Group #30666682 — subgroups

Subgroups — #30666682

Order 16768.

248 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 109 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×Dic₁₃₁⋊C₈	C₂	1	2	yes	maximal
2	C₂×#16862157	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Dic₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#22553066	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₃₁	C₄	1	2	yes	—
9	C₂×C₄×Dic₁₃₁	C₂²	1	2	yes	—
10	Dic₁₃₁⋊C₈	—	2	4	no	—
11	#16862157	—	2	2	no	—
12	C₁₃₁⋊#95	C₄	1	1	yes	characteristic
13	C₁₃₁⋊#95	C₄	1	1	yes	characteristic
14	C₂²×C₈×C₁₃₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₃₁	—	2	2	no	—
17	C₂²×Dic₁₃₁	C₂×C₄	1	2	yes	—
18	C₂²×Dic₁₃₁	C₂×C₄	1	2	yes	—
19	C₂×C₁₃₁⋊C₈	—	2	1	no	—
20	C₂×C₁₃₁⋊C₈	—	2	1	no	—
21	C₂×C₁₃₁⋊C₈	—	2	1	no	—
22	C₂×C₁₃₁⋊C₈	—	2	1	no	—
23	C₂×C₁₃₁⋊C₈	D₄	1	2	yes	—
24	C₂×C₁₃₁⋊C₈	Q₈	1	2	yes	—
25	C₄⋊Dic₁₃₁	—	2	4	no	—
26	C₄×Dic₁₃₁	—	2	2	no	—
27	C₄×Dic₁₃₁	—	2	2	no	—
28	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₁₃₁	—	2	2	no	—
32	C₂×C₈×C₁₃₁	—	2	2	no	—
33	C₂×C₈×C₁₃₁	—	2	2	no	—
34	C₄²×C₁₃₁	D₄	1	2	yes	—
35	C₄²×C₁₃₁	Q₈	1	2	yes	—
36	C₂×Dic₁₃₁	—	2	2	no	—
37	C₂×Dic₁₃₁	—	2	2	no	—
38	C₂×Dic₁₃₁	—	2	2	no	—
39	C₂×Dic₁₃₁	—	2	2	no	—
40	C₂×Dic₁₃₁	—	2	2	no	—
41	C₂×Dic₁₃₁	M₁₆	1	4	yes	—
42	C₁₃₁⋊C₈	—	2	2	no	—
43	C₁₃₁⋊C₈	—	2	2	no	—
44	C₂³×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₃₁	—	2	2	no	—
46	C₂×C₄×C₁₃₁	—	2	2	no	—
47	C₂×C₄×C₁₃₁	C₄⋊C₄	1	2	yes	—
48	C₂×C₄×C₁₃₁	C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₁₃₁	#41	1	1	yes	characteristic
54	C₂×C₄×C₁₃₁	#41	1	1	yes	characteristic
55	C₈×C₁₃₁	—	4	2	no	—
56	C₂²×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₃₁	C₄×D₄	1	1	yes	characteristic
58	C₂²×C₁₃₁	C₄×Q₈	1	1	yes	characteristic
59	C₂²×C₁₃₁	C₂×M₁₆	1	2	yes	—
60	C₂²×C₁₃₁	#131	1	2	yes	—
61	C₄×C₁₃₁	—	2	2	no	—
62	C₄×C₁₃₁	—	2	2	no	—
63	C₄×C₁₃₁	#121	1	1	yes	characteristic
64	C₄×C₁₃₁	C₂²⋊Q₈	1	1	yes	characteristic
65	C₄×C₁₃₁	#125	1	1	yes	characteristic
66	C₄×C₁₃₁	C₄⋊Q₈	1	1	yes	characteristic
67	Dic₁₃₁	—	2	4	no	—
68	Dic₁₃₁	—	4	2	no	—
69	C₂×C₁₃₁	#389	1	1	yes	characteristic
70	C₂×C₁₃₁	#423	1	1	yes	characteristic
71	C₂×C₁₃₁	#424	1	1	yes	characteristic
72	C₂×C₁₃₁	#435	1	2	yes	—
