Group #30665118 — subgroups

Subgroups — #30665118

Order 16768.

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

normal subgroups only

This group has 188 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	#22553662	C₂	1	1	yes	maximal, characteristic
3	#22553604	C₂	1	1	yes	maximal, characteristic
4	#22553057	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊(C₁₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#22553042	C₂	1	1	yes	maximal, characteristic
7	#477×C₁₃₁	C₂	1	1	yes	maximal, characteristic
8	#16862246	—	2	1	no	—
9	#16862246	C₂²	1	1	yes	characteristic
10	#16862237	—	2	2	no	—
11	#16862237	—	2	1	no	—
12	Q₈×Dic₁₃₁	—	2	1	no	—
13	C₄×Dic₂₆₂	—	2	1	no	—
14	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic
15	#16862179	—	2	1	no	—
16	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
17	#16862170	—	2	2	no	—
18	#16862169	—	2	1	no	—
19	C₄×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
22	#107×C₁₃₁	—	2	1	no	—
23	C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
24	#103×C₁₃₁	—	2	1	no	—
25	C₂×Dic₂₆₂	—	2	1	no	—
26	C₂×Dic₂₆₂	D₄	1	1	yes	characteristic
27	C₂×C₁₃₁⋊C₈	—	2	1	no	—
28	C₂×C₁₃₁⋊C₈	—	2	1	no	—
29	C₄⋊Dic₁₃₁	—	2	2	no	—
30	C₄⋊Dic₁₃₁	—	2	2	no	—
31	C₄⋊Dic₁₃₁	—	2	1	no	—
32	C₄⋊Dic₁₃₁	D₄	1	1	yes	characteristic
33	C₄⋊Dic₁₃₁	D₄	1	2	yes	—
34	Dic₁₃₁⋊C₄	—	2	2	no	—
35	Dic₁₃₁⋊C₄	—	2	1	no	—
36	C₄×Dic₁₃₁	—	2	2	no	—
37	C₄×Dic₁₃₁	—	2	1	no	—
38	C₂×Q₈×C₁₃₁	—	2	1	no	—
39	C₂×Q₈×C₁₃₁	D₄	1	1	yes	characteristic
40	C₂×C₈×C₁₃₁	—	2	1	no	—
41	C₄⋊C₄×C₁₃₁	—	2	1	no	—
42	C₄⋊C₄×C₁₃₁	—	2	1	no	—
43	C₄⋊C₄×C₁₃₁	—	2	1	no	—
44	C₄⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
45	C₄²×C₁₃₁	—	2	1	no	—
46	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
47	Dic₂₆₂	—	2	1	no	—
48	Dic₂₆₂	—	2	1	no	—
49	Dic₂₆₂	—	4	1	no	—
50	Dic₂₆₂	—	4	1	no	—
51	C₂×Dic₁₃₁	—	2	1	no	—
52	C₂×Dic₁₃₁	—	2	1	no	—
53	C₂×Dic₁₃₁	—	2	2	no	—
54	C₁₃₁⋊C₈	—	4	1	no	—
55	C₁₃₁⋊C₈	—	4	1	no	—
56	Q₈×C₁₃₁	—	2	1	no	—
57	Q₈×C₁₃₁	—	4	2	no	—
58	Q₈×C₁₃₁	Q₁₆	1	2	yes	—
59	C₂×C₄×C₁₃₁	—	2	1	no	—
60	C₂×C₄×C₁₃₁	—	2	1	no	—
61	C₂×C₄×C₁₃₁	—	4	1	no	—
62	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
65	C₈×C₁₃₁	—	4	1	no	—
66	C₂²×C₁₃₁	#120	1	1	yes	characteristic
67	C₄×C₁₃₁	—	2	2	no	—
68	C₄×C₁₃₁	—	2	1	no	—
69	C₄×C₁₃₁	—	4	1	no	—
70	C₄×C₁₃₁	—	8	1	no	—
71	C₄×C₁₃₁	C₂×Q₁₆	1	1	yes	characteristic
72	C₄×C₁₃₁	#135	1	1	yes	characteristic
73	C₄×C₁₃₁	#137	1	1	yes	characteristic
74	C₄×C₁₃₁	#137	1	1	yes	characteristic
