Subgroups — #30641024

Order 16256.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549710	C₂	1	3	yes	maximal, complemented
2	C₂×C₁₂₇⋊C₃₂	C₂	1	3	yes	maximal, complemented
3	#504×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇⋊C₃₂	C₂²	1	1	yes	characteristic
5	C₁₂₇⋊C₃₂	C₂²	1	3	yes	—
6	#131×C₁₂₇	C₄	1	1	yes	characteristic
7	M₃₂×C₁₂₇	C₄	1	3	yes	—
8	C₂×C₁₆×C₁₂₇	C₂²	1	3	yes	—
9	#41×C₁₂₇	C₈	1	1	yes	characteristic
10	M₁₆×C₁₂₇	C₈	1	3	yes	—
11	C₂×C₈×C₁₂₇	C₂×C₄	1	3	yes	—
12	C₁₆×C₁₂₇	C₂×C₄	1	3	yes	—
13	C₁₆×C₁₂₇	C₂³	1	1	yes	characteristic
14	Q₈×C₁₂₇	C₁₆	1	1	yes	characteristic
15	D₄×C₁₂₇	C₁₆	1	3	yes	—
16	C₂×C₄×C₁₂₇	C₂×C₈	1	3	yes	—
17	C₈×C₁₂₇	C₂×C₈	1	3	yes	—
18	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
19	C₂²×C₁₂₇	C₂×C₁₆	1	3	yes	—
20	C₄×C₁₂₇	C₂×C₁₆	1	3	yes	—
21	C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
22	C₂×C₁₂₇	—	2	3	no	—
23	C₂×C₁₂₇	C₂²×C₁₆	1	1	yes	characteristic
24	#2258	—	127	1	no	maximal, Sylow
25	C₁₂₇	#2258	1	1	yes	characteristic, complemented, Sylow
26	#504	D₁₂₇	1	1	yes	characteristic
27	M₆₄	—	127	3	no	—
28	C₂×C₃₂	—	127	3	no	—
29	#131	Dic₁₂₇	1	1	yes	characteristic
30	M₃₂	Dic₁₂₇	1	3	yes	—
31	C₂×C₁₆	C₂×D₁₂₇	1	3	yes	—
32	C₃₂	—	127	1	no	—
33	C₃₂	—	127	3	no	—
34	#41	C₁₂₇⋊C₈	1	1	yes	characteristic
35	M₁₆	C₁₂₇⋊C₈	1	3	yes	—
36	C₂×C₈	C₂×Dic₁₂₇	1	3	yes	—
37	C₁₆	C₂×Dic₁₂₇	1	3	yes	—
38	C₁₆	C₂²×D₁₂₇	1	1	yes	characteristic
39	Q₈	C₁₂₇⋊C₁₆	1	1	yes	characteristic
40	D₄	C₁₂₇⋊C₁₆	1	3	yes	—
41	C₂×C₄	C₂×C₁₂₇⋊C₈	1	3	yes	—
42	C₈	C₂×C₁₂₇⋊C₈	1	3	yes	—
43	C₈	C₂²×Dic₁₂₇	1	1	yes	characteristic
44	C₂²	C₂×C₁₂₇⋊C₁₆	1	3	yes	—
45	C₄	C₂×C₁₂₇⋊C₁₆	1	3	yes	—
46	C₄	C₂²×C₁₂₇⋊C₈	1	1	yes	characteristic
47	C₂	—	2	3	no	—
48	C₂	C₂²×C₁₂₇⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₁₆	C₂²×D₁₂₇
commutator, socle	C₂×C₁₂₇	C₂²×C₁₆
Fitting	#504×C₁₂₇	C₂
prime radical	#41×C₁₂₇	C₈
cosocle	C₁₆×C₁₂₇	C₂³
hypercenter	#504	D₁₂₇
hypocenter	C₁₂₇	#2258
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₁₆ ◁ #504

Complements

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

N	H ≅ G/N	type
1. #22549710	C₂	semidirect
2. C₂×C₁₂₇⋊C₃₂	C₂	semidirect
25. C₁₂₇	#2258	semidirect

Maximal subgroups

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

subgroup	index
#22549710	2
C₂×C₁₂₇⋊C₃₂	2
#504×C₁₂₇	2
#2258	127