Subgroups — #22577739

Order 8768.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984067	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#17984022	C₂	1	2	yes	maximal, complemented
4	#17984013	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₇₄	C₂²	1	1	yes	characteristic
7	C₂²×Dic₁₃₇	—	2	1	no	—
8	C₄⋊Dic₁₃₇	—	2	1	no	—
9	C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
10	C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
11	#14555489	—	2	1	no	—
12	C₂²⋊Dic₁₃₇	—	2	1	no	—
13	C₂²×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₁₃₇	C₂²	1	2	yes	—
15	Dic₂₇₄	—	2	2	no	—
16	C₂×Dic₁₃₇	—	2	1	no	—
17	C₂×Dic₁₃₇	—	2	1	no	—
18	C₂×Dic₁₃₇	—	2	2	no	—
19	C₂×Dic₁₃₇	—	4	2	no	—
20	C₂³×C₁₃₇	D₄	1	1	yes	characteristic
21	C₂×C₄×C₁₃₇	—	2	2	no	—
22	C₂×C₄×C₁₃₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
24	C₈×C₁₃₇	—	2	2	no	—
25	C₂²×C₁₃₇	—	2	1	no	—
26	C₂²×C₁₃₇	—	2	1	no	—
27	C₂²×C₁₃₇	Q₁₆	1	2	yes	—
28	C₂²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
29	C₄×C₁₃₇	—	2	1	no	—
30	C₄×C₁₃₇	#41	1	2	yes	—
31	Dic₁₃₇	—	4	1	no	—
32	Dic₁₃₇	—	4	1	no	—
33	Dic₁₃₇	—	4	2	no	—
34	C₂×C₁₃₇	—	2	2	no	—
35	C₂×C₁₃₇	#123	1	1	yes	characteristic
36	C₂×C₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₁₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
38	C₁₃₇	#484	1	1	yes	characteristic, complemented, Sylow
39	#484	—	137	1	no	maximal, Sylow
40	C₂²⋊Q₈	—	137	1	no	—
41	C₂×C₄⋊C₄	—	137	1	no	—
42	#107	—	137	2	no	—
43	#103	—	137	2	no	—
44	C₂²⋊C₈	D₁₃₇	1	1	yes	characteristic
45	C₂×Q₈	—	137	1	no	—
46	C₂²×C₄	—	274	1	no	—
47	C₂²×C₄	C₂×D₁₃₇	1	1	yes	characteristic
48	C₂×C₈	C₂×D₁₃₇	1	2	yes	—
49	C₄⋊C₄	—	274	1	no	—
50	C₄⋊C₄	—	274	1	no	—
51	C₄⋊C₄	—	137	2	no	—
52	C₄⋊C₄	—	137	1	no	—
53	C₂²⋊C₄	—	274	1	no	—
54	C₂³	D₅₄₈	1	1	yes	characteristic
55	Q₈	—	274	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	274	1	no	—
58	C₂×C₄	—	274	1	no	—
59	C₂×C₄	—	274	2	no	—
60	C₂×C₄	—	548	2	no	—
61	C₂×C₄	D₅₄₈	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
63	C₈	—	2	2	no	—
64	C₂²	—	2	1	no	—
65	C₂²	—	2	1	no	—
66	C₂²	Dic₅₄₈	1	2	yes	—
67	C₂²	C₂×D₅₄₈	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	548	1	no	—
70	C₄	—	548	2	no	—
71	C₄	—	548	1	no	—
72	C₄	#14555509	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	#17984072	1	1	yes	characteristic
75	C₂	C₂×Dic₅₄₈	1	1	yes	characteristic
76	C₂	#17984116	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₅₄₈
commutator	C₂×C₄×C₁₃₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₇
Fitting	C₂²⋊C₈×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×D₄
hypercenter	C₂²⋊C₈	D₁₃₇
hypocenter	C₁₃₇	#484
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
3. #17984022	C₂	semidirect
4. #17984013	C₂	semidirect
38. C₁₃₇	#484	semidirect