Subgroups — C₂₃×#18596 #19166222

Order 5888.

930 conjugacy classes of subgroups.

The subgroup lattice has 748 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3515	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3251	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3182	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3035	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1648	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1647	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1644	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1643	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1626	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1625	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1622	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1622	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1550	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1489	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1483	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×#135	—	2	1	no	—
17	C₂₃×#557	C₂²	1	1	yes	characteristic
18	C₂₃×#554	C₂²	1	1	yes	characteristic
19	C₂₃×#554	C₂²	1	1	yes	characteristic
20	C₂₃×#553	—	2	1	no	—
21	C₂₃×#552	—	2	1	no	—
22	C₂₃×#552	—	2	2	no	—
23	C₂₃×#552	—	2	1	no	—
24	C₂₃×#547	—	2	1	no	—
25	C₂₃×#547	C₂²	1	1	yes	characteristic
26	C₂₃×#541	—	2	1	no	—
27	C₂₃×#539	—	2	2	no	—
28	C₂₃×#533	C₂²	1	1	yes	characteristic
29	C₂₃×#532	C₂²	1	1	yes	characteristic
30	C₂₃×#529	—	2	1	no	—
31	C₂₃×#529	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×#125	—	2	1	no	—
33	C₂₃×#520	—	2	1	no	—
34	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
35	C₂₃×#500	C₂²	1	1	yes	characteristic
36	C₂₃×#492	C₂²	1	1	yes	characteristic
37	C₂₃×#487	C₂²	1	1	yes	characteristic
38	C₂₃×#486	C₂²	1	1	yes	characteristic
39	C₂₃×#478	—	2	1	no	—
40	C₂₃×#475	C₂²	1	1	yes	characteristic
41	C₂₃×#474	C₂²	1	1	yes	characteristic
42	C₂₃×#469	—	2	1	no	—
43	C₂₃×#468	—	2	2	no	—
44	C₂₃×#468	—	2	1	no	—
45	C₂₃×#464	—	2	1	no	—
46	C₂₃×#463	—	2	1	no	—
47	C₂₃×#463	—	2	1	no	—
48	C₂₃×#462	C₂²	1	1	yes	characteristic
49	C₂₃×#461	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×#459	C₂²	1	1	yes	characteristic
54	C₂₃×#440	—	2	2	no	—
55	C₂₃×#423	C₂²	1	1	yes	characteristic
56	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
57	C₂₃×#416	—	2	1	no	—
58	C₄×M₁₆×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	—	2	1	no	—
72	C₂×#41×C₂₃	—	2	1	no	—
73	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
74	C₂₃×#135	—	4	4	no	—
75	C₂×Q₁₆×C₂₃	—	2	1	no	—
76	C₂×SD₁₆×C₂₃	—	2	1	no	—
77	C₂×SD₁₆×C₂₃	—	2	2	no	—
78	C₂×SD₁₆×C₂₃	—	2	1	no	—
79	C₂×SD₁₆×C₂₃	—	2	1	no	—
80	C₂×D₈×C₂₃	—	2	1	no	—
