Subgroups — #30636767

Order 16256.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549697	C₂	1	2	yes	maximal, complemented
2	#493×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#16861149	—	2	2	no	—
4	C₂×D₈×C₁₂₇	—	2	2	no	—
5	#127×C₁₂₇	C₄	1	2	yes	—
6	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₁₂₇⋊C₁₆	—	4	2	no	—
8	C₂×D₄×C₁₂₇	—	2	2	no	—
9	C₂×D₄×C₁₂₇	—	4	2	no	—
10	D₈×C₁₂₇	—	4	2	no	—
11	D₈×C₁₂₇	—	4	2	no	—
12	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
13	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
14	C₂³×C₁₂₇	—	4	2	no	—
15	D₄×C₁₂₇	—	4	4	no	—
16	D₄×C₁₂₇	—	8	4	no	—
17	C₂×C₄×C₁₂₇	—	2	1	no	—
18	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
19	C₈×C₁₂₇	D₈	1	2	yes	—
20	C₈×C₁₂₇	SD₁₆	1	2	yes	—
21	C₂²×C₁₂₇	—	8	4	no	—
22	C₂²×C₁₂₇	—	8	2	no	—
23	C₂²×C₁₂₇	#100	1	1	yes	characteristic
24	C₄×C₁₂₇	—	2	2	no	—
25	C₄×C₁₂₇	#102	1	2	yes	—
26	C₂×C₁₂₇	—	2	1	no	—
27	C₂×C₁₂₇	—	16	2	no	—
28	C₂×C₁₂₇	#331	1	1	yes	characteristic
29	#1357	—	127	1	no	maximal, Sylow
30	C₁₂₇	#1357	1	1	yes	characteristic, complemented, Sylow
31	#493	D₁₂₇	1	1	yes	characteristic
32	#364	—	127	2	no	—
33	C₂×D₈	—	2	2	no	—
34	#127	Dic₁₂₇	1	2	yes	—
35	M₃₂	—	254	2	no	—
36	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
37	C₂×D₄	—	2	2	no	—
38	C₂×D₄	—	4	2	no	—
39	D₈	—	4	2	no	—
40	D₈	—	4	2	no	—
41	C₂×C₈	C₂²⋊D₁₂₇	1	2	yes	—
42	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
43	C₁₆	—	508	2	no	—
44	C₂³	—	4	2	no	—
45	D₄	—	4	4	no	—
46	D₄	—	8	4	no	—
47	C₂×C₄	—	2	1	no	—
48	C₂×C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
49	C₈	#14523233	1	2	yes	—
50	C₈	#14523236	1	2	yes	—
51	C₂²	—	8	4	no	—
52	C₂²	—	8	2	no	—
53	C₂²	#16861126	1	1	yes	characteristic
54	C₄	—	2	2	no	—
55	C₄	#16861130	1	2	yes	—
56	C₂	—	2	1	no	—
57	C₂	—	16	2	no	—
58	C₂	#22549620	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22549620
commutator	C₄²×C₁₂₇	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₂₇
Fitting, prime radical	#493×C₁₂₇	C₂
socle	C₂×C₁₂₇	#331
cosocle	C₄×C₈×C₁₂₇	C₂²
hypercenter	#493	D₁₂₇
hypocenter	C₁₂₇	#1357
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #493

Complements

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

N	H ≅ G/N	type
1. #22549697	C₂	semidirect
30. C₁₂₇	#1357	semidirect

Maximal subgroups

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

subgroup	index
#22549697	2
#493×C₁₂₇	2
#1357	127