Group #30638588 — subgroups

Subgroups — #30638588

Order 16256.

260 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 129 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	C₂×Dic₁₂₇⋊C₈	C₂	1	2	yes	maximal
2	C₂×#16861120	C₂	1	1	yes	maximal, characteristic
3	#22549722	C₂	1	1	yes	maximal, characteristic
4	#22549630	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
9	Dic₁₂₇⋊C₈	C₄	1	8	yes	complemented
10	#16861120	—	2	2	no	—
11	C₁₂₇⋊#95	C₄	1	4	yes	—
12	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₂₇	—	2	2	no	—
15	C₂²×Dic₁₂₇	C₂×C₄	1	2	yes	—
16	C₂²×Dic₁₂₇	C₂×C₄	1	2	yes	—
17	C₂×C₁₂₇⋊C₈	—	2	2	no	—
18	C₂×C₁₂₇⋊C₈	—	2	2	no	—
19	C₂×C₁₂₇⋊C₈	C₂×C₄	1	4	yes	—
20	C₄×Dic₁₂₇	—	2	2	no	—
21	C₄×Dic₁₂₇	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₁₂₇	—	2	2	no	—
25	C₂×C₈×C₁₂₇	—	2	2	no	—
26	C₂×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
27	C₄²×C₁₂₇	D₄	1	2	yes	—
28	C₄²×C₁₂₇	Q₈	1	2	yes	—
29	C₂×Dic₁₂₇	—	2	4	no	—
30	C₂×Dic₁₂₇	—	2	2	no	—
31	C₂×Dic₁₂₇	—	2	4	no	—
32	C₂×Dic₁₂₇	C₄²	1	4	yes	—
33	C₁₂₇⋊C₈	—	2	4	no	—
34	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₂₇	—	2	4	no	—
36	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
37	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₁₂₇	#41	1	2	yes	—
41	C₈×C₁₂₇	—	2	4	no	—
42	C₂²×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
43	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₁₂₇	#117	1	1	yes	characteristic
45	C₂²×C₁₂₇	#131	1	4	yes	—
46	C₄×C₁₂₇	—	2	4	no	—
47	C₄×C₁₂₇	C₄×D₄	1	2	yes	—
48	C₄×C₁₂₇	C₄×Q₈	1	2	yes	—
49	Dic₁₂₇	—	2	4	no	—
50	Dic₁₂₇	—	4	2	no	—
51	C₂×C₁₂₇	C₄×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₁₂₇	#405	1	2	yes	—
53	C₂×C₁₂₇	#424	1	2	yes	—
54	C₂×C₁₂₇	#433	1	2	yes	—
55	#1768	—	127	1	no	maximal, Sylow
56	C₁₂₇	#1768	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₄⋊C₈	—	127	2	no	—
58	C₂×#97	—	127	1	no	—
59	C₂×C₄×C₈	D₁₂₇	1	1	yes	characteristic
60	#376	—	127	1	no	—
61	#336	—	127	2	no	—
62	C₂²×C₈	—	127	2	no	—
63	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
64	C₂×C₄²	—	127	1	no	—
65	C₂×C₄²	—	127	1	no	—
66	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
67	C₄⋊C₈	—	127	8	no	—
68	#97	—	254	2	no	—
69	C₄×C₈	—	2	2	no	—
70	#95	—	127	4	no	—
71	C₂²×C₄	—	127	2	no	—
72	C₂²×C₄	—	127	2	no	—
73	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
74	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	127	4	no	—
78	C₂×C₈	—	254	2	no	—
79	C₂×C₈	—	254	2	no	—
80	C₂×C₈	C₄×D₁₂₇	1	4	yes	—
81	C₄²	—	127	4	no	—
82	C₄²	—	254	2	no	—
83	C₄²	C₂²⋊D₁₂₇	1	2	yes	—
84	C₄²	Dic₂₅₄	1	2	yes	—
85	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	127	4	no	—
88	C₂×C₄	—	254	4	no	—
89	C₂×C₄	—	254	2	no	—
90	C₂×C₄	—	254	4	no	—
91	C₂×C₄	Dic₁₂₇⋊C₄	1	4	yes	—
92	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
95	C₂×C₄	#14523248	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	254	4	no	—
98	C₂²	C₄²×D₁₂₇	1	1	yes	characteristic
99	C₂²	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
100	C₂²	#16861166	1	1	yes	characteristic
101	C₂²	#16861215	1	4	yes	—
102	C₄	—	2	4	no	—
103	C₄	—	508	2	no	—
104	C₄	—	254	4	no	—
105	C₄	C₄×C₂²⋊D₁₂₇	1	2	yes	—
106	C₄	C₄×Dic₂₅₄	1	2	yes	—
107	C₂	C₄×Dic₁₂₇⋊C₄	1	1	yes	characteristic
108	C₂	#22549830	1	2	yes	—
109	C₂	#22549902	1	2	yes	—
110	C₂	#22549935	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₂₇⋊C₈

