Subgroups — C₂₃×#18765 #19166391

Order 5888.

902 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3482	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3231	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3116	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3076	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1652	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1651	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1648	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1647	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1638	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1637	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1636	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1635	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1552	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1488	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1477	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#557	C₂²	1	1	yes	characteristic
17	C₂₃×#556	—	2	1	no	—
18	C₂₃×#556	C₂²	1	1	yes	characteristic
19	C₂₃×#552	—	2	2	no	—
20	C₂₃×#552	—	2	1	no	—
21	D₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₂₃×#548	—	2	1	no	—
23	C₂₃×#547	—	2	1	no	—
24	C₂₃×#547	C₂²	1	1	yes	characteristic
25	C₂₃×#533	—	2	1	no	—
26	C₂₃×#533	C₂²	1	1	yes	characteristic
27	C₂₃×#533	C₂²	1	1	yes	characteristic
28	C₂₃×#529	—	2	1	no	—
29	C₂₃×#529	—	2	2	no	—
30	C₄×#41×C₂₃	—	2	1	no	—
31	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
32	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
34	C₂₃×#500	C₂²	1	1	yes	characteristic
35	C₂₃×#498	C₂²	1	1	yes	characteristic
36	C₂₃×#487	C₂²	1	1	yes	characteristic
37	C₂₃×#487	C₂²	1	1	yes	characteristic
38	C₂₃×#485	—	2	2	no	—
39	C₂₃×#485	—	2	1	no	—
40	C₂₃×#479	—	2	2	no	—
41	C₂₃×#479	—	2	1	no	—
42	C₂₃×#478	—	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×#462	C₂²	1	1	yes	characteristic
50	C₂₃×#461	C₂²	1	1	yes	characteristic
51	C₂₃×#452	—	2	1	no	—
52	C₂₃×#441	—	2	1	no	—
53	C₂₃×#440	—	2	1	no	—
54	C₂₃×#432	C₂²	1	1	yes	characteristic
55	C₂₃×#431	C₂²	1	1	yes	characteristic
56	C₂₃×#427	—	2	1	no	—
57	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
58	C₂₃×#416	—	2	1	no	—
59	C₂₃×#332	C₂²	1	1	yes	characteristic
60	C₂₃×#332	C₂²	1	1	yes	characteristic
61	C₂₃×#332	C₂²	1	1	yes	characteristic
62	C₂₃×#332	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	D₄	1	1	yes	characteristic
73	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
74	C₂²×Q₈×C₂₃	—	2	1	no	—
75	C₂×Q₁₆×C₂₃	—	2	1	no	—
76	C₂×SD₁₆×C₂₃	—	2	1	no	—
77	C₂²×C₈×C₂₃	—	2	1	no	—
78	C₂₃×C₄⋊Q₈	—	2	1	no	—
79	C₂₃×C₄⋊Q₈	—	2	1	no	—
80	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
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×#126	—	2	2	no	—
85	C₂₃×#126	—	2	2	no	—
86	C₂₃×#126	—	2	2	no	—
87	C₂₃×#125	—	2	2	no	—
88	C₂₃×#125	—	2	2	no	—
89	C₂₃×#125	—	2	1	no	—
