Subgroups — C₂×#17984028 #22577818

Order 8768.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984028	C₂	1	4	yes	maximal, direct product factor
2	C₂×#14555485	C₂	1	2	yes	maximal, complemented
3	C₂²×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#14555485	C₂²	1	4	yes	complemented
5	C₂²×C₄×C₁₃₇	C₄	1	1	yes	characteristic
6	C₂×C₈×C₁₃₇	C₄	1	1	yes	characteristic
7	C₂×C₈×C₁₃₇	C₄	1	2	yes	—
8	C₂×C₈×C₁₃₇	C₂²	1	2	yes	—
9	C₂×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂³×C₁₃₇	C₈	1	1	yes	characteristic
11	C₂×C₄×C₁₃₇	C₈	1	2	yes	—
12	C₂×C₄×C₁₃₇	C₈	1	1	yes	characteristic
13	C₂×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
15	C₈×C₁₃₇	C₂×C₄	1	2	yes	—
16	C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
17	C₈×C₁₃₇	C₂³	1	1	yes	characteristic
18	C₂²×C₁₃₇	—	2	2	no	—
19	C₂²×C₁₃₇	C₂×C₈	1	2	yes	—
20	C₂²×C₁₃₇	C₂×C₈	1	1	yes	characteristic
21	C₄×C₁₃₇	C₂×C₈	1	2	yes	—
22	C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
23	C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
24	C₂×C₁₃₇	—	2	2	no	—
25	C₂×C₁₃₇	M₃₂	1	2	yes	complemented
26	C₂×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
27	C₁₃₇	C₂×M₃₂	1	1	yes	characteristic, complemented, Sylow
28	C₂×M₃₂	—	137	1	no	maximal, Sylow
29	C₂²×C₈	D₁₃₇	1	1	yes	characteristic
30	M₃₂	—	137	4	no	—
31	C₂×C₁₆	—	137	2	no	—
32	C₂²×C₄	Dic₁₃₇	1	1	yes	characteristic
33	C₂×C₈	Dic₁₃₇	1	1	yes	characteristic
34	C₂×C₈	Dic₁₃₇	1	2	yes	—
35	C₂×C₈	C₂×D₁₃₇	1	2	yes	—
36	C₂×C₈	C₂×D₁₃₇	1	1	yes	characteristic
37	C₁₆	—	137	4	no	—
38	C₂³	#11764040	1	1	yes	characteristic
39	C₂×C₄	#11764040	1	2	yes	—
40	C₂×C₄	#11764040	1	1	yes	characteristic
41	C₂×C₄	C₂×Dic₁₃₇	1	2	yes	—
42	C₂×C₄	C₂×Dic₁₃₇	1	1	yes	characteristic
43	C₈	C₂×Dic₁₃₇	1	2	yes	—
44	C₈	C₂×Dic₁₃₇	1	1	yes	characteristic
45	C₈	C₂²×D₁₃₇	1	1	yes	characteristic
46	C₂²	—	2	2	no	—
47	C₂²	C₂×#11764040	1	2	yes	—
48	C₂²	C₂×#11764040	1	1	yes	characteristic
49	C₄	C₂×#11764040	1	2	yes	—
50	C₄	C₂×#11764040	1	1	yes	characteristic
51	C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
52	C₂	—	2	2	no	—
53	C₂	#17984028	1	2	yes	direct product factor
54	C₂	C₂²×#11764040	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₁₃₇
commutator	C₂×C₁₃₇	C₂²×C₈
Frattini	C₈	C₂²×D₁₃₇
Fitting	C₂²×C₈×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×C₈
hypercenter	C₂²×C₈	D₁₃₇
hypocenter	C₁₃₇	C₂×M₃₂
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
1. #17984028	C₂	direct (N × H)
2. C₂×#14555485	C₂	semidirect
4. #14555485	C₂²	semidirect
25. C₂×C₁₃₇	M₃₂	semidirect
27. C₁₃₇	C₂×M₃₂	semidirect
53. C₂	#17984028	direct (N × H)

Subgroups — C2×#17984028 #22577818

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#17984028 #22577818