Group #30765168 — subgroups

Subgroups — #30765168

Order 17152.

342 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 155 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	#22562008	C₂	1	2	yes	maximal
2	#22558149	C₂	1	4	yes	maximal
3	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₆₇⋊C₁₆)	—	2	2	no	—
5	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	4	yes	—
6	C₄×C₆₇⋊C₁₆	—	2	2	no	—
7	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
8	C₈×Q₈×C₆₇	—	2	2	no	—
9	C₈×Q₈×C₆₇	—	2	2	no	—
10	C₂×C₄⋊C₈×C₆₇	C₄	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₆₇⋊C₁₆	—	2	4	no	—
15	C₂²×Q₈×C₆₇	C₈	1	1	yes	characteristic
16	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
19	C₄×Q₈×C₆₇	—	2	2	no	—
20	C₄×Q₈×C₆₇	—	2	2	no	—
21	C₂×C₄⋊C₄×C₆₇	C₈	1	2	yes	—
22	C₂×C₄⋊C₄×C₆₇	C₈	1	1	yes	characteristic
23	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₄⋊C₈×C₆₇	—	2	4	no	—
26	C₄⋊C₈×C₆₇	—	2	2	no	—
27	C₄×C₈×C₆₇	—	2	2	no	—
28	C₄×C₈×C₆₇	—	2	2	no	—
29	C₄×C₈×C₆₇	D₄	1	2	yes	—
30	C₄×C₈×C₆₇	D₄	1	2	yes	—
31	C₆₇⋊C₁₆	—	4	4	no	—
32	C₂×Q₈×C₆₇	—	2	2	no	—
33	C₂×Q₈×C₆₇	—	2	2	no	—
34	C₂×Q₈×C₆₇	C₁₆	1	4	yes	complemented
35	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
36	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
37	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₆₇	—	2	2	no	—
41	C₂×C₈×C₆₇	—	2	2	no	—
42	C₂×C₈×C₆₇	—	4	1	no	—
43	C₂×C₈×C₆₇	—	4	2	no	—
44	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
47	C₄⋊C₄×C₆₇	—	2	4	no	—
48	C₄⋊C₄×C₆₇	—	2	2	no	—
49	C₄²×C₆₇	—	2	2	no	—
50	C₄²×C₆₇	—	2	2	no	—
51	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
52	C₄²×C₆₇	M₁₆	1	2	yes	—
53	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
54	Q₈×C₆₇	—	2	8	no	—
55	C₂×C₄×C₆₇	—	2	2	no	—
56	C₂×C₄×C₆₇	—	2	2	no	—
57	C₂×C₄×C₆₇	—	2	1	no	—
58	C₂×C₄×C₆₇	—	2	1	no	—
59	C₂×C₄×C₆₇	—	2	2	no	—
60	C₂×C₄×C₆₇	—	4	1	no	—
61	C₂×C₄×C₆₇	—	4	2	no	—
62	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
63	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
64	C₂×C₄×C₆₇	C₂×C₁₆	1	4	yes	—
65	C₂×C₄×C₆₇	C₂×C₁₆	1	2	yes	—
66	C₂×C₄×C₆₇	M₃₂	1	2	yes	—
67	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
69	C₈×C₆₇	—	2	2	no	—
70	C₈×C₆₇	—	2	1	no	—
71	C₈×C₆₇	—	4	2	no	—
72	C₈×C₆₇	#101	1	2	yes	—
73	C₂²×C₆₇	—	2	1	no	—
74	C₂²×C₆₇	—	2	1	no	—
75	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
76	C₂²×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
77	C₂²×C₆₇	C₂×M₃₂	1	1	yes	characteristic
78	C₄×C₆₇	—	2	4	no	—
79	C₄×C₆₇	—	2	2	no	—
80	C₄×C₆₇	—	2	1	no	—
81	C₄×C₆₇	—	4	2	no	—
82	C₄×C₆₇	C₂²⋊C₁₆	1	4	yes	—
83	C₄×C₆₇	#410	1	1	yes	characteristic
84	C₄×C₆₇	C₂×#101	1	1	yes	characteristic
85	#11493	—	67	1	no	maximal, Sylow
86	C₂×C₆₇	—	2	1	no	—
87	C₂×C₆₇	—	2	1	no	—
88	C₂×C₆₇	#1461	1	1	yes	characteristic
89	C₂×C₆₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
90	C₂×C₆₇	#2115	1	1	yes	characteristic