90	C₂₃×#125	—	2	1	no	—
91	C₂₃×#125	D₄	1	2	yes	—
92	C₂₃×#124	—	2	2	no	—
93	C₂₃×#124	—	2	1	no	—
94	C₂₃×#123	—	2	2	no	—
95	C₂₃×#123	—	2	2	no	—
96	C₂₃×#123	—	2	2	no	—
97	C₂₃×C₂²⋊Q₈	—	2	2	no	—
98	C₂₃×C₂²⋊Q₈	—	2	1	no	—
99	C₂₃×C₂²⋊Q₈	—	2	1	no	—
100	C₂₃×C₂²⋊Q₈	—	2	1	no	—
101	C₂₃×C₂²⋊Q₈	—	2	1	no	—
102	C₂₃×C₂²⋊Q₈	—	2	1	no	—
103	C₂₃×C₂²⋊Q₈	—	2	2	no	—
104	C₂₃×C₂²⋊Q₈	—	2	1	no	—
105	C₂₃×C₂²⋊Q₈	—	2	1	no	—
106	C₂₃×#121	—	2	2	no	—
107	C₂₃×#121	—	2	1	no	—
108	C₄×Q₈×C₂₃	—	2	2	no	—
109	C₄×Q₈×C₂₃	—	2	2	no	—
110	C₄×Q₈×C₂₃	—	2	1	no	—
111	C₄×Q₈×C₂₃	—	2	1	no	—
112	C₄×Q₈×C₂₃	—	2	1	no	—
113	C₄×Q₈×C₂₃	—	2	1	no	—
114	C₄×Q₈×C₂₃	—	2	1	no	—
115	C₄×Q₈×C₂₃	—	2	1	no	—
116	C₄×Q₈×C₂₃	—	2	1	no	—
117	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
118	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
119	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
120	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
121	C₄×D₄×C₂₃	—	2	2	no	—
122	C₄×D₄×C₂₃	—	2	2	no	—
123	C₄×D₄×C₂₃	—	2	1	no	—
124	C₄×D₄×C₂₃	—	2	1	no	—
125	C₄×D₄×C₂₃	—	2	1	no	—
126	C₄×D₄×C₂₃	—	2	1	no	—
127	C₄×D₄×C₂₃	—	2	1	no	—
128	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
129	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
130	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
131	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
132	C₂₃×#117	—	2	2	no	—
133	C₂₃×#117	—	2	2	no	—
134	C₂₃×#117	—	2	1	no	—
135	C₂₃×#117	—	2	1	no	—
136	C₂₃×#117	—	2	1	no	—
137	C₂₃×#117	—	2	1	no	—
138	C₂₃×#117	D₄	1	1	yes	characteristic
139	C₂₃×#117	D₄	1	1	yes	characteristic
140	C₂₃×#117	D₄	1	2	yes	—
141	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
142	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
143	C₂×C₄²×C₂₃	—	2	2	no	—
144	C₂×C₄²×C₂₃	—	2	1	no	—
145	C₂×C₄²×C₂₃	—	2	1	no	—
146	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
147	C₂₃×#107	—	2	1	no	—
148	C₂₃×#107	—	2	1	no	—
149	C₂₃×#107	—	2	1	no	—
150	C₂₃×#107	—	2	2	no	—
151	C₂₃×#106	—	2	1	no	—
152	C₂₃×#106	—	2	1	no	—
153	C₂₃×#106	—	2	1	no	—
154	C₂₃×#106	—	2	2	no	—
155	C₂₃×C₄⋊C₈	—	2	1	no	—
156	C₂₃×C₄⋊C₈	—	2	1	no	—
157	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
158	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
159	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
160	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
161	C₂₃×#103	—	2	1	no	—
162	C₂₃×#103	—	2	1	no	—
163	C₂₃×#103	—	2	1	no	—
164	C₂₃×#103	—	2	1	no	—
165	C₂₃×#103	—	2	1	no	—
166	C₂₃×#103	—	2	2	no	—
167	C₂₃×#103	—	2	2	no	—
168	C₂₃×#102	—	2	1	no	—
169	C₂₃×#102	—	2	1	no	—
170	C₂₃×#102	—	2	1	no	—
171	C₂₃×C₂²⋊C₈	—	2	1	no	—
172	C₂₃×C₂²⋊C₈	—	2	1	no	—
173	C₂₃×#97	—	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	—	2	2	no	—
178	#41×C₂₃	—	4	1	no	—
179	#41×C₂₃	—	4	1	no	—
180	#41×C₂₃	—	4	2	no	—
181	#41×C₂₃	—	4	2	no	—
182	C₂×Q₈×C₂₃	—	2	2	no	—
183	C₂×Q₈×C₂₃	—	2	1	no	—
184	C₂×Q₈×C₂₃	—	2	1	no	—
