Subgroups — C₂₃×#18673 #19166299

Order 5888.

872 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3518	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3233	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3116	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1650	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1649	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1646	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1645	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×#1634	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1633	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1554	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	Q₈²×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#556	—	2	1	no	—
18	C₂₃×#556	—	2	2	no	—
19	C₂₃×#556	—	2	1	no	—
20	C₂₃×#556	C₂²	1	1	yes	characteristic
21	C₂₃×#555	—	2	1	no	—
22	C₂₃×#555	C₂²	1	1	yes	characteristic
23	D₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#548	—	2	1	no	—
25	C₂₃×#542	—	2	2	no	—
26	C₂₃×#542	—	2	1	no	—
27	C₂₃×#533	—	2	1	no	—
28	C₂₃×#533	C₂²	1	1	yes	characteristic
29	C₂₃×#533	C₂²	1	1	yes	characteristic
30	C₂₃×#529	—	2	1	no	—
31	C₂₃×#529	C₂²	1	1	yes	characteristic
32	C₂₃×#519	—	2	1	no	—
33	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
34	C₂₃×#500	C₂²	1	1	yes	characteristic
35	C₂₃×#499	—	2	1	no	—
36	C₂₃×#494	C₂²	1	1	yes	characteristic
37	C₂₃×#488	—	2	1	no	—
38	C₂₃×#487	C₂²	1	1	yes	characteristic
39	C₂₃×#487	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×#467	—	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×Q₁₆×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×#333	C₂²	1	1	yes	characteristic
60	C₂₃×#333	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	D₄	1	1	yes	characteristic
72	C₂²×Q₈×C₂₃	—	2	1	no	—
73	C₂×Q₁₆×C₂₃	—	2	2	no	—
74	C₂×Q₁₆×C₂₃	—	2	1	no	—
75	C₂×SD₁₆×C₂₃	—	2	1	no	—
76	C₂²×C₈×C₂₃	—	2	1	no	—
77	C₂₃×C₄⋊Q₈	—	2	2	no	—
78	C₂₃×C₄⋊Q₈	—	2	1	no	—
79	C₂₃×C₄⋊Q₈	—	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×C₄⋊Q₈	C₂³	1	1	yes	characteristic
85	C₂₃×#126	—	2	2	no	—
86	C₂₃×#126	—	2	2	no	—
87	C₂₃×#126	—	2	2	no	—
88	C₂₃×#126	—	2	2	no	—
89	C₂₃×#126	—	2	1	no	—
90	C₂₃×#125	—	2	1	no	—
91	C₂₃×#125	—	2	1	no	—
92	C₂₃×#125	D₄	1	1	yes	characteristic
93	C₂₃×#124	—	2	2	no	—
94	C₂₃×#124	—	2	2	no	—
95	C₂₃×#124	—	2	2	no	—
96	C₂₃×#124	—	2	2	no	—
97	C₂₃×#124	—	2	1	no	—
98	C₂₃×#124	D₄	1	1	yes	characteristic
99	C₂₃×#123	—	2	2	no	—
100	C₂₃×#123	—	2	1	no	—
101	C₂₃×C₂²⋊Q₈	—	2	2	no	—
102	C₂₃×C₂²⋊Q₈	—	2	2	no	—
103	C₂₃×C₂²⋊Q₈	—	2	1	no	—
104	C₂₃×C₂²⋊Q₈	—	2	1	no	—
105	C₂₃×C₂²⋊Q₈	—	2	1	no	—
106	C₂₃×C₂²⋊Q₈	—	2	2	no	—
107	C₂₃×C₂²⋊Q₈	—	2	1	no	—
