Subgroups — C₂₃×#18671 #19166297

Order 5888.

928 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3517	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3233	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3167	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3116	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1654	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1653	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1642	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1641	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1636	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1635	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1631	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1631	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1553	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1494	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1476	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#557	C₂²	1	1	yes	characteristic
17	C₂₃×#553	—	2	2	no	—
18	C₂₃×#553	—	2	1	no	—
19	C₂₃×#550	—	2	1	no	—
20	C₂₃×#550	—	2	1	no	—
21	C₂₃×#550	C₂²	1	1	yes	characteristic
22	C₂₃×#550	C₂²	1	1	yes	characteristic
23	C₂₃×#547	—	2	1	no	—
24	C₂₃×#547	C₂²	1	1	yes	characteristic
25	C₂₃×#540	—	2	2	no	—
26	C₂₃×#540	—	2	1	no	—
27	C₂₃×#533	C₂²	1	1	yes	characteristic
28	C₂₃×#532	—	2	1	no	—
29	C₂₃×#532	C₂²	1	1	yes	characteristic
30	C₂₃×#529	—	2	1	no	—
31	C₂₃×#529	C₂²	1	1	yes	characteristic
32	C₂₃×#520	—	2	1	no	—
33	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
34	C₂₃×#499	—	2	1	no	—
35	C₂₃×#498	C₂²	1	1	yes	characteristic
36	C₂₃×#492	C₂²	1	1	yes	characteristic
37	C₂₃×#488	—	2	1	no	—
38	C₂₃×#487	C₂²	1	1	yes	characteristic
39	C₂₃×#486	C₂²	1	1	yes	characteristic
40	C₂₃×#485	—	2	1	no	—
41	C₂₃×#485	—	2	1	no	—
42	C₂₃×#482	—	2	1	no	—
43	C₂₃×#477	C₂²	1	1	yes	characteristic
44	C₂₃×#477	C₂²	1	1	yes	characteristic
45	C₂₃×#476	C₂²	1	1	yes	characteristic
46	C₂₃×#476	C₂²	1	1	yes	characteristic
47	C₂₃×#475	C₂²	1	1	yes	characteristic
48	C₂₃×#474	C₂²	1	1	yes	characteristic
49	C₂₃×#465	—	2	2	no	—
50	C₂₃×#462	C₂²	1	1	yes	characteristic
51	C₂₃×#461	C₂²	1	1	yes	characteristic
52	C₂₃×#452	—	2	1	no	—
53	C₂₃×#441	—	2	1	no	—
54	C₂₃×#440	—	2	1	no	—
55	C₄×SD₁₆×C₂₃	—	2	2	no	—
56	C₂₃×#432	C₂²	1	1	yes	characteristic
57	C₂₃×#431	C₂²	1	1	yes	characteristic
58	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
59	C₂₃×#332	C₂²	1	1	yes	characteristic
60	C₂₃×#332	C₂²	1	1	yes	characteristic
61	C₂₃×#331	C₂²	1	1	yes	characteristic
62	C₂₃×#331	C₂²	1	1	yes	characteristic
63	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
64	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
65	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
66	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
67	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
68	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
69	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
70	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
71	C₂×#41×C₂₃	—	2	1	no	—
