Group #30775384 — subgroups

Subgroups — #30775384

Order 17152.

378 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 91 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	C₂²×Dic₅₃₆	C₂	1	1	yes	maximal, characteristic
2	#22561967	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863211	C₂	1	2	yes	maximal
4	#22561122	C₂	1	2	yes	maximal
5	C₆₇×#1381	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₆₈	C₂²	1	2	yes	—
7	C₂×Dic₅₃₆	—	2	4	no	—
8	C₂×Dic₅₃₆	—	2	2	no	—
9	C₂×#14523983	C₂²	1	1	yes	characteristic
10	#16863406	—	2	2	no	—
11	C₂×#14523961	—	2	2	no	—
12	#16863351	C₂²	1	1	yes	characteristic
13	Dic₆₇⋊C₁₆	—	2	2	no	—
14	#16863211	—	2	4	no	—
15	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#108×C₆₇	C₂²	1	2	yes	—
17	C₂²×Dic₁₃₄	—	2	2	no	—
18	C₂×Dic₂₆₈	—	2	2	no	—
19	C₂×Dic₂₆₈	—	2	2	no	—
20	C₂×Dic₂₆₈	—	2	2	no	—
21	C₂×Dic₂₆₈	—	2	2	no	—
22	C₂×Dic₂₆₈	—	2	2	no	—
23	C₂×Dic₂₆₈	D₄	1	4	yes	—
24	C₂×#11761612	D₄	1	1	yes	characteristic
25	#14524002	D₄	1	1	yes	characteristic
26	Dic₅₃₆	—	2	8	no	—
27	#14523983	—	2	2	no	—
28	C₂×C₆₇⋊C₁₆	—	2	1	no	—
29	#14523968	—	4	4	no	—
30	#14523961	—	4	4	no	—
31	#14523953	—	2	1	no	—
32	C₈×Dic₆₇	—	2	1	no	—
33	C₂×M₁₆×C₆₇	—	2	2	no	—
34	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
35	C₂×C₁₆×C₆₇	—	2	2	no	—
36	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
37	#108×C₆₇	—	2	2	no	—
38	#108×C₆₇	—	2	2	no	—
39	C₂×Dic₁₃₄	—	2	4	no	—
40	C₂×Dic₁₃₄	—	4	2	no	—
41	C₂×Dic₁₃₄	—	4	2	no	—
42	C₂×Dic₁₃₄	—	4	2	no	—
43	C₂×Dic₁₃₄	—	4	2	no	—
44	C₂×Dic₁₃₄	—	4	2	no	—
45	C₂²×Dic₆₇	—	4	2	no	—
46	Dic₂₆₈	—	2	4	no	—
47	Dic₂₆₈	—	2	4	no	—
48	Dic₂₆₈	—	4	2	no	—
49	Dic₂₆₈	—	4	2	no	—
50	#11761612	—	2	2	no	—
51	C₂×C₆₇⋊C₈	—	2	1	no	—
52	C₄⋊Dic₆₇	D₈	1	2	yes	—
53	C₂²⋊Dic₆₇	—	2	1	no	—
54	C₄×Dic₆₇	—	2	1	no	—
55	C₆₇⋊C₁₆	—	4	1	no	—
56	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
57	M₁₆×C₆₇	—	4	2	no	—
58	M₁₆×C₆₇	—	4	2	no	—
59	C₂×C₈×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	2	yes	—
63	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
64	C₁₆×C₆₇	—	2	2	no	—
65	C₂³×C₆₇	#135	1	1	yes	characteristic
66	C₂×C₄×C₆₇	#121	1	2	yes	—
67	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
70	C₂×Dic₆₇	—	2	2	no	—
71	C₂×Dic₆₇	—	4	4	no	—
72	C₂×Dic₆₇	—	8	2	no	—
73	C₂×Dic₆₇	—	8	2	no	—
74	Dic₁₃₄	—	4	4	no	—
75	Dic₁₃₄	—	4	4	no	—
76	Dic₁₃₄	—	8	2	no	—
77	Dic₁₃₄	—	8	2	no	—
78	C₈×C₆₇	—	8	2	no	—
79	C₈×C₆₇	#120	1	1	yes	characteristic
80	C₈×C₆₇	#123	1	2	yes	—
81	C₈×C₆₇	#124	1	1	yes	characteristic
82	C₆₇⋊C₈	—	4	1	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	#394	1	1	yes	characteristic
87	C₄×C₆₇	#459	1	2	yes	—