81	C₂×D₈×C₂₃	—	2	1	no	—
82	C₂×M₁₆×C₂₃	—	2	1	no	—
83	C₂×M₁₆×C₂₃	—	2	1	no	—
84	C₂²×C₈×C₂₃	—	2	1	no	—
85	C₂₃×C₄⋊Q₈	—	2	1	no	—
86	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
87	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
88	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
89	C₂₃×#127	C₂³	1	1	yes	characteristic
90	C₂₃×#126	—	2	2	no	—
91	C₂₃×#126	—	2	2	no	—
92	C₂₃×#126	—	2	2	no	—
93	C₂₃×#126	—	2	1	no	—
94	C₂₃×#125	—	2	2	no	—
95	C₂₃×#125	—	2	2	no	—
96	C₂₃×#125	—	2	2	no	—
97	C₂₃×#125	—	2	1	no	—
98	C₂₃×#125	—	2	1	no	—
99	C₂₃×#125	—	2	2	no	—
100	C₂₃×#125	—	2	1	no	—
101	C₂₃×#125	—	2	1	no	—
102	C₂₃×#125	D₄	1	1	yes	characteristic
103	C₂₃×#124	—	2	1	no	—
104	C₂₃×#124	D₄	1	1	yes	characteristic
105	C₂₃×#123	—	2	2	no	—
106	C₂₃×#123	—	2	2	no	—
107	C₂₃×#123	—	2	2	no	—
108	C₂₃×#123	—	2	1	no	—
109	C₂₃×#123	—	2	2	no	—
110	C₂₃×C₂²⋊Q₈	—	2	1	no	—
111	C₂₃×C₂²⋊Q₈	—	2	1	no	—
112	C₂₃×C₂²⋊Q₈	—	2	2	no	—
113	C₂₃×C₂²⋊Q₈	—	2	2	no	—
114	C₂₃×C₂²⋊Q₈	—	2	1	no	—
115	C₂₃×C₂²⋊Q₈	—	2	1	no	—
116	C₂₃×C₂²⋊Q₈	D₄	1	1	yes	characteristic
117	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
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	D₄	1	1	yes	characteristic
123	C₄×Q₈×C₂₃	—	2	1	no	—
124	C₄×Q₈×C₂₃	—	2	1	no	—
125	C₄×Q₈×C₂₃	—	2	1	no	—
126	C₄×Q₈×C₂₃	D₄	1	2	yes	—
127	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
128	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
129	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
130	C₄×D₄×C₂₃	—	2	2	no	—
131	C₄×D₄×C₂₃	—	2	2	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	—	2	1	no	—
136	C₄×D₄×C₂₃	—	2	1	no	—
137	C₄×D₄×C₂₃	—	2	1	no	—
138	C₄×D₄×C₂₃	—	2	1	no	—
139	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
140	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
141	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
142	C₂₃×#117	—	2	2	no	—
143	C₂₃×#117	—	2	2	no	—
144	C₂₃×#117	—	2	1	no	—
145	C₂₃×#117	—	2	1	no	—
146	C₂₃×#117	—	2	1	no	—
147	C₂₃×#117	D₄	1	1	yes	characteristic
148	C₂×C₄⋊C₄×C₂₃	—	2	2	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	—	2	1	no	—
153	C₂×C₄²×C₂₃	—	2	1	no	—
154	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
155	C₂₃×#107	—	2	2	no	—
156	C₂₃×#107	—	2	1	no	—
157	C₂₃×#106	—	2	1	no	—
158	C₂₃×C₄⋊C₈	—	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	2	no	—
165	C₂₃×#103	—	2	1	no	—
166	C₂₃×#103	—	2	1	no	—
167	C₂₃×#103	—	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	2	no	—
172	C₂₃×#102	—	2	1	no	—
173	C₂₃×#102	—	2	1	no	—
174	C₂₃×#97	—	2	1	no	—
175	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
176	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
177	#41×C₂₃	—	2	2	no	—
