Group #30755693 — subgroups

Subgroups — #30755693

Order 17152.

510 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 199 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	#22561171	C₂	1	2	yes	maximal
2	#22561012	C₂	1	2	yes	maximal
3	#22560369	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863117	C₂	1	1	yes	maximal, characteristic
5	C₂×#498×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523977	—	2	2	no	—
7	C₂×#14523967	—	2	4	no	—
8	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
9	#16863266	C₂²	1	1	yes	characteristic
10	#16863266	C₂²	1	1	yes	characteristic
11	#16863158	—	2	4	no	—
12	#16863117	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
14	#498×C₆₇	—	2	2	no	—
15	#498×C₆₇	C₄	1	4	yes	complemented
16	C₂×#106×C₆₇	—	2	2	no	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₆₇⋊C₈	D₄	1	4	yes	—
19	C₂×C₄⋊Dic₆₇	—	2	2	no	—
20	C₂×C₄⋊Dic₆₇	—	2	2	no	—
21	C₂×C₄⋊Dic₆₇	—	2	1	no	—
22	C₂×C₄⋊Dic₆₇	—	2	1	no	—
23	#14523977	—	4	4	no	—
24	#14523967	—	4	8	no	—
25	C₄×C₆₇⋊C₈	—	2	2	no	—
26	C₄×C₆₇⋊C₈	—	2	2	no	—
27	C₂²×Q₈×C₆₇	—	2	2	no	—
28	C₂²×C₈×C₆₇	D₄	1	2	yes	—
29	C₄⋊Q₈×C₆₇	—	2	2	no	—
30	C₄⋊Q₈×C₆₇	—	2	2	no	—
31	C₄⋊Q₈×C₆₇	—	2	2	no	—
32	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
35	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
36	#106×C₆₇	—	2	8	no	—
37	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₆₇	D₄	1	2	yes	—
39	C₂²×Dic₆₇	—	4	1	no	—
40	C₂²×Dic₆₇	—	4	1	no	—
41	C₂×C₆₇⋊C₈	—	2	4	no	—
42	C₂×C₆₇⋊C₈	—	2	4	no	—
43	C₂×C₆₇⋊C₈	—	2	4	no	—
44	C₄⋊Dic₆₇	—	4	2	no	—
45	C₄⋊Dic₆₇	—	4	2	no	—
46	C₄⋊Dic₆₇	—	4	4	no	—
47	C₄⋊Dic₆₇	—	4	4	no	—
48	C₂×Q₈×C₆₇	—	2	4	no	—
49	C₂×Q₈×C₆₇	—	4	2	no	—
50	C₂×Q₈×C₆₇	—	4	4	no	—
51	C₂×Q₈×C₆₇	—	4	4	no	—
52	C₂²×C₄×C₆₇	—	4	2	no	—
53	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
56	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
57	C₂×C₈×C₆₇	Q₁₆	1	4	yes	—
58	C₄⋊C₄×C₆₇	—	2	4	no	—
59	C₄⋊C₄×C₆₇	—	2	4	no	—
60	C₄⋊C₄×C₆₇	—	4	2	no	—
61	C₄⋊C₄×C₆₇	—	4	2	no	—
62	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₆₇	#41	1	2	yes	—
65	C₂³×C₆₇	#127	1	1	yes	characteristic
66	Q₈×C₆₇	—	4	8	no	—
67	Q₈×C₆₇	—	8	4	no	—
68	C₂×C₄×C₆₇	—	4	4	no	—
69	C₂×C₄×C₆₇	—	8	2	no	—
70	C₂×C₄×C₆₇	—	8	2	no	—
71	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
73	C₂×C₄×C₆₇	#121	1	2	yes	—
74	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
76	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
77	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
78	C₂×C₄×C₆₇	#137	1	4	yes	—
79	C₂×Dic₆₇	—	8	1	no	—
80	C₂×Dic₆₇	—	8	1	no	—
81	C₂×Dic₆₇	—	8	2	no	—
82	C₂×Dic₆₇	—	8	2	no	—
83	C₈×C₆₇	#103	1	8	yes	complemented
84	C₆₇⋊C₈	—	4	4	no	—
85	C₂²×C₆₇	#390	1	1	yes	characteristic
86	C₂²×C₆₇	#492	1	1	yes	characteristic
87	C₂²×C₆₇	#494	1	1	yes	characteristic
88	C₂²×C₆₇	#496	1	2	yes	—
89	C₂²×C₆₇	#497	1	2	yes	—
90	C₄×C₆₇	—	8	4	no	—
91	C₄×C₆₇	C₂×#103	1	2	yes	—
92	C₄×C₆₇	#440	1	4	yes	—
93	C₄×C₆₇	#468	1	4	yes	—
94	C₄×C₆₇	#487	1	2	yes	—
95	Dic₆₇	—	16	1	no	—
96	Dic₆₇	—	16	1	no	—
97	#9841	—	67	1	no	maximal, Sylow
98	C₂×C₆₇	#1715	1	1	yes	characteristic
99	C₂×C₆₇	#1716	1	1	yes	characteristic
