Subgroups — #22577802

Order 8768.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984080	C₂	1	1	yes	maximal, characteristic
2	#17984021	C₂	1	2	yes	maximal
3	#17984019	C₂	1	2	yes	maximal
4	C₄×#11764040	C₂	1	1	yes	maximal, characteristic
5	#125×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#11764040	C₂²	1	1	yes	characteristic
7	C₂×#11764040	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
9	#14555489	—	2	2	no	—
10	C₄⋊C₄×C₁₃₇	—	2	2	no	—
11	C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
12	C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₃₇	—	2	2	no	—
14	#11764040	Q₈	1	2	yes	—
15	#11764040	Q₈	1	2	yes	—
16	C₂×C₄×C₁₃₇	—	2	2	no	—
17	C₂×C₄×C₁₃₇	D₄	1	2	yes	—
18	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
19	C₂²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
20	C₄×C₁₃₇	—	2	2	no	—
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	—	4	2	no	—
25	C₂×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
26	C₂×C₁₃₇	#135	1	2	yes	—
27	C₁₃₇	#499	1	1	yes	characteristic, complemented, Sylow
28	#499	—	137	1	no	maximal, Sylow
29	#125	—	137	1	no	—
30	#125	D₁₃₇	1	1	yes	characteristic
31	#107	—	137	2	no	—
32	#106	—	137	2	no	—
33	C₄×C₈	—	137	1	no	—
34	C₂×C₈	—	137	1	no	—
35	C₂×C₈	—	137	1	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	—	274	2	no	—
38	C₄⋊C₄	—	137	2	no	—
39	C₄⋊C₄	C₂×D₁₃₇	1	2	yes	—
40	C₄²	C₂×D₁₃₇	1	1	yes	characteristic
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	274	2	no	—
43	C₂×C₄	C₂²⋊D₁₃₇	1	2	yes	—
44	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
45	C₈	—	137	2	no	—
46	C₈	—	137	2	no	—
47	C₂²	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
48	C₄	—	2	2	no	—
49	C₄	—	4	2	no	—
50	C₄	—	548	2	no	—
51	C₄	Q₈×D₁₃₇	1	1	yes	characteristic
52	C₄	Q₈×D₁₃₇	1	1	yes	characteristic
53	C₂	#17984086	1	1	yes	characteristic
54	C₂	#17984108	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₃₇
commutator	C₂×C₄×C₁₃₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₇
Fitting	#125×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×D₄
hypercenter	#125	D₁₃₇
hypocenter	C₁₃₇	#499
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

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