73	C₂×C₁₃₁	#446	1	2	yes	—
74	C₁₃₁	#1940	1	1	yes	characteristic, complemented, Sylow
75	#1940	—	131	1	no	maximal, Sylow
76	C₂×C₄⋊C₈	—	131	2	no	—
77	C₂×C₄⋊C₈	D₁₃₁	1	1	yes	characteristic
78	C₂×#97	—	131	1	no	—
79	C₄×C₄⋊C₄	—	131	1	no	—
80	#336	—	131	2	no	—
81	C₂²×C₈	—	131	1	no	—
82	C₂²×C₈	—	131	1	no	—
83	C₂²×C₈	C₂×D₁₃₁	1	2	yes	—
84	C₂×C₄⋊C₄	—	131	2	no	—
85	C₂×C₄²	—	131	2	no	—
86	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
87	C₄⋊C₈	—	2	2	no	—
88	C₄⋊C₈	—	262	4	no	—
89	#97	—	262	2	no	—
90	#95	—	131	1	no	—
91	#95	—	131	1	no	—
92	C₂²×C₄	—	131	2	no	—
93	C₂²×C₄	—	131	2	no	—
94	C₂²×C₄	C₄×D₁₃₁	1	1	yes	characteristic
95	C₂²×C₄	C₄×D₁₃₁	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	262	1	no	—
101	C₂×C₈	—	262	1	no	—
102	C₂×C₈	—	262	1	no	—
103	C₂×C₈	—	262	1	no	—
104	C₂×C₈	—	131	2	no	—
105	C₂×C₈	—	131	2	no	—
106	C₄⋊C₄	—	262	4	no	—
107	C₄²	—	262	2	no	—
108	C₄²	—	262	2	no	—
109	C₄²	C₂²⋊D₁₃₁	1	2	yes	—
110	C₄²	Dic₂₆₂	1	2	yes	—
111	C₂³	C₂×C₄×D₁₃₁	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	262	2	no	—
115	C₂×C₄	—	262	2	no	—
116	C₂×C₄	—	262	2	no	—
117	C₂×C₄	—	262	2	no	—
118	C₂×C₄	—	262	2	no	—
119	C₂×C₄	—	131	4	no	—
120	C₂×C₄	Dic₁₃₁⋊C₄	1	2	yes	—
121	C₂×C₄	Dic₁₃₁⋊C₄	1	2	yes	—
122	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
123	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
124	C₂×C₄	Q₈×D₁₃₁	1	1	yes	characteristic
125	C₂×C₄	C₂×Dic₂₆₂	1	1	yes	characteristic
126	C₂×C₄	#14523512	1	1	yes	characteristic
127	C₂×C₄	#14523514	1	1	yes	characteristic
128	C₈	—	4	2	no	—
129	C₈	—	262	2	no	—
130	C₈	—	262	2	no	—
131	C₂²	C₂×Dic₁₃₁⋊C₄	1	1	yes	characteristic
132	C₂²	#16862208	1	1	yes	characteristic
133	C₂²	#16862211	1	1	yes	characteristic
134	C₂²	M₁₆×D₁₃₁	1	2	yes	—
135	C₂²	#16862252	1	2	yes	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	262	4	no	—
139	C₄	—	524	2	no	—
140	C₄	#16862217	1	1	yes	characteristic
141	C₄	#16862222	1	1	yes	characteristic
142	C₄	#16862237	1	1	yes	characteristic
143	C₄	#16862245	1	1	yes	characteristic
144	C₂	#22553208	1	1	yes	characteristic
145	C₂	#22553335	1	1	yes	characteristic
146	C₂	#22553338	1	1	yes	characteristic
147	C₂	#22553382	1	2	yes	—
148	C₂	#22553441	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₃₁⋊C₈

