Group #30748855 — subgroups

Subgroups — #30748855

Order 17152.

404 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 109 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	#22568921	C₂	1	1	yes	maximal, characteristic
2	#22558967	C₂	1	2	yes	maximal, complemented
3	#22558944	C₂	1	1	yes	maximal, characteristic
4	C₆₇⋊#1309	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1560	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524050	C₄	1	2	yes	—
7	#16863474	—	2	2	no	—
8	D₄×C₆₇⋊C₈	—	2	2	no	—
9	#16863420	C₂²	1	2	yes	—
10	#16863420	C₂²	1	1	yes	characteristic
11	#16863363	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	#16863145	—	2	1	no	—
16	#16863140	—	2	1	no	—
17	#532×C₆₇	C₂²	1	1	yes	characteristic
18	#423×C₆₇	C₂²	1	2	yes	—
19	#333×C₆₇	—	2	1	no	—
20	#331×C₆₇	—	2	1	no	—
21	#14524050	—	4	4	no	—
22	C₂×#11761612	—	2	2	no	—
23	C₂×#11761612	—	2	2	no	—
24	C₂×#11761612	C₂×C₄	1	2	yes	—
25	C₂²×C₆₇⋊C₈	—	2	2	no	—
26	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
29	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
30	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
31	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
32	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
33	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
34	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
35	C₂×M₁₆×C₆₇	—	2	2	no	—
36	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
37	#127×C₆₇	D₄	1	1	yes	characteristic
38	#124×C₆₇	Q₈	1	2	yes	—
39	#121×C₆₇	—	2	2	no	—
40	C₄×D₄×C₆₇	—	2	2	no	—
41	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₆₇	—	2	2	no	—
43	C₄⋊C₈×C₆₇	—	2	2	no	—
44	#11761612	—	4	4	no	—
45	#11761612	—	4	2	no	—
46	#11761612	—	8	2	no	—
47	C₂×C₆₇⋊C₈	—	2	2	no	—
48	C₂×C₆₇⋊C₈	—	2	2	no	—
49	C₂×C₆₇⋊C₈	—	4	1	no	—
50	C₂×C₆₇⋊C₈	—	4	4	no	—
51	C₂×C₆₇⋊C₈	—	4	2	no	—
52	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
53	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
55	#41×C₆₇	—	4	2	no	—
56	#41×C₆₇	—	4	2	no	—
57	C₂×Q₈×C₆₇	C₄⋊C₄	1	2	yes	—
58	C₂×D₄×C₆₇	—	2	2	no	—
59	C₂×D₄×C₆₇	—	2	2	no	—
60	C₂×D₄×C₆₇	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₆₇	—	2	2	no	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	#41	1	2	yes	—
64	M₁₆×C₆₇	—	8	2	no	—
65	C₂×C₈×C₆₇	—	4	2	no	—
66	C₄⋊C₄×C₆₇	—	2	2	no	—
67	C₂²⋊C₄×C₆₇	—	2	4	no	—
68	C₄²×C₆₇	C₂×D₄	1	2	yes	—
69	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
71	C₂³×C₆₇	—	2	2	no	—
72	C₂³×C₆₇	#117	1	1	yes	characteristic
73	Q₈×C₆₇	—	4	2	no	—
74	D₄×C₆₇	—	4	2	no	—
75	D₄×C₆₇	—	4	4	no	—
76	D₄×C₆₇	—	4	2	no	—
77	D₄×C₆₇	—	4	2	no	—
78	C₂×C₄×C₆₇	—	2	1	no	—
79	C₂×C₄×C₆₇	—	2	1	no	—
80	C₂×C₄×C₆₇	—	2	2	no	—
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	—	2	2	no	—
83	C₂×C₄×C₆₇	—	4	2	no	—
84	C₂×C₄×C₆₇	—	4	2	no	—
85	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₆₇	#121	1	2	yes	—
88	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
89	C₈×C₆₇	—	8	2	no	—
90	C₆₇⋊C₈	—	2	1	no	—
91	C₆₇⋊C₈	—	2	2	no	—
92	C₆₇⋊C₈	—	4	2	no	—
93	C₆₇⋊C₈	—	8	2	no	—
94	C₆₇⋊C₈	#100	1	2	yes	—
95	C₂²×C₆₇	—	2	1	no	—
96	C₂²×C₆₇	—	2	2	no	—
97	C₂²×C₆₇	—	4	2	no	—
98	C₂²×C₆₇	—	4	4	no	—
