Subgroups — #22577804

Order 8768.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984079	C₂	1	2	yes	maximal
2	#17984022	C₂	1	1	yes	maximal, characteristic
3	#17984019	C₂	1	2	yes	maximal
4	C₈×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#107×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#11764040	C₂²	1	1	yes	characteristic
7	C₄⋊Dic₁₃₇	—	2	2	no	—
8	C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
9	#14555489	—	2	2	no	—
10	C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
13	C₂×Dic₁₃₇	—	2	2	no	—
14	C₂×Dic₁₃₇	D₄	1	1	yes	characteristic
15	C₂×Dic₁₃₇	D₄	1	1	yes	characteristic
16	#11764040	Q₈	1	2	yes	—
17	C₂×C₄×C₁₃₇	—	2	2	no	—
18	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
19	C₈×C₁₃₇	Q₈	1	2	yes	—
20	C₂²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
21	C₄×C₁₃₇	—	4	2	no	—
22	C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
23	C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
24	Dic₁₃₇	—	2	1	no	—
25	Dic₁₃₇	—	2	1	no	—
26	Dic₁₃₇	—	4	2	no	—
27	C₂×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
28	C₂×C₁₃₇	#135	1	1	yes	characteristic
29	C₂×C₁₃₇	#135	1	1	yes	characteristic
30	C₁₃₇	#499	1	1	yes	characteristic, complemented, Sylow
31	#499	—	137	1	no	maximal, Sylow
32	#125	—	137	2	no	—
33	#107	—	137	1	no	—
34	#107	D₁₃₇	1	1	yes	characteristic
35	#106	—	137	2	no	—
36	C₄×C₈	—	137	1	no	—
37	C₂×C₈	—	137	1	no	—
38	C₂×C₈	C₂×D₁₃₇	1	1	yes	characteristic
39	C₄⋊C₄	—	274	2	no	—
40	C₄⋊C₄	—	274	2	no	—
41	C₄⋊C₄	—	137	2	no	—
42	C₄⋊C₄	C₂×D₁₃₇	1	2	yes	—
43	C₄²	—	137	1	no	—
44	C₂×C₄	—	2	2	no	—
45	C₂×C₄	—	274	2	no	—
46	C₂×C₄	—	137	1	no	—
47	C₂×C₄	—	137	1	no	—
48	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
49	C₈	—	137	2	no	—
50	C₈	Dic₂₇₄	1	2	yes	—
51	C₂²	D₄×D₁₃₇	1	1	yes	characteristic
52	C₄	—	4	2	no	—
53	C₄	—	274	1	no	—
54	C₄	—	274	1	no	—
55	C₄	—	548	2	no	—
56	C₄	Q₈×D₁₃₇	1	1	yes	characteristic
57	C₄	C₂×Dic₂₇₄	1	1	yes	characteristic
58	C₂	#17984087	1	1	yes	characteristic
59	C₂	#17984107	1	1	yes	characteristic
60	C₂	#17984111	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₃₇
commutator	C₂×C₄×C₁₃₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₇
Fitting	#107×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×D₄
hypercenter	#107	D₁₃₇
hypocenter	C₁₃₇	#499
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
30. C₁₃₇	#499	semidirect