Group #30675055 — subgroups

Subgroups — #30675055

Order 16768.

322 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 161 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	#22553869	C₂	1	1	yes	maximal, characteristic
2	#22553442	C₂	1	2	yes	maximal, complemented
3	#22553441	C₂	1	4	yes	maximal, complemented
4	C₈×Dic₂₆₂	C₂	1	2	yes	maximal, complemented
5	#22553370	C₂	1	4	yes	maximal
6	#22553369	C₂	1	1	yes	maximal, characteristic
7	#431×C₁₃₁	C₂	1	1	yes	maximal, characteristic
8	#16862246	C₄	1	1	yes	characteristic
9	#16862246	C₄	1	1	yes	characteristic
10	#16862238	C₄	1	2	yes	—
11	#16862225	C₄	1	2	yes	—
12	#16862225	C₄	1	2	yes	—
13	C₄×Dic₂₆₂	C₂²	1	2	yes	—
14	C₄×Dic₂₆₂	C₂²	1	2	yes	—
15	#16862205	C₂²	1	2	yes	—
16	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
17	Dic₁₃₁⋊C₈	C₂²	1	4	yes	—
18	Dic₁₃₁⋊C₈	C₂²	1	4	yes	—
19	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
20	#16862157	C₂²	1	2	yes	—
21	#16862156	C₂²	1	4	yes	—
22	C₈×Dic₁₃₁	C₂²	1	4	yes	—
23	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
25	C₂²⋊C₈×C₁₃₁	C₂²	1	2	yes	—
26	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
28	C₂×Dic₂₆₂	C₂×C₄	1	2	yes	—
29	C₂×Dic₂₆₂	C₂×C₄	1	2	yes	—
30	C₂×C₁₃₁⋊C₈	C₂³	1	4	yes	—
31	C₄⋊Dic₁₃₁	C₂×C₄	1	2	yes	—
32	C₄⋊Dic₁₃₁	C₂×C₄	1	2	yes	—
33	C₄⋊Dic₁₃₁	C₂×C₄	1	2	yes	—
34	C₄⋊Dic₁₃₁	C₂×C₄	1	2	yes	—
35	Dic₁₃₁⋊C₄	C₂×C₄	1	4	yes	—
36	Dic₁₃₁⋊C₄	C₂×C₄	1	4	yes	—
37	C₂²⋊Dic₁₃₁	C₂×C₄	1	4	yes	—
38	C₄×Dic₁₃₁	C₂³	1	4	yes	—
39	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
42	C₂×C₈×C₁₃₁	C₂³	1	2	yes	—
43	C₂×C₈×C₁₃₁	C₂³	1	2	yes	—
44	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
45	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
47	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
48	Dic₂₆₂	—	2	4	no	—
49	Dic₂₆₂	—	4	2	no	—
50	C₂×Dic₁₃₁	C₂²×C₄	1	4	yes	—
51	C₂×Dic₁₃₁	C₂²×C₄	1	4	yes	—
52	C₁₃₁⋊C₈	—	2	4	no	—
53	C₂³×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃₁	—	2	2	no	—
55	C₂×C₄×C₁₃₁	—	2	2	no	—
56	C₂×C₄×C₁₃₁	—	4	1	no	—
57	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
63	C₈×C₁₃₁	—	2	2	no	—
64	C₈×C₁₃₁	—	2	2	no	—
65	C₂²×C₁₃₁	—	2	1	no	—
66	C₂²×C₁₃₁	—	2	1	no	—
67	C₂²×C₁₃₁	—	2	1	no	—
68	C₂²×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₁₃₁	—	2	1	no	—
70	C₄×C₁₃₁	—	2	1	no	—
71	C₄×C₁₃₁	—	2	1	no	—
72	C₄×C₁₃₁	—	2	1	no	—
73	C₄×C₁₃₁	#131	1	2	yes	—
74	C₄×C₁₃₁	2₋¹⁺⁴	1	2	yes	—
