Group #30765046 — subgroups

Subgroups — #30765046

Order 17152.

306 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 115 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	#22562012	C₂	1	2	yes	maximal, complemented
2	#22558151	C₂	1	2	yes	maximal, complemented
3	#22558135	C₂	1	2	yes	maximal
4	C₆₇×#2922	C₂	1	1	yes	maximal, characteristic
5	C₄⋊(C₆₇⋊C₁₆)	—	2	2	no	—
6	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	4	yes	—
7	#16863172	—	2	2	no	—
8	C₄×#41×C₆₇	C₄	1	1	yes	characteristic
9	#433×C₆₇	—	2	2	no	—
10	#431×C₆₇	—	2	2	no	—
11	#424×C₆₇	C₄	1	1	yes	characteristic
12	#424×C₆₇	C₂²	1	1	yes	characteristic
13	#408×C₆₇	C₄	1	1	yes	characteristic
14	#408×C₆₇	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₆₇	C₄	1	1	yes	characteristic
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₆₇⋊C₁₆	—	2	4	no	—
18	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
19	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
20	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
23	C₄×Q₈×C₆₇	C₈	1	2	yes	—
24	C₄×D₄×C₆₇	—	2	2	no	—
25	C₄×D₄×C₆₇	C₈	1	2	yes	—
26	#117×C₆₇	—	2	1	no	—
27	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₆₇	—	2	1	no	—
29	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₄⋊C₈×C₆₇	—	2	2	no	—
31	C₄⋊C₈×C₆₇	—	2	2	no	—
32	C₂²⋊C₈×C₆₇	—	2	2	no	—
33	C₂²⋊C₈×C₆₇	—	2	2	no	—
34	#97×C₆₇	—	2	1	no	—
35	C₄×C₈×C₆₇	—	2	1	no	—
36	C₄×C₈×C₆₇	D₄	1	2	yes	—
37	C₄×C₈×C₆₇	D₄	1	2	yes	—
38	C₆₇⋊C₁₆	—	4	4	no	—
39	#41×C₆₇	—	4	1	no	—
40	#41×C₆₇	—	4	1	no	—
41	C₂×Q₈×C₆₇	C₂×C₈	1	1	yes	characteristic
42	C₂×D₄×C₆₇	—	2	1	no	—
43	C₂×D₄×C₆₇	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	—	2	2	no	—
45	C₂²×C₄×C₆₇	—	2	1	no	—
46	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
47	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
48	M₁₆×C₆₇	—	4	1	no	—
49	M₁₆×C₆₇	—	4	1	no	—
50	C₂×C₈×C₆₇	—	2	1	no	—
51	C₂×C₈×C₆₇	—	2	1	no	—
52	C₂×C₈×C₆₇	—	4	1	no	—
53	C₂×C₈×C₆₇	—	4	1	no	—
54	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
56	C₄⋊C₄×C₆₇	—	2	2	no	—
57	C₄⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
58	C₂²⋊C₄×C₆₇	—	2	1	no	—
59	C₂²⋊C₄×C₆₇	—	2	2	no	—
60	C₄²×C₆₇	—	2	2	no	—
61	C₄²×C₆₇	—	2	1	no	—
62	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₆₇	C₂×C₈	1	2	yes	—
65	C₂³×C₆₇	—	2	1	no	—
66	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
67	Q₈×C₆₇	—	4	1	no	—
68	D₄×C₆₇	—	4	1	no	—
69	D₄×C₆₇	—	4	2	no	—
70	C₂×C₄×C₆₇	—	2	1	no	—
71	C₂×C₄×C₆₇	—	2	1	no	—
72	C₂×C₄×C₆₇	—	2	2	no	—
73	C₂×C₄×C₆₇	—	2	1	no	—
74	C₂×C₄×C₆₇	—	4	2	no	—
75	C₂×C₄×C₆₇	—	4	4	no	—
76	C₂×C₄×C₆₇	—	4	1	no	—
77	C₂×C₄×C₆₇	—	4	1	no	—
78	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
79	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
80	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
82	C₈×C₆₇	—	2	2	no	—
83	C₈×C₆₇	—	2	2	no	—
84	C₈×C₆₇	—	4	1	no	—
85	C₈×C₆₇	—	4	1	no	—
86	C₂²×C₆₇	—	2	2	no	—
87	C₂²×C₆₇	—	2	1	no	—
88	C₂²×C₆₇	—	4	2	no	—
89	C₂²×C₆₇	—	4	1	no	—
90	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
91	C₄×C₆₇	—	2	2	no	—
92	C₄×C₆₇	—	2	2	no	—
93	C₄×C₆₇	—	2	1	no	—
94	C₄×C₆₇	—	4	2	no	—
95	C₄×C₆₇	—	4	1	no	—
96	C₄×C₆₇	#410	1	1	yes	characteristic
97	C₄×C₆₇	#413	1	1	yes	characteristic
