Subgroups — C₂₃×#18619 #19166245

Order 5888.

1010 conjugacy classes of subgroups.

The subgroup lattice has 758 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3522	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3247	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3147	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3112	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1633	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1632	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1631	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1630	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1623	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1622	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1621	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1620	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1551	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1494	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₂₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×#135	—	2	2	no	—
17	C₂₃×#559	—	2	1	no	—
18	C₂₃×#559	C₂²	1	1	yes	characteristic
19	Q₈²×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#554	—	2	1	no	—
21	C₂₃×#554	C₂²	1	1	yes	characteristic
22	C₂₃×#550	C₂²	1	1	yes	characteristic
23	C₂₃×#548	—	2	1	no	—
24	C₂₃×#547	—	2	2	no	—
25	C₂₃×#547	—	2	1	no	—
26	C₂₃×#542	—	2	2	no	—
27	C₂₃×#542	—	2	1	no	—
28	C₂₃×#532	—	2	1	no	—
29	C₂₃×#532	C₂²	1	1	yes	characteristic
30	C₂₃×#532	C₂²	1	1	yes	characteristic
31	C₂₃×#529	—	2	1	no	—
32	C₂₃×#529	C₂²	1	1	yes	characteristic
33	C₂₃×#520	—	2	1	no	—
34	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
35	C₂₃×#495	—	2	2	no	—
36	C₂₃×#494	C₂²	1	1	yes	characteristic
37	C₂₃×#493	C₂²	1	1	yes	characteristic
38	C₂₃×#492	C₂²	1	1	yes	characteristic
39	C₂₃×#492	C₂²	1	1	yes	characteristic
40	C₂₃×#480	—	2	1	no	—
41	C₂₃×#478	—	2	1	no	—
42	C₂₃×#477	C₂²	1	1	yes	characteristic
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×#466	—	2	1	no	—
47	C₂₃×#465	—	2	1	no	—
48	C₂₃×#463	—	2	1	no	—
49	C₂₃×#460	C₂²	1	1	yes	characteristic
50	C₂₃×#460	C₂²	1	1	yes	characteristic
51	C₂₃×#460	C₂²	1	1	yes	characteristic
52	C₂₃×#459	C₂²	1	1	yes	characteristic
53	C₂₃×#448	—	2	1	no	—
54	C₂₃×#440	—	2	1	no	—
55	C₂₃×#440	—	2	1	no	—
56	C₂₃×#432	C₂²	1	1	yes	characteristic
57	C₂×C₂₃×C₄⋊C₈	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×#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×Q₈⋊C₈	C₂²	1	1	yes	characteristic
68	C₂₃×Q₈⋊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	2	no	—
72	C₂×#41×C₂₃	—	2	1	no	—
73	C₂×#41×C₂₃	—	2	1	no	—
74	C₂×#41×C₂₃	D₄	1	2	yes	—
75	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
76	C₂₃×#135	—	4	8	no	—
77	C₂×Q₁₆×C₂₃	—	2	2	no	—
78	C₂×SD₁₆×C₂₃	—	2	2	no	—
79	C₂×SD₁₆×C₂₃	—	2	1	no	—
80	C₂×SD₁₆×C₂₃	—	2	1	no	—
81	C₂×SD₁₆×C₂₃	—	2	2	no	—
82	C₂×SD₁₆×C₂₃	—	2	1	no	—
83	C₂×D₈×C₂₃	—	2	2	no	—
84	C₂×D₈×C₂₃	—	2	1	no	—
85	C₂²×C₈×C₂₃	—	2	1	no	—
86	C₂²×C₈×C₂₃	—	2	1	no	—
87	C₂₃×C₄⋊Q₈	—	2	2	no	—
88	C₂₃×C₄⋊Q₈	—	2	1	no	—
89	C₂₃×C₄⋊Q₈	—	2	1	no	—
