Subgroups — #30636931

Order 16256.

56 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
2	#22549702	C₂	1	1	yes	maximal, characteristic
3	#366×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#16861143	—	2	1	no	—
5	C₁₂₇⋊C₃₂	C₄	1	2	yes	complemented
6	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
7	#107×C₁₂₇	—	2	1	no	—
8	C₄⋊Dic₁₂₇	—	4	1	no	—
9	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
10	C₄⋊C₄×C₁₂₇	—	4	1	no	—
11	C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₁₂₇	Q₈	1	1	yes	characteristic
13	C₂×Dic₁₂₇	—	8	1	no	—
14	C₂×C₄×C₁₂₇	—	8	1	no	—
15	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
16	C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₁₂₇	Q₁₆	1	1	yes	characteristic
18	C₂²×C₁₂₇	D₁₆	1	1	yes	characteristic
19	C₄×C₁₂₇	—	16	1	no	—
20	C₄×C₁₂₇	#107	1	1	yes	characteristic
21	C₄×C₁₂₇	Q₃₂	1	1	yes	characteristic
22	Dic₁₂₇	—	16	1	no	—
23	C₂×C₁₂₇	#366	1	1	yes	characteristic
24	C₂×C₁₂₇	SD₆₄	1	1	yes	characteristic
25	C₂×C₁₂₇	SD₆₄	1	1	yes	characteristic
26	#1424	—	127	1	no	maximal, Sylow
27	C₁₂₇	#1424	1	1	yes	characteristic, complemented, Sylow
28	C₂×C₃₂	—	127	1	no	—
29	#366	—	127	1	no	—
30	#366	D₁₂₇	1	1	yes	characteristic
31	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
32	#107	—	2	1	no	—
33	#107	—	254	1	no	—
34	C₃₂	—	127	2	no	—
35	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
36	C₄⋊C₄	—	4	1	no	—
37	C₄⋊C₄	—	508	1	no	—
38	C₁₆	C₄×D₁₂₇	1	1	yes	characteristic
39	C₁₆	Dic₂₅₄	1	1	yes	characteristic
40	C₂×C₄	—	8	1	no	—
41	C₂×C₄	—	1016	1	no	—
42	C₂×C₄	#14523233	1	1	yes	characteristic
43	C₈	Dic₁₂₇⋊C₄	1	1	yes	characteristic
44	C₈	#14523238	1	1	yes	characteristic
45	C₂²	#16861150	1	1	yes	characteristic
46	C₄	—	16	1	no	—
47	C₄	—	2032	1	no	—
48	C₄	#16861142	1	1	yes	characteristic
49	C₄	#16861155	1	1	yes	characteristic
50	C₂	#22549701	1	1	yes	characteristic
51	C₂	#22549713	1	1	yes	characteristic
52	C₂	#22549714	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16861150
commutator	C₁₆×C₁₂₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₂₇
Fitting	#366×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	D₁₆
cosocle	C₂×C₁₆×C₁₂₇	C₂²
hypercenter	#366	D₁₂₇
hypocenter	C₁₂₇	#1424
LCS	G ▷ C₁₆×C₁₂₇ ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #366

Complements

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

N	H ≅ G/N	type
5. C₁₂₇⋊C₃₂	C₄	semidirect
27. C₁₂₇	#1424	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
#366×C₁₂₇	2
#1424	127