	4	1	no	—
188	C₄²×C₂₃	—	4	1	no	—
189	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
190	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
191	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
192	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
193	#19752	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
194	C₂³×C₂₃	—	4	1	no	—
195	C₂³×C₂₃	#127	1	1	yes	characteristic
196	Q₈×C₂₃	—	4	2	no	—
197	Q₈×C₂₃	—	4	2	no	—
198	Q₈×C₂₃	—	4	2	no	—
199	Q₈×C₂₃	—	4	2	no	—
200	Q₈×C₂₃	—	4	2	no	—
201	Q₈×C₂₃	—	4	2	no	—
202	Q₈×C₂₃	—	4	2	no	—
203	Q₈×C₂₃	—	8	2	no	—
204	D₄×C₂₃	—	4	2	no	—
205	D₄×C₂₃	—	4	2	no	—
206	D₄×C₂₃	—	4	2	no	—
207	D₄×C₂₃	—	8	2	no	—
208	C₂×C₄×C₂₃	—	2	2	no	—
209	C₂×C₄×C₂₃	—	2	2	no	—
210	C₂×C₄×C₂₃	—	2	2	no	—
211	C₂×C₄×C₂₃	—	4	1	no	—
212	C₂×C₄×C₂₃	—	4	1	no	—
213	C₂×C₄×C₂₃	—	4	1	no	—
214	C₂×C₄×C₂₃	—	4	1	no	—
215	C₂×C₄×C₂₃	—	4	1	no	—
216	C₂×C₄×C₂₃	—	4	1	no	—
217	C₂×C₄×C₂₃	—	4	1	no	—
218	C₂×C₄×C₂₃	—	8	2	no	—
219	C₂×C₄×C₂₃	—	8	2	no	—
220	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
221	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
222	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
223	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
226	C₈×C₂₃	—	2	2	no	—
227	C₈×C₂₃	—	2	2	no	—
228	C₈×C₂₃	—	2	2	no	—
229	C₈×C₂₃	—	2	2	no	—
230	C₈×C₂₃	—	2	2	no	—
231	C₈×C₂₃	#137	1	4	yes	—
232	#3185	C₂×C₂₃	1	1	yes	characteristic, complemented
233	#3182	C₂×C₂₃	1	1	yes	characteristic
234	#3182	C₂×C₂₃	1	1	yes	characteristic
235	#3165	C₂×C₂₃	1	1	yes	characteristic, complemented
236	#3141	C₂×C₂₃	1	1	yes	characteristic
237	#3139	C₂×C₂₃	1	1	yes	characteristic, complemented
238	#1723	C₂×C₂₃	1	1	yes	characteristic, complemented
239	#1720	C₂×C₂₃	1	1	yes	characteristic, complemented
240	#1719	C₂×C₂₃	1	1	yes	characteristic, complemented
241	#1718	C₂×C₂₃	1	1	yes	characteristic, complemented
242	#1718	C₂×C₂₃	1	1	yes	characteristic, complemented
243	#1714	C₂×C₂₃	1	1	yes	characteristic, complemented
244	#1710	C₂×C₂₃	1	1	yes	characteristic, complemented
245	#1708	C₂×C₂₃	1	1	yes	characteristic, complemented
246	#1455	C₂×C₂₃	1	1	yes	characteristic, complemented
247	C₂²×C₂₃	—	2	1	no	—
248	C₂²×C₂₃	—	2	1	no	—
249	C₂²×C₂₃	—	2	1	no	—
250	C₂²×C₂₃	—	8	1	no	—
251	C₂²×C₂₃	—	8	1	no	—
252	C₂²×C₂₃	—	8	1	no	—
253	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
254	C₄×C₂₃	—	2	1	no	—
255	C₄×C₂₃	—	2	1	no	—
256	C₄×C₂₃	—	2	1	no	—
257	C₄×C₂₃	—	8	1	no	—
258	C₄×C₂₃	—	8	1	no	—
259	C₄×C₂₃	—	8	1	no	—
260	C₄×C₂₃	—	8	1	no	—
261	C₄×C₂₃	—	8	1	no	—
262	C₄×C₂₃	—	8	1	no	—
263	C₄×C₂₃	—	8	1	no	—
264	C₄×C₂₃	C₂×#137	1	2	yes	—
265	C₄×C₂₃	#575	1	2	yes	—
266	C₄×C₂₃	#575	1	2	yes	—
267	#533	C₂²×C₂₃	1	1	yes	characteristic
268	#533	C₂²×C₂₃	1	1	yes	characteristic
269	#532	C₂²×C₂₃	1	1	yes	characteristic
270	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