100	C₂×C₆₇	#1723	1	2	yes	—
101	C₂×C₆₇	#1956	1	1	yes	characteristic
102	C₂×C₆₇	#1973	1	2	yes	—
103	C₂×#498	D₆₇	1	1	yes	characteristic
104	#1990	—	67	2	no	—
105	#1956	—	67	2	no	—
106	#1845	—	67	1	no	—
107	C₂×#322	—	67	1	no	—
108	C₆₇	#9841	1	1	yes	characteristic, complemented, Sylow
109	C₂×C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
110	#498	—	2	2	no	—
111	#498	Dic₆₇	1	4	yes	complemented
112	C₂×#107	—	134	2	no	—
113	C₂×#106	—	2	2	no	—
114	C₂×#103	—	134	4	no	—
115	C₂×C₄×C₈	—	67	2	no	—
116	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
117	#384	—	67	1	no	—
118	#384	—	67	1	no	—
119	#340	—	134	4	no	—
120	#322	—	134	2	no	—
121	C₂²×Q₈	—	2	2	no	—
122	C₂²×C₈	—	67	4	no	—
123	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	C₂×Dic₆₇	1	4	yes	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	134	2	no	—
131	C₂×C₄⋊C₄	—	134	2	no	—
132	C₂×C₄⋊C₄	—	134	1	no	—
133	C₂×C₄⋊C₄	—	134	1	no	—
134	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
135	#107	—	268	4	no	—
136	#106	—	2	8	no	—
137	#103	—	268	8	no	—
138	C₄×C₈	—	134	2	no	—
139	C₄×C₈	—	134	2	no	—
140	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
141	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
142	C₂×Q₈	—	2	4	no	—
143	C₂×Q₈	—	4	2	no	—
144	C₂×Q₈	—	4	4	no	—
145	C₂×Q₈	—	4	4	no	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	—	268	1	no	—
148	C₂²×C₄	—	268	1	no	—
149	C₂²×C₄	D₄×D₆₇	1	2	yes	—
150	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
151	C₂×C₈	—	134	4	no	—
152	C₂×C₈	—	134	4	no	—
153	C₂×C₈	—	134	4	no	—
154	C₂×C₈	C₂²⋊Dic₆₇	1	4	yes	—
155	C₂×C₈	#11761615	1	4	yes	—
156	C₂×C₈	#11761618	1	4	yes	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	268	2	no	—
162	C₄⋊C₄	—	268	2	no	—
163	C₄⋊C₄	—	268	4	no	—
164	C₄⋊C₄	—	268	4	no	—
165	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
166	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
167	C₄²	#11761627	1	2	yes	—
168	C₂³	#14524039	1	1	yes	characteristic
169	Q₈	—	4	8	no	—
170	Q₈	—	8	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	536	1	no	—
175	C₂×C₄	—	536	1	no	—
176	C₂×C₄	—	536	2	no	—
177	C₂×C₄	—	536	2	no	—
178	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
179	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
180	C₂×C₄	#14524014	1	2	yes	—
181	C₂×C₄	#14524030	1	1	yes	characteristic
182	C₂×C₄	C₂×#11761615	1	2	yes	—
183	C₂×C₄	C₂×#11761618	1	2	yes	—
184	C₂×C₄	#14524070	1	4	yes	—
185	C₂×C₄	#14524072	1	4	yes	—
186	C₈	—	268	4	no	—
187	C₈	#14523967	1	8	yes	complemented
188	C₂²	#16863299	1	1	yes	characteristic
189	C₂²	#16863770	1	1	yes	characteristic
190	C₂²	#16863775	1	1	yes	characteristic
191	C₂²	#16863782	1	2	yes	—
192	C₂²	#16863787	1	2	yes	—
193	C₄	—	8	4	no	—
194	C₄	—	1072	1	no	—
195	C₄	—	1072	1	no	—
196	C₄	C₂×#14523967	1	2	yes	—
197	C₄	#16863497	1	4	yes	—
198	C₄	#16863639	1	4	yes	—
199	C₄	#16863746	1	2	yes	—
200	C₂	#22559838	1	1	yes	characteristic
201	C₂	#22559843	1	1	yes	characteristic
202	C₂	#22559883	1	2	yes	—
203	C₂	#22561012	1	1	yes	characteristic
204	C₂	#22561092	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22561171

