Subgroups — C₂₃×#18670 #19166296

Order 5888.

962 conjugacy classes of subgroups.

The subgroup lattice has 780 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, complemented
2	C₂₃×#3212	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3169	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3115	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×#1642	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1641	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×#1621	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1621	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×#532	—	2	1	no	—
28	C₂₃×#532	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	C₂₃×#520	—	2	1	no	—
33	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
34	C₂₃×#500	C₂²	1	1	yes	characteristic
35	C₂₃×#495	—	2	1	no	—
36	C₂₃×#493	C₂²	1	1	yes	characteristic
37	C₂₃×#488	—	2	1	no	—
38	C₂₃×#486	C₂²	1	1	yes	characteristic
39	C₂₃×#486	C₂²	1	1	yes	characteristic
40	C₂₃×#482	—	2	1	no	—
41	C₂₃×#475	C₂²	1	1	yes	characteristic
42	C₂₃×#474	C₂²	1	1	yes	characteristic
43	C₂₃×#466	—	2	2	no	—
44	C₂₃×#462	C₂²	1	1	yes	characteristic
45	C₂₃×#461	C₂²	1	1	yes	characteristic
46	C₂₃×#460	C₂²	1	1	yes	characteristic
47	C₂₃×#460	C₂²	1	1	yes	characteristic
48	C₂₃×#459	C₂²	1	1	yes	characteristic
49	C₂₃×#459	C₂²	1	1	yes	characteristic
50	C₂₃×#452	—	2	1	no	—
51	C₂₃×#449	—	2	1	no	—
52	C₂₃×#449	—	2	1	no	—
53	C₂₃×#442	—	2	1	no	—
54	C₂₃×#440	—	2	1	no	—
55	C₄×D₈×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	—	2	1	no	—
76	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
77	C₂×Q₁₆×C₂₃	—	2	1	no	—
78	C₂×SD₁₆×C₂₃	—	2	1	no	—
79	C₂×SD₁₆×C₂₃	—	2	1	no	—
80	C₂×SD₁₆×C₂₃	—	2	1	no	—
81	C₂×D₈×C₂₃	—	2	2	no	—
82	C₂×D₈×C₂₃	—	2	1	no	—
83	C₂×D₈×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×#127	—	2	2	no	—
89	C₂₃×#127	—	2	1	no	—
90	C₂₃×#127	—	2	1	no	—
91	C₂₃×#127	C₂³	1	1	yes	characteristic
92	C₂₃×#127	C₂³	1	1	yes	characteristic
93	C₂₃×#126	—	2	2	no	—
94	C₂₃×#126	—	2	1	no	—
95	C₂₃×#125	—	2	2	no	—
96	C₂₃×#125	—	2	1	no	—
97	C₂₃×#125	D₄	1	1	yes	characteristic
98	C₂₃×#124	—	2	1	no	—
99	C₂₃×#124	—	2	1	no	—
100	C₂₃×#124	D₄	1	1	yes	characteristic
101	C₂₃×#123	—	2	2	no	—
102	C₂₃×#123	—	2	2	no	—
103	C₂₃×#123	—	2	2	no	—
104	C₂₃×#123	—	2	1	no	—
105	C₂₃×#123	—	2	2	no	—
106	C₂₃×C₂²⋊Q₈	—	2	1	no	—
107	C₂₃×C₂²⋊Q₈	D₄	1	1	yes	characteristic
108	C₂₃×#121	—	2	2	no	—
109	C₂₃×#121	—	2	2	no	—
110	C₂₃×#121	—	2	2	no	—
111	C₂₃×#121	—	2	2	no	—
112	C₂₃×#121	—	2	2	no	—
113	C₂₃×#121	—	2	1	no	—
114	C₂₃×#121	—	2	2	no	—
115	C₂₃×#121	—	2	2	no	—
116	C₂₃×#121	—	2	1	no	—
117	C₂₃×#121	—	2	1	no	—
118	C₂₃×#121	—	2	1	no	—
119	C₂₃×#121	—	2	1	no	—
120	C₂₃×#121	—	2	1	no	—