72	C₂×#41×C₂₃	—	2	1	no	—
73	C₂×#41×C₂₃	—	2	1	no	—
74	C₂×#41×C₂₃	—	2	1	no	—
75	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
76	C₂×Q₁₆×C₂₃	—	2	1	no	—
77	C₂×SD₁₆×C₂₃	—	2	2	no	—
78	C₂×SD₁₆×C₂₃	—	2	1	no	—
79	C₂²×C₈×C₂₃	—	2	1	no	—
80	C₂₃×C₄⋊Q₈	—	2	1	no	—
81	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
82	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
83	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
84	C₂₃×#127	—	2	2	no	—
85	C₂₃×#127	—	2	1	no	—
86	C₂₃×#127	—	2	1	no	—
87	C₂₃×#127	C₂³	1	1	yes	characteristic
88	C₂₃×#126	—	2	2	no	—
89	C₂₃×#126	—	2	1	no	—
90	C₂₃×#125	—	2	2	no	—
91	C₂₃×#125	—	2	1	no	—
92	C₂₃×#125	—	2	1	no	—
93	C₂₃×#125	D₄	1	1	yes	characteristic
94	C₂₃×#124	—	2	1	no	—
95	C₂₃×#124	D₄	1	1	yes	characteristic
96	C₂₃×#123	—	2	2	no	—
97	C₂₃×#123	—	2	2	no	—
98	C₂₃×#123	—	2	2	no	—
99	C₂₃×#123	—	2	1	no	—
100	C₂₃×#123	—	2	2	no	—
101	C₂₃×C₂²⋊Q₈	—	2	1	no	—
102	C₂₃×C₂²⋊Q₈	—	2	1	no	—
103	C₂₃×C₂²⋊Q₈	—	2	1	no	—
104	C₂₃×C₂²⋊Q₈	D₄	1	1	yes	characteristic
105	C₂₃×#121	—	2	2	no	—
106	C₂₃×#121	—	2	2	no	—
107	C₂₃×#121	—	2	2	no	—
108	C₂₃×#121	—	2	2	no	—
109	C₂₃×#121	—	2	2	no	—
110	C₂₃×#121	—	2	1	no	—
111	C₂₃×#121	—	2	2	no	—
112	C₂₃×#121	—	2	2	no	—
113	C₂₃×#121	—	2	1	no	—
114	C₂₃×#121	—	2	1	no	—
115	C₂₃×#121	—	2	1	no	—
116	C₂₃×#121	D₄	1	1	yes	characteristic
117	C₂₃×#121	D₄	1	2	yes	—
118	C₄×Q₈×C₂₃	—	2	1	no	—
119	C₄×Q₈×C₂₃	—	2	1	no	—
120	C₄×Q₈×C₂₃	—	2	1	no	—
121	C₄×Q₈×C₂₃	—	2	1	no	—
122	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
123	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
124	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
125	C₄×D₄×C₂₃	—	2	2	no	—
126	C₄×D₄×C₂₃	—	2	2	no	—
127	C₄×D₄×C₂₃	—	2	2	no	—
128	C₄×D₄×C₂₃	—	2	2	no	—
129	C₄×D₄×C₂₃	—	2	1	no	—
130	C₄×D₄×C₂₃	—	2	1	no	—
131	C₄×D₄×C₂₃	—	2	1	no	—
132	C₄×D₄×C₂₃	—	2	1	no	—
133	C₄×D₄×C₂₃	—	2	1	no	—
134	C₄×D₄×C₂₃	—	2	1	no	—
135	C₄×D₄×C₂₃	D₄	1	2	yes	—
136	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
137	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
138	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
139	C₂₃×#117	—	2	2	no	—
140	C₂₃×#117	—	2	1	no	—
141	C₂₃×#117	—	2	1	no	—
142	C₂₃×#117	—	2	1	no	—
143	C₂₃×#117	D₄	1	1	yes	characteristic
144	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
145	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
146	C₂×C₄²×C₂₃	—	2	1	no	—
147	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
148	C₂₃×#107	—	2	1	no	—
149	C₂₃×#107	—	2	1	no	—
150	C₂₃×#107	—	2	1	no	—
151	C₂₃×#106	—	2	2	no	—
152	C₂₃×#106	—	2	1	no	—
153	C₂₃×#106	—	2	1	no	—
154	C₂₃×#106	—	2	1	no	—
155	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
156	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
157	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
158	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
159	C₂₃×#103	—	2	1	no	—
160	C₂₃×#103	—	2	1	no	—
161	C₂₃×#103	—	2	1	no	—
162	C₂₃×#103	—	2	2	no	—
163	C₂₃×#103	—	2	1	no	—
164	C₂₃×#103	—	2	1	no	—
165	C₂₃×#102	—	2	2	no	—
166	C₂₃×#102	—	2	1	no	—
167	C₂₃×#102	—	2	1	no	—
168	C₂₃×#102	—	2	1	no	—