75	Dic₁₃₁	—	2	4	no	—
76	Dic₁₃₁	—	2	4	no	—
77	C₂×C₁₃₁	—	4	1	no	—
78	C₂×C₁₃₁	#519	1	1	yes	characteristic
79	C₂×C₁₃₁	C₂×#131	1	1	yes	characteristic
80	C₂×C₁₃₁	#568	1	1	yes	characteristic
81	C₁₃₁	#2985	1	1	yes	characteristic, complemented, Sylow
82	#2985	—	131	1	no	maximal, Sylow
83	#533	—	131	1	no	—
84	#446	—	131	4	no	—
85	#446	—	131	2	no	—
86	C₈×Q₈	—	131	2	no	—
87	#433	—	131	4	no	—
88	#432	—	131	1	no	—
89	#431	D₁₃₁	1	1	yes	characteristic
90	C₄⋊Q₈	—	131	1	no	—
91	C₄⋊Q₈	—	131	1	no	—
92	#125	—	131	2	no	—
93	C₂²⋊Q₈	—	131	2	no	—
94	C₂²⋊Q₈	—	131	2	no	—
95	C₄×Q₈	—	131	2	no	—
96	C₄×Q₈	—	131	2	no	—
97	#117	—	131	2	no	—
98	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
99	C₄⋊C₈	—	131	4	no	—
100	C₄⋊C₈	—	131	4	no	—
101	C₄⋊C₈	—	131	2	no	—
102	C₄⋊C₈	C₂×D₁₃₁	1	2	yes	—
103	C₂²⋊C₈	—	131	2	no	—
104	C₂²⋊C₈	C₂×D₁₃₁	1	2	yes	—
105	#97	—	131	4	no	—
106	#97	—	131	2	no	—
107	C₄×C₈	—	131	4	no	—
108	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
109	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
110	C₂×Q₈	—	131	2	no	—
111	C₂×Q₈	—	131	2	no	—
112	C₂²×C₄	C₄×D₁₃₁	1	1	yes	characteristic
113	C₂²×C₄	C₄×D₁₃₁	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
115	C₂×C₈	—	131	4	no	—
116	C₂×C₈	C₂²×D₁₃₁	1	2	yes	—
117	C₂×C₈	C₂²×D₁₃₁	1	2	yes	—
118	C₄⋊C₄	—	131	4	no	—
119	C₄⋊C₄	—	131	4	no	—
120	C₄⋊C₄	—	131	2	no	—
121	C₄⋊C₄	—	131	2	no	—
122	C₄⋊C₄	—	131	2	no	—
123	C₄⋊C₄	—	131	2	no	—
124	C₂²⋊C₄	—	131	4	no	—
125	C₄²	—	131	4	no	—
126	C₄²	C₄×D₁₃₁	1	1	yes	characteristic
127	C₄²	C₄×D₁₃₁	1	1	yes	characteristic
128	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
129	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
130	C₂³	C₂×C₄×D₁₃₁	1	1	yes	characteristic
131	Q₈	—	262	4	no	—
132	Q₈	—	524	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	131	4	no	—
137	C₂×C₄	—	131	4	no	—
138	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
143	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	—	2	2	no	—
146	C₈	—	262	4	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	C₂²×C₄×D₁₃₁	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	262	4	no	—
156	C₄	—	262	4	no	—
157	C₄	#16862252	1	2	yes	—
158	C₄	#16862293	1	2	yes	—
159	C₂	—	4	1	no	—
160	C₂	#22553789	1	1	yes	characteristic
161	C₂	C₂×#16862252	1	1	yes	characteristic
162	C₂	#22554078	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22553869

