Group #30772011 — subgroups

Subgroups — #30772011

Order 17152.

394 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 119 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	C₂²×C₁₆×D₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863225	C₂	1	2	yes	maximal, complemented
3	#22560244	C₂	1	2	yes	maximal
4	#22558303	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1381	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₆₇	C₂²	1	1	yes	characteristic
7	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₁₆×D₆₇	—	2	4	no	—
9	C₂×C₁₆×D₆₇	C₄	1	4	yes	—
10	C₂×#14523979	C₂²	1	2	yes	—
11	#16863359	—	2	2	no	—
12	#16863225	—	2	4	no	—
13	#16863160	—	2	2	no	—
14	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#108×C₆₇	C₂²	1	2	yes	—
16	C₂²×C₄×D₆₇	D₄	1	1	yes	characteristic
17	C₂×#11761612	—	2	2	no	—
18	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
19	C₂×C₈×D₆₇	—	2	2	no	—
20	C₂×C₈×D₆₇	—	2	2	no	—
21	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
22	C₂×C₈×D₆₇	Q₈	1	2	yes	—
23	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₆₇⋊C₁₆	D₄	1	4	yes	—
25	C₁₆×D₆₇	—	2	8	no	—
26	#14523979	—	2	2	no	—
27	#14523979	—	2	2	no	—
28	(C₂×D₆₇)⋊C₈	—	4	4	no	—
29	C₂×M₁₆×C₆₇	—	2	2	no	—
30	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
31	C₂×C₁₆×C₆₇	—	2	2	no	—
32	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
33	#108×C₆₇	—	2	2	no	—
34	#108×C₆₇	—	2	2	no	—
35	C₂³×D₆₇	Q₁₆	1	1	yes	characteristic
36	C₂²×Dic₆₇	Q₁₆	1	1	yes	characteristic
37	C₂×C₄×D₆₇	—	2	2	no	—
38	C₂×C₄×D₆₇	—	2	2	no	—
39	C₂×C₄×D₆₇	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×D₆₇	D₈	1	2	yes	—
41	#11761612	—	4	2	no	—
42	#11761612	—	4	2	no	—
43	C₂×C₆₇⋊C₈	—	2	2	no	—
44	C₂×C₆₇⋊C₈	—	4	2	no	—
45	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
46	C₈×D₆₇	—	2	2	no	—
47	C₈×D₆₇	—	2	2	no	—
48	C₈×D₆₇	—	2	4	no	—
49	C₆₇⋊C₁₆	C₂²⋊C₄	1	4	yes	—
50	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
51	M₁₆×C₆₇	—	4	2	no	—
52	M₁₆×C₆₇	—	4	2	no	—
53	C₂×C₈×C₆₇	—	4	2	no	—
54	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₆₇	#41	1	2	yes	—
58	C₁₆×C₆₇	—	2	2	no	—
59	C₂³×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
60	C₂²×D₆₇	—	2	2	no	—
61	C₂²×D₆₇	—	2	4	no	—
62	C₂²×D₆₇	#107	1	2	yes	—
63	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₇	#121	1	2	yes	—
65	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₆₇	#135	1	2	yes	—
67	C₂×Dic₆₇	—	2	2	no	—
68	C₂×Dic₆₇	#107	1	2	yes	—
69	C₄×D₆₇	—	2	2	no	—
70	C₄×D₆₇	—	2	2	no	—
71	C₄×D₆₇	—	2	4	no	—
72	C₈×C₆₇	—	8	2	no	—
73	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₈×C₆₇	#117	1	1	yes	characteristic
75	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
76	C₆₇⋊C₈	—	2	2	no	—
77	C₆₇⋊C₈	—	8	2	no	—
78	C₂²×C₆₇	—	2	2	no	—
79	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
80	C₂²×C₆₇	#467	1	2	yes	—
81	C₂×D₆₇	—	2	8	no	—
82	C₂×D₆₇	—	2	2	no	—
83	C₂×D₆₇	—	2	2	no	—
84	C₂×D₆₇	#368	1	4	yes	complemented