121	C₂₃×#121	D₄	1	1	yes	characteristic
122	C₂₃×#121	D₄	1	2	yes	—
123	C₄×Q₈×C₂₃	—	2	1	no	—
124	C₄×Q₈×C₂₃	—	2	1	no	—
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	2	no	—
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	—	2	1	no	—
140	C₄×D₄×C₂₃	D₄	1	2	yes	—
141	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
142	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
143	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
144	C₂₃×#117	—	2	2	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	1	no	—
150	C₂×C₄²×C₂₃	—	2	1	no	—
151	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
152	C₂₃×#107	—	2	2	no	—
153	C₂₃×#107	—	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×#102	—	2	2	no	—
163	C₂₃×#102	—	2	1	no	—
164	C₂₃×#102	—	2	2	no	—
165	C₂₃×#102	—	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	#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	#41×C₂₃	—	8	1	no	—
187	#41×C₂₃	—	8	1	no	—
188	C₂×Q₈×C₂₃	—	2	1	no	—
189	C₂×Q₈×C₂₃	—	2	1	no	—
190	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
191	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
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	2	no	—
196	C₂×D₄×C₂₃	—	2	2	no	—
197	C₂×D₄×C₂₃	—	2	2	no	—
198	C₂×D₄×C₂₃	—	2	1	no	—
199	C₂×D₄×C₂₃	—	2	2	no	—
200	C₂×D₄×C₂₃	—	2	1	no	—
201	C₂×D₄×C₂₃	—	2	1	no	—
202	C₂×D₄×C₂₃	—	2	1	no	—
203	C₂×D₄×C₂₃	—	2	1	no	—
204	C₂×D₄×C₂₃	—	2	1	no	—
205	C₂×D₄×C₂₃	—	4	1	no	—
206	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
207	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
208	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
209	C₂²×C₄×C₂₃	—	2	2	no	—
210	C₂²×C₄×C₂₃	—	2	2	no	—
211	C₂²×C₄×C₂₃	—	2	2	no	—
212	C₂²×C₄×C₂₃	—	2	1	no	—
213	C₂²×C₄×C₂₃	—	2	1	no	—
214	C₂²×C₄×C₂₃	—	2	1	no	—
215	C₂²×C₄×C₂₃	—	2	1	no	—
216	C₂²×C₄×C₂₃	—	2	2	no	—
217	C₂²×C₄×C₂₃	—	4	1	no	—
218	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
219	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
220	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
221	Q₁₆×C₂₃	—	8	1	no	—
222	SD₁₆×C₂₃	—	4	2	no	—
223	SD₁₆×C₂₃	—	8	1	no	—
224	SD₁₆×C₂₃	—	8	1	no	—
225	D₈×C₂₃	—	4	4	no	—
226	D₈×C₂₃	—	4	2	no	—
227	D₈×C₂₃	—	8	1	no	—
228	C₂×C₈×C₂₃	—	2	1	no	—
229	C₂×C₈×C₂₃	—	2	1	no	—
230	C₂×C₈×C₂₃	—	2	1	no	—
231	C₂×C₈×C₂₃	—	2	1	no	—
232	C₂×C₈×C₂₃	—	2	1	no	—
233	C₂×C₈×C₂₃	—	2	1	no	—
234	C₂×C₈×C₂₃	—	4	2	no	—
235	C₄⋊C₄×C₂₃	—	2	2	no	—
236	C₄⋊C₄×C₂₃	—	2	2	no	—
237	C₄⋊C₄×C₂₃	—	2	2	no	—
238	C₄⋊C₄×C₂₃	—	2	2	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	—	2	1	no	—
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	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
252	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
