Group #30774353 — subgroups

Subgroups — #30774353

Order 17152.

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

normal subgroups only

This group has 386 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	C₂×C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863172	C₂	1	1	yes	maximal, characteristic
3	#22558286	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₁₃₄	C₄	1	1	yes	characteristic
6	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
7	C₈×Dic₁₃₄	—	2	2	no	—
8	C₈×Dic₁₃₄	—	2	2	no	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	1	yes	characteristic
10	C₂×Dic₆₇⋊C₈	C₄	1	2	yes	—
11	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
12	#16863172	—	2	2	no	—
13	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₆₇	—	2	2	no	—
16	C₂²×Dic₁₃₄	C₈	1	1	yes	characteristic
17	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
18	C₄×Dic₁₃₄	—	2	2	no	—
19	C₄×Dic₁₃₄	—	2	2	no	—
20	C₂×C₄⋊Dic₆₇	C₈	1	1	yes	characteristic
21	C₂×Dic₆₇⋊C₄	C₈	1	2	yes	—
22	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
23	C₂×C₆₇⋊C₁₆	—	2	2	no	—
24	C₂×C₆₇⋊C₁₆	—	2	2	no	—
25	C₂×C₆₇⋊C₁₆	Q₈	1	4	yes	complemented
26	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
27	Dic₆₇⋊C₈	—	2	4	no	—
28	C₈×Dic₆₇	—	2	2	no	—
29	C₈×Dic₆₇	—	2	2	no	—
30	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₆₇	—	2	2	no	—
34	C₂×C₁₆×C₆₇	—	2	4	no	—
35	C₄×C₈×C₆₇	D₄	1	2	yes	—
36	C₄×C₈×C₆₇	D₄	1	2	yes	—
37	C₂×Dic₁₃₄	—	2	2	no	—
38	C₂×Dic₁₃₄	—	2	4	no	—
39	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
40	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
41	C₂×C₆₇⋊C₈	—	2	2	no	—
42	C₂×C₆₇⋊C₈	—	2	4	no	—
43	C₄⋊Dic₆₇	—	2	2	no	—
44	Dic₆₇⋊C₄	—	2	4	no	—
45	C₄×Dic₆₇	—	2	2	no	—
46	C₄×Dic₆₇	—	2	2	no	—
47	C₆₇⋊C₁₆	—	2	4	no	—
48	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	—	2	2	no	—
52	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
53	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
56	C₂×C₈×C₆₇	#41	1	2	yes	—
57	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
58	C₄²×C₆₇	M₁₆	1	2	yes	—
59	C₁₆×C₆₇	—	4	2	no	—
60	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₆₇	—	2	2	no	—
62	C₂×C₄×C₆₇	—	2	2	no	—
63	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
64	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
65	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
67	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
68	C₂×C₄×C₆₇	#131	1	2	yes	—
69	C₂×Dic₆₇	—	2	2	no	—
70	C₂×Dic₆₇	—	2	2	no	—
71	C₂×Dic₆₇	—	2	2	no	—
72	C₂×Dic₆₇	—	2	4	no	—
73	C₂×Dic₆₇	M₃₂	1	4	yes	—
74	Dic₁₃₄	—	2	8	no	—
75	C₈×C₆₇	—	2	2	no	—
76	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
77	C₈×C₆₇	#124	1	1	yes	characteristic
78	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
79	C₆₇⋊C₈	—	4	2	no	—
80	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
81	C₂²×C₆₇	C₈×Q₈	1	2	yes	—
82	C₂²×C₆₇	C₂×M₃₂	1	2	yes	—
83	C₂²×C₆₇	#504	1	2	yes	—
84	C₄×C₆₇	—	2	2	no	—
85	C₄×C₆₇	—	2	2	no	—
86	C₄×C₆₇	#391	1	1	yes	characteristic
87	C₄×C₆₇	#408	1	1	yes	characteristic
88	C₄×C₆₇	#446	1	2	yes	—
89	Dic₆₇	—	2	4	no	—
90	Dic₆₇	—	4	2	no	—
91	#13264	—	67	1	no	maximal, Sylow