C₂

yes

maximal, characteristic

#22553442

C₂

yes

maximal, complemented

#22553441

C₂

yes

maximal, complemented

C₈×Dic₂₆₂

C₂

yes

maximal, complemented

#22553370

C₂

yes

maximal

#22553369

C₂

yes

maximal, characteristic

#431×C₁₃₁

C₂

yes

maximal, characteristic

#16862246

C₄

yes

characteristic

#16862246

C₄

yes

characteristic

#16862238

C₄

yes

—

#16862225

C₄

yes

—

#16862225

C₄

yes

—

C₄×Dic₂₆₂

C₂²

yes

—

C₄×Dic₂₆₂

C₂²

yes

—

#16862205

C₂²

yes

—

C₄⋊(C₁₃₁⋊C₈)

C₂²

yes

—

Dic₁₃₁⋊C₈

C₂²

yes

—

Dic₁₃₁⋊C₈

C₂²

yes

—

C₂²⋊(C₁₃₁⋊C₈)

C₂²

yes

—

#16862157

C₂²

yes

—

#16862156

C₂²

yes

—

C₈×Dic₁₃₁

C₂²

yes

—

C₂×C₄²×C₁₃₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₁

C₂²

yes

—

C₂²⋊C₈×C₁₃₁

C₂²

yes

—

C₄×C₈×C₁₃₁

C₂²

yes

characteristic

C₄×C₈×C₁₃₁

C₂²

yes

characteristic

C₂×Dic₂₆₂

C₂×C₄

yes

—

C₂×Dic₂₆₂

C₂×C₄

yes

—

C₂×C₁₃₁⋊C₈

C₂³

yes

—

C₄⋊Dic₁₃₁

C₂×C₄

yes

—

C₄⋊Dic₁₃₁

C₂×C₄

yes

—

C₄⋊Dic₁₃₁

C₂×C₄

yes

—

C₄⋊Dic₁₃₁

C₂×C₄

yes

—

Dic₁₃₁⋊C₄

C₂×C₄

yes

—

Dic₁₃₁⋊C₄

C₂×C₄

yes

—

C₂²⋊Dic₁₃₁

C₂×C₄

yes

—

C₄×Dic₁₃₁

C₂³

yes

—

C₂²×C₄×C₁₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₁

C₂³

yes

characteristic

C₂×C₈×C₁₃₁

C₂³

yes

—

C₂×C₈×C₁₃₁

C₂³

yes

—

C₄²×C₁₃₁

C₂×C₄

yes

characteristic

C₄²×C₁₃₁

C₂×C₄

yes

characteristic

C₄²×C₁₃₁

C₂³

yes

characteristic

C₄²×C₁₃₁

C₂³

yes

characteristic

Dic₂₆₂

—

Dic₂₆₂

—

C₂×Dic₁₃₁

C₂²×C₄

yes

—

C₂×Dic₁₃₁

C₂²×C₄

yes

—

C₁₃₁⋊C₈

—

C₂³×C₁₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂⁴

yes

characteristic

C₈×C₁₃₁

—

C₈×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

C₂³×C₄

yes

characteristic

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

#131

yes

—

C₄×C₁₃₁

2₋¹⁺⁴

yes

—

Dic₁₃₁

—

Dic₁₃₁

—

C₂×C₁₃₁

—

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₁₃₁

yes

characteristic, complemented, Sylow

#2985

—

131

maximal, Sylow

#533

—

131

—

#446

—

131

—

#446

—

131

—

C₈×Q₈

—

131

—

#433

—

131

—

#432

—

131

—

#431

D₁₃₁

yes

characteristic

C₄⋊Q₈

—

131

—

C₄⋊Q₈

—

131

—

#125

—

131

—

C₂²⋊Q₈

—

131

—

C₂²⋊Q₈

—

131

—

C₄×Q₈

—

131

—

C₄×Q₈

—

131

—

#117

—

131

—

C₂×C₄²

C₂×D₁₃₁

yes

characteristic

C₄⋊C₈

—

131

—

100

C₄⋊C₈

—

131

—

101

C₄⋊C₈

—

131

—

102

C₄⋊C₈

C₂×D₁₃₁

yes

—

103

C₂²⋊C₈

—

131

—

104

C₂²⋊C₈

C₂×D₁₃₁

yes

—

105

#97

—

131

—

106

#97

—

131

—

107

C₄×C₈

—

131

—

108

