Subgroups — #22577193

Order 8768.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984005	C₂	1	1	yes	maximal, characteristic
2	#17984003	C₂	1	1	yes	maximal, characteristic
3	#123×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#14555494	—	2	1	no	—
5	C₂²⋊Dic₁₃₇	—	2	1	no	—
6	C₂×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₁₃₇	C₄	1	1	yes	characteristic, complemented
8	C₄⋊C₄×C₁₃₇	—	2	1	no	—
9	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
10	C₂²⋊C₄×C₁₃₇	C₄	1	1	yes	characteristic, complemented
11	C₂×Dic₁₃₇	—	4	1	no	—
12	#11764040	—	4	1	no	—
13	C₂³×C₁₃₇	C₂×C₄	1	1	yes	characteristic
14	C₂³×C₁₃₇	D₄	1	1	yes	characteristic
15	D₄×C₁₃₇	—	4	1	no	—
16	C₂×C₄×C₁₃₇	—	2	1	no	—
17	C₂×C₄×C₁₃₇	—	2	1	no	—
18	C₂×C₄×C₁₃₇	—	4	1	no	—
19	C₂×C₄×C₁₃₇	D₄	1	1	yes	characteristic
20	C₂²×C₁₃₇	—	2	1	no	—
21	C₂²×C₁₃₇	—	2	1	no	—
22	C₂²×C₁₃₇	—	4	1	no	—
23	C₂²×C₁₃₇	—	4	1	no	—
24	C₂²×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₄×C₁₃₇	—	2	1	no	—
26	C₄×C₁₃₇	—	4	1	no	—
27	C₄×C₁₃₇	—	4	1	no	—
28	Dic₁₃₇	—	8	1	no	—
29	C₂×C₁₃₇	—	2	1	no	—
30	C₂×C₁₃₇	—	4	1	no	—
31	C₂×C₁₃₇	—	4	1	no	—
32	C₂×C₁₃₇	#99	1	1	yes	characteristic
33	C₁₃₇	#352	1	1	yes	characteristic, complemented, Sylow
34	#352	—	137	1	no	maximal, Sylow
35	#123	D₁₃₇	1	1	yes	characteristic
36	#100	—	137	1	no	—
37	#99	—	137	1	no	—
38	C₂×D₄	C₂×D₁₃₇	1	1	yes	characteristic
39	C₂²×C₄	Dic₁₃₇	1	1	yes	characteristic, complemented
40	M₁₆	—	274	1	no	—
41	C₄⋊C₄	—	2	1	no	—
42	C₂²⋊C₄	—	2	1	no	—
43	C₂²⋊C₄	—	274	1	no	—
44	C₂²⋊C₄	Dic₁₃₇	1	1	yes	characteristic, complemented
45	C₂³	C₂×Dic₁₃₇	1	1	yes	characteristic
46	C₂³	C₂²⋊D₁₃₇	1	1	yes	characteristic
47	D₄	—	4	1	no	—
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	—	2	1	no	—
50	C₂×C₄	—	4	1	no	—
51	C₂×C₄	—	548	1	no	—
52	C₂×C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
53	C₈	—	548	1	no	—
54	C₂²	—	2	1	no	—
55	C₂²	—	2	1	no	—
56	C₂²	—	4	1	no	—
57	C₂²	—	4	1	no	—
58	C₂²	C₂²⋊Dic₁₃₇	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	—	4	1	no	—
61	C₄	—	4	1	no	—
62	C₄	—	1096	1	no	—
63	C₂	—	2	1	no	—
64	C₂	—	4	1	no	—
65	C₂	—	4	1	no	—
66	C₂	#17984003	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#17984003
commutator	C₂³×C₁₃₇	C₂×C₄
Frattini	C₂×D₄	C₂×D₁₃₇
Fitting	#123×C₁₃₇	C₂
socle	C₂×C₁₃₇	#99
hypercenter	#123	D₁₃₇
hypocenter	C₁₃₇	#352
LCS	G ▷ C₂³×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #123

Complements

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

N	H ≅ G/N	type
7. C₂²×C₄×C₁₃₇	C₄	semidirect
10. C₂²⋊C₄×C₁₃₇	C₄	semidirect
33. C₁₃₇	#352	semidirect
39. C₂²×C₄	Dic₁₃₇	semidirect
44. C₂²⋊C₄	Dic₁₃₇	semidirect