178	#41×C₂₃	—	4	1	no	—
179	#41×C₂₃	—	4	2	no	—
180	#41×C₂₃	—	4	2	no	—
181	#41×C₂₃	—	4	2	no	—
182	#41×C₂₃	—	4	2	no	—
183	C₂×Q₈×C₂₃	—	2	1	no	—
184	C₂×Q₈×C₂₃	—	2	1	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	2	no	—
189	C₂×D₄×C₂₃	—	2	1	no	—
190	C₂×D₄×C₂₃	—	2	1	no	—
191	C₂×D₄×C₂₃	—	2	1	no	—
192	C₂×D₄×C₂₃	—	2	1	no	—
193	C₂×D₄×C₂₃	—	2	1	no	—
194	C₂×D₄×C₂₃	—	4	1	no	—
195	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
196	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
197	C₂²×C₄×C₂₃	—	2	2	no	—
198	C₂²×C₄×C₂₃	—	2	2	no	—
199	C₂²×C₄×C₂₃	—	2	2	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	—	4	1	no	—
205	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
206	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
207	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
208	Q₁₆×C₂₃	—	4	2	no	—
209	SD₁₆×C₂₃	—	4	4	no	—
210	SD₁₆×C₂₃	—	4	2	no	—
211	SD₁₆×C₂₃	—	4	2	no	—
212	SD₁₆×C₂₃	—	8	1	no	—
213	D₈×C₂₃	—	4	2	no	—
214	D₈×C₂₃	—	8	1	no	—
215	M₁₆×C₂₃	—	4	2	no	—
216	M₁₆×C₂₃	—	8	1	no	—
217	C₂×C₈×C₂₃	—	2	1	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	—	4	2	no	—
224	C₄⋊C₄×C₂₃	—	2	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	2	no	—
230	C₄⋊C₄×C₂₃	—	2	2	no	—
231	C₄⋊C₄×C₂₃	—	2	2	no	—
232	C₄⋊C₄×C₂₃	—	2	1	no	—
233	C₄⋊C₄×C₂₃	—	2	1	no	—
234	C₄⋊C₄×C₂₃	—	2	2	no	—
235	C₄⋊C₄×C₂₃	—	2	2	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	C₂×D₄	1	1	yes	characteristic
246	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
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	2	yes	—
250	C₂²⋊C₄×C₂₃	—	2	2	no	—
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	1	no	—
255	C₂²⋊C₄×C₂₃	—	2	1	no	—
256	C₂²⋊C₄×C₂₃	—	2	1	no	—
257	C₂²⋊C₄×C₂₃	—	2	1	no	—
258	C₂²⋊C₄×C₂₃	—	2	1	no	—
259	C₂²⋊C₄×C₂₃	—	2	1	no	—
260	C₂²⋊C₄×C₂₃	—	2	1	no	—
261	C₂²⋊C₄×C₂₃	—	2	1	no	—
262	C₂²⋊C₄×C₂₃	—	4	1	no	—
263	C₂²⋊C₄×C₂₃	—	4	1	no	—
264	C₄²×C₂₃	—	2	2	no	—
265	C₄²×C₂₃	—	2	1	no	—
266	C₄²×C₂₃	—	2	1	no	—
267	C₄²×C₂₃	—	2	2	no	—
268	C₄²×C₂₃	—	2	1	no	—
269	C₄²×C₂₃	—	2	1	no	—
270	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
271	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
272	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
273	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
274	#18596	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
275	C₂³×C₂₃	—	2	1	no	—
276	C₂³×C₂₃	—	2	1	no	—
277	C₂³×C₂₃	—	4	1	no	—
278	C₂³×C₂₃	#120	1	1	yes	characteristic
279	Q₈×C₂₃	—	2	1	no	—
280	Q₈×C₂₃	—	2	1	no	—
281	Q₈×C₂₃	—	2	1	no	—
282	Q₈×C₂₃	—	2	1	no	—
283	Q₈×C₂₃	—	4	2	no	—
284	Q₈×C₂₃	—	4	1	no	—
285	Q₈×C₂₃	—	4	1	no	—
286	Q₈×C₂₃	—	4	1	no	—
287	Q₈×C₂₃	—	4	1	no	—
288	D₄×C₂₃	—	2	1	no	—
289	D₄×C₂₃	—	2	1	no	—
