Group #30755653 — subgroups

Subgroups — #30755653

Order 17152.

494 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 163 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	C₂×#16863801	C₂	1	1	yes	maximal, characteristic
2	#22561169	C₂	1	2	yes	maximal
3	#22561010	C₂	1	2	yes	maximal
4	C₂×#16863118	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1834	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524043	C₂²	1	2	yes	—
7	#16863801	—	2	2	no	—
8	#16863801	C₄	1	4	yes	complemented
9	C₂×#14523977	—	2	2	no	—
10	C₂×#14523966	—	2	4	no	—
11	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	#16863158	—	2	4	no	—
14	#16863118	—	2	2	no	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	#384×C₆₇	C₂²	1	2	yes	—
17	#340×C₆₇	—	2	2	no	—
18	C₂²×Dic₁₃₄	—	2	2	no	—
19	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
20	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
21	#14524043	—	2	2	no	—
22	#14524043	—	2	2	no	—
23	#14524043	—	2	2	no	—
24	#14524043	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊Dic₆₇	—	2	2	no	—
26	C₂×C₄⋊Dic₆₇	—	2	2	no	—
27	#14523977	—	2	8	no	—
28	#14523966	—	4	8	no	—
29	C₄×C₆₇⋊C₈	—	2	1	no	—
30	C₄×C₆₇⋊C₈	—	2	1	no	—
31	C₄×C₆₇⋊C₈	—	2	1	no	—
32	C₄×C₆₇⋊C₈	—	2	1	no	—
33	C₂²×C₈×C₆₇	D₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
37	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
38	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₆₇	D₄	1	2	yes	—
40	C₂×Dic₁₃₄	—	2	4	no	—
41	C₂×Dic₁₃₄	—	4	4	no	—
42	C₂×Dic₁₃₄	—	4	4	no	—
43	C₂×Dic₁₃₄	—	4	2	no	—
44	C₂²×Dic₆₇	—	4	2	no	—
45	C₂×C₆₇⋊C₈	—	2	2	no	—
46	C₂×C₆₇⋊C₈	—	2	2	no	—
47	C₂×C₆₇⋊C₈	—	2	2	no	—
48	C₂×C₆₇⋊C₈	—	2	4	no	—
49	C₂×C₆₇⋊C₈	Q₁₆	1	4	yes	—
50	C₄⋊Dic₆₇	—	2	4	no	—
51	C₄⋊Dic₆₇	—	2	4	no	—
52	C₄⋊Dic₆₇	—	4	2	no	—
53	C₄⋊Dic₆₇	—	4	2	no	—
54	C₂²×C₄×C₆₇	—	4	2	no	—
55	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₆₇	—	2	4	no	—
59	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
60	C₄⋊C₄×C₆₇	—	4	4	no	—
61	C₄⋊C₄×C₆₇	—	4	4	no	—
62	C₄⋊C₄×C₆₇	—	4	4	no	—
63	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₆₇	#41	1	1	yes	characteristic
66	C₄²×C₆₇	#41	1	1	yes	characteristic
67	C₂³×C₆₇	#127	1	1	yes	characteristic
68	C₂×C₄×C₆₇	—	8	2	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	1	yes	characteristic
73	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
75	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
76	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
78	C₂×C₄×C₆₇	#135	1	2	yes	—
79	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
80	C₂×C₄×C₆₇	#137	1	4	yes	—
81	C₂×Dic₆₇	—	4	4	no	—
82	C₂×Dic₆₇	—	8	2	no	—
83	C₂×Dic₆₇	—	8	2	no	—
84	Dic₁₃₄	—	4	8	no	—
85	Dic₁₃₄	—	8	4	no	—
86	C₈×C₆₇	—	2	4	no	—
87	C₆₇⋊C₈	—	2	4	no	—
88	C₆₇⋊C₈	—	4	2	no	—
89	C₂²×C₆₇	#390	1	1	yes	characteristic
90	C₂²×C₆₇	#494	1	1	yes	characteristic
91	C₂²×C₆₇	#495	1	1	yes	characteristic
92	C₂²×C₆₇	#496	1	2	yes	—
93	C₂²×C₆₇	#497	1	2	yes	—
94	C₄×C₆₇	—	16	2	no	—
95	C₄×C₆₇	#419	1	2	yes	—
96	C₄×C₆₇	C₄×Q₁₆	1	2	yes	—
97	C₄×C₆₇	#442	1	2	yes	—
98	C₄×C₆₇	#467	1	2	yes	—
99	C₄×C₆₇	#469	1	2	yes	—
100	C₄×C₆₇	#491	1	2	yes	—
101	Dic₆₇	—	8	4	no	—
102	#9834	—	67	1	no	maximal, Sylow
103	C₂×C₆₇	#1719	1	2	yes	—
104	C₂×C₆₇	#1723	1	2	yes	—
105	C₂×C₆₇	#1960	1	1	yes	characteristic