271	#501	—	2	2	no	—
272	#500	C₂²×C₂₃	1	1	yes	characteristic
273	#500	C₂²×C₂₃	1	1	yes	characteristic
274	#500	C₂²×C₂₃	1	1	yes	characteristic
275	#500	C₂²×C₂₃	1	1	yes	characteristic
276	#498	C₂²×C₂₃	1	1	yes	characteristic
277	#498	C₂²×C₂₃	1	1	yes	characteristic
278	#498	C₂²×C₂₃	1	1	yes	characteristic
279	#494	C₂²×C₂₃	1	1	yes	characteristic
280	#492	C₂²×C₂₃	1	1	yes	characteristic
281	#492	C₂²×C₂₃	1	1	yes	characteristic
282	#491	—	2	2	no	—
283	#489	—	2	2	no	—
284	#487	C₂²×C₂₃	1	1	yes	characteristic
285	#487	C₂²×C₂₃	1	1	yes	characteristic
286	#487	C₂²×C₂₃	1	1	yes	characteristic
287	#487	C₂²×C₂₃	1	1	yes	characteristic
288	#487	C₂²×C₂₃	1	1	yes	characteristic
289	#487	C₂²×C₂₃	1	1	yes	characteristic
290	#487	C₂²×C₂₃	1	1	yes	characteristic
291	#487	C₂²×C₂₃	1	1	yes	characteristic
292	#487	C₂²×C₂₃	1	1	yes	characteristic
293	#487	C₂²×C₂₃	1	1	yes	characteristic
294	#486	C₂²×C₂₃	1	1	yes	characteristic
295	#486	C₂²×C₂₃	1	1	yes	characteristic
296	#486	C₂²×C₂₃	1	1	yes	characteristic
297	#486	C₂²×C₂₃	1	1	yes	characteristic
298	#470	—	2	2	no	—
299	#468	—	2	2	no	—
300	#468	—	2	2	no	—
301	#441	—	2	2	no	—
302	#440	—	2	2	no	—
303	#440	—	2	2	no	—
304	#428	—	2	2	no	—
305	#419	—	2	2	no	—
306	#417	—	2	2	no	—
307	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
308	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
309	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
310	#322	C₂²×C₂₃	1	1	yes	characteristic
311	#322	C₂²×C₂₃	1	1	yes	characteristic
312	#322	C₂²×C₂₃	1	1	yes	characteristic
313	C₈²	C₂²×C₂₃	1	1	yes	characteristic
314	C₂×C₂₃	—	4	1	no	—
315	C₂×C₂₃	—	16	1	no	—
316	C₂×C₂₃	#3152	1	1	yes	characteristic
317	C₂×C₂₃	#3153	1	1	yes	characteristic
318	C₂×C₂₃	#3153	1	1	yes	characteristic
319	C₂×#41	—	2	1	no	—
320	C₂²×Q₈	—	2	1	no	—
321	C₂×Q₁₆	—	2	2	no	—
322	C₂×SD₁₆	—	2	2	no	—
323	C₂×SD₁₆	—	2	2	no	—
324	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
325	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
326	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
327	C₄⋊Q₈	—	2	1	no	—
328	C₄⋊Q₈	—	2	1	no	—
329	C₄⋊Q₈	—	2	1	no	—
330	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
331	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
332	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
333	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
334	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
335	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
336	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
337	#127	C₂³×C₂₃	1	1	yes	characteristic
338	#125	—	2	1	no	—
339	#125	—	2	1	no	—
340	#124	—	2	1	no	—
341	C₂²⋊Q₈	—	2	1	no	—
342	C₂²⋊Q₈	—	2	1	no	—
343	C₂²⋊Q₈	—	2	1	no	—
344	C₂²⋊Q₈	—	2	1	no	—
345	#121	—	2	1	no	—
346	#121	—	2	1	no	—
347	C₄×Q₈	—	2	1	no	—
348	C₄×Q₈	—	2	1	no	—
349	C₄×Q₈	—	2	1	no	—