90	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
91	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
92	C₂₃×#127	—	2	1	no	—
93	C₂₃×#127	C₂³	1	1	yes	characteristic
94	C₂₃×#127	C₂³	1	1	yes	characteristic
95	C₂₃×#126	—	2	2	no	—
96	C₂₃×#126	—	2	1	no	—
97	C₂₃×#125	—	2	1	no	—
98	C₂₃×#125	D₄	1	1	yes	characteristic
99	C₂₃×#124	—	2	2	no	—
100	C₂₃×#124	—	2	2	no	—
101	C₂₃×#124	—	2	1	no	—
102	C₂₃×#124	—	2	1	no	—
103	C₂₃×#124	—	2	1	no	—
104	C₂₃×#124	D₄	1	1	yes	characteristic
105	C₂₃×#124	D₄	1	2	yes	—
106	C₂₃×#123	—	2	2	no	—
107	C₂₃×#123	—	2	2	no	—
108	C₂₃×#123	—	2	2	no	—
109	C₂₃×#123	—	2	2	no	—
110	C₂₃×#123	—	2	1	no	—
111	C₂₃×C₂²⋊Q₈	—	2	2	no	—
112	C₂₃×C₂²⋊Q₈	—	2	2	no	—
113	C₂₃×C₂²⋊Q₈	—	2	2	no	—
114	C₂₃×C₂²⋊Q₈	—	2	2	no	—
115	C₂₃×C₂²⋊Q₈	—	2	1	no	—
116	C₂₃×C₂²⋊Q₈	D₄	1	1	yes	characteristic
117	C₂₃×#121	—	2	2	no	—
118	C₂₃×#121	—	2	2	no	—
119	C₂₃×#121	—	2	1	no	—
120	C₂₃×#121	—	2	1	no	—
121	C₂₃×#121	—	2	1	no	—
122	C₂₃×#121	—	2	1	no	—
123	C₂₃×#121	—	2	1	no	—
124	C₂₃×#121	D₄	1	1	yes	characteristic
125	C₄×Q₈×C₂₃	—	2	1	no	—
126	C₄×Q₈×C₂₃	—	2	1	no	—
127	C₄×Q₈×C₂₃	—	2	1	no	—
128	C₄×Q₈×C₂₃	—	2	1	no	—
129	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
130	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
131	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
132	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
133	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
134	C₄×D₄×C₂₃	—	2	2	no	—
135	C₄×D₄×C₂₃	—	2	2	no	—
136	C₄×D₄×C₂₃	—	2	2	no	—
137	C₄×D₄×C₂₃	—	2	1	no	—
138	C₄×D₄×C₂₃	—	2	1	no	—
139	C₄×D₄×C₂₃	—	2	1	no	—
140	C₄×D₄×C₂₃	—	2	1	no	—
141	C₄×D₄×C₂₃	—	2	1	no	—
142	C₄×D₄×C₂₃	—	2	1	no	—
143	C₄×D₄×C₂₃	—	2	1	no	—
144	C₄×D₄×C₂₃	—	2	1	no	—
145	C₄×D₄×C₂₃	—	2	1	no	—
146	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
147	C₂₃×#117	—	2	2	no	—
148	C₂₃×#117	—	2	1	no	—
149	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
150	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
151	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
152	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
153	C₂×C₄²×C₂₃	—	2	1	no	—
154	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
155	C₂₃×#107	—	2	1	no	—
156	C₂₃×#107	—	2	1	no	—
157	C₂₃×#107	—	2	1	no	—
158	C₂₃×#106	—	2	1	no	—
159	C₂₃×C₄⋊C₈	—	2	1	no	—
160	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
161	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
162	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
163	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
164	C₂₃×#103	—	2	1	no	—
165	C₂₃×#103	—	2	1	no	—
166	C₂₃×#103	—	2	1	no	—
167	C₂₃×#102	—	2	1	no	—
168	C₂₃×#102	—	2	1	no	—
169	C₂₃×#102	—	2	1	no	—
170	C₂₃×#102	—	2	1	no	—
171	C₂₃×#102	—	2	1	no	—
172	C₂₃×C₂²⋊C₈	—	2	1	no	—
173	C₂₃×#97	—	2	1	no	—
174	C₄×C₈×C₂₃	—	2	1	no	—
175	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
176	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
177	#41×C₂₃	—	4	4	no	—
178	#41×C₂₃	—	4	4	no	—
179	#41×C₂₃	—	4	2	no	—
180	#41×C₂₃	—	4	2	no	—
181	#41×C₂₃	—	8	4	no	—