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	1	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	—	2	2	no	—
264	C₂²⋊C₄×C₂₃	—	2	1	no	—
265	C₂²⋊C₄×C₂₃	—	2	1	no	—
266	C₂²⋊C₄×C₂₃	—	4	1	no	—
267	C₂²⋊C₄×C₂₃	—	4	1	no	—
268	C₄²×C₂₃	—	2	2	no	—
269	C₄²×C₂₃	—	2	1	no	—
270	C₄²×C₂₃	—	2	1	no	—
271	C₄²×C₂₃	—	2	1	no	—
272	C₄²×C₂₃	—	2	1	no	—
273	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
274	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
275	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
276	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
277	#18670	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
278	C₂³×C₂₃	—	2	1	no	—
279	C₂³×C₂₃	—	2	1	no	—
280	C₂³×C₂₃	—	2	2	no	—
281	C₂³×C₂₃	—	4	1	no	—
282	C₂³×C₂₃	#120	1	1	yes	characteristic
283	Q₈×C₂₃	—	2	1	no	—
284	Q₈×C₂₃	—	2	1	no	—
285	Q₈×C₂₃	—	2	1	no	—
286	Q₈×C₂₃	—	4	1	no	—
287	Q₈×C₂₃	—	4	1	no	—
288	Q₈×C₂₃	—	4	1	no	—
289	Q₈×C₂₃	—	4	1	no	—
290	Q₈×C₂₃	#135	1	2	yes	—
291	D₄×C₂₃	—	2	1	no	—
292	D₄×C₂₃	—	2	1	no	—
293	D₄×C₂₃	—	2	1	no	—
294	D₄×C₂₃	—	2	1	no	—
295	D₄×C₂₃	—	4	2	no	—
296	D₄×C₂₃	—	4	2	no	—
297	D₄×C₂₃	—	4	4	no	—
298	D₄×C₂₃	—	4	2	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	—	4	1	no	—
306	D₄×C₂₃	—	4	1	no	—
307	D₄×C₂₃	—	8	1	no	—
308	D₄×C₂₃	—	8	2	no	—
309	D₄×C₂₃	—	8	2	no	—
310	D₄×C₂₃	—	8	2	no	—
311	D₄×C₂₃	—	8	2	no	—
312	D₄×C₂₃	—	8	2	no	—
313	D₄×C₂₃	—	8	2	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	1	no	—
318	C₂×C₄×C₂₃	—	2	1	no	—
319	C₂×C₄×C₂₃	—	2	1	no	—
320	C₂×C₄×C₂₃	—	2	1	no	—
321	C₂×C₄×C₂₃	—	2	1	no	—
322	C₂×C₄×C₂₃	—	2	1	no	—
323	C₂×C₄×C₂₃	—	2	2	no	—
324	C₂×C₄×C₂₃	—	2	1	no	—
325	C₂×C₄×C₂₃	—	4	1	no	—
326	C₂×C₄×C₂₃	—	4	2	no	—
327	C₂×C₄×C₂₃	—	4	4	no	—
328	C₂×C₄×C₂₃	—	4	1	no	—
329	C₂×C₄×C₂₃	—	4	1	no	—
330	C₂×C₄×C₂₃	—	4	1	no	—
331	C₂×C₄×C₂₃	—	4	1	no	—
332	C₂×C₄×C₂₃	—	4	1	no	—
333	C₂×C₄×C₂₃	—	8	1	no	—
334	C₂×C₄×C₂₃	—	8	2	no	—
335	C₂×C₄×C₂₃	—	8	2	no	—
336	C₂×C₄×C₂₃	—	8	2	no	—
337	C₂×C₄×C₂₃	—	8	2	no	—
338	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
339	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
340	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
341	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
342	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
343	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
344	C₈×C₂₃	—	2	2	no	—
345	C₈×C₂₃	—	4	1	no	—
346	C₈×C₂₃	—	4	1	no	—
347	C₈×C₂₃	—	4	1	no	—
348	C₈×C₂₃	—	4	1	no	—
349	C₈×C₂₃	—	4	1	no	—
350	#3517	C₂×C₂₃	1	1	yes	characteristic, complemented
351	#3212	C₂×C₂₃	1	1	yes	characteristic, complemented
352	#3169	C₂×C₂₃	1	1	yes	characteristic, complemented
353	#3115	C₂×C₂₃	1	1	yes	characteristic, complemented