C₂

yes

maximal

C₂×#16861120

C₂

yes

maximal, characteristic

#22549722

C₂

yes

maximal, characteristic

#22549630

C₂

yes

maximal

C₂×C₄×C₈×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

C₂×C₄×Dic₁₂₇

C₂²

yes

characteristic

C₂×C₄×Dic₁₂₇

C₂²

yes

characteristic

Dic₁₂₇⋊C₈

C₄

yes

complemented

#16861120

—

C₁₂₇⋊#95

C₄

yes

—

C₂²×C₈×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₄×C₈×C₁₂₇

—

C₂²×Dic₁₂₇

C₂×C₄

yes

—

C₂²×Dic₁₂₇

C₂×C₄

yes

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

D₄

yes

—

C₄²×C₁₂₇

Q₈

yes

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₄²

yes

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

C₂×C₄²

yes

characteristic

C₂²×C₁₂₇

C₂×C₄⋊C₄

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₄×D₄

yes

—

C₄×C₁₂₇

C₄×Q₈

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₂₇

#405

yes

—

C₂×C₁₂₇

#424

yes

—

C₂×C₁₂₇

#433

yes

—

#1768

—

127

maximal, Sylow

C₁₂₇

#1768

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

127

—

C₂×#97

—

127

—

C₂×C₄×C₈

D₁₂₇

yes

characteristic

#376

—

127

—

#336

—

127

—

C₂²×C₈

—

127

—

C₂²×C₈

C₂×D₁₂₇

yes

—

C₂×C₄²

—

127

—

C₂×C₄²

—

127

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₄⋊C₈

—

127

—

#97

—

254

—

C₄×C₈

—

#95

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₄×D₁₂₇

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₂×C₈

C₄×D₁₂₇

yes

—

C₄²

—

127

—

C₄²

—

254

—

C₄²

C₂²⋊D₁₂₇

yes

—

C₄²

Dic₂₅₄

yes

—

C₂³

C₂×C₄×D₁₂₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

Dic₁₂₇⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₁₂₇

yes

—

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

C₂×C₄

C₂×Dic₂₅₄

yes

characteristic

C₂×C₄

#14523248

yes

—

C₈

—

C₈

—

254

—

C₂²

C₄²×D₁₂₇

yes

characteristic

C₂²

C₂×Dic₁₂₇⋊C₄

yes

characteristic

100

C₂²

#16861166

yes

characteristic

101

C₂²

#16861215

yes

—

102

C₄

—

103

C₄

—

508

—

104

C₄

—

254

—

105

C₄

C₄×C₂²⋊D₁₂₇

yes

—

106

C₄

C₄×Dic₂₅₄

yes

—

107

C₂

C₄×Dic₁₂₇⋊C₄

yes

characteristic

108

C₂

#22549830

yes

—

109

C₂

#22549902

yes

—

110

C₂

#22549935

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₂₇
commutator	C₂²×C₁₂₇	C₂×C₄²
Fitting	C₂×C₄×C₈×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂²×C₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₂₇
hypocenter	C₁₂₇	#1768
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₂₇

commutator

C₂²×C₁₂₇

C₂×C₄²

Fitting

C₂×C₄×C₈×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂²×C₄

cosocle

C₂²×C₄×C₁₂₇

C₂³

hypercenter

C₂×C₄×C₈

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
9. Dic₁₂₇⋊C₈	C₄	semidirect
56. C₁₂₇	#1768	semidirect

H ≅ G/N

type

9. Dic₁₂₇⋊C₈

C₄

semidirect

56. C₁₂₇

#1768

semidirect

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

subgroup	index
C₂×Dic₁₂₇⋊C₈	2
C₂×#16861120	2
#22549722	2
#22549630	2
C₂×C₄×C₈×C₁₂₇	2
#1768	127

subgroup

index

C₂×Dic₁₂₇⋊C₈

C₂×C₄×C₈×C₁₂₇

127

Subgroups — #30638588

Conjugacy classes of subgroups

Complements

Maximal subgroups