Group #30748860 — subgroups

Subgroups — #30748860

Order 17152.

336 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 101 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	#22569139	C₂	1	1	yes	maximal, characteristic
2	#22558967	C₂	1	2	yes	maximal
3	#22558946	C₂	1	2	yes	maximal
4	C₆₇⋊#1309	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1309	C₂	1	1	yes	maximal, characteristic
6	#16863953	C₂²	1	1	yes	characteristic
7	#16863523	—	2	2	no	—
8	#16863520	—	2	2	no	—
9	#16863424	C₄	1	2	yes	—
10	#16863420	C₂²	1	2	yes	—
11	#16863354	C₄	1	2	yes	—
12	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	#16863148	—	2	2	no	—
14	#16863148	—	2	2	no	—
15	#16863143	—	2	2	no	—
16	#16863141	—	2	2	no	—
17	#423×C₆₇	C₂²	1	2	yes	—
18	#423×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#11761612	—	2	2	no	—
20	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
22	#14524043	D₄	1	2	yes	—
23	#14524035	Q₈	1	2	yes	—
24	#14524022	—	2	2	no	—
25	C₄×Dic₁₃₄	—	2	2	no	—
26	#14524002	C₂×C₄	1	2	yes	—
27	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
28	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
29	Dic₆₇⋊C₈	—	2	4	no	—
30	#14523953	—	2	2	no	—
31	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
32	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
33	C₈×Dic₆₇	—	2	2	no	—
34	C₂×M₁₆×C₆₇	—	2	2	no	—
35	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₆₇	—	2	2	no	—
38	C₄⋊C₈×C₆₇	—	2	2	no	—
39	C₄⋊C₈×C₆₇	—	2	2	no	—
40	C₂×Dic₁₃₄	—	2	2	no	—
41	#11761612	—	8	2	no	—
42	C₂×C₆₇⋊C₈	—	2	2	no	—
43	C₂×C₆₇⋊C₈	—	4	1	no	—
44	C₂×C₆₇⋊C₈	—	4	2	no	—
45	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
47	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
48	C₄⋊Dic₆₇	—	2	2	no	—
49	C₄⋊Dic₆₇	C₄⋊C₄	1	4	yes	—
50	Dic₆₇⋊C₄	—	2	4	no	—
51	C₂²⋊Dic₆₇	—	2	2	no	—
52	C₄×Dic₆₇	—	2	2	no	—
53	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	#41	1	2	yes	—
55	M₁₆×C₆₇	—	4	2	no	—
56	M₁₆×C₆₇	—	8	2	no	—
57	C₂×C₈×C₆₇	—	2	2	no	—
58	C₂×C₈×C₆₇	—	4	2	no	—
59	C₄²×C₆₇	C₂×D₄	1	2	yes	—
60	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
62	C₂³×C₆₇	#117	1	1	yes	characteristic
63	C₂×C₄×C₆₇	—	2	1	no	—
64	C₂×C₄×C₆₇	—	2	1	no	—
65	C₂×C₄×C₆₇	—	2	2	no	—
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₆₇	#121	1	2	yes	—
70	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
71	C₂×Dic₆₇	—	2	4	no	—
72	Dic₁₃₄	—	4	2	no	—
73	Dic₁₃₄	—	4	2	no	—
74	C₈×C₆₇	—	4	2	no	—
75	C₈×C₆₇	—	8	2	no	—
76	C₆₇⋊C₈	—	2	1	no	—
77	C₆₇⋊C₈	—	2	2	no	—
78	C₆₇⋊C₈	—	8	2	no	—
79	C₆₇⋊C₈	#101	1	2	yes	—
80	C₂²×C₆₇	—	2	1	no	—
81	C₂²×C₆₇	—	2	2	no	—
82	C₂²×C₆₇	#380	1	1	yes	characteristic
83	C₄×C₆₇	—	2	1	no	—