290	D₄×C₂₃	—	2	1	no	—
291	D₄×C₂₃	—	4	2	no	—
292	D₄×C₂₃	—	4	2	no	—
293	D₄×C₂₃	—	4	4	no	—
294	D₄×C₂₃	—	4	1	no	—
295	D₄×C₂₃	—	4	1	no	—
296	D₄×C₂₃	—	4	1	no	—
297	D₄×C₂₃	—	4	1	no	—
298	D₄×C₂₃	—	8	1	no	—
299	D₄×C₂₃	—	8	2	no	—
300	D₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
301	C₂×C₄×C₂₃	—	2	1	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	2	no	—
310	C₂×C₄×C₂₃	—	2	2	no	—
311	C₂×C₄×C₂₃	—	2	1	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	—	8	1	no	—
322	C₂×C₄×C₂₃	—	8	2	no	—
323	C₂×C₄×C₂₃	—	8	2	no	—
324	C₂×C₄×C₂₃	—	8	2	no	—
325	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
326	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
327	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
328	C₂×C₄×C₂₃	C₂²×D₄	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	C₈×C₂₃	—	2	2	no	—
332	C₈×C₂₃	—	4	1	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	#3515	C₂×C₂₃	1	1	yes	characteristic, complemented
338	#3251	C₂×C₂₃	1	1	yes	characteristic, complemented
339	#3182	C₂×C₂₃	1	1	yes	characteristic, complemented
340	#3035	C₂×C₂₃	1	1	yes	characteristic, complemented
341	#1648	C₂×C₂₃	1	1	yes	characteristic, complemented
342	#1647	C₂×C₂₃	1	1	yes	characteristic, complemented
343	#1644	C₂×C₂₃	1	1	yes	characteristic, complemented
344	#1643	C₂×C₂₃	1	1	yes	characteristic, complemented
345	#1626	C₂×C₂₃	1	1	yes	characteristic, complemented
346	#1625	C₂×C₂₃	1	1	yes	characteristic, complemented
347	#1622	C₂×C₂₃	1	1	yes	characteristic, complemented
348	#1622	C₂×C₂₃	1	1	yes	characteristic, complemented
349	#1550	C₂×C₂₃	1	1	yes	characteristic, complemented
350	#1489	C₂×C₂₃	1	1	yes	characteristic, complemented
351	#1483	C₂×C₂₃	1	1	yes	characteristic, complemented
352	C₂²×C₂₃	—	2	1	no	—
353	C₂²×C₂₃	—	2	2	no	—
354	C₂²×C₂₃	—	2	1	no	—
355	C₂²×C₂₃	—	2	1	no	—
356	C₂²×C₂₃	—	4	1	no	—
357	C₂²×C₂₃	—	4	1	no	—
358	C₂²×C₂₃	—	4	1	no	—
359	C₂²×C₂₃	—	4	1	no	—
360	C₂²×C₂₃	—	4	1	no	—
361	C₂²×C₂₃	—	8	1	no	—
362	C₂²×C₂₃	—	8	1	no	—
363	C₂²×C₂₃	—	8	1	no	—
364	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
365	C₄×C₂₃	—	2	1	no	—
366	C₄×C₂₃	—	2	1	no	—
367	C₄×C₂₃	—	2	1	no	—
368	C₄×C₂₃	—	2	1	no	—
369	C₄×C₂₃	—	4	1	no	—
370	C₄×C₂₃	—	4	1	no	—
371	C₄×C₂₃	—	4	2	no	—
372	C₄×C₂₃	—	4	1	no	—
373	C₄×C₂₃	—	4	1	no	—
374	C₄×C₂₃	—	4	1	no	—
375	C₄×C₂₃	—	4	1	no	—
376	C₄×C₂₃	—	4	2	no	—
377	C₄×C₂₃	—	8	1	no	—
378	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
379	C₄×C₂₃	#575	1	1	yes	characteristic
380	C₄×C₂₃	#577	1	1	yes	characteristic
381	C₄×C₂₃	#578	1	1	yes	characteristic
382	C₂×#135	—	2	1	no	—
383	#557	C₂²×C₂₃	1	1	yes	characteristic
384	#554	C₂²×C₂₃	1	1	yes	characteristic
385	#554	C₂²×C₂₃	1	1	yes	characteristic
386	#553	—	2	1	no	—
