Subgroups — #30636908

Order 16256.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₂₇⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#22549702	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#16861143	—	2	1	no	—
5	C₁₂₇⋊C₃₂	—	2	1	no	—
6	C₂×D₈×C₁₂₇	—	2	1	no	—
7	D₁₆×C₁₂₇	—	2	1	no	—
8	D₁₆×C₁₂₇	C₄	1	2	yes	complemented
9	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₁₂₇	—	4	1	no	—
11	C₂×D₄×C₁₂₇	—	4	1	no	—
12	D₈×C₁₂₇	—	2	2	no	—
13	D₈×C₁₂₇	—	4	1	no	—
14	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
15	C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
17	C₂×Dic₁₂₇	—	8	1	no	—
18	C₂³×C₁₂₇	—	8	1	no	—
19	D₄×C₁₂₇	—	4	2	no	—
20	D₄×C₁₂₇	—	8	1	no	—
21	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
22	C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
24	C₂²×C₁₂₇	—	8	2	no	—
25	C₂²×C₁₂₇	—	16	1	no	—
26	C₂²×C₁₂₇	—	16	1	no	—
27	C₂²×C₁₂₇	D₁₆	1	1	yes	characteristic
28	C₄×C₁₂₇	#102	1	1	yes	characteristic
29	C₄×C₁₂₇	SD₃₂	1	1	yes	characteristic
30	Dic₁₂₇	—	16	1	no	—
31	C₂×C₁₂₇	—	16	2	no	—
32	C₂×C₁₂₇	#357	1	1	yes	characteristic
33	C₂×C₁₂₇	D₃₂	1	1	yes	characteristic
34	C₂×C₁₂₇	SD₆₄	1	1	yes	characteristic
35	#1415	—	127	1	no	maximal, Sylow
36	C₁₂₇	#1415	1	1	yes	characteristic, complemented, Sylow
37	C₂×D₁₆	D₁₂₇	1	1	yes	characteristic
38	C₂×C₃₂	—	127	1	no	—
39	#366	—	127	1	no	—
40	C₂×D₈	—	2	1	no	—
41	D₁₆	—	2	1	no	—
42	D₁₆	Dic₁₂₇	1	2	yes	complemented
43	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
44	#107	—	254	1	no	—
45	C₃₂	—	254	1	no	—
46	C₂×D₄	—	4	1	no	—
47	D₈	—	2	2	no	—
48	D₈	—	4	1	no	—
49	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
50	C₄⋊C₄	—	508	1	no	—
51	C₁₆	C₂×Dic₁₂₇	1	1	yes	characteristic
52	C₁₆	C₂²⋊D₁₂₇	1	1	yes	characteristic
53	C₂³	—	8	1	no	—
54	D₄	—	4	2	no	—
55	D₄	—	8	1	no	—
56	C₂×C₄	—	1016	1	no	—
57	C₂×C₄	#14523233	1	1	yes	characteristic
58	C₈	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
59	C₈	#14523236	1	1	yes	characteristic
60	C₂²	—	8	2	no	—
61	C₂²	—	16	1	no	—
62	C₂²	—	16	1	no	—
63	C₂²	#16861150	1	1	yes	characteristic
64	C₄	—	2032	1	no	—
65	C₄	#16861130	1	1	yes	characteristic
66	C₄	#16861153	1	1	yes	characteristic
67	C₂	—	16	2	no	—
68	C₂	#22549677	1	1	yes	characteristic
69	C₂	#22549711	1	1	yes	characteristic
70	C₂	#22549714	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16861150
commutator	C₁₆×C₁₂₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₂₇
Fitting, prime radical	C₂×D₁₆×C₁₂₇	C₂
socle	C₂²×C₁₂₇	D₁₆
cosocle	C₂×C₁₆×C₁₂₇	C₂²
hypercenter	C₂×D₁₆	D₁₂₇
hypocenter	C₁₂₇	#1415
LCS	G ▷ C₁₆×C₁₂₇ ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×D₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₂₇⋊C₃₂	C₂	semidirect
2. #22549702	C₂	semidirect
8. D₁₆×C₁₂₇	C₄	semidirect
36. C₁₂₇	#1415	semidirect
42. D₁₆	Dic₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₂₇⋊C₃₂	2
#22549702	2
C₂×D₁₆×C₁₂₇	2
#1415	127