Subgroups — #22578524

Order 8768.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984185	C₂	1	1	yes	maximal, characteristic, complemented
2	#17984184	C₂	1	2	yes	maximal, complemented
3	#17984183	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14555522	C₂	1	2	yes	maximal, complemented
5	#17984175	C₂	1	1	yes	maximal, characteristic
6	#14555522	C₂²	1	1	yes	characteristic
7	#14555522	C₂²	1	2	yes	—
8	#14555522	C₂²	1	1	yes	characteristic
9	#14555520	C₄	1	1	yes	characteristic
10	#14555519	C₄	1	2	yes	—
11	#14555518	C₂²	1	1	yes	characteristic
12	C₂×#11764047	C₂²	1	2	yes	—
13	#14555509	C₄	1	1	yes	characteristic
14	#11764047	C₂×C₄	1	2	yes	—
15	#11764047	C₂×C₄	1	1	yes	characteristic
16	#11764047	C₂³	1	1	yes	characteristic
17	Dic₂₇₄	C₈	1	1	yes	characteristic
18	C₂²⋊D₁₃₇	C₈	1	2	yes	—
19	C₂×Dic₁₃₇	C₂×C₄	1	2	yes	—
20	C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
21	D₄×C₁₃₇	C₈	1	1	yes	characteristic
22	C₂²×C₁₃₇	C₂×C₈	1	2	yes	—
23	C₂×D₁₃₇	C₂×C₈	1	1	yes	characteristic
24	C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
25	Dic₁₃₇	C₂×C₈	1	2	yes	—
26	Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₁₃₇	—	2	2	no	—
28	C₂×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
29	D₁₃₇	—	2	1	no	—
30	C₁₃₇	#504	1	1	yes	characteristic, complemented, Sylow
31	#504	—	137	1	no	maximal, Sylow
32	#131	—	137	1	no	—
33	M₃₂	—	137	2	no	—
34	M₃₂	—	137	1	no	—
35	C₂×C₁₆	—	137	2	no	—
36	C₂×C₁₆	—	137	1	no	—
37	#41	—	137	1	no	—
38	M₁₆	—	137	2	no	—
39	M₁₆	—	137	1	no	—
40	C₂×C₈	—	137	2	no	—
41	C₂×C₈	—	137	1	no	—
42	C₁₆	—	137	1	no	—
43	C₁₆	—	137	2	no	—
44	C₁₆	—	137	1	no	—
45	Q₈	—	137	1	no	—
46	D₄	—	137	2	no	—
47	D₄	#11764049	1	1	yes	characteristic
48	C₂×C₄	—	137	2	no	—
49	C₂×C₄	—	137	1	no	—
50	C₈	—	137	1	no	—
51	C₈	—	137	2	no	—
52	C₈	—	137	1	no	—
53	C₂²	—	137	1	no	—
54	C₂²	C₂×#11764049	1	2	yes	—
55	C₄	—	137	1	no	—
56	C₄	—	137	2	no	—
57	C₄	C₂×#11764049	1	1	yes	characteristic
58	C₂	—	2	2	no	—
59	C₂	—	274	1	no	—
60	C₂	C₂²×#11764049	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×#11764049
commutator, socle	C₂×C₁₃₇	C₂²×C₈
Fitting	D₄×C₁₃₇	C₈
hypercenter	D₄	#11764049
hypocenter	C₁₃₇	#504
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ D₄

Complements

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

N	H ≅ G/N	type
1. #17984185	C₂	semidirect
2. #17984184	C₂	semidirect
3. #17984183	C₂	semidirect
4. C₂×#14555522	C₂	semidirect
30. C₁₃₇	#504	semidirect