Group #30748616 — subgroups

Subgroups — #30748616

Order 17152.

456 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 143 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	#22560062	C₂	1	2	yes	maximal
2	C₂×#16863138	C₂	1	2	yes	maximal
3	#22558090	C₂	1	2	yes	maximal
4	C₄×#125×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
6	C₂×#14523954	C₂²	1	4	yes	—
7	#16863151	—	2	2	no	—
8	#16863138	—	2	4	no	—
9	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
10	#389×C₆₇	C₄	1	2	yes	—
11	#387×C₆₇	—	2	2	no	—
12	#382×C₆₇	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
14	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
15	C₄³×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₆₇⋊C₈	—	2	4	no	—
17	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
18	#14523954	—	2	4	no	—
19	#14523954	—	2	4	no	—
20	#125×C₆₇	—	2	2	no	—
21	#125×C₆₇	D₄	1	2	yes	—
22	#125×C₆₇	Q₈	1	2	yes	—
23	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
24	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
25	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
26	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₆₇	—	2	2	no	—
28	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₆₇	D₄	1	2	yes	—
31	C₂×C₄²×C₆₇	D₄	1	2	yes	—
32	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
33	#95×C₆₇	—	2	2	no	—
34	#95×C₆₇	—	2	2	no	—
35	#95×C₆₇	—	2	2	no	—
36	C₂×C₆₇⋊C₈	—	2	8	no	—
37	C₂×C₆₇⋊C₈	—	4	4	no	—
38	C₂×C₆₇⋊C₈	—	4	4	no	—
39	C₂²×C₄×C₆₇	—	2	2	no	—
40	C₂²×C₄×C₆₇	—	2	2	no	—
41	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
46	C₄⋊C₄×C₆₇	—	2	2	no	—
47	C₄⋊C₄×C₆₇	—	2	4	no	—
48	C₄⋊C₄×C₆₇	—	2	4	no	—
49	C₄⋊C₄×C₆₇	—	4	2	no	—
50	C₄⋊C₄×C₆₇	—	4	4	no	—
51	C₄⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
52	C₄²×C₆₇	—	2	2	no	—
53	C₄²×C₆₇	—	2	4	no	—
54	C₄²×C₆₇	—	4	1	no	—
55	C₄²×C₆₇	—	4	2	no	—
56	C₄²×C₆₇	—	4	4	no	—
57	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₆₇	#41	1	2	yes	—
60	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	2	no	—
65	C₂×C₄×C₆₇	—	2	2	no	—
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	—	2	2	no	—
68	C₂×C₄×C₆₇	—	2	4	no	—
69	C₂×C₄×C₆₇	—	2	1	no	—
70	C₂×C₄×C₆₇	—	2	2	no	—
71	C₂×C₄×C₆₇	—	4	2	no	—
72	C₂×C₄×C₆₇	—	4	2	no	—
73	C₂×C₄×C₆₇	—	4	2	no	—
74	C₂×C₄×C₆₇	—	4	2	no	—
75	C₂×C₄×C₆₇	—	4	2	no	—
76	C₂×C₄×C₆₇	#101	1	2	yes	—
77	C₂×C₄×C₆₇	#104	1	4	yes	—
78	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
80	C₂×C₄×C₆₇	#121	1	2	yes	—
81	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
82	C₆₇⋊C₈	—	4	8	no	—
83	C₂²×C₆₇	#380	1	1	yes	characteristic
84	C₂²×C₆₇	C₂×#101	1	1	yes	characteristic
85	C₂²×C₆₇	#413	1	1	yes	characteristic
86	C₂²×C₆₇	C₂×#104	1	2	yes	—
87	C₂²×C₆₇	#421	1	2	yes	—
88	C₄×C₆₇	—	2	1	no	—
89	C₄×C₆₇	—	2	1	no	—
90	C₄×C₆₇	—	2	2	no	—
91	C₄×C₆₇	—	2	2	no	—
92	C₄×C₆₇	—	2	4	no	—
93	C₄×C₆₇	—	2	2	no	—
94	C₄×C₆₇	—	4	4	no	—
95	C₄×C₆₇	—	4	1	no	—