169	C₂₃×C₂²⋊C₈	—	2	1	no	—
170	C₂₃×C₂²⋊C₈	—	2	1	no	—
171	C₂₃×#97	—	2	1	no	—
172	C₄×C₈×C₂₃	—	2	1	no	—
173	C₄×C₈×C₂₃	—	2	1	no	—
174	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
175	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
176	#41×C₂₃	—	2	2	no	—
177	#41×C₂₃	—	4	1	no	—
178	#41×C₂₃	—	4	2	no	—
179	#41×C₂₃	—	4	2	no	—
180	#41×C₂₃	—	8	1	no	—
181	#41×C₂₃	—	8	1	no	—
182	#41×C₂₃	—	8	1	no	—
183	#41×C₂₃	—	8	1	no	—
184	#41×C₂₃	—	8	1	no	—
185	#41×C₂₃	—	8	1	no	—
186	C₂×Q₈×C₂₃	—	2	1	no	—
187	C₂×Q₈×C₂₃	—	2	1	no	—
188	C₂×Q₈×C₂₃	—	2	1	no	—
189	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
190	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
191	C₂×D₄×C₂₃	—	2	2	no	—
192	C₂×D₄×C₂₃	—	2	1	no	—
193	C₂×D₄×C₂₃	—	2	2	no	—
194	C₂×D₄×C₂₃	—	2	2	no	—
195	C₂×D₄×C₂₃	—	2	2	no	—
196	C₂×D₄×C₂₃	—	2	2	no	—
197	C₂×D₄×C₂₃	—	2	1	no	—
198	C₂×D₄×C₂₃	—	2	2	no	—
199	C₂×D₄×C₂₃	—	2	1	no	—
200	C₂×D₄×C₂₃	—	2	1	no	—
201	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
202	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
203	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
204	C₂²×C₄×C₂₃	—	2	2	no	—
205	C₂²×C₄×C₂₃	—	2	2	no	—
206	C₂²×C₄×C₂₃	—	2	2	no	—
207	C₂²×C₄×C₂₃	—	2	1	no	—
208	C₂²×C₄×C₂₃	—	2	1	no	—
209	C₂²×C₄×C₂₃	—	2	1	no	—
210	C₂²×C₄×C₂₃	—	2	1	no	—
211	C₂²×C₄×C₂₃	—	2	2	no	—
212	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
213	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
214	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
215	Q₁₆×C₂₃	—	4	2	no	—
216	SD₁₆×C₂₃	—	4	4	no	—
217	SD₁₆×C₂₃	—	4	2	no	—
218	C₂×C₈×C₂₃	—	2	1	no	—
219	C₂×C₈×C₂₃	—	2	1	no	—
220	C₂×C₈×C₂₃	—	2	1	no	—
221	C₂×C₈×C₂₃	—	2	1	no	—
222	C₂×C₈×C₂₃	—	2	1	no	—
223	C₂×C₈×C₂₃	—	2	1	no	—
224	C₂×C₈×C₂₃	—	4	2	no	—
225	C₄⋊C₄×C₂₃	—	2	2	no	—
226	C₄⋊C₄×C₂₃	—	2	2	no	—
227	C₄⋊C₄×C₂₃	—	2	2	no	—
228	C₄⋊C₄×C₂₃	—	2	2	no	—
229	C₄⋊C₄×C₂₃	—	2	1	no	—
230	C₄⋊C₄×C₂₃	—	2	1	no	—
231	C₄⋊C₄×C₂₃	—	2	1	no	—
232	C₄⋊C₄×C₂₃	—	2	1	no	—
233	C₄⋊C₄×C₂₃	—	2	1	no	—
234	C₄⋊C₄×C₂₃	—	2	1	no	—
235	C₄⋊C₄×C₂₃	—	2	1	no	—
236	C₄⋊C₄×C₂₃	—	2	1	no	—
237	C₄⋊C₄×C₂₃	—	2	1	no	—
238	C₄⋊C₄×C₂₃	—	2	1	no	—
239	C₄⋊C₄×C₂₃	—	2	1	no	—
240	C₄⋊C₄×C₂₃	—	4	1	no	—
241	C₄⋊C₄×C₂₃	—	4	1	no	—
242	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
243	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
244	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
245	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
246	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
247	C₂²⋊C₄×C₂₃	—	2	2	no	—
248	C₂²⋊C₄×C₂₃	—	2	2	no	—
249	C₂²⋊C₄×C₂₃	—	2	2	no	—
250	C₂²⋊C₄×C₂₃	—	2	1	no	—
251	C₂²⋊C₄×C₂₃	—	2	1	no	—
252	C₂²⋊C₄×C₂₃	—	2	1	no	—
253	C₂²⋊C₄×C₂₃	—	2	1	no	—
254	C₂²⋊C₄×C₂₃	—	2	2	no	—
255	C₂²⋊C₄×C₂₃	—	2	1	no	—
256	C₂²⋊C₄×C₂₃	—	2	1	no	—
257	C₂²⋊C₄×C₂₃	—	2	2	no	—
258	C₂²⋊C₄×C₂₃	—	2	1	no	—
259	C₂²⋊C₄×C₂₃	—	2	1	no	—
260	C₂²⋊C₄×C₂₃	—	4	1	no	—
261	C₄²×C₂₃	—	2	2	no	—
262	C₄²×C₂₃	—	2	1	no	—
263	C₄²×C₂₃	—	2	1	no	—
264	C₄²×C₂₃	—	2	1	no	—
265	C₄²×C₂₃	—	2	1	no	—
266	C₄²×C₂₃	—	4	1	no	—
267	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
268	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