106	C₂×C₆₇	#1969	1	1	yes	characteristic
107	C₂×C₆₇	#1972	1	1	yes	characteristic
108	C₂×#500	—	67	1	no	—
109	#1990	—	67	2	no	—
110	#1956	—	67	2	no	—
111	#1834	D₆₇	1	1	yes	characteristic
112	C₂×#322	—	67	1	no	—
113	C₆₇	#9834	1	1	yes	characteristic, complemented, Sylow
114	C₂×C₄⋊Q₈	—	67	2	no	—
115	#500	—	67	4	no	—
116	#500	—	134	2	no	—
117	C₂×#107	—	134	2	no	—
118	C₂×#103	—	134	4	no	—
119	C₂×C₄×C₈	—	67	1	no	—
120	C₂×C₄×C₈	—	67	1	no	—
121	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
122	#384	C₂×D₆₇	1	2	yes	—
123	#340	—	2	2	no	—
124	#340	—	134	4	no	—
125	#322	—	134	2	no	—
126	C₂²×Q₈	—	134	2	no	—
127	C₂²×C₈	—	67	2	no	—
128	C₂²×C₈	—	67	2	no	—
129	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
130	C₄⋊Q₈	—	67	4	no	—
131	C₄⋊Q₈	—	134	2	no	—
132	C₄⋊Q₈	—	134	2	no	—
133	C₄⋊Q₈	—	134	2	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	134	2	no	—
138	C₂×C₄⋊C₄	—	134	2	no	—
139	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
140	#107	—	134	8	no	—
141	#103	—	268	8	no	—
142	C₄×C₈	—	134	1	no	—
143	C₄×C₈	—	134	1	no	—
144	C₄×C₈	—	134	1	no	—
145	C₄×C₈	—	134	1	no	—
146	C₄×C₈	C₄×D₆₇	1	2	yes	—
147	C₄×C₈	D₂₆₈	1	2	yes	—
148	C₂×Q₈	—	268	2	no	—
149	C₂×Q₈	—	268	4	no	—
150	C₂×Q₈	—	268	4	no	—
151	C₂×Q₈	—	134	4	no	—
152	C₂²×C₄	—	4	2	no	—
153	C₂²×C₄	—	268	2	no	—
154	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
155	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
156	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	67	4	no	—
159	C₂×C₈	—	134	2	no	—
160	C₂×C₈	—	134	4	no	—
161	C₂×C₈	—	134	2	no	—
162	C₂×C₈	—	134	2	no	—
163	C₂×C₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄⋊C₄	—	4	4	no	—
166	C₄⋊C₄	—	4	4	no	—
167	C₄⋊C₄	—	268	2	no	—
168	C₄⋊C₄	—	268	2	no	—
169	C₄⋊C₄	—	134	4	no	—
170	C₄⋊C₄	—	134	4	no	—
171	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
172	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
173	C₄²	#11761629	1	1	yes	characteristic
174	C₄²	#11761629	1	1	yes	characteristic
175	C₂³	#14524039	1	1	yes	characteristic
176	Q₈	—	268	8	no	—
177	Q₈	—	536	4	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	—	8	2	no	—
181	C₂×C₄	—	268	4	no	—
182	C₂×C₄	—	536	2	no	—
183	C₂×C₄	—	536	2	no	—
184	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
185	C₂×C₄	#14524005	1	1	yes	characteristic
186	C₂×C₄	#14524005	1	1	yes	characteristic
187	C₂×C₄	#14524018	1	1	yes	characteristic
188	C₂×C₄	#14524018	1	1	yes	characteristic
189	C₂×C₄	#14524032	1	1	yes	characteristic
190	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
191	C₂×C₄	#14524062	1	2	yes	—
192	C₂×C₄	#14524070	1	4	yes	—
193	C₂×C₄	#14524073	1	4	yes	—
194	C₈	—	2	4	no	—
195	C₈	—	268	2	no	—
196	C₈	—	134	4	no	—
197	C₂²	#16863297	1	1	yes	characteristic
198	C₂²	#16863774	1	1	yes	characteristic
199	C₂²	#16863779	1	1	yes	characteristic
200	C₂²	#16863782	1	2	yes	—
201	C₂²	#16863791	1	2	yes	—
202	C₄	—	16	2	no	—
203	C₄	—	536	4	no	—
204	C₄	#16863404	1	2	yes	—
205	C₄	#16863490	1	2	yes	—
206	C₄	#16863507	1	2	yes	—
207	C₄	#16863638	1	2	yes	—
208	C₄	#16863650	1	2	yes	—
209	C₄	#16863766	1	2	yes	—
210	C₂	#22559861	1	2	yes	—
211	C₂	#22559889	1	2	yes	—
212	C₂	#22561028	1	1	yes	characteristic
213	C₂	#22561068	1	1	yes	characteristic
214	C₂	#22561083	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863801

