Group #30775722 — subgroups

Subgroups — #30775722

Order 17152.

292 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 83 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	#22561819	C₂	1	1	yes	maximal, characteristic
2	#22560244	C₂	1	2	yes	maximal
3	#22558320	C₂	1	2	yes	maximal, complemented
4	#22558313	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2161	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₆₇	C₂²	1	1	yes	characteristic
7	C₂×#14523983	C₂²	1	1	yes	characteristic
8	C₂×#14523979	C₂²	1	2	yes	—
9	#16863359	—	2	2	no	—
10	(C₂×D₆₇)⋊C₁₆	—	2	2	no	—
11	#16863160	—	2	2	no	—
12	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#108×C₆₇	C₂²	1	2	yes	—
14	#368×C₆₇	—	2	2	no	—
15	C₂²×C₄×D₆₇	D₄	1	1	yes	characteristic
16	C₂×#11761612	—	2	2	no	—
17	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
18	C₂×C₈×D₆₇	—	2	1	no	—
19	C₂×C₈×D₆₇	—	2	1	no	—
20	C₂×C₈×D₆₇	—	2	2	no	—
21	C₂×C₈×D₆₇	—	2	2	no	—
22	#14523983	—	2	2	no	—
23	C₂×C₆₇⋊C₁₆	—	2	1	no	—
24	#14523979	—	2	2	no	—
25	#14523979	—	2	2	no	—
26	(C₂×D₆₇)⋊C₈	—	4	4	no	—
27	C₂×M₁₆×C₆₇	—	2	2	no	—
28	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	M₃₂×C₆₇	—	2	2	no	—
30	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
31	#108×C₆₇	—	2	2	no	—
32	#108×C₆₇	—	2	2	no	—
33	C₂³×D₆₇	Q₁₆	1	1	yes	characteristic
34	C₂²×Dic₆₇	Q₁₆	1	1	yes	characteristic
35	C₂×C₄×D₆₇	—	2	1	no	—
36	C₂×C₄×D₆₇	—	2	2	no	—
37	C₂×C₄×D₆₇	—	2	1	no	—
38	C₂×C₄×D₆₇	—	2	2	no	—
39	#11761612	—	4	2	no	—
40	#11761612	—	4	2	no	—
41	C₂×C₆₇⋊C₈	—	2	1	no	—
42	C₂×C₆₇⋊C₈	—	2	2	no	—
43	C₂×C₆₇⋊C₈	—	4	2	no	—
44	C₈×D₆₇	—	4	1	no	—
45	C₈×D₆₇	—	4	1	no	—
46	C₈×D₆₇	—	4	2	no	—
47	C₆₇⋊C₁₆	—	4	1	no	—
48	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	M₁₆×C₆₇	—	4	2	no	—
50	M₁₆×C₆₇	—	4	2	no	—
51	C₂×C₈×C₆₇	—	4	2	no	—
52	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
54	C₂×C₈×C₆₇	#41	1	2	yes	—
55	C₂×C₈×C₆₇	#41	1	2	yes	—
56	C₁₆×C₆₇	—	2	2	no	—
57	C₂³×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
58	C₂²×D₆₇	—	2	1	no	—
59	C₂²×D₆₇	—	2	2	no	—
60	C₂²×D₆₇	—	4	2	no	—
61	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
62	C₂×C₄×C₆₇	#123	1	2	yes	—
63	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
64	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
65	C₂×Dic₆₇	—	2	1	no	—
66	C₂×Dic₆₇	—	2	2	no	—
67	C₄×D₆₇	—	4	1	no	—
68	C₄×D₆₇	—	4	1	no	—
69	C₄×D₆₇	—	4	2	no	—
70	C₈×C₆₇	—	8	2	no	—
71	C₈×C₆₇	#123	1	1	yes	characteristic
72	C₈×C₆₇	#124	1	1	yes	characteristic
73	C₈×C₆₇	#126	1	2	yes	—
74	C₆₇⋊C₈	—	4	1	no	—
75	C₆₇⋊C₈	—	8	2	no	—
76	C₂²×C₆₇	—	4	1	no	—
77	C₂²×C₆₇	#467	1	1	yes	characteristic
78	C₂²×C₆₇	#484	1	2	yes	—
79	C₂×D₆₇	—	2	2	no	—
80	C₂×D₆₇	—	2	4	no	—
81	C₂×D₆₇	—	4	1	no	—
82	C₂×D₆₇	—	4	2	no	—
83	C₂×D₆₇	—	4	1	no	—
84	C₄×C₆₇	#397	1	1	yes	characteristic
85	C₄×C₆₇	#469	1	1	yes	characteristic
