Group #30755291 — subgroups

Subgroups — #30755291

Order 17152.

414 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 159 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	#22560381	C₂	1	1	yes	maximal, characteristic
2	#22558894	C₂	1	2	yes	maximal
3	#22558890	C₂	1	2	yes	maximal
4	#22558492	C₂	1	1	yes	maximal, characteristic
5	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	1	yes	characteristic
7	#16863449	C₂²	1	1	yes	characteristic
8	C₂×#14523979	—	2	2	no	—
9	#16863420	C₂²	1	2	yes	—
10	#16863145	—	2	4	no	—
11	#16863142	—	2	4	no	—
12	#16863117	—	2	1	no	—
13	C₈×C₆₇⋊C₈	—	2	1	no	—
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
15	#500×C₆₇	—	2	1	no	—
16	#500×C₆₇	—	2	1	no	—
17	#500×C₆₇	C₄	1	4	yes	—
18	C₂×#107×C₆₇	—	2	2	no	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#11761612	—	2	2	no	—
21	#14523979	—	4	4	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
25	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
26	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
27	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
28	C₄×C₆₇⋊C₈	—	2	1	no	—
29	C₄×C₆₇⋊C₈	—	2	1	no	—
30	C₂²×Q₈×C₆₇	—	2	2	no	—
31	C₂²×C₈×C₆₇	D₄	1	2	yes	—
32	C₄⋊Q₈×C₆₇	—	2	2	no	—
33	C₄⋊Q₈×C₆₇	—	2	2	no	—
34	C₄⋊Q₈×C₆₇	—	2	2	no	—
35	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
38	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
39	#107×C₆₇	—	2	4	no	—
40	#107×C₆₇	—	2	4	no	—
41	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
43	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
44	#11761612	—	4	4	no	—
45	C₂×C₆₇⋊C₈	—	2	2	no	—
46	C₂×C₆₇⋊C₈	—	2	2	no	—
47	C₂×C₆₇⋊C₈	—	4	2	no	—
48	C₂×Q₈×C₆₇	—	2	4	no	—
49	C₂×Q₈×C₆₇	—	4	2	no	—
50	C₂×Q₈×C₆₇	—	8	4	no	—
51	C₂²×C₄×C₆₇	—	4	2	no	—
52	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
55	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
58	C₂×C₈×C₆₇	Q₁₆	1	4	yes	—
59	C₄⋊C₄×C₆₇	—	2	4	no	—
60	C₄⋊C₄×C₆₇	—	2	4	no	—
61	C₄⋊C₄×C₆₇	—	4	2	no	—
62	C₄⋊C₄×C₆₇	—	4	2	no	—
63	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂³×C₆₇	#124	1	1	yes	characteristic
68	Q₈×C₆₇	—	8	4	no	—
69	Q₈×C₆₇	—	8	4	no	—
70	C₂×C₄×C₆₇	—	2	2	no	—
71	C₂×C₄×C₆₇	—	2	2	no	—
72	C₂×C₄×C₆₇	—	4	4	no	—
73	C₂×C₄×C₆₇	—	8	4	no	—
74	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
78	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
79	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
81	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
82	C₈×C₆₇	#103	1	4	yes	—
83	C₈×C₆₇	#103	1	4	yes	—
84	C₆₇⋊C₈	—	4	2	no	—
85	C₆₇⋊C₈	—	4	2	no	—
86	C₆₇⋊C₈	—	8	2	no	—
87	C₂²×C₆₇	—	2	1	no	—
88	C₂²×C₆₇	—	2	1	no	—
89	C₂²×C₆₇	#390	1	1	yes	characteristic
90	C₂²×C₆₇	#487	1	2	yes	—
91	C₄×C₆₇	—	2	1	no	—
92	C₄×C₆₇	—	2	1	no	—
93	C₄×C₆₇	—	8	4	no	—
94	C₄×C₆₇	C₂×#103	1	2	yes	—
95	C₄×C₆₇	#472	1	2	yes	—
96	C₄×C₆₇	#473	1	2	yes	—
97	C₄×C₆₇	#492	1	1	yes	characteristic
98	C₄×C₆₇	#494	1	1	yes	characteristic
99	#9771	—	67	1	no	maximal, Sylow
100	C₂×C₆₇	—	2	2	no	—
101	C₂×C₆₇	#1956	1	1	yes	characteristic
102	C₂×C₆₇	#1978	1	1	yes	characteristic
103	C₂×C₆₇	#1979	1	1	yes	characteristic