185	C₂×Q₈×C₂₃	—	2	1	no	—
186	C₂×Q₈×C₂₃	—	2	1	no	—
187	C₂×Q₈×C₂₃	—	4	2	no	—
188	C₂×Q₈×C₂₃	—	4	2	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	1	no	—
194	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
195	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
196	C₂²×C₄×C₂₃	—	2	2	no	—
197	C₂²×C₄×C₂₃	—	2	2	no	—
198	C₂²×C₄×C₂₃	—	2	1	no	—
199	C₂²×C₄×C₂₃	—	2	1	no	—
200	C₂²×C₄×C₂₃	—	2	1	no	—
201	C₂²×C₄×C₂₃	—	2	1	no	—
202	C₂²×C₄×C₂₃	—	2	1	no	—
203	C₂²×C₄×C₂₃	—	2	1	no	—
204	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
205	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
206	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
207	Q₁₆×C₂₃	—	4	2	no	—
208	SD₁₆×C₂₃	—	4	2	no	—
209	C₂×C₈×C₂₃	—	2	1	no	—
210	C₂×C₈×C₂₃	—	2	1	no	—
211	C₂×C₈×C₂₃	—	2	1	no	—
212	C₂×C₈×C₂₃	—	2	1	no	—
213	C₂×C₈×C₂₃	—	2	1	no	—
214	C₂×C₈×C₂₃	—	2	1	no	—
215	C₂×C₈×C₂₃	—	4	2	no	—
216	C₄⋊C₄×C₂₃	—	2	2	no	—
217	C₄⋊C₄×C₂₃	—	2	2	no	—
218	C₄⋊C₄×C₂₃	—	2	2	no	—
219	C₄⋊C₄×C₂₃	—	2	2	no	—
220	C₄⋊C₄×C₂₃	—	2	2	no	—
221	C₄⋊C₄×C₂₃	—	2	2	no	—
222	C₄⋊C₄×C₂₃	—	2	1	no	—
223	C₄⋊C₄×C₂₃	—	2	2	no	—
224	C₄⋊C₄×C₂₃	—	2	2	no	—
225	C₄⋊C₄×C₂₃	—	2	1	no	—
226	C₄⋊C₄×C₂₃	—	2	1	no	—
227	C₄⋊C₄×C₂₃	—	2	1	no	—
228	C₄⋊C₄×C₂₃	—	2	1	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	—	4	1	no	—
235	C₄⋊C₄×C₂₃	—	4	1	no	—
236	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
237	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
238	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
239	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
240	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
241	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
242	C₂²⋊C₄×C₂₃	—	2	2	no	—
243	C₂²⋊C₄×C₂₃	—	2	2	no	—
244	C₂²⋊C₄×C₂₃	—	2	2	no	—
245	C₂²⋊C₄×C₂₃	—	2	2	no	—
246	C₂²⋊C₄×C₂₃	—	2	1	no	—
247	C₂²⋊C₄×C₂₃	—	2	1	no	—
248	C₂²⋊C₄×C₂₃	—	2	2	no	—
249	C₂²⋊C₄×C₂₃	—	2	1	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	—	4	1	no	—
254	C₄²×C₂₃	—	2	2	no	—
255	C₄²×C₂₃	—	2	2	no	—
256	C₄²×C₂₃	—	2	1	no	—
257	C₄²×C₂₃	—	2	1	no	—
258	C₄²×C₂₃	—	2	1	no	—
259	C₄²×C₂₃	—	2	2	no	—
260	C₄²×C₂₃	—	2	1	no	—
261	C₄²×C₂₃	—	2	1	no	—
262	C₄²×C₂₃	—	2	1	no	—
263	C₄²×C₂₃	—	4	1	no	—
264	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
265	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
266	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
267	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
268	#18765	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
269	C₂³×C₂₃	—	2	1	no	—
270	C₂³×C₂₃	—	2	1	no	—
271	C₂³×C₂₃	#120	1	1	yes	characteristic
272	Q₈×C₂₃	—	2	1	no	—
273	Q₈×C₂₃	—	2	1	no	—
274	Q₈×C₂₃	—	2	1	no	—
275	Q₈×C₂₃	—	4	4	no	—
276	Q₈×C₂₃	—	4	2	no	—
277	Q₈×C₂₃	—	4	1	no	—
278	Q₈×C₂₃	—	4	1	no	—
279	Q₈×C₂₃	—	4	1	no	—
280	Q₈×C₂₃	—	4	1	no	—
281	Q₈×C₂₃	—	4	1	no	—
282	Q₈×C₂₃	—	4	1	no	—
283	Q₈×C₂₃	#137	1	2	yes	—
284	D₄×C₂₃	—	2	1	no	—
285	D₄×C₂₃	—	2	1	no	—