99	C₂²×C₆₇	#380	1	1	yes	characteristic
100	C₄×C₆₇	—	2	1	no	—
101	C₄×C₆₇	—	2	2	no	—
102	C₄×C₆₇	—	4	2	no	—
103	C₄×C₆₇	C₂×#100	1	1	yes	characteristic
104	C₄×C₆₇	#413	1	1	yes	characteristic
105	C₄×C₆₇	#472	1	2	yes	—
106	C₄×C₆₇	#473	1	2	yes	—
107	#8638	—	67	1	no	maximal, Sylow
108	C₂×C₆₇	—	4	1	no	—
109	C₂×C₆₇	—	8	2	no	—
110	C₂×C₆₇	#1800	1	1	yes	characteristic
111	C₂×C₆₇	#1816	1	2	yes	—
112	#2931	—	67	1	no	—
113	#1567	—	67	2	no	—
114	#1560	—	67	1	no	—
115	#1560	D₆₇	1	1	yes	characteristic
116	#1309	—	67	2	no	—
117	C₆₇	#8638	1	1	yes	characteristic, complemented, Sylow
118	C₂×#131	—	67	2	no	—
119	#532	C₂×D₆₇	1	1	yes	characteristic
120	#436	—	134	2	no	—
121	C₈×D₄	—	134	2	no	—
122	#423	—	67	1	no	—
123	#423	—	67	2	no	—
124	#423	C₂×D₆₇	1	2	yes	—
125	#408	—	67	2	no	—
126	C₂×C₄×C₈	—	67	1	no	—
127	#335	—	134	2	no	—
128	#335	—	134	2	no	—
129	#333	—	2	1	no	—
130	#333	—	134	1	no	—
131	#331	—	2	1	no	—
132	#331	—	134	1	no	—
133	C₂×#41	C₄×D₆₇	1	2	yes	—
134	#131	—	268	4	no	—
135	C₂×M₁₆	—	2	2	no	—
136	C₂×M₁₆	—	67	2	no	—
137	C₂×M₁₆	—	134	2	no	—
138	C₂×M₁₆	—	134	2	no	—
139	C₂²×C₈	—	67	1	no	—
140	C₂²×C₈	—	67	1	no	—
141	C₂²×C₈	—	134	2	no	—
142	C₄⋊Q₈	C₂²⋊D₆₇	1	1	yes	characteristic
143	#127	C₂²⋊D₆₇	1	1	yes	characteristic
144	#124	Dic₁₃₄	1	2	yes	—
145	#121	—	2	2	no	—
146	C₄×D₄	—	2	2	no	—
147	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	134	2	no	—
151	C₄⋊C₈	—	134	2	no	—
152	C₄⋊C₈	—	134	2	no	—
153	C₂²⋊C₈	—	134	4	no	—
154	C₄×C₈	—	67	2	no	—
155	C₄×C₈	—	67	2	no	—
156	#41	—	4	2	no	—
157	#41	—	4	2	no	—
158	C₂×Q₈	Dic₆₇⋊C₄	1	2	yes	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	Dic₆₇⋊C₄	1	2	yes	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
164	C₂²×C₄	#11761627	1	2	yes	—
165	M₁₆	—	8	2	no	—
166	M₁₆	—	268	4	no	—
167	M₁₆	—	268	2	no	—
168	M₁₆	—	536	2	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	268	4	no	—
171	C₂×C₈	—	268	2	no	—
172	C₂×C₈	—	268	1	no	—
173	C₂×C₈	—	67	1	no	—
174	C₂×C₈	—	67	2	no	—
175	C₂×C₈	—	67	1	no	—
176	C₂×C₈	—	134	2	no	—
177	C₂×C₈	—	134	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₂²⋊C₄	—	2	4	no	—
180	C₄²	D₄×D₆₇	1	2	yes	—
181	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
182	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
183	C₂³	—	2	2	no	—
184	C₂³	#14523999	1	1	yes	characteristic
185	Q₈	—	4	2	no	—
186	D₄	—	4	2	no	—
187	D₄	—	4	4	no	—
188	D₄	—	4	2	no	—
189	D₄	—	4	2	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
198	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
199	C₂×C₄	#14524015	1	2	yes	—
200	C₂×C₄	#14524020	1	2	yes	—
201	C₈	—	8	2	no	—
202	C₈	—	268	2	no	—
203	C₈	—	536	2	no	—
204	C₈	—	67	2	no	—
205	C₈	—	134	1	no	—
206	C₈	—	134	2	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	4	4	no	—
210	C₂²	—	4	2	no	—
211	C₂²	#16863254	1	1	yes	characteristic
212	C₄	—	2	2	no	—
213	C₄	—	2	1	no	—
214	C₄	—	4	2	no	—
215	C₄	#100×D₆₇	1	1	yes	characteristic
216	C₄	#16863377	1	1	yes	characteristic
217	C₄	#16863664	1	2	yes	—
218	C₄	#16863669	1	2	yes	—
219	C₂	—	4	1	no	—
220	C₂	—	8	2	no	—
221	C₂	#22560199	1	1	yes	characteristic
222	C₂	#22560271	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22568921

