Subgroups — #20298997

Order 7232.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813614	C₂	1	1	yes	maximal, characteristic
2	#16813555	C₂	1	2	yes	maximal
3	#16813553	C₂	1	2	yes	maximal
4	C₄×#11744751	C₂	1	1	yes	maximal, characteristic
5	#125×C₁₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11744751	C₂²	1	1	yes	characteristic
7	C₂×#11744751	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
9	#14268626	—	2	2	no	—
10	C₄⋊C₄×C₁₁₃	—	2	2	no	—
11	C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
12	C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₁₁₃	—	2	2	no	—
14	C₂×C₄×C₁₁₃	D₄	1	2	yes	—
15	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
16	C₂×Dic₁₁₃	—	2	2	no	—
17	#11744751	Q₈	1	2	yes	—
18	#11744751	Q₈	1	2	yes	—
19	C₂²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
20	C₄×C₁₁₃	—	2	2	no	—
21	C₄×C₁₁₃	—	4	2	no	—
22	C₄×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
23	C₄×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
24	Dic₁₁₃	—	4	2	no	—
25	C₂×C₁₁₃	C₄⋊Q₈	1	1	yes	characteristic
26	C₂×C₁₁₃	#135	1	2	yes	—
27	C₁₁₃	#499	1	1	yes	characteristic, complemented, Sylow
28	#499	—	113	1	no	maximal, Sylow
29	#125	—	113	1	no	—
30	#125	D₁₁₃	1	1	yes	characteristic
31	#107	—	113	2	no	—
32	#106	—	113	2	no	—
33	C₄×C₈	—	113	1	no	—
34	C₂×C₈	—	113	1	no	—
35	C₂×C₈	—	113	1	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	—	113	2	no	—
38	C₄⋊C₄	—	226	2	no	—
39	C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
40	C₄²	C₂×D₁₁₃	1	1	yes	characteristic
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	226	2	no	—
43	C₂×C₄	C₂²⋊D₁₁₃	1	2	yes	—
44	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
45	C₈	—	113	2	no	—
46	C₈	—	113	2	no	—
47	C₂²	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
48	C₄	—	2	2	no	—
49	C₄	—	4	2	no	—
50	C₄	—	452	2	no	—
51	C₄	Q₈×D₁₁₃	1	1	yes	characteristic
52	C₄	Q₈×D₁₁₃	1	1	yes	characteristic
53	C₂	#16813620	1	1	yes	characteristic
54	C₂	#16813642	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₁₃
commutator	C₂×C₄×C₁₁₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₁₃
Fitting	#125×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂×D₄
hypercenter	#125	D₁₁₃
hypocenter	C₁₁₃	#499
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
27. C₁₁₃	#499	semidirect