84	C₄×C₆₇	—	2	2	no	—
85	C₄×C₆₇	C₂×#101	1	1	yes	characteristic
86	C₄×C₆₇	#413	1	1	yes	characteristic
87	C₄×C₆₇	#472	1	2	yes	—
88	C₄×C₆₇	#473	1	2	yes	—
89	Dic₆₇	—	4	4	no	—
90	#8639	—	67	1	no	maximal, Sylow
91	C₂×C₆₇	—	4	1	no	—
92	C₂×C₆₇	#1801	1	1	yes	characteristic
93	C₂×C₆₇	#1816	1	2	yes	—
94	#2960	—	67	1	no	—
95	#1567	—	67	2	no	—
96	#1561	—	67	2	no	—
97	#1309	—	67	1	no	—
98	#1309	D₆₇	1	1	yes	characteristic
99	C₆₇	#8639	1	1	yes	characteristic, complemented, Sylow
100	#533	—	67	1	no	—
101	#446	—	134	2	no	—
102	C₈×Q₈	—	134	2	no	—
103	#424	—	67	2	no	—
104	#423	—	67	2	no	—
105	#423	C₂×D₆₇	1	2	yes	—
106	#423	C₂×D₆₇	1	1	yes	characteristic
107	#405	—	67	2	no	—
108	C₂×C₄×C₈	—	67	1	no	—
109	#335	—	134	2	no	—
110	#335	—	134	2	no	—
111	#332	—	134	2	no	—
112	#332	—	134	2	no	—
113	C₂×M₁₆	—	2	2	no	—
114	C₂×M₁₆	—	134	2	no	—
115	C₂×M₁₆	C₄×D₆₇	1	2	yes	—
116	C₂²×C₈	—	67	1	no	—
117	C₂²×C₈	—	67	1	no	—
118	C₄⋊Q₈	—	67	2	no	—
119	#125	—	67	2	no	—
120	C₂²⋊Q₈	—	134	2	no	—
121	C₄×Q₈	—	134	2	no	—
122	#117	—	67	2	no	—
123	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	134	4	no	—
128	C₄⋊C₈	—	134	2	no	—
129	C₄⋊C₈	—	134	2	no	—
130	#97	—	134	2	no	—
131	C₄×C₈	—	67	2	no	—
132	C₄×C₈	—	67	2	no	—
133	C₄×C₈	—	134	2	no	—
134	C₂×Q₈	—	134	2	no	—
135	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
136	C₂²×C₄	#11761628	1	2	yes	—
137	M₁₆	—	4	2	no	—
138	M₁₆	—	8	2	no	—
139	M₁₆	—	536	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	268	2	no	—
143	C₂×C₈	—	268	1	no	—
144	C₂×C₈	—	67	1	no	—
145	C₂×C₈	—	67	1	no	—
146	C₂×C₈	—	67	2	no	—
147	C₂×C₈	—	134	2	no	—
148	C₄⋊C₄	—	67	4	no	—
149	C₄⋊C₄	—	134	2	no	—
150	C₄⋊C₄	—	134	4	no	—
151	C₂²⋊C₄	—	134	2	no	—
152	C₄²	—	134	2	no	—
153	C₄²	D₄×D₆₇	1	2	yes	—
154	C₄²	D₄×D₆₇	1	1	yes	characteristic
155	C₄²	Q₈×D₆₇	1	1	yes	characteristic
156	C₂³	#14524000	1	1	yes	characteristic
157	Q₈	—	268	2	no	—
158	Q₈	—	268	2	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	134	4	no	—
164	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
165	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
166	C₂×C₄	#14524016	1	2	yes	—
167	C₂×C₄	#14524021	1	2	yes	—
168	C₈	—	4	2	no	—
169	C₈	—	8	2	no	—
170	C₈	—	536	2	no	—
171	C₈	—	67	2	no	—
172	C₈	—	134	1	no	—
173	C₈	—	134	2	no	—
174	C₂²	—	2	2	no	—
175	C₂²	—	2	1	no	—
176	C₂²	#16863255	1	1	yes	characteristic
177	C₄	—	2	2	no	—
178	C₄	—	2	1	no	—
179	C₄	—	268	4	no	—
180	C₄	#101×D₆₇	1	1	yes	characteristic
181	C₄	#16863377	1	1	yes	characteristic
182	C₄	#16863665	1	2	yes	—
183	C₄	#16863670	1	2	yes	—
184	C₂	—	4	1	no	—
185	C₂	#22560204	1	1	yes	characteristic
186	C₂	#22560272	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22569139