88	C₄×C₆₇	#469	1	1	yes	characteristic
89	Dic₆₇	—	4	2	no	—
90	Dic₆₇	—	8	4	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	—	67	1	no	—
96	#2151	—	67	1	no	—
97	C₂×#362	—	67	2	no	—
98	#1979	—	67	2	no	—
99	#1381	D₆₇	1	1	yes	characteristic
100	C₆₇	#13441	1	1	yes	characteristic, complemented, Sylow
101	C₂²×Q₁₆	—	67	2	no	—
102	C₂×Q₃₂	—	134	4	no	—
103	C₂×Q₃₂	—	134	2	no	—
104	C₂×M₃₂	—	67	1	no	—
105	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
106	C₂×#108	C₂×D₆₇	1	2	yes	—
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	—	134	2	no	—
113	C₂×Q₁₆	—	67	4	no	—
114	C₂×Q₁₆	—	134	2	no	—
115	C₂×Q₁₆	—	134	2	no	—
116	C₂×Q₁₆	—	134	2	no	—
117	C₂×Q₁₆	—	134	2	no	—
118	C₂×Q₁₆	—	134	2	no	—
119	C₂×M₁₆	—	2	2	no	—
120	C₂×M₁₆	—	67	1	no	—
121	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
122	#117	—	67	1	no	—
123	Q₃₂	—	134	8	no	—
124	M₃₂	—	134	2	no	—
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	134	1	no	—
127	C₂×C₁₆	D₂₆₈	1	2	yes	—
128	#108	—	2	2	no	—
129	#108	—	2	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	—	268	2	no	—
135	C₂×Q₈	—	268	2	no	—
136	C₂×Q₈	—	268	2	no	—
137	C₂×Q₈	—	268	2	no	—
138	C₂×Q₈	—	268	2	no	—
139	C₂×Q₈	—	134	4	no	—
140	C₂²×C₄	—	268	2	no	—
141	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
142	Q₁₆	—	268	2	no	—
143	Q₁₆	—	268	2	no	—
144	Q₁₆	—	134	4	no	—
145	Q₁₆	—	134	4	no	—
146	M₁₆	—	4	2	no	—
147	M₁₆	—	4	2	no	—
148	M₁₆	—	134	2	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	134	1	no	—
151	C₂×C₈	D₄×D₆₇	1	2	yes	—
152	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
153	C₂×C₈	#11761628	1	2	yes	—
154	C₂×C₈	#11761629	1	1	yes	characteristic
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	#14524066	1	1	yes	characteristic
161	Q₈	—	268	4	no	—
162	Q₈	—	268	4	no	—
163	Q₈	—	536	2	no	—
164	Q₈	—	536	2	no	—
165	C₂×C₄	—	268	4	no	—
166	C₂×C₄	—	536	2	no	—
167	C₂×C₄	—	536	2	no	—
168	C₂×C₄	—	134	2	no	—
169	C₂×C₄	#14524016	1	2	yes	—
170	C₂×C₄	#14524018	1	1	yes	characteristic
171	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
172	C₂×C₄	#14524070	1	2	yes	—
173	C₈	—	8	2	no	—
174	C₈	—	268	1	no	—
175	C₈	#14524013	1	1	yes	characteristic
176	C₈	#14524028	1	2	yes	—
177	C₈	#14524031	1	1	yes	characteristic
178	C₂²	—	2	2	no	—
179	C₂²	#16863614	1	2	yes	—
180	C₂²	#16863633	1	1	yes	characteristic
181	C₄	—	268	2	no	—
182	C₄	—	536	4	no	—
183	C₄	#16863314	1	1	yes	characteristic
184	C₄	#16863590	1	2	yes	—
185	C₄	#16863650	1	1	yes	characteristic
186	C₂	—	2	2	no	—
187	C₂	#22561521	1	1	yes	characteristic
188	C₂	#22562335	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₅₃₆