C₂

yes

maximal

C₂×#16862157

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₁₃₁

C₂

yes

maximal, characteristic

#22553066

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₃₁

C₂

yes

maximal, characteristic

C₂²×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₂²×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃₁

C₄

yes

—

C₂×C₄×Dic₁₃₁

C₂²

yes

—

Dic₁₃₁⋊C₈

—

#16862157

—

C₁₃₁⋊#95

C₄

yes

characteristic

C₁₃₁⋊#95

C₄

yes

characteristic

C₂²×C₈×C₁₃₁

C₂²

yes

—

C₂×C₄²×C₁₃₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₁

—

C₂²×Dic₁₃₁

C₂×C₄

yes

—

C₂²×Dic₁₃₁

C₂×C₄

yes

—

C₂×C₁₃₁⋊C₈

—

C₂×C₁₃₁⋊C₈

—

C₂×C₁₃₁⋊C₈

—

C₂×C₁₃₁⋊C₈

—

C₂×C₁₃₁⋊C₈

D₄

yes

—

C₂×C₁₃₁⋊C₈

Q₈

yes

—

C₄⋊Dic₁₃₁

—

C₄×Dic₁₃₁

—

C₄×Dic₁₃₁

—

C₂²×C₄×C₁₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₁

C₂³

yes

characteristic

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₄²×C₁₃₁

D₄

yes

—

C₄²×C₁₃₁

Q₈

yes

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

M₁₆

yes

—

C₁₃₁⋊C₈

—

C₁₃₁⋊C₈

—

C₂³×C₁₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₁

C₂×Q₈

yes

C₂×C₄×C₁₃₁

yes

C₂×C₄×C₁₃₁

yes

C₈×C₁₃₁

—

C₂²×C₁₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₃₁

C₄×D₄

yes

characteristic

C₂²×C₁₃₁

C₄×Q₈

yes

characteristic

C₂²×C₁₃₁

C₂×M₁₆

yes

—

C₂²×C₁₃₁

#131

yes

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

#121

yes

characteristic

C₄×C₁₃₁

C₂²⋊Q₈

yes

characteristic

C₄×C₁₃₁

#125

yes

characteristic

C₄×C₁₃₁

C₄⋊Q₈

yes

characteristic

Dic₁₃₁

—

Dic₁₃₁

—

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

—

C₂×C₁₃₁

#446

yes

—

C₁₃₁

#1940

yes

characteristic, complemented, Sylow

#1940

—

131

maximal, Sylow

C₂×C₄⋊C₈

—

131

—

C₂×C₄⋊C₈

D₁₃₁

yes

characteristic

C₂×#97

—

131

—

C₄×C₄⋊C₄

—

131

—

#336

—

131

—

C₂²×C₈

—

131

—

C₂²×C₈

—

131

—

C₂²×C₈

C₂×D₁₃₁

yes

—

C₂×C₄⋊C₄

—

131

—

C₂×C₄²

—

131

—

C₂×C₄²

C₂×D₁₃₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

262

—

#97

—

262

—

#95

—

131

—

#95

—

131

—

C₂²×C₄

—

131

—

C₂²×C₄

—

131

—

C₂²×C₄

C₄×D₁₃₁

yes

characteristic

C₂²×C₄

C₄×D₁₃₁

yes

characteristic

C₂²×C₄

C₂²×D₁₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

262

—

101

C₂×C₈

—

262

—

102

C₂×C₈

—

262

—

103

C₂×C₈

—

262

—

104

C₂×C₈

—

131

—

105

C₂×C₈

—

131

—

106

C₄⋊C₄

—

262

—

107

C₄²

—

262

—

108

C₄²

—

262

—

109

C₄²

C₂²⋊D₁₃₁

yes

—

110

C₄²

Dic₂₆₂

yes

—

111

C₂³

C₂×C₄×D₁₃₁

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

262

—

115

C₂×C₄

—

262

—

116

C₂×C₄

—

262

—

117

C₂×C₄

—

262

—

118

C₂×C₄

—

262

—

119

C₂×C₄

—

131

—

120

C₂×C₄

Dic₁₃₁⋊C₄

yes

—

121

C₂×C₄

Dic₁₃₁⋊C₄

yes

—

122

C₂×C₄

D₄×D₁₃₁

yes

characteristic

123

C₂×C₄

C₂×C₂²⋊D₁₃₁

yes

characteristic

124

C₂×C₄

Q₈×D₁₃₁

yes

characteristic

125

C₂×C₄

C₂×Dic₂₆₂

yes

characteristic

126

C₂×C₄

#14523512

yes

characteristic

127

C₂×C₄

#14523514

yes

characteristic

128

C₈

—

129

C₈

—

262

—

130

C₈

—

262

—

131

C₂²

C₂×Dic₁₃₁⋊C₄

yes

characteristic

132

C₂²

#16862208

yes

characteristic

133

C₂²

#16862211

yes

characteristic

134

C₂²

M₁₆×D₁₃₁

yes

—

135

C₂²

#16862252

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

262

—

139

C₄

—

524

—

140

C₄

#16862217

yes

characteristic

141

C₄

#16862222

yes

characteristic

142

C₄

#16862237

yes

characteristic

143

C₄

#16862245

yes

characteristic

144

C₂

#22553208

yes

characteristic

145

C₂

#22553335

yes

characteristic

146

C₂

#22553338

yes

characteristic

147

C₂

#22553382

yes

—

148

C₂

#22553441

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₃₁
commutator, socle, prime radical	C₂³×C₁₃₁	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₁₃₁	C₂
cosocle	C₂²×C₄×C₁₃₁	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₃₁
hypocenter	C₁₃₁	#1940
LCS	G ▷ C₂³×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₃₁

commutator, socle, prime radical

C₂³×C₁₃₁

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₁₃₁

C₂

cosocle

C₂²×C₄×C₁₃₁

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₃₁

C₁₃₁

G ▷ C₂³×C₁₃₁ ▷ C₁₃₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
74. C₁₃₁	#1940	semidirect

H ≅ G/N

type

74. C₁₃₁

#1940

semidirect

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

subgroup	index
C₂×Dic₁₃₁⋊C₈	2
C₂×#16862157	2
C₄×C₄⋊Dic₁₃₁	2
#22553066	2
C₂×C₄⋊C₈×C₁₃₁	2
#1940	131

subgroup

index

C₂×Dic₁₃₁⋊C₈

C₂×#16862157

C₄×C₄⋊Dic₁₃₁

#22553066

C₂×C₄⋊C₈×C₁₃₁

#1940

131

Subgroups — #30666682

Conjugacy classes of subgroups

Complements

Maximal subgroups