286	D₄×C₂₃	—	2	1	no	—
287	D₄×C₂₃	—	4	2	no	—
288	D₄×C₂₃	—	4	2	no	—
289	D₄×C₂₃	—	4	4	no	—
290	D₄×C₂₃	#135	1	2	yes	—
291	C₂×C₄×C₂₃	—	2	1	no	—
292	C₂×C₄×C₂₃	—	2	1	no	—
293	C₂×C₄×C₂₃	—	2	1	no	—
294	C₂×C₄×C₂₃	—	2	1	no	—
295	C₂×C₄×C₂₃	—	2	1	no	—
296	C₂×C₄×C₂₃	—	2	1	no	—
297	C₂×C₄×C₂₃	—	2	1	no	—
298	C₂×C₄×C₂₃	—	2	1	no	—
299	C₂×C₄×C₂₃	—	2	2	no	—
300	C₂×C₄×C₂₃	—	2	1	no	—
301	C₂×C₄×C₂₃	—	2	1	no	—
302	C₂×C₄×C₂₃	—	2	2	no	—
303	C₂×C₄×C₂₃	—	4	1	no	—
304	C₂×C₄×C₂₃	—	4	2	no	—
305	C₂×C₄×C₂₃	—	4	2	no	—
306	C₂×C₄×C₂₃	—	4	2	no	—
307	C₂×C₄×C₂₃	—	4	2	no	—
308	C₂×C₄×C₂₃	—	4	4	no	—
309	C₂×C₄×C₂₃	—	4	4	no	—
310	C₂×C₄×C₂₃	—	4	1	no	—
311	C₂×C₄×C₂₃	—	4	1	no	—
312	C₂×C₄×C₂₃	—	4	1	no	—
313	C₂×C₄×C₂₃	—	4	1	no	—
314	C₂×C₄×C₂₃	—	4	1	no	—
315	C₂×C₄×C₂₃	—	4	1	no	—
316	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
317	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
318	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
319	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
320	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
321	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
322	C₈×C₂₃	—	4	1	no	—
323	C₈×C₂₃	—	4	1	no	—
324	C₈×C₂₃	—	4	1	no	—
325	C₈×C₂₃	—	4	1	no	—
326	C₈×C₂₃	—	4	1	no	—
327	C₈×C₂₃	—	4	1	no	—
328	#3482	C₂×C₂₃	1	1	yes	characteristic
329	#3231	C₂×C₂₃	1	1	yes	characteristic, complemented
330	#3116	C₂×C₂₃	1	1	yes	characteristic, complemented
331	#3076	C₂×C₂₃	1	1	yes	characteristic, complemented
332	#1652	C₂×C₂₃	1	1	yes	characteristic, complemented
333	#1651	C₂×C₂₃	1	1	yes	characteristic, complemented
334	#1648	C₂×C₂₃	1	1	yes	characteristic, complemented
335	#1647	C₂×C₂₃	1	1	yes	characteristic, complemented
336	#1638	C₂×C₂₃	1	1	yes	characteristic, complemented
337	#1637	C₂×C₂₃	1	1	yes	characteristic, complemented
338	#1636	C₂×C₂₃	1	1	yes	characteristic, complemented
339	#1635	C₂×C₂₃	1	1	yes	characteristic, complemented
340	#1552	C₂×C₂₃	1	1	yes	characteristic, complemented
341	#1488	C₂×C₂₃	1	1	yes	characteristic, complemented
342	#1477	C₂×C₂₃	1	1	yes	characteristic, complemented
343	C₂²×C₂₃	—	2	1	no	—
344	C₂²×C₂₃	—	2	2	no	—
345	C₂²×C₂₃	—	2	1	no	—
346	C₂²×C₂₃	—	2	1	no	—
347	C₂²×C₂₃	—	4	1	no	—
348	C₂²×C₂₃	—	4	1	no	—
349	C₂²×C₂₃	—	4	1	no	—
350	C₂²×C₂₃	—	4	1	no	—
351	C₂²×C₂₃	—	4	1	no	—
352	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
353	C₄×C₂₃	—	2	1	no	—
354	C₄×C₂₃	—	2	1	no	—
355	C₄×C₂₃	—	2	2	no	—
356	C₄×C₂₃	—	2	1	no	—
357	C₄×C₂₃	—	2	1	no	—
358	C₄×C₂₃	—	4	1	no	—
359	C₄×C₂₃	—	4	1	no	—
360	C₄×C₂₃	—	4	2	no	—
361	C₄×C₂₃	—	4	2	no	—
362	C₄×C₂₃	—	4	1	no	—
363	C₄×C₂₃	—	4	1	no	—
364	C₄×C₂₃	—	4	1	no	—
365	C₄×C₂₃	—	8	1	no	—
366	C₄×C₂₃	—	8	1	no	—
367	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
368	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
369	C₄×C₂₃	#575	1	1	yes	characteristic
370	C₄×C₂₃	#575	1	1	yes	characteristic
371	#557	C₂²×C₂₃	1	1	yes	characteristic
372	#556	—	2	1	no	—