C₂

yes

maximal, characteristic

#22558967

C₂

yes

maximal, complemented

#22558944

C₂

yes

maximal, characteristic

C₆₇⋊#1309

C₂

yes

maximal, complemented

C₆₇×#1560

C₂

yes

maximal, characteristic

C₂×#14524050

C₄

yes

—

#16863474

—

D₄×C₆₇⋊C₈

—

#16863420

C₂²

yes

—

#16863420

C₂²

yes

characteristic

#16863363

C₄

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

—

—

—

#532×C₆₇

C₂²

yes

characteristic

#423×C₆₇

C₂²

yes

—

#333×C₆₇

—

#331×C₆₇

—

—

—

—

C₂×C₄

yes

—

C₂²×C₆₇⋊C₈

—

C₂²×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₂×#41×C₆₇

C₂×C₄

yes

—

C₂×M₁₆×C₆₇

—

C₄⋊Q₈×C₆₇

D₄

yes

characteristic

#127×C₆₇

D₄

yes

characteristic

#124×C₆₇

Q₈

yes

—

#121×C₆₇

—

C₄×D₄×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

#11761612

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂²⋊C₄

yes

—

C₂×C₆₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₆₇⋊C₈

C₂²⋊C₄

yes

characteristic

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

C₄⋊C₄

yes

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₂³×C₆₇

—

C₂³×C₆₇

#117

yes

characteristic

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×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×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

