Subgroups — #20298508

Order 7232.

322 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄₅₂	C₂	1	3	yes	maximal, complemented
2	C₂×#14268624	C₂	1	3	yes	maximal, complemented
3	#95×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₁₁₃	C₂²	1	3	yes	—
5	C₂×D₄₅₂	—	2	12	no	—
6	C₂×D₄₅₂	—	2	6	no	—
7	C₂²×Dic₁₁₃	C₂²	1	1	yes	characteristic
8	#14268624	—	2	6	no	—
9	C₂²×C₄×C₁₁₃	C₂²	1	3	yes	—
10	C₂³×C₁₁₃	C₂³	1	1	yes	characteristic
11	C₂²×D₁₁₃	—	2	3	no	—
12	C₂²×D₁₁₃	—	2	6	no	—
13	C₂²×D₁₁₃	—	2	6	no	—
14	C₂²×D₁₁₃	—	2	3	no	—
15	C₂²×D₁₁₃	D₄	1	6	yes	complemented
16	C₂×C₄×C₁₁₃	—	2	6	no	—
17	C₂×C₄×C₁₁₃	D₄	1	6	yes	complemented
18	C₂×Dic₁₁₃	—	2	3	no	—
19	D₄₅₂	—	2	24	no	—
20	C₂²×C₁₁₃	C₂×D₄	1	3	yes	—
21	C₂²×C₁₁₃	C₂×D₄	1	3	yes	—
22	C₂²×C₁₁₃	#41	1	1	yes	characteristic
23	C₂×D₁₁₃	—	2	12	no	—
24	C₂×D₁₁₃	—	2	6	no	—
25	C₂×D₁₁₃	—	4	3	no	—
26	C₂×D₁₁₃	—	4	6	no	—
27	C₂×D₁₁₃	—	4	3	no	—
28	C₄×C₁₁₃	—	2	6	no	—
29	Dic₁₁₃	—	4	1	no	—
30	C₂×C₁₁₃	#120	1	3	yes	—
31	C₂×C₁₁₃	#121	1	3	yes	—
32	C₂×C₁₁₃	#127	1	1	yes	characteristic
33	D₁₁₃	—	4	6	no	—
34	C₁₁₃	#392	1	1	yes	characteristic, complemented, Sylow
35	#392	—	113	1	no	maximal, Sylow
36	C₂²×D₄	—	113	3	no	—
37	C₂×C₂²⋊C₄	—	113	3	no	—
38	#95	D₁₁₃	1	1	yes	characteristic, complemented
39	C₂⁴	—	113	3	no	—
40	C₂×D₄	—	226	12	no	—
41	C₂×D₄	—	226	6	no	—
42	C₂²×C₄	—	113	1	no	—
43	C₂²×C₄	C₂×D₁₁₃	1	3	yes	—
44	C₂²⋊C₄	—	226	6	no	—
45	C₂³	—	113	6	no	—
46	C₂³	—	226	6	no	—
47	C₂³	—	226	3	no	—
48	C₂³	—	226	6	no	—
49	C₂³	—	226	3	no	—
50	C₂³	C₂²×D₁₁₃	1	1	yes	characteristic
51	D₄	—	226	24	no	—
52	C₂×C₄	—	2	6	no	—
53	C₂×C₄	—	226	3	no	—
54	C₂×C₄	D₄₅₂	1	6	yes	complemented
55	C₂²	—	452	3	no	—
56	C₂²	—	452	6	no	—
57	C₂²	—	452	3	no	—
58	C₂²	—	226	12	no	—
59	C₂²	—	226	6	no	—
60	C₂²	D₄×D₁₁₃	1	3	yes	—
61	C₂²	C₂×D₄₅₂	1	3	yes	—
62	C₂²	#14268648	1	1	yes	characteristic
63	C₄	—	2	6	no	—
64	C₄	—	452	1	no	—
65	C₂	—	452	6	no	—
66	C₂	#16813592	1	3	yes	—
67	C₂	#16813597	1	3	yes	—
68	C₂	#16813619	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₁₃
commutator, socle	C₂³×C₁₁₃	C₂³
Fitting	#95×C₁₁₃	C₂
hypercenter	#95	D₁₁₃
hypocenter	C₁₁₃	#392
LCS	G ▷ C₂³×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂³ ◁ #95

Complements

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

N	H ≅ G/N	type
1. C₂²×D₄₅₂	C₂	semidirect
2. C₂×#14268624	C₂	semidirect
3. #95×C₁₁₃	C₂	semidirect
15. C₂²×D₁₁₃	D₄	semidirect
17. C₂×C₄×C₁₁₃	D₄	semidirect
34. C₁₁₃	#392	semidirect
38. #95	D₁₁₃	semidirect
54. C₂×C₄	D₄₅₂	semidirect