Group #30772023 — subgroups

Subgroups — #30772023

Order 17152.

354 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 111 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×#16863814	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863225	C₂	1	2	yes	maximal, complemented
3	#22560256	C₂	1	2	yes	maximal
4	#22558313	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1385	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14524049	C₂²	1	1	yes	characteristic
7	#16863814	—	2	4	no	—
8	C₂×#14523982	C₄	1	2	yes	—
9	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×D₆₇	C₄	1	2	yes	—
11	C₂×#14523979	C₂²	1	2	yes	—
12	#16863378	—	2	2	no	—
13	#16863225	—	2	4	no	—
14	C₆₇⋊#338	—	2	2	no	—
15	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#108×C₆₇	C₂²	1	2	yes	—
17	C₂×#11761628	D₄	1	1	yes	characteristic
18	#14524049	—	2	2	no	—
19	#14524049	—	2	2	no	—
20	C₂×#11761612	—	2	2	no	—
21	C₂×#11761612	D₄	1	1	yes	characteristic
22	C₂×#11761611	Q₈	1	2	yes	—
23	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
24	#14523982	—	2	4	no	—
25	C₂×C₆₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₆₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₆₇⋊C₁₆	D₄	1	4	yes	—
28	C₁₆×D₆₇	—	2	4	no	—
29	#14523979	—	2	2	no	—
30	#14523979	—	2	2	no	—
31	#14523961	—	4	2	no	—
32	#14523959	—	4	2	no	—
33	C₂×M₁₆×C₆₇	—	2	2	no	—
34	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
35	M₃₂×C₆₇	—	2	2	no	—
36	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
37	#108×C₆₇	—	2	2	no	—
38	#108×C₆₇	—	2	2	no	—
39	#11761628	—	2	2	no	—
40	#11761628	—	2	2	no	—
41	C₂×Dic₁₃₄	D₈	1	1	yes	characteristic
42	C₂×D₂₆₈	D₈	1	1	yes	characteristic
43	C₂×C₂²⋊D₆₇	Q₁₆	1	2	yes	—
44	C₂×C₄×D₆₇	C₄⋊C₄	1	2	yes	—
45	#11761612	—	2	2	no	—
46	#11761612	—	4	2	no	—
47	#11761612	—	4	2	no	—
48	#11761611	—	2	4	no	—
49	C₂×C₆₇⋊C₈	—	4	2	no	—
50	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
51	C₈×D₆₇	—	2	2	no	—
52	C₈×D₆₇	—	2	2	no	—
53	C₆₇⋊C₁₆	C₂²⋊C₄	1	2	yes	—
54	C₆₇⋊C₁₆	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
56	M₁₆×C₆₇	—	4	2	no	—
57	M₁₆×C₆₇	—	4	2	no	—
58	C₂×C₈×C₆₇	—	4	2	no	—
59	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₆₇	#41	1	2	yes	—
63	C₁₆×C₆₇	—	2	2	no	—
64	C₂³×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
65	C₂²×D₆₇	#107	1	2	yes	—
66	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₆₇	#121	1	2	yes	—
68	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₆₇	#135	1	2	yes	—
70	C₂²⋊D₆₇	—	2	4	no	—
71	C₂×Dic₆₇	#107	1	2	yes	—
72	D₂₆₈	—	2	2	no	—
73	C₄×D₆₇	—	2	2	no	—
74	C₄×D₆₇	—	2	2	no	—
75	Dic₁₃₄	—	2	2	no	—
76	C₈×C₆₇	—	8	2	no	—
77	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₈×C₆₇	#117	1	1	yes	characteristic
79	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