354	#1648	C₂×C₂₃	1	1	yes	characteristic, complemented
355	#1647	C₂×C₂₃	1	1	yes	characteristic, complemented
356	#1642	C₂×C₂₃	1	1	yes	characteristic, complemented
357	#1641	C₂×C₂₃	1	1	yes	characteristic, complemented
358	#1626	C₂×C₂₃	1	1	yes	characteristic, complemented
359	#1625	C₂×C₂₃	1	1	yes	characteristic, complemented
360	#1621	C₂×C₂₃	1	1	yes	characteristic, complemented
361	#1621	C₂×C₂₃	1	1	yes	characteristic, complemented
362	#1553	C₂×C₂₃	1	1	yes	characteristic, complemented
363	#1494	C₂×C₂₃	1	1	yes	characteristic, complemented
364	#1476	C₂×C₂₃	1	1	yes	characteristic, complemented
365	C₂²×C₂₃	—	2	1	no	—
366	C₂²×C₂₃	—	2	1	no	—
367	C₂²×C₂₃	—	2	1	no	—
368	C₂²×C₂₃	—	4	1	no	—
369	C₂²×C₂₃	—	4	2	no	—
370	C₂²×C₂₃	—	4	1	no	—
371	C₂²×C₂₃	—	4	1	no	—
372	C₂²×C₂₃	—	4	1	no	—
373	C₂²×C₂₃	—	4	2	no	—
374	C₂²×C₂₃	—	4	1	no	—
375	C₂²×C₂₃	—	4	2	no	—
376	C₂²×C₂₃	—	4	1	no	—
377	C₂²×C₂₃	—	8	1	no	—
378	C₂²×C₂₃	—	8	1	no	—
379	C₂²×C₂₃	—	8	1	no	—
380	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
381	C₄×C₂₃	—	2	1	no	—
382	C₄×C₂₃	—	2	2	no	—
383	C₄×C₂₃	—	2	1	no	—
384	C₄×C₂₃	—	2	1	no	—
385	C₄×C₂₃	—	2	1	no	—
386	C₄×C₂₃	—	4	1	no	—
387	C₄×C₂₃	—	4	1	no	—
388	C₄×C₂₃	—	4	1	no	—
389	C₄×C₂₃	—	4	2	no	—
390	C₄×C₂₃	—	4	1	no	—
391	C₄×C₂₃	—	4	1	no	—
392	C₄×C₂₃	—	8	1	no	—
393	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
394	C₄×C₂₃	#575	1	1	yes	characteristic
395	C₄×C₂₃	#576	1	1	yes	characteristic
396	C₄×C₂₃	#576	1	1	yes	characteristic
397	#557	C₂²×C₂₃	1	1	yes	characteristic
398	#553	—	2	2	no	—
399	#553	—	2	1	no	—
400	#550	—	2	1	no	—
401	#550	—	2	1	no	—
402	#550	C₂²×C₂₃	1	1	yes	characteristic
403	#550	C₂²×C₂₃	1	1	yes	characteristic
404	#547	—	2	1	no	—
405	#547	C₂²×C₂₃	1	1	yes	characteristic
406	#540	—	2	2	no	—
407	#540	—	2	1	no	—
408	#532	—	2	1	no	—
409	#532	C₂²×C₂₃	1	1	yes	characteristic
410	#532	C₂²×C₂₃	1	1	yes	characteristic
411	#529	—	2	1	no	—
412	#529	C₂²×C₂₃	1	1	yes	characteristic
413	#520	—	2	1	no	—
414	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
415	#500	C₂²×C₂₃	1	1	yes	characteristic
416	#495	—	2	1	no	—
417	#493	C₂²×C₂₃	1	1	yes	characteristic
418	#488	—	2	1	no	—
419	#486	C₂²×C₂₃	1	1	yes	characteristic
420	#486	C₂²×C₂₃	1	1	yes	characteristic
421	#482	—	2	1	no	—
422	#475	C₂²×C₂₃	1	1	yes	characteristic
423	#474	C₂²×C₂₃	1	1	yes	characteristic
424	#466	—	2	2	no	—
425	#462	C₂²×C₂₃	1	1	yes	characteristic
426	#461	C₂²×C₂₃	1	1	yes	characteristic
427	#460	C₂²×C₂₃	1	1	yes	characteristic
428	#460	C₂²×C₂₃	1	1	yes	characteristic
429	#459	C₂²×C₂₃	1	1	yes	characteristic
430	#459	C₂²×C₂₃	1	1	yes	characteristic
431	#452	—	2	1	no	—
432	#449	—	2	1	no	—
433	#449	—	2	1	no	—
434	#442	—	2	1	no	—