75	Dic₁₃₁	—	4	1	no	—
76	Dic₁₃₁	—	4	2	no	—
77	Dic₁₃₁	—	4	1	no	—
78	C₂×C₁₃₁	#451	1	1	yes	characteristic
79	C₂×C₁₃₁	#452	1	1	yes	characteristic
80	C₂×C₁₃₁	#454	1	1	yes	characteristic
81	C₁₃₁	#1636	1	1	yes	characteristic, complemented, Sylow
82	#1636	—	131	1	no	maximal, Sylow
83	#557	—	131	1	no	—
84	#487	—	131	1	no	—
85	#477	—	131	1	no	—
86	#477	D₁₃₁	1	1	yes	characteristic
87	#332	—	131	1	no	—
88	Q₈⋊C₈	—	131	1	no	—
89	Q₈⋊C₈	—	131	1	no	—
90	C₄⋊Q₈	—	262	1	no	—
91	C₄⋊Q₈	—	131	1	no	—
92	C₄⋊Q₈	C₂×D₁₃₁	1	1	yes	characteristic
93	#125	—	262	2	no	—
94	#125	—	262	1	no	—
95	C₄×Q₈	—	262	1	no	—
96	C₄×Q₈	—	262	1	no	—
97	C₄×Q₈	—	131	1	no	—
98	C₄×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
99	#107	—	2	1	no	—
100	#107	—	262	1	no	—
101	C₄⋊C₈	—	131	1	no	—
102	C₄⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
103	#103	—	2	1	no	—
104	#103	—	262	1	no	—
105	#103	—	262	2	no	—
106	C₄×C₈	—	131	1	no	—
107	C₂×Q₈	—	2	1	no	—
108	C₂×Q₈	—	262	1	no	—
109	C₂×Q₈	—	131	1	no	—
110	C₂×Q₈	C₂²⋊D₁₃₁	1	1	yes	characteristic
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	262	1	no	—
113	C₂×C₈	—	262	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	262	2	no	—
118	C₄⋊C₄	—	262	2	no	—
119	C₄⋊C₄	—	262	2	no	—
120	C₄⋊C₄	—	262	1	no	—
121	C₄⋊C₄	—	262	1	no	—
122	C₄⋊C₄	—	131	2	no	—
123	C₄⋊C₄	—	131	1	no	—
124	C₄⋊C₄	C₂²⋊D₁₃₁	1	1	yes	characteristic
125	C₄²	—	2	1	no	—
126	C₄²	—	262	2	no	—
127	C₄²	—	262	1	no	—
128	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
129	Q₈	—	2	1	no	—
130	Q₈	—	4	2	no	—
131	Q₈	—	262	1	no	—
132	Q₈	—	262	1	no	—
133	Q₈	—	524	1	no	—
134	Q₈	—	524	1	no	—
135	Q₈	#14523503	1	2	yes	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	262	1	no	—
140	C₂×C₄	—	262	1	no	—
141	C₂×C₄	—	262	2	no	—
142	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
143	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
145	C₈	—	4	1	no	—
146	C₈	—	524	1	no	—
147	C₈	—	524	1	no	—
148	C₂²	#16862214	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	1	no	—
151	C₄	—	4	1	no	—
152	C₄	—	8	1	no	—
153	C₄	—	524	1	no	—
154	C₄	—	524	2	no	—
155	C₄	—	524	1	no	—
156	C₄	C₂×#14523503	1	1	yes	characteristic
157	C₄	#16862269	1	1	yes	characteristic
158	C₄	#16862276	1	1	yes	characteristic
159	C₄	#16862278	1	1	yes	characteristic
160	C₂	#22553467	1	1	yes	characteristic
161	C₂	#22553473	1	1	yes	characteristic
162	C₂	#22553483	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22554041

