Subgroups — #15682121

Order 2944.

502 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625072	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×M₁₆×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092929	C₂	1	1	yes	maximal, characteristic
4	#13092494	C₂	1	2	yes	maximal, complemented
5	#13092487	C₂	1	2	yes	maximal, complemented
6	#13092487	C₂	1	2	yes	maximal, complemented
7	#13092482	C₂	1	2	yes	maximal, complemented
8	#13092385	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×(C₂×D₂₃)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092373	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#67288	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#117	C₂²	1	1	yes	characteristic
17	C₂×#67267	C₂²	1	1	yes	characteristic
18	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
19	M₁₆×D₂₃	—	2	4	no	—
20	#10625072	—	2	4	no	—
21	C₂×#67262	C₂²	1	2	yes	—
22	C₂×#67262	C₂²	1	2	yes	complemented
23	C₂×C₈×D₂₃	C₂²	1	2	yes	—
24	C₂×C₈×D₂₃	C₂²	1	2	yes	complemented
25	#10625068	C₄	1	1	yes	characteristic
26	#10625068	C₄	1	1	yes	characteristic
27	#10625066	C₄	1	1	yes	characteristic
28	#10625066	C₄	1	1	yes	characteristic
29	#10625057	C₄	1	2	yes	—
30	#10625053	C₄	1	2	yes	—
31	C₄×D₉₂	C₂²	1	2	yes	—
32	C₄×D₉₂	C₂²	1	2	yes	—
33	C₂₃×C₄⋊C₈	C₂²	1	2	yes	complemented
34	C₂₃×C₄⋊C₈	C₂²	1	2	yes	complemented
35	(C₂×D₂₃)⋊C₈	C₂²	1	1	yes	characteristic
36	(C₂×D₂₃)⋊C₈	C₂²	1	1	yes	characteristic
37	(C₂×D₂₃)⋊C₈	C₂²	1	1	yes	characteristic
38	(C₂×D₂₃)⋊C₈	C₂²	1	1	yes	characteristic
39	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	complemented
40	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	complemented
41	#10624985	C₂²	1	2	yes	complemented
42	C₄×C₂₃⋊C₈	C₂²	1	2	yes	complemented
43	C₂³×D₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	C₂×C₂²⋊D₂₃	C₂×C₄	1	2	yes	—
46	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
47	#67288	—	2	2	no	—
48	#67288	—	2	2	no	—
49	C₂×D₉₂	C₂×C₄	1	2	yes	—
50	C₂×D₉₂	C₂×C₄	1	1	yes	characteristic
51	C₂×D₉₂	C₂×C₄	1	2	yes	—
52	C₂×C₄×D₂₃	—	2	2	no	—
53	C₂×C₄×D₂₃	—	2	2	no	—
54	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₂₃	C₂³	1	2	yes	—
58	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
59	C₂×Dic₄₆	C₂×C₄	1	1	yes	characteristic
60	M₁₆×C₂₃	—	2	2	no	—
61	C₂×C₈×C₂₃	—	2	2	no	—
62	C₂×C₈×C₂₃	C₂³	1	2	yes	—
63	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
66	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
67	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
68	C₄²×C₂₃	C₂³	1	2	yes	—
69	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
70	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
71	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
72	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
73	C₄⋊Dic₂₃	C₂×C₄	1	2	yes	—
74	C₄⋊Dic₂₃	C₂×C₄	1	2	yes	—
75	#67267	D₄	1	4	yes	complemented
76	C₂×C₂₃⋊C₈	—	2	2	no	—
77	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
78	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
79	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
80	#67262	—	2	4	no	—
81	#67262	—	2	4	no	—
82	C₈×D₂₃	—	2	4	no	—
83	C₈×D₂₃	—	2	4	no	—
84	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×D₂₃	—	2	2	no	—
86	C₂²×D₂₃	—	2	2	no	—
87	C₂²×D₂₃	—	4	1	no	—
88	C₂²×D₂₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×D₂₃	C₂²×C₄	1	2	yes	—
