Subgroups — #30636804

Order 16256.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549629	C₂	1	2	yes	maximal
2	C₄×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₁₂₇⋊C₈)	—	2	2	no	—
4	C₂×C₁₆×C₁₂₇	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₁₂₇	C₄	1	1	yes	characteristic
6	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂×C₁₂₇⋊C₈	—	4	2	no	—
8	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
9	C₂×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
10	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
11	C₁₆×C₁₂₇	—	2	1	no	—
12	C₁₆×C₁₂₇	C₈	1	2	yes	complemented
13	C₁₂₇⋊C₈	—	8	2	no	—
14	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
16	C₂×C₄×C₁₂₇	Q₁₆	1	1	yes	characteristic
17	C₈×C₁₂₇	—	2	1	no	—
18	C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
19	C₈×C₁₂₇	M₁₆	1	1	yes	characteristic
20	C₂²×C₁₂₇	#107	1	1	yes	characteristic
21	C₄×C₁₂₇	—	2	1	no	—
22	C₄×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
23	C₄×C₁₂₇	#108	1	1	yes	characteristic
24	C₄×C₁₂₇	SD₃₂	1	2	yes	—
25	C₂×C₁₂₇	#335	1	1	yes	characteristic
26	C₂×C₁₂₇	#367	1	1	yes	characteristic
27	C₂×C₁₂₇	#368	1	1	yes	characteristic
28	#1372	—	127	1	no	maximal, Sylow
29	C₁₂₇	#1372	1	1	yes	characteristic, complemented, Sylow
30	C₄×C₁₆	D₁₂₇	1	1	yes	characteristic
31	#335	—	127	2	no	—
32	C₂×C₁₆	Dic₁₂₇	1	1	yes	characteristic
33	C₂×C₁₆	Dic₁₂₇	1	1	yes	characteristic
34	C₄⋊C₈	—	254	2	no	—
35	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
36	C₂×C₈	—	508	2	no	—
37	C₂×C₈	C₂×Dic₁₂₇	1	1	yes	characteristic
38	C₂×C₈	Dic₂₅₄	1	1	yes	characteristic
39	C₄²	D₅₀₈	1	1	yes	characteristic
40	C₁₆	—	2	1	no	—
41	C₁₆	C₁₂₇⋊C₈	1	2	yes	complemented
42	C₂×C₄	C₄⋊Dic₁₂₇	1	1	yes	characteristic
43	C₂×C₄	D₁₀₁₆	1	1	yes	characteristic
44	C₂×C₄	Dic₅₀₈	1	1	yes	characteristic
45	C₈	—	2	1	no	—
46	C₈	—	1016	2	no	—
47	C₈	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
48	C₈	#14523232	1	1	yes	characteristic
49	C₂²	#16861143	1	1	yes	characteristic
50	C₄	—	2	1	no	—
51	C₄	C₄⋊(C₁₂₇⋊C₈)	1	1	yes	characteristic
52	C₄	#16861145	1	1	yes	characteristic
53	C₄	#16861154	1	2	yes	—
54	C₂	#22549629	1	1	yes	characteristic
55	C₂	#22549704	1	1	yes	characteristic
56	C₂	#22549706	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₀₁₆
commutator	C₈×C₁₂₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₂₇
Fitting	C₄×C₁₆×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	#107
cosocle	C₄×C₈×C₁₂₇	C₂²
hypercenter	C₄×C₁₆	D₁₂₇
hypocenter	C₁₂₇	#1372
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
12. C₁₆×C₁₂₇	C₈	semidirect
29. C₁₂₇	#1372	semidirect
41. C₁₆	C₁₂₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22549629	2
C₄×C₁₆×C₁₂₇	2
#1372	127