Group #30775380 — subgroups

Subgroups — #30775380

Order 17152.

378 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 95 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	#22561967	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863212	C₂	1	2	yes	maximal
3	#22561125	C₂	1	2	yes	maximal
4	#22558303	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	1	yes	characteristic
7	C₂×#14523979	C₂²	1	2	yes	—
8	#16863405	—	2	2	no	—
9	C₂×#14523962	—	2	2	no	—
10	#16863351	C₂²	1	1	yes	characteristic
11	Dic₆₇⋊C₁₆	—	2	2	no	—
12	#16863212	—	2	4	no	—
13	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂×Q₃₂×C₆₇	—	2	4	no	—
15	C₂×Q₃₂×C₆₇	—	2	2	no	—
16	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#11761612	—	2	2	no	—
18	C₂×#11761612	D₄	1	1	yes	characteristic
19	#14524002	D₄	1	1	yes	characteristic
20	#14523983	—	2	2	no	—
21	C₂×C₆₇⋊C₁₆	—	2	1	no	—
22	#14523979	—	2	2	no	—
23	#14523979	—	2	2	no	—
24	#14523967	—	4	4	no	—
25	#14523962	—	4	4	no	—
26	#14523953	—	2	1	no	—
27	C₈×Dic₆₇	—	2	1	no	—
28	C₂²×Q₈×C₆₇	—	2	2	no	—
29	C₂×Q₁₆×C₆₇	—	2	2	no	—
30	C₂×Q₁₆×C₆₇	—	2	2	no	—
31	C₂×Q₁₆×C₆₇	—	2	2	no	—
32	C₂×Q₁₆×C₆₇	—	2	2	no	—
33	C₂×Q₁₆×C₆₇	—	2	2	no	—
34	C₂×Q₁₆×C₆₇	D₄	1	4	yes	—
35	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
36	Q₃₂×C₆₇	—	2	8	no	—
37	C₂×C₁₆×C₆₇	—	2	2	no	—
38	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
39	#11761612	—	2	2	no	—
40	#11761612	—	4	2	no	—
41	#11761612	—	4	2	no	—
42	C₂×C₆₇⋊C₈	—	2	1	no	—
43	C₂×C₆₇⋊C₈	—	4	2	no	—
44	C₄⋊Dic₆₇	D₈	1	2	yes	—
45	C₂²⋊Dic₆₇	—	2	1	no	—
46	C₄×Dic₆₇	—	2	1	no	—
47	C₆₇⋊C₁₆	—	4	1	no	—
48	C₂×Q₈×C₆₇	—	2	4	no	—
49	C₂×Q₈×C₆₇	—	4	2	no	—
50	C₂×Q₈×C₆₇	—	4	2	no	—
51	C₂×Q₈×C₆₇	—	4	2	no	—
52	C₂×Q₈×C₆₇	—	4	2	no	—
53	C₂×Q₈×C₆₇	—	4	2	no	—
54	C₂²×C₄×C₆₇	—	4	2	no	—
55	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
56	Q₁₆×C₆₇	—	2	4	no	—
57	Q₁₆×C₆₇	—	2	4	no	—
58	Q₁₆×C₆₇	—	4	2	no	—
59	Q₁₆×C₆₇	—	4	2	no	—
60	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
63	C₂×C₈×C₆₇	#41	1	2	yes	—
64	C₁₆×C₆₇	—	2	2	no	—
65	C₂³×C₆₇	#135	1	1	yes	characteristic
66	Q₈×C₆₇	—	4	4	no	—
67	Q₈×C₆₇	—	4	4	no	—
68	Q₈×C₆₇	—	8	2	no	—
69	Q₈×C₆₇	—	8	2	no	—
70	C₂×C₄×C₆₇	—	4	4	no	—
71	C₂×C₄×C₆₇	—	8	2	no	—
72	C₂×C₄×C₆₇	—	8	2	no	—
73	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
74	C₂×C₄×C₆₇	#121	1	2	yes	—
75	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×Dic₆₇	—	2	2	no	—
78	C₈×C₆₇	#120	1	1	yes	characteristic
79	C₈×C₆₇	#123	1	2	yes	—
80	C₈×C₆₇	#124	1	1	yes	characteristic
81	C₆₇⋊C₈	—	4	1	no	—
82	C₆₇⋊C₈	—	8	2	no	—
83	C₂²×C₆₇	—	2	2	no	—
84	C₂²×C₆₇	#463	1	2	yes	—
85	C₂²×C₆₇	#466	1	1	yes	characteristic
86	C₄×C₆₇	—	8	4	no	—
87	C₄×C₆₇	#394	1	1	yes	characteristic
88	C₄×C₆₇	#459	1	2	yes	—
89	C₄×C₆₇	#469	1	1	yes	characteristic
90	Dic₆₇	—	4	2	no	—
