Group #30665201 — subgroups

Subgroups — #30665201

Order 16768.

194 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 55 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	#22554040	C₂	1	1	yes	maximal, characteristic
2	#22553715	C₂	1	1	yes	maximal, characteristic
3	#22553594	C₂	1	1	yes	maximal, characteristic
4	#22553059	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	#462×C₁₃₁	C₂	1	1	yes	maximal, characteristic
8	#16862246	C₂²	1	1	yes	characteristic
9	#16862244	—	2	1	no	—
10	#16862238	—	2	1	no	—
11	#16862236	—	2	2	no	—
12	C₄×Dic₂₆₂	—	2	1	no	—
13	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic
14	#16862211	—	2	1	no	—
15	#16862179	—	2	2	no	—
16	#16862177	—	2	1	no	—
17	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
18	#16862169	—	2	1	no	—
19	C₄×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×Q₁₆×C₁₃₁	—	2	1	no	—
21	C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
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	1	no	—
30	C₄⋊Dic₁₃₁	D₄	1	1	yes	characteristic
31	C₄⋊Dic₁₃₁	D₄	1	2	yes	—
32	Dic₁₃₁⋊C₄	—	2	2	no	—
33	Dic₁₃₁⋊C₄	—	2	2	no	—
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	Q₁₆×C₁₃₁	—	4	2	no	—
41	C₂×C₈×C₁₃₁	—	2	1	no	—
42	C₄⋊C₄×C₁₃₁	—	2	1	no	—
43	C₄⋊C₄×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	—	4	1	no	—
50	Dic₂₆₂	—	4	1	no	—
51	Dic₂₆₂	Q₁₆	1	2	yes	complemented
52	C₂×Dic₁₃₁	—	2	1	no	—
53	C₂×Dic₁₃₁	—	2	1	no	—
54	C₂×Dic₁₃₁	—	2	2	no	—
55	C₁₃₁⋊C₈	—	2	2	no	—
56	C₁₃₁⋊C₈	—	4	1	no	—
57	Q₈×C₁₃₁	—	2	1	no	—
58	Q₈×C₁₃₁	—	2	1	no	—
59	Q₈×C₁₃₁	—	4	1	no	—
60	Q₈×C₁₃₁	—	4	1	no	—
61	C₂×C₄×C₁₃₁	—	2	1	no	—
62	C₂×C₄×C₁₃₁	—	2	1	no	—
63	C₂×C₄×C₁₃₁	—	4	1	no	—
64	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
67	C₈×C₁₃₁	—	4	1	no	—
68	C₂²×C₁₃₁	#120	1	1	yes	characteristic
69	C₄×C₁₃₁	—	2	1	no	—
70	C₄×C₁₃₁	—	4	1	no	—
71	C₄×C₁₃₁	—	4	1	no	—
72	C₄×C₁₃₁	—	8	1	no	—
73	C₄×C₁₃₁	C₂×Q₁₆	1	1	yes	characteristic
74	C₄×C₁₃₁	#135	1	1	yes	characteristic
75	C₄×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₄×C₁₃₁	#137	1	1	yes	characteristic
77	Dic₁₃₁	—	2	2	no	—
78	Dic₁₃₁	—	4	2	no	—
79	Dic₁₃₁	—	4	1	no	—
