Group #30640779 — subgroups

Subgroups — #30640779

Order 16256.

154 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 49 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×Dic₁₀₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16861155	C₂	1	1	yes	maximal, characteristic
3	#22550254	C₂	1	1	yes	maximal, characteristic
4	#22549970	C₂	1	1	yes	maximal, characteristic
5	Dic₁₂₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	#22549679	C₂	1	1	yes	maximal, characteristic
7	#358×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₅₀₈	C₂²	1	1	yes	characteristic
9	C₂×Dic₅₀₈	C₂²	1	1	yes	characteristic
10	C₂×#14523238	—	2	1	no	—
11	#16861207	—	2	1	no	—
12	#16861174	—	2	1	no	—
13	Dic₁₀₁₆	—	2	2	no	—
14	#16861155	—	2	2	no	—
15	C₂×C₁₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	#16861132	—	2	1	no	—
17	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
20	#107×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₅₄	—	2	1	no	—
22	C₂×Dic₂₅₄	—	2	1	no	—
23	Dic₅₀₈	—	2	1	no	—
24	Dic₅₀₈	—	2	1	no	—
25	Dic₅₀₈	—	2	1	no	—
26	Dic₅₀₈	D₄	1	2	yes	—
27	#14523238	—	4	2	no	—
28	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
29	Dic₁₂₇⋊C₄	—	4	1	no	—
30	Dic₁₂₇⋊C₄	—	4	1	no	—
31	Dic₁₂₇⋊C₄	—	4	1	no	—
32	C₄×Dic₁₂₇	—	4	1	no	—
33	C₄×Dic₁₂₇	D₄	1	1	yes	characteristic
34	C₁₂₇⋊C₁₆	—	2	1	no	—
35	C₂×Q₈×C₁₂₇	—	2	1	no	—
36	Q₁₆×C₁₂₇	—	2	1	no	—
37	Q₁₆×C₁₂₇	—	2	1	no	—
38	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₁₂₇	—	2	1	no	—
40	C₁₆×C₁₂₇	—	2	1	no	—
41	Dic₂₅₄	—	2	2	no	—
42	Dic₂₅₄	—	4	1	no	—
43	Dic₂₅₄	—	4	1	no	—
44	Dic₂₅₄	—	4	1	no	—
45	C₂×Dic₁₂₇	—	4	1	no	—
46	C₂×Dic₁₂₇	—	4	1	no	—
47	C₂×Dic₁₂₇	D₈	1	1	yes	characteristic
48	C₂×Dic₁₂₇	D₈	1	1	yes	characteristic
49	C₁₂₇⋊C₈	—	2	1	no	—
50	Q₈×C₁₂₇	—	4	1	no	—
51	Q₈×C₁₂₇	—	4	1	no	—
52	C₂×C₄×C₁₂₇	—	4	1	no	—
53	C₂×C₄×C₁₂₇	—	4	1	no	—
54	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
55	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
56	C₈×C₁₂₇	#41	1	1	yes	characteristic
57	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
58	C₄×C₁₂₇	—	8	1	no	—
59	C₄×C₁₂₇	—	8	1	no	—
60	C₄×C₁₂₇	#121	1	1	yes	characteristic