C₂

yes

maximal

#22561012

C₂

yes

maximal

#22560369

C₂

yes

maximal, characteristic

C₂×#16863117

C₂

yes

maximal, characteristic

C₂×#498×C₆₇

C₂

yes

maximal, characteristic

C₂×#14523977

—

C₂×#14523967

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

#16863266

C₂²

yes

characteristic

#16863266

C₂²

yes

characteristic

#16863158

—

#16863117

—

C₂×C₄⋊Q₈×C₆₇

C₂²

yes

—

#498×C₆₇

—

#498×C₆₇

C₄

yes

complemented

C₂×#106×C₆₇

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

D₄

yes

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

#14523977

—

#14523967

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₂²×Q₈×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#106×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₂×C₈×C₆₇

Q₁₆

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

#127

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

#124

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

—

C₂×C₄×C₆₇

C₂×Q₁₆

yes

—

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

#103

yes

complemented

C₆₇⋊C₈

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₂²×C₆₇

#497

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₂×#103

yes

—

C₄×C₆₇

#440

yes

—

C₄×C₆₇

#468

yes

—

C₄×C₆₇

#487

yes

—

Dic₆₇

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

100

C₂×C₆₇

#1723

yes

—

101

C₂×C₆₇

#1956

yes

characteristic

102

C₂×C₆₇

#1973

yes

—

103

C₂×#498

D₆₇

yes

characteristic

104

#1990

—

105

#1956

—

106

#1845

—

107

C₂×#322

—

108

C₆₇

#9841

yes

characteristic, complemented, Sylow

109

C₂×C₄⋊Q₈

C₂×D₆₇

yes

—

110

#498

—

111

#498

Dic₆₇

yes

complemented

112

C₂×#107

—

134

—

113

C₂×#106

—

114

C₂×#103

—

134

—

115

C₂×C₄×C₈

—

116

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

117

#384

—

118

#384

—

119

#340

—

134

—

120

#322

—

134

—

121

C₂²×Q₈

—

122

C₂²×C₈

—

123

C₂²×C₈

C₂²⋊D₆₇

yes

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

C₂×Dic₆₇

yes

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

134

—

131

C₂×C₄⋊C₄

—

134

—

132

C₂×C₄⋊C₄

—

134

—

133

C₂×C₄⋊C₄

—

134

—

134

C₂×C₄²

C₂²×D₆₇

yes

characteristic

135

#107

—

268

—

136

#106

—

137

#103

—

268

—

138

C₄×C₈

—

134

—

139

C₄×C₈

—

134

—

140

C₄×C₈

C₂×Dic₆₇

yes

—

141

C₄×C₈

C₂²⋊D₆₇

yes

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂²×C₄

—

147

C₂²×C₄

—

268

—

148

C₂²×C₄

—

268

—

149

C₂²×C₄

D₄×D₆₇

yes

—

150

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