C₂

yes

maximal, characteristic

#22553662

C₂

yes

maximal, characteristic

#22553604

C₂

yes

maximal, characteristic

#22553057

C₂

yes

maximal, characteristic

Q₈⋊(C₁₃₁⋊C₈)

C₂

yes

maximal, characteristic

#22553042

C₂

yes

maximal, characteristic

#477×C₁₃₁

C₂

yes

maximal, characteristic

#16862246

—

#16862246

C₂²

yes

characteristic

#16862237

—

#16862237

—

Q₈×Dic₁₃₁

—

C₄×Dic₂₆₂

—

C₄×Dic₂₆₂

C₂²

yes

characteristic

#16862179

—

C₄⋊(C₁₃₁⋊C₈)

C₂²

yes

characteristic

#16862170

—

#16862169

—

C₄×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₃₁

C₂²

yes

characteristic

C₄×Q₈×C₁₃₁

C₂²

yes

characteristic

#107×C₁₃₁

—

C₄⋊C₈×C₁₃₁

C₂²

yes

characteristic

#103×C₁₃₁

—

C₂×Dic₂₆₂

—

C₂×Dic₂₆₂

D₄

yes

characteristic

C₂×C₁₃₁⋊C₈

—

C₂×C₁₃₁⋊C₈

—

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

—

C₂×Q₈×C₁₃₁

D₄

yes

characteristic

C₂×C₈×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

D₄

yes

characteristic

C₄²×C₁₃₁

—

C₄²×C₁₃₁

C₂³

yes

characteristic

Dic₂₆₂

—

Dic₂₆₂

—

Dic₂₆₂

—

Dic₂₆₂

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₁₃₁⋊C₈

—

C₁₃₁⋊C₈

—

Q₈×C₁₃₁

—

Q₈×C₁₃₁

—

Q₈×C₁₃₁

Q₁₆

yes

—

C₂×C₄×C₁₃₁

—

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

—

C₂²×C₁₃₁

#120

yes

characteristic

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

C₂×Q₁₆

yes

C₄×C₁₃₁

yes

C₄×C₁₃₁

yes

C₄×C₁₃₁

yes

Dic₁₃₁

—

Dic₁₃₁

—

Dic₁₃₁

—

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₁₃₁

yes

characteristic, complemented, Sylow

#1636

—

131

maximal, Sylow

#557

—

131

—

#487

—

131

—

#477

—

131

—

#477

D₁₃₁

yes

characteristic

#332

—

131

—

Q₈⋊C₈

—

131

—

Q₈⋊C₈

—

131

—

C₄⋊Q₈

—

262

—

C₄⋊Q₈

—

131

—

C₄⋊Q₈

C₂×D₁₃₁

yes

characteristic

#125

—

262

—

#125

—

262

—

C₄×Q₈

—

262

—

C₄×Q₈

—

262

—

C₄×Q₈

—

131

—

C₄×Q₈

C₂×D₁₃₁

yes

characteristic

#107

—

100

#107

—

262

—

101

C₄⋊C₈

—

131

—

102

C₄⋊C₈

C₂×D₁₃₁

yes

characteristic

103

#103

—

104

#103

—

262

—

105

#103

—

262

—

106

C₄×C₈

—

131

—

107

C₂×Q₈

—

108

C₂×Q₈

—

262

—

109

C₂×Q₈

—

131

—

110

C₂×Q₈

C₂²⋊D₁₃₁

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

262

—

113

C₂×C₈

—

262

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

262

—

118

C₄⋊C₄

—

262

—

119

C₄⋊C₄

—

262

—

120

C₄⋊C₄

—

262

—

121

C₄⋊C₄

—

262

—

122

C₄⋊C₄

—

131

—

123

C₄⋊C₄

—

131

—

124

C₄⋊C₄

C₂²⋊D₁₃₁

yes

characteristic

125

C₄²

—

126

C₄²

—

262

—

127

C₄²

—

262

—

128

C₄²

C₂²×D₁₃₁

yes

characteristic

129

Q₈

—

130

Q₈

—

131

Q₈

—

262

—

132

Q₈

—

262

—

133

Q₈

—

524

—

134

Q₈

—

524

—

135

Q₈

#14523503

yes

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

262

—

140

C₂×C₄

—

262

—

141

C₂×C₄

—

262

—

142

C₂×C₄

D₄×D₁₃₁

yes

characteristic

143

C₂×C₄

D₄×D₁₃₁

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₁₃₁

yes

characteristic

145

C₈

—

146

C₈

—

524

—

147

C₈

—

524

—

148

C₂²

#16862214

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

524

—

154

C₄

—

524

—

155

C₄

—

524

—

156

C₄

C₂×#14523503

yes

characteristic

157

C₄

#16862269

yes

characteristic

158

C₄

#16862276

yes

characteristic

159

C₄

#16862278

yes

characteristic

160

C₂

#22553467

yes

characteristic

161

C₂

#22553473

yes

characteristic

162

C₂

#22553483

yes

characteristic

	subgroup	G/H
center	C₂²	#16862214
commutator, cosocle	C₄²×C₁₃₁	C₂³
Frattini	C₄²	C₂²×D₁₃₁
Fitting	#477×C₁₃₁	C₂
socle, prime radical	C₂²×C₁₃₁	#120
hypercenter	#477	D₁₃₁
hypocenter	C₁₃₁	#1636
LCS	G ▷ C₄²×C₁₃₁ ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

subgroup

G/H

center

C₂²

#16862214

commutator, cosocle

C₄²×C₁₃₁

C₂³

Frattini

C₄²

C₂²×D₁₃₁

Fitting

#477×C₁₃₁

C₂

C₂²×C₁₃₁

D₁₃₁

C₁₃₁

G ▷ C₄²×C₁₃₁ ▷ C₂²×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #477

N	H ≅ G/N	type
81. C₁₃₁	#1636	semidirect

H ≅ G/N

type

81. C₁₃₁

#1636

semidirect

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

subgroup	index
#22554041	2
#22553662	2
#22553604	2
#22553057	2
Q₈⋊(C₁₃₁⋊C₈)	2
#22553042	2
#477×C₁₃₁	2
#1636	131

subgroup

index

Q₈⋊(C₁₃₁⋊C₈)

#477×C₁₃₁

131

Subgroups — #30665118

Conjugacy classes of subgroups

Complements

Maximal subgroups