Subgroups — #30636932

Order 16256.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549702	C₂	1	2	yes	maximal
2	C₂×C₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#16861143	—	2	2	no	—
4	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
5	C₃₂×C₁₂₇	C₄	1	2	yes	complemented
6	C₄⋊Dic₁₂₇	—	4	2	no	—
7	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
8	C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
9	C₁₆×C₁₂₇	Q₈	1	1	yes	characteristic
10	C₂×Dic₁₂₇	—	8	2	no	—
11	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
12	C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
13	C₈×C₁₂₇	Q₁₆	1	1	yes	characteristic
14	C₂²×C₁₂₇	D₁₆	1	1	yes	characteristic
15	C₄×C₁₂₇	#107	1	1	yes	characteristic
16	C₄×C₁₂₇	Q₃₂	1	1	yes	characteristic
17	Dic₁₂₇	—	16	2	no	—
18	C₂×C₁₂₇	#366	1	1	yes	characteristic
19	C₂×C₁₂₇	SD₆₄	1	2	yes	—
20	#1424	—	127	1	no	maximal, Sylow
21	C₁₂₇	#1424	1	1	yes	characteristic, complemented, Sylow
22	C₂×C₃₂	D₁₂₇	1	1	yes	characteristic
23	#366	—	127	2	no	—
24	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
25	#107	—	254	2	no	—
26	C₃₂	Dic₁₂₇	1	2	yes	complemented
27	C₂×C₈	D₅₀₈	1	1	yes	characteristic
28	C₄⋊C₄	—	508	2	no	—
29	C₁₆	C₂×Dic₁₂₇	1	1	yes	characteristic
30	C₁₆	Dic₂₅₄	1	1	yes	characteristic
31	C₂×C₄	—	1016	2	no	—
32	C₂×C₄	D₁₀₁₆	1	1	yes	characteristic
33	C₈	C₄⋊Dic₁₂₇	1	1	yes	characteristic
34	C₈	Dic₅₀₈	1	1	yes	characteristic
35	C₂²	D₂₀₃₂	1	1	yes	characteristic
36	C₄	—	2032	2	no	—
37	C₄	#16861143	1	1	yes	characteristic
38	C₄	Dic₁₀₁₆	1	1	yes	characteristic
39	C₂	#22549702	1	1	yes	characteristic
40	C₂	#22549715	1	2	yes	—

	subgroup	G/H
center	C₂²	D₂₀₃₂
commutator	C₁₆×C₁₂₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₂₇
Fitting	C₂×C₃₂×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	D₁₆
cosocle	C₂×C₁₆×C₁₂₇	C₂²
hypercenter	C₂×C₃₂	D₁₂₇
hypocenter	C₁₂₇	#1424
LCS	G ▷ C₁₆×C₁₂₇ ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
5. C₃₂×C₁₂₇	C₄	semidirect
21. C₁₂₇	#1424	semidirect
26. C₃₂	Dic₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22549702	2
C₂×C₃₂×C₁₂₇	2
#1424	127