92	C₂×C₆₇	#1984	1	1	yes	characteristic
93	C₂×C₆₇	#2112	1	1	yes	characteristic
94	C₂×C₆₇	#2113	1	1	yes	characteristic
95	C₂×C₆₇	#2183	1	4	yes	—
96	C₂×C₈×Q₈	—	67	1	no	—
97	C₂×C₄⋊C₁₆	D₆₇	1	1	yes	characteristic
98	C₂×#346	—	67	1	no	—
99	#1374	—	67	4	no	—
100	C₆₇	#13264	1	1	yes	characteristic, complemented, Sylow
101	C₂×C₄×Q₈	—	67	1	no	—
102	C₂²×C₁₆	—	67	2	no	—
103	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
104	C₈×Q₈	—	134	2	no	—
105	C₈×Q₈	—	134	2	no	—
106	C₂×C₄⋊C₈	—	67	2	no	—
107	C₂×C₄⋊C₈	—	67	1	no	—
108	C₂×C₄×C₈	—	67	2	no	—
109	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
110	C₄⋊C₁₆	—	2	2	no	—
111	#346	—	134	2	no	—
112	C₂²×Q₈	—	67	1	no	—
113	C₂²×C₈	—	67	2	no	—
114	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
116	C₄×Q₈	—	134	2	no	—
117	C₄×Q₈	—	134	2	no	—
118	C₂×C₄⋊C₄	—	67	2	no	—
119	C₂×C₄⋊C₄	—	67	1	no	—
120	C₂×C₄²	—	67	2	no	—
121	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
122	C₂×C₁₆	—	2	2	no	—
123	C₂×C₁₆	—	2	4	no	—
124	C₂×C₁₆	—	67	4	no	—
125	C₂×C₁₆	—	134	2	no	—
126	C₂×C₁₆	—	134	2	no	—
127	C₄⋊C₈	—	134	4	no	—
128	C₄⋊C₈	—	134	2	no	—
129	C₄×C₈	—	134	2	no	—
130	C₄×C₈	—	134	2	no	—
131	C₄×C₈	D₂₆₈	1	2	yes	—
132	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
133	C₂×Q₈	—	134	4	no	—
134	C₂×Q₈	—	134	2	no	—
135	C₂²×C₄	—	67	2	no	—
136	C₂²×C₄	—	67	2	no	—
137	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
138	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	134	2	no	—
142	C₂×C₈	—	134	4	no	—
143	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
144	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
145	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
146	C₂×C₈	Q₈×D₆₇	1	2	yes	—
147	C₂×C₈	#11761627	1	2	yes	—
148	C₄⋊C₄	—	134	4	no	—
149	C₄⋊C₄	—	134	2	no	—
150	C₄²	—	134	2	no	—
151	C₄²	—	134	2	no	—
152	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
153	C₄²	#11761611	1	2	yes	—
154	C₁₆	—	4	2	no	—
155	C₁₆	—	134	4	no	—
156	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
157	Q₈	—	134	8	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	67	4	no	—
161	C₂×C₄	—	134	4	no	—
162	C₂×C₄	—	134	2	no	—
163	C₂×C₄	—	134	2	no	—
164	C₂×C₄	—	134	2	no	—
165	C₂×C₄	(C₂×D₆₇)⋊C₈	1	2	yes	—
166	C₂×C₄	(C₂×D₆₇)⋊C₈	1	2	yes	—
167	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
168	C₂×C₄	#14524008	1	2	yes	—
169	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
170	C₂×C₄	#14524051	1	2	yes	—
171	C₈	—	2	2	no	—
172	C₈	—	268	2	no	—
173	C₈	#14524023	1	2	yes	—
174	C₈	#14524030	1	1	yes	characteristic
175	C₈	#14524041	1	1	yes	characteristic
176	C₂²	C₂×(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
177	C₂²	#16863520	1	2	yes	—
178	C₂²	M₃₂×D₆₇	1	2	yes	—
179	C₂²	#16863812	1	2	yes	—
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	268	2	no	—
183	C₄	—	134	4	no	—
184	C₄	#16863301	1	1	yes	characteristic
185	C₄	#16863364	1	1	yes	characteristic
186	C₄	#16863523	1	2	yes	—
187	C₂	#22561144	1	1	yes	characteristic
188	C₂	#22561819	1	1	yes	characteristic
189	C₂	#22561822	1	1	yes	characteristic
190	C₂	#22562116	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₁₃₄