91	#13441	—	67	1	no	maximal, Sylow
92	C₂×C₆₇	—	2	2	no	—
93	C₂×C₆₇	#2054	1	1	yes	characteristic
94	C₂×C₆₇	#2222	1	2	yes	—
95	C₂²×Q₃₂	D₆₇	1	1	yes	characteristic
96	#2151	—	67	1	no	—
97	C₂×#362	—	67	2	no	—
98	#1979	—	67	2	no	—
99	#1381	—	67	1	no	—
100	C₆₇	#13441	1	1	yes	characteristic, complemented, Sylow
101	C₂²×Q₁₆	C₂×D₆₇	1	2	yes	—
102	C₂×Q₃₂	—	2	4	no	—
103	C₂×Q₃₂	—	2	2	no	—
104	C₂×M₃₂	—	67	1	no	—
105	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
106	C₂×#108	—	67	2	no	—
107	#419	—	134	2	no	—
108	C₂×#101	—	134	2	no	—
109	#405	—	67	1	no	—
110	C₄⋊C₁₆	—	134	2	no	—
111	#362	—	134	4	no	—
112	C₂²×Q₈	—	2	2	no	—
113	C₂×Q₁₆	—	2	2	no	—
114	C₂×Q₁₆	—	2	2	no	—
115	C₂×Q₁₆	—	2	2	no	—
116	C₂×Q₁₆	—	2	2	no	—
117	C₂×Q₁₆	—	2	2	no	—
118	C₂×Q₁₆	C₂²⋊D₆₇	1	4	yes	—
119	C₂×M₁₆	—	67	1	no	—
120	C₂×M₁₆	—	134	2	no	—
121	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
122	#117	—	67	1	no	—
123	Q₃₂	—	2	8	no	—
124	M₃₂	—	134	2	no	—
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	134	1	no	—
127	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
128	#108	—	134	2	no	—
129	#108	—	134	2	no	—
130	#103	—	268	4	no	—
131	#101	—	268	4	no	—
132	#97	—	134	1	no	—
133	C₄×C₈	—	134	1	no	—
134	C₂×Q₈	—	2	4	no	—
135	C₂×Q₈	—	4	2	no	—
136	C₂×Q₈	—	4	2	no	—
137	C₂×Q₈	—	4	2	no	—
138	C₂×Q₈	—	4	2	no	—
139	C₂×Q₈	—	4	2	no	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
142	Q₁₆	—	2	4	no	—
143	Q₁₆	—	2	4	no	—
144	Q₁₆	—	4	2	no	—
145	Q₁₆	—	4	2	no	—
146	M₁₆	—	268	2	no	—
147	M₁₆	—	268	2	no	—
148	M₁₆	—	134	2	no	—
149	C₂×C₈	—	268	2	no	—
150	C₂×C₈	—	134	1	no	—
151	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
152	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
153	C₂×C₈	#11761627	1	1	yes	characteristic
154	C₂×C₈	#11761627	1	2	yes	—
155	C₄⋊C₄	—	67	2	no	—
156	C₂²⋊C₄	—	134	1	no	—
157	C₄²	—	134	1	no	—
158	C₁₆	—	2	2	no	—
159	C₁₆	—	268	1	no	—
160	C₂³	#14524062	1	1	yes	characteristic
161	Q₈	—	4	4	no	—
162	Q₈	—	4	4	no	—
163	Q₈	—	8	2	no	—
164	Q₈	—	8	2	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	—	134	2	no	—
169	C₂×C₄	#14524014	1	1	yes	characteristic
170	C₂×C₄	#14524015	1	2	yes	—
171	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
172	C₂×C₄	#14524067	1	2	yes	—
173	C₈	—	268	1	no	—
174	C₈	—	536	2	no	—
175	C₈	#14524012	1	1	yes	characteristic
176	C₈	#14524024	1	2	yes	—
177	C₈	#14524030	1	1	yes	characteristic
178	C₂²	—	2	2	no	—
179	C₂²	#16863609	1	2	yes	—
180	C₂²	#16863629	1	1	yes	characteristic
181	C₄	—	8	4	no	—
182	C₄	—	268	2	no	—
183	C₄	#16863310	1	1	yes	characteristic
184	C₄	#16863589	1	2	yes	—
185	C₄	#16863646	1	1	yes	characteristic
186	C₂	—	2	2	no	—
187	C₂	#22561517	1	1	yes	characteristic
188	C₂	#22562331	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22561967

