Subgroups — #30636917

Order 16256.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549710	C₂	1	1	yes	maximal, characteristic, complemented
2	#22549704	C₂	1	1	yes	maximal, characteristic, complemented
3	#508×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#16861143	—	2	1	no	—
5	C₁₂₇⋊C₃₂	—	2	1	no	—
6	#135×C₁₂₇	—	2	1	no	—
7	Q₃₂×C₁₂₇	C₄	1	1	yes	characteristic, complemented
8	SD₃₂×C₁₂₇	—	2	1	no	—
9	D₁₆×C₁₂₇	C₄	1	1	yes	characteristic, complemented
10	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₂₇	—	4	1	no	—
12	#41×C₁₂₇	—	4	1	no	—
13	Q₁₆×C₁₂₇	—	2	1	no	—
14	SD₁₆×C₁₂₇	—	4	1	no	—
15	D₈×C₁₂₇	—	2	1	no	—
16	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
17	C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
19	C₂×Dic₁₂₇	—	8	1	no	—
20	Q₈×C₁₂₇	—	4	1	no	—
21	D₄×C₁₂₇	—	4	1	no	—
22	D₄×C₁₂₇	—	8	1	no	—
23	C₂×C₄×C₁₂₇	—	8	1	no	—
24	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
25	C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
27	C₂²×C₁₂₇	—	8	1	no	—
28	C₂²×C₁₂₇	D₁₆	1	1	yes	characteristic
29	C₄×C₁₂₇	—	8	1	no	—
30	C₄×C₁₂₇	#102	1	1	yes	characteristic
31	C₄×C₁₂₇	SD₃₂	1	1	yes	characteristic
32	Dic₁₂₇	—	16	1	no	—
33	C₂×C₁₂₇	—	2	1	no	—
34	C₂×C₁₂₇	—	16	1	no	—
35	C₂×C₁₂₇	#357	1	1	yes	characteristic
36	#1418	—	127	1	no	maximal, Sylow
37	C₁₂₇	#1418	1	1	yes	characteristic, complemented, Sylow
38	#508	D₁₂₇	1	1	yes	characteristic
39	M₆₄	—	127	1	no	—
40	#367	—	127	1	no	—
41	#135	—	2	1	no	—
42	Q₃₂	Dic₁₂₇	1	1	yes	characteristic, complemented
43	SD₃₂	—	2	1	no	—
44	D₁₆	Dic₁₂₇	1	1	yes	characteristic, complemented
45	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
46	#107	—	254	1	no	—
47	C₃₂	—	254	1	no	—
48	#41	—	4	1	no	—
49	Q₁₆	—	2	1	no	—
50	SD₁₆	—	4	1	no	—
51	D₈	—	2	1	no	—
52	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
53	C₄⋊C₄	—	508	1	no	—
54	C₁₆	C₂×Dic₁₂₇	1	1	yes	characteristic
55	C₁₆	C₂²⋊D₁₂₇	1	1	yes	characteristic
56	Q₈	—	4	1	no	—
57	D₄	—	4	1	no	—
58	D₄	—	8	1	no	—
59	C₂×C₄	—	8	1	no	—
60	C₂×C₄	—	1016	1	no	—
61	C₂×C₄	#14523233	1	1	yes	characteristic
62	C₈	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
63	C₈	#14523236	1	1	yes	characteristic
64	C₂²	—	8	1	no	—
65	C₂²	#16861150	1	1	yes	characteristic
66	C₄	—	8	1	no	—
67	C₄	—	2032	1	no	—
68	C₄	#16861130	1	1	yes	characteristic
69	C₄	#16861153	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	16	1	no	—
72	C₂	#22549677	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22549677
commutator	C₁₆×C₁₂₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₂₇
Fitting, prime radical	#508×C₁₂₇	C₂
socle	C₂×C₁₂₇	#357
cosocle	C₂×C₁₆×C₁₂₇	C₂²
hypercenter	#508	D₁₂₇
hypocenter	C₁₂₇	#1418
LCS	G ▷ C₁₆×C₁₂₇ ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #508

Complements

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

N	H ≅ G/N	type
1. #22549710	C₂	semidirect
2. #22549704	C₂	semidirect
7. Q₃₂×C₁₂₇	C₄	semidirect
9. D₁₆×C₁₂₇	C₄	semidirect
37. C₁₂₇	#1418	semidirect
42. Q₃₂	Dic₁₂₇	semidirect
44. D₁₆	Dic₁₂₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22549710	2
#22549704	2
#508×C₁₂₇	2
#1418	127