Group #30640526 — subgroups

Subgroups — #30640526

Order 16256.

160 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 81 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×C₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
2	Dic₁₂₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₂²⋊(C₁₂₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	C₁₆×Dic₁₂₇	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₂₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₁₂₇	C₄	1	1	yes	characteristic
8	C₂×C₁₂₇⋊C₁₆	C₂²	1	2	yes	—
9	C₈×Dic₁₂₇	C₄	1	1	yes	characteristic
10	C₈×Dic₁₂₇	C₄	1	1	yes	characteristic
11	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
12	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₁₂₇	C₂²	1	2	yes	—
15	C₂²×Dic₁₂₇	C₈	1	1	yes	characteristic
16	C₂²×Dic₁₂₇	C₈	1	1	yes	characteristic
17	C₂×C₁₂₇⋊C₈	—	2	2	no	—
18	C₂×C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₄×Dic₁₂₇	C₈	1	1	yes	characteristic
21	C₄×Dic₁₂₇	C₈	1	1	yes	characteristic
22	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
23	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₁₂₇⋊C₁₆	—	2	2	no	—
25	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₁₂₇	—	2	2	no	—
27	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
29	C₁₆×C₁₂₇	—	2	2	no	—
30	C₂×Dic₁₂₇	—	2	2	no	—
31	C₂×Dic₁₂₇	C₁₆	1	4	yes	complemented
32	C₂×Dic₁₂₇	C₂×C₈	1	1	yes	characteristic
33	C₂×Dic₁₂₇	C₂×C₈	1	1	yes	characteristic
34	C₂×Dic₁₂₇	C₂×C₈	1	1	yes	characteristic
35	C₂×Dic₁₂₇	C₂×C₈	1	1	yes	characteristic
36	C₁₂₇⋊C₈	—	2	1	no	—
37	C₁₂₇⋊C₈	M₁₆	1	2	yes	—
38	C₂³×C₁₂₇	C₂×C₈	1	1	yes	characteristic
39	C₂×C₄×C₁₂₇	—	2	1	no	—
40	C₂×C₄×C₁₂₇	—	2	1	no	—
41	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
42	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
43	C₈×C₁₂₇	—	2	1	no	—
44	C₈×C₁₂₇	#41	1	2	yes	—
45	C₂²×C₁₂₇	—	2	1	no	—
46	C₂²×C₁₂₇	—	2	1	no	—
47	C₂²×C₁₂₇	C₂×C₁₆	1	2	yes	—
48	C₂²×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
49	C₄×C₁₂₇	—	2	1	no	—
50	C₄×C₁₂₇	#117	1	1	yes	characteristic
51	C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
52	Dic₁₂₇	—	2	1	no	—
53	Dic₁₂₇	C₂×C₁₆	1	2	yes	—
54	Dic₁₂₇	C₂×C₁₆	1	2	yes	—
55	Dic₁₂₇	M₃₂	1	2	yes	—
56	C₂×C₁₂₇	—	2	2	no	—
57	C₂×C₁₂₇	#431	1	1	yes	characteristic
58	C₂×C₁₂₇	C₂²×C₁₆	1	1	yes	characteristic
59	C₂×C₁₂₇	C₂×M₃₂	1	1	yes	characteristic
60	#2162	—	127	1	no	maximal, Sylow
61	C₁₂₇	#2162	1	1	yes	characteristic, complemented, Sylow
62	C₂×C₄×C₈	—	127	1	no	—
63	C₄⋊C₁₆	—	127	2	no	—
64	C₂²⋊C₁₆	—	127	1	no	—
65	C₂²⋊C₁₆	D₁₂₇	1	1	yes	characteristic
66	C₄×C₁₆	—	127	2	no	—
67	C₂²×C₈	—	127	1	no	—
68	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
69	C₂×C₄²	—	127	1	no	—
70	C₂×C₁₆	—	127	2	no	—
71	C₂×C₁₆	C₂×D₁₂₇	1	2	yes	—
72	C₄×C₈	—	127	1	no	—
73	C₄×C₈	—	127	1	no	—
74	C₄×C₈	—	127	1	no	—
75	C₄×C₈	—	127	1	no	—
76	C₂²×C₄	—	127	1	no	—
77	C₂²×C₄	—	127	1	no	—
78	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	127	1	no	—
81	C₂×C₈	—	127	1	no	—
82	C₂×C₈	—	254	2	no	—
83	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
84	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
85	C₄²	—	127	1	no	—
86	C₄²	—	127	1	no	—
87	C₄²	—	127	1	no	—
88	C₄²	—	127	1	no	—
89	C₁₆	—	2	2	no	—
90	C₁₆	—	254	2	no	—
91	C₂³	C₈×D₁₂₇	1	1	yes	characteristic
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	127	1	no	—
95	C₂×C₄	—	127	1	no	—
96	C₂×C₄	—	127	1	no	—
97	C₂×C₄	—	127	4	no	—
98	C₂×C₄	—	127	1	no	—
99	C₂×C₄	—	254	2	no	—
100	C₂×C₄	C₈×D₁₂₇	1	1	yes	characteristic
101	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	—	127	2	no	—
104	C₈	—	254	1	no	—
105	C₈	#14523247	1	2	yes	—
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	C₁₆×D₁₂₇	1	2	yes	—
109	C₂²	C₂×C₈×D₁₂₇	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	127	2	no	—
112	C₄	—	127	2	no	—
113	C₄	—	127	2	no	—
114	C₄	—	254	1	no	—
115	C₄	#16861165	1	1	yes	characteristic
116	C₄	M₁₆×D₁₂₇	1	1	yes	characteristic
117	C₂	—	2	2	no	—
118	C₂	#22549926	1	1	yes	characteristic
119	C₂	C₂×C₁₆×D₁₂₇	1	1	yes	characteristic
120	C₂	M₃₂×D₁₂₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₁₂₇