182	C₂×Q₈×C₂₃	—	2	2	no	—
183	C₂×Q₈×C₂₃	—	2	1	no	—
184	C₂×Q₈×C₂₃	—	2	2	no	—
185	C₂×Q₈×C₂₃	—	2	1	no	—
186	C₂×Q₈×C₂₃	—	4	1	no	—
187	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
188	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
189	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
190	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
191	C₂×D₄×C₂₃	—	2	2	no	—
192	C₂×D₄×C₂₃	—	2	2	no	—
193	C₂×D₄×C₂₃	—	2	1	no	—
194	C₂×D₄×C₂₃	—	2	2	no	—
195	C₂×D₄×C₂₃	—	2	1	no	—
196	C₂×D₄×C₂₃	—	2	1	no	—
197	C₂×D₄×C₂₃	—	2	1	no	—
198	C₂×D₄×C₂₃	—	2	1	no	—
199	C₂×D₄×C₂₃	—	2	1	no	—
200	C₂×D₄×C₂₃	—	4	1	no	—
201	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
202	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
203	C₂²×C₄×C₂₃	—	2	2	no	—
204	C₂²×C₄×C₂₃	—	2	1	no	—
205	C₂²×C₄×C₂₃	—	2	1	no	—
206	C₂²×C₄×C₂₃	—	2	2	no	—
207	C₂²×C₄×C₂₃	—	2	2	no	—
208	C₂²×C₄×C₂₃	—	2	1	no	—
209	C₂²×C₄×C₂₃	—	2	2	no	—
210	C₂²×C₄×C₂₃	—	4	1	no	—
211	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
212	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
213	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
214	Q₁₆×C₂₃	—	4	4	no	—
215	SD₁₆×C₂₃	—	4	4	no	—
216	SD₁₆×C₂₃	—	4	4	no	—
217	SD₁₆×C₂₃	—	4	2	no	—
218	SD₁₆×C₂₃	—	4	2	no	—
219	SD₁₆×C₂₃	—	8	1	no	—
220	D₈×C₂₃	—	4	4	no	—
221	D₈×C₂₃	—	8	1	no	—
222	C₂×C₈×C₂₃	—	2	4	no	—
223	C₂×C₈×C₂₃	—	2	1	no	—
224	C₂×C₈×C₂₃	—	2	1	no	—
225	C₂×C₈×C₂₃	—	2	1	no	—
226	C₂×C₈×C₂₃	—	2	1	no	—
227	C₂×C₈×C₂₃	—	2	1	no	—
228	C₂×C₈×C₂₃	—	2	1	no	—
229	C₂×C₈×C₂₃	—	8	1	no	—
230	C₄⋊C₄×C₂₃	—	2	2	no	—
231	C₄⋊C₄×C₂₃	—	2	2	no	—
232	C₄⋊C₄×C₂₃	—	2	2	no	—
233	C₄⋊C₄×C₂₃	—	2	2	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	—	2	1	no	—
241	C₄⋊C₄×C₂₃	—	2	1	no	—
242	C₄⋊C₄×C₂₃	—	2	1	no	—
243	C₄⋊C₄×C₂₃	—	2	1	no	—
244	C₄⋊C₄×C₂₃	—	2	1	no	—
245	C₄⋊C₄×C₂₃	—	2	1	no	—
246	C₄⋊C₄×C₂₃	—	2	1	no	—
247	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
248	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
249	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
250	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
251	C₂²⋊C₄×C₂₃	—	2	2	no	—
252	C₂²⋊C₄×C₂₃	—	2	2	no	—
253	C₂²⋊C₄×C₂₃	—	2	2	no	—
254	C₂²⋊C₄×C₂₃	—	2	2	no	—
255	C₂²⋊C₄×C₂₃	—	2	2	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	—	2	2	no	—
261	C₂²⋊C₄×C₂₃	—	2	1	no	—
262	C₂²⋊C₄×C₂₃	—	2	1	no	—
263	C₂²⋊C₄×C₂₃	—	4	1	no	—
264	C₂²⋊C₄×C₂₃	—	4	1	no	—
265	C₄²×C₂₃	—	2	2	no	—
266	C₄²×C₂₃	—	2	1	no	—
267	C₄²×C₂₃	—	2	1	no	—
268	C₄²×C₂₃	—	2	1	no	—
269	C₄²×C₂₃	—	2	1	no	—
270	C₄²×C₂₃	—	2	1	no	—
271	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
272	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
273	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
274	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
275	#18619	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
276	C₂³×C₂₃	—	2	1	no	—
277	C₂³×C₂₃	—	2	2	no	—
278	C₂³×C₂₃	—	4	1	no	—
279	C₂³×C₂₃	#120	1	1	yes	characteristic
280	Q₈×C₂₃	—	2	1	no	—
281	Q₈×C₂₃	—	2	1	no	—
282	Q₈×C₂₃	—	2	1	no	—
283	Q₈×C₂₃	—	2	1	no	—
284	Q₈×C₂₃	—	2	1	no	—