373	#556	C₂²×C₂₃	1	1	yes	characteristic
374	#552	—	2	2	no	—
375	#552	—	2	1	no	—
376	D₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
377	#548	—	2	1	no	—
378	#547	—	2	1	no	—
379	#547	C₂²×C₂₃	1	1	yes	characteristic
380	#533	—	2	1	no	—
381	#533	C₂²×C₂₃	1	1	yes	characteristic
382	#533	C₂²×C₂₃	1	1	yes	characteristic
383	#529	—	2	1	no	—
384	#529	—	2	2	no	—
385	C₄×#41	—	2	1	no	—
386	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
387	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
388	C₂×C₄×Q₈	—	2	1	no	—
389	#500	C₂²×C₂₃	1	1	yes	characteristic
390	#498	C₂²×C₂₃	1	1	yes	characteristic
391	#487	C₂²×C₂₃	1	1	yes	characteristic
392	#487	C₂²×C₂₃	1	1	yes	characteristic
393	#485	—	2	1	no	—
394	#485	—	2	2	no	—
395	#479	—	2	2	no	—
396	#479	—	2	1	no	—
397	#478	—	2	1	no	—
398	#477	C₂²×C₂₃	1	1	yes	characteristic
399	#477	C₂²×C₂₃	1	1	yes	characteristic
400	#476	C₂²×C₂₃	1	1	yes	characteristic
401	#476	C₂²×C₂₃	1	1	yes	characteristic
402	#475	C₂²×C₂₃	1	1	yes	characteristic
403	#474	C₂²×C₂₃	1	1	yes	characteristic
404	#462	C₂²×C₂₃	1	1	yes	characteristic
405	#461	C₂²×C₂₃	1	1	yes	characteristic
406	#452	—	2	1	no	—
407	#441	—	2	1	no	—
408	#440	—	2	1	no	—
409	#432	C₂²×C₂₃	1	1	yes	characteristic
410	#431	C₂²×C₂₃	1	1	yes	characteristic
411	#427	—	2	1	no	—
412	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
413	#416	—	2	1	no	—
414	#332	C₂²×C₂₃	1	1	yes	characteristic
415	#332	C₂²×C₂₃	1	1	yes	characteristic
416	#332	C₂²×C₂₃	1	1	yes	characteristic
417	#332	C₂²×C₂₃	1	1	yes	characteristic
418	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
419	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
420	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
421	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
422	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
423	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
424	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
425	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
426	C₂×C₂₃	—	4	1	no	—
427	C₂×C₂₃	—	4	2	no	—
428	C₂×C₂₃	—	8	1	no	—
429	C₂×C₂₃	#3002	1	1	yes	characteristic
430	C₂×C₂₃	#3007	1	1	yes	characteristic
431	C₂×C₂₃	#3020	1	1	yes	characteristic
432	C₂×#41	—	2	1	no	—
433	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
434	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
435	C₂²×Q₈	—	2	1	no	—
436	C₂×Q₁₆	—	2	1	no	—
437	C₂×SD₁₆	—	2	1	no	—
438	C₂²×C₈	—	2	1	no	—
439	C₄⋊Q₈	—	2	1	no	—
440	C₄⋊Q₈	—	2	1	no	—
441	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
442	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
443	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
444	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
445	#126	—	2	2	no	—
446	#126	—	2	2	no	—
447	#126	—	2	2	no	—
448	#125	—	2	2	no	—
449	#125	—	2	2	no	—
450	#125	—	2	1	no	—
451	#125	—	2	1	no	—
452	#125	D₄×C₂₃	1	2	yes	—
