Group #30748789 — subgroups

Subgroups — #30748789

Order 17152.

362 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 131 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	#22558972	C₂	1	2	yes	maximal
2	#22558553	C₂	1	2	yes	maximal
3	#22558079	C₂	1	2	yes	maximal
4	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
5	#16863453	C₂²	1	4	yes	—
6	#16863149	—	2	2	no	—
7	#16863147	—	2	2	no	—
8	#16863122	—	2	4	no	—
9	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₈×Q₈×C₆₇	—	2	2	no	—
11	C₈×Q₈×C₆₇	—	2	2	no	—
12	C₂×C₄⋊C₈×C₆₇	C₄	1	2	yes	—
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
17	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
18	#14523954	—	2	4	no	—
19	C₂²×Q₈×C₆₇	Q₈	1	1	yes	characteristic
20	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₄×Q₈×C₆₇	—	2	2	no	—
24	C₄×Q₈×C₆₇	—	2	2	no	—
25	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
26	C₂×C₄⋊C₄×C₆₇	Q₈	1	1	yes	characteristic
27	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₆₇	—	2	4	no	—
30	C₄⋊C₈×C₆₇	—	2	2	no	—
31	C₄×C₈×C₆₇	—	2	2	no	—
32	C₄×C₈×C₆₇	—	2	2	no	—
33	C₄×C₈×C₆₇	D₄	1	2	yes	—
34	C₄×C₈×C₆₇	D₄	1	2	yes	—
35	C₂×C₆₇⋊C₈	—	4	4	no	—
36	C₂×Q₈×C₆₇	—	2	4	no	—
37	C₂×Q₈×C₆₇	—	2	2	no	—
38	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₆₇	—	2	2	no	—
44	C₂×C₈×C₆₇	—	2	2	no	—
45	C₂×C₈×C₆₇	—	4	1	no	—
46	C₂×C₈×C₆₇	—	4	2	no	—
47	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
48	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₄⋊C₄×C₆₇	—	2	4	no	—
51	C₄⋊C₄×C₆₇	—	2	2	no	—
52	C₄²×C₆₇	—	2	2	no	—
53	C₄²×C₆₇	—	2	2	no	—
54	C₄²×C₆₇	C₂×D₄	1	2	yes	—
55	C₄²×C₆₇	#41	1	2	yes	—
56	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	Q₈×C₆₇	—	2	8	no	—
58	C₂×C₄×C₆₇	—	2	2	no	—
59	C₂×C₄×C₆₇	—	2	2	no	—
60	C₂×C₄×C₆₇	—	2	2	no	—
61	C₂×C₄×C₆₇	—	2	1	no	—
62	C₂×C₄×C₆₇	—	2	1	no	—
63	C₂×C₄×C₆₇	—	2	2	no	—
64	C₂×C₄×C₆₇	—	4	1	no	—
65	C₂×C₄×C₆₇	—	4	2	no	—
66	C₂×C₄×C₆₇	#108	1	4	yes	—
67	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
69	C₂×C₄×C₆₇	#121	1	2	yes	—
70	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
71	C₈×C₆₇	—	2	1	no	—
72	C₈×C₆₇	—	2	2	no	—
73	C₈×C₆₇	—	4	2	no	—
74	C₈×C₆₇	#101	1	2	yes	—
75	C₆₇⋊C₈	—	8	4	no	—
76	C₂²×C₆₇	—	2	1	no	—
77	C₂²×C₆₇	—	2	1	no	—
78	C₂²×C₆₇	#380	1	1	yes	characteristic
79	C₂²×C₆₇	C₂×#108	1	2	yes	—
80	C₄×C₆₇	—	2	1	no	—
81	C₄×C₆₇	—	2	4	no	—
82	C₄×C₆₇	—	2	2	no	—
83	C₄×C₆₇	—	2	2	no	—
84	C₄×C₆₇	—	4	2	no	—
85	C₄×C₆₇	C₂×#101	1	1	yes	characteristic
86	C₄×C₆₇	#413	1	1	yes	characteristic
87	#8623	—	67	1	no	maximal, Sylow
88	C₂×C₆₇	—	2	2	no	—
89	C₂×C₆₇	#1801	1	1	yes	characteristic
90	C₂×C₆₇	#1810	1	1	yes	characteristic
91	C₂×C₆₇	#1812	1	1	yes	characteristic
92	C₂×C₈×Q₈	D₆₇	1	1	yes	characteristic