350	C₄×Q₈	—	2	1	no	—
351	C₄×D₄	—	2	1	no	—
352	C₄×D₄	—	2	1	no	—
353	#117	—	2	1	no	—
354	#117	—	2	1	no	—
355	C₂×C₄⋊C₄	—	2	1	no	—
356	C₂×C₄⋊C₄	—	2	1	no	—
357	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
358	#107	—	2	2	no	—
359	#107	—	2	2	no	—
360	#107	—	2	2	no	—
361	#107	—	2	2	no	—
362	#106	—	2	2	no	—
363	#106	—	2	2	no	—
364	#106	—	2	2	no	—
365	#103	—	2	2	no	—
366	#103	—	2	2	no	—
367	#103	—	2	2	no	—
368	#103	—	2	2	no	—
369	#103	—	2	2	no	—
370	#103	—	2	2	no	—
371	#103	—	2	2	no	—
372	#103	—	2	2	no	—
373	#103	—	2	2	no	—
374	#103	—	2	2	no	—
375	#102	—	2	2	no	—
376	#102	—	2	2	no	—
377	#102	—	2	2	no	—
378	#102	—	2	2	no	—
379	#97	D₄×C₂₃	1	2	yes	—
380	#97	D₄×C₂₃	1	2	yes	—
381	#97	D₄×C₂₃	1	2	yes	—
382	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
383	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
384	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
385	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
386	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
387	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
388	C₂₃	#19752	1	1	yes	characteristic, direct product factor, Sylow
389	#41	—	4	4	no	—
390	C₂×Q₈	—	2	1	no	—
391	C₂×Q₈	—	2	1	no	—
392	C₂×Q₈	—	2	1	no	—
393	C₂×Q₈	—	2	1	no	—
394	C₂×Q₈	—	2	1	no	—
395	C₂×Q₈	—	2	1	no	—
396	C₂×Q₈	—	2	1	no	—
397	C₂×Q₈	—	4	4	no	—
398	C₂×Q₈	—	4	1	no	—
399	C₂×D₄	—	2	1	no	—
400	C₂×D₄	—	2	1	no	—
401	C₂×D₄	—	2	1	no	—
402	C₂×D₄	—	4	1	no	—
403	C₂²×C₄	—	4	1	no	—
404	C₂²×C₄	—	4	1	no	—
405	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
406	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
407	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
408	Q₁₆	—	4	4	no	—
409	SD₁₆	—	4	4	no	—
410	SD₁₆	—	4	4	no	—
411	M₁₆	—	2	4	no	—
412	M₁₆	—	4	2	no	—
413	M₁₆	—	4	2	no	—
414	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
415	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
416	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
417	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
418	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
419	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
420	C₄⋊C₄	—	2	1	no	—
421	C₄⋊C₄	—	2	1	no	—
422	C₄⋊C₄	—	2	1	no	—
423	C₄⋊C₄	—	2	1	no	—
424	C₄⋊C₄	—	2	1	no	—
425	C₄⋊C₄	—	2	1	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	1	no	—
429	C₄⋊C₄	—	2	1	no	—
430	C₄⋊C₄	—	2	1	no	—
431	C₄⋊C₄	—	2	1	no	—
432	C₄⋊C₄	—	2	1	no	—
433	C₄⋊C₄	—	2	1	no	—
434	C₄⋊C₄	—	4	1	no	—
435	C₄⋊C₄	—	4	1	no	—
436	C₄⋊C₄	—	4	1	no	—
437	C₄⋊C₄	—	4	1	no	—
438	C₄⋊C₄	—	4	1	no	—
439	C₄⋊C₄	—	4	1	no	—
440	C₂²⋊C₄	—	4	1	no	—
441	C₂²⋊C₄	—	4	1	no	—
442	C₂²⋊C₄	—	4	1	no	—
443	C₂²⋊C₄	—	4	1	no	—
444	C₄²	—	4	1	no	—
445	C₄²	—	4	1	no	—
446	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
447	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