86	C₄×C₆₇	#482	1	2	yes	—
87	Dic₆₇	—	4	1	no	—
88	#13498	—	67	1	no	maximal, Sylow
89	C₂×C₆₇	—	2	1	no	—
90	C₂×C₆₇	—	2	2	no	—
91	C₂×C₆₇	#2076	1	1	yes	characteristic
92	D₆₇	—	4	2	no	—
93	#2161	D₆₇	1	1	yes	characteristic, complemented
94	#2112	—	67	1	no	—
95	#1810	—	67	2	no	—
96	#1388	—	67	2	no	—
97	#1385	—	67	1	no	—
98	C₆₇	#13498	1	1	yes	characteristic, complemented, Sylow
99	C₂³×C₈	—	67	1	no	—
100	C₂×M₃₂	—	67	1	no	—
101	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
102	C₂×#108	—	67	2	no	—
103	C₂×#108	C₂×D₆₇	1	2	yes	—
104	#407	—	134	2	no	—
105	#368	—	2	2	no	—
106	C₂²⋊C₁₆	—	134	2	no	—
107	#341	—	134	2	no	—
108	C₂³×C₄	—	67	1	no	—
109	C₂×M₁₆	—	2	2	no	—
110	C₂×M₁₆	—	134	2	no	—
111	C₂²×C₈	—	67	1	no	—
112	C₂²×C₈	—	134	1	no	—
113	C₂²×C₈	—	134	1	no	—
114	C₂²×C₈	—	134	2	no	—
115	C₂²×C₈	—	134	2	no	—
116	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
117	M₃₂	—	2	2	no	—
118	M₃₂	—	134	2	no	—
119	C₂×C₁₆	—	134	1	no	—
120	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
121	#108	—	2	2	no	—
122	#108	—	2	2	no	—
123	#108	—	134	2	no	—
124	#108	—	134	2	no	—
125	C₂²⋊C₈	—	268	4	no	—
126	C₂⁴	—	67	1	no	—
127	C₂²×C₄	—	67	1	no	—
128	C₂²×C₄	—	134	1	no	—
129	C₂²×C₄	—	134	2	no	—
130	C₂²×C₄	—	134	1	no	—
131	C₂²×C₄	—	134	2	no	—
132	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
133	M₁₆	—	4	2	no	—
134	M₁₆	—	4	2	no	—
135	M₁₆	—	268	2	no	—
136	M₁₆	—	268	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	268	1	no	—
139	C₂×C₈	—	268	1	no	—
140	C₂×C₈	—	268	2	no	—
141	C₂×C₈	—	268	2	no	—
142	C₂×C₈	—	134	1	no	—
143	C₂×C₈	—	134	2	no	—
144	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
145	C₂×C₈	#11761627	1	1	yes	characteristic
146	C₂×C₈	#11761628	1	2	yes	—
147	C₂×C₈	#11761629	1	2	yes	—
148	C₁₆	—	2	2	no	—
149	C₁₆	—	268	1	no	—
150	C₂³	—	268	2	no	—
151	C₂³	—	134	1	no	—
152	C₂³	—	134	2	no	—
153	C₂³	Q₁₆×D₆₇	1	1	yes	characteristic
154	C₂×C₄	—	268	1	no	—
155	C₂×C₄	—	268	1	no	—
156	C₂×C₄	—	268	2	no	—
157	C₂×C₄	—	134	1	no	—
158	C₂×C₄	—	134	2	no	—
159	C₂×C₄	#14524014	1	1	yes	characteristic
160	C₂×C₄	#14524028	1	2	yes	—
161	C₂×C₄	#14524066	1	1	yes	characteristic
162	C₂×C₄	#14524067	1	2	yes	—
163	C₈	—	8	2	no	—
164	C₈	—	268	1	no	—
165	C₈	—	536	2	no	—
166	C₈	#14524026	1	1	yes	characteristic
167	C₈	#14524032	1	1	yes	characteristic
168	C₈	#14524038	1	2	yes	—
169	C₂²	—	4	1	no	—
170	C₂²	—	268	1	no	—
171	C₂²	—	268	2	no	—
172	C₂²	—	268	1	no	—
173	C₂²	—	134	2	no	—
174	C₂²	—	134	4	no	—
175	C₂²	#16863634	1	1	yes	characteristic
176	C₂²	#16863733	1	2	yes	—
177	C₄	—	268	1	no	—
178	C₄	#16863322	1	1	yes	characteristic
179	C₄	#16863646	1	1	yes	characteristic
180	C₄	#16863723	1	2	yes	—
181	C₂	—	2	1	no	—
182	C₂	—	2	2	no	—
183	C₂	—	268	2	no	—
184	C₂	#22561638	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22561819