104	C₂×#500	D₆₇	1	1	yes	characteristic
105	#1848	—	67	1	no	—
106	#1549	—	67	2	no	—
107	#1548	—	67	2	no	—
108	#1450	—	67	1	no	—
109	C₆₇	#9771	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
111	#500	—	2	1	no	—
112	#500	—	2	1	no	—
113	#500	Dic₆₇	1	4	yes	—
114	#431	—	67	1	no	—
115	#431	—	67	1	no	—
116	C₂×#108	—	134	2	no	—
117	C₂×#107	—	2	2	no	—
118	#423	—	67	2	no	—
119	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
120	#333	—	134	4	no	—
121	#332	—	134	4	no	—
122	#322	—	134	1	no	—
123	C₈²	—	134	1	no	—
124	C₂²×Q₈	—	2	2	no	—
125	C₂×M₁₆	—	134	2	no	—
126	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
127	C₄⋊Q₈	—	2	2	no	—
128	C₄⋊Q₈	—	2	2	no	—
129	C₄⋊Q₈	—	2	2	no	—
130	C₄⋊Q₈	C₂×Dic₆₇	1	4	yes	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
134	#108	—	268	4	no	—
135	#107	—	2	4	no	—
136	#107	—	2	4	no	—
137	C₄⋊C₈	—	67	2	no	—
138	C₄⋊C₈	—	67	2	no	—
139	C₄⋊C₈	—	134	2	no	—
140	C₄⋊C₈	—	134	2	no	—
141	C₂²⋊C₈	—	134	1	no	—
142	C₂²⋊C₈	—	134	1	no	—
143	C₄×C₈	—	134	1	no	—
144	C₄×C₈	—	134	1	no	—
145	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
146	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
147	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
148	C₂×Q₈	—	2	4	no	—
149	C₂×Q₈	—	4	2	no	—
150	C₂×Q₈	—	8	4	no	—
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
153	C₂²×C₄	#11761627	1	1	yes	characteristic
154	C₂²×C₄	#11761627	1	1	yes	characteristic
155	M₁₆	—	268	4	no	—
156	C₂×C₈	—	268	2	no	—
157	C₂×C₈	—	134	2	no	—
158	C₂×C₈	—	134	2	no	—
159	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
160	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
161	C₂×C₈	#11761615	1	4	yes	—
162	C₂×C₈	#11761618	1	4	yes	—
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
168	C₄²	D₄×D₆₇	1	1	yes	characteristic
169	C₄²	D₄×D₆₇	1	1	yes	characteristic
170	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
171	C₂³	#14524030	1	1	yes	characteristic
172	Q₈	—	8	4	no	—
173	Q₈	—	8	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	—	8	4	no	—
178	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
179	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
180	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
181	C₂×C₄	#14524014	1	1	yes	characteristic
182	C₂×C₄	#14524014	1	1	yes	characteristic
183	C₂×C₄	#14524039	1	1	yes	characteristic
184	C₂×C₄	C₂×#11761615	1	2	yes	—
185	C₂×C₄	C₂×#11761618	1	2	yes	—
186	C₈	—	268	2	no	—
187	C₈	—	268	2	no	—
188	C₈	—	536	2	no	—
189	C₈	#14523967	1	4	yes	—
190	C₈	#14523967	1	4	yes	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	#16863299	1	1	yes	characteristic
194	C₂²	#16863746	1	2	yes	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	8	4	no	—
198	C₄	C₂×#14523967	1	2	yes	—
199	C₄	#16863663	1	2	yes	—
200	C₄	#16863668	1	2	yes	—
201	C₄	#16863770	1	1	yes	characteristic
202	C₄	#16863775	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	#22561012	1	1	yes	characteristic
205	C₂	#22561120	1	1	yes	characteristic
206	C₂	#22561125	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560381