98	#11463	—	67	1	no	maximal, Sylow
99	C₂×C₆₇	—	4	1	no	—
100	C₂×C₆₇	—	8	1	no	—
101	C₂×C₆₇	#1458	1	1	yes	characteristic
102	C₂×C₆₇	#2117	1	2	yes	—
103	#2922	D₆₇	1	1	yes	characteristic
104	#2163	—	67	2	no	—
105	#1323	—	67	2	no	—
106	#1315	—	67	2	no	—
107	C₆₇	#11463	1	1	yes	characteristic, complemented, Sylow
108	C₄×#41	Dic₆₇	1	1	yes	characteristic
109	#433	—	2	2	no	—
110	#431	—	2	2	no	—
111	#424	Dic₆₇	1	1	yes	characteristic
112	#424	C₂×D₆₇	1	1	yes	characteristic
113	#408	Dic₆₇	1	1	yes	characteristic
114	#408	C₂×D₆₇	1	1	yes	characteristic
115	C₄×M₁₆	Dic₆₇	1	1	yes	characteristic
116	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
117	C₄⋊C₁₆	—	134	2	no	—
118	C₂²⋊C₁₆	—	67	4	no	—
119	#346	—	134	2	no	—
120	C₂×#41	C₂×Dic₆₇	1	1	yes	characteristic
121	C₂×M₁₆	C₂×Dic₆₇	1	1	yes	characteristic
122	C₂×M₁₆	C₂×Dic₆₇	1	1	yes	characteristic
123	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
124	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
125	C₄×Q₈	C₆₇⋊C₈	1	2	yes	—
126	C₄×D₄	—	2	2	no	—
127	C₄×D₄	C₆₇⋊C₈	1	2	yes	—
128	#117	—	2	1	no	—
129	#117	C₂×Dic₆₇	1	1	yes	characteristic
130	C₂×C₄²	—	2	1	no	—
131	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
132	C₂×C₁₆	—	134	4	no	—
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₂²⋊C₈	—	2	2	no	—
136	C₂²⋊C₈	—	2	2	no	—
137	#97	—	2	1	no	—
138	C₄×C₈	—	2	1	no	—
139	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
140	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
141	#41	—	4	1	no	—
142	#41	—	4	1	no	—
143	C₂×Q₈	C₂×C₆₇⋊C₈	1	1	yes	characteristic
144	C₂×D₄	—	2	1	no	—
145	C₂×D₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	#11761612	1	2	yes	—
149	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
150	M₁₆	—	4	1	no	—
151	M₁₆	—	4	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	4	1	no	—
155	C₂×C₈	—	4	1	no	—
156	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
157	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	C₂×C₆₇⋊C₈	1	2	yes	—
160	C₂²⋊C₄	—	2	1	no	—
161	C₂²⋊C₄	—	2	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	1	no	—
164	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
165	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
166	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
167	C₁₆	—	268	4	no	—
168	C₂³	—	2	1	no	—
169	C₂³	C₂×#11761612	1	1	yes	characteristic
170	Q₈	—	4	1	no	—
171	D₄	—	4	1	no	—
172	D₄	—	4	2	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	4	1	no	—
180	C₂×C₄	—	4	1	no	—
181	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
182	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
183	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
184	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	2	2	no	—
187	C₈	—	4	1	no	—
188	C₈	—	4	1	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	4	2	no	—
192	C₂²	—	4	1	no	—
193	C₂²	C₂×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	2	1	no	—
197	C₄	—	4	2	no	—
198	C₄	—	4	1	no	—
199	C₄	#16863370	1	1	yes	characteristic
200	C₄	#16863380	1	1	yes	characteristic
201	C₂	—	4	1	no	—
202	C₂	—	8	1	no	—
203	C₂	#22558523	1	1	yes	characteristic
204	C₂	#22561837	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22562012