453	#124	—	2	2	no	—
454	#124	—	2	1	no	—
455	#123	—	2	2	no	—
456	#123	—	2	2	no	—
457	#123	—	2	2	no	—
458	C₂²⋊Q₈	—	2	2	no	—
459	C₂²⋊Q₈	—	2	1	no	—
460	C₂²⋊Q₈	—	2	1	no	—
461	C₂²⋊Q₈	—	2	1	no	—
462	C₂²⋊Q₈	—	2	1	no	—
463	C₂²⋊Q₈	—	2	2	no	—
464	C₂²⋊Q₈	—	2	1	no	—
465	C₂²⋊Q₈	—	2	1	no	—
466	C₂²⋊Q₈	—	2	1	no	—
467	#121	—	2	2	no	—
468	#121	—	2	1	no	—
469	C₄×Q₈	—	2	2	no	—
470	C₄×Q₈	—	2	2	no	—
471	C₄×Q₈	—	2	1	no	—
472	C₄×Q₈	—	2	1	no	—
473	C₄×Q₈	—	2	1	no	—
474	C₄×Q₈	—	2	1	no	—
475	C₄×Q₈	—	2	1	no	—
476	C₄×Q₈	—	2	1	no	—
477	C₄×Q₈	—	2	1	no	—
478	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
479	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
480	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
481	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
482	C₄×D₄	—	2	2	no	—
483	C₄×D₄	—	2	2	no	—
484	C₄×D₄	—	2	1	no	—
485	C₄×D₄	—	2	1	no	—
486	C₄×D₄	—	2	1	no	—
487	C₄×D₄	—	2	1	no	—
488	C₄×D₄	—	2	1	no	—
489	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
490	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
491	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
492	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
493	#117	—	2	2	no	—
494	#117	—	2	2	no	—
495	#117	—	2	1	no	—
496	#117	—	2	1	no	—
497	#117	—	2	1	no	—
498	#117	—	2	1	no	—
499	#117	D₄×C₂₃	1	1	yes	characteristic
500	#117	D₄×C₂₃	1	1	yes	characteristic
501	#117	D₄×C₂₃	1	2	yes	—
502	C₂×C₄⋊C₄	—	2	2	no	—
503	C₂×C₄⋊C₄	—	2	1	no	—
504	C₂×C₄²	—	2	2	no	—
505	C₂×C₄²	—	2	1	no	—
506	C₂×C₄²	—	2	1	no	—
507	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
508	#107	—	2	1	no	—
509	#107	—	2	1	no	—
510	#107	—	2	1	no	—
511	#107	—	2	2	no	—
512	#106	—	2	1	no	—
513	#106	—	2	1	no	—
514	#106	—	2	1	no	—
515	#106	—	2	2	no	—
516	C₄⋊C₈	—	2	1	no	—
517	C₄⋊C₈	—	2	1	no	—
518	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
519	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
520	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
521	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
522	#103	—	2	1	no	—
523	#103	—	2	1	no	—
524	#103	—	2	1	no	—
525	#103	—	2	1	no	—
526	#103	—	2	1	no	—
527	#103	—	2	2	no	—
528	#103	—	2	2	no	—
529	#102	—	2	1	no	—
530	#102	—	2	1	no	—
531	#102	—	2	1	no	—
532	C₂²⋊C₈	—	2	1	no	—
533	C₂²⋊C₈	—	2	1	no	—
534	#97	—	2	1	no	—
535	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
536	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
537	C₂₃	#18765	1	1	yes	characteristic, direct product factor, Sylow
538	#41	—	2	2	no	—
539	#41	—	2	2	no	—
540	#41	—	4	1	no	—
541	#41	—	4	1	no	—
542	#41	—	4	2	no	—
543	#41	—	4	2	no	—
544	C₂×Q₈	—	2	2	no	—
545	C₂×Q₈	—	2	1	no	—
546	C₂×Q₈	—	2	1	no	—
547	C₂×Q₈	—	2	1	no	—
548	C₂×Q₈	—	2	1	no	—
549	C₂×Q₈	—	4	2	no	—
550	C₂×Q₈	—	4	2	no	—
551	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
552	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
553	C₂×D₄	—	2	2	no	—
554	C₂×D₄	—	2	1	no	—