269	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
270	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
271	#18671	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
272	C₂³×C₂₃	—	2	1	no	—
273	C₂³×C₂₃	—	2	1	no	—
274	C₂³×C₂₃	—	2	2	no	—
275	C₂³×C₂₃	#120	1	1	yes	characteristic
276	Q₈×C₂₃	—	2	1	no	—
277	Q₈×C₂₃	—	2	1	no	—
278	Q₈×C₂₃	—	2	1	no	—
279	Q₈×C₂₃	—	4	2	no	—
280	Q₈×C₂₃	—	4	1	no	—
281	Q₈×C₂₃	—	4	1	no	—
282	Q₈×C₂₃	—	4	1	no	—
283	Q₈×C₂₃	—	4	1	no	—
284	Q₈×C₂₃	#135	1	2	yes	—
285	D₄×C₂₃	—	2	1	no	—
286	D₄×C₂₃	—	2	1	no	—
287	D₄×C₂₃	—	2	1	no	—
288	D₄×C₂₃	—	2	1	no	—
289	D₄×C₂₃	—	4	2	no	—
290	D₄×C₂₃	—	4	2	no	—
291	D₄×C₂₃	—	4	4	no	—
292	D₄×C₂₃	—	4	1	no	—
293	D₄×C₂₃	—	4	1	no	—
294	D₄×C₂₃	—	4	1	no	—
295	D₄×C₂₃	—	4	1	no	—
296	D₄×C₂₃	—	8	1	no	—
297	D₄×C₂₃	—	8	2	no	—
298	D₄×C₂₃	—	8	2	no	—
299	D₄×C₂₃	—	8	2	no	—
300	D₄×C₂₃	—	8	2	no	—
301	D₄×C₂₃	—	8	2	no	—
302	C₂×C₄×C₂₃	—	2	1	no	—
303	C₂×C₄×C₂₃	—	2	1	no	—
304	C₂×C₄×C₂₃	—	2	1	no	—
305	C₂×C₄×C₂₃	—	2	1	no	—
306	C₂×C₄×C₂₃	—	2	1	no	—
307	C₂×C₄×C₂₃	—	2	1	no	—
308	C₂×C₄×C₂₃	—	2	1	no	—
309	C₂×C₄×C₂₃	—	2	1	no	—
310	C₂×C₄×C₂₃	—	2	1	no	—
311	C₂×C₄×C₂₃	—	2	2	no	—
312	C₂×C₄×C₂₃	—	2	1	no	—
313	C₂×C₄×C₂₃	—	4	1	no	—
314	C₂×C₄×C₂₃	—	4	2	no	—
315	C₂×C₄×C₂₃	—	4	4	no	—
316	C₂×C₄×C₂₃	—	4	1	no	—
317	C₂×C₄×C₂₃	—	4	1	no	—
318	C₂×C₄×C₂₃	—	4	1	no	—
319	C₂×C₄×C₂₃	—	4	1	no	—
320	C₂×C₄×C₂₃	—	4	1	no	—
321	C₂×C₄×C₂₃	—	4	1	no	—
322	C₂×C₄×C₂₃	—	8	1	no	—
323	C₂×C₄×C₂₃	—	8	2	no	—
324	C₂×C₄×C₂₃	—	8	2	no	—
325	C₂×C₄×C₂₃	—	8	2	no	—
326	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
327	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
328	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
329	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
330	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
331	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
332	C₈×C₂₃	—	2	2	no	—
333	C₈×C₂₃	—	4	1	no	—
334	C₈×C₂₃	—	4	1	no	—
335	C₈×C₂₃	—	4	1	no	—
336	C₈×C₂₃	—	4	1	no	—
337	C₈×C₂₃	—	4	1	no	—
338	#3517	C₂×C₂₃	1	1	yes	characteristic
339	#3233	C₂×C₂₃	1	1	yes	characteristic, complemented
340	#3167	C₂×C₂₃	1	1	yes	characteristic, complemented
341	#3116	C₂×C₂₃	1	1	yes	characteristic, complemented
342	#1654	C₂×C₂₃	1	1	yes	characteristic, complemented
343	#1653	C₂×C₂₃	1	1	yes	characteristic, complemented
344	#1642	C₂×C₂₃	1	1	yes	characteristic, complemented
345	#1641	C₂×C₂₃	1	1	yes	characteristic, complemented
346	#1636	C₂×C₂₃	1	1	yes	characteristic, complemented
347	#1635	C₂×C₂₃	1	1	yes	characteristic, complemented
348	#1631	C₂×C₂₃	1	1	yes	characteristic, complemented
349	#1631	C₂×C₂₃	1	1	yes	characteristic, complemented
350	#1553	C₂×C₂₃	1	1	yes	characteristic, complemented
351	#1494	C₂×C₂₃	1	1	yes	characteristic, complemented
352	#1476	C₂×C₂₃	1	1	yes	characteristic, complemented
353	C₂²×C₂₃	—	2	1	no	—
354	C₂²×C₂₃	—	2	1	no	—
355	C₂²×C₂₃	—	2	1	no	—
356	C₂²×C₂₃	—	4	1	no	—
357	C₂²×C₂₃	—	4	2	no	—
358	C₂²×C₂₃	—	4	1	no	—
359	C₂²×C₂₃	—	4	1	no	—
360	C₂²×C₂₃	—	4	1	no	—
361	C₂²×C₂₃	—	4	2	no	—
362	C₂²×C₂₃	—	4	1	no	—
363	C₂²×C₂₃	—	4	2	no	—