85	C₄×C₆₇	#380	1	1	yes	characteristic
86	C₄×C₆₇	#427	1	1	yes	characteristic
87	C₄×C₆₇	#466	1	2	yes	—
88	Dic₆₇	—	2	2	no	—
89	#12837	—	67	1	no	maximal, Sylow
90	C₂×C₆₇	—	2	2	no	—
91	C₂×C₆₇	#1808	1	1	yes	characteristic
92	C₂×C₆₇	C₂×#368	1	2	yes	—
93	D₆₇	—	2	4	no	—
94	C₂³×C₁₆	—	67	1	no	—
95	C₂×#368	—	67	2	no	—
96	#1810	—	67	2	no	—
97	#1381	—	67	1	no	—
98	#1381	D₆₇	1	1	yes	characteristic, complemented
99	C₆₇	#12837	1	1	yes	characteristic, complemented, Sylow
100	C₂³×C₈	—	67	1	no	—
101	C₂²×C₁₆	—	67	1	no	—
102	C₂²×C₁₆	—	67	4	no	—
103	C₂²×C₁₆	—	134	4	no	—
104	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
105	C₂×#108	—	67	2	no	—
106	C₂×#108	C₂×D₆₇	1	2	yes	—
107	#407	—	134	2	no	—
108	#368	—	134	4	no	—
109	#341	—	134	2	no	—
110	C₂³×C₄	—	67	1	no	—
111	C₂×M₁₆	—	2	2	no	—
112	C₂×M₁₆	—	134	2	no	—
113	C₂²×C₈	—	67	1	no	—
114	C₂²×C₈	—	67	2	no	—
115	C₂²×C₈	—	67	2	no	—
116	C₂²×C₈	—	134	2	no	—
117	C₂²×C₈	—	134	2	no	—
118	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	—	67	2	no	—
121	C₂×C₁₆	—	67	4	no	—
122	C₂×C₁₆	—	134	8	no	—
123	C₂×C₁₆	C₄×D₆₇	1	2	yes	—
124	#108	—	2	2	no	—
125	#108	—	2	2	no	—
126	#108	—	134	2	no	—
127	#108	—	134	2	no	—
128	C₂²⋊C₈	—	268	4	no	—
129	C₂⁴	—	67	1	no	—
130	C₂²×C₄	—	67	1	no	—
131	C₂²×C₄	—	67	2	no	—
132	C₂²×C₄	—	67	2	no	—
133	C₂²×C₄	—	134	2	no	—
134	C₂²×C₄	—	134	2	no	—
135	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
136	M₁₆	—	4	2	no	—
137	M₁₆	—	4	2	no	—
138	M₁₆	—	268	2	no	—
139	M₁₆	—	268	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	268	2	no	—
142	C₂×C₈	—	67	2	no	—
143	C₂×C₈	—	134	2	no	—
144	C₂×C₈	—	134	2	no	—
145	C₂×C₈	—	134	2	no	—
146	C₂×C₈	—	134	4	no	—
147	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
148	C₂×C₈	D₄×D₆₇	1	2	yes	—
149	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
150	C₂×C₈	#11761628	1	2	yes	—
151	C₁₆	—	2	2	no	—
152	C₁₆	—	67	4	no	—
153	C₂³	—	67	2	no	—
154	C₂³	—	134	4	no	—
155	C₂³	—	134	2	no	—
156	C₂³	Q₁₆×D₆₇	1	1	yes	characteristic
157	C₂×C₄	—	67	2	no	—
158	C₂×C₄	—	134	2	no	—
159	C₂×C₄	—	134	2	no	—
160	C₂×C₄	—	134	4	no	—
161	C₂×C₄	—	134	2	no	—
162	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
163	C₂×C₄	#14524016	1	2	yes	—
164	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
165	C₂×C₄	#14524064	1	2	yes	—
166	C₈	—	8	2	no	—
167	C₈	—	536	2	no	—
168	C₈	—	134	2	no	—
169	C₈	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
170	C₈	#14524000	1	1	yes	characteristic
171	C₈	#14524021	1	2	yes	—
172	C₂²	—	2	2	no	—
173	C₂²	—	67	4	no	—
174	C₂²	—	134	8	no	—
175	C₂²	—	134	2	no	—
176	C₂²	—	134	2	no	—
177	C₂²	#107×D₆₇	1	1	yes	characteristic
178	C₂²	#16863636	1	2	yes	—
179	C₄	—	134	2	no	—
180	C₄	#16863255	1	1	yes	characteristic
181	C₄	#16863431	1	1	yes	characteristic
182	C₄	#16863631	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	—	134	4	no	—
185	C₂	#22560237	1	1	yes	characteristic
186	C₂	#368×D₆₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₆×D₆₇