151

C₂×C₈

—

134

—

152

C₂×C₈

—

134

—

153

C₂×C₈

—

134

—

154

C₂×C₈

C₂²⋊Dic₆₇

yes

—

155

C₂×C₈

#11761615

yes

—

156

C₂×C₈

#11761618

yes

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

268

—

162

C₄⋊C₄

—

268

—

163

C₄⋊C₄

—

268

—

164

C₄⋊C₄

—

268

—

165

C₄²

C₂²×Dic₆₇

yes

characteristic

166

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

167

C₄²

#11761627

yes

—

168

C₂³

#14524039

yes

characteristic

169

Q₈

—

170

Q₈

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

536

—

175

C₂×C₄

—

536

—

176

C₂×C₄

—

536

—

177

C₂×C₄

—

536

—

178

C₂×C₄

C₂×C₂²⋊Dic₆₇

yes

characteristic

179

C₂×C₄

D₄×Dic₆₇

yes

—

180

C₂×C₄

#14524014

yes

—

181

C₂×C₄

#14524030

yes

characteristic

182

C₂×C₄

C₂×#11761615

yes

—

183

C₂×C₄

C₂×#11761618

yes

—

184

C₂×C₄

#14524070

yes

—

185

C₂×C₄

#14524072

yes

—

186

C₈

—

268

—

187

C₈

#14523967

yes

complemented

188

C₂²

#16863299

yes

characteristic

189

C₂²

#16863770

yes

characteristic

190

C₂²

#16863775

yes

characteristic

191

C₂²

#16863782

yes

—

192

C₂²

#16863787

yes

—

193

C₄

—

194

C₄

—

1072

—

195

C₄

—

1072

—

196

C₄

C₂×#14523967

yes

—

197

C₄

#16863497

yes

—

198

C₄

#16863639

yes

—

199

C₄

#16863746

yes

—

200

C₂

#22559838

yes

characteristic

201

C₂

#22559843

yes

characteristic

202

C₂

#22559883

yes

—

203

C₂

#22561012

yes

characteristic

204

C₂

#22561092

yes

—

	subgroup	G/H
center	C₂³	#14524039
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#498×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#127
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#498	D₆₇
hypocenter	C₆₇	#9841
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

subgroup

G/H

center

C₂³

#14524039

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₂×#498×C₆₇

C₂

socle, prime radical

C₂³×C₆₇

#127

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

N	H ≅ G/N	type
15. #498×C₆₇	C₄	semidirect
83. C₈×C₆₇	#103	semidirect
108. C₆₇	#9841	semidirect
111. #498	Dic₆₇	semidirect
187. C₈	#14523967	semidirect

H ≅ G/N

type

15. #498×C₆₇

C₄

semidirect

83. C₈×C₆₇

#103

semidirect

108. C₆₇

#9841

semidirect

111. #498

Dic₆₇

semidirect

187. C₈

#14523967

semidirect

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

subgroup	index
#22561171	2
#22561012	2
#22560369	2
C₂×#16863117	2
C₂×#498×C₆₇	2
#9841	67

subgroup

index

C₂×#498×C₆₇

Subgroups — #30755693

Conjugacy classes of subgroups

Complements

Maximal subgroups