364	C₂²×C₂₃	—	4	1	no	—
365	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
366	C₄×C₂₃	—	2	1	no	—
367	C₄×C₂₃	—	2	2	no	—
368	C₄×C₂₃	—	2	1	no	—
369	C₄×C₂₃	—	2	1	no	—
370	C₄×C₂₃	—	2	1	no	—
371	C₄×C₂₃	—	4	1	no	—
372	C₄×C₂₃	—	4	1	no	—
373	C₄×C₂₃	—	4	1	no	—
374	C₄×C₂₃	—	4	2	no	—
375	C₄×C₂₃	—	4	1	no	—
376	C₄×C₂₃	—	4	1	no	—
377	C₄×C₂₃	—	8	1	no	—
378	C₄×C₂₃	—	8	1	no	—
379	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
380	C₄×C₂₃	#575	1	1	yes	characteristic
381	C₄×C₂₃	#577	1	1	yes	characteristic
382	C₄×C₂₃	#577	1	1	yes	characteristic
383	#557	C₂²×C₂₃	1	1	yes	characteristic
384	#553	—	2	2	no	—
385	#553	—	2	1	no	—
386	#550	—	2	1	no	—
387	#550	—	2	1	no	—
388	#550	C₂²×C₂₃	1	1	yes	characteristic
389	#550	C₂²×C₂₃	1	1	yes	characteristic
390	#547	—	2	1	no	—
391	#547	C₂²×C₂₃	1	1	yes	characteristic
392	#540	—	2	2	no	—
393	#540	—	2	1	no	—
394	#533	C₂²×C₂₃	1	1	yes	characteristic
395	#532	—	2	1	no	—
396	#532	C₂²×C₂₃	1	1	yes	characteristic
397	#529	—	2	1	no	—
398	#529	C₂²×C₂₃	1	1	yes	characteristic
399	#520	—	2	1	no	—
400	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
401	#499	—	2	1	no	—
402	#498	C₂²×C₂₃	1	1	yes	characteristic
403	#492	C₂²×C₂₃	1	1	yes	characteristic
404	#488	—	2	1	no	—
405	#487	C₂²×C₂₃	1	1	yes	characteristic
406	#486	C₂²×C₂₃	1	1	yes	characteristic
407	#485	—	2	1	no	—
408	#485	—	2	1	no	—
409	#482	—	2	1	no	—
410	#477	C₂²×C₂₃	1	1	yes	characteristic
411	#477	C₂²×C₂₃	1	1	yes	characteristic
412	#476	C₂²×C₂₃	1	1	yes	characteristic
413	#476	C₂²×C₂₃	1	1	yes	characteristic
414	#475	C₂²×C₂₃	1	1	yes	characteristic
415	#474	C₂²×C₂₃	1	1	yes	characteristic
416	#465	—	2	2	no	—
417	#462	C₂²×C₂₃	1	1	yes	characteristic
418	#461	C₂²×C₂₃	1	1	yes	characteristic
419	#452	—	2	1	no	—
420	#441	—	2	1	no	—
421	#440	—	2	1	no	—
422	C₄×SD₁₆	—	2	2	no	—
423	#432	C₂²×C₂₃	1	1	yes	characteristic
424	#431	C₂²×C₂₃	1	1	yes	characteristic
425	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
426	#332	C₂²×C₂₃	1	1	yes	characteristic
427	#332	C₂²×C₂₃	1	1	yes	characteristic
428	#331	C₂²×C₂₃	1	1	yes	characteristic
429	#331	C₂²×C₂₃	1	1	yes	characteristic
430	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
431	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
432	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
433	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
434	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
435	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
436	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
437	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
438	C₂×C₂₃	—	4	1	no	—
439	C₂×C₂₃	—	8	1	no	—
440	C₂×C₂₃	—	8	1	no	—
441	C₂×C₂₃	—	8	2	no	—
442	C₂×C₂₃	#3002	1	1	yes	characteristic
443	C₂×C₂₃	#3009	1	1	yes	characteristic
444	C₂×C₂₃	#3018	1	1	yes	characteristic
445	C₂×#41	—	2	1	no	—
446	C₂×#41	—	2	1	no	—
447	C₂×#41	—	2	1	no	—
448	C₂×#41	—	2	1	no	—
449	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
450	C₂×Q₁₆	—	2	1	no	—
451	C₂×SD₁₆	—	2	2	no	—
452	C₂×SD₁₆	—	2	1	no	—
453	C₂²×C₈	—	2	1	no	—
454	C₄⋊Q₈	—	2	1	no	—