C₂

yes

maximal, characteristic

C₂×#16863225

C₂

yes

maximal, complemented

#22560244

C₂

yes

maximal

#22558303

C₂

yes

maximal, characteristic, complemented

C₆₇×#1381

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₆₇

C₂²

yes

characteristic

C₂²×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₂×C₁₆×D₆₇

—

C₂×C₁₆×D₆₇

C₄

yes

—

C₂×#14523979

C₂²

yes

—

#16863359

—

#16863225

—

#16863160

—

C₂²×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×#108×C₆₇

C₂²

yes

—

C₂²×C₄×D₆₇

D₄

yes

characteristic

C₂×#11761612

—

C₂²×C₆₇⋊C₈

D₄

yes

characteristic

C₂×C₈×D₆₇

—

C₂×C₈×D₆₇

—

C₂×C₈×D₆₇

C₂×C₄

yes

—

C₂×C₈×D₆₇

Q₈

yes

—

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

—

C₂×C₆₇⋊C₁₆

D₄

yes

—

C₁₆×D₆₇

—

#14523979

—

#14523979

—

(C₂×D₆₇)⋊C₈

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

C₂×C₄

yes

—

#108×C₆₇

—

#108×C₆₇

—

C₂³×D₆₇

Q₁₆

yes

characteristic

C₂²×Dic₆₇

Q₁₆

yes

characteristic

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

C₄⋊C₄

yes

—

C₂×C₄×D₆₇

D₈

yes

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

—

C₈×D₆₇

—

C₈×D₆₇

—

C₈×D₆₇

—

C₆₇⋊C₁₆

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₁₆×C₆₇

—

C₂³×C₆₇

C₂×Q₁₆

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

#107

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

#135

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

#107

yes

—

C₄×D₆₇

—

C₄×D₆₇

—

C₄×D₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₆₇

#117

yes

characteristic

C₈×C₆₇

C₂²⋊Q₈

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×#107

yes

characteristic

C₂²×C₆₇

#467

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

#12837

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1808

yes

characteristic

C₂×C₆₇

C₂×#368

yes

—

D₆₇

—

C₂³×C₁₆

—

—

—

—

D₆₇

yes

characteristic, complemented

C₆₇

#12837

yes

characteristic, complemented, Sylow

100

C₂³×C₈

—

101

C₂²×C₁₆

—

102

C₂²×C₁₆

—

103

C₂²×C₁₆

—

134

—

104

C₂²×C₁₆

C₂×D₆₇

yes

characteristic

105

C₂×#108

—

106

C₂×#108

C₂×D₆₇

yes

—

107

#407

—

134

—

108

#368

—

134

—

109

#341

—

134

—

110

C₂³×C₄

—

111

C₂×M₁₆

—

112

C₂×M₁₆

—

134

—

113

C₂²×C₈

—

114

C₂²×C₈

—

115

C₂²×C₈

—

116

C₂²×C₈

—

134

—

117

C₂²×C₈

—

134

—

118

C₂²×C₈

C₂²×D₆₇

yes

characteristic

119

C₂×C₁₆

—

120

C₂×C₁₆

—

121

C₂×C₁₆

—

122

C₂×C₁₆

—

134

—

123

C₂×C₁₆

C₄×D₆₇

yes

—

124

#108

—

125

#108

—

126

#108

—

134

—

127

#108

—

134

—

128

C₂²⋊C₈