555	C₂×D₄	—	2	1	no	—
556	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
557	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
558	C₂²×C₄	—	2	2	no	—
559	C₂²×C₄	—	2	2	no	—
560	C₂²×C₄	—	2	1	no	—
561	C₂²×C₄	—	2	1	no	—
562	C₂²×C₄	—	2	1	no	—
563	C₂²×C₄	—	2	1	no	—
564	C₂²×C₄	—	2	1	no	—
565	C₂²×C₄	—	2	1	no	—
566	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
567	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
568	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
569	Q₁₆	—	4	2	no	—
570	SD₁₆	—	4	2	no	—
571	C₂×C₈	—	2	1	no	—
572	C₂×C₈	—	2	1	no	—
573	C₂×C₈	—	2	1	no	—
574	C₂×C₈	—	2	1	no	—
575	C₂×C₈	—	2	1	no	—
576	C₂×C₈	—	2	1	no	—
577	C₂×C₈	—	4	2	no	—
578	C₄⋊C₄	—	2	2	no	—
579	C₄⋊C₄	—	2	2	no	—
580	C₄⋊C₄	—	2	2	no	—
581	C₄⋊C₄	—	2	2	no	—
582	C₄⋊C₄	—	2	2	no	—
583	C₄⋊C₄	—	2	2	no	—
584	C₄⋊C₄	—	2	1	no	—
585	C₄⋊C₄	—	2	2	no	—
586	C₄⋊C₄	—	2	2	no	—
587	C₄⋊C₄	—	2	1	no	—
588	C₄⋊C₄	—	2	1	no	—
589	C₄⋊C₄	—	2	1	no	—
590	C₄⋊C₄	—	2	1	no	—
591	C₄⋊C₄	—	2	1	no	—
592	C₄⋊C₄	—	2	1	no	—
593	C₄⋊C₄	—	2	1	no	—
594	C₄⋊C₄	—	2	1	no	—
595	C₄⋊C₄	—	2	1	no	—
596	C₄⋊C₄	—	4	1	no	—
597	C₄⋊C₄	—	4	1	no	—
598	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
599	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
600	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
601	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
602	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
603	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
604	C₂²⋊C₄	—	2	2	no	—
605	C₂²⋊C₄	—	2	2	no	—
606	C₂²⋊C₄	—	2	2	no	—
607	C₂²⋊C₄	—	2	2	no	—
608	C₂²⋊C₄	—	2	1	no	—
609	C₂²⋊C₄	—	2	1	no	—
610	C₂²⋊C₄	—	2	2	no	—
611	C₂²⋊C₄	—	2	1	no	—
612	C₂²⋊C₄	—	2	1	no	—
613	C₂²⋊C₄	—	2	1	no	—
614	C₂²⋊C₄	—	2	1	no	—
615	C₂²⋊C₄	—	4	1	no	—
616	C₄²	—	2	2	no	—
617	C₄²	—	2	2	no	—
618	C₄²	—	2	1	no	—
619	C₄²	—	2	1	no	—
620	C₄²	—	2	1	no	—
621	C₄²	—	2	2	no	—
622	C₄²	—	2	1	no	—
623	C₄²	—	2	1	no	—
624	C₄²	—	2	1	no	—
625	C₄²	—	4	1	no	—
626	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
627	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
628	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
629	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
630	C₂³	—	2	1	no	—
631	C₂³	—	2	1	no	—
632	C₂³	C₂₃×#120	1	1	yes	characteristic
633	Q₈	—	2	1	no	—
634	Q₈	—	2	1	no	—
635	Q₈	—	2	1	no	—
636	Q₈	—	4	4	no	—
637	Q₈	—	4	2	no	—
638	Q₈	—	4	1	no	—
639	Q₈	—	4	1	no	—
640	Q₈	—	4	1	no	—
641	Q₈	—	4	1	no	—
642	Q₈	—	4	1	no	—
643	Q₈	—	4	1	no	—
644	Q₈	C₂₃×#137	1	2	yes	—
645	D₄	—	2	1	no	—
646	D₄	—	2	1	no	—
647	D₄	—	2	1	no	—
648	D₄	—	4	2	no	—
649	D₄	—	4	2	no	—
650	D₄	—	4	4	no	—
651	D₄	C₂₃×#135	1	2	yes	—
652	C₂×C₄	—	2	1	no	—
653	C₂×C₄	—	2	1	no	—
654	C₂×C₄	—	2	1	no	—
655	C₂×C₄	—	2	1	no	—
656	C₂×C₄	—	2	1	no	—
657	C₂×C₄	—	2	1	no	—
658	C₂×C₄	—	2	1	no	—
659	C₂×C₄	—	2	1	no	—
660	C₂×C₄	—	2	2	no	—
661	C₂×C₄	—	2	1	no	—
662	C₂×C₄	—	2	1	no	—
663	C₂×C₄	—	2	2	no	—
664	C₂×C₄	—	4	1	no	—