435	#440	—	2	1	no	—
436	C₄×D₈	—	2	2	no	—
437	#432	C₂²×C₂₃	1	1	yes	characteristic
438	#431	C₂²×C₂₃	1	1	yes	characteristic
439	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
440	#332	C₂²×C₂₃	1	1	yes	characteristic
441	#332	C₂²×C₂₃	1	1	yes	characteristic
442	#331	C₂²×C₂₃	1	1	yes	characteristic
443	#331	C₂²×C₂₃	1	1	yes	characteristic
444	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
445	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
446	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
447	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
448	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
449	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
450	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
451	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
452	C₂×C₂₃	—	4	1	no	—
453	C₂×C₂₃	—	8	1	no	—
454	C₂×C₂₃	—	8	1	no	—
455	C₂×C₂₃	—	8	2	no	—
456	C₂×C₂₃	—	16	1	no	—
457	C₂×C₂₃	#3002	1	1	yes	characteristic
458	C₂×C₂₃	#3008	1	1	yes	characteristic
459	C₂×C₂₃	#3018	1	1	yes	characteristic
460	C₂×#41	—	2	1	no	—
461	C₂×#41	—	2	1	no	—
462	C₂×#41	—	2	1	no	—
463	C₂×#41	—	2	1	no	—
464	C₂×#41	—	2	1	no	—
465	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
466	C₂×Q₁₆	—	2	1	no	—
467	C₂×SD₁₆	—	2	1	no	—
468	C₂×SD₁₆	—	2	1	no	—
469	C₂×SD₁₆	—	2	1	no	—
470	C₂×D₈	—	2	2	no	—
471	C₂×D₈	—	2	1	no	—
472	C₂×D₈	—	2	1	no	—
473	C₂²×C₈	—	2	1	no	—
474	C₄⋊Q₈	—	2	1	no	—
475	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
476	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
477	#127	—	2	2	no	—
478	#127	—	2	1	no	—
479	#127	—	2	1	no	—
480	#127	C₂³×C₂₃	1	1	yes	characteristic
481	#127	C₂³×C₂₃	1	1	yes	characteristic
482	#126	—	2	2	no	—
483	#126	—	2	1	no	—
484	#125	—	2	2	no	—
485	#125	—	2	1	no	—
486	#125	D₄×C₂₃	1	1	yes	characteristic
487	#124	—	2	1	no	—
488	#124	—	2	1	no	—
489	#124	D₄×C₂₃	1	1	yes	characteristic
490	#123	—	2	2	no	—
491	#123	—	2	2	no	—
492	#123	—	2	2	no	—
493	#123	—	2	1	no	—
494	#123	—	2	2	no	—
495	C₂²⋊Q₈	—	2	1	no	—
496	C₂²⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
497	#121	—	2	2	no	—
498	#121	—	2	2	no	—
499	#121	—	2	2	no	—
500	#121	—	2	2	no	—
501	#121	—	2	2	no	—
502	#121	—	2	1	no	—
503	#121	—	2	2	no	—
504	#121	—	2	2	no	—
505	#121	—	2	1	no	—
506	#121	—	2	1	no	—
507	#121	—	2	1	no	—
508	#121	—	2	1	no	—
509	#121	—	2	1	no	—
510	#121	D₄×C₂₃	1	1	yes	characteristic
511	#121	D₄×C₂₃	1	2	yes	—
512	C₄×Q₈	—	2	1	no	—
513	C₄×Q₈	—	2	1	no	—
514	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
515	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
516	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
517	C₄×D₄	—	2	2	no	—
518	C₄×D₄	—	2	2	no	—
519	C₄×D₄	—	2	2	no	—