91	C₂×C₈×Q₈	D₆₇	1	1	yes	characteristic
92	#2162	—	67	2	no	—
93	#1322	—	67	4	no	—
94	C₆₇	#11493	1	1	yes	characteristic, complemented, Sylow
95	C₂×C₄×Q₈	Dic₆₇	1	1	yes	characteristic
96	C₈×Q₈	—	2	2	no	—
97	C₈×Q₈	—	2	2	no	—
98	C₂×C₄⋊C₈	Dic₆₇	1	1	yes	characteristic
99	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
100	C₂×C₄×C₈	Dic₆₇	1	2	yes	—
101	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
102	C₄⋊C₁₆	—	134	2	no	—
103	C₂²⋊C₁₆	—	67	4	no	—
104	C₄×C₁₆	—	134	2	no	—
105	C₂²×Q₈	C₆₇⋊C₈	1	1	yes	characteristic
106	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
107	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
108	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
109	C₄×Q₈	—	2	2	no	—
110	C₄×Q₈	—	2	2	no	—
111	C₂×C₄⋊C₄	C₆₇⋊C₈	1	2	yes	—
112	C₂×C₄⋊C₄	C₆₇⋊C₈	1	1	yes	characteristic
113	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
114	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
115	C₂×C₁₆	—	134	4	no	—
116	C₄⋊C₈	—	2	4	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
121	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
122	C₂×Q₈	—	2	2	no	—
123	C₂×Q₈	—	2	2	no	—
124	C₂×Q₈	C₆₇⋊C₁₆	1	4	yes	complemented
125	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
126	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
127	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
128	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
129	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	4	1	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
135	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
136	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
137	C₄⋊C₄	—	2	4	no	—
138	C₄⋊C₄	—	2	2	no	—
139	C₄²	—	2	2	no	—
140	C₄²	—	2	2	no	—
141	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
142	C₄²	#11761612	1	2	yes	—
143	C₁₆	—	268	4	no	—
144	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
145	Q₈	—	2	8	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
154	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
155	C₂×C₄	C₂×C₆₇⋊C₁₆	1	4	yes	—
156	C₂×C₄	C₂×C₆₇⋊C₁₆	1	2	yes	—
157	C₂×C₄	#14523983	1	2	yes	—
158	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
159	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	2	1	no	—
162	C₈	—	4	2	no	—
163	C₈	#14523962	1	2	yes	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	C₂×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
167	C₂²	C₂²×C₆₇⋊C₁₆	1	1	yes	characteristic
168	C₂²	C₂×#14523983	1	1	yes	characteristic
169	C₄	—	2	4	no	—
170	C₄	—	2	2	no	—
171	C₄	—	2	1	no	—
172	C₄	—	4	2	no	—
173	C₄	C₂²⋊(C₆₇⋊C₁₆)	1	4	yes	—
174	C₄	#16863370	1	1	yes	characteristic
175	C₄	C₂×#14523962	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	#22558533	1	1	yes	characteristic
179	C₂	C₂×C₂²⋊(C₆₇⋊C₁₆)	1	1	yes	characteristic
180	C₂	#22561830	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562008