90	C₂²×D₂₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₂₃	#41	1	2	yes	—
96	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₂₃	—	2	4	no	—
98	C₂×Dic₂₃	—	2	2	no	—
99	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
100	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
102	D₉₂	—	2	4	no	—
103	D₉₂	—	2	2	no	—
104	D₉₂	—	4	2	no	—
105	C₄×D₂₃	—	2	2	no	—
106	C₄×D₂₃	—	2	2	no	—
107	C₄×D₂₃	—	2	1	no	—
108	C₄×D₂₃	—	2	2	no	—
109	C₄×D₂₃	—	2	1	no	—
110	C₄×D₂₃	—	2	1	no	—
111	C₄×D₂₃	—	2	1	no	—
112	C₄×D₂₃	—	2	2	no	—
113	Dic₄₆	—	2	2	no	—
114	C₈×C₂₃	—	2	1	no	—
115	C₈×C₂₃	—	2	2	no	—
116	C₈×C₂₃	—	2	1	no	—
117	C₂₃⋊C₈	—	2	1	no	—
118	C₂₃⋊C₈	—	2	1	no	—
119	C₂₃⋊C₈	C₂×D₄	1	4	yes	—
120	#2933	—	23	1	no	maximal, Sylow
121	C₂²×C₂₃	—	2	1	no	—
122	C₂²×C₂₃	—	2	1	no	—
123	C₂²×C₂₃	C₄×D₄	1	2	yes	—
124	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×D₂₃	—	2	1	no	—
126	C₂×D₂₃	—	2	1	no	—
127	C₂×D₂₃	—	2	1	no	—
128	C₂×D₂₃	—	2	2	no	—
129	C₂×D₂₃	—	2	2	no	—
130	C₂×D₂₃	—	2	1	no	—
131	C₂×D₂₃	—	2	1	no	—
132	C₂×D₂₃	—	2	2	no	—
133	C₂×D₂₃	—	4	4	no	—
134	C₂×D₂₃	#131	1	2	yes	complemented
135	C₄×C₂₃	—	2	2	no	—
136	C₄×C₂₃	—	2	2	no	—
137	C₄×C₂₃	C₄×D₄	1	2	yes	—
138	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
139	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
140	Dic₂₃	—	2	2	no	—
141	Dic₂₃	—	2	1	no	—
142	Dic₂₃	—	2	1	no	—
143	C₂×#131	—	23	1	no	—
144	C₂²×M₁₆	—	23	1	no	—
145	#525	—	23	1	no	—
146	#436	—	23	2	no	—
147	#435	—	23	2	no	—
148	#435	—	23	2	no	—
149	C₈×D₄	—	23	2	no	—
150	#424	D₂₃	1	1	yes	characteristic, complemented
151	#408	—	23	1	no	—
152	C₂×C₂²⋊C₈	—	23	1	no	—
153	#405	—	23	1	no	—
154	C₂×C₂₃	—	2	2	no	—
155	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
156	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
157	C₂×C₂₃	#568	1	1	yes	characteristic
158	D₂₃	—	2	2	no	—
159	D₂₃	—	4	2	no	—
160	D₂₃	—	4	1	no	—
161	C₂×#41	—	23	1	no	—
162	C₂³×C₄	—	23	1	no	—
163	#131	—	46	4	no	—
164	C₂×M₁₆	—	23	1	no	—
165	C₂×M₁₆	—	23	2	no	—
166	C₂×M₁₆	—	23	2	no	—
167	C₂×M₁₆	—	46	4	no	—
168	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
169	C₂²×C₈	—	23	1	no	—
170	C₂²×C₈	—	23	2	no	—
171	C₂²×C₈	—	23	2	no	—
172	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
173	#123	—	23	2	no	—
174	C₂²⋊Q₈	—	23	1	no	—
175	C₂²⋊Q₈	—	23	1	no	—
176	#121	—	23	1	no	—
177	#121	—	23	1	no	—
178	#120	—	23	2	no	—
179	C₄×D₄	—	23	2	no	—
180	C₄×D₄	—	23	2	no	—
181	#117	C₂×D₂₃	1	1	yes	characteristic
182	C₄⋊C₈	C₂×D₂₃	1	2	yes	complemented
183	C₄⋊C₈	C₂×D₂₃	1	2	yes	complemented
184	C₂²⋊C₈	—	23	1	no	—
185	C₂²⋊C₈	—	23	1	no	—
186	C₂²⋊C₈	—	23	1	no	—
187	C₂²⋊C₈	—	23	1	no	—
188	C₂²⋊C₈	—	23	2	no	—
189	C₂²⋊C₈	—	23	2	no	—
190	#97	—	23	2	no	—
191	C₄×C₈	—	23	2	no	—
192	C₂₃	#2933	1	1	yes	characteristic, complemented, Sylow
193	C₂⁴	—	23	1	no	—
194	#41	—	46	2	no	—
195	#41	—	46	2	no	—
196	C₂×Q₈	—	23	1	no	—
197	C₂×D₄	—	23	2	no	—
198	C₂×D₄	—	23	2	no	—
199	C₂×D₄	—	23	1	no	—
200	C₂×D₄	—	23	2	no	—
201	C₂²×C₄	—	23	1	no	—
202	C₂²×C₄	—	23	1	no	—
203	C₂²×C₄	—	23	1	no	—
204	C₂²×C₄	—	23	1	no	—
205	C₂²×C₄	—	23	2	no	—
206	C₂²×C₄	—	23	1	no	—
207	C₂²×C₄	—	46	2	no	—
208	C₂²×C₄	—	46	2	no	—
209	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
210	M₁₆	—	2	2	no	—
211	M₁₆	—	23	4	no	—
212	M₁₆	—	46	4	no	—
