Subgroups — #22578500

Order 8768.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, Frattini	C₂²	C₂×#11764049
commutator	C₂×C₁₃₇	C₂×C₁₆
Fitting	C₂×C₄×C₁₃₇	C₈
socle	C₂²×C₁₃₇	C₂×C₈
hypercenter	C₂×C₄	#11764049
hypocenter	C₁₃₇	C₂²⋊C₁₆
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#14555522	C₂	semidirect
18. C₂×D₁₃₇	C₁₆	semidirect
26. C₁₃₇	C₂²⋊C₁₆	semidirect