61	C₄×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₁₂₇	—	2	1	no	—
63	Dic₁₂₇	—	4	2	no	—
64	Dic₁₂₇	—	8	1	no	—
65	Dic₁₂₇	Q₃₂	1	2	yes	complemented
66	C₂×C₁₂₇	#459	1	1	yes	characteristic
67	C₂×C₁₂₇	C₂×Q₃₂	1	1	yes	characteristic
68	C₂×C₁₂₇	#510	1	1	yes	characteristic
69	#2209	—	127	1	no	maximal, Sylow
70	C₁₂₇	#2209	1	1	yes	characteristic, complemented, Sylow
71	C₂×Q₃₂	—	127	1	no	—
72	C₂×Q₃₂	—	127	1	no	—
73	#494	—	127	1	no	—
74	C₄×Q₁₆	—	127	1	no	—
75	C₄⋊C₁₆	—	127	1	no	—
76	#358	—	127	1	no	—
77	#358	D₁₂₇	1	1	yes	characteristic
78	C₂×Q₁₆	—	127	1	no	—
79	C₂×Q₁₆	—	127	1	no	—
80	C₂×Q₁₆	—	254	1	no	—
81	C₂×Q₁₆	C₂×D₁₂₇	1	1	yes	characteristic
82	C₄⋊Q₈	—	254	1	no	—
83	C₄×Q₈	—	254	1	no	—
84	Q₃₂	—	254	2	no	—
85	Q₃₂	—	254	2	no	—
86	C₂×C₁₆	—	127	1	no	—
87	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
88	#107	C₂×D₁₂₇	1	1	yes	characteristic
89	#103	—	254	1	no	—
90	C₄×C₈	—	127	1	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	254	1	no	—
93	C₂×Q₈	—	254	1	no	—
94	Q₁₆	—	2	1	no	—
95	Q₁₆	—	2	1	no	—
96	Q₁₆	—	127	2	no	—
97	Q₁₆	—	508	2	no	—
98	Q₁₆	—	254	1	no	—
99	Q₁₆	—	254	1	no	—
100	Q₁₆	—	254	1	no	—
101	C₂×C₈	—	127	1	no	—
102	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	508	1	no	—
105	C₄⋊C₄	—	508	1	no	—
106	C₄⋊C₄	—	508	1	no	—
107	C₄²	—	127	1	no	—
108	C₄²	—	508	1	no	—
109	C₁₆	—	2	1	no	—
110	C₁₆	—	254	1	no	—
111	Q₈	—	4	1	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	508	1	no	—
114	Q₈	—	508	1	no	—
115	Q₈	—	508	1	no	—
116	Q₈	—	254	2	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	127	1	no	—
120	C₂×C₄	—	127	1	no	—
121	C₂×C₄	—	508	1	no	—
122	C₂×C₄	—	508	1	no	—
123	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
124	C₈	—	254	1	no	—
125	C₈	D₄×D₁₂₇	1	1	yes	characteristic
126	C₈	#14523248	1	1	yes	characteristic
127	C₂²	D₈×D₁₂₇	1	1	yes	characteristic
128	C₄	—	8	1	no	—
129	C₄	—	8	1	no	—
130	C₄	—	127	2	no	—
131	C₄	—	1016	1	no	—
132	C₄	—	508	2	no	—
133	C₄	—	254	1	no	—
134	C₄	#16861182	1	1	yes	characteristic
135	C₄	#16861236	1	1	yes	characteristic
136	C₂	#22550070	1	1	yes	characteristic
137	C₂	Q₃₂×D₁₂₇	1	1	yes	characteristic
138	C₂	#22550315	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₀₁₆