213	M₁₆	—	46	4	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	23	2	no	—
216	C₂×C₈	—	23	1	no	—
217	C₂×C₈	—	23	1	no	—
218	C₂×C₈	—	46	4	no	—
219	C₂×C₈	—	46	4	no	—
220	C₂×C₈	—	46	2	no	—
221	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
222	C₂×C₈	C₂²×D₂₃	1	2	yes	—
223	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
224	C₄⋊C₄	—	23	2	no	—
225	C₄⋊C₄	—	23	2	no	—
226	C₄⋊C₄	C₄×D₂₃	1	1	yes	characteristic
227	C₄⋊C₄	C₄×D₂₃	1	1	yes	characteristic
228	C₂²⋊C₄	—	23	2	no	—
229	C₂²⋊C₄	—	23	2	no	—
230	C₂²⋊C₄	—	23	2	no	—
231	C₂²⋊C₄	—	23	2	no	—
232	C₂²⋊C₄	C₄×D₂₃	1	2	yes	—
233	C₄²	C₂²×D₂₃	1	2	yes	—
234	C₂³	—	23	1	no	—
235	C₂³	—	23	1	no	—
236	C₂³	—	23	2	no	—
237	C₂³	—	46	2	no	—
238	C₂³	—	46	2	no	—
239	C₂³	—	92	1	no	—
240	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
241	Q₈	—	46	2	no	—
242	D₄	—	46	4	no	—
243	D₄	—	46	4	no	—
244	D₄	—	46	2	no	—
245	D₄	—	92	2	no	—
246	C₂×C₄	—	23	1	no	—
247	C₂×C₄	—	23	1	no	—
248	C₂×C₄	—	23	2	no	—
249	C₂×C₄	—	46	2	no	—
250	C₂×C₄	—	46	2	no	—
251	C₂×C₄	—	46	2	no	—
252	C₂×C₄	—	46	1	no	—
253	C₂×C₄	—	46	2	no	—
254	C₂×C₄	—	46	1	no	—
255	C₂×C₄	—	46	1	no	—
256	C₂×C₄	—	46	1	no	—
257	C₂×C₄	—	46	2	no	—
258	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
259	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
260	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
261	C₂×C₄	D₄×D₂₃	1	2	yes	—
262	C₂×C₄	#67292	1	2	yes	—
263	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	2	2	no	—
267	C₈	—	23	4	no	—
268	C₈	—	46	1	no	—
269	C₈	—	46	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	23	2	no	—
273	C₂²	—	46	1	no	—
274	C₂²	—	46	1	no	—
275	C₂²	—	46	1	no	—
276	C₂²	—	46	2	no	—
277	C₂²	—	46	2	no	—
278	C₂²	—	46	1	no	—
279	C₂²	—	46	1	no	—
280	C₂²	—	46	2	no	—
281	C₂²	—	92	4	no	—
282	C₂²	#10625064	1	2	yes	—
283	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	46	1	no	—
287	C₄	—	46	1	no	—
288	C₄	—	46	2	no	—
289	C₄	#10625064	1	2	yes	—
290	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
291	C₄	C₂×#67292	1	1	yes	characteristic
292	C₂	—	2	2	no	—
293	C₂	—	46	2	no	—
294	C₂	—	92	1	no	—
295	C₂	—	92	2	no	—
296	C₂	C₂×#10625064	1	1	yes	characteristic
297	C₂	#131×D₂₃	1	1	yes	characteristic
298	C₂	#13093182	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting	C₂₃×#424	C₂
prime radical	#13092929	C₂
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#424	D₂₃
hypocenter	C₂₃	#2933
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
1. C₂×#10625072	C₂	semidirect
2. C₂×M₁₆×D₂₃	C₂	semidirect
4. #13092494	C₂	semidirect
5. #13092487	C₂	semidirect
6. #13092487	C₂	semidirect
7. #13092482	C₂	semidirect
8. #13092385	C₂	semidirect
9. C₂×(C₂×D₂₃)⋊C₈	C₂	semidirect
10. #13092373	C₂	semidirect
11. C₂₃×#424	C₂	semidirect
22. C₂×#67262	C₂²	semidirect
24. C₂×C₈×D₂₃	C₂²	semidirect
33. C₂₃×C₄⋊C₈	C₂²	semidirect
34. C₂₃×C₄⋊C₈	C₂²	semidirect
39. (C₂×D₂₃)⋊C₈	C₂²	semidirect
40. (C₂×D₂₃)⋊C₈	C₂²	semidirect
41. #10624985	C₂²	semidirect
42. C₄×C₂₃⋊C₈	C₂²	semidirect
75. #67267	D₄	semidirect
134. C₂×D₂₃	#131	semidirect
150. #424	D₂₃	semidirect
182. C₄⋊C₈	C₂×D₂₃	semidirect
183. C₄⋊C₈	C₂×D₂₃	semidirect
192. C₂₃	#2933	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625072	2
C₂×M₁₆×D₂₃	2
#13092929	2
#13092494	2
#13092487	2
#13092482	2
#13092385	2
C₂×(C₂×D₂₃)⋊C₈	2
#13092373	2
C₂₃×#424	2
#2933	23