387	#552	—	2	1	no	—
388	#552	—	2	2	no	—
389	#552	—	2	1	no	—
390	#547	—	2	1	no	—
391	#547	C₂²×C₂₃	1	1	yes	characteristic
392	#541	—	2	1	no	—
393	#539	—	2	2	no	—
394	#533	C₂²×C₂₃	1	1	yes	characteristic
395	#532	C₂²×C₂₃	1	1	yes	characteristic
396	#529	—	2	1	no	—
397	#529	C₂²×C₂₃	1	1	yes	characteristic
398	C₂×#125	—	2	1	no	—
399	#520	—	2	1	no	—
400	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
401	#500	C₂²×C₂₃	1	1	yes	characteristic
402	#492	C₂²×C₂₃	1	1	yes	characteristic
403	#487	C₂²×C₂₃	1	1	yes	characteristic
404	#486	C₂²×C₂₃	1	1	yes	characteristic
405	#478	—	2	1	no	—
406	#475	C₂²×C₂₃	1	1	yes	characteristic
407	#474	C₂²×C₂₃	1	1	yes	characteristic
408	#469	—	2	1	no	—
409	#468	—	2	2	no	—
410	#468	—	2	1	no	—
411	#464	—	2	1	no	—
412	#463	—	2	1	no	—
413	#463	—	2	1	no	—
414	#462	C₂²×C₂₃	1	1	yes	characteristic
415	#461	C₂²×C₂₃	1	1	yes	characteristic
416	#460	C₂²×C₂₃	1	1	yes	characteristic
417	#460	C₂²×C₂₃	1	1	yes	characteristic
418	#459	C₂²×C₂₃	1	1	yes	characteristic
419	#459	C₂²×C₂₃	1	1	yes	characteristic
420	#440	—	2	2	no	—
421	#423	C₂²×C₂₃	1	1	yes	characteristic
422	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
423	#416	—	2	1	no	—
424	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
425	#333	C₂²×C₂₃	1	1	yes	characteristic
426	#333	C₂²×C₂₃	1	1	yes	characteristic
427	#332	C₂²×C₂₃	1	1	yes	characteristic
428	#332	C₂²×C₂₃	1	1	yes	characteristic
429	Q₈⋊C₈	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	D₄⋊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	C₂×C₂₃	—	4	1	no	—
438	C₂×C₂₃	—	4	2	no	—
439	C₂×C₂₃	—	8	1	no	—
440	C₂×C₂₃	—	16	1	no	—
441	C₂×C₂₃	#3006	1	1	yes	characteristic
442	C₂×C₂₃	#3013	1	1	yes	characteristic
443	C₂×C₂₃	#3018	1	1	yes	characteristic
444	C₂×#41	—	2	1	no	—
445	C₂×#41	—	2	1	no	—
446	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
447	#135	—	4	4	no	—
448	C₂×Q₁₆	—	2	1	no	—
449	C₂×SD₁₆	—	2	1	no	—
450	C₂×SD₁₆	—	2	2	no	—
451	C₂×SD₁₆	—	2	1	no	—
452	C₂×SD₁₆	—	2	1	no	—
453	C₂×D₈	—	2	1	no	—
454	C₂×D₈	—	2	1	no	—
455	C₂×M₁₆	—	2	1	no	—
456	C₂×M₁₆	—	2	1	no	—
457	C₂²×C₈	—	2	1	no	—
458	C₄⋊Q₈	—	2	1	no	—
459	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
460	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
461	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
462	#127	C₂³×C₂₃	1	1	yes	characteristic
463	#126	—	2	2	no	—
464	#126	—	2	2	no	—
465	#126	—	2	2	no	—
466	#126	—	2	1	no	—
467	#125	—	2	2	no	—
468	#125	—	2	2	no	—
469	#125	—	2	2	no	—
470	#125	—	2	1	no	—
471	#125	—	2	1	no	—
472	#125	—	2	2	no	—
473	#125	—	2	1	no	—
474	#125	—	2	1	no	—
475	#125	D₄×C₂₃	1	1	yes	characteristic
476	#124	—	2	1	no	—
477	#124	D₄×C₂₃	1	1	yes	characteristic