C₂

yes

maximal, characteristic

#22558967

C₂

yes

maximal

#22558946

C₂

yes

maximal

C₆₇⋊#1309

C₂

yes

maximal, characteristic

C₆₇×#1309

C₂

yes

maximal, characteristic

#16863953

C₂²

yes

—

—

C₄

yes

—

#16863420

C₂²

yes

—

#16863354

C₄

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

—

—

—

#423×C₆₇

C₂²

yes

—

#423×C₆₇

C₂²

yes

characteristic

C₂×#11761612

—

C₂²×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₆₇⋊C₈

C₂×C₄

yes

characteristic

#14524043

D₄

yes

—

#14524035

Q₈

yes

—

#14524022

—

C₄×Dic₁₃₄

—

#14524002

C₂×C₄

yes

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

Dic₆₇⋊C₈

—

#14523953

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₈×Dic₆₇

—

C₂×M₁₆×C₆₇

—

C₂×M₁₆×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₂×Dic₁₃₄

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₆₇⋊C₈

C₂²⋊C₄

yes

—

C₂×C₆₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₄⋊C₄

yes

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₂³×C₆₇

#117

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

#101

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#380

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₄×C₆₇

#473

yes

—

Dic₆₇

—

#8639

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1801

yes

characteristic

C₂×C₆₇

#1816

yes

—

—

—

—

—

D₆₇

yes

characteristic

C₆₇

#8639

yes

characteristic, complemented, Sylow

100

#533

—

101

#446

—

134

—

102

C₈×Q₈

—

134

—

103

#424

—

104

#423

—

105

#423

C₂×D₆₇

yes

—

106

#423

C₂×D₆₇

yes

characteristic

107

#405

—

108

C₂×C₄×C₈

—

109

#335

—

134

—

110

#335

—

134

—

111

#332

—

134

—

112

#332

—

134

—

113

C₂×M₁₆

—

114

C₂×M₁₆

—

134

—

115

C₂×M₁₆

C₄×D₆₇

yes

—

116

C₂²×C₈

—

117

C₂²×C₈

—

118

C₄⋊Q₈

—

119

#125

—

120

C₂²⋊Q₈

—

134

—

121

C₄×Q₈

—

134

—

122

#117

—

123

C₂×C₄²

C₂²×D₆₇

yes

characteristic

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₄⋊C₈

—

134

—

128

C₄⋊C₈

—

134

—

129

C₄⋊C₈

—

134

—

130

#97

—

134

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

—

134

—

134

C₂×Q₈

—

134

—

135

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

136

C₂²×C₄

#11761628

yes

—

137

M₁₆

—

138

M₁₆

—

139

M₁₆

—

536

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

268

—

143

C₂×C₈

—

268

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

134

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

134

—

150

C₄⋊C₄

—

134

—

151

C₂²⋊C₄

—

134

—

152

C₄²

—

134

—

153

C₄²

D₄×D₆₇

yes

—

154

C₄²

D₄×D₆₇

yes

characteristic

155

C₄²

Q₈×D₆₇

yes

characteristic

156

C₂³

#14524000

yes

characteristic

157

Q₈

—

268

—

158

Q₈

—

268

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

134

—

164

C₂×C₄

C₂²⋊C₄×D₆₇

yes

characteristic

165

C₂×C₄

C₄⋊C₄×D₆₇

yes

characteristic

166

C₂×C₄

#14524016

yes

—

167

C₂×C₄

#14524021

yes

—

168

C₈

—

169

C₈

—

170

C₈

—

536

—

171

C₈

—

172

C₈

—

134

—

173

C₈

—

134

—

174

C₂²

—

175

C₂²

—

176

C₂²

#16863255

yes

characteristic

177

C₄

—

178

C₄

—

179

C₄

—

268

—

180

C₄

#101×D₆₇

yes

characteristic

181

C₄

#16863377

yes

characteristic

182

C₄

#16863665

yes

—

183

C₄

#16863670

yes

—

184

C₂

—

185

C₂

#22560204

yes

characteristic

186

C₂

#22560272

yes

—

	subgroup	G/H
center	C₂²	#16863255
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1309	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂³×C₆₇	#117
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1309	D₆₇
hypocenter	C₆₇	#8639
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

#16863255

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1309

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

N	H ≅ G/N	type
99. C₆₇	#8639	semidirect

H ≅ G/N

type

99. C₆₇

#8639

semidirect

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

subgroup	index
#22569139	2
#22558967	2
#22558946	2
C₆₇⋊#1309	2
C₆₇×#1309	2
#8639	67

subgroup

index

#22569139

#22558967

#22558946

C₆₇⋊#1309

C₆₇×#1309

#8639

Subgroups — #30748860

Conjugacy classes of subgroups

Complements

Maximal subgroups