520	C₄×D₄	—	2	2	no	—
521	C₄×D₄	—	2	1	no	—
522	C₄×D₄	—	2	1	no	—
523	C₄×D₄	—	2	1	no	—
524	C₄×D₄	—	2	1	no	—
525	C₄×D₄	—	2	1	no	—
526	C₄×D₄	—	2	1	no	—
527	C₄×D₄	—	2	1	no	—
528	C₄×D₄	—	2	1	no	—
529	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
530	C₄×D₄	D₄×C₂₃	1	2	yes	—
531	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
532	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
533	#117	—	2	2	no	—
534	#117	—	2	1	no	—
535	#117	—	2	1	no	—
536	#117	D₄×C₂₃	1	1	yes	characteristic
537	C₂×C₄⋊C₄	—	2	2	no	—
538	C₂×C₄⋊C₄	—	2	1	no	—
539	C₂×C₄²	—	2	1	no	—
540	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
541	#107	—	2	2	no	—
542	#107	—	2	1	no	—
543	#106	—	2	1	no	—
544	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
545	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
546	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
547	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
548	#103	—	2	1	no	—
549	#103	—	2	1	no	—
550	#103	—	2	1	no	—
551	#102	—	2	2	no	—
552	#102	—	2	1	no	—
553	#102	—	2	2	no	—
554	#102	—	2	1	no	—
555	#102	—	2	1	no	—
556	#102	—	2	1	no	—
557	#102	—	2	1	no	—
558	C₂²⋊C₈	—	2	1	no	—
559	C₂²⋊C₈	—	2	1	no	—
560	#97	—	2	1	no	—
561	C₄×C₈	—	2	1	no	—
562	C₄×C₈	—	2	1	no	—
563	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
564	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
565	C₂₃	#18670	1	1	yes	characteristic, direct product factor, Sylow
566	#41	—	2	2	no	—
567	#41	—	4	1	no	—
568	#41	—	4	2	no	—
569	#41	—	4	2	no	—
570	#41	—	8	1	no	—
571	#41	—	8	1	no	—
572	#41	—	8	1	no	—
573	#41	—	8	1	no	—
574	#41	—	8	1	no	—
575	#41	—	8	1	no	—
576	#41	—	8	1	no	—
577	#41	—	8	1	no	—
578	C₂×Q₈	—	2	1	no	—
579	C₂×Q₈	—	2	1	no	—
580	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
581	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
582	C₂×D₄	—	2	2	no	—
583	C₂×D₄	—	2	1	no	—
584	C₂×D₄	—	2	2	no	—
585	C₂×D₄	—	2	2	no	—
586	C₂×D₄	—	2	2	no	—
587	C₂×D₄	—	2	2	no	—
588	C₂×D₄	—	2	1	no	—
589	C₂×D₄	—	2	2	no	—
590	C₂×D₄	—	2	1	no	—
591	C₂×D₄	—	2	1	no	—
592	C₂×D₄	—	2	1	no	—
593	C₂×D₄	—	2	1	no	—
594	C₂×D₄	—	2	1	no	—
595	C₂×D₄	—	4	1	no	—
596	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
597	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
598	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
599	C₂²×C₄	—	2	2	no	—
600	C₂²×C₄	—	2	2	no	—
601	C₂²×C₄	—	2	2	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	2	no	—
607	C₂²×C₄	—	4	1	no	—
608	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
609	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
610	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
611	Q₁₆	—	8	1	no	—
612	SD₁₆	—	4	2	no	—
613	SD₁₆	—	8	1	no	—