478	#123	—	2	2	no	—
479	#123	—	2	2	no	—
480	#123	—	2	2	no	—
481	#123	—	2	1	no	—
482	#123	—	2	2	no	—
483	C₂²⋊Q₈	—	2	1	no	—
484	C₂²⋊Q₈	—	2	2	no	—
485	C₂²⋊Q₈	—	2	2	no	—
486	C₂²⋊Q₈	—	2	1	no	—
487	C₂²⋊Q₈	—	2	1	no	—
488	C₂²⋊Q₈	—	2	1	no	—
489	C₂²⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
490	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
491	#121	—	2	2	no	—
492	#121	—	2	1	no	—
493	#121	—	2	1	no	—
494	#121	—	2	1	no	—
495	#121	D₄×C₂₃	1	1	yes	characteristic
496	C₄×Q₈	—	2	1	no	—
497	C₄×Q₈	—	2	1	no	—
498	C₄×Q₈	—	2	1	no	—
499	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
500	C₄×Q₈	D₄×C₂₃	1	2	yes	—
501	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
502	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
503	C₄×D₄	—	2	2	no	—
504	C₄×D₄	—	2	2	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	—	2	1	no	—
510	C₄×D₄	—	2	1	no	—
511	C₄×D₄	—	2	1	no	—
512	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
513	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
514	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
515	#117	—	2	2	no	—
516	#117	—	2	2	no	—
517	#117	—	2	1	no	—
518	#117	—	2	1	no	—
519	#117	—	2	1	no	—
520	#117	D₄×C₂₃	1	1	yes	characteristic
521	C₂×C₄⋊C₄	—	2	2	no	—
522	C₂×C₄⋊C₄	—	2	2	no	—
523	C₂×C₄⋊C₄	—	2	1	no	—
524	C₂×C₄⋊C₄	—	2	1	no	—
525	C₂×C₄⋊C₄	—	2	1	no	—
526	C₂×C₄²	—	2	1	no	—
527	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
528	#107	—	2	2	no	—
529	#107	—	2	1	no	—
530	#106	—	2	1	no	—
531	C₄⋊C₈	—	2	1	no	—
532	C₄⋊C₈	—	2	1	no	—
533	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
534	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
535	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
536	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
537	#103	—	2	2	no	—
538	#103	—	2	1	no	—
539	#103	—	2	1	no	—
540	#103	—	2	1	no	—
541	#102	—	2	1	no	—
542	#102	—	2	1	no	—
543	#102	—	2	1	no	—
544	#102	—	2	2	no	—
545	#102	—	2	1	no	—
546	#102	—	2	1	no	—
547	#97	—	2	1	no	—
548	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
549	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
550	C₂₃	#18596	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	—	4	2	no	—
556	#41	—	4	2	no	—
557	C₂×Q₈	—	2	1	no	—
558	C₂×Q₈	—	2	1	no	—
559	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
560	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
561	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
562	C₂×D₄	—	2	2	no	—
563	C₂×D₄	—	2	1	no	—
564	C₂×D₄	—	2	1	no	—
565	C₂×D₄	—	2	1	no	—
566	C₂×D₄	—	2	1	no	—
567	C₂×D₄	—	2	1	no	—
568	C₂×D₄	—	4	1	no	—
569	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
570	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
