Subgroups — #15682133

Order 2944.

544 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₂₃	C₂	1	2	yes	maximal, complemented
2	C₄×D₄×D₂₃	C₂	1	1	yes	maximal, characteristic
3	#13092495	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092491	C₂	1	2	yes	maximal, complemented
5	#13092486	C₂	1	2	yes	maximal, complemented
6	C₈×D₉₂	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092482	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092439	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092379	C₂	1	2	yes	maximal, complemented
10	C₄×#67262	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₂₃	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₂₃	C₂²	1	2	yes	—
14	C₂×M₁₆×C₂₃	C₂²	1	2	yes	complemented
15	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
16	D₄×Dic₂₃	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₂₃	C₂²	1	2	yes	—
18	C₂×#67267	C₂²	1	2	yes	complemented
19	M₁₆×D₂₃	—	2	8	no	—
20	C₂×#67262	C₂²	1	1	yes	characteristic
21	C₂×#67262	C₂²	1	2	yes	—
22	C₂×#67262	C₂²	1	1	yes	characteristic
23	C₂×C₈×D₂₃	C₂²	1	2	yes	complemented
24	C₂×C₈×D₂₃	C₂²	1	2	yes	complemented
25	#10625064	C₄	1	1	yes	characteristic
26	C₄⋊C₄×D₂₃	C₄	1	1	yes	characteristic
27	(C₂²⋊D₂₃)⋊C₄	C₄	1	2	yes	—
28	C₂²⋊C₄×D₂₃	C₄	1	2	yes	—
29	C₄×D₉₂	C₂²	1	1	yes	characteristic
30	C₄²×D₂₃	C₂²	1	1	yes	characteristic
31	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
32	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
33	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
34	C₂²⋊(C₂₃⋊C₈)	C₂²	1	2	yes	complemented
35	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	complemented
36	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	complemented
37	#10624997	C₂²	1	1	yes	characteristic, complemented
38	#10624997	C₂²	1	1	yes	characteristic, complemented
39	Dic₂₃⋊C₈	C₂²	1	2	yes	complemented
40	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic, complemented
41	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
42	C₂³×D₂₃	C₂×C₄	1	2	yes	—
43	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
45	C₂×C₂²⋊D₂₃	C₂×C₄	1	2	yes	—
46	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
47	D₄×D₂₃	—	2	4	no	—
48	C₂×D₉₂	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₂₃	—	2	4	no	—
50	C₂×C₄×D₂₃	—	2	4	no	—
51	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
54	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₂₃	C₂³	1	2	yes	—
56	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
57	M₁₆×C₂₃	—	2	4	no	—
58	C₂×C₈×C₂₃	C₂³	1	2	yes	—
59	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
62	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
63	C₄²×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²⋊Dic₂₃	C₂×C₄	1	2	yes	—
65	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
66	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
67	C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
68	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
69	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
72	#67267	—	2	4	no	—
73	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
74	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
75	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
76	#67262	—	2	4	no	—
77	#67262	—	2	2	no	—
78	#67262	D₄	1	4	yes	complemented
79	C₈×D₂₃	—	2	4	no	—
80	C₈×D₂₃	—	2	4	no	—
81	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
82	C₂²×D₂₃	—	2	4	no	—