108	C₂₃×C₂²⋊Q₈	—	2	1	no	—
109	C₂₃×C₂²⋊Q₈	D₄	1	1	yes	characteristic
110	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
111	C₂₃×#121	—	2	2	no	—
112	C₂₃×#121	—	2	1	no	—
113	C₂₃×#121	—	2	1	no	—
114	C₂₃×#121	D₄	1	1	yes	characteristic
115	C₄×Q₈×C₂₃	—	2	2	no	—
116	C₄×Q₈×C₂₃	—	2	2	no	—
117	C₄×Q₈×C₂₃	—	2	2	no	—
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	—	2	1	no	—
123	C₄×Q₈×C₂₃	—	2	1	no	—
124	C₄×Q₈×C₂₃	D₄	1	2	yes	—
125	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
126	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
127	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
134	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
135	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
136	C₂₃×#117	—	2	2	no	—
137	C₂₃×#117	—	2	2	no	—
138	C₂₃×#117	—	2	1	no	—
139	C₂₃×#117	—	2	1	no	—
140	C₂₃×#117	—	2	1	no	—
141	C₂₃×#117	—	2	1	no	—
142	C₂₃×#117	—	2	1	no	—
143	C₂₃×#117	—	2	1	no	—
144	C₂₃×#117	—	2	1	no	—
145	C₂₃×#117	D₄	1	1	yes	characteristic
146	C₂×C₄²×C₂₃	—	2	1	no	—
147	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
148	C₂₃×#107	—	2	2	no	—
149	C₂₃×#107	—	2	1	no	—
150	C₂₃×#107	—	2	1	no	—
151	C₂₃×#107	—	2	1	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	2	no	—
160	C₂₃×#103	—	2	1	no	—
161	C₂₃×#103	—	2	1	no	—
162	C₂₃×#103	—	2	1	no	—
163	C₂₃×#103	—	2	2	no	—
164	C₂₃×#103	—	2	1	no	—
165	C₂₃×#103	—	2	1	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	C₂×Q₈×C₂₃	—	2	2	no	—
179	C₂×Q₈×C₂₃	—	2	2	no	—
180	C₂×Q₈×C₂₃	—	2	1	no	—
181	C₂×Q₈×C₂₃	—	2	1	no	—
182	C₂×Q₈×C₂₃	—	2	1	no	—
183	C₂×Q₈×C₂₃	—	4	2	no	—
184	C₂×Q₈×C₂₃	—	4	2	no	—
185	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
186	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
187	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
188	C₂×D₄×C₂₃	—	2	1	no	—
189	C₂×D₄×C₂₃	—	2	2	no	—
190	C₂×D₄×C₂₃	—	2	1	no	—
191	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
192	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
193	C₂²×C₄×C₂₃	—	2	2	no	—
194	C₂²×C₄×C₂₃	—	2	1	no	—
195	C₂²×C₄×C₂₃	—	2	1	no	—
196	C₂²×C₄×C₂₃	—	2	1	no	—
197	C₂²×C₄×C₂₃	—	2	1	no	—
198	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
199	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
200	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
201	Q₁₆×C₂₃	—	4	4	no	—
202	Q₁₆×C₂₃	—	4	2	no	—
203	SD₁₆×C₂₃	—	4	2	no	—
204	C₂×C₈×C₂₃	—	2	1	no	—
205	C₂×C₈×C₂₃	—	2	1	no	—
206	C₂×C₈×C₂₃	—	2	1	no	—
207	C₂×C₈×C₂₃	—	2	1	no	—
208	C₂×C₈×C₂₃	—	2	1	no	—
209	C₂×C₈×C₂₃	—	2	1	no	—
210	C₂×C₈×C₂₃	—	4	2	no	—
211	C₄⋊C₄×C₂₃	—	2	2	no	—
212	C₄⋊C₄×C₂₃	—	2	2	no	—
213	C₄⋊C₄×C₂₃	—	2	2	no	—
214	C₄⋊C₄×C₂₃	—	2	2	no	—