C₂

yes

maximal, characteristic

C₂×#16861155

C₂

yes

maximal, characteristic

#22550254

C₂

yes

maximal, characteristic

#22549970

C₂

yes

maximal, characteristic

Dic₁₂₇⋊C₁₆

C₂

yes

maximal, characteristic

#22549679

C₂

yes

maximal, characteristic

#358×C₁₂₇

C₂

yes

maximal, characteristic

C₂×Dic₅₀₈

C₂²

yes

characteristic

C₂×Dic₅₀₈

C₂²

yes

—

—

—

Dic₁₀₁₆

—

#16861155

—

C₂×C₁₂₇⋊C₁₆

C₂²

yes

characteristic

#16861132

—

C₈×Dic₁₂₇

C₂²

yes

characteristic

C₂×Q₁₆×C₁₂₇

C₂²

yes

characteristic

C₂×C₁₆×C₁₂₇

C₂²

yes

characteristic

#107×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

Dic₅₀₈

—

Dic₅₀₈

—

Dic₅₀₈

—

Dic₅₀₈

D₄

yes

—

#14523238

—

C₂×C₁₂₇⋊C₈

D₄

yes

characteristic

Dic₁₂₇⋊C₄

—

Dic₁₂₇⋊C₄

—

Dic₁₂₇⋊C₄

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

D₄

yes

characteristic

C₁₂₇⋊C₁₆

—

C₂×Q₈×C₁₂₇

—

Q₁₆×C₁₂₇

—

Q₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₁₆×C₁₂₇

—

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

D₈

yes

characteristic

C₂×Dic₁₂₇

D₈

yes

characteristic

C₁₂₇⋊C₈

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

#41

yes

characteristic

C₂²×C₁₂₇

C₂×D₈

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

Q₃₂

yes

complemented

C₂×C₁₂₇

#459

yes

characteristic

C₂×C₁₂₇

C₂×Q₃₂

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#2209

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

127

—

C₂×Q₃₂

—

127

—

#494

—

127

—

C₄×Q₁₆

—

127

—

C₄⋊C₁₆

—

127

—

#358

—

127

—

#358

D₁₂₇

yes

characteristic

C₂×Q₁₆

—

127

—

C₂×Q₁₆

—

127

—

C₂×Q₁₆

—

254

—

C₂×Q₁₆

C₂×D₁₂₇

yes

characteristic

C₄⋊Q₈

—

254

—

C₄×Q₈

—

254

—

Q₃₂

—

254

—

Q₃₂

—

254

—

C₂×C₁₆

—

127

—

C₂×C₁₆

C₂×D₁₂₇

yes

characteristic

#107

C₂×D₁₂₇

yes

characteristic

#103

—

254

—

C₄×C₈

—

127

—

C₂×Q₈

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

127

—

Q₁₆

—

508

—

Q₁₆

—

254

—

Q₁₆

—

254

—

100

Q₁₆

—

254

—

101

C₂×C₈

—

127

—

102

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

508

—

105

C₄⋊C₄

—

508

—

106

C₄⋊C₄

—

508

—

107

C₄²

—

127

—

108

C₄²

—

508

—

109

C₁₆

—

110

C₁₆

—

254

—

111

Q₈

—

112

Q₈

—

113

Q₈

—

508

—

114

Q₈

—

508

—

115

Q₈

—

508

—

116

Q₈

—

254

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

127

—

120

C₂×C₄

—

127

—

121

C₂×C₄

—

508

—

122

C₂×C₄

—

508

—

123

C₂×C₄

D₄×D₁₂₇

yes

characteristic

124

C₈

—

254

—

125

C₈

D₄×D₁₂₇

yes

characteristic

126

C₈

#14523248

yes

characteristic

127

C₂²

D₈×D₁₂₇

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

127

—

131

C₄

—

1016

—

132

C₄

—

508

—

133

C₄

—

254

—

134

C₄

#16861182

yes

characteristic

135

C₄

#16861236

yes

characteristic

136

C₂

#22550070

yes

characteristic

137

C₂

Q₃₂×D₁₂₇

yes

characteristic

138

C₂

#22550315

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₂₇
commutator, cosocle	C₂×C₈×C₁₂₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#358×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	C₂×D₈
hypercenter	#358	D₁₂₇
hypocenter	C₁₂₇	#2209
LCS	G ▷ C₂×C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₁₂₇

commutator, cosocle

C₂×C₈×C₁₂₇

C₂³

Frattini

C₂×C₈

C₂²×D₁₂₇

Fitting

#358×C₁₂₇

C₂

socle, prime radical

C₂²×C₁₂₇

C₂×D₈

D₁₂₇

C₁₂₇

G ▷ C₂×C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
65. Dic₁₂₇	Q₃₂	semidirect
70. C₁₂₇	#2209	semidirect

H ≅ G/N

type

65. Dic₁₂₇

Q₃₂

semidirect

70. C₁₂₇

#2209

semidirect

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

subgroup	index
C₂×Dic₁₀₁₆	2
C₂×#16861155	2
#22550254	2
#22549970	2
Dic₁₂₇⋊C₁₆	2
#22549679	2
#358×C₁₂₇	2
#2209	127

subgroup

index

C₂×Dic₁₀₁₆

Dic₁₂₇⋊C₁₆

#358×C₁₂₇

127

Subgroups — #30640779

Conjugacy classes of subgroups

Complements

Maximal subgroups