Subgroups — #22578483

Order 8768.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984176	C₂	1	8	yes	maximal, complemented
2	D₁₃₇⋊M₁₆	C₂	1	6	yes	maximal, complemented
3	C₂×#14555511	C₂	1	1	yes	maximal, characteristic
4	#14555520	C₂²	1	12	yes	complemented
5	#14555519	C₂²	1	4	yes	—
6	#14555518	C₂²	1	12	yes	complemented
7	#14555511	C₄	1	4	yes	—
8	#14555511	C₂²	1	4	yes	—
9	C₂×D₅₄₈	C₄	1	3	yes	—
10	C₂×C₄×D₁₃₇	C₂²	1	3	yes	—
11	C₂×Q₈×C₁₃₇	C₄	1	1	yes	characteristic
12	#11764047	C₂³	1	8	yes	—
13	C₂²×D₁₃₇	C₂×C₄	1	3	yes	—
14	D₅₄₈	C₂×C₄	1	12	yes	—
15	C₂×Dic₁₃₇	C₂³	1	1	yes	characteristic
16	C₄×D₁₃₇	C₂×C₄	1	6	yes	—
17	C₄×D₁₃₇	C₂³	1	6	yes	—
18	Q₈×C₁₃₇	C₂×C₄	1	4	yes	—
19	C₂×C₄×C₁₃₇	C₂×C₄	1	3	yes	—
20	C₂²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
21	C₂×D₁₃₇	—	4	3	no	—
22	C₂×D₁₃₇	C₂²×C₄	1	6	yes	—
23	C₄×C₁₃₇	C₂²×C₄	1	6	yes	—
24	Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
25	Dic₁₃₇	C₂⁴	1	1	yes	characteristic
26	C₂×C₁₃₇	—	2	1	no	—
27	C₂×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
28	D₁₃₇	—	2	6	no	—
29	C₁₃₇	#568	1	1	yes	characteristic, complemented, Sylow
30	#568	—	137	1	no	maximal, Sylow
31	C₂×#41	—	137	1	no	—
32	#131	—	137	8	no	—
33	C₂×M₁₆	—	137	6	no	—
34	#41	—	137	4	no	—
35	#41	—	137	4	no	—
36	C₂×Q₈	#10587739	1	1	yes	characteristic
37	C₂×D₄	—	137	3	no	—
38	C₂²×C₄	—	137	3	no	—
39	M₁₆	—	137	4	no	—
40	M₁₆	—	137	12	no	—
41	C₂×C₈	—	137	12	no	—
42	C₂³	—	137	3	no	—
43	Q₈	C₂×#10587739	1	4	yes	—
44	D₄	—	137	12	no	—
45	C₂×C₄	—	137	1	no	—
46	C₂×C₄	—	137	6	no	—
47	C₂×C₄	—	137	6	no	—
48	C₂×C₄	C₂×#10587739	1	3	yes	—
49	C₈	—	137	8	no	—
50	C₂²	—	548	3	no	—
51	C₂²	—	137	6	no	—
52	C₂²	C₂²×#10587739	1	1	yes	characteristic
53	C₄	—	137	1	no	—
54	C₄	—	137	1	no	—
55	C₄	C₂²×#10587739	1	6	yes	—
56	C₂	—	2	1	no	—
57	C₂	—	274	6	no	—
58	C₂	C₂³×#10587739	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×#10587739
commutator, socle	C₂×C₁₃₇	C₂³×C₄
Fitting	C₂×Q₈×C₁₃₇	C₄
hypercenter	C₂×Q₈	#10587739
hypocenter	C₁₃₇	#568
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. #17984176	C₂	semidirect
2. D₁₃₇⋊M₁₆	C₂	semidirect
4. #14555520	C₂²	semidirect
6. #14555518	C₂²	semidirect
29. C₁₃₇	#568	semidirect