96	C₄×C₆₇	—	8	2	no	—
97	#8584	—	67	1	no	maximal, Sylow
98	C₂×C₆₇	#1526	1	1	yes	characteristic
99	C₂×C₆₇	#1532	1	2	yes	—
100	C₂×C₆₇	#1534	1	1	yes	characteristic
101	C₂×C₆₇	#1801	1	1	yes	characteristic
102	C₂×C₆₇	#1806	1	2	yes	—
103	C₄×#125	D₆₇	1	1	yes	characteristic
104	#1767	—	67	2	no	—
105	C₂×#330	—	67	2	no	—
106	#1295	—	67	2	no	—
107	C₆₇	#8584	1	1	yes	characteristic, complemented, Sylow
108	C₂×#125	C₂×D₆₇	1	1	yes	characteristic
109	C₂×C₄⋊C₈	—	134	2	no	—
110	C₂×#97	—	67	4	no	—
111	#389	Dic₆₇	1	2	yes	—
112	#387	—	2	2	no	—
113	#382	C₂×D₆₇	1	1	yes	characteristic
114	C₄×C₄⋊C₄	—	2	2	no	—
115	C₄×C₄⋊C₄	Dic₆₇	1	2	yes	—
116	C₄³	C₂×D₆₇	1	1	yes	characteristic
117	#336	—	134	2	no	—
118	#330	—	134	4	no	—
119	C₂²×C₈	—	134	4	no	—
120	#125	—	2	2	no	—
121	#125	Dic₁₃₄	1	2	yes	—
122	#125	D₂₆₈	1	2	yes	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
126	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
127	C₂×C₄²	—	2	2	no	—
128	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
129	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
130	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
131	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
132	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
133	C₄⋊C₈	—	134	8	no	—
134	#97	—	134	4	no	—
135	#97	—	134	4	no	—
136	#95	—	2	2	no	—
137	#95	—	2	2	no	—
138	#95	—	2	2	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
142	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
143	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
144	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
146	C₂×C₈	—	268	4	no	—
147	C₂×C₈	—	268	4	no	—
148	C₂×C₈	—	134	8	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	4	no	—
154	C₄⋊C₄	C₄⋊Dic₆₇	1	4	yes	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	4	no	—
157	C₄²	—	4	1	no	—
158	C₄²	—	4	2	no	—
159	C₄²	—	4	4	no	—
160	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
161	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
162	C₄²	#11761628	1	2	yes	—
163	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	#14523962	1	2	yes	—
180	C₂×C₄	#14523970	1	4	yes	—
181	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
182	C₂×C₄	#14524002	1	1	yes	characteristic
183	C₂×C₄	#14524017	1	2	yes	—
184	C₂×C₄	#14524022	1	2	yes	—
185	C₈	—	268	8	no	—
186	C₂²	#16863256	1	1	yes	characteristic
187	C₂²	C₂×#14523962	1	1	yes	characteristic
188	C₂²	#16863380	1	1	yes	characteristic
189	C₂²	C₂×#14523970	1	2	yes	—
190	C₂²	#16863413	1	2	yes	—
191	C₄	—	2	4	no	—
192	C₄	—	2	2	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	4	1	no	—
198	C₄	—	4	4	no	—
199	C₄	—	8	2	no	—
200	C₂	#22558804	1	1	yes	characteristic
201	C₂	#22558829	1	2	yes	—
202	C₂	#22558836	1	1	yes	characteristic
203	C₂	#22560205	1	1	yes	characteristic
204	C₂	#22560229	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22560062

