Subgroups — #22578522

Order 8768.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984185	C₂	1	2	yes	maximal, complemented
2	#17984184	C₂	1	2	yes	maximal, complemented
3	#17984183	C₂	1	2	yes	maximal, complemented
4	C₂×#14555518	C₂	1	1	yes	maximal, characteristic
5	#14555522	C₂²	1	4	yes	complemented
6	#14555518	C₄	1	1	yes	characteristic
7	#14555518	C₄	1	1	yes	characteristic
8	#14555518	C₂²	1	1	yes	characteristic
9	#14555518	C₂²	1	1	yes	characteristic
10	C₂×#11764047	C₄	1	1	yes	characteristic
11	C₂×#11764047	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₁₃₇	C₄	1	1	yes	characteristic
13	#11764047	C₂×C₄	1	1	yes	characteristic
14	#11764047	C₂×C₄	1	1	yes	characteristic
15	#11764047	C₂×C₄	1	1	yes	characteristic
16	#11764047	C₂³	1	1	yes	characteristic
17	C₂²×D₁₃₇	C₈	1	1	yes	characteristic
18	C₂×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₄×D₁₃₇	C₈	1	1	yes	characteristic
20	C₄×D₁₃₇	C₈	1	1	yes	characteristic
21	C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
22	C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₁₃₇	C₈	1	1	yes	characteristic
24	C₂²×C₁₃₇	C₂×C₈	1	1	yes	characteristic
25	C₂×D₁₃₇	—	2	2	no	—
26	C₂×D₁₃₇	C₂×C₈	1	1	yes	characteristic
27	C₂×D₁₃₇	C₂×C₈	1	1	yes	characteristic
28	C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
29	C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
30	Dic₁₃₇	C₂×C₈	1	1	yes	characteristic
31	Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₁₃₇	—	2	1	no	—
33	C₂×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
34	D₁₃₇	—	2	1	no	—
35	D₁₃₇	M₃₂	1	2	yes	complemented
36	C₁₃₇	C₂×M₃₂	1	1	yes	characteristic, complemented, Sylow
37	C₂×M₃₂	—	137	1	no	maximal, Sylow
38	C₂²×C₈	—	137	1	no	—
39	M₃₂	—	137	2	no	—
40	M₃₂	—	137	2	no	—
41	C₂×C₁₆	—	137	2	no	—
42	C₂²×C₄	—	137	1	no	—
43	C₂×C₈	—	137	1	no	—
44	C₂×C₈	—	137	1	no	—
45	C₂×C₈	—	137	1	no	—
46	C₂×C₈	—	137	1	no	—
47	C₂×C₈	—	137	1	no	—
48	C₂×C₈	—	137	1	no	—
49	C₁₆	—	137	4	no	—
50	C₂³	—	137	1	no	—
51	C₂×C₄	—	137	1	no	—
52	C₂×C₄	—	137	1	no	—
53	C₂×C₄	—	137	1	no	—
54	C₂×C₄	—	137	1	no	—
55	C₂×C₄	—	137	1	no	—
56	C₂×C₄	#11764049	1	1	yes	characteristic
57	C₈	—	137	1	no	—
58	C₈	—	137	1	no	—
59	C₈	—	137	1	no	—
60	C₈	—	137	1	no	—
61	C₂²	—	274	2	no	—
62	C₂²	—	137	1	no	—
63	C₂²	—	137	1	no	—
64	C₂²	C₂×#11764049	1	1	yes	characteristic
65	C₄	—	137	1	no	—
66	C₄	—	137	1	no	—
67	C₄	C₂×#11764049	1	1	yes	characteristic
68	C₄	C₂×#11764049	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	—	274	1	no	—
71	C₂	—	137	2	no	—
72	C₂	C₂²×#11764049	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#11764049
commutator, socle	C₂×C₁₃₇	C₂²×C₈
Frattini	C₂	C₂²×#11764049
Fitting	C₂×C₄×C₁₃₇	C₈
hypercenter	C₂×C₄	#11764049
hypocenter	C₁₃₇	C₂×M₃₂
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
1. #17984185	C₂	semidirect
2. #17984184	C₂	semidirect
3. #17984183	C₂	semidirect
5. #14555522	C₂²	semidirect
35. D₁₃₇	M₃₂	semidirect
36. C₁₃₇	C₂×M₃₂	semidirect