215	C₄⋊C₄×C₂₃	—	2	1	no	—
216	C₄⋊C₄×C₂₃	—	2	1	no	—
217	C₄⋊C₄×C₂₃	—	2	2	no	—
218	C₄⋊C₄×C₂₃	—	2	1	no	—
219	C₄⋊C₄×C₂₃	—	2	1	no	—
220	C₄⋊C₄×C₂₃	—	2	2	no	—
221	C₄⋊C₄×C₂₃	—	2	1	no	—
222	C₄⋊C₄×C₂₃	—	2	1	no	—
223	C₄⋊C₄×C₂₃	—	2	1	no	—
224	C₄⋊C₄×C₂₃	—	2	1	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	—	4	1	no	—
231	C₄⋊C₄×C₂₃	—	4	1	no	—
232	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
233	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
234	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
235	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
236	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
237	C₂²⋊C₄×C₂₃	—	2	2	no	—
238	C₂²⋊C₄×C₂₃	—	2	2	no	—
239	C₂²⋊C₄×C₂₃	—	2	2	no	—
240	C₂²⋊C₄×C₂₃	—	2	2	no	—
241	C₂²⋊C₄×C₂₃	—	2	2	no	—
242	C₂²⋊C₄×C₂₃	—	2	2	no	—
243	C₂²⋊C₄×C₂₃	—	2	1	no	—
244	C₂²⋊C₄×C₂₃	—	2	1	no	—
245	C₂²⋊C₄×C₂₃	—	2	1	no	—
246	C₂²⋊C₄×C₂₃	—	2	1	no	—
247	C₂²⋊C₄×C₂₃	—	2	1	no	—
248	C₂²⋊C₄×C₂₃	—	2	1	no	—
249	C₂²⋊C₄×C₂₃	—	2	1	no	—
250	C₂²⋊C₄×C₂₃	—	2	1	no	—
251	C₂²⋊C₄×C₂₃	—	4	1	no	—
252	C₄²×C₂₃	—	2	2	no	—
253	C₄²×C₂₃	—	2	2	no	—
254	C₄²×C₂₃	—	2	2	no	—
255	C₄²×C₂₃	—	2	1	no	—
256	C₄²×C₂₃	—	2	1	no	—
257	C₄²×C₂₃	—	2	2	no	—
258	C₄²×C₂₃	—	2	1	no	—
259	C₄²×C₂₃	—	2	1	no	—
260	C₄²×C₂₃	—	4	1	no	—
261	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
262	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
263	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
264	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
265	#18673	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
266	C₂³×C₂₃	—	2	1	no	—
267	C₂³×C₂₃	—	2	1	no	—
268	C₂³×C₂₃	#120	1	1	yes	characteristic
269	Q₈×C₂₃	—	2	1	no	—
270	Q₈×C₂₃	—	2	1	no	—
271	Q₈×C₂₃	—	2	1	no	—
272	Q₈×C₂₃	—	4	2	no	—
273	Q₈×C₂₃	—	4	4	no	—
274	Q₈×C₂₃	—	4	2	no	—
275	Q₈×C₂₃	—	4	1	no	—
276	Q₈×C₂₃	—	4	1	no	—
277	Q₈×C₂₃	—	4	1	no	—
278	Q₈×C₂₃	—	4	1	no	—
279	Q₈×C₂₃	—	8	2	no	—
280	Q₈×C₂₃	#135	1	2	yes	—
281	D₄×C₂₃	—	2	1	no	—
282	D₄×C₂₃	—	2	1	no	—
283	D₄×C₂₃	—	2	1	no	—
284	D₄×C₂₃	—	2	1	no	—
285	D₄×C₂₃	—	4	2	no	—
286	D₄×C₂₃	—	8	1	no	—
287	D₄×C₂₃	—	8	2	no	—
288	C₂×C₄×C₂₃	—	2	1	no	—
289	C₂×C₄×C₂₃	—	2	1	no	—
290	C₂×C₄×C₂₃	—	2	1	no	—
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	2	no	—
296	C₂×C₄×C₂₃	—	2	1	no	—
297	C₂×C₄×C₂₃	—	2	1	no	—
298	C₂×C₄×C₂₃	—	2	2	no	—
299	C₂×C₄×C₂₃	—	2	1	no	—
300	C₂×C₄×C₂₃	—	4	1	no	—
301	C₂×C₄×C₂₃	—	4	2	no	—
302	C₂×C₄×C₂₃	—	4	4	no	—
303	C₂×C₄×C₂₃	—	4	1	no	—
304	C₂×C₄×C₂₃	—	4	1	no	—
305	C₂×C₄×C₂₃	—	4	1	no	—
306	C₂×C₄×C₂₃	—	4	1	no	—
307	C₂×C₄×C₂₃	—	4	1	no	—
308	C₂×C₄×C₂₃	—	4	1	no	—
309	C₂×C₄×C₂₃	—	8	1	no	—
310	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
