Subgroups — #20299699

Order 7232.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813719	C₂	1	1	yes	maximal, characteristic, complemented
2	#16813717	C₂	1	1	yes	maximal, characteristic, complemented
3	#16813710	C₂	1	1	yes	maximal, characteristic
4	#14268659	—	2	1	no	—
5	#14268659	—	2	1	no	—
6	#14268657	—	2	1	no	—
7	#14268657	C₄	1	1	yes	characteristic
8	#14268655	—	2	1	no	—
9	#14268655	C₂²	1	1	yes	characteristic
10	#14268648	C₄	1	1	yes	characteristic
11	Q₈×C₁₁₃	C₈	1	1	yes	characteristic
12	D₄₅₂	—	2	1	no	—
13	D₄₅₂	C₈	1	1	yes	characteristic
14	C₄×D₁₁₃	—	2	1	no	—
15	C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
16	#11744753	—	2	1	no	—
17	#11744753	D₄	1	1	yes	characteristic
18	#11744753	D₄	1	1	yes	characteristic
19	C₂×D₁₁₃	—	2	1	no	—
20	C₂×D₁₁₃	M₁₆	1	1	yes	characteristic
21	C₄×C₁₁₃	—	2	1	no	—
22	C₄×C₁₁₃	C₂×C₈	1	1	yes	characteristic
23	Dic₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₁₁₃	C₂²⋊C₈	1	1	yes	characteristic
25	D₁₁₃	—	2	1	no	—
26	D₁₁₃	—	4	1	no	—
27	C₁₁₃	#350	1	1	yes	characteristic, complemented, Sylow
28	#350	—	113	1	no	maximal, Sylow
29	#131	—	113	1	no	—
30	M₃₂	—	113	1	no	—
31	C₂×C₁₆	—	113	1	no	—
32	#41	—	113	1	no	—
33	M₁₆	—	113	1	no	—
34	M₁₆	—	226	1	no	—
35	C₂×C₈	—	113	1	no	—
36	C₂×C₈	—	226	1	no	—
37	C₁₆	—	226	1	no	—
38	C₁₆	—	226	1	no	—
39	Q₈	#11744762	1	1	yes	characteristic
40	D₄	—	113	1	no	—
41	D₄	—	226	1	no	—
42	C₂×C₄	—	113	1	no	—
43	C₂×C₄	—	226	1	no	—
44	C₈	—	113	1	no	—
45	C₈	—	113	1	no	—
46	C₈	—	226	1	no	—
47	C₂²	—	113	1	no	—
48	C₂²	—	226	1	no	—
49	C₄	—	2	1	no	—
50	C₄	—	113	1	no	—
51	C₄	C₂×#11744762	1	1	yes	characteristic
52	C₂	—	452	1	no	—
53	C₂	—	226	1	no	—
54	C₂	C₂²⋊#11744762	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²⋊#11744762
commutator	C₄×C₁₁₃	C₂×C₈
Frattini	C₄	C₂×#11744762
Fitting	Q₈×C₁₁₃	C₈
socle	C₂×C₁₁₃	C₂²⋊C₈
hypercenter	Q₈	#11744762
hypocenter	C₁₁₃	#350
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₈

Complements

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

N	H ≅ G/N	type
1. #16813719	C₂	semidirect
2. #16813717	C₂	semidirect
27. C₁₁₃	#350	semidirect