571	C₂²×C₄	—	2	2	no	—
572	C₂²×C₄	—	2	2	no	—
573	C₂²×C₄	—	2	2	no	—
574	C₂²×C₄	—	2	1	no	—
575	C₂²×C₄	—	2	1	no	—
576	C₂²×C₄	—	2	1	no	—
577	C₂²×C₄	—	2	1	no	—
578	C₂²×C₄	—	4	1	no	—
579	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
580	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
581	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
582	Q₁₆	—	4	2	no	—
583	SD₁₆	—	4	4	no	—
584	SD₁₆	—	4	2	no	—
585	SD₁₆	—	4	2	no	—
586	SD₁₆	—	8	1	no	—
587	D₈	—	4	2	no	—
588	D₈	—	8	1	no	—
589	M₁₆	—	4	2	no	—
590	M₁₆	—	8	1	no	—
591	C₂×C₈	—	2	1	no	—
592	C₂×C₈	—	2	1	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	—	4	2	no	—
598	C₄⋊C₄	—	2	2	no	—
599	C₄⋊C₄	—	2	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	2	no	—
605	C₄⋊C₄	—	2	2	no	—
606	C₄⋊C₄	—	2	1	no	—
607	C₄⋊C₄	—	2	1	no	—
608	C₄⋊C₄	—	2	2	no	—
609	C₄⋊C₄	—	2	2	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	—	2	1	no	—
616	C₄⋊C₄	—	2	1	no	—
617	C₄⋊C₄	—	2	1	no	—
618	C₄⋊C₄	—	2	1	no	—
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	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
623	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
624	C₂²⋊C₄	—	2	2	no	—
625	C₂²⋊C₄	—	2	2	no	—
626	C₂²⋊C₄	—	2	2	no	—
627	C₂²⋊C₄	—	2	2	no	—
628	C₂²⋊C₄	—	2	1	no	—
629	C₂²⋊C₄	—	2	1	no	—
630	C₂²⋊C₄	—	2	1	no	—
631	C₂²⋊C₄	—	2	1	no	—
632	C₂²⋊C₄	—	2	1	no	—
633	C₂²⋊C₄	—	2	1	no	—
634	C₂²⋊C₄	—	2	1	no	—
635	C₂²⋊C₄	—	2	1	no	—
636	C₂²⋊C₄	—	4	1	no	—
637	C₂²⋊C₄	—	4	1	no	—
638	C₄²	—	2	2	no	—
639	C₄²	—	2	1	no	—
640	C₄²	—	2	1	no	—
641	C₄²	—	2	2	no	—
642	C₄²	—	2	1	no	—
643	C₄²	—	2	1	no	—
644	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
645	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
646	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
647	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
648	C₂³	—	2	1	no	—
649	C₂³	—	2	1	no	—
650	C₂³	—	4	1	no	—
651	C₂³	C₂₃×#120	1	1	yes	characteristic
652	Q₈	—	2	1	no	—
653	Q₈	—	2	1	no	—
654	Q₈	—	2	1	no	—
655	Q₈	—	2	1	no	—
656	Q₈	—	4	2	no	—
657	Q₈	—	4	1	no	—
658	Q₈	—	4	1	no	—
659	Q₈	—	4	1	no	—
660	Q₈	—	4	1	no	—
661	D₄	—	2	1	no	—
662	D₄	—	2	1	no	—
663	D₄	—	2	1	no	—
664	D₄	—	4	2	no	—
665	D₄	—	4	2	no	—
666	D₄	—	4	4	no	—
667	D₄	—	4	1	no	—
668	D₄	—	4	1	no	—
669	D₄	—	4	1	no	—
670	D₄	—	4	1	no	—
671	D₄	—	8	1	no	—
672	D₄	—	8	2	no	—
673	D₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
674	C₂×C₄	—	2	1	no	—
675	C₂×C₄	—	2	1	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	2	no	—
683	C₂×C₄	—	2	2	no	—
684	C₂×C₄	—	2	1	no	—
685	C₂×C₄	—	2	1	no	—
686	C₂×C₄	—	4	1	no	—
687	C₂×C₄	—	4	2	no	—
688	C₂×C₄	—	4	4	no	—
689	C₂×C₄	—	4	1	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	—	8	1	no	—
695	C₂×C₄	—	8	2	no	—