311	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
312	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
313	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
314	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
315	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
316	C₈×C₂₃	—	2	2	no	—
317	C₈×C₂₃	—	4	1	no	—
318	C₈×C₂₃	—	4	1	no	—
319	C₈×C₂₃	—	4	1	no	—
320	C₈×C₂₃	—	4	1	no	—
321	C₈×C₂₃	—	4	1	no	—
322	#3518	C₂×C₂₃	1	1	yes	characteristic
323	#3233	C₂×C₂₃	1	1	yes	characteristic, complemented
324	#3170	C₂×C₂₃	1	1	yes	characteristic, complemented
325	#3116	C₂×C₂₃	1	1	yes	characteristic, complemented
326	#1650	C₂×C₂₃	1	1	yes	characteristic, complemented
327	#1649	C₂×C₂₃	1	1	yes	characteristic, complemented
328	#1646	C₂×C₂₃	1	1	yes	characteristic, complemented
329	#1645	C₂×C₂₃	1	1	yes	characteristic, complemented
330	#1638	C₂×C₂₃	1	1	yes	characteristic, complemented
331	#1637	C₂×C₂₃	1	1	yes	characteristic, complemented
332	#1634	C₂×C₂₃	1	1	yes	characteristic, complemented
333	#1633	C₂×C₂₃	1	1	yes	characteristic, complemented
334	#1554	C₂×C₂₃	1	1	yes	characteristic, complemented
335	#1494	C₂×C₂₃	1	1	yes	characteristic, complemented
336	#1476	C₂×C₂₃	1	1	yes	characteristic, complemented
337	C₂²×C₂₃	—	2	1	no	—
338	C₂²×C₂₃	—	2	1	no	—
339	C₂²×C₂₃	—	2	1	no	—
340	C₂²×C₂₃	—	4	1	no	—
341	C₂²×C₂₃	—	4	1	no	—
342	C₂²×C₂₃	—	4	1	no	—
343	C₂²×C₂₃	—	4	1	no	—
344	C₂²×C₂₃	—	4	1	no	—
345	C₂²×C₂₃	—	4	1	no	—
346	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
347	C₄×C₂₃	—	2	1	no	—
348	C₄×C₂₃	—	2	2	no	—
349	C₄×C₂₃	—	2	1	no	—
350	C₄×C₂₃	—	2	1	no	—
351	C₄×C₂₃	—	2	1	no	—
352	C₄×C₂₃	—	4	2	no	—
353	C₄×C₂₃	—	4	1	no	—
354	C₄×C₂₃	—	4	1	no	—
355	C₄×C₂₃	—	4	1	no	—
356	C₄×C₂₃	—	4	1	no	—
357	C₄×C₂₃	—	4	2	no	—
358	C₄×C₂₃	—	4	1	no	—
359	C₄×C₂₃	—	8	1	no	—
360	C₄×C₂₃	—	8	1	no	—
361	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
362	C₄×C₂₃	#575	1	1	yes	characteristic
363	C₄×C₂₃	#578	1	1	yes	characteristic
364	C₄×C₂₃	#578	1	1	yes	characteristic
365	Q₈²	C₂²×C₂₃	1	1	yes	characteristic
366	#556	—	2	1	no	—
367	#556	—	2	2	no	—
368	#556	—	2	1	no	—
369	#556	C₂²×C₂₃	1	1	yes	characteristic
370	#555	—	2	1	no	—
371	#555	C₂²×C₂₃	1	1	yes	characteristic
372	D₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
373	#548	—	2	1	no	—
374	#542	—	2	2	no	—
375	#542	—	2	1	no	—
376	#533	—	2	1	no	—
377	#533	C₂²×C₂₃	1	1	yes	characteristic
378	#533	C₂²×C₂₃	1	1	yes	characteristic
379	#529	—	2	1	no	—
380	#529	C₂²×C₂₃	1	1	yes	characteristic
381	#519	—	2	1	no	—
382	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
383	#500	C₂²×C₂₃	1	1	yes	characteristic
384	#499	—	2	1	no	—
385	#494	C₂²×C₂₃	1	1	yes	characteristic