C₂

yes

maximal, characteristic

#22560244

C₂

yes

maximal

#22558320

C₂

yes

maximal, complemented

#22558313

C₂

yes

maximal, characteristic, complemented

C₆₇×#2161

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₆₇

C₂²

yes

characteristic

C₂×#14523983

C₂²

yes

characteristic

C₂×#14523979

C₂²

yes

—

#16863359

—

(C₂×D₆₇)⋊C₁₆

—

#16863160

—

C₂×M₃₂×C₆₇

C₂²

yes

characteristic

C₂×#108×C₆₇

C₂²

yes

—

#368×C₆₇

—

C₂²×C₄×D₆₇

D₄

yes

characteristic

C₂×#11761612

—

C₂²×C₆₇⋊C₈

D₄

yes

characteristic

C₂×C₈×D₆₇

—

C₂×C₈×D₆₇

—

C₂×C₈×D₆₇

—

C₂×C₈×D₆₇

—

#14523983

—

C₂×C₆₇⋊C₁₆

—

#14523979

—

#14523979

—

(C₂×D₆₇)⋊C₈

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

—

#108×C₆₇

—

#108×C₆₇

—

C₂³×D₆₇

Q₁₆

yes

characteristic

C₂²×Dic₆₇

Q₁₆

yes

characteristic

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₈×D₆₇

—

C₈×D₆₇

—

C₈×D₆₇

—

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

yes

C₂×C₈×C₆₇

yes

—

C₂×C₈×C₆₇

#41

yes

—

C₁₆×C₆₇

—

C₂³×C₆₇

C₂×Q₁₆

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂×C₄×C₆₇

#121

yes

characteristic

C₂×C₄×C₆₇

#123

yes

—

C₂×C₄×C₆₇

#135

yes

characteristic

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₄×D₆₇

—

C₄×D₆₇

—

C₄×D₆₇

—

C₈×C₆₇

—

C₈×C₆₇

yes

C₈×C₆₇

yes

C₈×C₆₇

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#467

yes

characteristic

C₂²×C₆₇

#484

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

#13498

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#2076

yes

characteristic

D₆₇

—

#2161

D₆₇

yes

characteristic, complemented

—

—

—

—

C₆₇

#13498

yes

characteristic, complemented, Sylow

C₂³×C₈

—

100

C₂×M₃₂

—

101

C₂×M₃₂

C₂×D₆₇

yes

characteristic

102

C₂×#108

—

103

C₂×#108

C₂×D₆₇

yes

—

104

#407

—

134

—

105

#368

—

106

C₂²⋊C₁₆

—

134

—

107

#341

—

134

—

108

C₂³×C₄

—

109

C₂×M₁₆

—

110

C₂×M₁₆

—

134

—

111

C₂²×C₈

—

112

C₂²×C₈

—

134

—

113

C₂²×C₈

—

134

—

114

C₂²×C₈

—

134

—

115

C₂²×C₈

—

134

—

116

C₂²×C₈

C₂²×D₆₇

yes

characteristic

117

M₃₂

—

118

M₃₂

—

134

—

119

C₂×C₁₆

—

134

—

120

C₂×C₁₆

C₂²⋊D₆₇

yes

—

121

#108

—

122

#108

—

123

#108

—

134

—

124

#108

—

134

—

125

C₂²⋊C₈

—

268

—

126

C₂⁴

—

127

C₂²×C₄

—

128

C₂²×C₄

—

134

—

129

C₂²×C₄

—

134

—

130

C₂²×C₄

—

134

—

131

C₂²×C₄

—

134

—

132

C₂²×C₄

D₄×D₆₇

yes

characteristic

133

M₁₆

—

134