285	Q₈×C₂₃	—	4	2	no	—
286	Q₈×C₂₃	—	4	2	no	—
287	Q₈×C₂₃	—	4	2	no	—
288	Q₈×C₂₃	—	8	2	no	—
289	Q₈×C₂₃	#135	1	2	yes	complemented
290	D₄×C₂₃	—	2	1	no	—
291	D₄×C₂₃	—	2	1	no	—
292	D₄×C₂₃	—	4	2	no	—
293	D₄×C₂₃	—	4	4	no	—
294	D₄×C₂₃	—	4	2	no	—
295	D₄×C₂₃	—	4	4	no	—
296	D₄×C₂₃	—	4	2	no	—
297	D₄×C₂₃	—	4	1	no	—
298	D₄×C₂₃	—	4	1	no	—
299	D₄×C₂₃	—	4	1	no	—
300	D₄×C₂₃	—	4	1	no	—
301	D₄×C₂₃	—	4	1	no	—
302	D₄×C₂₃	—	4	1	no	—
303	D₄×C₂₃	—	4	1	no	—
304	D₄×C₂₃	—	4	1	no	—
305	D₄×C₂₃	—	8	2	no	—
306	D₄×C₂₃	—	8	2	no	—
307	C₂×C₄×C₂₃	—	2	1	no	—
308	C₂×C₄×C₂₃	—	2	2	no	—
309	C₂×C₄×C₂₃	—	2	1	no	—
310	C₂×C₄×C₂₃	—	2	1	no	—
311	C₂×C₄×C₂₃	—	2	1	no	—
312	C₂×C₄×C₂₃	—	2	1	no	—
313	C₂×C₄×C₂₃	—	2	1	no	—
314	C₂×C₄×C₂₃	—	2	1	no	—
315	C₂×C₄×C₂₃	—	2	1	no	—
316	C₂×C₄×C₂₃	—	2	1	no	—
317	C₂×C₄×C₂₃	—	2	2	no	—
318	C₂×C₄×C₂₃	—	2	1	no	—
319	C₂×C₄×C₂₃	—	2	1	no	—
320	C₂×C₄×C₂₃	—	4	2	no	—
321	C₂×C₄×C₂₃	—	4	4	no	—
322	C₂×C₄×C₂₃	—	4	2	no	—
323	C₂×C₄×C₂₃	—	4	1	no	—
324	C₂×C₄×C₂₃	—	4	1	no	—
325	C₂×C₄×C₂₃	—	4	1	no	—
326	C₂×C₄×C₂₃	—	8	2	no	—
327	C₂×C₄×C₂₃	—	8	2	no	—
328	C₂×C₄×C₂₃	—	8	2	no	—
329	C₂×C₄×C₂₃	—	8	2	no	—
330	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
331	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
332	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
333	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
334	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
335	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
336	C₈×C₂₃	—	2	2	no	—
337	C₈×C₂₃	—	2	2	no	—
338	C₈×C₂₃	—	4	1	no	—
339	C₈×C₂₃	—	4	1	no	—
340	C₈×C₂₃	—	4	1	no	—
341	C₈×C₂₃	—	4	1	no	—
342	#3522	C₂×C₂₃	1	1	yes	characteristic, complemented
343	#3247	C₂×C₂₃	1	1	yes	characteristic, complemented
344	#3147	C₂×C₂₃	1	1	yes	characteristic, complemented
345	#3112	C₂×C₂₃	1	1	yes	characteristic, complemented
346	#1633	C₂×C₂₃	1	1	yes	characteristic, complemented
347	#1632	C₂×C₂₃	1	1	yes	characteristic, complemented
348	#1631	C₂×C₂₃	1	1	yes	characteristic, complemented
349	#1630	C₂×C₂₃	1	1	yes	characteristic, complemented
350	#1623	C₂×C₂₃	1	1	yes	characteristic, complemented
351	#1622	C₂×C₂₃	1	1	yes	characteristic, complemented
352	#1621	C₂×C₂₃	1	1	yes	characteristic, complemented
353	#1620	C₂×C₂₃	1	1	yes	characteristic, complemented
354	#1551	C₂×C₂₃	1	1	yes	characteristic, complemented
355	#1494	C₂×C₂₃	1	1	yes	characteristic, complemented
356	C₂×Q₈⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
357	C₂²×C₂₃	—	2	1	no	—
358	C₂²×C₂₃	—	2	1	no	—
359	C₂²×C₂₃	—	4	2	no	—
360	C₂²×C₂₃	—	4	1	no	—
361	C₂²×C₂₃	—	4	1	no	—
362	C₂²×C₂₃	—	4	2	no	—
363	C₂²×C₂₃	—	4	2	no	—
364	C₂²×C₂₃	—	4	1	no	—
365	C₂²×C₂₃	—	8	1	no	—
366	C₂²×C₂₃	—	8	1	no	—
367	C₂²×C₂₃	—	8	1	no	—
368	C₂²×C₂₃	#453	1	2	yes	—
369	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
370	C₄×C₂₃	—	2	1	no	—
371	C₄×C₂₃	—	2	2	no	—
372	C₄×C₂₃	—	2	1	no	—
373	C₄×C₂₃	—	2	1	no	—
374	C₄×C₂₃	—	2	1	no	—
375	C₄×C₂₃	—	4	1	no	—
376	C₄×C₂₃	—	4	1	no	—
377	C₄×C₂₃	—	4	1	no	—
378	C₄×C₂₃	—	4	1	no	—
379	C₄×C₂₃	—	4	1	no	—
380	C₄×C₂₃	—	4	2	no	—