386	#488	—	2	1	no	—
387	#487	C₂²×C₂₃	1	1	yes	characteristic
388	#487	C₂²×C₂₃	1	1	yes	characteristic
389	#485	—	2	1	no	—
390	#485	—	2	1	no	—
391	#482	—	2	1	no	—
392	#477	C₂²×C₂₃	1	1	yes	characteristic
393	#477	C₂²×C₂₃	1	1	yes	characteristic
394	#476	C₂²×C₂₃	1	1	yes	characteristic
395	#476	C₂²×C₂₃	1	1	yes	characteristic
396	#475	C₂²×C₂₃	1	1	yes	characteristic
397	#474	C₂²×C₂₃	1	1	yes	characteristic
398	#467	—	2	2	no	—
399	#462	C₂²×C₂₃	1	1	yes	characteristic
400	#461	C₂²×C₂₃	1	1	yes	characteristic
401	#452	—	2	1	no	—
402	#441	—	2	1	no	—
403	#440	—	2	1	no	—
404	C₄×Q₁₆	—	2	2	no	—
405	#432	C₂²×C₂₃	1	1	yes	characteristic
406	#431	C₂²×C₂₃	1	1	yes	characteristic
407	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
408	#333	C₂²×C₂₃	1	1	yes	characteristic
409	#333	C₂²×C₂₃	1	1	yes	characteristic
410	#332	C₂²×C₂₃	1	1	yes	characteristic
411	#332	C₂²×C₂₃	1	1	yes	characteristic
412	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
413	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
414	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
415	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
416	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
417	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
418	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
419	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
420	C₂×C₂₃	—	4	1	no	—
421	C₂×C₂₃	—	8	1	no	—
422	C₂×C₂₃	—	8	1	no	—
423	C₂×C₂₃	#3002	1	1	yes	characteristic
424	C₂×C₂₃	#3011	1	1	yes	characteristic
425	C₂×C₂₃	#3018	1	1	yes	characteristic
426	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
427	C₂²×Q₈	—	2	1	no	—
428	C₂×Q₁₆	—	2	2	no	—
429	C₂×Q₁₆	—	2	1	no	—
430	C₂×SD₁₆	—	2	1	no	—
431	C₂²×C₈	—	2	1	no	—
432	C₄⋊Q₈	—	2	2	no	—
433	C₄⋊Q₈	—	2	1	no	—
434	C₄⋊Q₈	—	2	1	no	—
435	C₄⋊Q₈	—	2	1	no	—
436	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
437	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
438	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
439	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
440	#126	—	2	2	no	—
441	#126	—	2	2	no	—
442	#126	—	2	2	no	—
443	#126	—	2	2	no	—
444	#126	—	2	1	no	—
445	#125	—	2	1	no	—
446	#125	—	2	1	no	—
447	#125	D₄×C₂₃	1	1	yes	characteristic
448	#124	—	2	2	no	—
449	#124	—	2	2	no	—
450	#124	—	2	2	no	—
451	#124	—	2	2	no	—
452	#124	—	2	1	no	—
453	#124	D₄×C₂₃	1	1	yes	characteristic
454	#123	—	2	2	no	—
455	#123	—	2	1	no	—
456	C₂²⋊Q₈	—	2	2	no	—
457	C₂²⋊Q₈	—	2	2	no	—
458	C₂²⋊Q₈	—	2	1	no	—
459	C₂²⋊Q₈	—	2	1	no	—
460	C₂²⋊Q₈	—	2	1	no	—
461	C₂²⋊Q₈	—	2	2	no	—
462	C₂²⋊Q₈	—	2	1	no	—
463	C₂²⋊Q₈	—	2	1	no	—
