Subgroups — #30636910

Order 16256.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₀₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₂₇⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	#366×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₅₀₈	—	2	1	no	—
5	Dic₁₀₁₆	—	2	1	no	—
6	Dic₁₀₁₆	C₄	1	2	yes	complemented
7	C₁₂₇⋊C₃₂	—	2	1	no	—
8	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
9	#107×C₁₂₇	—	2	1	no	—
10	C₂×Dic₂₅₄	—	4	1	no	—
11	Dic₅₀₈	—	2	2	no	—
12	Dic₅₀₈	—	4	1	no	—
13	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
14	C₄⋊C₄×C₁₂₇	—	4	1	no	—
15	C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
17	Dic₂₅₄	—	4	2	no	—
18	Dic₂₅₄	—	8	1	no	—
19	C₂×Dic₁₂₇	—	8	1	no	—
20	C₂×C₄×C₁₂₇	—	8	1	no	—
21	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
22	C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
24	C₂²×C₁₂₇	D₁₆	1	1	yes	characteristic
25	C₄×C₁₂₇	—	16	1	no	—
26	C₄×C₁₂₇	#102	1	1	yes	characteristic
27	C₄×C₁₂₇	SD₃₂	1	1	yes	characteristic
28	Dic₁₂₇	—	8	2	no	—
29	C₂×C₁₂₇	#357	1	1	yes	characteristic
30	C₂×C₁₂₇	SD₆₄	1	1	yes	characteristic
31	C₂×C₁₂₇	Q₆₄	1	1	yes	characteristic
32	#1416	—	127	1	no	maximal, Sylow
33	C₁₂₇	#1416	1	1	yes	characteristic, complemented, Sylow
34	C₂×Q₃₂	—	127	1	no	—
35	C₂×C₃₂	—	127	1	no	—
36	#366	D₁₂₇	1	1	yes	characteristic
37	C₂×Q₁₆	—	254	1	no	—
38	Q₃₂	—	127	2	no	—
39	Q₃₂	—	254	1	no	—
40	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
41	#107	—	2	1	no	—
42	C₃₂	—	254	1	no	—
43	C₂×Q₈	—	508	1	no	—
44	Q₁₆	—	508	1	no	—
45	Q₁₆	—	254	2	no	—
46	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
47	C₄⋊C₄	—	4	1	no	—
48	C₁₆	C₄×D₁₂₇	1	1	yes	characteristic
49	C₁₆	D₅₀₈	1	1	yes	characteristic
50	Q₈	—	1016	1	no	—
51	Q₈	—	508	2	no	—
52	C₂×C₄	—	8	1	no	—
53	C₂×C₄	—	1016	1	no	—
54	C₂×C₄	#14523233	1	1	yes	characteristic
55	C₈	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
56	C₈	#14523235	1	1	yes	characteristic
57	C₂²	#16861150	1	1	yes	characteristic
58	C₄	—	16	1	no	—
59	C₄	—	1016	2	no	—
60	C₄	#16861129	1	1	yes	characteristic
61	C₄	#16861152	1	1	yes	characteristic
62	C₂	#22549676	1	1	yes	characteristic
63	C₂	#22549714	1	1	yes	characteristic
64	C₂	#22549716	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	#1416
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
6. Dic₁₀₁₆	C₄	semidirect
33. C₁₂₇	#1416	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₀₁₆	2
C₂×C₁₂₇⋊C₃₂	2
#366×C₁₂₇	2
#1416	127