Subgroups — #22578504

Order 8768.

56 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	#17984183	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984176	C₂	1	1	yes	maximal, characteristic
4	#14555522	—	2	1	no	—
5	#14555522	—	2	1	no	—
6	#14555520	—	2	1	no	—
7	#14555520	C₄	1	1	yes	characteristic
8	#14555518	—	2	1	no	—
9	#14555518	C₂²	1	1	yes	characteristic
10	#14555511	C₄	1	1	yes	characteristic
11	#11764047	—	2	1	no	—
12	#11764047	D₄	1	1	yes	characteristic
13	#11764047	D₄	1	1	yes	characteristic
14	D₅₄₈	—	2	1	no	—
15	D₅₄₈	C₈	1	1	yes	characteristic
16	C₄×D₁₃₇	—	2	1	no	—
17	C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
18	Q₈×C₁₃₇	C₈	1	1	yes	characteristic
19	C₂×D₁₃₇	—	2	1	no	—
20	C₂×D₁₃₇	M₁₆	1	1	yes	characteristic
21	C₄×C₁₃₇	—	2	1	no	—
22	C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
23	Dic₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
25	D₁₃₇	—	2	1	no	—
26	D₁₃₇	—	4	1	no	—
27	C₁₃₇	#350	1	1	yes	characteristic, complemented, Sylow
28	#350	—	137	1	no	maximal, Sylow
29	#131	—	137	1	no	—
30	M₃₂	—	137	1	no	—
31	C₂×C₁₆	—	137	1	no	—
32	#41	—	137	1	no	—
33	M₁₆	—	274	1	no	—
34	M₁₆	—	137	1	no	—
35	C₂×C₈	—	274	1	no	—
36	C₂×C₈	—	137	1	no	—
37	C₁₆	—	274	1	no	—
38	C₁₆	—	274	1	no	—
39	Q₈	#11764049	1	1	yes	characteristic
40	D₄	—	274	1	no	—
41	D₄	—	137	1	no	—
42	C₂×C₄	—	274	1	no	—
43	C₂×C₄	—	137	1	no	—
44	C₈	—	274	1	no	—
45	C₈	—	137	1	no	—
46	C₈	—	137	1	no	—
47	C₂²	—	274	1	no	—
48	C₂²	—	137	1	no	—
49	C₄	—	2	1	no	—
50	C₄	—	137	1	no	—
51	C₄	C₂×#11764049	1	1	yes	characteristic
52	C₂	—	274	1	no	—
53	C₂	—	548	1	no	—
54	C₂	C₂²⋊#11764049	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²⋊#11764049
commutator	C₄×C₁₃₇	C₂×C₈
Frattini	C₄	C₂×#11764049
Fitting	Q₈×C₁₃₇	C₈
socle	C₂×C₁₃₇	C₂²⋊C₈
hypercenter	Q₈	#11764049
hypocenter	C₁₃₇	#350
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₈

Complements

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

N	H ≅ G/N	type
1. #17984185	C₂	semidirect
2. #17984183	C₂	semidirect
27. C₁₃₇	#350	semidirect