C₂

yes

maximal, complemented

#22558151

C₂

yes

maximal, complemented

#22558135

C₂

yes

maximal

C₆₇×#2922

C₂

yes

maximal, characteristic

C₄⋊(C₆₇⋊C₁₆)

—

C₂²⋊(C₆₇⋊C₁₆)

C₂²

yes

—

#16863172

—

C₄×#41×C₆₇

C₄

yes

characteristic

#433×C₆₇

—

#431×C₆₇

—

#424×C₆₇

C₄

yes

characteristic

#424×C₆₇

C₂²

yes

characteristic

#408×C₆₇

C₄

yes

characteristic

#408×C₆₇

C₂²

yes

characteristic

C₄×M₁₆×C₆₇

C₄

yes

characteristic

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₆₇⋊C₁₆

—

C₂×#41×C₆₇

C₂×C₄

yes

characteristic

C₂×M₁₆×C₆₇

C₂×C₄

yes

characteristic

C₂×M₁₆×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₄×Q₈×C₆₇

C₈

yes

—

C₄×D₄×C₆₇

—

C₄×D₄×C₆₇

C₈

yes

—

#117×C₆₇

—

#117×C₆₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

#97×C₆₇

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₆₇⋊C₁₆

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

C₂×C₈

yes

characteristic

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

M₁₆

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂×C₈

yes

—

C₂²⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂²⋊C₄

yes

characteristic

C₄²×C₆₇

C₂²⋊C₄

yes

characteristic

C₄²×C₆₇

C₂×C₈

yes

—

C₂³×C₆₇

—

C₂³×C₆₇

C₂×M₁₆

yes

characteristic

Q₈×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×C₄×C₆₇

C₂²⋊C₈

yes

—

C₂×C₄×C₆₇

C₂²⋊C₈

yes

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₈

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₂×C₆₇

—

100

C₂×C₆₇

—

101

C₂×C₆₇

#1458

yes

characteristic

102

C₂×C₆₇

#2117

yes

—

103

#2922

D₆₇

yes

characteristic

104

#2163

—

105

#1323

—

106

#1315

—

107

C₆₇

#11463

yes

characteristic, complemented, Sylow

108

C₄×#41

Dic₆₇

yes

characteristic

109

#433

—

110

#431

—

111

#424

Dic₆₇

yes

characteristic

112

#424

C₂×D₆₇

yes

characteristic

113

#408

Dic₆₇

yes

characteristic

114

#408

C₂×D₆₇

yes

characteristic

115

C₄×M₁₆

Dic₆₇

yes

characteristic

116

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

117

C₄⋊C₁₆

—

134

—

118

C₂²⋊C₁₆

—

119

#346

—

134

—

120

C₂×#41

C₂×Dic₆₇

yes

characteristic

121

C₂×M₁₆

C₂×Dic₆₇

yes

characteristic

122

C₂×M₁₆

C₂×Dic₆₇

yes

characteristic

123

C₂²×C₈

C₂×Dic₆₇

yes

characteristic

124

C₂²×C₈

C₂²×D₆₇

yes

characteristic

125

C₄×Q₈

C₆₇⋊C₈

yes

—

126

C₄×D₄

—

127

C₄×D₄

C₆₇⋊C₈

yes

—

128

#117

—

129

#117

C₂×Dic₆₇

yes

characteristic

130

C₂×C₄²

—

131

C₂×C₄²

C₂×Dic₆₇

yes

characteristic

132

C₂×C₁₆

—

134

—

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

C₂²⋊C₈

—

136

C₂²⋊C₈

—

137

#97

—

138

C₄×C₈

—

139

C₄×C₈

C₂²⋊D₆₇

yes

—

140

C₄×C₈

C₂²⋊D₆₇

yes

—

141

#41

—

142

#41

—

143

C₂×Q₈

C₂×C₆₇⋊C₈

yes

characteristic

144

C₂×D₄

—

145

C₂×D₄

C₂×C₆₇⋊C₈

yes

characteristic

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

#11761612

yes

—

149

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

150

M₁₆

—

151

M₁₆

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

C₂²⋊Dic₆₇

yes

—

157

C₂×C₈

C₂×C₂²⋊D₆₇

yes

—

158

C₄⋊C₄

—

159

C₄⋊C₄

C₂×C₆₇⋊C₈

yes

—

160

C₂²⋊C₄

—

161

C₂²⋊C₄

—

162

C₄²

—

163

C₄²

—

164

C₄²

C₂²⋊Dic₆₇

yes

characteristic

165

C₄²

C₂²⋊Dic₆₇

yes

characteristic

166

C₄²

C₂×C₆₇⋊C₈

yes

—

167

C₁₆

—

268

—

168

C₂³

—

169

C₂³

C₂×#11761612

yes

characteristic

170

Q₈

—

171

D₄

—

172

D₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

C₂²⋊(C₆₇⋊C₈)

yes

—

182

C₂×C₄

C₂²⋊(C₆₇⋊C₈)

yes

—

183

C₂×C₄

C₂×C₂²⋊Dic₆₇

yes

characteristic

184

C₂×C₄

C₂²×C₆₇⋊C₈

yes

characteristic

185

C₈

—

186

C₈

—

187

C₈

—

188

C₈

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

—

193

C₂²

C₂×C₂²⋊(C₆₇⋊C₈)

yes

characteristic

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

#16863370

yes

characteristic

200

C₄

#16863380

yes

characteristic

201

C₂

—

202

C₂

—

203

C₂

#22558523

yes

characteristic

204

C₂

#22561837

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#2922	C₂
socle	C₂²×C₆₇	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₆₇	C₂×C₄
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#2922	D₆₇
hypocenter	C₆₇	#11463
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2922

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊Dic₆₇

commutator

C₂×C₄×C₆₇

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#2922

C₂

socle

C₂²×C₆₇

C₂×C₂²⋊C₈

prime radical

C₂×#41×C₆₇

C₂×C₄

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2922

N	H ≅ G/N	type
1. #22562012	C₂	semidirect
2. #22558151	C₂	semidirect
107. C₆₇	#11463	semidirect

H ≅ G/N

type

1. #22562012

C₂

semidirect

2. #22558151

C₂

semidirect

107. C₆₇

#11463

semidirect

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

subgroup	index
#22562012	2
#22558151	2
#22558135	2
C₆₇×#2922	2
#11463	67

subgroup

index

C₆₇×#2922

Subgroups — #30765046

Conjugacy classes of subgroups

Complements

Maximal subgroups