455	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
456	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
457	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
458	#127	—	2	2	no	—
459	#127	—	2	1	no	—
460	#127	—	2	1	no	—
461	#127	C₂³×C₂₃	1	1	yes	characteristic
462	#126	—	2	2	no	—
463	#126	—	2	1	no	—
464	#125	—	2	2	no	—
465	#125	—	2	1	no	—
466	#125	—	2	1	no	—
467	#125	D₄×C₂₃	1	1	yes	characteristic
468	#124	—	2	1	no	—
469	#124	D₄×C₂₃	1	1	yes	characteristic
470	#123	—	2	2	no	—
471	#123	—	2	2	no	—
472	#123	—	2	2	no	—
473	#123	—	2	1	no	—
474	#123	—	2	2	no	—
475	C₂²⋊Q₈	—	2	1	no	—
476	C₂²⋊Q₈	—	2	1	no	—
477	C₂²⋊Q₈	—	2	1	no	—
478	C₂²⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
479	#121	—	2	2	no	—
480	#121	—	2	2	no	—
481	#121	—	2	2	no	—
482	#121	—	2	2	no	—
483	#121	—	2	2	no	—
484	#121	—	2	1	no	—
485	#121	—	2	2	no	—
486	#121	—	2	2	no	—
487	#121	—	2	1	no	—
488	#121	—	2	1	no	—
489	#121	—	2	1	no	—
490	#121	D₄×C₂₃	1	1	yes	characteristic
491	#121	D₄×C₂₃	1	2	yes	—
492	C₄×Q₈	—	2	1	no	—
493	C₄×Q₈	—	2	1	no	—
494	C₄×Q₈	—	2	1	no	—
495	C₄×Q₈	—	2	1	no	—
496	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
497	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
498	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
499	C₄×D₄	—	2	2	no	—
500	C₄×D₄	—	2	2	no	—
501	C₄×D₄	—	2	2	no	—
502	C₄×D₄	—	2	2	no	—
503	C₄×D₄	—	2	1	no	—
504	C₄×D₄	—	2	1	no	—
505	C₄×D₄	—	2	1	no	—
506	C₄×D₄	—	2	1	no	—
507	C₄×D₄	—	2	1	no	—
508	C₄×D₄	—	2	1	no	—
509	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
510	C₄×D₄	D₄×C₂₃	1	2	yes	—
511	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
512	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
513	#117	—	2	2	no	—
514	#117	—	2	1	no	—
515	#117	—	2	1	no	—
516	#117	—	2	1	no	—
517	#117	D₄×C₂₃	1	1	yes	characteristic
518	C₂×C₄⋊C₄	—	2	2	no	—
519	C₂×C₄⋊C₄	—	2	1	no	—
520	C₂×C₄²	—	2	1	no	—
521	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
522	#107	—	2	1	no	—
523	#107	—	2	1	no	—
524	#107	—	2	1	no	—
525	#106	—	2	2	no	—
526	#106	—	2	1	no	—
527	#106	—	2	1	no	—
528	#106	—	2	1	no	—
529	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
530	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
531	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
532	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
533	#103	—	2	1	no	—
534	#103	—	2	1	no	—
535	#103	—	2	1	no	—
536	#103	—	2	2	no	—
537	#103	—	2	1	no	—
538	#103	—	2	1	no	—
539	#102	—	2	2	no	—
540	#102	—	2	1	no	—
541	#102	—	2	1	no	—
542	#102	—	2	1	no	—
543	C₂²⋊C₈	—	2	1	no	—
544	C₂²⋊C₈	—	2	1	no	—
545	#97	—	2	1	no	—
546	C₄×C₈	—	2	1	no	—
547	C₄×C₈	—	2	1	no	—
548	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
549	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
550	C₂₃	#18671	1	1	yes	characteristic, direct product factor, Sylow
551	#41	—	2	2	no	—
552	#41	—	4	1	no	—
553	#41	—	4	2	no	—
554	#41	—	4	2	no	—
555	#41	—	8	1	no	—
556	#41	—	8	1	no	—
557	#41	—	8	1	no	—
558	#41	—	8	1	no	—
559	#41	—	8	1	no	—
560	#41	—	8	1	no	—
561	C₂×Q₈	—	2	1	no	—
562	C₂×Q₈	—	2	1	no	—
563	C₂×Q₈	—	2	1	no	—
