Subgroups — #22578302

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	#17984096	C₂	1	1	yes	maximal, characteristic
4	#14555513	—	2	1	no	—
5	#14555513	—	2	1	no	—
6	#14555510	C₄	1	1	yes	characteristic
7	#14555493	—	2	1	no	—
8	C₂×#11764040	C₂²	1	1	yes	characteristic
9	C₈×D₁₃₇	—	2	1	no	—
10	M₁₆×C₁₃₇	C₄	1	1	yes	characteristic
11	Dic₂₇₄	C₈	1	1	yes	characteristic
12	D₅₄₈	C₈	1	1	yes	characteristic
13	C₂²⋊D₁₃₇	—	2	1	no	—
14	C₄×D₁₃₇	—	2	1	no	—
15	#11764040	D₄	1	1	yes	characteristic
16	#11764040	D₄	1	1	yes	characteristic
17	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
18	C₈×C₁₃₇	—	2	1	no	—
19	C₂²×C₁₃₇	M₁₆	1	1	yes	characteristic
20	C₂×D₁₃₇	—	2	1	no	—
21	C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
23	Dic₁₃₇	—	2	1	no	—
24	C₂×C₁₃₇	—	2	1	no	—
25	C₂×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
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	#10587739	1	1	yes	characteristic
35	C₂×C₈	—	274	1	no	—
36	C₂×C₈	—	137	1	no	—
37	C₁₆	—	274	1	no	—
38	C₁₆	—	274	1	no	—
39	Q₈	—	137	1	no	—
40	D₄	—	274	1	no	—
41	D₄	—	137	1	no	—
42	C₂×C₄	—	274	1	no	—
43	C₂×C₄	C₂×#10587739	1	1	yes	characteristic
44	C₈	—	2	1	no	—
45	C₈	—	137	1	no	—
46	C₈	—	137	1	no	—
47	C₂²	—	274	1	no	—
48	C₂²	#14555520	1	1	yes	characteristic
49	C₄	—	274	1	no	—
50	C₄	C₂²⋊#10587739	1	1	yes	characteristic
51	C₄	#14555518	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	548	1	no	—
54	C₂	#17984145	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊#10587739
commutator	C₄×C₁₃₇	C₂×C₈
Frattini	C₂×C₄	C₂×#10587739
Fitting	M₁₆×C₁₃₇	C₄
socle	C₂×C₁₃₇	C₂²⋊C₈
hypercenter	M₁₆	#10587739
hypocenter	C₁₃₇	#350
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ M₁₆

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