614	SD₁₆	—	8	1	no	—
615	D₈	—	4	4	no	—
616	D₈	—	4	2	no	—
617	D₈	—	8	1	no	—
618	C₂×C₈	—	2	1	no	—
619	C₂×C₈	—	2	1	no	—
620	C₂×C₈	—	2	1	no	—
621	C₂×C₈	—	2	1	no	—
622	C₂×C₈	—	2	1	no	—
623	C₂×C₈	—	2	1	no	—
624	C₂×C₈	—	4	2	no	—
625	C₄⋊C₄	—	2	2	no	—
626	C₄⋊C₄	—	2	2	no	—
627	C₄⋊C₄	—	2	2	no	—
628	C₄⋊C₄	—	2	2	no	—
629	C₄⋊C₄	—	2	1	no	—
630	C₄⋊C₄	—	2	1	no	—
631	C₄⋊C₄	—	2	1	no	—
632	C₄⋊C₄	—	2	1	no	—
633	C₄⋊C₄	—	2	1	no	—
634	C₄⋊C₄	—	2	1	no	—
635	C₄⋊C₄	—	2	1	no	—
636	C₄⋊C₄	—	2	1	no	—
637	C₄⋊C₄	—	2	1	no	—
638	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
639	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
640	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
641	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
642	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
643	C₂²⋊C₄	—	2	2	no	—
644	C₂²⋊C₄	—	2	2	no	—
645	C₂²⋊C₄	—	2	2	no	—
646	C₂²⋊C₄	—	2	1	no	—
647	C₂²⋊C₄	—	2	1	no	—
648	C₂²⋊C₄	—	2	1	no	—
649	C₂²⋊C₄	—	2	1	no	—
650	C₂²⋊C₄	—	2	2	no	—
651	C₂²⋊C₄	—	2	1	no	—
652	C₂²⋊C₄	—	2	1	no	—
653	C₂²⋊C₄	—	2	2	no	—
654	C₂²⋊C₄	—	2	1	no	—
655	C₂²⋊C₄	—	2	1	no	—
656	C₂²⋊C₄	—	4	1	no	—
657	C₂²⋊C₄	—	4	1	no	—
658	C₄²	—	2	2	no	—
659	C₄²	—	2	1	no	—
660	C₄²	—	2	1	no	—
661	C₄²	—	2	1	no	—
662	C₄²	—	2	1	no	—
663	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
664	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
665	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
666	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
667	C₂³	—	2	1	no	—
668	C₂³	—	2	1	no	—
669	C₂³	—	2	2	no	—
670	C₂³	—	4	1	no	—
671	C₂³	C₂₃×#120	1	1	yes	characteristic
672	Q₈	—	2	1	no	—
673	Q₈	—	2	1	no	—
674	Q₈	—	2	1	no	—
675	Q₈	—	4	1	no	—
676	Q₈	—	4	1	no	—
677	Q₈	—	4	1	no	—
678	Q₈	—	4	1	no	—
679	Q₈	C₂₃×#135	1	2	yes	—
680	D₄	—	2	1	no	—
681	D₄	—	2	1	no	—
682	D₄	—	2	1	no	—
683	D₄	—	2	1	no	—
684	D₄	—	4	2	no	—
685	D₄	—	4	2	no	—
686	D₄	—	4	4	no	—
687	D₄	—	4	2	no	—
688	D₄	—	4	1	no	—
689	D₄	—	4	1	no	—
690	D₄	—	4	1	no	—
691	D₄	—	4	1	no	—
692	D₄	—	4	1	no	—
693	D₄	—	4	1	no	—
694	D₄	—	4	1	no	—
695	D₄	—	4	1	no	—
696	D₄	—	8	2	no	—
697	D₄	—	8	1	no	—
698	D₄	—	8	2	no	—
699	D₄	—	8	2	no	—
700	D₄	—	8	2	no	—
701	D₄	—	8	2	no	—
702	D₄	—	8	2	no	—
703	C₂×C₄	—	2	1	no	—
704	C₂×C₄	—	2	1	no	—
705	C₂×C₄	—	2	1	no	—
706	C₂×C₄	—	2	1	no	—
707	C₂×C₄	—	2	1	no	—
708	C₂×C₄	—	2	1	no	—
709	C₂×C₄	—	2	1	no	—
710	C₂×C₄	—	2	1	no	—
711	C₂×C₄	—	2	1	no	—
712	C₂×C₄	—	2	2	no	—
713	C₂×C₄	—	2	1	no	—
714	C₂×C₄	—	4	1	no	—
715	C₂×C₄	—	4	2	no	—
716	C₂×C₄	—	4	4	no	—
717	C₂×C₄	—	4	1	no	—
718	C₂×C₄	—	4	1	no	—
719	C₂×C₄	—	4	1	no	—
720	C₂×C₄	—	4	1	no	—
721	C₂×C₄	—	4	1	no	—
722	C₂×C₄	—	8	2	no	—
723	C₂×C₄	—	8	1	no	—
724	C₂×C₄	—	8	2	no	—
725	C₂×C₄	—	8	2	no	—
726	C₂×C₄	—	8	2	no	—
