Subgroups — #22578481

Order 8768.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984175	C₂	1	8	yes	maximal, complemented
2	D₁₃₇⋊M₁₆	C₂	1	2	yes	maximal, complemented
3	C₂×#14555519	C₂	1	4	yes	maximal, complemented
4	C₂×#14555509	C₂	1	1	yes	maximal, characteristic
5	#14555520	C₂²	1	4	yes	complemented
6	#14555519	C₂²	1	4	yes	—
7	#14555519	C₂²	1	8	yes	complemented
8	#14555518	C₂²	1	4	yes	complemented
9	C₂×#11764047	C₂²	1	8	yes	complemented
10	#14555509	C₄	1	4	yes	—
11	#14555509	C₂²	1	4	yes	—
12	C₂×Dic₂₇₄	C₄	1	1	yes	characteristic
13	C₂×C₂²⋊D₁₃₇	C₄	1	2	yes	—
14	C₂²×Dic₁₃₇	C₂²	1	2	yes	—
15	C₂×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×D₄×C₁₃₇	C₄	1	1	yes	characteristic
17	#11764047	C₂³	1	8	yes	—
18	C₂²×D₁₃₇	C₂×C₄	1	1	yes	characteristic
19	Dic₂₇₄	C₂×C₄	1	4	yes	—
20	C₂²⋊D₁₃₇	C₂×C₄	1	8	yes	—
21	C₂×Dic₁₃₇	C₂×C₄	1	2	yes	—
22	C₂×Dic₁₃₇	C₂×C₄	1	4	yes	—
23	C₂×Dic₁₃₇	C₂³	1	1	yes	characteristic
24	C₂×Dic₁₃₇	C₂³	1	4	yes	—
25	C₄×D₁₃₇	C₂×C₄	1	2	yes	—
26	C₄×D₁₃₇	C₂³	1	2	yes	—
27	C₂³×C₁₃₇	C₂×C₄	1	2	yes	—
28	D₄×C₁₃₇	C₂×C₄	1	4	yes	—
29	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₁₃₇	—	4	2	no	—
31	C₂²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₁₃₇	C₂²×C₄	1	4	yes	—
33	C₂×D₁₃₇	—	4	1	no	—
34	C₂×D₁₃₇	C₂²×C₄	1	2	yes	—
35	C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
36	Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
37	Dic₁₃₇	C₂²×C₄	1	4	yes	—
38	Dic₁₃₇	C₂⁴	1	1	yes	characteristic
39	C₂×C₁₃₇	—	2	1	no	—
40	C₂×C₁₃₇	—	2	4	no	—
41	C₂×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
42	D₁₃₇	—	2	2	no	—
43	C₁₃₇	#568	1	1	yes	characteristic, complemented, Sylow
44	#568	—	137	1	no	maximal, Sylow
45	C₂×#41	—	137	1	no	—
46	#131	—	137	8	no	—
47	C₂×M₁₆	—	137	4	no	—
48	C₂×M₁₆	—	137	2	no	—
49	#41	—	137	4	no	—
50	#41	—	137	4	no	—
51	C₂×Q₈	—	137	1	no	—
52	C₂×D₄	—	137	2	no	—
53	C₂×D₄	#10587739	1	1	yes	characteristic
54	C₂²×C₄	—	137	2	no	—
55	C₂²×C₄	—	137	1	no	—
56	M₁₆	—	137	4	no	—
57	M₁₆	—	137	8	no	—
58	M₁₆	—	137	4	no	—
59	C₂×C₈	—	137	8	no	—
60	C₂×C₈	—	137	4	no	—
61	C₂³	—	137	1	no	—
62	C₂³	C₂×#10587739	1	2	yes	—
63	Q₈	—	137	4	no	—
64	D₄	—	137	8	no	—
65	D₄	C₂×#10587739	1	4	yes	—
66	C₂×C₄	—	137	4	no	—
67	C₂×C₄	—	137	1	no	—
68	C₂×C₄	—	137	2	no	—
69	C₂×C₄	—	137	4	no	—
70	C₂×C₄	—	137	2	no	—
71	C₂×C₄	—	137	2	no	—
72	C₂×C₄	C₂×#10587739	1	1	yes	characteristic
73	C₈	—	137	8	no	—
74	C₂²	—	4	2	no	—
75	C₂²	—	548	1	no	—
76	C₂²	—	137	2	no	—
77	C₂²	C₂²×#10587739	1	1	yes	characteristic
78	C₂²	C₂²×#10587739	1	4	yes	—
79	C₄	—	137	1	no	—
80	C₄	—	137	1	no	—
81	C₄	—	137	4	no	—
82	C₄	C₂²×#10587739	1	2	yes	—
83	C₂	—	2	1	no	—
84	C₂	—	2	4	no	—
85	C₂	—	274	2	no	—
86	C₂	C₂³×#10587739	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×#10587739
commutator, socle	C₂×C₁₃₇	C₂³×C₄
Fitting	C₂×D₄×C₁₃₇	C₄
hypercenter	C₂×D₄	#10587739
hypocenter	C₁₃₇	#568
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. #17984175	C₂	semidirect
2. D₁₃₇⋊M₁₆	C₂	semidirect
3. C₂×#14555519	C₂	semidirect
5. #14555520	C₂²	semidirect
7. #14555519	C₂²	semidirect
8. #14555518	C₂²	semidirect
9. C₂×#11764047	C₂²	semidirect
43. C₁₃₇	#568	semidirect