C₂

yes

maximal, characteristic

#22561169

C₂

yes

maximal

#22561010

C₂

yes

maximal

C₂×#16863118

C₂

yes

maximal, characteristic

C₆₇×#1834

C₂

yes

maximal, characteristic

C₂×#14524043

C₂²

yes

—

#16863801

—

#16863801

C₄

yes

complemented

C₂×#14523977

—

C₂×#14523966

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₆₇⋊C₈

C₂²

yes

characteristic

#16863158

—

#16863118

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#384×C₆₇

C₂²

yes

—

#340×C₆₇

—

C₂²×Dic₁₃₄

—

C₂²×C₆₇⋊C₈

D₄

yes

—

C₂²×C₆₇⋊C₈

D₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

#14523977

—

#14523966

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

Q₁₆

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

C₄²×C₆₇

yes

C₄²×C₆₇

yes

C₂³×C₆₇

yes

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×Q₁₆

yes

—

C₂×C₄×C₆₇

#135

yes

—

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₂²×C₆₇

#497

yes

—

C₄×C₆₇

—

C₄×C₆₇

#419

yes

—

C₄×C₆₇

C₄×Q₁₆

yes

—

C₄×C₆₇

#442

yes

—

C₄×C₆₇

#467

yes

—

C₄×C₆₇

#469

yes

—

100

C₄×C₆₇

#491

yes

—

101

Dic₆₇

—

102

#9834

—

maximal, Sylow

103

C₂×C₆₇

#1719

yes

—

104

C₂×C₆₇

#1723

yes

—

105

C₂×C₆₇

#1960

yes

characteristic

106

C₂×C₆₇

#1969

yes

characteristic

107

C₂×C₆₇

#1972

yes

characteristic

108

C₂×#500

—

109

#1990

—

110

#1956

—

111

#1834

D₆₇

yes

characteristic

112

C₂×#322

—

113

C₆₇

#9834

yes

characteristic, complemented, Sylow

114

C₂×C₄⋊Q₈

—

115

#500

—

116

#500

—

134

—

117

C₂×#107

—

134

—

118

C₂×#103

—

134

—

119

C₂×C₄×C₈

—

120

C₂×C₄×C₈

—

121

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

122

#384

C₂×D₆₇

yes

—

123

#340

—

124

#340

—

134

—

125

#322

—

134

—

126

C₂²×Q₈

—

134

—

127

C₂²×C₈

—

128

C₂²×C₈

—

129

C₂²×C₈

C₂²⋊D₆₇

yes

—

130

C₄⋊Q₈

—

131

C₄⋊Q₈

—

134

—

132

C₄⋊Q₈

—

134

—

133

C₄⋊Q₈

—

134

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄⋊C₄

—

136

C₂×C₄⋊C₄

—

137

C₂×C₄⋊C₄

—

134

—

138

C₂×C₄⋊C₄

—

134

—

139

C₂×C₄²

C₂²×D₆₇

yes

characteristic

140

#107

—

134

—

141

#103

—

268

—

142

C₄×C₈

—

134

—

143

C₄×C₈

—

134

—

144

C₄×C₈

—

134

—

145

C₄×C₈

—

134

—

146

C₄×C₈

C₄×D₆₇

yes

—

147

C₄×C₈

D₂₆₈

yes

—

148

C₂×Q₈

—

268

—

149

C₂×Q₈

—

268

—

150

C₂×Q₈

—

268

—

151

C₂×Q₈

—

134

—

152

C₂²×C₄

—

153

C₂²×C₄

—

268

—

154

C₂²×C₄

D₄×D₆₇

yes

characteristic

155

C₂²×C₄

D₄×D₆₇