464	C₂²⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
465	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
466	#121	—	2	2	no	—
467	#121	—	2	1	no	—
468	#121	—	2	1	no	—
469	#121	D₄×C₂₃	1	1	yes	characteristic
470	C₄×Q₈	—	2	2	no	—
471	C₄×Q₈	—	2	2	no	—
472	C₄×Q₈	—	2	2	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	—	2	1	no	—
479	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
480	C₄×Q₈	D₄×C₂₃	1	2	yes	—
481	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
482	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
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	D₄×C₂₃	1	1	yes	characteristic
489	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
490	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
491	#117	—	2	2	no	—
492	#117	—	2	2	no	—
493	#117	—	2	1	no	—
494	#117	—	2	1	no	—
495	#117	—	2	1	no	—
496	#117	—	2	1	no	—
497	#117	—	2	1	no	—
498	#117	—	2	1	no	—
499	#117	—	2	1	no	—
500	#117	D₄×C₂₃	1	1	yes	characteristic
501	C₂×C₄²	—	2	1	no	—
502	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
503	#107	—	2	2	no	—
504	#107	—	2	1	no	—
505	#107	—	2	1	no	—
506	#107	—	2	1	no	—
507	#106	—	2	1	no	—
508	#106	—	2	1	no	—
509	#106	—	2	1	no	—
510	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
511	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
512	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
513	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
514	#103	—	2	2	no	—
515	#103	—	2	1	no	—
516	#103	—	2	1	no	—
517	#103	—	2	1	no	—
518	#103	—	2	2	no	—
519	#103	—	2	1	no	—
520	#103	—	2	1	no	—
521	#102	—	2	1	no	—
522	#102	—	2	1	no	—
523	#102	—	2	1	no	—
524	C₂²⋊C₈	—	2	1	no	—
525	C₂²⋊C₈	—	2	1	no	—
526	#97	—	2	1	no	—
527	C₄×C₈	—	2	1	no	—
528	C₄×C₈	—	2	1	no	—
529	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
530	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
531	C₂₃	#18673	1	1	yes	characteristic, direct product factor, Sylow
532	#41	—	2	2	no	—
533	#41	—	4	1	no	—
534	C₂×Q₈	—	2	2	no	—
535	C₂×Q₈	—	2	2	no	—
536	C₂×Q₈	—	2	1	no	—
537	C₂×Q₈	—	2	1	no	—
538	C₂×Q₈	—	2	1	no	—
539	C₂×Q₈	—	4	2	no	—
540	C₂×Q₈	—	4	2	no	—
541	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
542	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
543	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
544	C₂×D₄	—	2	1	no	—
545	C₂×D₄	—	2	2	no	—
546	C₂×D₄	—	2	1	no	—
547	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
548	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
549	C₂²×C₄	—	2	2	no	—
550	C₂²×C₄	—	2	1	no	—
551	C₂²×C₄	—	2	1	no	—
552	C₂²×C₄	—	2	1	no	—
553	C₂²×C₄	—	2	1	no	—
554	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