93	#1568	—	67	2	no	—
94	#1467	—	67	2	no	—
95	#1290	—	67	2	no	—
96	C₆₇	#8623	1	1	yes	characteristic, complemented, Sylow
97	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
98	C₈×Q₈	—	2	2	no	—
99	C₈×Q₈	—	2	2	no	—
100	#432	—	67	4	no	—
101	C₂×C₄⋊C₈	Dic₆₇	1	2	yes	—
102	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
103	C₂×C₄×C₈	Dic₆₇	1	2	yes	—
104	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
105	#335	—	134	2	no	—
106	#334	—	134	2	no	—
107	#324	—	134	4	no	—
108	C₂²×Q₈	Dic₁₃₄	1	1	yes	characteristic
109	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
110	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
111	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
112	C₄×Q₈	—	2	2	no	—
113	C₄×Q₈	—	2	2	no	—
114	C₂×C₄⋊C₄	Dic₁₃₄	1	1	yes	characteristic
115	C₂×C₄⋊C₄	D₂₆₈	1	2	yes	—
116	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
117	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
118	C₄⋊C₈	—	2	4	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	134	4	no	—
121	C₂²⋊C₈	—	134	4	no	—
122	#97	—	134	4	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
126	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
127	C₂×Q₈	—	2	4	no	—
128	C₂×Q₈	—	2	2	no	—
129	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
130	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
131	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
132	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
133	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	4	1	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	268	4	no	—
139	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
140	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
141	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
147	C₄²	#11761628	1	2	yes	—
148	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
149	Q₈	—	2	8	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	#14523979	1	4	yes	—
159	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
160	C₂×C₄	#14524002	1	1	yes	characteristic
161	C₂×C₄	#14524017	1	2	yes	—
162	C₂×C₄	#14524022	1	2	yes	—
163	C₈	—	2	1	no	—
164	C₈	—	2	2	no	—
165	C₈	—	4	2	no	—
166	C₈	—	536	4	no	—
167	C₈	#14523962	1	2	yes	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	#16863256	1	1	yes	characteristic
171	C₂²	C₂×#14523979	1	2	yes	—
172	C₄	—	2	1	no	—
173	C₄	—	2	4	no	—
174	C₄	—	2	2	no	—
175	C₄	—	2	2	no	—
176	C₄	—	4	2	no	—
177	C₄	C₂×#14523962	1	1	yes	characteristic
178	C₄	#16863380	1	1	yes	characteristic
179	C₂	—	2	2	no	—
180	C₂	#22560205	1	1	yes	characteristic
181	C₂	#22560247	1	1	yes	characteristic
182	C₂	#22560255	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22558972