80	C₂×C₁₃₁	#449	1	1	yes	characteristic
81	C₂×C₁₃₁	#451	1	1	yes	characteristic
82	C₂×C₁₃₁	#456	1	1	yes	characteristic
83	C₁₃₁	#1648	1	1	yes	characteristic, complemented, Sylow
84	#1648	—	131	1	no	maximal, Sylow
85	#557	—	131	1	no	—
86	#500	—	131	1	no	—
87	#475	—	131	1	no	—
88	#462	D₁₃₁	1	1	yes	characteristic
89	#332	—	131	1	no	—
90	Q₈⋊C₈	—	131	1	no	—
91	Q₈⋊C₈	—	131	1	no	—
92	C₂×Q₁₆	—	2	1	no	—
93	C₄⋊Q₈	—	262	1	no	—
94	C₄⋊Q₈	—	131	1	no	—
95	C₄⋊Q₈	C₂×D₁₃₁	1	1	yes	characteristic
96	#125	—	262	1	no	—
97	#125	—	262	2	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	#107	—	262	2	no	—
103	#106	—	262	1	no	—
104	C₄⋊C₈	—	131	1	no	—
105	C₄⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
106	#103	—	2	1	no	—
107	#103	—	262	1	no	—
108	C₄×C₈	—	131	1	no	—
109	C₂×Q₈	—	2	1	no	—
110	C₂×Q₈	—	262	1	no	—
111	C₂×Q₈	—	131	1	no	—
112	C₂×Q₈	C₂²⋊D₁₃₁	1	1	yes	characteristic
113	Q₁₆	—	4	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	262	1	no	—
116	C₂×C₈	—	262	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	262	2	no	—
121	C₄⋊C₄	—	262	2	no	—
122	C₄⋊C₄	—	262	2	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	C₂²⋊D₁₃₁	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	—	2	1	no	—
134	Q₈	—	4	1	no	—
135	Q₈	—	4	1	no	—
136	Q₈	—	262	1	no	—
137	Q₈	—	524	1	no	—
138	Q₈	—	524	1	no	—
139	Q₈	—	131	2	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	262	1	no	—
144	C₂×C₄	—	262	1	no	—
145	C₂×C₄	—	262	2	no	—
146	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
147	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
148	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	262	2	no	—
151	C₈	—	524	1	no	—
152	C₂²	#16862214	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	4	1	no	—
156	C₄	—	262	2	no	—
157	C₄	—	8	1	no	—
158	C₄	—	524	2	no	—
159	C₄	—	524	1	no	—
160	C₄	Q₁₆×D₁₃₁	1	1	yes	characteristic
161	C₄	#16862266	1	1	yes	characteristic
162	C₄	#16862272	1	1	yes	characteristic
163	C₄	#16862278	1	1	yes	characteristic
164	C₂	#22553455	1	1	yes	characteristic
165	C₂	#22553466	1	1	yes	characteristic
166	C₂	#22553493	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22554040

