Group #30647832 — subgroups

Subgroups — #30647832

Order 16256.

330 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 205 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550007	C₂	1	4	yes	maximal, complemented
2	Q₈×C₁₂₇⋊C₈	C₂	1	4	yes	maximal, complemented
3	#22549937	C₂	1	4	yes	maximal
4	#22549929	C₂	1	2	yes	maximal
5	#533×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
7	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
8	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	8	yes	—
9	C₂²⋊(C₁₂₇⋊C₈)	C₂²	1	4	yes	—
10	#16861120	C₂²	1	4	yes	—
11	C₄×C₁₂₇⋊C₈	C₂²	1	4	yes	—
12	C₄×C₁₂₇⋊C₈	C₂²	1	4	yes	—
13	C₄⋊Q₈×C₁₂₇	C₄	1	2	yes	—
14	#125×C₁₂₇	C₄	1	2	yes	—
15	C₂²⋊Q₈×C₁₂₇	C₄	1	4	yes	—
16	C₄×Q₈×C₁₂₇	C₂²	1	4	yes	—
17	#117×C₁₂₇	C₂²	1	2	yes	—
18	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
20	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
21	C₂×Q₈×C₁₂₇	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
25	C₄⋊C₄×C₁₂₇	C₂×C₄	1	8	yes	—
26	C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
27	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
28	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₁₂₇	C₂³	1	2	yes	—
31	C₄²×C₁₂₇	C₂³	1	4	yes	—
32	C₁₂₇⋊C₈	—	2	4	no	—
33	C₁₂₇⋊C₈	—	2	4	no	—
34	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₁₂₇	—	2	8	no	—
36	C₂×C₄×C₁₂₇	—	2	4	no	—
37	C₂×C₄×C₁₂₇	—	2	2	no	—
38	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
40	C₂×C₄×C₁₂₇	C₂²×C₄	1	8	yes	—
41	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
43	C₂²×C₁₂₇	—	2	1	no	—
44	C₂²×C₁₂₇	—	2	2	no	—
45	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
46	C₄×C₁₂₇	—	2	1	no	—
47	C₄×C₁₂₇	—	2	2	no	—
48	C₄×C₁₂₇	—	2	8	no	—
49	C₄×C₁₂₇	#131	1	4	yes	—
50	C₄×C₁₂₇	2₋¹⁺⁴	1	2	yes	—
51	C₂×C₁₂₇	—	4	1	no	—
52	C₂×C₁₂₇	#519	1	1	yes	characteristic
53	C₂×C₁₂₇	C₂×#131	1	2	yes	—
54	#2986	—	127	1	no	maximal, Sylow
55	C₁₂₇	#2986	1	1	yes	characteristic, complemented, Sylow
56	#533	D₁₂₇	1	1	yes	characteristic
57	#446	—	127	4	no	—
58	C₈×Q₈	—	127	4	no	—
59	#433	—	127	4	no	—
60	#431	—	127	2	no	—
61	C₄⋊Q₈	Dic₁₂₇	1	2	yes	—
62	#125	Dic₁₂₇	1	2	yes	—
63	C₂²⋊Q₈	Dic₁₂₇	1	4	yes	—
64	C₄×Q₈	C₂×D₁₂₇	1	4	yes	—
65	#117	C₂×D₁₂₇	1	2	yes	—
66	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
67	C₄⋊C₈	—	127	2	no	—
68	C₄⋊C₈	—	127	2	no	—
69	C₄⋊C₈	—	127	8	no	—
70	C₂²⋊C₈	—	127	4	no	—
71	#97	—	127	4	no	—
72	C₄×C₈	—	127	4	no	—
73	C₄×C₈	—	127	4	no	—
74	C₂×Q₈	C₂×Dic₁₂₇	1	4	yes	—
75	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
76	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
77	C₂×C₈	—	127	4	no	—
78	C₂×C₈	—	127	4	no	—
79	C₄⋊C₄	C₂×Dic₁₂₇	1	4	yes	—
80	C₄⋊C₄	C₂×Dic₁₂₇	1	8	yes	—
81	C₄⋊C₄	C₂×Dic₁₂₇	1	4	yes	—
82	C₂²⋊C₄	C₂×Dic₁₂₇	1	4	yes	—
83	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
84	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
85	C₄²	C₂²×D₁₂₇	1	2	yes	—
86	C₄²	C₂²×D₁₂₇	1	4	yes	—
87	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
88	Q₈	—	2	8	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
92	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
93	C₂×C₄	C₂²×Dic₁₂₇	1	8	yes	—
94	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
95	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
96	C₈	—	254	4	no	—
97	C₈	—	254	4	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂³×Dic₁₂₇	1	1	yes	characteristic
101	C₄	—	2	1	no	—
102	C₄	—	2	2	no	—
103	C₄	—	2	8	no	—
104	C₄	#16861216	1	4	yes	—
105	C₄	#16861257	1	2	yes	—
106	C₂	—	4	1	no	—
107	C₂	#22550354	1	1	yes	characteristic
108	C₂	C₂×#16861216	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22550007