83	C₂²×D₂₃	—	2	4	no	—
84	C₂²×D₂₃	—	4	2	no	—
85	C₂²×D₂₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×D₂₃	C₂²×C₄	1	2	yes	—
87	D₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	—	2	2	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₂₃	—	2	4	no	—
95	C₂×Dic₂₃	—	2	4	no	—
96	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
99	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
100	D₉₂	M₁₆	1	4	yes	complemented
101	C₄×D₂₃	—	2	4	no	—
102	C₄×D₂₃	—	2	2	no	—
103	C₄×D₂₃	—	2	2	no	—
104	C₄×D₂₃	—	2	2	no	—
105	C₄×D₂₃	—	2	4	no	—
106	C₄×D₂₃	—	4	1	no	—
107	C₄×D₂₃	C₂×D₄	1	2	yes	—
108	C₄×D₂₃	#41	1	2	yes	—
109	C₈×C₂₃	—	2	2	no	—
110	C₈×C₂₃	—	2	1	no	—
111	C₈×C₂₃	C₂×D₄	1	2	yes	complemented
112	C₂₃⋊C₈	—	2	2	no	—
113	C₂₃⋊C₈	—	2	1	no	—
114	C₂₃⋊C₈	C₂×D₄	1	2	yes	complemented
115	D₄×M₁₆	—	23	1	no	maximal, Sylow
116	C₂²×C₂₃	—	2	2	no	—
117	C₂²×C₂₃	—	2	2	no	—
118	C₂²×C₂₃	—	2	2	no	—
119	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×D₂₃	—	2	2	no	—
121	C₂×D₂₃	—	2	1	no	—
122	C₂×D₂₃	—	2	1	no	—
123	C₂×D₂₃	—	2	2	no	—
124	C₂×D₂₃	—	4	8	no	—
125	C₂×D₂₃	C₄×D₄	1	2	yes	—
126	C₂×D₂₃	C₂×M₁₆	1	4	yes	complemented
127	C₄×C₂₃	—	2	2	no	—
128	C₄×C₂₃	—	2	1	no	—
129	C₄×C₂₃	C₂×M₁₆	1	2	yes	complemented
130	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
131	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
132	Dic₂₃	—	2	2	no	—
133	Dic₂₃	—	2	1	no	—
134	Dic₂₃	—	2	1	no	—
135	Dic₂₃	C₄×D₄	1	2	yes	—
136	C₂²×M₁₆	—	23	2	no	—
137	C₂×C₄×D₄	—	23	1	no	—
138	#436	—	23	1	no	—
139	#436	D₂₃	1	1	yes	characteristic, complemented
140	#435	—	23	2	no	—
141	#435	—	23	2	no	—
142	C₈×D₄	—	23	1	no	—
143	C₈×D₄	—	23	1	no	—
144	#423	—	23	1	no	—
145	#407	—	23	2	no	—
146	C₄×M₁₆	—	23	1	no	—
147	C₂×C₂₃	—	4	2	no	—
148	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
149	C₂×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
150	C₂×C₂₃	#568	1	1	yes	characteristic
151	D₂₃	—	2	2	no	—
152	D₂₃	—	2	4	no	—
153	C₂²×D₄	—	23	1	no	—
154	C₂³×C₄	—	23	2	no	—
155	C₂×M₁₆	—	23	2	no	—
156	C₂×M₁₆	—	23	1	no	—
157	C₂×M₁₆	—	23	2	no	—
158	C₂×M₁₆	—	23	1	no	—
159	C₂×M₁₆	—	46	8	no	—
160	C₂×M₁₆	C₂×D₂₃	1	2	yes	complemented
161	C₂²×C₈	—	23	2	no	—
162	C₂²×C₈	—	23	2	no	—
163	C₄×D₄	—	23	1	no	—
164	C₄×D₄	—	23	2	no	—
165	C₄×D₄	—	23	1	no	—
166	C₄×D₄	—	23	2	no	—
167	C₄×D₄	—	23	1	no	—
168	C₄×D₄	C₂×D₂₃	1	1	yes	characteristic
169	C₂×C₄⋊C₄	—	23	1	no	—
170	C₂×C₂²⋊C₄	—	23	2	no	—
171	C₂×C₄²	—	23	1	no	—
172	C₄⋊C₈	—	23	2	no	—
173	C₄⋊C₈	—	23	1	no	—
174	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic, complemented
175	C₂²⋊C₈	—	23	2	no	—
176	C₂²⋊C₈	—	23	2	no	—
177	C₂²⋊C₈	—	23	2	no	—
178	C₂²⋊C₈	C₂×D₂₃	1	2	yes	complemented
179	#97	—	23	1	no	—
180	#97	—	23	1	no	—
181	C₄×C₈	—	23	1	no	—
182	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic, complemented
183	C₂₃	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
184	C₂⁴	—	23	2	no	—
185	C₂×D₄	—	23	2	no	—
186	C₂×D₄	—	23	1	no	—
187	C₂×D₄	—	46	4	no	—
188	C₂×D₄	C₄×D₂₃	1	1	yes	characteristic
189	C₂²×C₄	—	23	1	no	—
190	C₂²×C₄	—	23	2	no	—
191	C₂²×C₄	—	23	1	no	—
192	C₂²×C₄	—	23	2	no	—
193	C₂²×C₄	—	23	2	no	—
194	C₂²×C₄	—	23	2	no	—
195	C₂²×C₄	—	23	1	no	—
196	C₂²×C₄	—	46	4	no	—
197	C₂²×C₄	—	46	4	no	—
198	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
199	M₁₆	—	2	4	no	—
200	M₁₆	—	23	4	no	—
201	M₁₆	—	46	4	no	—
202	M₁₆	—	46	2	no	—
203	M₁₆	—	46	4	no	—
204	C₂×C₈	—	23	1	no	—
205	C₂×C₈	—	23	1	no	—
206	C₂×C₈	—	23	2	no	—