564	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
565	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
566	C₂×D₄	—	2	2	no	—
567	C₂×D₄	—	2	1	no	—
568	C₂×D₄	—	2	2	no	—
569	C₂×D₄	—	2	2	no	—
570	C₂×D₄	—	2	2	no	—
571	C₂×D₄	—	2	2	no	—
572	C₂×D₄	—	2	1	no	—
573	C₂×D₄	—	2	2	no	—
574	C₂×D₄	—	2	1	no	—
575	C₂×D₄	—	2	1	no	—
576	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
577	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
578	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
579	C₂²×C₄	—	2	2	no	—
580	C₂²×C₄	—	2	2	no	—
581	C₂²×C₄	—	2	2	no	—
582	C₂²×C₄	—	2	1	no	—
583	C₂²×C₄	—	2	1	no	—
584	C₂²×C₄	—	2	1	no	—
585	C₂²×C₄	—	2	1	no	—
586	C₂²×C₄	—	2	2	no	—
587	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
588	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
589	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
590	Q₁₆	—	4	2	no	—
591	SD₁₆	—	4	4	no	—
592	SD₁₆	—	4	2	no	—
593	C₂×C₈	—	2	1	no	—
594	C₂×C₈	—	2	1	no	—
595	C₂×C₈	—	2	1	no	—
596	C₂×C₈	—	2	1	no	—
597	C₂×C₈	—	2	1	no	—
598	C₂×C₈	—	2	1	no	—
599	C₂×C₈	—	4	2	no	—
600	C₄⋊C₄	—	2	2	no	—
601	C₄⋊C₄	—	2	2	no	—
602	C₄⋊C₄	—	2	2	no	—
603	C₄⋊C₄	—	2	2	no	—
604	C₄⋊C₄	—	2	1	no	—
605	C₄⋊C₄	—	2	1	no	—
606	C₄⋊C₄	—	2	1	no	—
607	C₄⋊C₄	—	2	1	no	—
608	C₄⋊C₄	—	2	1	no	—
609	C₄⋊C₄	—	2	1	no	—
610	C₄⋊C₄	—	2	1	no	—
611	C₄⋊C₄	—	2	1	no	—
612	C₄⋊C₄	—	2	1	no	—
613	C₄⋊C₄	—	2	1	no	—
614	C₄⋊C₄	—	2	1	no	—
615	C₄⋊C₄	—	4	1	no	—
616	C₄⋊C₄	—	4	1	no	—
617	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
618	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
619	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
620	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
621	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
622	C₂²⋊C₄	—	2	2	no	—
623	C₂²⋊C₄	—	2	2	no	—
624	C₂²⋊C₄	—	2	2	no	—
625	C₂²⋊C₄	—	2	1	no	—
626	C₂²⋊C₄	—	2	1	no	—
627	C₂²⋊C₄	—	2	1	no	—
628	C₂²⋊C₄	—	2	1	no	—
629	C₂²⋊C₄	—	2	2	no	—
630	C₂²⋊C₄	—	2	1	no	—
631	C₂²⋊C₄	—	2	1	no	—
632	C₂²⋊C₄	—	2	2	no	—
633	C₂²⋊C₄	—	2	1	no	—
634	C₂²⋊C₄	—	2	1	no	—
635	C₂²⋊C₄	—	4	1	no	—
636	C₄²	—	2	2	no	—
637	C₄²	—	2	1	no	—
638	C₄²	—	2	1	no	—
639	C₄²	—	2	1	no	—
640	C₄²	—	2	1	no	—
641	C₄²	—	4	1	no	—
642	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
643	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
644	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
645	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
646	C₂³	—	2	1	no	—
647	C₂³	—	2	1	no	—
648	C₂³	—	2	2	no	—
649	C₂³	C₂₃×#120	1	1	yes	characteristic
650	Q₈	—	2	1	no	—
651	Q₈	—	2	1	no	—
652	Q₈	—	2	1	no	—
653	Q₈	—	4	2	no	—
654	Q₈	—	4	1	no	—
655	Q₈	—	4	1	no	—
656	Q₈	—	4	1	no	—
657	Q₈	—	4	1	no	—
658	Q₈	C₂₃×#135	1	2	yes	—
659	D₄	—	2	1	no	—
660	D₄	—	2	1	no	—
661	D₄	—	2	1	no	—
662	D₄	—	2	1	no	—
663	D₄	—	4	2	no	—
664	D₄	—	4	2	no	—
665	D₄	—	4	4	no	—
666	D₄	—	4	1	no	—
667	D₄	—	4	1	no	—
668	D₄	—	4	1	no	—
669	D₄	—	4	1	no	—
670	D₄	—	8	2	no	—
671	D₄	—	8	1	no	—
672	D₄	—	8	2	no	—
673	D₄	—	8	2	no	—
674	D₄	—	8	2	no	—
675	D₄	—	8	2	no	—
676	C₂×C₄	—	2	1	no	—
677	C₂×C₄	—	2	1	no	—
678	C₂×C₄	—	2	1	no	—
679	C₂×C₄	—	2	1	no	—