381	C₄×C₂₃	—	4	1	no	—
382	C₄×C₂₃	—	8	1	no	—
383	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
384	C₄×C₂₃	#575	1	1	yes	characteristic
385	C₄×C₂₃	#577	1	1	yes	characteristic
386	C₄×C₂₃	#578	1	1	yes	characteristic
387	C₂×#135	—	2	2	no	—
388	#559	—	2	1	no	—
389	#559	C₂²×C₂₃	1	1	yes	characteristic
390	Q₈²	C₂²×C₂₃	1	1	yes	characteristic
391	#554	—	2	1	no	—
392	#554	C₂²×C₂₃	1	1	yes	characteristic
393	#550	C₂²×C₂₃	1	1	yes	characteristic
394	#548	—	2	1	no	—
395	#547	—	2	2	no	—
396	#547	—	2	1	no	—
397	#542	—	2	2	no	—
398	#542	—	2	1	no	—
399	#532	—	2	1	no	—
400	#532	C₂²×C₂₃	1	1	yes	characteristic
401	#532	C₂²×C₂₃	1	1	yes	characteristic
402	#529	—	2	1	no	—
403	#529	C₂²×C₂₃	1	1	yes	characteristic
404	#520	—	2	1	no	—
405	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
406	#495	—	2	2	no	—
407	#494	C₂²×C₂₃	1	1	yes	characteristic
408	#493	C₂²×C₂₃	1	1	yes	characteristic
409	#492	C₂²×C₂₃	1	1	yes	characteristic
410	#492	C₂²×C₂₃	1	1	yes	characteristic
411	#480	—	2	1	no	—
412	#478	—	2	1	no	—
413	#477	C₂²×C₂₃	1	1	yes	characteristic
414	#477	C₂²×C₂₃	1	1	yes	characteristic
415	#477	C₂²×C₂₃	1	1	yes	characteristic
416	#476	C₂²×C₂₃	1	1	yes	characteristic
417	#466	—	2	1	no	—
418	#465	—	2	1	no	—
419	#463	—	2	1	no	—
420	#460	C₂²×C₂₃	1	1	yes	characteristic
421	#460	C₂²×C₂₃	1	1	yes	characteristic
422	#460	C₂²×C₂₃	1	1	yes	characteristic
423	#459	C₂²×C₂₃	1	1	yes	characteristic
424	#448	—	2	1	no	—
425	#440	—	2	1	no	—
426	#440	—	2	1	no	—
427	#432	C₂²×C₂₃	1	1	yes	characteristic
428	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
429	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
430	#333	C₂²×C₂₃	1	1	yes	characteristic
431	#333	C₂²×C₂₃	1	1	yes	characteristic
432	#331	C₂²×C₂₃	1	1	yes	characteristic
433	#331	C₂²×C₂₃	1	1	yes	characteristic
434	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
435	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
436	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
437	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
438	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
439	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
440	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
441	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
442	C₂×C₂₃	—	2	2	no	—
443	C₂×C₂₃	—	8	1	no	—
444	C₂×C₂₃	—	8	2	no	—
445	C₂×C₂₃	—	16	1	no	—
446	C₂×C₂₃	#3002	1	1	yes	characteristic
447	C₂×C₂₃	#3013	1	1	yes	characteristic
448	C₂×C₂₃	C₂×#453	1	1	yes	characteristic
449	C₂×#41	—	2	2	no	—
450	C₂×#41	—	2	1	no	—
451	C₂×#41	—	2	1	no	—
452	C₂×#41	D₄×C₂₃	1	2	yes	—
453	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
454	#135	—	4	8	no	—
455	C₂×Q₁₆	—	2	2	no	—
456	C₂×SD₁₆	—	2	2	no	—
457	C₂×SD₁₆	—	2	1	no	—
458	C₂×SD₁₆	—	2	1	no	—
459	C₂×SD₁₆	—	2	2	no	—
460	C₂×SD₁₆	—	2	1	no	—
461	C₂×D₈	—	2	2	no	—
462	C₂×D₈	—	2	1	no	—
463	C₂²×C₈	—	2	1	no	—
464	C₂²×C₈	—	2	1	no	—
465	C₄⋊Q₈	—	2	2	no	—
466	C₄⋊Q₈	—	2	1	no	—
467	C₄⋊Q₈	—	2	1	no	—
468	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
469	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