C₂

yes

maximal

#22558553

C₂

yes

maximal

#22558079

C₂

yes

maximal

C₂×C₈×Q₈×C₆₇

C₂

yes

maximal, characteristic

#16863453

C₂²

yes

—

#16863149

—

#16863147

—

#16863122

—

C₂×C₄×Q₈×C₆₇

C₂²

yes

characteristic

C₈×Q₈×C₆₇

—

C₈×Q₈×C₆₇

—

C₂×C₄⋊C₈×C₆₇

C₄

yes

—

C₂×C₄⋊C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₆₇

C₄

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

#14523954

—

C₂²×Q₈×C₆₇

Q₈

yes

characteristic

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

C₂×C₄⋊C₄×C₆₇

D₄

yes

—

C₂×C₄⋊C₄×C₆₇

Q₈

yes

characteristic

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₂×C₆₇⋊C₈

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

C₂×C₄⋊C₄

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#108

yes

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#101

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#380

yes

characteristic

C₂²×C₆₇

C₂×#108

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₈×Q₈

D₆₇

yes

—

—

—

C₆₇

#8623

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

C₂×D₆₇

yes

characteristic

C₈×Q₈

—

C₈×Q₈

—

100

#432

—

101

C₂×C₄⋊C₈

Dic₆₇

yes

—

102

C₂×C₄⋊C₈

C₂×D₆₇

yes

characteristic

103

C₂×C₄×C₈

Dic₆₇

yes

—

104

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

105

#335

—

134

—

106

#334

—

134

—

107

#324

—

134

—

108

C₂²×Q₈

Dic₁₃₄

yes

characteristic

109

C₂²×C₈

C₂×Dic₆₇

yes

characteristic

110

C₂²×C₈

C₂×Dic₆₇

yes

—

111

C₂²×C₈

C₂×Dic₆₇

yes

characteristic

112

C₄×Q₈

—

113

C₄×Q₈

—

114

C₂×C₄⋊C₄

Dic₁₃₄

yes

characteristic

115

C₂×C₄⋊C₄

D₂₆₈

yes

—

116

C₂×C₄²

C₂×Dic₆₇

yes

—

117

C₂×C₄²

C₂²×D₆₇

yes

characteristic

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

134

—

121

C₂²⋊C₈

—

134

—

122

#97

—

134

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₄×C₈

C₂²⋊D₆₇

yes

—

126

C₄×C₈

C₂²⋊D₆₇

yes

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂²×C₄

C₄⋊Dic₆₇

yes

—

130

C₂²×C₄

C₄⋊Dic₆₇

yes

—

131

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

132

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

133

C₂²×C₄

C₂×Dic₁₃₄

yes

characteristic

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

268

—

139

C₂×C₈

C₂²⋊Dic₆₇

yes

characteristic

140

C₂×C₈

C₂²⋊Dic₆₇

yes

—

141

C₂×C₈

C₂²⋊Dic₆₇

yes

characteristic

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄²

—

145

C₄²

—

146

C₄²

C₂×C₂²⋊D₆₇

yes

—

147

C₄²

#11761628

yes

—

148

C₂³

C₂×C₄⋊Dic₆₇

yes

characteristic

149

Q₈

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

#14523979

yes

—

159

C₂×C₄

C₂×C₂²⋊Dic₆₇

yes

characteristic

160

C₂×C₄

#14524002

yes

characteristic

161

C₂×C₄

#14524017

yes

—

162

C₂×C₄

#14524022

yes

—

163

C₈

—

164

C₈

—

165

C₈

—

166

C₈

—

536

—

167

C₈

#14523962

yes

—

168

C₂²

—

169

C₂²

—

170

C₂²

#16863256

yes

characteristic

171

C₂²

C₂×#14523979

yes

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

C₂×#14523962

yes

characteristic

178

C₄

#16863380

yes

characteristic

179

C₂

—

180

C₂

#22560205

yes

characteristic

181

C₂

#22560247

yes

characteristic

182

C₂

#22560255

yes

characteristic

	subgroup	G/H
center	C₂²	#16863256
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×C₈×Q₈×C₆₇	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×C₈×Q₈	D₆₇
hypocenter	C₆₇	#8623
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈

subgroup

G/H

center

C₂²

#16863256

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₂×C₈×Q₈×C₆₇

C₂

socle

C₂²×C₆₇

#380

prime radical

C₂³×C₆₇

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₆₇

C₂³

hypercenter

C₂×C₈×Q₈

D₆₇

hypocenter

C₆₇

#8623

LCS

G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈

N	H ≅ G/N	type
96. C₆₇	#8623	semidirect

H ≅ G/N

type

96. C₆₇

#8623

semidirect

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

subgroup	index
#22558972	2
#22558553	2
#22558079	2
C₂×C₈×Q₈×C₆₇	2
#8623	67

subgroup

index

C₂×C₈×Q₈×C₆₇

Subgroups — #30748789

Conjugacy classes of subgroups

Complements

Maximal subgroups