M₁₆

—

135

M₁₆

—

268

—

136

M₁₆

—

268

—

137

C₂×C₈

—

138

C₂×C₈

—

268

—

139

C₂×C₈

—

268

—

140

C₂×C₈

—

268

—

141

C₂×C₈

—

268

—

142

C₂×C₈

—

134

—

143

C₂×C₈

—

134

—

144

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

145

C₂×C₈

#11761627

yes

characteristic

146

C₂×C₈

#11761628

yes

—

147

C₂×C₈

#11761629

yes

—

148

C₁₆

—

149

C₁₆

—

268

—

150

C₂³

—

268

—

151

C₂³

—

134

—

152

C₂³

—

134

—

153

C₂³

Q₁₆×D₆₇

yes

characteristic

154

C₂×C₄

—

268

—

155

C₂×C₄

—

268

—

156

C₂×C₄

—

268

—

157

C₂×C₄

—

134

—

158

C₂×C₄

—

134

—

159

C₂×C₄

#14524014

yes

characteristic

160

C₂×C₄

#14524028

yes

—

161

C₂×C₄

#14524066

yes

characteristic

162

C₂×C₄

#14524067

yes

—

163

C₈

—

164

C₈

—

268

—

165

C₈

—

536

—

166

C₈

#14524026

yes

characteristic

167

C₈

#14524032

yes

characteristic

168

C₈

#14524038

yes

—

169

C₂²

—

170

C₂²

—

268

—

171

C₂²

—

268

—

172

C₂²

—

268

—

173

C₂²

—

134

—

174

C₂²

—

134

—

175

C₂²

#16863634

yes

characteristic

176

C₂²

#16863733

yes

—

177

C₄

—

268

—

178

C₄

#16863322

yes

characteristic

179

C₄

#16863646

yes

characteristic

180

C₄

#16863723

yes

—

181

C₂

—

182

C₂

—

183

C₂

—

268

—

184

C₂

#22561638

yes

characteristic

	subgroup	G/H
center	C₄	#16863646
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#2161	C₂
socle	C₂×C₆₇	#2076
prime radical	C₂³×D₆₇	Q₁₆
hypercenter	#2161	D₆₇
hypocenter	C₆₇	#13498
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2161

subgroup

G/H

center

C₄

#16863646

commutator, cosocle

C₂²×C₈×C₆₇

C₂³

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#2161

C₂

socle

C₂×C₆₇

#2076

prime radical

C₂³×D₆₇

Q₁₆

D₆₇

C₆₇

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

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2161

N	H ≅ G/N	type
3. #22558320	C₂	semidirect
4. #22558313	C₂	semidirect
5. C₆₇×#2161	C₂	semidirect
93. #2161	D₆₇	semidirect
98. C₆₇	#13498	semidirect

H ≅ G/N

type

3. #22558320

C₂

semidirect

4. #22558313

C₂

semidirect

5. C₆₇×#2161

C₂

semidirect

93. #2161

D₆₇

semidirect

98. C₆₇

#13498

semidirect

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

subgroup	index
#22561819	2
#22560244	2
#22558320	2
#22558313	2
C₆₇×#2161	2
#13498	67

subgroup

index

C₆₇×#2161

Subgroups — #30775722

Conjugacy classes of subgroups

Complements

Maximal subgroups