C₂

yes

maximal

#22558149

C₂

yes

maximal

C₂×C₈×Q₈×C₆₇

C₂

yes

maximal, characteristic

C₄⋊(C₆₇⋊C₁₆)

—

C₂²⋊(C₆₇⋊C₁₆)

C₂²

yes

—

C₄×C₆₇⋊C₁₆

—

C₂×C₄×Q₈×C₆₇

C₄

yes

characteristic

C₈×Q₈×C₆₇

—

C₈×Q₈×C₆₇

—

C₂×C₄⋊C₈×C₆₇

C₄

yes

characteristic

C₂×C₄⋊C₈×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₄

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₆₇⋊C₁₆

—

C₂²×Q₈×C₆₇

C₈

yes

characteristic

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₈

yes

—

C₂×C₄⋊C₄×C₆₇

C₈

yes

characteristic

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₆₇⋊C₁₆

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

C₁₆

yes

complemented

C₂²×C₄×C₆₇

C₂×C₈

yes

—

C₂²×C₄×C₆₇

C₂×C₈

yes

—

C₂²×C₄×C₆₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₆₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂²⋊C₄

yes

—

C₄²×C₆₇

M₁₆

yes

—

C₂³×C₆₇

C₂²×C₈

yes

characteristic

Q₈×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×C₁₆

yes

—

C₂×C₄×C₆₇

C₂×C₁₆

yes

—

C₂×C₄×C₆₇

M₃₂

yes

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×M₁₆

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#101

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₆₇

C₂²×C₁₆

yes

characteristic

C₂²×C₆₇

C₂×M₃₂

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊C₁₆

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#1461

yes

characteristic

C₂×C₆₇

C₂×C₂²⋊C₁₆

yes

characteristic

C₂×C₆₇

#2115

yes

characteristic

C₂×C₈×Q₈

D₆₇

yes

characteristic

#2162

—

#1322

—

C₆₇

#11493

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

Dic₆₇

yes

characteristic

C₈×Q₈

—

C₈×Q₈

—

C₂×C₄⋊C₈

Dic₆₇

yes

characteristic

C₂×C₄⋊C₈

C₂×D₆₇

yes

—

100

C₂×C₄×C₈

Dic₆₇

yes

—

101

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

102

C₄⋊C₁₆

—

134

—

103

C₂²⋊C₁₆

—

104

C₄×C₁₆

—

134

—

105

C₂²×Q₈

C₆₇⋊C₈

yes

characteristic

106

C₂²×C₈

C₂×Dic₆₇

yes

characteristic

107

C₂²×C₈

C₂×Dic₆₇

yes

—

108

C₂²×C₈

C₂²×D₆₇

yes

characteristic

109

C₄×Q₈

—

110

C₄×Q₈

—

111

C₂×C₄⋊C₄

C₆₇⋊C₈

yes

—

112

C₂×C₄⋊C₄

C₆₇⋊C₈

yes

characteristic

113

C₂×C₄²

C₂×Dic₆₇

yes

—

114

C₂×C₄²

C₂×Dic₆₇

yes

characteristic

115

C₂×C₁₆

—

134

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

C₂²⋊D₆₇

yes

—

121

C₄×C₈

C₂²⋊D₆₇

yes

—

122

C₂×Q₈

—

123

C₂×Q₈

—

124

C₂×Q₈

C₆₇⋊C₁₆

yes

complemented

125

C₂²×C₄

C₂×C₆₇⋊C₈

yes

—

126

C₂²×C₄

C₂×C₆₇⋊C₈

yes

—

127

C₂²×C₄

C₂×C₆₇⋊C₈

yes

characteristic

128

C₂²×C₄

C₂×C₆₇⋊C₈

yes

characteristic

129

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

C₂²⋊Dic₆₇

yes

characteristic

135

C₂×C₈

C₂²⋊Dic₆₇

yes

characteristic

136

C₂×C₈

C₂×C₂²⋊D₆₇

yes

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄²

—

140

C₄²

—

141

C₄²

C₂²⋊Dic₆₇

yes

—

142

C₄²

#11761612

yes

—

143

C₁₆

—

268

—

144

C₂³

C₂²×C₆₇⋊C₈

yes

characteristic

145

Q₈

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

C₂²⋊(C₆₇⋊C₈)

yes

—

154

C₂×C₄

C₂²⋊(C₆₇⋊C₈)

yes

—

155

C₂×C₄

C₂×C₆₇⋊C₁₆

yes

—

156

C₂×C₄

C₂×C₆₇⋊C₁₆

yes

—

157

C₂×C₄

#14523983

yes

—

158

C₂×C₄

C₂×C₂²⋊Dic₆₇

yes

characteristic

159

C₂×C₄

C₂×#11761612

yes

characteristic

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

#14523962

yes

—

164

C₂²

—

165

C₂²

—

166

C₂²

C₂×C₂²⋊(C₆₇⋊C₈)

yes

characteristic

167

C₂²

C₂²×C₆₇⋊C₁₆

yes

characteristic

168

C₂²

C₂×#14523983

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

C₂²⋊(C₆₇⋊C₁₆)

yes

—

174

C₄

#16863370

yes

characteristic

175

C₄

C₂×#14523962

yes

characteristic

176

C₂

—

177

C₂

—

178

C₂

#22558533

yes

characteristic

179

C₂

C₂×C₂²⋊(C₆₇⋊C₁₆)

yes

characteristic

180

C₂

#22561830

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₆₇
commutator	C₂²×C₆₇	C₂²×C₁₆
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×C₈×Q₈×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₂²⋊C₈
prime radical	C₂³×C₆₇	C₂²×C₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×C₈×Q₈	D₆₇
hypocenter	C₆₇	#11493
LCS	G ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈×Q₈

subgroup

G/H

center

C₂×C₈

C₂²⋊Dic₆₇

commutator

C₂²×C₆₇

C₂²×C₁₆

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₂×C₈×Q₈×C₆₇

C₂

socle

C₂²×C₆₇

C₂×C₂²⋊C₈

prime radical

C₂³×C₆₇

C₂²×C₈

cosocle

C₂²×C₈×C₆₇

C₂³

hypercenter

C₂×C₈×Q₈

D₆₇

hypocenter

C₆₇

#11493

LCS

G ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈×Q₈

N	H ≅ G/N	type
34. C₂×Q₈×C₆₇	C₁₆	semidirect
94. C₆₇	#11493	semidirect
124. C₂×Q₈	C₆₇⋊C₁₆	semidirect

H ≅ G/N

type

34. C₂×Q₈×C₆₇

C₁₆

semidirect

94. C₆₇

#11493

semidirect

124. C₂×Q₈

C₆₇⋊C₁₆

semidirect

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

subgroup	index
#22562008	2
#22558149	2
C₂×C₈×Q₈×C₆₇	2
#11493	67

subgroup

index

#22562008

#22558149

C₂×C₈×Q₈×C₆₇

#11493

Subgroups — #30765168

Conjugacy classes of subgroups

Complements

Maximal subgroups