727	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
728	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
729	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
730	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
731	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
732	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
733	C₈	—	2	2	no	—
734	C₈	—	4	1	no	—
735	C₈	—	4	1	no	—
736	C₈	—	4	1	no	—
737	C₈	—	4	1	no	—
738	C₈	—	4	1	no	—
739	C₂²	—	2	1	no	—
740	C₂²	—	2	1	no	—
741	C₂²	—	2	1	no	—
742	C₂²	—	4	1	no	—
743	C₂²	—	4	2	no	—
744	C₂²	—	4	1	no	—
745	C₂²	—	4	1	no	—
746	C₂²	—	4	1	no	—
747	C₂²	—	4	2	no	—
748	C₂²	—	4	1	no	—
749	C₂²	—	4	2	no	—
750	C₂²	—	4	1	no	—
751	C₂²	—	8	1	no	—
752	C₂²	—	8	1	no	—
753	C₂²	—	8	1	no	—
754	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
755	C₄	—	2	1	no	—
756	C₄	—	2	2	no	—
757	C₄	—	2	1	no	—
758	C₄	—	2	1	no	—
759	C₄	—	2	1	no	—
760	C₄	—	4	1	no	—
761	C₄	—	4	1	no	—
762	C₄	—	4	1	no	—
763	C₄	—	4	2	no	—
764	C₄	—	4	1	no	—
765	C₄	—	4	1	no	—
766	C₄	—	8	1	no	—
767	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
768	C₄	C₂₃×#575	1	1	yes	characteristic
769	C₄	C₂₃×#576	1	1	yes	characteristic
770	C₄	C₂₃×#576	1	1	yes	characteristic
771	C₂	—	4	1	no	—
772	C₂	—	8	1	no	—
773	C₂	—	8	1	no	—
774	C₂	—	8	2	no	—
775	C₂	—	16	1	no	—
776	C₂	C₂₃×#3002	1	1	yes	characteristic
777	C₂	C₂₃×#3008	1	1	yes	characteristic
778	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×#3517	C₂	semidirect
2. C₂₃×#3212	C₂	semidirect
3. C₂₃×#3169	C₂	semidirect
4. C₂₃×#3115	C₂	semidirect
5. C₂₃×#1648	C₂	semidirect
6. C₂₃×#1647	C₂	semidirect
7. C₂₃×#1642	C₂	semidirect
8. C₂₃×#1641	C₂	semidirect
9. C₂₃×#1626	C₂	semidirect
10. C₂₃×#1625	C₂	semidirect
11. C₂₃×#1621	C₂	semidirect
12. C₂₃×#1621	C₂	semidirect
13. C₂₃×#1553	C₂	semidirect
14. C₂₃×#1494	C₂	semidirect
15. C₂₃×#1476	C₂	semidirect
277. #18670	C₂₃	direct (N × H)
350. #3517	C₂×C₂₃	semidirect
351. #3212	C₂×C₂₃	semidirect
352. #3169	C₂×C₂₃	semidirect
353. #3115	C₂×C₂₃	semidirect
354. #1648	C₂×C₂₃	semidirect
355. #1647	C₂×C₂₃	semidirect
356. #1642	C₂×C₂₃	semidirect
357. #1641	C₂×C₂₃	semidirect
358. #1626	C₂×C₂₃	semidirect
359. #1625	C₂×C₂₃	semidirect
360. #1621	C₂×C₂₃	semidirect
361. #1621	C₂×C₂₃	semidirect
362. #1553	C₂×C₂₃	semidirect
363. #1494	C₂×C₂₃	semidirect
364. #1476	C₂×C₂₃	semidirect
565. C₂₃	#18670	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×#3212	2
C₂₃×#3169	2
C₂₃×#3115	2
C₂₃×#1648	2
C₂₃×#1647	2
C₂₃×#1642	2
C₂₃×#1641	2
C₂₃×#1626	2
C₂₃×#1625	2
C₂₃×#1621	2
C₂₃×#1553	2
C₂₃×#1494	2
C₂₃×#1476	2
#18670	23

Subgroups — C23×#18670 #19166296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18670 #19166296