Subgroups — #22578301

Order 8768.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984162	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14555513	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984095	C₂	1	1	yes	maximal, characteristic
4	#14555513	—	2	1	no	—
5	#14555513	—	2	1	no	—
6	#14555494	—	2	1	no	—
7	#14555494	C₄	1	1	yes	characteristic
8	C₂×#11764040	—	2	1	no	—
9	C₂×#11764040	C₂²	1	1	yes	characteristic
10	#41×C₁₃₇	C₄	1	1	yes	characteristic
11	#11764040	—	2	1	no	—
12	#11764040	D₄	1	1	yes	characteristic
13	#11764040	D₄	1	1	yes	characteristic
14	Q₈×C₁₃₇	C₈	1	1	yes	characteristic
15	D₄×C₁₃₇	—	2	1	no	—
16	D₄×C₁₃₇	C₈	1	1	yes	characteristic
17	C₂×C₄×C₁₃₇	—	2	1	no	—
18	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₁₃₇	—	2	1	no	—
20	C₂²×C₁₃₇	M₁₆	1	1	yes	characteristic
21	C₄×C₁₃₇	—	2	1	no	—
22	C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
24	C₂×C₁₃₇	—	2	1	no	—
25	C₂×C₁₃₇	—	4	1	no	—
26	C₂×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
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	#10587739	1	1	yes	characteristic
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	#11764047	1	1	yes	characteristic
40	D₄	—	2	1	no	—
41	D₄	#11764047	1	1	yes	characteristic
42	C₂×C₄	—	2	1	no	—
43	C₂×C₄	C₂×#10587739	1	1	yes	characteristic
44	C₈	—	274	1	no	—
45	C₈	—	137	1	no	—
46	C₈	—	137	1	no	—
47	C₂²	—	2	1	no	—
48	C₂²	#14555519	1	1	yes	characteristic
49	C₄	—	2	1	no	—
50	C₄	C₂²⋊#10587739	1	1	yes	characteristic
51	C₄	C₂×#11764047	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	4	1	no	—
54	C₂	C₂²⋊#11764047	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊#10587739
commutator	C₄×C₁₃₇	C₂×C₈
Frattini	C₂×C₄	C₂×#10587739
Fitting	#41×C₁₃₇	C₄
socle	C₂×C₁₃₇	C₂²⋊C₈
hypercenter	#41	#10587739
hypocenter	C₁₃₇	#350
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #17984162	C₂	semidirect
2. C₂×#14555513	C₂	semidirect
27. C₁₃₇	#350	semidirect