Group #30665240 — subgroups

Subgroups — #30665240

Order 16768.

194 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 57 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	#22554041	C₂	1	1	yes	maximal, characteristic
2	#22553710	C₂	1	1	yes	maximal, characteristic
3	#22553598	C₂	1	1	yes	maximal, characteristic
4	#22553522	C₂	1	1	yes	maximal, characteristic
5	#22553059	C₂	1	1	yes	maximal, characteristic
6	#22553044	C₂	1	1	yes	maximal, characteristic
7	Q₈⋊C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#14523503	—	2	1	no	—
9	#16862246	—	2	1	no	—
10	#16862246	C₂²	1	1	yes	characteristic
11	#16862246	C₂²	1	1	yes	characteristic
12	#16862237	—	2	2	no	—
13	#16862237	—	2	1	no	—
14	Q₈×Dic₁₃₁	—	2	1	no	—
15	C₄×Dic₂₆₂	—	2	1	no	—
16	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic
17	#16862178	—	2	2	no	—
18	#16862178	—	2	1	no	—
19	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
20	#16862171	—	2	1	no	—
21	#16862169	—	2	1	no	—
22	C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
24	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
25	C₂×Dic₂₆₂	—	2	1	no	—
26	C₂×Dic₂₆₂	—	2	1	no	—
27	C₂×Dic₂₆₂	D₄	1	1	yes	characteristic
28	#14523503	—	4	2	no	—
29	C₂×C₁₃₁⋊C₈	—	2	1	no	—
30	C₄⋊Dic₁₃₁	—	2	2	no	—
31	C₄⋊Dic₁₃₁	—	2	2	no	—
32	C₄⋊Dic₁₃₁	—	2	1	no	—
33	C₄⋊Dic₁₃₁	—	2	1	no	—
34	C₄⋊Dic₁₃₁	—	2	1	no	—
35	C₄⋊Dic₁₃₁	D₄	1	1	yes	characteristic
36	C₄⋊Dic₁₃₁	D₄	1	2	yes	—
37	Dic₁₃₁⋊C₄	—	2	2	no	—
38	Dic₁₃₁⋊C₄	—	2	1	no	—
39	C₄×Dic₁₃₁	—	2	2	no	—
40	C₄×Dic₁₃₁	—	2	1	no	—
41	C₂×Q₈×C₁₃₁	D₄	1	1	yes	characteristic
42	C₂×C₈×C₁₃₁	—	2	1	no	—
43	C₂×C₈×C₁₃₁	—	2	1	no	—
44	C₄⋊C₄×C₁₃₁	—	2	1	no	—
45	C₄⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
46	C₄²×C₁₃₁	—	2	1	no	—
47	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
48	Dic₂₆₂	—	2	1	no	—
49	Dic₂₆₂	—	2	1	no	—
50	Dic₂₆₂	—	4	1	no	—
51	Dic₂₆₂	—	4	1	no	—
52	Dic₂₆₂	—	4	1	no	—
53	Dic₂₆₂	—	4	1	no	—
54	C₂×Dic₁₃₁	—	2	1	no	—
55	C₂×Dic₁₃₁	—	2	1	no	—
56	C₂×Dic₁₃₁	—	2	2	no	—
57	C₂×Dic₁₃₁	—	4	1	no	—
58	C₁₃₁⋊C₈	—	4	1	no	—
59	Q₈×C₁₃₁	—	2	1	no	—
60	Q₈×C₁₃₁	SD₁₆	1	2	yes	—
61	C₂×C₄×C₁₃₁	—	2	1	no	—
62	C₂×C₄×C₁₃₁	—	2	1	no	—
63	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
66	C₈×C₁₃₁	—	2	2	no	—
67	C₈×C₁₃₁	—	4	1	no	—
68	C₂²×C₁₃₁	#120	1	1	yes	characteristic
69	C₄×C₁₃₁	—	2	2	no	—
70	C₄×C₁₃₁	—	2	1	no	—
71	C₄×C₁₃₁	—	4	1	no	—
72	C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
73	C₄×C₁₃₁	#135	1	1	yes	characteristic
74	C₄×C₁₃₁	#137	1	1	yes	characteristic
75	C₄×C₁₃₁	#137	1	1	yes	characteristic
76	Dic₁₃₁	—	4	1	no	—
77	Dic₁₃₁	—	4	2	no	—