C₂

yes

maximal, characteristic

Dic₁₂₇⋊C₁₆

C₂

yes

maximal, complemented

C₂²⋊(C₁₂₇⋊C₁₆)

C₂

yes

maximal, characteristic

C₁₆×Dic₁₂₇

C₂

yes

maximal, complemented

C₂²⋊C₁₆×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₈

C₄

yes

characteristic

C₂×C₄×Dic₁₂₇

C₄

yes

characteristic

C₂×C₁₂₇⋊C₁₆

C₂²

yes

—

C₈×Dic₁₂₇

C₄

yes

characteristic

C₈×Dic₁₂₇

C₄

yes

characteristic

C₈×Dic₁₂₇

C₂²

yes

characteristic

C₈×Dic₁₂₇

C₂²

yes

characteristic

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₁₆×C₁₂₇

C₂²

yes

—

C₂²×Dic₁₂₇

C₈

yes

characteristic

C₂²×Dic₁₂₇

C₈

yes

characteristic

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

characteristic

C₄×Dic₁₂₇

C₈

yes

characteristic

C₄×Dic₁₂₇

C₈

yes

characteristic

C₄×Dic₁₂₇

C₂×C₄

yes

characteristic

C₄×Dic₁₂₇

C₂×C₄

yes

characteristic

C₁₂₇⋊C₁₆

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₁₆×C₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₁₆

