Group #30638059 — subgroups

Subgroups — #30638059

Order 16256.

172 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 45 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	C₁₂₇⋊#564	C₂	1	1	yes	maximal, characteristic
2	#22550285	C₂	1	2	yes	maximal
3	#22550246	C₂	1	1	yes	maximal, characteristic
4	#22549617	C₂	1	2	yes	maximal
5	#330×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	#16861209	C₂²	1	1	yes	characteristic
7	#16861201	C₂²	1	2	yes	—
8	#16861200	—	2	2	no	—
9	#16861199	—	2	2	no	—
10	#16861199	—	2	2	no	—
11	#16861143	—	2	2	no	—
12	#16861141	—	2	2	no	—
13	#16861132	—	2	2	no	—
14	#16861120	C₂²	1	1	yes	characteristic
15	#125×C₁₂₇	C₂²	1	1	yes	characteristic
16	#97×C₁₂₇	C₂²	1	2	yes	—
17	C₂×Dic₂₅₄	—	2	1	no	—
18	C₂×C₁₂₇⋊C₈	—	2	1	no	—
19	C₄⋊Dic₁₂₇	—	2	2	no	—
20	C₄⋊Dic₁₂₇	—	2	2	no	—
21	C₄⋊Dic₁₂₇	D₄	1	2	yes	—
22	C₄⋊Dic₁₂₇	D₄	1	2	yes	—
23	Dic₁₂₇⋊C₄	—	2	2	no	—
24	Dic₁₂₇⋊C₄	—	2	2	no	—
25	Dic₁₂₇⋊C₄	—	2	2	no	—
26	Dic₁₂₇⋊C₄	—	2	2	no	—
27	C₄×Dic₁₂₇	—	2	2	no	—
28	C₂×C₈×C₁₂₇	—	2	2	no	—
29	C₄⋊C₄×C₁₂₇	—	2	2	no	—
30	C₄⋊C₄×C₁₂₇	D₄	1	2	yes	—
31	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
32	Dic₂₅₄	—	4	2	no	—
33	C₂×Dic₁₂₇	—	2	2	no	—
34	C₂×Dic₁₂₇	—	2	2	no	—
35	C₂×Dic₁₂₇	—	4	1	no	—
36	C₁₂₇⋊C₈	—	4	1	no	—
37	C₂×C₄×C₁₂₇	—	2	2	no	—
38	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
40	C₈×C₁₂₇	—	2	4	no	—
41	C₂²×C₁₂₇	#120	1	1	yes	characteristic
42	C₄×C₁₂₇	—	2	1	no	—
43	C₄×C₁₂₇	—	2	2	no	—
44	C₄×C₁₂₇	—	4	2	no	—
45	Dic₁₂₇	—	4	2	no	—
46	Dic₁₂₇	—	4	2	no	—
47	Dic₁₂₇	—	8	1	no	—
48	C₂×C₁₂₇	#454	1	1	yes	characteristic
49	C₂×C₁₂₇	#456	1	2	yes	—
50	#1663	—	127	1	no	maximal, Sylow
51	C₁₂₇	#1663	1	1	yes	characteristic, complemented, Sylow
52	#564	—	127	1	no	—
53	#501	—	127	2	no	—
54	#491	—	127	1	no	—
55	#330	—	127	2	no	—
56	#330	D₁₂₇	1	1	yes	characteristic
57	C₄⋊Q₈	—	127	1	no	—
58	#125	—	127	2	no	—
59	#125	—	254	2	no	—
60	#125	—	254	2	no	—
61	#125	—	254	2	no	—
62	#125	C₂×D₁₂₇	1	1	yes	characteristic
63	#107	—	254	2	no	—
64	#106	—	254	2	no	—
65	#103	—	254	2	no	—
66	#97	—	127	1	no	—
67	#97	C₂×D₁₂₇	1	2	yes	—
68	C₂×Q₈	—	254	1	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	254	1	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	127	2	no	—
73	C₄⋊C₄	—	127	2	no	—
74	C₄⋊C₄	—	254	2	no	—
75	C₄⋊C₄	—	254	2	no	—
76	C₄⋊C₄	—	254	2	no	—
77	C₄⋊C₄	—	254	2	no	—
78	C₄⋊C₄	—	254	2	no	—
79	C₄⋊C₄	—	254	2	no	—
80	C₄⋊C₄	D₅₀₈	1	2	yes	—
81	C₄²	—	254	2	no	—
82	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
83	Q₈	—	508	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	508	1	no	—
86	C₂×C₄	—	254	2	no	—
87	C₂×C₄	—	254	2	no	—
88	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
89	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
90	C₈	—	2	4	no	—
91	C₈	—	508	1	no	—
92	C₂²	#16861179	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	4	2	no	—
96	C₄	—	1016	1	no	—
97	C₄	—	508	2	no	—
98	C₄	—	508	2	no	—
99	C₂	#22550051	1	1	yes	characteristic
100	C₂	#22550060	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₂₇⋊#564