555	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
556	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
557	Q₁₆	—	4	4	no	—
558	Q₁₆	—	4	2	no	—
559	SD₁₆	—	4	2	no	—
560	C₂×C₈	—	2	1	no	—
561	C₂×C₈	—	2	1	no	—
562	C₂×C₈	—	2	1	no	—
563	C₂×C₈	—	2	1	no	—
564	C₂×C₈	—	2	1	no	—
565	C₂×C₈	—	2	1	no	—
566	C₂×C₈	—	4	2	no	—
567	C₄⋊C₄	—	2	2	no	—
568	C₄⋊C₄	—	2	2	no	—
569	C₄⋊C₄	—	2	2	no	—
570	C₄⋊C₄	—	2	2	no	—
571	C₄⋊C₄	—	2	1	no	—
572	C₄⋊C₄	—	2	1	no	—
573	C₄⋊C₄	—	2	2	no	—
574	C₄⋊C₄	—	2	1	no	—
575	C₄⋊C₄	—	2	1	no	—
576	C₄⋊C₄	—	2	2	no	—
577	C₄⋊C₄	—	2	1	no	—
578	C₄⋊C₄	—	2	1	no	—
579	C₄⋊C₄	—	2	1	no	—
580	C₄⋊C₄	—	2	1	no	—
581	C₄⋊C₄	—	2	1	no	—
582	C₄⋊C₄	—	2	1	no	—
583	C₄⋊C₄	—	2	1	no	—
584	C₄⋊C₄	—	2	1	no	—
585	C₄⋊C₄	—	2	1	no	—
586	C₄⋊C₄	—	4	1	no	—
587	C₄⋊C₄	—	4	1	no	—
588	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
589	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
590	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
591	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
592	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
593	C₂²⋊C₄	—	2	2	no	—
594	C₂²⋊C₄	—	2	2	no	—
595	C₂²⋊C₄	—	2	2	no	—
596	C₂²⋊C₄	—	2	2	no	—
597	C₂²⋊C₄	—	2	2	no	—
598	C₂²⋊C₄	—	2	2	no	—
599	C₂²⋊C₄	—	2	1	no	—
600	C₂²⋊C₄	—	2	1	no	—
601	C₂²⋊C₄	—	2	1	no	—
602	C₂²⋊C₄	—	2	1	no	—
603	C₂²⋊C₄	—	2	1	no	—
604	C₂²⋊C₄	—	2	1	no	—
605	C₂²⋊C₄	—	2	1	no	—
606	C₂²⋊C₄	—	2	1	no	—
607	C₂²⋊C₄	—	4	1	no	—
608	C₄²	—	2	2	no	—
609	C₄²	—	2	2	no	—
610	C₄²	—	2	2	no	—
611	C₄²	—	2	1	no	—
612	C₄²	—	2	1	no	—
613	C₄²	—	2	2	no	—
614	C₄²	—	2	1	no	—
615	C₄²	—	2	1	no	—
616	C₄²	—	4	1	no	—
617	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
618	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
619	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
620	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
621	C₂³	—	2	1	no	—
622	C₂³	—	2	1	no	—
623	C₂³	C₂₃×#120	1	1	yes	characteristic
624	Q₈	—	2	1	no	—
625	Q₈	—	2	1	no	—
626	Q₈	—	2	1	no	—
627	Q₈	—	4	2	no	—
628	Q₈	—	4	4	no	—
629	Q₈	—	4	2	no	—
630	Q₈	—	4	1	no	—
631	Q₈	—	4	1	no	—
632	Q₈	—	4	1	no	—
633	Q₈	—	4	1	no	—
634	Q₈	—	8	2	no	—
635	Q₈	C₂₃×#135	1	2	yes	—
636	D₄	—	2	1	no	—
637	D₄	—	2	1	no	—
638	D₄	—	2	1	no	—
639	D₄	—	2	1	no	—
640	D₄	—	4	2	no	—
641	D₄	—	8	1	no	—
642	D₄	—	8	2	no	—
643	C₂×C₄	—	2	1	no	—
644	C₂×C₄	—	2	1	no	—
645	C₂×C₄	—	2	1	no	—
646	C₂×C₄	—	2	1	no	—
647	C₂×C₄	—	2	1	no	—
648	C₂×C₄	—	2	1	no	—
649	C₂×C₄	—	2	1	no	—
650	C₂×C₄	—	2	2	no	—
651	C₂×C₄	—	2	1	no	—
652	C₂×C₄	—	2	1	no	—
653	C₂×C₄	—	2	2	no	—
654	C₂×C₄	—	2	1	no	—
655	C₂×C₄	—	4	1	no	—
656	C₂×C₄	—	4	2	no	—
657	C₂×C₄	—	4	4	no	—
658	C₂×C₄	—	4	1	no	—
659	C₂×C₄	—	4	1	no	—
660	C₂×C₄	—	4	1	no	—