—

268

—

129

C₂⁴

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

—

134

C₂²×C₄

—

134

—

135

C₂²×C₄

D₄×D₆₇

yes

characteristic

136

M₁₆

—

137

M₁₆

—

138

M₁₆

—

268

—

139

M₁₆

—

268

—

140

C₂×C₈

—

141

C₂×C₈

—

268

—

142

C₂×C₈

—

143

C₂×C₈

—

134

—

144

C₂×C₈

—

134

—

145

C₂×C₈

—

134

—

146

C₂×C₈

—

134

—

147

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

148

C₂×C₈

D₄×D₆₇

yes

—

149

C₂×C₈

Q₈×D₆₇

yes

characteristic

150

C₂×C₈

#11761628

yes

—

151

C₁₆

—

152

C₁₆

—

153

C₂³

—

154

C₂³

—

134

—

155

C₂³

—

134

—

156

C₂³

Q₁₆×D₆₇

yes

characteristic

157

C₂×C₄

—

158

C₂×C₄

—

134

—

159

C₂×C₄

—

134

—

160

C₂×C₄

—

134

—

161

C₂×C₄

—

134

—

162

C₂×C₄

C₄⋊C₄×D₆₇

yes

characteristic

163

C₂×C₄

#14524016

yes

—

164

C₂×C₄

D₈×D₆₇

yes

characteristic

165

C₂×C₄

#14524064

yes

—

166

C₈

—

167

C₈

—

536

—

168

C₈

—

134

—

169

C₈

C₂²⋊C₄×D₆₇

yes

characteristic

170

C₈

#14524000

yes

characteristic

171

C₈

#14524021

yes

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

134

—

175

C₂²

—

134

—

176

C₂²

—

134

—

177

C₂²

#107×D₆₇

yes

characteristic

178

C₂²

#16863636

yes

—

179

C₄

—

134

—

180

C₄

#16863255

yes

characteristic

181

C₄

#16863431

yes

characteristic

182

C₄

#16863631

yes

—

183

C₂

—

184

C₂

—

134

—

185

C₂

#22560237

yes

characteristic

186

C₂

#368×D₆₇

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₆₇
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1381	C₂
socle	C₂²×C₆₇	C₂×#107
prime radical	C₂³×D₆₇	Q₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1381	D₆₇
hypocenter	C₆₇	#12837
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂×C₄

D₈×D₆₇

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1381

C₂

socle

C₂²×C₆₇

C₂×#107

prime radical

C₂³×D₆₇

Q₁₆

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
2. C₂×#16863225	C₂	semidirect
4. #22558303	C₂	semidirect
5. C₆₇×#1381	C₂	semidirect
84. C₂×D₆₇	#368	semidirect
98. #1381	D₆₇	semidirect
99. C₆₇	#12837	semidirect

H ≅ G/N

type

2. C₂×#16863225

C₂

semidirect

4. #22558303

C₂

semidirect

5. C₆₇×#1381

C₂

semidirect

84. C₂×D₆₇

#368

semidirect

98. #1381

D₆₇

semidirect

99. C₆₇

#12837

semidirect

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

subgroup	index
C₂²×C₁₆×D₆₇	2
C₂×#16863225	2
#22560244	2
#22558303	2
C₆₇×#1381	2
#12837	67

subgroup

index

C₂²×C₁₆×D₆₇

C₆₇×#1381

Subgroups — #30772011

Conjugacy classes of subgroups

Complements

Maximal subgroups