Subgroups — #20298974

Order 7232.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14268632	C₂	1	4	yes	maximal, complemented
2	#16813619	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#11744751	C₂	1	1	yes	maximal, characteristic, complemented
4	#127×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄₅₂	—	2	2	no	—
6	C₂×D₄₅₂	C₂²	1	2	yes	—
7	#14268632	—	2	8	no	—
8	C₂×#11744751	C₂²	1	2	yes	—
9	C₂×D₄×C₁₁₃	—	2	2	no	—
10	C₂×D₄×C₁₁₃	C₂²	1	2	yes	—
11	C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
12	C₂³×C₁₁₃	—	2	2	no	—
13	C₂²×D₁₁₃	—	2	2	no	—
14	D₄×C₁₁₃	—	2	4	no	—
15	D₄×C₁₁₃	—	4	4	no	—
16	C₂×C₄×C₁₁₃	D₄	1	2	yes	—
17	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
18	D₄₅₂	—	2	4	no	—
19	D₄₅₂	—	4	4	no	—
20	#11744751	D₄	1	4	yes	complemented
21	C₂²×C₁₁₃	—	4	2	no	—
22	C₂²×C₁₁₃	—	4	4	no	—
23	C₂²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
24	C₂×D₁₁₃	—	4	2	no	—
25	C₂×D₁₁₃	—	4	4	no	—
26	C₄×C₁₁₃	D₈	1	4	yes	complemented
27	C₄×C₁₁₃	C₂×D₄	1	2	yes	—
28	C₂×C₁₁₃	—	8	2	no	—
29	C₂×C₁₁₃	#127	1	1	yes	characteristic
30	C₂×C₁₁₃	C₂×D₈	1	2	yes	—
31	D₁₁₃	—	8	2	no	—
32	C₁₁₃	#493	1	1	yes	characteristic, complemented, Sylow
33	#493	—	113	1	no	maximal, Sylow
34	C₂×D₈	—	113	4	no	—
35	#127	—	113	1	no	—
36	#127	D₁₁₃	1	1	yes	characteristic, complemented
37	C₄×C₈	—	113	1	no	—
38	C₂×D₄	—	2	2	no	—
39	C₂×D₄	—	113	2	no	—
40	C₂×D₄	—	226	2	no	—
41	C₂×D₄	C₂×D₁₁₃	1	2	yes	—
42	D₈	—	226	8	no	—
43	C₂×C₈	—	113	2	no	—
44	C₄²	C₂×D₁₁₃	1	1	yes	characteristic
45	C₂³	—	2	2	no	—
46	C₂³	—	226	2	no	—
47	D₄	—	2	4	no	—
48	D₄	—	4	4	no	—
49	D₄	—	452	4	no	—
50	D₄	—	226	4	no	—
51	C₂×C₄	C₂²⋊D₁₁₃	1	2	yes	—
52	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
53	C₈	—	113	4	no	—
54	C₂²	—	4	2	no	—
55	C₂²	—	4	4	no	—
56	C₂²	—	452	2	no	—
57	C₂²	—	452	4	no	—
58	C₂²	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
59	C₄	#14268632	1	4	yes	complemented
60	C₄	D₄×D₁₁₃	1	2	yes	—
61	C₂	—	8	2	no	—
62	C₂	—	904	2	no	—
63	C₂	#16813618	1	1	yes	characteristic
64	C₂	C₂×#14268632	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₁₃
commutator	C₂×C₄×C₁₁₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₁₃
Fitting	#127×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂×D₄
hypercenter	#127	D₁₁₃
hypocenter	C₁₁₃	#493
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂×#14268632	C₂	semidirect
2. #16813619	C₂	semidirect
3. C₄×#11744751	C₂	semidirect
4. #127×C₁₁₃	C₂	semidirect
20. #11744751	D₄	semidirect
26. C₄×C₁₁₃	D₈	semidirect
32. C₁₁₃	#493	semidirect
36. #127	D₁₁₃	semidirect
59. C₄	#14268632	semidirect