C₂

yes

maximal, characteristic

#22553715

C₂

yes

maximal, characteristic

#22553594

C₂

yes

maximal, characteristic

#22553059

C₂

yes

maximal, characteristic

Q₈⋊(C₁₃₁⋊C₈)

C₂

yes

maximal, characteristic

#22553042

C₂

yes

maximal, characteristic

#462×C₁₃₁

C₂

yes

maximal, characteristic

#16862246

C₂²

yes

—

—

—

C₄×Dic₂₆₂

—

C₄×Dic₂₆₂

C₂²

yes

—

—

—

C₄⋊(C₁₃₁⋊C₈)

C₂²

yes

characteristic

#16862169

—

C₄×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₂×Q₁₆×C₁₃₁

—

C₄⋊Q₈×C₁₃₁

C₂²

yes

characteristic

C₄×Q₈×C₁₃₁

C₂²

yes

characteristic

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

D₄

yes

characteristic

C₄⋊Dic₁₃₁

D₄

yes

—

Dic₁₃₁⋊C₄

—

Dic₁₃₁⋊C₄

—

Dic₁₃₁⋊C₄

—

Dic₁₃₁⋊C₄

—

C₄×Dic₁₃₁

—

C₄×Dic₁₃₁

—

C₂×Q₈×C₁₃₁

—

C₂×Q₈×C₁₃₁

D₄

yes

characteristic

Q₁₆×C₁₃₁

—

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

Q₁₆

yes

complemented

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₁₃₁⋊C₈

—

C₁₃₁⋊C₈

—

Q₈×C₁₃₁

—

Q₈×C₁₃₁

—

Q₈×C₁₃₁

—

Q₈×C₁₃₁

—

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

#1648

—

131

maximal, Sylow

#557

—

131

—

#500

—

131

—

#475

—

131

—

#462

D₁₃₁

yes

characteristic

#332

—

131

—

Q₈⋊C₈

—

131

—

Q₈⋊C₈

—

131

—

C₂×Q₁₆

—

C₄⋊Q₈

—

262

—

C₄⋊Q₈

—

131

—

C₄⋊Q₈

C₂×D₁₃₁

yes

characteristic

#125

—

262

—

#125

—

262

—

C₄×Q₈

—

262

—

C₄×Q₈

—

262

—

100

C₄×Q₈

—

131

—

101

C₄×Q₈

C₂×D₁₃₁

yes

characteristic

102

#107

—

262

—

103

#106

—

262

—

104

C₄⋊C₈

—

131

—

105

C₄⋊C₈

C₂×D₁₃₁

yes

characteristic

106

#103

—

107

#103

—

262

—

108

C₄×C₈

—

131

—

109

C₂×Q₈

—

110

C₂×Q₈

—

262

—

111

C₂×Q₈

—

131

—

112

C₂×Q₈

C₂²⋊D₁₃₁

yes

characteristic

113

Q₁₆

—

114

C₂×C₈

—

115

C₂×C₈

—

262

—

116

C₂×C₈

—

262

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

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

C₂²⋊D₁₃₁

yes

characteristic

128

C₄²

—

129

C₄²

—

262

—

130

C₄²

—

262

—

131

C₄²

C₂²×D₁₃₁

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

Q₈

—

262

—

137

Q₈

—

524

—

138

Q₈

—

524

—

139

Q₈

—

131

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

262

—

144

C₂×C₄

—

262

—

145

C₂×C₄

—

262

—

146

C₂×C₄

D₄×D₁₃₁

yes

characteristic

147

C₂×C₄

D₄×D₁₃₁

yes

characteristic

148

C₂×C₄

C₂×C₂²⋊D₁₃₁

yes

characteristic

149

C₈

—

150

C₈

—

262

—

151

C₈

—

524

—

152

C₂²

#16862214

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

262

—

157

C₄

—

158

C₄

—

524

—

159

C₄

—

524

—

160

C₄

Q₁₆×D₁₃₁

yes

characteristic

161

C₄

#16862266

yes

characteristic

162

C₄

#16862272

yes

characteristic

163

C₄

#16862278

yes

characteristic

164

C₂

#22553455

yes

characteristic

165

C₂

#22553466

yes

characteristic

166

C₂

#22553493

yes

characteristic

	subgroup	G/H
center	C₂²	#16862214
commutator, cosocle	C₄²×C₁₃₁	C₂³
Frattini	C₄²	C₂²×D₁₃₁
Fitting	#462×C₁₃₁	C₂
socle, prime radical	C₂²×C₁₃₁	#120
hypercenter	#462	D₁₃₁
hypocenter	C₁₃₁	#1648
LCS	G ▷ C₄²×C₁₃₁ ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

subgroup

G/H

center

C₂²

#16862214

commutator, cosocle

C₄²×C₁₃₁

C₂³

Frattini

C₄²

C₂²×D₁₃₁

Fitting

#462×C₁₃₁

C₂

C₂²×C₁₃₁

D₁₃₁

C₁₃₁

G ▷ C₄²×C₁₃₁ ▷ C₂²×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #462

N	H ≅ G/N	type
51. Dic₂₆₂	Q₁₆	semidirect
83. C₁₃₁	#1648	semidirect

H ≅ G/N

type

51. Dic₂₆₂

Q₁₆

semidirect

83. C₁₃₁

#1648

semidirect

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

subgroup	index
#22554040	2
#22553715	2
#22553594	2
#22553059	2
Q₈⋊(C₁₃₁⋊C₈)	2
#22553042	2
#462×C₁₃₁	2
#1648	131

subgroup

index

Q₈⋊(C₁₃₁⋊C₈)

#462×C₁₃₁

131

Subgroups — #30665201

Conjugacy classes of subgroups

Complements

Maximal subgroups