#100

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#380

yes

characteristic

100

C₄×C₆₇

—

101

C₄×C₆₇

—

102

C₄×C₆₇

—

103

C₄×C₆₇

C₂×#100

yes

characteristic

104

C₄×C₆₇

#413

yes

characteristic

105

C₄×C₆₇

#472

yes

—

106

C₄×C₆₇

#473

yes

—

107

#8638

—

maximal, Sylow

108

C₂×C₆₇

—

109

C₂×C₆₇

—

110

C₂×C₆₇

#1800

yes

characteristic

111

C₂×C₆₇

#1816

yes

—

112

#2931

—

113

#1567

—

114

#1560

—

115

#1560

D₆₇

yes

characteristic

116

#1309

—

117

C₆₇

#8638

yes

characteristic, complemented, Sylow

118

C₂×#131

—

119

#532

C₂×D₆₇

yes

characteristic

120

#436

—

134

—

121

C₈×D₄

—

134

—

122

#423

—

123

#423

—

124

#423

C₂×D₆₇

yes

—

125

#408

—

126

C₂×C₄×C₈

—

127

#335

—

134

—

128

#335

—

134

—

129

#333

—

130

#333

—

134

—

131

#331

—

132

#331

—

134

—

133

C₂×#41

C₄×D₆₇

yes

—

134

#131

—

268

—

135

C₂×M₁₆

—

136

C₂×M₁₆

—

137

C₂×M₁₆

—

134

—

138

C₂×M₁₆

—

134

—

139

C₂²×C₈

—

140

C₂²×C₈

—

141

C₂²×C₈

—

134

—

142

C₄⋊Q₈

C₂²⋊D₆₇

yes

characteristic

143

#127

C₂²⋊D₆₇

yes

characteristic

144

#124

Dic₁₃₄

yes

—

145

#121

—

146

C₄×D₄

—

147

C₂×C₄²

C₂²×D₆₇

yes

characteristic

148

C₄⋊C₈

—

149

C₄⋊C₈

—

150

C₄⋊C₈

—

134

—

151

C₄⋊C₈

—

134

—

152

C₄⋊C₈

—

134

—

153

C₂²⋊C₈

—

134

—

154

C₄×C₈

—

155

C₄×C₈

—

156

#41

—

157

#41

—

158

C₂×Q₈

Dic₆₇⋊C₄

yes

—

159

C₂×D₄

—

160

C₂×D₄

—

161

C₂×D₄

Dic₆₇⋊C₄

yes

—

162

C₂²×C₄

—

163

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

164

C₂²×C₄

#11761627

yes

—

165

M₁₆

—

166

M₁₆

—

268

—

167

M₁₆

—

268

—

168

M₁₆

—

536

—

169

C₂×C₈

—

170

C₂×C₈

—

268

—

171

C₂×C₈

—

268

—

172

C₂×C₈

—

268

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

176

C₂×C₈

—

134

—

177

C₂×C₈

—

134

—

178

C₄⋊C₄

—

179

C₂²⋊C₄

—

180

C₄²

D₄×D₆₇

yes

—

181

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

182

C₄²

C₂×Dic₁₃₄

yes

characteristic

183

C₂³

—

184

C₂³

#14523999

yes

characteristic

185

Q₈

—

186

D₄

—

187

D₄

—

188

D₄

—

189

D₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

C₂²⋊C₄×D₆₇

yes

characteristic

198

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

199

C₂×C₄

#14524015

yes

—

200

C₂×C₄

#14524020

yes

—

201

C₈

—

202

C₈

—

268

—

203

C₈

—

536

—

204

C₈

—

205

C₈

—

134

—

206

C₈

—

134

—

207

C₂²

—

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

#16863254

yes

characteristic

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

#100×D₆₇

yes

characteristic

216

C₄

#16863377

yes

characteristic

217

C₄

#16863664

yes

—

218

C₄

#16863669

yes

—

219

C₂

—

220

C₂

—

221

C₂

#22560199

yes

characteristic

222

C₂

#22560271

yes

—

	subgroup	G/H
center	C₂²	#16863254
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1560	C₂
socle	C₂²×C₆₇	#380
prime radical	#532×C₆₇	C₂²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1560	D₆₇
hypocenter	C₆₇	#8638
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1560

subgroup

G/H

center

C₂²

#16863254

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1560

C₂

socle

C₂²×C₆₇

#380

prime radical

#532×C₆₇

C₂²

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1560

N	H ≅ G/N	type
2. #22558967	C₂	semidirect
4. C₆₇⋊#1309	C₂	semidirect
117. C₆₇	#8638	semidirect

H ≅ G/N

type

2. #22558967

C₂

semidirect

4. C₆₇⋊#1309

C₂

semidirect

117. C₆₇

#8638

semidirect

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

subgroup	index
#22568921	2
#22558967	2
#22558944	2
C₆₇⋊#1309	2
C₆₇×#1560	2
#8638	67

subgroup

index

#22568921

#22558967

#22558944

C₆₇⋊#1309

C₆₇×#1560

#8638

Subgroups — #30748855

Conjugacy classes of subgroups

Complements

Maximal subgroups