470	#127	—	2	1	no	—
471	#127	C₂³×C₂₃	1	1	yes	characteristic
472	#127	C₂³×C₂₃	1	1	yes	characteristic
473	#126	—	2	2	no	—
474	#126	—	2	1	no	—
475	#125	—	2	1	no	—
476	#125	D₄×C₂₃	1	1	yes	characteristic
477	#124	—	2	2	no	—
478	#124	—	2	2	no	—
479	#124	—	2	1	no	—
480	#124	—	2	1	no	—
481	#124	—	2	1	no	—
482	#124	D₄×C₂₃	1	1	yes	characteristic
483	#124	D₄×C₂₃	1	2	yes	—
484	#123	—	2	2	no	—
485	#123	—	2	2	no	—
486	#123	—	2	2	no	—
487	#123	—	2	2	no	—
488	#123	—	2	1	no	—
489	C₂²⋊Q₈	—	2	2	no	—
490	C₂²⋊Q₈	—	2	2	no	—
491	C₂²⋊Q₈	—	2	2	no	—
492	C₂²⋊Q₈	—	2	2	no	—
493	C₂²⋊Q₈	—	2	1	no	—
494	C₂²⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
495	#121	—	2	2	no	—
496	#121	—	2	2	no	—
497	#121	—	2	1	no	—
498	#121	—	2	1	no	—
499	#121	—	2	1	no	—
500	#121	—	2	1	no	—
501	#121	—	2	1	no	—
502	#121	D₄×C₂₃	1	1	yes	characteristic
503	C₄×Q₈	—	2	1	no	—
504	C₄×Q₈	—	2	1	no	—
505	C₄×Q₈	—	2	1	no	—
506	C₄×Q₈	—	2	1	no	—
507	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
508	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
509	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
510	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
511	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
512	C₄×D₄	—	2	2	no	—
513	C₄×D₄	—	2	2	no	—
514	C₄×D₄	—	2	2	no	—
515	C₄×D₄	—	2	1	no	—
516	C₄×D₄	—	2	1	no	—
517	C₄×D₄	—	2	1	no	—
518	C₄×D₄	—	2	1	no	—
519	C₄×D₄	—	2	1	no	—
520	C₄×D₄	—	2	1	no	—
521	C₄×D₄	—	2	1	no	—
522	C₄×D₄	—	2	1	no	—
523	C₄×D₄	—	2	1	no	—
524	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
525	#117	—	2	2	no	—
526	#117	—	2	1	no	—
527	C₂×C₄⋊C₄	—	2	2	no	—
528	C₂×C₄⋊C₄	—	2	1	no	—
529	C₂×C₄⋊C₄	—	2	1	no	—
530	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
531	C₂×C₄²	—	2	1	no	—
532	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
533	#107	—	2	1	no	—
534	#107	—	2	1	no	—
535	#107	—	2	1	no	—
536	#106	—	2	1	no	—
537	C₄⋊C₈	—	2	1	no	—
538	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
539	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
540	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
541	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
542	#103	—	2	1	no	—
543	#103	—	2	1	no	—
544	#103	—	2	1	no	—
545	#102	—	2	1	no	—
546	#102	—	2	1	no	—
547	#102	—	2	1	no	—
548	#102	—	2	1	no	—
549	#102	—	2	1	no	—
550	C₂²⋊C₈	—	2	1	no	—
551	#97	—	2	1	no	—
552	C₄×C₈	—	2	1	no	—
553	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
554	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
555	C₂₃	#18619	1	1	yes	characteristic, direct product factor, Sylow
556	#41	—	4	4	no	—
557	#41	—	4	4	no	—
558	#41	—	4	2	no	—
559	#41	—	4	2	no	—
560	#41	—	8	4	no	—
561	C₂×Q₈	—	2	2	no	—
562	C₂×Q₈	—	2	1	no	—
563	C₂×Q₈	—	2	2	no	—
564	C₂×Q₈	—	2	1	no	—
565	C₂×Q₈	—	4	1	no	—
566	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
567	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
568	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
569	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
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	—	2	1	no	—