C₂

yes

maximal, characteristic

C₂×#16863212

C₂

yes

maximal

#22561125

C₂

yes

maximal

#22558303

C₂

yes

maximal, characteristic

C₂²×Q₃₂×C₆₇

C₂

yes

maximal, characteristic

C₂×#14523983

C₂²

yes

characteristic

C₂×#14523979

C₂²

yes

—

#16863405

—

C₂×#14523962

—

#16863351

C₂²

yes

characteristic

Dic₆₇⋊C₁₆

—

#16863212

—

C₂²×Q₁₆×C₆₇

C₂²

yes

—

C₂×Q₃₂×C₆₇

—

C₂×Q₃₂×C₆₇

—

C₂²×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×#11761612

—

C₂×#11761612

D₄

yes

characteristic

#14524002

D₄

yes

characteristic

#14523983

—

C₂×C₆₇⋊C₁₆

—

—

—

—

—

—

C₈×Dic₆₇

—

C₂²×Q₈×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

D₄

yes

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

Q₃₂×C₆₇

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

—

#11761612

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

D₈

yes

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

Q₁₆×C₆₇

—

Q₁₆×C₆₇

—

Q₁₆×C₆₇

—

Q₁₆×C₆₇

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

C₂×C₈×C₆₇

yes

C₂×C₈×C₆₇

yes

—

C₁₆×C₆₇

—

C₂³×C₆₇

#135

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#121

yes

characteristic

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₇

—

C₈×C₆₇

#120

yes

characteristic

C₈×C₆₇

#123

yes

—

C₈×C₆₇

#124

yes

characteristic

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#463

yes

—

C₂²×C₆₇

#466

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

#394

yes

characteristic

C₄×C₆₇

#459

yes

—

C₄×C₆₇

#469

yes

characteristic

Dic₆₇

—

#13441

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#2054

yes

characteristic

C₂×C₆₇

#2222

yes

—

C₂²×Q₃₂

D₆₇

yes

—

—

—

—

100

C₆₇

#13441

yes

characteristic, complemented, Sylow

101

C₂²×Q₁₆

C₂×D₆₇

yes

—

102

C₂×Q₃₂

—

103

C₂×Q₃₂

—

104

C₂×M₃₂

—

105

C₂²×C₁₆

C₂×D₆₇

yes

characteristic

106

C₂×#108

—

107

#419

—

134

—

108

C₂×#101

—

134

—

109

#405

—

110

C₄⋊C₁₆

—

134

—

111

#362

—

134

—

112

C₂²×Q₈

—

113

C₂×Q₁₆

—

114

C₂×Q₁₆

—

115

C₂×Q₁₆

—

116

C₂×Q₁₆

—

117

C₂×Q₁₆

—

118

C₂×Q₁₆

C₂²⋊D₆₇

yes

—

119

C₂×M₁₆

—

120

C₂×M₁₆

—

134

—

121

C₂²×C₈

C₂²×D₆₇

yes

characteristic

122

#117

—

123

Q₃₂

—

124

M₃₂

—

134

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

134

—

127

C₂×C₁₆

C₂²⋊D₆₇

yes

—

128

#108

—

134

—

129

#108

—

134

—

130

#103

—

268

—

131

#101

—

268

—

132

#97

—

134

—

133

C₄×C₈

—

134

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂²×C₄

—

141

C₂²×C₄

D₄×D₆₇

yes

characteristic

142

Q₁₆

—

143

Q₁₆

—

144

Q₁₆

—

145

Q₁₆

—

146

M₁₆

—

268

—

147

M₁₆

—

268

—

148

M₁₆

—

134

—

149

C₂×C₈

—

268

—

150

C₂×C₈

—

134

—

151

C₂×C₈

C₂×C₂²⋊D₆₇

yes

—

152

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

153

C₂×C₈

#11761627

yes

characteristic

154

C₂×C₈

#11761627

yes

—

155

C₄⋊C₄

—

156

C₂²⋊C₄

—

134

—

157

C₄²

—

134

—

158

C₁₆

—

159

C₁₆

—

268

—

160

C₂³

#14524062

yes

characteristic

161

Q₈

—

162

Q₈

—

163

Q₈

—

164

Q₈

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

134

—

169

C₂×C₄

#14524014

yes

characteristic

170

C₂×C₄

#14524015

yes

—

171

C₂×C₄

D₈×D₆₇

yes

characteristic

172

C₂×C₄

#14524067

yes

—

173

C₈

—

268

—

174

C₈

—

536

—

175

C₈

#14524012

yes

characteristic

176

C₈

#14524024

yes

—

177

C₈

#14524030

yes

characteristic

178

C₂²

—

179

C₂²

#16863609

yes

—

180

C₂²

#16863629

yes

characteristic

181

C₄

—

182

C₄

—

268

—

183

C₄

#16863310

yes

characteristic

184

C₄

#16863589

yes

—

185

C₄

#16863646

yes

characteristic

186

C₂

—

187

C₂

#22561517

yes

characteristic

188

C₂

#22562331

yes

—

	subgroup	G/H
center	C₂²	#16863629
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂²×Q₃₂×C₆₇	C₂
socle	C₂²×C₆₇	#466
prime radical	C₂³×C₆₇	#135
hypercenter	C₂²×Q₃₂	D₆₇
hypocenter	C₆₇	#13441
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

subgroup

G/H

center

C₂²

#16863629

commutator, cosocle

C₂²×C₈×C₆₇

C₂³

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₂²×Q₃₂×C₆₇

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂²×Q₃₂

D₆₇

hypocenter

C₆₇

#13441

LCS

G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

N	H ≅ G/N	type
100. C₆₇	#13441	semidirect

H ≅ G/N

type

100. C₆₇

#13441

semidirect

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

subgroup	index
#22561967	2
C₂×#16863212	2
#22561125	2
#22558303	2
C₂²×Q₃₂×C₆₇	2
#13441	67

subgroup

index

C₂²×Q₃₂×C₆₇

Subgroups — #30775380

Conjugacy classes of subgroups

Complements

Maximal subgroups