Subgroups — #20299334

Order 7232.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813609	C₂	1	4	yes	maximal, complemented
2	#16813603	C₂	1	4	yes	maximal, complemented
3	C₄×Dic₂₂₆	C₂	1	2	yes	maximal, complemented
4	D₄×Dic₁₁₃	C₂	1	4	yes	maximal, complemented
5	#127×C₁₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₂₆	C₂²	1	2	yes	—
7	C₂²×Dic₁₁₃	C₂²	1	4	yes	—
8	C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
9	#14268626	C₂²	1	4	yes	—
10	C₂²⋊Dic₁₁₃	C₂²	1	8	yes	—
11	C₂²⋊Dic₁₁₃	C₂²	1	4	yes	—
12	C₄×Dic₁₁₃	C₂²	1	4	yes	—
13	C₂×D₄×C₁₁₃	C₂²	1	4	yes	—
14	C₂×D₄×C₁₁₃	C₂²	1	2	yes	—
15	C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
16	C₂³×C₁₁₃	C₂³	1	4	yes	—
17	D₄×C₁₁₃	—	2	8	no	—
18	D₄×C₁₁₃	—	4	2	no	—
19	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₁₁₃	C₂³	1	2	yes	—
21	C₂×Dic₁₁₃	—	4	4	no	—
22	C₂×Dic₁₁₃	C₂³	1	4	yes	—
23	C₂×Dic₁₁₃	C₂³	1	4	yes	—
24	Dic₂₂₆	—	2	4	no	—
25	C₂²×C₁₁₃	—	2	8	no	—
26	C₂²×C₁₁₃	—	2	4	no	—
27	C₂²×C₁₁₃	C₂⁴	1	1	yes	characteristic
28	C₄×C₁₁₃	—	2	1	no	—
29	C₄×C₁₁₃	#41	1	4	yes	—
30	Dic₁₁₃	—	2	4	no	—
31	Dic₁₁₃	—	2	4	no	—
32	C₂×C₁₁₃	—	4	4	no	—
33	C₂×C₁₁₃	C₂×#41	1	2	yes	—
34	C₂×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
35	C₁₁₃	#559	1	1	yes	characteristic, complemented, Sylow
36	#559	—	113	1	no	maximal, Sylow
37	#127	D₁₁₃	1	1	yes	characteristic
38	#124	—	113	4	no	—
39	#123	—	113	4	no	—
40	C₄×Q₈	—	113	2	no	—
41	C₄×D₄	—	113	4	no	—
42	C₂×Q₈	—	113	2	no	—
43	C₂×D₄	C₂×D₁₁₃	1	4	yes	—
44	C₂×D₄	C₂×D₁₁₃	1	2	yes	—
45	C₂²×C₄	—	113	4	no	—
46	C₄⋊C₄	—	113	2	no	—
47	C₄⋊C₄	—	113	4	no	—
48	C₂²⋊C₄	—	113	8	no	—
49	C₂²⋊C₄	—	113	4	no	—
50	C₄²	—	113	4	no	—
51	C₄²	C₂×D₁₁₃	1	1	yes	characteristic
52	C₂³	C₂²×D₁₁₃	1	4	yes	—
53	Q₈	—	226	4	no	—
54	D₄	—	2	8	no	—
55	D₄	—	4	2	no	—
56	C₂×C₄	—	113	4	no	—
57	C₂×C₄	—	113	4	no	—
58	C₂×C₄	—	452	4	no	—
59	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₁₁₃	1	2	yes	—
61	C₂²	—	2	8	no	—
62	C₂²	—	2	4	no	—
63	C₂²	C₂³×D₁₁₃	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	226	4	no	—
66	C₄	—	226	4	no	—
67	C₄	#14268646	1	4	yes	—
68	C₂	—	4	4	no	—
69	C₂	C₂×#14268646	1	2	yes	—
70	C₂	#16813667	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₁₃
commutator, socle	C₂²×C₁₁₃	C₂⁴
Fitting	#127×C₁₁₃	C₂
hypercenter	#127	D₁₁₃
hypocenter	C₁₁₃	#559
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #16813609	C₂	semidirect
2. #16813603	C₂	semidirect
3. C₄×Dic₂₂₆	C₂	semidirect
4. D₄×Dic₁₁₃	C₂	semidirect
35. C₁₁₃	#559	semidirect