yes

complemented

C₂×Dic₁₂₇

C₂×C₈

yes

characteristic

C₂×Dic₁₂₇

C₂×C₈

yes

characteristic

C₂×Dic₁₂₇

C₂×C₈

yes

characteristic

C₂×Dic₁₂₇

C₂×C₈

yes

characteristic

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

M₁₆

yes

—

C₂³×C₁₂₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

#41

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×C₁₆

yes

—

C₂²×C₁₂₇

C₂²×C₈

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

#117

yes

characteristic

C₄×C₁₂₇

C₂×M₁₆

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

C₂×C₁₆

yes

—

Dic₁₂₇

C₂×C₁₆

yes

—

Dic₁₂₇

M₃₂

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#431

yes

characteristic

C₂×C₁₂₇

C₂²×C₁₆

yes

characteristic

C₂×C₁₂₇

C₂×M₃₂

yes

characteristic

#2162

—

127

maximal, Sylow

C₁₂₇

#2162

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

127

—

C₄⋊C₁₆

—

127

—

C₂²⋊C₁₆

—

127

—

C₂²⋊C₁₆

D₁₂₇

yes

characteristic

C₄×C₁₆

—

127

—

C₂²×C₈

—

127

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

C₂×C₄²

—

127

—

C₂×C₁₆

—

127

—

C₂×C₁₆

C₂×D₁₂₇

yes

—

C₄×C₈

—

127

—

C₄×C₈

—

127

—

C₄×C₈

—

127

—

C₄×C₈

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₄×D₁₂₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

C₄×D₁₂₇

yes

characteristic

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

C₄²

—

127

—

C₄²

—

127

—

C₄²

—

127

—

C₄²

—

127

—

C₁₆

—

C₁₆

—

254

—

C₂³

C₈×D₁₂₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

100

C₂×C₄

C₈×D₁₂₇

yes

characteristic

101

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

102

C₈

—

103

C₈

—

127

—

104

C₈

—

254

—

105

C₈

#14523247

yes

—

106

C₂²

—

107

C₂²

—

108

C₂²

C₁₆×D₁₂₇

yes

—

109

C₂²

C₂×C₈×D₁₂₇

yes

characteristic

110

C₄

—

111

C₄

—

127

—

112

C₄

—

127

—

113

C₄

—

127

—

114

C₄

—

254

—

115

C₄

#16861165

yes

characteristic

116

C₄

M₁₆×D₁₂₇

yes

characteristic

117

C₂

—

118

C₂

#22549926

yes

characteristic

119

C₂

C₂×C₁₆×D₁₂₇

yes

characteristic

120

C₂

M₃₂×D₁₂₇

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₂₇
commutator	C₂×C₁₂₇	C₂²×C₁₆
Fitting	C₂²⋊C₁₆×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×C₈
prime radical	C₂³×C₁₂₇	C₂×C₈
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	C₂²⋊C₁₆	D₁₂₇
hypocenter	C₁₂₇	#2162
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₂₇

commutator

C₂×C₁₂₇

C₂²×C₁₆

Fitting

C₂²⋊C₁₆×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂²×C₈

prime radical

C₂³×C₁₂₇

C₂×C₈

cosocle

C₂×C₈×C₁₂₇

C₂³

hypercenter

C₂²⋊C₁₆

D₁₂₇

C₁₂₇

G ▷ C₂×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
2. Dic₁₂₇⋊C₁₆	C₂	semidirect
4. C₁₆×Dic₁₂₇	C₂	semidirect
31. C₂×Dic₁₂₇	C₁₆	semidirect
61. C₁₂₇	#2162	semidirect

H ≅ G/N

type

2. Dic₁₂₇⋊C₁₆

C₂

semidirect

4. C₁₆×Dic₁₂₇

C₂

semidirect

31. C₂×Dic₁₂₇

C₁₆

semidirect

61. C₁₂₇

#2162

semidirect

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

subgroup	index
C₂×C₈×Dic₁₂₇	2
Dic₁₂₇⋊C₁₆	2
C₂²⋊(C₁₂₇⋊C₁₆)	2
C₁₆×Dic₁₂₇	2
C₂²⋊C₁₆×C₁₂₇	2
#2162	127

subgroup

index

C₂×C₈×Dic₁₂₇

Dic₁₂₇⋊C₁₆

C₂²⋊(C₁₂₇⋊C₁₆)

C₁₆×Dic₁₂₇

C₂²⋊C₁₆×C₁₂₇

#2162

127

Subgroups — #30640526

Conjugacy classes of subgroups

Complements

Maximal subgroups