C₂

yes

maximal, characteristic

C₂×#16863172

C₂

yes

maximal, characteristic

#22558286

C₂

yes

maximal

C₂×C₄⋊C₁₆×C₆₇

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₃₄

C₄

yes

characteristic

C₂²×C₆₇⋊C₁₆

C₂²

yes

—

C₈×Dic₁₃₄

—

C₈×Dic₁₃₄

—

C₂×C₄⋊(C₆₇⋊C₈)

C₄

yes

characteristic

C₂×Dic₆₇⋊C₈

C₄

yes

—

C₂×C₈×Dic₆₇

C₂²

yes

—

#16863172

—

C₂²×C₁₆×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₁₆×C₆₇

—

C₂²×Dic₁₃₄

C₈

yes

characteristic

C₂²×C₆₇⋊C₈

C₂×C₄

yes

—

C₄×Dic₁₃₄

—

C₄×Dic₁₃₄

—

C₂×C₄⋊Dic₆₇

C₈

yes

characteristic

C₂×Dic₆₇⋊C₄

C₈

yes

—

C₂×C₄×Dic₆₇

C₂×C₄

yes

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

Q₈

yes

complemented

C₄⋊(C₆₇⋊C₈)

—

Dic₆₇⋊C₈

—

C₈×Dic₆₇

—

C₈×Dic₆₇

—

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

C₂×C₈

yes

—

C₂²×Dic₆₇

C₂×C₈

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

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

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

—

C₂×C₈×C₆₇

#41

yes

—

C₄²×C₆₇

C₂²⋊C₄

yes

—

C₄²×C₆₇

M₁₆

yes

—

C₁₆×C₆₇

—

C₂³×C₆₇

C₂²×C₈

yes

characteristic

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

C₄×Q₈

yes

—

C₂×C₄×C₆₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₆₇

#131

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

M₃₂

yes

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

C₂²⋊Q₈

yes

—

C₈×C₆₇

#124

yes

characteristic

C₈×C₆₇

C₄⋊Q₈

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₆₇

C₈×Q₈

yes

—

C₂²×C₆₇

C₂×M₃₂

yes

—

C₂²×C₆₇

#504

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×C₈×Q₈

—

C₂×C₄⋊C₁₆

D₆₇

yes

characteristic

C₂×#346

—

#1374

—

100

C₆₇

#13264

yes

characteristic, complemented, Sylow

101

C₂×C₄×Q₈

—

102

C₂²×C₁₆

—

103

C₂²×C₁₆

C₂×D₆₇

yes

—

104

C₈×Q₈

—

134

—

105

C₈×Q₈

—

134

—

106

C₂×C₄⋊C₈

—

107

C₂×C₄⋊C₈

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

110

C₄⋊C₁₆

—

111

#346

—

134

—

112

C₂²×Q₈

—

113

C₂²×C₈

—

114

C₂²×C₈

C₄×D₆₇

yes

characteristic

115

C₂²×C₈

C₂²×D₆₇

yes

characteristic

116

C₄×Q₈

—

134

—

117

C₄×Q₈

—

134

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄²

—

121

C₂×C₄²

C₄×D₆₇

yes

characteristic

122

C₂×C₁₆

—

123

C₂×C₁₆

—

124

C₂×C₁₆

—

125

C₂×C₁₆

—

134

—

126

C₂×C₁₆

—

134

—

127

C₄⋊C₈

—

134

—

128

C₄⋊C₈

—

134

—

129

C₄×C₈

—

134

—

130

C₄×C₈

—

134

—

131

C₄×C₈

D₂₆₈

yes

—

132

C₄×C₈

C₂²⋊D₆₇

yes

—

133

C₂×Q₈

—

134

—

134