C₄×C₈

C₂×D₁₃₁

yes

characteristic

109

C₄×C₈

C₂×D₁₃₁

yes

characteristic

110

C₂×Q₈

—

131

—

111

C₂×Q₈

—

131

—

112

C₂²×C₄

C₄×D₁₃₁

yes

characteristic

113

C₂²×C₄

C₄×D₁₃₁

yes

characteristic

114

C₂²×C₄

C₂²×D₁₃₁

yes

characteristic

115

C₂×C₈

—

131

—

116

C₂×C₈

C₂²×D₁₃₁

yes

—

117

C₂×C₈

C₂²×D₁₃₁

yes

—

118

C₄⋊C₄

—

131

—

119

C₄⋊C₄

—

131

—

120

C₄⋊C₄

—

131

—

121

C₄⋊C₄

—

131

—

122

C₄⋊C₄

—

131

—

123

C₄⋊C₄

—

131

—

124

C₂²⋊C₄

—

131

—

125

C₄²

—

131

—

126

C₄²

C₄×D₁₃₁

yes

characteristic

127

C₄²

C₄×D₁₃₁

yes

characteristic

128

C₄²

C₂²×D₁₃₁

yes

characteristic

129

C₄²

C₂²×D₁₃₁

yes

characteristic

130

C₂³

C₂×C₄×D₁₃₁

yes

characteristic

131

Q₈

—

262

—

132

Q₈

—

524

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

131

—

137

C₂×C₄

—

131

—

138

C₂×C₄

C₂×C₄×D₁₃₁

yes

characteristic

139

C₂×C₄

C₂×C₄×D₁₃₁

yes

characteristic

140

C₂×C₄

C₂×C₄×D₁₃₁

yes

characteristic

141

C₂×C₄

C₂×C₄×D₁₃₁

yes

characteristic

142

C₂×C₄

C₂×C₄×D₁₃₁

yes

characteristic

143

C₂×C₄

C₂³×D₁₃₁

yes

characteristic

144

C₈

—

145

C₈

—

146

C₈

—

262

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

C₂²×C₄×D₁₃₁

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

262

—

156

C₄

—

262

—

157

C₄

#16862252

yes

—

158

C₄

#16862293

yes

—

159

C₂

—

160

C₂

#22553789

yes

characteristic

161

C₂

C₂×#16862252

yes

characteristic

162

C₂

#22554078

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₁
commutator, socle	C₂²×C₁₃₁	C₂³×C₄
Fitting	#431×C₁₃₁	C₂
prime radical	C₂³×C₁₃₁	C₂²×C₄
cosocle	C₂×C₄×C₁₃₁	C₂⁴
hypercenter	#431	D₁₃₁
hypocenter	C₁₃₁	#2985
LCS	G ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₃₁

commutator, socle

C₂²×C₁₃₁

C₂³×C₄

Fitting

#431×C₁₃₁

C₂

prime radical

C₂³×C₁₃₁

C₂²×C₄

cosocle

C₂×C₄×C₁₃₁

C₂⁴

D₁₃₁

C₁₃₁

G ▷ C₂²×C₁₃₁ ▷ C₁₃₁

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
2. #22553442	C₂	semidirect
3. #22553441	C₂	semidirect
4. C₈×Dic₂₆₂	C₂	semidirect
81. C₁₃₁	#2985	semidirect

H ≅ G/N

type

2. #22553442

C₂

semidirect

3. #22553441

C₂

semidirect

4. C₈×Dic₂₆₂

C₂

semidirect

81. C₁₃₁

#2985

semidirect

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

subgroup	index
#22553869	2
#22553442	2
#22553441	2
C₈×Dic₂₆₂	2
#22553370	2
#22553369	2
#431×C₁₃₁	2
#2985	131

subgroup

index

C₈×Dic₂₆₂

#431×C₁₃₁

131

Subgroups — #30675055

Conjugacy classes of subgroups

Complements

Maximal subgroups