yes

characteristic

156

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

134

—

160

C₂×C₈

—

134

—

161

C₂×C₈

—

134

—

162

C₂×C₈

—

134

—

163

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

268

—

168

C₄⋊C₄

—

268

—

169

C₄⋊C₄

—

134

—

170

C₄⋊C₄

—

134

—

171

C₄²

C₂×C₄×D₆₇

yes

characteristic

172

C₄²

C₂×D₂₆₈

yes

characteristic

173

C₄²

#11761629

yes

characteristic

174

C₄²

#11761629

yes

characteristic

175

C₂³

#14524039

yes

characteristic

176

Q₈

—

268

—

177

Q₈

—

536

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

268

—

182

C₂×C₄

—

536

—

183

C₂×C₄

—

536

—

184

C₂×C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

185

C₂×C₄

#14524005

yes

characteristic

186

C₂×C₄

#14524005

yes

characteristic

187

C₂×C₄

#14524018

yes

characteristic

188

C₂×C₄

#14524018

yes

characteristic

189

C₂×C₄

#14524032

yes

characteristic

190

C₂×C₄

Q₁₆×D₆₇

yes

—

191

C₂×C₄

#14524062

yes

—

192

C₂×C₄

#14524070

yes

—

193

C₂×C₄

#14524073

yes

—

194

C₈

—

195

C₈

—

268

—

196

C₈

—

134

—

197

C₂²

#16863297

yes

characteristic

198

C₂²

#16863774

yes

characteristic

199

C₂²

#16863779

yes

characteristic

200

C₂²

#16863782

yes

—

201

C₂²

#16863791

yes

—

202

C₄

—

203

C₄

—

536

—

204

C₄

#16863404

yes

—

205

C₄

#16863490

yes

—

206

C₄

#16863507

yes

—

207

C₄

#16863638

yes

—

208

C₄

#16863650

yes

—

209

C₄

#16863766

yes

—

210

C₂

#22559861

yes

—

211

C₂

#22559889

yes

—

212

C₂

#22561028

yes

characteristic

213

C₂

#22561068

yes

characteristic

214

C₂

#22561083

yes

characteristic

	subgroup	G/H
center	C₂³	#14524039
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1834	C₂
socle, prime radical	C₂³×C₆₇	#127
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1834	D₆₇
hypocenter	C₆₇	#9834
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

subgroup

G/H

center

C₂³

#14524039

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1834

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 ◁ #1834

N	H ≅ G/N	type
8. #16863801	C₄	semidirect
113. C₆₇	#9834	semidirect

H ≅ G/N

type

8. #16863801

C₄

semidirect

113. C₆₇

#9834

semidirect

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

subgroup	index
C₂×#16863801	2
#22561169	2
#22561010	2
C₂×#16863118	2
C₆₇×#1834	2
#9834	67

subgroup

index

C₆₇×#1834

Subgroups — #30755653

Conjugacy classes of subgroups

Complements

Maximal subgroups