Subgroups — #20299678

Order 7232.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813710	C₂	1	8	yes	maximal, complemented
2	D₁₁₃⋊M₁₆	C₂	1	6	yes	maximal, complemented
3	C₂×#14268648	C₂	1	1	yes	maximal, characteristic
4	#14268657	C₂²	1	12	yes	complemented
5	#14268656	C₂²	1	4	yes	—
6	#14268655	C₂²	1	12	yes	complemented
7	#14268648	C₄	1	4	yes	—
8	#14268648	C₂²	1	4	yes	—
9	C₂×D₄₅₂	C₄	1	3	yes	—
10	C₂×C₄×D₁₁₃	C₂²	1	3	yes	—
11	C₂×Q₈×C₁₁₃	C₄	1	1	yes	characteristic
12	C₂²×D₁₁₃	C₂×C₄	1	3	yes	—
13	Q₈×C₁₁₃	C₂×C₄	1	4	yes	—
14	C₂×C₄×C₁₁₃	C₂×C₄	1	3	yes	—
15	C₂×Dic₁₁₃	C₂³	1	1	yes	characteristic
16	D₄₅₂	C₂×C₄	1	12	yes	—
17	C₄×D₁₁₃	C₂×C₄	1	6	yes	—
18	C₄×D₁₁₃	C₂³	1	6	yes	—
19	#11744753	C₂³	1	8	yes	—
20	C₂²×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
21	C₂×D₁₁₃	—	4	3	no	—
22	C₂×D₁₁₃	C₂²×C₄	1	6	yes	—
23	C₄×C₁₁₃	C₂²×C₄	1	6	yes	—
24	Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
25	Dic₁₁₃	C₂⁴	1	1	yes	characteristic
26	C₂×C₁₁₃	—	2	1	no	—
27	C₂×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
28	D₁₁₃	—	2	6	no	—
29	C₁₁₃	#568	1	1	yes	characteristic, complemented, Sylow
30	#568	—	113	1	no	maximal, Sylow
31	C₂×#41	—	113	1	no	—
32	#131	—	113	8	no	—
33	C₂×M₁₆	—	113	6	no	—
34	#41	—	113	4	no	—
35	#41	—	113	4	no	—
36	C₂×Q₈	#90673	1	1	yes	characteristic
37	C₂×D₄	—	113	3	no	—
38	C₂²×C₄	—	113	3	no	—
39	M₁₆	—	113	4	no	—
40	M₁₆	—	113	12	no	—
41	C₂×C₈	—	113	12	no	—
42	C₂³	—	113	3	no	—
43	Q₈	C₂×#90673	1	4	yes	—
44	D₄	—	113	12	no	—
45	C₂×C₄	—	113	1	no	—
46	C₂×C₄	—	113	6	no	—
47	C₂×C₄	—	113	6	no	—
48	C₂×C₄	C₂×#90673	1	3	yes	—
49	C₈	—	113	8	no	—
50	C₂²	—	113	6	no	—
51	C₂²	—	452	3	no	—
52	C₂²	C₂²×#90673	1	1	yes	characteristic
53	C₄	—	113	1	no	—
54	C₄	—	113	1	no	—
55	C₄	C₂²×#90673	1	6	yes	—
56	C₂	—	2	1	no	—
57	C₂	—	226	6	no	—
58	C₂	C₂³×#90673	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×#90673
commutator, socle	C₂×C₁₁₃	C₂³×C₄
Fitting	C₂×Q₈×C₁₁₃	C₄
hypercenter	C₂×Q₈	#90673
hypocenter	C₁₁₃	#568
LCS	G ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. #16813710	C₂	semidirect
2. D₁₁₃⋊M₁₆	C₂	semidirect
4. #14268657	C₂²	semidirect
6. #14268655	C₂²	semidirect
29. C₁₁₃	#568	semidirect