665	C₂×C₄	—	4	2	no	—
666	C₂×C₄	—	4	2	no	—
667	C₂×C₄	—	4	4	no	—
668	C₂×C₄	—	4	2	no	—
669	C₂×C₄	—	4	2	no	—
670	C₂×C₄	—	4	4	no	—
671	C₂×C₄	—	4	1	no	—
672	C₂×C₄	—	4	1	no	—
673	C₂×C₄	—	4	1	no	—
674	C₂×C₄	—	4	1	no	—
675	C₂×C₄	—	4	1	no	—
676	C₂×C₄	—	4	1	no	—
677	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
678	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
679	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
680	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
681	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
682	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
683	C₈	—	4	1	no	—
684	C₈	—	4	1	no	—
685	C₈	—	4	1	no	—
686	C₈	—	4	1	no	—
687	C₈	—	4	1	no	—
688	C₈	—	4	1	no	—
689	C₂²	—	2	1	no	—
690	C₂²	—	2	2	no	—
691	C₂²	—	2	1	no	—
692	C₂²	—	2	1	no	—
693	C₂²	—	4	1	no	—
694	C₂²	—	4	1	no	—
695	C₂²	—	4	1	no	—
696	C₂²	—	4	1	no	—
697	C₂²	—	4	1	no	—
698	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
699	C₄	—	2	1	no	—
700	C₄	—	2	1	no	—
701	C₄	—	2	2	no	—
702	C₄	—	2	1	no	—
703	C₄	—	2	1	no	—
704	C₄	—	4	1	no	—
705	C₄	—	4	1	no	—
706	C₄	—	4	2	no	—
707	C₄	—	4	2	no	—
708	C₄	—	4	1	no	—
709	C₄	—	4	1	no	—
710	C₄	—	4	1	no	—
711	C₄	—	8	1	no	—
712	C₄	—	8	1	no	—
713	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
714	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
715	C₄	C₂₃×#575	1	1	yes	characteristic
716	C₄	C₂₃×#575	1	1	yes	characteristic
717	C₂	—	4	1	no	—
718	C₂	—	4	2	no	—
719	C₂	—	8	1	no	—
720	C₂	C₂₃×#3002	1	1	yes	characteristic
721	C₂	C₂₃×#3007	1	1	yes	characteristic
722	C₂	C₂₃×#3020	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3482	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×#3231	C₂	semidirect
3. C₂₃×#3116	C₂	semidirect
4. C₂₃×#3076	C₂	semidirect
5. C₂₃×#1652	C₂	semidirect
6. C₂₃×#1651	C₂	semidirect
7. C₂₃×#1648	C₂	semidirect
8. C₂₃×#1647	C₂	semidirect
9. C₂₃×#1638	C₂	semidirect
10. C₂₃×#1637	C₂	semidirect
11. C₂₃×#1636	C₂	semidirect
12. C₂₃×#1635	C₂	semidirect
13. C₂₃×#1552	C₂	semidirect
14. C₂₃×#1488	C₂	semidirect
15. C₂₃×#1477	C₂	semidirect
268. #18765	C₂₃	direct (N × H)
329. #3231	C₂×C₂₃	semidirect
330. #3116	C₂×C₂₃	semidirect
331. #3076	C₂×C₂₃	semidirect
332. #1652	C₂×C₂₃	semidirect
333. #1651	C₂×C₂₃	semidirect
334. #1648	C₂×C₂₃	semidirect
335. #1647	C₂×C₂₃	semidirect
336. #1638	C₂×C₂₃	semidirect
337. #1637	C₂×C₂₃	semidirect
338. #1636	C₂×C₂₃	semidirect
339. #1635	C₂×C₂₃	semidirect
340. #1552	C₂×C₂₃	semidirect
341. #1488	C₂×C₂₃	semidirect
342. #1477	C₂×C₂₃	semidirect
537. C₂₃	#18765	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×#3482	2
C₂₃×#3231	2
C₂₃×#3116	2
C₂₃×#3076	2
C₂₃×#1652	2
C₂₃×#1651	2
C₂₃×#1648	2
C₂₃×#1647	2
C₂₃×#1638	2
C₂₃×#1637	2
C₂₃×#1636	2
C₂₃×#1635	2
C₂₃×#1552	2
C₂₃×#1488	2
C₂₃×#1477	2
#18765	23

Subgroups — C23×#18765 #19166391

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18765 #19166391