696	C₂×C₄	—	8	2	no	—
697	C₂×C₄	—	8	2	no	—
698	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
699	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
700	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
701	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
702	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
703	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
704	C₈	—	2	2	no	—
705	C₈	—	4	1	no	—
706	C₈	—	4	1	no	—
707	C₈	—	4	1	no	—
708	C₈	—	4	1	no	—
709	C₈	—	4	1	no	—
710	C₂²	—	2	1	no	—
711	C₂²	—	2	2	no	—
712	C₂²	—	2	1	no	—
713	C₂²	—	2	1	no	—
714	C₂²	—	4	1	no	—
715	C₂²	—	4	1	no	—
716	C₂²	—	4	1	no	—
717	C₂²	—	4	1	no	—
718	C₂²	—	4	1	no	—
719	C₂²	—	8	1	no	—
720	C₂²	—	8	1	no	—
721	C₂²	—	8	1	no	—
722	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
723	C₄	—	2	1	no	—
724	C₄	—	2	1	no	—
725	C₄	—	2	1	no	—
726	C₄	—	2	1	no	—
727	C₄	—	4	1	no	—
728	C₄	—	4	1	no	—
729	C₄	—	4	2	no	—
730	C₄	—	4	1	no	—
731	C₄	—	4	1	no	—
732	C₄	—	4	1	no	—
733	C₄	—	4	1	no	—
734	C₄	—	4	2	no	—
735	C₄	—	8	1	no	—
736	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
737	C₄	C₂₃×#575	1	1	yes	characteristic
738	C₄	C₂₃×#577	1	1	yes	characteristic
739	C₄	C₂₃×#578	1	1	yes	characteristic
740	C₂	—	4	1	no	—
741	C₂	—	4	2	no	—
742	C₂	—	8	1	no	—
743	C₂	—	16	1	no	—
744	C₂	C₂₃×#3006	1	1	yes	characteristic
745	C₂	C₂₃×#3013	1	1	yes	characteristic
746	C₂	C₂₃×#3018	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×#3515	C₂	semidirect
2. C₂₃×#3251	C₂	semidirect
3. C₂₃×#3182	C₂	semidirect
4. C₂₃×#3035	C₂	semidirect
5. C₂₃×#1648	C₂	semidirect
6. C₂₃×#1647	C₂	semidirect
7. C₂₃×#1644	C₂	semidirect
8. C₂₃×#1643	C₂	semidirect
9. C₂₃×#1626	C₂	semidirect
10. C₂₃×#1625	C₂	semidirect
11. C₂₃×#1622	C₂	semidirect
12. C₂₃×#1622	C₂	semidirect
13. C₂₃×#1550	C₂	semidirect
14. C₂₃×#1489	C₂	semidirect
15. C₂₃×#1483	C₂	semidirect
274. #18596	C₂₃	direct (N × H)
337. #3515	C₂×C₂₃	semidirect
338. #3251	C₂×C₂₃	semidirect
339. #3182	C₂×C₂₃	semidirect
340. #3035	C₂×C₂₃	semidirect
341. #1648	C₂×C₂₃	semidirect
342. #1647	C₂×C₂₃	semidirect
343. #1644	C₂×C₂₃	semidirect
344. #1643	C₂×C₂₃	semidirect
345. #1626	C₂×C₂₃	semidirect
346. #1625	C₂×C₂₃	semidirect
347. #1622	C₂×C₂₃	semidirect
348. #1622	C₂×C₂₃	semidirect
349. #1550	C₂×C₂₃	semidirect
350. #1489	C₂×C₂₃	semidirect
351. #1483	C₂×C₂₃	semidirect
550. C₂₃	#18596	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×#3515	2
C₂₃×#3251	2
C₂₃×#3182	2
C₂₃×#3035	2
C₂₃×#1648	2
C₂₃×#1647	2
C₂₃×#1644	2
C₂₃×#1643	2
C₂₃×#1626	2
C₂₃×#1625	2
C₂₃×#1622	2
C₂₃×#1550	2
C₂₃×#1489	2
C₂₃×#1483	2
#18596	23

Subgroups — C23×#18596 #19166222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18596 #19166222