577	C₂×D₄	—	2	1	no	—
578	C₂×D₄	—	2	1	no	—
579	C₂×D₄	—	4	1	no	—
580	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
581	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
582	C₂²×C₄	—	2	2	no	—
583	C₂²×C₄	—	2	1	no	—
584	C₂²×C₄	—	2	1	no	—
585	C₂²×C₄	—	2	2	no	—
586	C₂²×C₄	—	2	2	no	—
587	C₂²×C₄	—	2	1	no	—
588	C₂²×C₄	—	2	2	no	—
589	C₂²×C₄	—	4	1	no	—
590	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
591	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
592	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
593	Q₁₆	—	4	4	no	—
594	SD₁₆	—	4	4	no	—
595	SD₁₆	—	4	4	no	—
596	SD₁₆	—	4	2	no	—
597	SD₁₆	—	4	2	no	—
598	SD₁₆	—	8	1	no	—
599	D₈	—	4	4	no	—
600	D₈	—	8	1	no	—
601	C₂×C₈	—	2	4	no	—
602	C₂×C₈	—	2	1	no	—
603	C₂×C₈	—	2	1	no	—
604	C₂×C₈	—	2	1	no	—
605	C₂×C₈	—	2	1	no	—
606	C₂×C₈	—	2	1	no	—
607	C₂×C₈	—	2	1	no	—
608	C₂×C₈	—	8	1	no	—
609	C₄⋊C₄	—	2	2	no	—
610	C₄⋊C₄	—	2	2	no	—
611	C₄⋊C₄	—	2	2	no	—
612	C₄⋊C₄	—	2	2	no	—
613	C₄⋊C₄	—	2	1	no	—
614	C₄⋊C₄	—	2	1	no	—
615	C₄⋊C₄	—	2	1	no	—
616	C₄⋊C₄	—	2	1	no	—
617	C₄⋊C₄	—	2	1	no	—
618	C₄⋊C₄	—	2	1	no	—
619	C₄⋊C₄	—	2	1	no	—
620	C₄⋊C₄	—	2	1	no	—
621	C₄⋊C₄	—	2	1	no	—
622	C₄⋊C₄	—	2	1	no	—
623	C₄⋊C₄	—	2	1	no	—
624	C₄⋊C₄	—	2	1	no	—
625	C₄⋊C₄	—	2	1	no	—
626	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
627	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
628	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
629	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
630	C₂²⋊C₄	—	2	2	no	—
631	C₂²⋊C₄	—	2	2	no	—
632	C₂²⋊C₄	—	2	2	no	—
633	C₂²⋊C₄	—	2	2	no	—
634	C₂²⋊C₄	—	2	2	no	—
635	C₂²⋊C₄	—	2	1	no	—
636	C₂²⋊C₄	—	2	2	no	—
637	C₂²⋊C₄	—	2	1	no	—
638	C₂²⋊C₄	—	2	1	no	—
639	C₂²⋊C₄	—	2	2	no	—
640	C₂²⋊C₄	—	2	1	no	—
641	C₂²⋊C₄	—	2	1	no	—
642	C₂²⋊C₄	—	4	1	no	—
643	C₂²⋊C₄	—	4	1	no	—
644	C₄²	—	2	2	no	—
645	C₄²	—	2	1	no	—
646	C₄²	—	2	1	no	—
647	C₄²	—	2	1	no	—
648	C₄²	—	2	1	no	—
649	C₄²	—	2	1	no	—
650	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
651	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
652	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
653	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
654	C₂³	—	2	1	no	—
655	C₂³	—	2	2	no	—
656	C₂³	—	4	1	no	—
657	C₂³	C₂₃×#120	1	1	yes	characteristic
658	Q₈	—	2	1	no	—
659	Q₈	—	2	1	no	—
660	Q₈	—	2	1	no	—
661	Q₈	—	2	1	no	—
662	Q₈	—	2	1	no	—
663	Q₈	—	4	2	no	—
664	Q₈	—	4	2	no	—
665	Q₈	—	4	2	no	—
666	Q₈	—	8	2	no	—
667	Q₈	C₂₃×#135	1	2	yes	complemented
668	D₄	—	2	1	no	—
669	D₄	—	2	1	no	—
670	D₄	—	4	2	no	—
671	D₄	—	4	4	no	—
672	D₄	—	4	2	no	—
673	D₄	—	4	4	no	—
674	D₄	—	4	2	no	—
675	D₄	—	4	1	no	—
676	D₄	—	4	1	no	—
677	D₄	—	4	1	no	—
678	D₄	—	4	1	no	—
679	D₄	—	4	1	no	—
680	D₄	—	4	1	no	—
681	D₄	—	4	1	no	—
682	D₄	—	4	1	no	—
683	D₄	—	8	2	no	—
684	D₄	—	8	2	no	—
685	C₂×C₄	—	2	2	no	—
686	C₂×C₄	—	2	1	no	—
687	C₂×C₄	—	2	1	no	—
688	C₂×C₄	—	2	1	no	—
689	C₂×C₄	—	2	1	no	—
690	C₂×C₄	—	2	1	no	—
691	C₂×C₄	—	2	1	no	—
692	C₂×C₄	—	2	1	no	—
693	C₂×C₄	—	2	1	no	—
694	C₂×C₄	—	2	1	no	—
695	C₂×C₄	—	2	2	no	—
696	C₂×C₄	—	2	1	no	—