C₂

yes

maximal

C₂×#16863138

C₂

yes

maximal

#22558090

C₂

yes

maximal

C₄×#125×C₆₇

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₆₇⋊C₈)

—

C₂×#14523954

C₂²

yes

—

#16863151

—

#16863138

—

C₂×#125×C₆₇

C₂²

yes

characteristic

#389×C₆₇

C₄

yes

—

#387×C₆₇

—

#382×C₆₇

C₂²

yes

characteristic

C₄×C₄⋊C₄×C₆₇

—

C₄×C₄⋊C₄×C₆₇

C₄

yes

—

C₄³×C₆₇

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

—

C₄⋊(C₆₇⋊C₈)

—

#14523954

—

#14523954

—

#125×C₆₇

—

#125×C₆₇

D₄

yes

—

#125×C₆₇

Q₈

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#95×C₆₇

—

#95×C₆₇

—

#95×C₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₄⋊C₄

yes

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

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

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#101

yes

—

C₂×C₄×C₆₇

#104

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₂²×C₆₇

#421

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

100

C₂×C₆₇

#1534

yes

characteristic

101

C₂×C₆₇

#1801

yes

characteristic

102

C₂×C₆₇

#1806

yes

—

103

C₄×#125

D₆₇

yes

characteristic

104

#1767

—

105

C₂×#330

—

106

#1295

—

107

C₆₇

#8584

yes

characteristic, complemented, Sylow

108

C₂×#125

C₂×D₆₇

yes

characteristic

109

C₂×C₄⋊C₈

—

134

—

110

C₂×#97

—

111

#389

Dic₆₇

yes

—

112

#387

—

113

#382

C₂×D₆₇

yes

characteristic

114

C₄×C₄⋊C₄

—

115

C₄×C₄⋊C₄

Dic₆₇

yes

—

116

C₄³

C₂×D₆₇

yes

characteristic

117

#336

—

134

—

118

#330

—

134

—

119

C₂²×C₈

—

134

—

120

#125

—

121

#125

Dic₁₃₄

yes

—

122

#125

D₂₆₈

yes

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

C₂×Dic₆₇

yes

—

126

C₂×C₄⋊C₄

C₂×Dic₆₇

yes

—

127

C₂×C₄²

—

128

C₂×C₄²

C₂×Dic₆₇

yes

characteristic

129

C₂×C₄²

C₂×Dic₆₇

yes

characteristic

130

C₂×C₄²

C₂²⋊D₆₇

yes

—

131

C₂×C₄²

C₂²⋊D₆₇

yes

—

132

C₂×C₄²

C₂²×D₆₇

yes

characteristic

133

C₄⋊C₈

—

134

—

134

#97

—

134

—

135

#97

—

134

—

136

#95

—

137

#95

—

138

#95

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

C₂²⋊Dic₆₇

yes

characteristic

142

C₂²×C₄

C₂²⋊Dic₆₇

yes

characteristic

143

C₂²×C₄

C₂²⋊Dic₆₇

yes

—

144

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

145

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

—

146

C₂×C₈

—

268

—

147

C₂×C₈

—

268

—

148

C₂×C₈

—

134

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

C₄⋊Dic₆₇

yes

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₄²

—

160

C₄²

C₂×D₂₆₈

yes

characteristic

161

C₄²

C₂×Dic₁₃₄

yes

characteristic

162

C₄²

#11761628

yes

—

163

C₂³

C₂×C₂²⋊Dic₆₇

yes

characteristic

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

#14523962

yes

—

180

C₂×C₄

#14523970

yes

—

181

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

182

C₂×C₄

#14524002

yes

characteristic

183

C₂×C₄

#14524017

yes

—

184

C₂×C₄

#14524022

yes

—

185

C₈

—

268

—

186

C₂²

#16863256

yes

characteristic

187

C₂²

C₂×#14523962

yes

characteristic

188

C₂²

#16863380

yes

characteristic

189

C₂²

C₂×#14523970

yes

—

190

C₂²

#16863413

yes

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₂

#22558804

yes

characteristic

201

C₂

#22558829

yes

—

202

C₂

#22558836

yes

characteristic

203

C₂

#22560205

yes

characteristic

204

C₂

#22560229

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×#125×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×#125	D₆₇
hypocenter	C₆₇	#8584
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₆₇

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₄×#125×C₆₇

C₂

socle, prime radical

C₂³×C₆₇

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
107. C₆₇	#8584	semidirect

H ≅ G/N

type

107. C₆₇

#8584

semidirect

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

subgroup	index
#22560062	2
C₂×#16863138	2
#22558090	2
C₄×#125×C₆₇	2
#8584	67

subgroup

index

C₄×#125×C₆₇

Subgroups — #30748616

Conjugacy classes of subgroups

Complements

Maximal subgroups