78	Dic₁₃₁	—	4	1	no	—
79	Dic₁₃₁	—	8	1	no	—
80	C₂×C₁₃₁	#448	1	1	yes	characteristic
81	C₂×C₁₃₁	#452	1	1	yes	characteristic
82	C₂×C₁₃₁	#456	1	1	yes	characteristic
83	C₁₃₁	#1653	1	1	yes	characteristic, complemented, Sylow
84	#1653	—	131	1	no	maximal, Sylow
85	#557	—	131	1	no	—
86	#498	—	131	1	no	—
87	#475	—	131	1	no	—
88	#462	—	131	1	no	—
89	#332	—	131	1	no	—
90	Q₈⋊C₈	—	131	1	no	—
91	Q₈⋊C₈	D₁₃₁	1	1	yes	characteristic
92	C₂×Q₁₆	—	262	1	no	—
93	C₄⋊Q₈	—	262	1	no	—
94	C₄⋊Q₈	—	131	1	no	—
95	C₄⋊Q₈	—	131	1	no	—
96	#125	—	262	2	no	—
97	#125	—	262	1	no	—
98	C₄×Q₈	—	262	1	no	—
99	C₄×Q₈	—	262	1	no	—
100	C₄×Q₈	—	131	1	no	—
101	C₄×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
102	#106	—	262	1	no	—
103	#106	—	262	2	no	—
104	C₄⋊C₈	—	131	1	no	—
105	C₄⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
106	#103	—	262	1	no	—
107	#103	—	262	1	no	—
108	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
109	C₂×Q₈	—	262	1	no	—
110	C₂×Q₈	—	262	1	no	—
111	C₂×Q₈	—	131	1	no	—
112	C₂×Q₈	D₅₂₄	1	1	yes	characteristic
113	Q₁₆	—	524	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	262	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	262	2	no	—
119	C₄⋊C₄	—	262	2	no	—
120	C₄⋊C₄	—	262	2	no	—
121	C₄⋊C₄	—	262	1	no	—
122	C₄⋊C₄	—	262	1	no	—
123	C₄⋊C₄	—	262	1	no	—
124	C₄⋊C₄	—	262	1	no	—
125	C₄⋊C₄	—	131	1	no	—
126	C₄⋊C₄	—	131	2	no	—
127	C₄⋊C₄	D₅₂₄	1	1	yes	characteristic
128	C₄²	—	2	1	no	—
129	C₄²	—	262	2	no	—
130	C₄²	—	262	1	no	—
131	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
132	Q₈	—	2	1	no	—
133	Q₈	—	262	1	no	—
134	Q₈	—	262	1	no	—
135	Q₈	—	524	1	no	—
136	Q₈	—	524	1	no	—
137	Q₈	—	524	1	no	—
138	Q₈	—	524	1	no	—
139	Q₈	#14523502	1	2	yes	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	262	1	no	—
143	C₂×C₄	—	262	1	no	—
144	C₂×C₄	—	262	2	no	—
145	C₂×C₄	—	524	1	no	—
146	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
147	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
148	C₂×C₄	C₂×D₅₂₄	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	4	1	no	—
151	C₈	—	524	1	no	—
152	C₂²	#16862216	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	1	no	—
155	C₄	—	4	1	no	—
156	C₄	—	524	1	no	—
157	C₄	—	524	2	no	—
158	C₄	—	524	1	no	—
159	C₄	—	1048	1	no	—
160	C₄	C₂×#14523502	1	1	yes	characteristic
161	C₄	#16862267	1	1	yes	characteristic
162	C₄	#16862278	1	1	yes	characteristic
163	C₄	#16862279	1	1	yes	characteristic
164	C₂	#22553453	1	1	yes	characteristic
165	C₂	#22553477	1	1	yes	characteristic
166	C₂	#22553496	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22554041

