Subgroups — #20298968

Order 7232.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813622	C₂	1	1	yes	maximal, characteristic
2	#16813545	C₂	1	4	yes	maximal
3	#16813533	C₂	1	1	yes	maximal, characteristic
4	#125×C₁₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₂₂₆	C₂²	1	2	yes	—
6	C₂×#11744751	C₂²	1	2	yes	—
7	C₄⋊Dic₁₁₃	—	2	2	no	—
8	C₄⋊C₄×C₁₁₃	—	2	2	no	—
9	C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
10	C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₁₃	—	2	2	no	—
12	C₂×C₄×C₁₁₃	D₄	1	2	yes	—
13	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
14	C₂×Dic₁₁₃	—	2	2	no	—
15	Dic₂₂₆	—	2	4	no	—
16	#11744751	—	2	2	no	—
17	C₂²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
18	C₄×C₁₁₃	—	2	2	no	—
19	C₄×C₁₁₃	—	4	2	no	—
20	C₄×C₁₁₃	#41	1	2	yes	—
21	Dic₁₁₃	—	4	2	no	—
22	C₂×C₁₁₃	#124	1	1	yes	characteristic
23	C₂×C₁₁₃	#137	1	2	yes	—
24	C₁₁₃	#491	1	1	yes	characteristic, complemented, Sylow
25	#491	—	113	1	no	maximal, Sylow
26	C₄⋊Q₈	—	113	1	no	—
27	#125	D₁₁₃	1	1	yes	characteristic
28	#103	—	113	4	no	—
29	#97	—	113	1	no	—
30	C₂×Q₈	—	113	2	no	—
31	C₂×C₈	—	113	2	no	—
32	C₄⋊C₄	—	2	2	no	—
33	C₄⋊C₄	—	226	2	no	—
34	C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
35	C₄²	C₂×D₁₁₃	1	1	yes	characteristic
36	Q₈	—	226	4	no	—
37	C₂×C₄	—	2	2	no	—
38	C₂×C₄	—	226	2	no	—
39	C₂×C₄	C₂²⋊D₁₁₃	1	2	yes	—
40	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
41	C₈	—	226	2	no	—
42	C₂²	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
43	C₄	—	2	2	no	—
44	C₄	—	4	2	no	—
45	C₄	—	452	2	no	—
46	C₄	#14268648	1	2	yes	—
47	C₂	#16813611	1	1	yes	characteristic
48	C₂	#16813652	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	#491
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
24. C₁₁₃	#491	semidirect