207	C₂×C₈	—	46	4	no	—
208	C₂×C₈	—	46	4	no	—
209	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
210	C₂×C₈	C₂²×D₂₃	1	2	yes	—
211	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
212	C₄⋊C₄	—	23	2	no	—
213	C₄⋊C₄	—	23	1	no	—
214	C₄⋊C₄	C₄×D₂₃	1	1	yes	characteristic
215	C₂²⋊C₄	—	23	2	no	—
216	C₂²⋊C₄	—	23	2	no	—
217	C₂²⋊C₄	—	23	2	no	—
218	C₂²⋊C₄	C₄×D₂₃	1	2	yes	—
219	C₄²	—	23	1	no	—
220	C₄²	—	23	1	no	—
221	C₄²	—	23	1	no	—
222	C₄²	C₂²×D₂₃	1	1	yes	characteristic
223	C₂³	—	23	2	no	—
224	C₂³	—	23	1	no	—
225	C₂³	—	46	4	no	—
226	C₂³	—	46	4	no	—
227	C₂³	—	92	2	no	—
228	C₂³	C₂×C₄×D₂₃	1	2	yes	—
229	D₄	—	2	2	no	—
230	D₄	—	23	4	no	—
231	D₄	—	46	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	23	1	no	—
235	C₂×C₄	—	23	2	no	—
236	C₂×C₄	—	23	2	no	—
237	C₂×C₄	—	23	1	no	—
238	C₂×C₄	—	23	1	no	—
239	C₂×C₄	—	23	2	no	—
240	C₂×C₄	—	46	4	no	—
241	C₂×C₄	—	46	4	no	—
242	C₂×C₄	—	46	4	no	—
243	C₂×C₄	—	46	2	no	—
244	C₂×C₄	—	46	2	no	—
245	C₂×C₄	—	46	2	no	—
246	C₂×C₄	—	92	1	no	—
247	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
249	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
250	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	2	1	no	—
253	C₈	—	23	2	no	—
254	C₈	—	46	2	no	—
255	C₈	—	46	1	no	—
256	C₈	D₄×D₂₃	1	2	yes	complemented
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	23	2	no	—
261	C₂²	—	23	4	no	—
262	C₂²	—	46	2	no	—
263	C₂²	—	46	1	no	—
264	C₂²	—	46	1	no	—
265	C₂²	—	46	2	no	—
266	C₂²	—	92	8	no	—
267	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	—	23	2	no	—
271	C₄	—	46	2	no	—
272	C₄	—	46	1	no	—
273	C₄	—	46	1	no	—
274	C₄	M₁₆×D₂₃	1	2	yes	complemented
275	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
276	C₄	#41×D₂₃	1	1	yes	characteristic
277	C₂	—	4	2	no	—
278	C₂	—	46	2	no	—
279	C₂	—	46	4	no	—
280	C₂	C₄×D₄×D₂₃	1	1	yes	characteristic
281	C₂	C₂×M₁₆×D₂₃	1	1	yes	characteristic
282	C₂	#13093182	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting	C₂₃×#436	C₂
prime radical	C₂×D₄×D₂₃	C₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#436	D₂₃
hypocenter	C₂₃	D₄×M₁₆
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×M₁₆×D₂₃	C₂	semidirect
3. #13092495	C₂	semidirect
4. #13092491	C₂	semidirect
5. #13092486	C₂	semidirect
6. C₈×D₉₂	C₂	semidirect
7. #13092482	C₂	semidirect
8. #13092439	C₂	semidirect
9. #13092379	C₂	semidirect
10. C₄×#67262	C₂	semidirect
11. C₂₃×#436	C₂	semidirect
14. C₂×M₁₆×C₂₃	C₂²	semidirect
18. C₂×#67267	C₂²	semidirect
23. C₂×C₈×D₂₃	C₂²	semidirect
24. C₂×C₈×D₂₃	C₂²	semidirect
31. C₂₃×C₄⋊C₈	C₂²	semidirect
32. C₂₃×C₂²⋊C₈	C₂²	semidirect
33. C₄×C₈×C₂₃	C₂²	semidirect
34. C₂²⋊(C₂₃⋊C₈)	C₂²	semidirect
35. (C₂×D₂₃)⋊C₈	C₂²	semidirect
36. (C₂×D₂₃)⋊C₈	C₂²	semidirect
37. #10624997	C₂²	semidirect
38. #10624997	C₂²	semidirect
39. Dic₂₃⋊C₈	C₂²	semidirect
40. C₄⋊(C₂₃⋊C₈)	C₂²	semidirect
41. C₄×C₂₃⋊C₈	C₂²	semidirect
78. #67262	D₄	semidirect
100. D₉₂	M₁₆	semidirect
111. C₈×C₂₃	C₂×D₄	semidirect
114. C₂₃⋊C₈	C₂×D₄	semidirect
126. C₂×D₂₃	C₂×M₁₆	semidirect
129. C₄×C₂₃	C₂×M₁₆	semidirect
139. #436	D₂₃	semidirect
160. C₂×M₁₆	C₂×D₂₃	semidirect
174. C₄⋊C₈	C₂×D₂₃	semidirect
178. C₂²⋊C₈	C₂×D₂₃	semidirect
182. C₄×C₈	C₂×D₂₃	semidirect
183. C₂₃	D₄×M₁₆	semidirect
256. C₈	D₄×D₂₃	semidirect
274. C₄	M₁₆×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₂₃	2
C₄×D₄×D₂₃	2
#13092495	2
#13092491	2
#13092486	2
C₈×D₉₂	2
#13092482	2
#13092439	2
#13092379	2
C₄×#67262	2
C₂₃×#436	2
D₄×M₁₆	23