C₂×Q₈

—

134

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₈×D₆₇

yes

characteristic

138

C₂²×C₄

C₈×D₆₇

yes

characteristic

139

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

140

C₂×C₈

—

141

C₂×C₈

—

134

—

142

C₂×C₈

—

134

—

143

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

144

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

145

C₂×C₈

C₂×D₂₆₈

yes

characteristic

146

C₂×C₈

Q₈×D₆₇

yes

—

147

C₂×C₈

#11761627

yes

—

148

C₄⋊C₄

—

134

—

149

C₄⋊C₄

—

134

—

150

C₄²

—

134

—

151

C₄²

—

134

—

152

C₄²

(C₂×D₆₇)⋊C₄

yes

—

153

C₄²

#11761611

yes

—

154

C₁₆

—

155

C₁₆

—

134

—

156

C₂³

C₂×C₈×D₆₇

yes

characteristic

157

Q₈

—

134

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

134

—

162

C₂×C₄

—

134

—

163

C₂×C₄

—

134

—

164

C₂×C₄

—

134

—

165

C₂×C₄

(C₂×D₆₇)⋊C₈

yes

—

166

C₂×C₄

(C₂×D₆₇)⋊C₈

yes

—

167

C₂×C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

168

C₂×C₄

#14524008

yes

—

169

C₂×C₄

C₂×#11761611

yes

characteristic

170

C₂×C₄

#14524051

yes

—

171

C₈

—

172

C₈

—

268

—

173

C₈

#14524023

yes

—

174

C₈

#14524030

yes

characteristic

175

C₈

#14524041

yes

characteristic

176

C₂²

C₂×(C₂×D₆₇)⋊C₈

yes

characteristic

177

C₂²

#16863520

yes

—

178

C₂²

M₃₂×D₆₇

yes

—

179

C₂²

#16863812

yes

—

180

C₄

—

181

C₄

—

182

C₄

—

268

—

183

C₄

—

134

—

184

C₄

#16863301

yes

characteristic

185

C₄

#16863364

yes

characteristic

186

C₄

#16863523

yes

—

187

C₂

#22561144

yes

characteristic

188

C₂

#22561819

yes

characteristic

189

C₂

#22561822

yes

characteristic

190

C₂

#22562116

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂²×C₈
Fitting	C₂×C₄⋊C₁₆×C₆₇	C₂
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₆₇
hypocenter	C₆₇	#13264
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₆₇

commutator, socle, prime radical

C₂³×C₆₇

C₂²×C₈

Fitting

C₂×C₄⋊C₁₆×C₆₇

C₂

cosocle

C₂²×C₈×C₆₇

C₂³

hypercenter

C₂×C₄⋊C₁₆

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
25. C₂×C₆₇⋊C₁₆	Q₈	semidirect
100. C₆₇	#13264	semidirect

H ≅ G/N

type

25. C₂×C₆₇⋊C₁₆

Q₈

semidirect

100. C₆₇

#13264

semidirect

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

subgroup	index
C₂×C₈×Dic₁₃₄	2
C₂×#16863172	2
#22558286	2
C₂×C₄⋊C₁₆×C₆₇	2
#13264	67

subgroup

index

C₂×C₈×Dic₁₃₄

C₂×#16863172

#22558286

C₂×C₄⋊C₁₆×C₆₇

#13264

Subgroups — #30774353

Conjugacy classes of subgroups

Complements

Maximal subgroups