Subgroups — #30636799

Order 16256.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₁₂₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₈×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₂₇⋊C₁₆	—	2	1	no	—
5	C₄×C₁₂₇⋊C₈	C₄	1	1	yes	characteristic
6	C₄×C₁₂₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₁₂₇	—	2	1	no	—
8	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₂×C₁₂₇⋊C₈	—	2	1	no	—
10	C₂×C₁₂₇⋊C₈	C₈	1	1	yes	characteristic
11	C₂×C₁₂₇⋊C₈	C₈	1	1	yes	characteristic
12	C₁₂₇⋊C₁₆	—	4	1	no	—
13	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
14	C₂×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
15	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₁₂₇	—	4	1	no	—
17	C₁₂₇⋊C₈	—	2	2	no	—
18	C₁₂₇⋊C₈	—	2	1	no	—
19	C₁₂₇⋊C₈	C₁₆	1	2	yes	complemented
20	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
22	C₂×C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
23	C₈×C₁₂₇	—	2	1	no	—
24	C₈×C₁₂₇	D₈	1	1	yes	characteristic
25	C₈×C₁₂₇	Q₁₆	1	1	yes	characteristic
26	C₂²×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
27	C₄×C₁₂₇	—	2	1	no	—
28	C₄×C₁₂₇	#107	1	1	yes	characteristic
29	C₄×C₁₂₇	#108	1	1	yes	characteristic
30	C₄×C₁₂₇	C₂×C₁₆	1	1	yes	characteristic
31	C₄×C₁₂₇	M₃₂	1	1	yes	characteristic
32	C₂×C₁₂₇	#335	1	1	yes	characteristic
33	C₂×C₁₂₇	C₄⋊C₁₆	1	1	yes	characteristic
34	C₂×C₁₂₇	#364	1	1	yes	characteristic
35	#1370	—	127	1	no	maximal, Sylow
36	C₁₂₇	#1370	1	1	yes	characteristic, complemented, Sylow
37	C₄⋊C₁₆	—	127	1	no	—
38	C₄⋊C₁₆	D₁₂₇	1	1	yes	characteristic
39	C₈²	—	127	1	no	—
40	C₂×C₁₆	—	2	1	no	—
41	C₂×C₁₆	—	254	1	no	—
42	C₄×C₈	—	127	1	no	—
43	C₄×C₈	—	127	1	no	—
44	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
45	C₂×C₈	—	127	1	no	—
46	C₂×C₈	—	127	1	no	—
47	C₂×C₈	—	254	1	no	—
48	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
49	C₂×C₈	Dic₂₅₄	1	1	yes	characteristic
50	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
51	C₁₆	—	4	1	no	—
52	C₁₆	—	508	1	no	—
53	C₂×C₄	Dic₁₂₇⋊C₄	1	1	yes	characteristic
54	C₂×C₄	C₈×D₁₂₇	1	1	yes	characteristic
55	C₂×C₄	#14523231	1	1	yes	characteristic
56	C₈	—	2	1	no	—
57	C₈	—	127	2	no	—
58	C₈	—	254	2	no	—
59	C₈	—	254	1	no	—
60	C₈	#14523233	1	1	yes	characteristic
61	C₈	#14523238	1	1	yes	characteristic
62	C₂²	Dic₁₂₇⋊C₈	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	#16861142	1	1	yes	characteristic
65	C₄	#16861144	1	1	yes	characteristic
66	C₄	C₁₆×D₁₂₇	1	1	yes	characteristic
67	C₄	#16861148	1	1	yes	characteristic
68	C₂	#22549628	1	1	yes	characteristic
69	C₂	Dic₁₂₇⋊C₁₆	1	1	yes	characteristic
70	C₂	#22549696	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊D₁₂₇
commutator	C₄×C₁₂₇	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₁₂₇
Fitting	C₄⋊C₁₆×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	C₄⋊C₈
cosocle	C₄×C₈×C₁₂₇	C₂²
hypercenter	C₄⋊C₁₆	D₁₂₇
hypocenter	C₁₂₇	#1370
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
19. C₁₂₇⋊C₈	C₁₆	semidirect
36. C₁₂₇	#1370	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₁₂₇⋊C₁₆)	2
C₈×C₁₂₇⋊C₈	2
C₄⋊C₁₆×C₁₂₇	2
#1370	127