C₂

yes

maximal, characteristic

#22553710

C₂

yes

maximal, characteristic

#22553598

C₂

yes

maximal, characteristic

#22553522

C₂

yes

maximal, characteristic

#22553059

C₂

yes

maximal, characteristic

#22553044

C₂

yes

maximal, characteristic

Q₈⋊C₈×C₁₃₁

C₂

yes

maximal, characteristic

C₂×#14523503

—

#16862246

—

#16862246

C₂²

yes

characteristic

#16862246

C₂²

yes

characteristic

#16862237

—

#16862237

—

Q₈×Dic₁₃₁

—

C₄×Dic₂₆₂

—

C₄×Dic₂₆₂

C₂²

yes

characteristic

#16862178

—

#16862178

—

C₄⋊(C₁₃₁⋊C₈)

C₂²

yes

characteristic

#16862171

—

#16862169

—

C₄×Q₈×C₁₃₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₁

C₂²

yes

characteristic

C₄×C₈×C₁₃₁

C₂²

yes

characteristic

C₂×Dic₂₆₂

—

C₂×Dic₂₆₂

—

C₂×Dic₂₆₂

D₄

yes

characteristic

#14523503

—

C₂×C₁₃₁⋊C₈

—

C₄⋊Dic₁₃₁

—

C₄⋊Dic₁₃₁

—

C₄⋊Dic₁₃₁

—

C₄⋊Dic₁₃₁

—

C₄⋊Dic₁₃₁

—

C₄⋊Dic₁₃₁

D₄

yes

characteristic

C₄⋊Dic₁₃₁

D₄

yes

—

Dic₁₃₁⋊C₄

—

Dic₁₃₁⋊C₄

—

C₄×Dic₁₃₁

—

C₄×Dic₁₃₁

—

C₂×Q₈×C₁₃₁

D₄

yes

characteristic

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

D₄

yes

characteristic

C₄²×C₁₃₁

—

C₄²×C₁₃₁

C₂³

yes

characteristic

Dic₂₆₂

—

Dic₂₆₂

—

Dic₂₆₂

—

Dic₂₆₂

—

Dic₂₆₂

—

Dic₂₆₂

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₁₃₁⋊C₈

—

Q₈×C₁₃₁

—

Q₈×C₁₃₁

SD₁₆

yes

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₈×C₁₃₁

—

C₈×C₁₃₁

—

C₂²×C₁₃₁

#120

yes

characteristic

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

C₂×SD₁₆

yes

C₄×C₁₃₁

yes

C₄×C₁₃₁

yes

C₄×C₁₃₁

yes

Dic₁₃₁

—

Dic₁₃₁

—

Dic₁₃₁

—

Dic₁₃₁

—

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₁₃₁

yes

characteristic, complemented, Sylow

#1653

—

131

maximal, Sylow

#557

—

131

—

#498

—

131

—

#475

—

131

—

#462

—

131

—

#332

—

131

—

Q₈⋊C₈

—

131

—

Q₈⋊C₈

D₁₃₁

yes

characteristic

C₂×Q₁₆

—

262

—

C₄⋊Q₈

—

262

—

C₄⋊Q₈

—

131

—

C₄⋊Q₈

—

131

—

#125

—

262

—

#125

—

262

—

C₄×Q₈

—

262

—

C₄×Q₈

—

262

—

100

C₄×Q₈

—

131

—

101

C₄×Q₈

C₂×D₁₃₁

yes

characteristic

102

#106

—

262

—

103

#106

—

262

—

104

C₄⋊C₈

—

131

—

105

C₄⋊C₈

C₂×D₁₃₁

yes

characteristic

106

#103

—

262

—

107

#103

—

262

—

108

C₄×C₈

C₂×D₁₃₁

yes

characteristic

109

C₂×Q₈

—

262

—

110

C₂×Q₈

—

262

—

111

C₂×Q₈

—

131

—

112

C₂×Q₈

D₅₂₄

yes

characteristic

113

Q₁₆

—

524

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

262

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

262

—

119

C₄⋊C₄

—

262

—

120

C₄⋊C₄

—

262

—

121

C₄⋊C₄

—

262

—

122

C₄⋊C₄

—

262

—

123

C₄⋊C₄

—

262

—

124

C₄⋊C₄

—

262

—

125

C₄⋊C₄

—

131

—

126

C₄⋊C₄

—

131

—

127

C₄⋊C₄

D₅₂₄

yes

characteristic

128

C₄²

—

129

C₄²

—

262

—

130

C₄²

—

262

—

131

C₄²

C₂²×D₁₃₁

yes

characteristic

132

Q₈

—

133

Q₈

—

262

—

134

Q₈

—

262

—

135

Q₈

—

524

—

136

Q₈

—

524

—

137

Q₈

—

524

—

138

Q₈

—

524

—

139

Q₈

#14523502

yes

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

262

—

143

C₂×C₄

—

262

—

144

C₂×C₄

—

262

—

145

C₂×C₄

—

524

—

146

C₂×C₄

D₄×D₁₃₁

yes

characteristic

147

C₂×C₄

D₄×D₁₃₁

yes

characteristic

148

C₂×C₄

C₂×D₅₂₄

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

524

—

152

C₂²

#16862216

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

524

—

157

C₄

—

524

—

158

C₄

—

524

—

159

C₄

—

1048

—

160

C₄

C₂×#14523502

yes

characteristic

161

C₄

#16862267

yes

characteristic

162

C₄

#16862278

yes

characteristic

163

C₄

#16862279

yes

characteristic

164

C₂

#22553453

yes

characteristic

165

C₂

#22553477

yes

characteristic

166

C₂

#22553496

yes

characteristic

	subgroup	G/H
center	C₂²	#16862216
commutator, cosocle	C₄²×C₁₃₁	C₂³
Frattini	C₄²	C₂²×D₁₃₁
Fitting	Q₈⋊C₈×C₁₃₁	C₂
socle, prime radical	C₂²×C₁₃₁	#120
hypercenter	Q₈⋊C₈	D₁₃₁
hypocenter	C₁₃₁	#1653
LCS	G ▷ C₄²×C₁₃₁ ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

subgroup

G/H

center

C₂²

#16862216

commutator, cosocle

C₄²×C₁₃₁

C₂³

Frattini

C₄²

C₂²×D₁₃₁

Fitting

Q₈⋊C₈×C₁₃₁

C₂

socle, prime radical

C₂²×C₁₃₁

#120

hypercenter

Q₈⋊C₈

D₁₃₁

hypocenter

C₁₃₁

#1653

LCS

G ▷ C₄²×C₁₃₁ ▷ C₂²×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

N	H ≅ G/N	type
83. C₁₃₁	#1653	semidirect

H ≅ G/N

type

83. C₁₃₁

#1653

semidirect

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

subgroup	index
#22554041	2
#22553710	2
#22553598	2
#22553522	2
#22553059	2
#22553044	2
Q₈⋊C₈×C₁₃₁	2
#1653	131

subgroup

index

Q₈⋊C₈×C₁₃₁

131

Subgroups — #30665240

Conjugacy classes of subgroups

Complements

Maximal subgroups