680	C₂×C₄	—	2	1	no	—
681	C₂×C₄	—	2	1	no	—
682	C₂×C₄	—	2	1	no	—
683	C₂×C₄	—	2	1	no	—
684	C₂×C₄	—	2	1	no	—
685	C₂×C₄	—	2	2	no	—
686	C₂×C₄	—	2	1	no	—
687	C₂×C₄	—	4	1	no	—
688	C₂×C₄	—	4	2	no	—
689	C₂×C₄	—	4	4	no	—
690	C₂×C₄	—	4	1	no	—
691	C₂×C₄	—	4	1	no	—
692	C₂×C₄	—	4	1	no	—
693	C₂×C₄	—	4	1	no	—
694	C₂×C₄	—	4	1	no	—
695	C₂×C₄	—	4	1	no	—
696	C₂×C₄	—	8	2	no	—
697	C₂×C₄	—	8	1	no	—
698	C₂×C₄	—	8	2	no	—
699	C₂×C₄	—	8	2	no	—
700	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
701	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
702	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
703	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
704	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
705	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
706	C₈	—	2	2	no	—
707	C₈	—	4	1	no	—
708	C₈	—	4	1	no	—
709	C₈	—	4	1	no	—
710	C₈	—	4	1	no	—
711	C₈	—	4	1	no	—
712	C₂²	—	2	1	no	—
713	C₂²	—	2	1	no	—
714	C₂²	—	2	1	no	—
715	C₂²	—	4	1	no	—
716	C₂²	—	4	2	no	—
717	C₂²	—	4	1	no	—
718	C₂²	—	4	1	no	—
719	C₂²	—	4	1	no	—
720	C₂²	—	4	2	no	—
721	C₂²	—	4	1	no	—
722	C₂²	—	4	2	no	—
723	C₂²	—	4	1	no	—
724	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
725	C₄	—	2	1	no	—
726	C₄	—	2	2	no	—
727	C₄	—	2	1	no	—
728	C₄	—	2	1	no	—
729	C₄	—	2	1	no	—
730	C₄	—	4	1	no	—
731	C₄	—	4	1	no	—
732	C₄	—	4	1	no	—
733	C₄	—	4	2	no	—
734	C₄	—	4	1	no	—
735	C₄	—	4	1	no	—
736	C₄	—	8	1	no	—
737	C₄	—	8	1	no	—
738	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
739	C₄	C₂₃×#575	1	1	yes	characteristic
740	C₄	C₂₃×#577	1	1	yes	characteristic
741	C₄	C₂₃×#577	1	1	yes	characteristic
742	C₂	—	4	1	no	—
743	C₂	—	8	1	no	—
744	C₂	—	8	1	no	—
745	C₂	—	8	2	no	—
746	C₂	C₂₃×#3002	1	1	yes	characteristic
747	C₂	C₂₃×#3009	1	1	yes	characteristic
748	C₂	C₂₃×#3018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3517	C₂
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
2. C₂₃×#3233	C₂	semidirect
3. C₂₃×#3167	C₂	semidirect
4. C₂₃×#3116	C₂	semidirect
5. C₂₃×#1654	C₂	semidirect
6. C₂₃×#1653	C₂	semidirect
7. C₂₃×#1642	C₂	semidirect
8. C₂₃×#1641	C₂	semidirect
9. C₂₃×#1636	C₂	semidirect
10. C₂₃×#1635	C₂	semidirect
11. C₂₃×#1631	C₂	semidirect
12. C₂₃×#1631	C₂	semidirect
13. C₂₃×#1553	C₂	semidirect
14. C₂₃×#1494	C₂	semidirect
15. C₂₃×#1476	C₂	semidirect
271. #18671	C₂₃	direct (N × H)
339. #3233	C₂×C₂₃	semidirect
340. #3167	C₂×C₂₃	semidirect
341. #3116	C₂×C₂₃	semidirect
342. #1654	C₂×C₂₃	semidirect
343. #1653	C₂×C₂₃	semidirect
344. #1642	C₂×C₂₃	semidirect
345. #1641	C₂×C₂₃	semidirect
346. #1636	C₂×C₂₃	semidirect
347. #1635	C₂×C₂₃	semidirect
348. #1631	C₂×C₂₃	semidirect
349. #1631	C₂×C₂₃	semidirect
350. #1553	C₂×C₂₃	semidirect
351. #1494	C₂×C₂₃	semidirect
352. #1476	C₂×C₂₃	semidirect
550. C₂₃	#18671	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×#3517	2
C₂₃×#3233	2
C₂₃×#3167	2
C₂₃×#3116	2
C₂₃×#1654	2
C₂₃×#1653	2
C₂₃×#1642	2
C₂₃×#1641	2
C₂₃×#1636	2
C₂₃×#1635	2
C₂₃×#1631	2
C₂₃×#1553	2
C₂₃×#1494	2
C₂₃×#1476	2
#18671	23

Subgroups — C23×#18671 #19166297

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18671 #19166297