C₂

yes

maximal, characteristic

#22558894

C₂

yes

maximal

#22558890

C₂

yes

maximal

#22558492

C₂

yes

maximal, characteristic

C₂×#500×C₆₇

C₂

yes

maximal, characteristic

#16863449

C₂²

yes

characteristic

#16863449

C₂²

yes

characteristic

C₂×#14523979

—

#16863420

C₂²

yes

—

#16863145

—

#16863142

—

#16863117

—

C₈×C₆₇⋊C₈

—

C₂×C₄⋊Q₈×C₆₇

C₂²

yes

—

#500×C₆₇

—

#500×C₆₇

—

#500×C₆₇

C₄

yes

—

C₂×#107×C₆₇

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×#11761612

—

#14523979

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

—

C₂²⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

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

#107×C₆₇

—

#107×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

characteristic

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

C₂²×C₄×C₆₇

yes

C₂²×C₄×C₆₇

yes

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

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

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₂³×C₆₇

#124

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

C₂×SD₁₆

yes

—

C₂×C₄×C₆₇

C₂×Q₁₆

yes

—

C₈×C₆₇

#103

yes

—

C₈×C₆₇

#103

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#390

yes

characteristic

C₂²×C₆₇

#487

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×#103

yes

—

C₄×C₆₇

#472

yes

—

C₄×C₆₇

#473

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

—

100

C₂×C₆₇

—

101

C₂×C₆₇

#1956

yes

characteristic

102

C₂×C₆₇

#1978

yes

characteristic

103

C₂×C₆₇

#1979

yes

characteristic

104

C₂×#500

D₆₇

yes

characteristic

105

#1848

—

106

#1549

—

107

#1548

—

108

#1450

—

109

C₆₇

#9771

yes

characteristic, complemented, Sylow

110

C₂×C₄⋊Q₈

C₂×D₆₇

yes

—

111

#500

—

112

#500

—

113

#500

Dic₆₇

yes

—

114

#431

—

115

#431

—

116

C₂×#108

—

134

—

117

C₂×#107

—

118

#423

—

119

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

120

#333

—

134

—

121

#332

—

134

—

122

#322

—

134

—

123

C₈²

—

134

—

124

C₂²×Q₈

—

125

C₂×M₁₆

—

134

—

126

C₂²×C₈

C₂²⋊D₆₇

yes

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

130

C₄⋊Q₈

C₂×Dic₆₇

yes

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄²

C₂²×D₆₇

yes

characteristic

134

#108

—

268

—

135

#107

—

136

#107

—

137

C₄⋊C₈

—

138

C₄⋊C₈

—

139

C₄⋊C₈

—

134

—

140

C₄⋊C₈

—

134

—

141

C₂²⋊C₈

—

134

—

142

C₂²⋊C₈

—

134

—

143

C₄×C₈

—

134

—

144

C₄×C₈

—

134

—

145

C₄×C₈

C₂×Dic₆₇

yes

—

146

C₄×C₈

C₂²⋊D₆₇

yes

characteristic

147

C₄×C₈

C₂²⋊D₆₇

yes

characteristic

148

C₂×Q₈

—

149

C₂×Q₈

—

150

C₂×Q₈

—

151

C₂²×C₄

—

152

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

153

C₂²×C₄

#11761627

yes

characteristic

154

C₂²×C₄

#11761627

yes

characteristic

155

M₁₆

—

268

—

156

C₂×C₈

—

268

—

157

C₂×C₈

—

134

—

158

C₂×C₈

—

134

—

159

C₂×C₈

C₂²⋊Dic₆₇

yes

—

160

C₂×C₈

C₂²⋊Dic₆₇

yes

—

161

C₂×C₈

#11761615

yes

—

162

C₂×C₈

#11761618

yes

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄²

C₂²×Dic₆₇

yes

characteristic

168

C₄²

D₄×D₆₇

yes

characteristic

169

C₄²

D₄×D₆₇

yes

characteristic

170

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

171

C₂³

#14524030

yes

characteristic

172

Q₈

—

173

Q₈

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

C₂×C₂²⋊Dic₆₇

yes

characteristic

179

C₂×C₄

D₄×Dic₆₇

yes

characteristic

180

C₂×C₄

D₄×Dic₆₇

yes

characteristic

181

C₂×C₄

#14524014

yes

characteristic

182

C₂×C₄

#14524014

yes

characteristic

183

C₂×C₄

#14524039

yes

characteristic

184

C₂×C₄

C₂×#11761615

yes

—

185

C₂×C₄

C₂×#11761618

yes

—

186

C₈

—

268

—

187

C₈

—

268

—

188

C₈

—

536

—

189

C₈

#14523967

yes

—

190

C₈

#14523967

yes

—

191

C₂²

—

192

C₂²

—

193

C₂²

#16863299

yes

characteristic

194

C₂²

#16863746

yes

—

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

C₂×#14523967

yes

—

199

C₄

#16863663

yes

—

200

C₄

#16863668

yes

—

201

C₄

#16863770

yes

characteristic

202

C₄

#16863775

yes

characteristic

203

C₂

—

204

C₂

#22561012

yes

characteristic

205

C₂

#22561120

yes

characteristic

206

C₂

#22561125

yes

characteristic

	subgroup	G/H
center	C₂²	#16863299
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#500×C₆₇	C₂
socle	C₂²×C₆₇	#390
prime radical	C₂³×C₆₇	#124
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#500	D₆₇
hypocenter	C₆₇	#9771
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

subgroup

G/H

center

C₂²

#16863299

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₂×#500×C₆₇

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

N	H ≅ G/N	type
109. C₆₇	#9771	semidirect

H ≅ G/N

type

109. C₆₇

#9771

semidirect

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

subgroup	index
#22560381	2
#22558894	2
#22558890	2
#22558492	2
C₂×#500×C₆₇	2
#9771	67

subgroup

index

C₂×#500×C₆₇

Subgroups — #30755291

Conjugacy classes of subgroups

Complements

Maximal subgroups