Subgroups — #30641006

Order 16256.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550278	C₂	1	1	yes	maximal, characteristic
2	#22549703	C₂	1	2	yes	maximal
3	#22549701	C₂	1	2	yes	maximal
4	C₄×C₁₂₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	#499×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	#16861201	—	2	1	no	—
7	C₂×C₁₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₁₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
9	#16861143	C₂²	1	2	yes	—
10	#16861141	—	2	1	no	—
11	#125×C₁₂₇	—	2	1	no	—
12	#107×C₁₂₇	C₂²	1	2	yes	—
13	#106×C₁₂₇	—	2	1	no	—
14	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₂₇	—	2	2	no	—
16	Dic₁₂₇⋊C₄	—	4	2	no	—
17	C₁₂₇⋊C₁₆	Q₈	1	2	yes	—
18	C₁₂₇⋊C₁₆	Q₈	1	2	yes	—
19	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
20	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₁₂₇	—	2	2	no	—
22	C₄⋊C₄×C₁₂₇	—	4	2	no	—
23	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
24	C₂×Dic₁₂₇	—	4	2	no	—
25	C₂×C₄×C₁₂₇	—	4	2	no	—
26	C₂×C₄×C₁₂₇	D₈	1	2	yes	—
27	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
28	C₈×C₁₂₇	Q₁₆	1	2	yes	—
29	C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
30	C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
31	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
32	C₄×C₁₂₇	—	2	2	no	—
33	C₄×C₁₂₇	—	8	2	no	—
34	C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
35	C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
36	Dic₁₂₇	—	8	2	no	—
37	C₂×C₁₂₇	#500	1	1	yes	characteristic
38	C₂×C₁₂₇	#508	1	2	yes	—
39	#2253	—	127	1	no	maximal, Sylow
40	C₁₂₇	#2253	1	1	yes	characteristic, complemented, Sylow
41	#499	—	127	1	no	—
42	#499	D₁₂₇	1	1	yes	characteristic
43	#367	—	127	2	no	—
44	#366	—	127	2	no	—
45	C₄×C₁₆	—	127	1	no	—
46	#125	—	2	1	no	—
47	#125	—	254	1	no	—
48	C₂×C₁₆	—	127	1	no	—
49	C₂×C₁₆	—	127	1	no	—
50	#107	—	127	2	no	—
51	#107	C₂×D₁₂₇	1	2	yes	—
52	#106	—	2	1	no	—
53	#106	—	254	1	no	—
54	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
55	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
56	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
57	C₄⋊C₄	—	2	2	no	—
58	C₄⋊C₄	—	4	2	no	—
59	C₄⋊C₄	—	508	2	no	—
60	C₄⋊C₄	—	254	2	no	—
61	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
62	C₁₆	—	127	2	no	—
63	C₁₆	—	127	2	no	—
64	C₂×C₄	—	4	2	no	—
65	C₂×C₄	—	508	2	no	—
66	C₂×C₄	#14523233	1	2	yes	—
67	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
68	C₈	#14523238	1	2	yes	—
69	C₈	Q₈×D₁₂₇	1	1	yes	characteristic
70	C₈	Q₈×D₁₂₇	1	1	yes	characteristic
71	C₂²	C₂×#14523233	1	1	yes	characteristic
72	C₄	—	2	2	no	—
73	C₄	—	8	2	no	—
74	C₄	—	1016	2	no	—
75	C₄	#16861207	1	1	yes	characteristic
76	C₄	C₂×#14523238	1	1	yes	characteristic
77	C₂	#22550279	1	1	yes	characteristic
78	C₂	#22550306	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14523233
commutator, cosocle	C₂×C₈×C₁₂₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#499×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	C₂×D₈
hypercenter	#499	D₁₂₇
hypocenter	C₁₂₇	#2253
LCS	G ▷ C₂×C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #499

Complements

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

N	H ≅ G/N	type
40. C₁₂₇	#2253	semidirect

Maximal subgroups

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

subgroup	index
#22550278	2
#22549703	2
#22549701	2
C₄×C₁₂₇⋊C₁₆	2
#499×C₁₂₇	2
#2253	127