Subgroups — #30636795

Order 16256.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549652	C₂	1	1	yes	maximal, characteristic
2	#22549627	C₂	1	1	yes	maximal, characteristic
3	#334×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₂₇⋊C₁₆	C₄	1	1	yes	characteristic
5	C₂×C₁₂₇⋊C₁₆	C₄	1	1	yes	characteristic
6	C₄⋊(C₁₂₇⋊C₈)	—	2	1	no	—
7	C₄⋊C₈×C₁₂₇	—	2	1	no	—
8	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₂×C₁₂₇⋊C₈	—	4	1	no	—
10	C₁₂₇⋊C₁₆	—	2	1	no	—
11	C₁₂₇⋊C₁₆	C₈	1	2	yes	complemented
12	C₂×C₈×C₁₂₇	—	4	1	no	—
13	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
15	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
16	C₁₂₇⋊C₈	—	8	1	no	—
17	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
19	C₂×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
20	C₈×C₁₂₇	—	2	1	no	—
21	C₈×C₁₂₇	—	8	1	no	—
22	C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
23	C₈×C₁₂₇	M₁₆	1	1	yes	characteristic
24	C₂²×C₁₂₇	#106	1	1	yes	characteristic
25	C₄×C₁₂₇	—	2	1	no	—
26	C₄×C₁₂₇	—	2	1	no	—
27	C₄×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
28	C₄×C₁₂₇	#108	1	1	yes	characteristic
29	C₂×C₁₂₇	#334	1	1	yes	characteristic
30	C₂×C₁₂₇	#365	1	1	yes	characteristic
31	C₂×C₁₂₇	#365	1	1	yes	characteristic
32	#1368	—	127	1	no	maximal, Sylow
33	C₁₂₇	#1368	1	1	yes	characteristic, complemented, Sylow
34	#346	—	127	1	no	—
35	#334	—	127	1	no	—
36	#334	D₁₂₇	1	1	yes	characteristic
37	C₂×C₁₆	—	127	1	no	—
38	C₂×C₁₆	—	127	1	no	—
39	C₄⋊C₈	—	2	1	no	—
40	C₄⋊C₈	—	254	1	no	—
41	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
42	C₂×C₈	—	4	1	no	—
43	C₂×C₈	—	508	1	no	—
44	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
45	C₂×C₈	Dic₂₅₄	1	1	yes	characteristic
46	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
47	C₁₆	—	127	2	no	—
48	C₁₆	—	254	1	no	—
49	C₂×C₄	Dic₁₂₇⋊C₄	1	1	yes	characteristic
50	C₂×C₄	#14523235	1	1	yes	characteristic
51	C₂×C₄	#14523236	1	1	yes	characteristic
52	C₈	—	2	1	no	—
53	C₈	—	8	1	no	—
54	C₈	—	1016	1	no	—
55	C₈	C₈×D₁₂₇	1	1	yes	characteristic
56	C₈	#14523231	1	1	yes	characteristic
57	C₂²	#16861140	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	Dic₁₂₇⋊C₈	1	1	yes	characteristic
61	C₄	#16861144	1	1	yes	characteristic
62	C₂	#22549626	1	1	yes	characteristic
63	C₂	#22549698	1	1	yes	characteristic
64	C₂	#22549700	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16861140
commutator	C₈×C₁₂₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₂₇
Fitting	#334×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	#106
cosocle	C₄×C₈×C₁₂₇	C₂²
hypercenter	#334	D₁₂₇
hypocenter	C₁₂₇	#1368
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

Complements

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

N	H ≅ G/N	type
11. C₁₂₇⋊C₁₆	C₈	semidirect
33. C₁₂₇	#1368	semidirect

Maximal subgroups

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

subgroup	index
#22549652	2
#22549627	2
#334×C₁₂₇	2
#1368	127