697	C₂×C₄	—	2	1	no	—
698	C₂×C₄	—	4	2	no	—
699	C₂×C₄	—	4	4	no	—
700	C₂×C₄	—	4	2	no	—
701	C₂×C₄	—	4	1	no	—
702	C₂×C₄	—	4	1	no	—
703	C₂×C₄	—	4	1	no	—
704	C₂×C₄	—	8	2	no	—
705	C₂×C₄	—	8	2	no	—
706	C₂×C₄	—	8	2	no	—
707	C₂×C₄	—	8	2	no	—
708	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
709	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
710	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
711	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
712	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
713	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
714	C₈	—	2	2	no	—
715	C₈	—	2	2	no	—
716	C₈	—	4	1	no	—
717	C₈	—	4	1	no	—
718	C₈	—	4	1	no	—
719	C₈	—	4	1	no	—
720	C₂²	—	2	1	no	—
721	C₂²	—	2	1	no	—
722	C₂²	—	4	2	no	—
723	C₂²	—	4	1	no	—
724	C₂²	—	4	1	no	—
725	C₂²	—	4	2	no	—
726	C₂²	—	4	2	no	—
727	C₂²	—	4	1	no	—
728	C₂²	—	8	1	no	—
729	C₂²	—	8	1	no	—
730	C₂²	—	8	1	no	—
731	C₂²	C₂₃×#453	1	2	yes	—
732	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
733	C₄	—	2	1	no	—
734	C₄	—	2	2	no	—
735	C₄	—	2	1	no	—
736	C₄	—	2	1	no	—
737	C₄	—	2	1	no	—
738	C₄	—	4	1	no	—
739	C₄	—	4	1	no	—
740	C₄	—	4	1	no	—
741	C₄	—	4	1	no	—
742	C₄	—	4	1	no	—
743	C₄	—	4	2	no	—
744	C₄	—	4	1	no	—
745	C₄	—	8	1	no	—
746	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
747	C₄	C₂₃×#575	1	1	yes	characteristic
748	C₄	C₂₃×#577	1	1	yes	characteristic
749	C₄	C₂₃×#578	1	1	yes	characteristic
750	C₂	—	2	2	no	—
751	C₂	—	8	1	no	—
752	C₂	—	8	2	no	—
753	C₂	—	16	1	no	—
754	C₂	C₂₃×#3002	1	1	yes	characteristic
755	C₂	C₂₃×#3013	1	1	yes	characteristic
756	C₂	C₂×C₂₃×#453	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
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×#3522	C₂	semidirect
2. C₂₃×#3247	C₂	semidirect
3. C₂₃×#3147	C₂	semidirect
4. C₂₃×#3112	C₂	semidirect
5. C₂₃×#1633	C₂	semidirect
6. C₂₃×#1632	C₂	semidirect
7. C₂₃×#1631	C₂	semidirect
8. C₂₃×#1630	C₂	semidirect
9. C₂₃×#1623	C₂	semidirect
10. C₂₃×#1622	C₂	semidirect
11. C₂₃×#1621	C₂	semidirect
12. C₂₃×#1620	C₂	semidirect
13. C₂₃×#1551	C₂	semidirect
14. C₂₃×#1494	C₂	semidirect
15. C₂×C₂₃×Q₈⋊C₈	C₂	semidirect
275. #18619	C₂₃	direct (N × H)
289. Q₈×C₂₃	#135	semidirect
342. #3522	C₂×C₂₃	semidirect
343. #3247	C₂×C₂₃	semidirect
344. #3147	C₂×C₂₃	semidirect
345. #3112	C₂×C₂₃	semidirect
346. #1633	C₂×C₂₃	semidirect
347. #1632	C₂×C₂₃	semidirect
348. #1631	C₂×C₂₃	semidirect
349. #1630	C₂×C₂₃	semidirect
350. #1623	C₂×C₂₃	semidirect
351. #1622	C₂×C₂₃	semidirect
352. #1621	C₂×C₂₃	semidirect
353. #1620	C₂×C₂₃	semidirect
354. #1551	C₂×C₂₃	semidirect
355. #1494	C₂×C₂₃	semidirect
356. C₂×Q₈⋊C₈	C₂×C₂₃	semidirect
555. C₂₃	#18619	direct (N × H)
667. Q₈	C₂₃×#135	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3522	2
C₂₃×#3247	2
C₂₃×#3147	2
C₂₃×#3112	2
C₂₃×#1633	2
C₂₃×#1632	2
C₂₃×#1631	2
C₂₃×#1630	2
C₂₃×#1623	2
C₂₃×#1622	2
C₂₃×#1621	2
C₂₃×#1620	2
C₂₃×#1551	2
C₂₃×#1494	2
C₂×C₂₃×Q₈⋊C₈	2
#18619	23

Subgroups — C23×#18619 #19166245

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18619 #19166245