C₂

yes

maximal, characteristic

#22561967

C₂

yes

maximal, characteristic

C₂×#16863211

C₂

yes

maximal

#22561122

C₂

yes

maximal

C₆₇×#1381

C₂

yes

maximal, characteristic

C₂²×Dic₂₆₈

C₂²

yes

—

C₂×Dic₅₃₆

—

C₂×Dic₅₃₆

—

C₂×#14523983

C₂²

yes

—

—

C₂²

yes

characteristic

Dic₆₇⋊C₁₆

—

#16863211

—

C₂²×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×#108×C₆₇

C₂²

yes

—

C₂²×Dic₁₃₄

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

D₄

yes

—

C₂×#11761612

D₄

yes

characteristic

#14524002

D₄

yes

characteristic

Dic₅₃₆

—

#14523983

—

C₂×C₆₇⋊C₁₆

—

#14523968

—

#14523961

—

#14523953

—

C₈×Dic₆₇

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

—

#108×C₆₇

—

#108×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

Dic₂₆₈

—

Dic₂₆₈

—

Dic₂₆₈

—

Dic₂₆₈

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

D₈

yes

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₂×C₈×C₆₇

#41

yes

characteristic

C₁₆×C₆₇

—

C₂³×C₆₇

#135

yes

characteristic

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

#121

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

#120

yes

characteristic

C₈×C₆₇

#123

yes

—

C₈×C₆₇

#124

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#463

yes

—

C₂²×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₄×C₆₇

#469

yes

characteristic

Dic₆₇

—

Dic₆₇

—

#13441

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#2054

yes

characteristic

C₂×C₆₇

#2222

yes

—

C₂²×Q₃₂

—

—

—

—

D₆₇

yes

characteristic

100

C₆₇

#13441

yes

characteristic, complemented, Sylow

101

C₂²×Q₁₆

—

102

C₂×Q₃₂

—

134

—

103

C₂×Q₃₂

—

134

—

104

C₂×M₃₂

—

105

C₂²×C₁₆

C₂×D₆₇

yes

characteristic

106

C₂×#108

C₂×D₆₇

yes

—

107

#419

—

134

—

108

C₂×#101

—

134

—

109

#405

—

110

C₄⋊C₁₆

—

134

—

111

#362

—

134

—

112

C₂²×Q₈

—

134

—

113

C₂×Q₁₆

—

114

C₂×Q₁₆

—

134

—

115

C₂×Q₁₆

—

134

—

116

C₂×Q₁₆

—

134

—

117

C₂×Q₁₆

—

134

—

118

C₂×Q₁₆

—

134

—

119

C₂×M₁₆

—

120

C₂×M₁₆

—

121

C₂²×C₈

C₂²×D₆₇

yes

characteristic

122

#117

—

123

Q₃₂

—

134

—

124

M₃₂

—

134

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

134

—

127

C₂×C₁₆

D₂₆₈

yes

—

128

#108

—

129

#108

—

130

#103

—

268

—

131

#101

—

268

—

132

#97

—

134

—

133

C₄×C₈

—

134

—

134

C₂×Q₈

—

268

—

135

C₂×Q₈

—

268

—

136

C₂×Q₈

—

268

—

137

C₂×Q₈

—

268

—

138

C₂×Q₈

—

268

—

139

C₂×Q₈

—

134

—

140

C₂²×C₄

—

268

—

141

C₂²×C₄

D₄×D₆₇

yes

characteristic

142

Q₁₆

—

268

—

143

Q₁₆

—

268

—

144

Q₁₆

—

134

—

145

Q₁₆

—

134

—

146

M₁₆

—

147

M₁₆

—

148

M₁₆

—

134

—

149

C₂×C₈

—

150

C₂×C₈

—

134

—

151

C₂×C₈

D₄×D₆₇

yes

—

152

C₂×C₈

C₂×D₂₆₈

yes

characteristic

153

C₂×C₈

#11761628

yes

—

154

C₂×C₈

#11761629

yes

characteristic

155

C₄⋊C₄

—

156

C₂²⋊C₄

—

134

—

157

C₄²

—

134

—

158

C₁₆

—

159

C₁₆

—

268

—

160

C₂³

#14524066

yes

characteristic

161

Q₈

—

268

—

162

Q₈

—

268

—

163

Q₈

—

536

—

164

Q₈

—

536

—

165

C₂×C₄

—

268

—

166

C₂×C₄

—

536

—

167

C₂×C₄

—

536

—

168

C₂×C₄

—

134

—

169

C₂×C₄

#14524016

yes

—

170

C₂×C₄

#14524018

yes

characteristic

171

C₂×C₄

D₈×D₆₇

yes

characteristic

172

C₂×C₄

#14524070

yes

—

173

C₈

—

174

C₈

—

268

—

175

C₈

#14524013

yes

characteristic

176

C₈

#14524028

yes

—

177

C₈

#14524031

yes

characteristic

178

C₂²

—

179

C₂²

#16863614

yes

—

180

C₂²

#16863633

yes

characteristic

181

C₄

—

268

—

182

C₄

—

536

—

183

C₄

#16863314

yes

characteristic

184

C₄

#16863590

yes

—

185

C₄

#16863650

yes

characteristic

186

C₂

—

187

C₂

#22561521

yes

characteristic

188

C₂

#22562335

yes

—

	subgroup	G/H
center	C₂²	#16863633
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1381	C₂
socle	C₂²×C₆₇	#466
prime radical	C₂³×C₆₇	#135
hypercenter	#1381	D₆₇
hypocenter	C₆₇	#13441
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂²

#16863633

commutator, cosocle

C₂²×C₈×C₆₇

C₂³

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1381

C₂

C₂²×C₆₇

C₂³×C₆₇

D₆₇

C₆₇

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

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
C₂²×Dic₅₃₆	2
#22561967	2
C₂×#16863211	2
#22561122	2
C₆₇×#1381	2
#13441	67

subgroup

index

C₂²×Dic₅₃₆

C₆₇×#1381

Subgroups — #30775384

Conjugacy classes of subgroups

Complements

Maximal subgroups