C₂

yes

maximal, characteristic

#22550285

C₂

yes

maximal

#22550246

C₂

yes

maximal, characteristic

#22549617

C₂

yes

maximal

#330×C₁₂₇

C₂

yes

maximal, characteristic

#16861209

C₂²

yes

characteristic

#16861201

C₂²

yes

—

—

—

—

—

—

—

C₂²

yes

characteristic

#125×C₁₂₇

C₂²

yes

characteristic

#97×C₁₂₇

C₂²

yes

—

C₂×Dic₂₅₄

—

C₂×C₁₂₇⋊C₈

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

D₄

yes

—

C₄⋊Dic₁₂₇

D₄

yes

—

Dic₁₂₇⋊C₄

—

Dic₁₂₇⋊C₄

—

Dic₁₂₇⋊C₄

—

Dic₁₂₇⋊C₄

—

C₄×Dic₁₂₇

—

C₂×C₈×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

D₄

yes

—

C₄²×C₁₂₇

C₂³

yes

characteristic

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

#120

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

#454

yes

characteristic

C₂×C₁₂₇

#456

yes

—

#1663

—

127

maximal, Sylow

C₁₂₇

#1663

yes

characteristic, complemented, Sylow

#564

—

127

—

#501

—

127

—

#491

—

127

—

#330

—

127

—

#330

D₁₂₇

yes

characteristic

C₄⋊Q₈

—

127

—

#125

—

127

—

#125

—

254

—

#125

—

254

—

#125

—

254

—

#125

C₂×D₁₂₇

yes

characteristic

#107

—

254

—

#106

—

254

—

#103

—

254

—

#97

—

127

—

#97

C₂×D₁₂₇

yes

—

C₂×Q₈

—

254

—

C₂×C₈

—

C₂×C₈

—

254

—

C₄⋊C₄

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄⋊C₄

D₅₀₈

yes

—

C₄²

—

254

—

C₄²

C₂²×D₁₂₇

yes

characteristic

Q₈

—

508

—

C₂×C₄

—

C₂×C₄

—

508

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

D₄×D₁₂₇

yes

—

C₂×C₄

C₂×D₅₀₈

yes

characteristic

C₈

—

C₈

—

508

—

C₂²

#16861179

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

1016

—

C₄

—

508

—

C₄

—

508

—

C₂

#22550051

yes

characteristic

100

C₂

#22550060

yes

—

	subgroup	G/H
center	C₂²	#16861179
commutator, cosocle	C₄²×C₁₂₇	C₂³
Frattini	C₄²	C₂²×D₁₂₇
Fitting	#330×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	#120
hypercenter	#330	D₁₂₇
hypocenter	C₁₂₇	#1663
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #330

subgroup

G/H

center

C₂²

#16861179

commutator, cosocle

C₄²×C₁₂₇

C₂³

Frattini

C₄²

C₂²×D₁₂₇

Fitting

#330×C₁₂₇

C₂

C₂²×C₁₂₇

D₁₂₇

C₁₂₇

G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #330

N	H ≅ G/N	type
51. C₁₂₇	#1663	semidirect

H ≅ G/N

type

51. C₁₂₇

#1663

semidirect

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

subgroup	index
C₁₂₇⋊#564	2
#22550285	2
#22550246	2
#22549617	2
#330×C₁₂₇	2
#1663	127

subgroup

index

C₁₂₇⋊#564

#330×C₁₂₇

127

Subgroups — #30638059

Conjugacy classes of subgroups

Complements

Maximal subgroups