661	C₂×C₄	—	4	1	no	—
662	C₂×C₄	—	4	1	no	—
663	C₂×C₄	—	4	1	no	—
664	C₂×C₄	—	8	1	no	—
665	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
666	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
667	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
668	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
669	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
670	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
671	C₈	—	2	2	no	—
672	C₈	—	4	1	no	—
673	C₈	—	4	1	no	—
674	C₈	—	4	1	no	—
675	C₈	—	4	1	no	—
676	C₈	—	4	1	no	—
677	C₂²	—	2	1	no	—
678	C₂²	—	2	1	no	—
679	C₂²	—	2	1	no	—
680	C₂²	—	4	1	no	—
681	C₂²	—	4	1	no	—
682	C₂²	—	4	1	no	—
683	C₂²	—	4	1	no	—
684	C₂²	—	4	1	no	—
685	C₂²	—	4	1	no	—
686	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
687	C₄	—	2	1	no	—
688	C₄	—	2	2	no	—
689	C₄	—	2	1	no	—
690	C₄	—	2	1	no	—
691	C₄	—	2	1	no	—
692	C₄	—	4	2	no	—
693	C₄	—	4	1	no	—
694	C₄	—	4	1	no	—
695	C₄	—	4	1	no	—
696	C₄	—	4	1	no	—
697	C₄	—	4	2	no	—
698	C₄	—	4	1	no	—
699	C₄	—	8	1	no	—
700	C₄	—	8	1	no	—
701	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
702	C₄	C₂₃×#575	1	1	yes	characteristic
703	C₄	C₂₃×#578	1	1	yes	characteristic
704	C₄	C₂₃×#578	1	1	yes	characteristic
705	C₂	—	4	1	no	—
706	C₂	—	8	1	no	—
707	C₂	—	8	1	no	—
708	C₂	C₂₃×#3002	1	1	yes	characteristic
709	C₂	C₂₃×#3011	1	1	yes	characteristic
710	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×#3518	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×#3170	C₂	semidirect
4. C₂₃×#3116	C₂	semidirect
5. C₂₃×#1650	C₂	semidirect
6. C₂₃×#1649	C₂	semidirect
7. C₂₃×#1646	C₂	semidirect
8. C₂₃×#1645	C₂	semidirect
9. C₂₃×#1638	C₂	semidirect
10. C₂₃×#1637	C₂	semidirect
11. C₂₃×#1634	C₂	semidirect
12. C₂₃×#1633	C₂	semidirect
13. C₂₃×#1554	C₂	semidirect
14. C₂₃×#1494	C₂	semidirect
15. C₂₃×#1476	C₂	semidirect
265. #18673	C₂₃	direct (N × H)
323. #3233	C₂×C₂₃	semidirect
324. #3170	C₂×C₂₃	semidirect
325. #3116	C₂×C₂₃	semidirect
326. #1650	C₂×C₂₃	semidirect
327. #1649	C₂×C₂₃	semidirect
328. #1646	C₂×C₂₃	semidirect
329. #1645	C₂×C₂₃	semidirect
330. #1638	C₂×C₂₃	semidirect
331. #1637	C₂×C₂₃	semidirect
332. #1634	C₂×C₂₃	semidirect
333. #1633	C₂×C₂₃	semidirect
334. #1554	C₂×C₂₃	semidirect
335. #1494	C₂×C₂₃	semidirect
336. #1476	C₂×C₂₃	semidirect
531. C₂₃	#18673	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×#3518	2
C₂₃×#3233	2
C₂₃×#3170	2
C₂₃×#3116	2
C₂₃×#1650	2
C₂₃×#1649	2
C₂₃×#1646	2
C₂₃×#1645	2
C₂₃×#1638	2
C₂₃×#1637	2
C₂₃×#1634	2
C₂₃×#1633	2
C₂₃×#1554	2
C₂₃×#1494	2
C₂₃×#1476	2
#18673	23

Subgroups — C23×#18673 #19166299

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18673 #19166299