C₂

yes

maximal, complemented

Q₈×C₁₂₇⋊C₈

C₂

yes

maximal, complemented

#22549937

C₂

yes

maximal

#22549929

C₂

yes

maximal

#533×C₁₂₇

C₂

yes

maximal, characteristic

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

—

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

—

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

—

C₂²⋊(C₁₂₇⋊C₈)

C₂²

yes

—

#16861120

C₂²

yes

—

C₄×C₁₂₇⋊C₈

C₂²

yes

—

C₄×C₁₂₇⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₁₂₇

C₄

yes

—

#125×C₁₂₇

C₄

yes

—

C₂²⋊Q₈×C₁₂₇

C₄

yes

—

C₄×Q₈×C₁₂₇

C₂²

yes

—

#117×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₂×Q₈×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂×C₄

yes

—

C₄⋊C₄×C₁₂₇

C₂×C₄

yes

—

C₄⋊C₄×C₁₂₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

C₂×C₄

yes

characteristic

C₄²×C₁₂₇

C₂×C₄

yes

characteristic

C₄²×C₁₂₇

C₂³

yes

—

C₄²×C₁₂₇

C₂³

yes

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂²×C₄

yes

characteristic

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂³×C₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#131

yes

—

C₄×C₁₂₇

2₋¹⁺⁴

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#519

yes

characteristic

C₂×C₁₂₇

C₂×#131

yes

—

#2986

—

127

maximal, Sylow

C₁₂₇

#2986

yes

characteristic, complemented, Sylow

#533

D₁₂₇

yes

characteristic

#446

—

127

—

C₈×Q₈

—

127

—

#433

—

127

—

#431

—

127

—

C₄⋊Q₈

Dic₁₂₇

yes

—

#125

Dic₁₂₇

yes

—

C₂²⋊Q₈

Dic₁₂₇

yes

—

C₄×Q₈

C₂×D₁₂₇

yes

—

#117

C₂×D₁₂₇

yes

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₄⋊C₈

—

127

—

C₄⋊C₈

—

127

—

C₄⋊C₈

—

127

—

C₂²⋊C₈

—

127

—

#97

—

127

—

C₄×C₈

—

127

—

C₄×C₈

—

127

—

C₂×Q₈

C₂×Dic₁₂₇

yes

—

C₂²×C₄

C₂×Dic₁₂₇

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₄⋊C₄

C₂×Dic₁₂₇

yes

—

C₄⋊C₄

C₂×Dic₁₂₇

yes

—

C₄⋊C₄

C₂×Dic₁₂₇

yes

—

C₂²⋊C₄

C₂×Dic₁₂₇

yes

—

C₄²

C₂×Dic₁₂₇

yes

characteristic

C₄²

C₂×Dic₁₂₇

yes

characteristic

C₄²

C₂²×D₁₂₇

yes

—

C₄²

C₂²×D₁₂₇

yes

—

C₂³

C₂²×Dic₁₂₇

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×Dic₁₂₇

yes

characteristic

C₂×C₄

C₂²×Dic₁₂₇

yes

—

C₂×C₄

C₂²×Dic₁₂₇

yes

—

C₂×C₄

C₂²×Dic₁₂₇

yes

—

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

C₈

—

254

—

C₈

—

254

—

C₂²

—

C₂²

—

100

C₂²

C₂³×Dic₁₂₇

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

#16861216

yes

—

105

C₄

#16861257

yes

—

106

C₂

—

107

C₂

#22550354

yes

characteristic

108

C₂

C₂×#16861216

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Fitting	#533×C₁₂₇	C₂
prime radical	C₂³×C₁₂₇	C₂²×C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#533	D₁₂₇
hypocenter	C₁₂₇	#2986
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #533

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂³×C₄

Fitting

#533×C₁₂₇

C₂

prime radical

C₂³×C₁₂₇

C₂²×C₄

cosocle

C₂×C₄×C₁₂₇

C₂⁴

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂×C₄ ◁ #533

N	H ≅ G/N	type
1. #22550007	C₂	semidirect
2. Q₈×C₁₂₇⋊C₈	C₂	semidirect
55. C₁₂₇	#2986	semidirect

H ≅ G/N

type

1. #22550007

C₂

semidirect

2. Q₈×C₁₂₇⋊C₈

C₂

semidirect

55. C₁₂₇

#2986

semidirect

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

subgroup	index
#22550007	2
Q₈×C₁₂₇⋊C₈	2
#22549937	2
#22549929	2
#533×C₁₂₇	2
#2986	127

subgroup

index

Q₈×C₁₂₇⋊C₈

#533×C₁₂₇

127

Subgroups — #30647832

Conjugacy classes of subgroups

Complements

Maximal subgroups