448	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
449	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
450	C₂³	—	4	1	no	—
451	C₂³	C₂₃×#127	1	1	yes	characteristic
452	Q₈	—	4	2	no	—
453	Q₈	—	4	2	no	—
454	Q₈	—	4	2	no	—
455	Q₈	—	4	2	no	—
456	Q₈	—	4	2	no	—
457	Q₈	—	4	2	no	—
458	Q₈	—	4	2	no	—
459	Q₈	—	8	2	no	—
460	D₄	—	4	2	no	—
461	D₄	—	4	2	no	—
462	D₄	—	4	2	no	—
463	D₄	—	8	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	4	1	no	—
468	C₂×C₄	—	4	1	no	—
469	C₂×C₄	—	4	1	no	—
470	C₂×C₄	—	4	1	no	—
471	C₂×C₄	—	4	1	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	1	no	—
474	C₂×C₄	—	8	2	no	—
475	C₂×C₄	—	8	2	no	—
476	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
477	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
478	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
479	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
480	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
481	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
482	C₈	—	2	2	no	—
483	C₈	—	2	2	no	—
484	C₈	—	2	2	no	—
485	C₈	—	2	2	no	—
486	C₈	—	2	2	no	—
487	C₈	C₂₃×#137	1	4	yes	—
488	C₂²	—	2	1	no	—
489	C₂²	—	2	1	no	—
490	C₂²	—	2	1	no	—
491	C₂²	—	8	1	no	—
492	C₂²	—	8	1	no	—
493	C₂²	—	8	1	no	—
494	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
495	C₄	—	2	1	no	—
496	C₄	—	2	1	no	—
497	C₄	—	2	1	no	—
498	C₄	—	8	1	no	—
499	C₄	—	8	1	no	—
500	C₄	—	8	1	no	—
501	C₄	—	8	1	no	—
502	C₄	—	8	1	no	—
503	C₄	—	8	1	no	—
504	C₄	—	8	1	no	—
505	C₄	C₂×C₂₃×#137	1	2	yes	—
506	C₄	C₂₃×#575	1	2	yes	—
507	C₄	C₂₃×#575	1	2	yes	—
508	C₂	—	4	1	no	—
509	C₂	—	16	1	no	—
510	C₂	C₂₃×#3152	1	1	yes	characteristic
511	C₂	C₂₃×#3153	1	1	yes	characteristic
512	C₂	C₂₃×#3153	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3185	C₂	semidirect
4. C₂₃×#3165	C₂	semidirect
6. C₂₃×#3139	C₂	semidirect
7. C₂₃×#1723	C₂	semidirect
8. C₂₃×#1720	C₂	semidirect
9. C₂₃×#1719	C₂	semidirect
10. C₂₃×#1718	C₂	semidirect
11. C₂₃×#1718	C₂	semidirect
12. C₂₃×#1714	C₂	semidirect
13. C₂₃×#1710	C₂	semidirect
14. C₂₃×#1708	C₂	semidirect
15. C₂₃×#1455	C₂	semidirect
193. #19752	C₂₃	direct (N × H)
232. #3185	C₂×C₂₃	semidirect
235. #3165	C₂×C₂₃	semidirect
237. #3139	C₂×C₂₃	semidirect
238. #1723	C₂×C₂₃	semidirect
239. #1720	C₂×C₂₃	semidirect
240. #1719	C₂×C₂₃	semidirect
241. #1718	C₂×C₂₃	semidirect
242. #1718	C₂×C₂₃	semidirect
243. #1714	C₂×C₂₃	semidirect
244. #1710	C₂×C₂₃	semidirect
245. #1708	C₂×C₂₃	semidirect
246. #1455	C₂×C₂₃	semidirect
388. C₂₃	#19752	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×#3185	2
C₂₃×#3182	2
C₂₃×#3165	2
C₂₃×#3141	2
C₂₃×#3139	2
C₂₃×#1723	2
C₂₃×#1720	2
C₂₃×#1719	2
C₂₃×#1718	2
C₂₃×#1714	2
C₂₃×#1710	2
C₂₃×#1708	2
C₂₃×#1455	2
#19752	23

Subgroups — C23×#19752 #19167378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19752 #19167378