80	C₆₇⋊C₈	—	2	2	no	—
81	C₆₇⋊C₈	—	8	2	no	—
82	C₂²×C₆₇	—	4	1	no	—
83	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
84	C₂²×C₆₇	#467	1	2	yes	—
85	C₂×D₆₇	—	2	2	no	—
86	C₂×D₆₇	—	4	2	no	—
87	C₄×C₆₇	#380	1	1	yes	characteristic
88	C₄×C₆₇	#427	1	1	yes	characteristic
89	C₄×C₆₇	#466	1	2	yes	—
90	Dic₆₇	—	2	2	no	—
91	#12840	—	67	1	no	maximal, Sylow
92	C₂×C₆₇	—	2	1	no	—
93	C₂×C₆₇	—	2	2	no	—
94	C₂×C₆₇	#1808	1	1	yes	characteristic
95	D₆₇	—	4	2	no	—
96	C₂×#504	—	67	1	no	—
97	C₂×#368	—	67	2	no	—
98	#1813	—	67	2	no	—
99	#1385	—	67	1	no	—
100	#1385	D₆₇	1	1	yes	characteristic, complemented
101	C₆₇	#12840	1	1	yes	characteristic, complemented, Sylow
102	C₂×#131	—	67	1	no	—
103	#504	—	134	4	no	—
104	C₂×M₃₂	—	67	2	no	—
105	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
106	C₂²×C₁₆	—	67	1	no	—
107	C₂²×C₁₆	—	67	2	no	—
108	C₂×#108	—	67	2	no	—
109	C₂×#108	C₂×D₆₇	1	2	yes	—
110	#413	—	134	2	no	—
111	#368	—	134	4	no	—
112	#338	—	134	2	no	—
113	C₂×#41	—	67	1	no	—
114	#131	—	134	2	no	—
115	#131	—	134	2	no	—
116	C₂×M₁₆	—	2	2	no	—
117	C₂×M₁₆	—	67	1	no	—
118	C₂×M₁₆	—	67	2	no	—
119	C₂×M₁₆	—	134	2	no	—
120	C₂²×C₈	—	67	2	no	—
121	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
122	M₃₂	—	2	2	no	—
123	M₃₂	—	134	4	no	—
124	C₂×C₁₆	—	67	1	no	—
125	C₂×C₁₆	—	67	1	no	—
126	C₂×C₁₆	—	67	4	no	—
127	C₂×C₁₆	—	134	4	no	—
128	C₂×C₁₆	C₄×D₆₇	1	2	yes	—
129	#108	—	2	2	no	—
130	#108	—	2	2	no	—
131	#108	—	134	2	no	—
132	#108	—	134	2	no	—
133	#101	—	268	2	no	—
134	#100	—	268	2	no	—
135	#41	—	134	2	no	—
136	#41	—	134	2	no	—
137	C₂×Q₈	—	67	1	no	—
138	C₂×D₄	—	67	2	no	—
139	C₂×D₄	—	67	1	no	—
140	C₂²×C₄	—	67	2	no	—
141	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
142	M₁₆	—	4	2	no	—
143	M₁₆	—	4	2	no	—
144	M₁₆	—	268	2	no	—
145	M₁₆	—	268	2	no	—
146	M₁₆	—	134	2	no	—
147	M₁₆	—	134	4	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	268	2	no	—
150	C₂×C₈	—	67	2	no	—
151	C₂×C₈	—	134	2	no	—
152	C₂×C₈	—	134	2	no	—
153	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
154	C₂×C₈	D₄×D₆₇	1	2	yes	—
155	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
156	C₂×C₈	#11761628	1	2	yes	—
157	C₁₆	—	2	2	no	—
158	C₁₆	—	67	2	no	—
159	C₁₆	—	67	2	no	—
160	C₂³	—	67	2	no	—
161	C₂³	Q₁₆×D₆₇	1	1	yes	characteristic
162	Q₈	—	134	2	no	—
163	D₄	—	134	4	no	—
164	D₄	—	134	2	no	—
165	C₂×C₄	—	67	2	no	—
166	C₂×C₄	—	134	2	no	—
167	C₂×C₄	—	134	2	no	—
168	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
169	C₂×C₄	#14524016	1	2	yes	—
170	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
171	C₂×C₄	#14524064	1	2	yes	—
172	C₈	—	8	2	no	—
173	C₈	—	536	2	no	—
174	C₈	—	134	2	no	—
175	C₈	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
176	C₈	#14524000	1	1	yes	characteristic
177	C₈	#14524021	1	2	yes	—
178	C₂²	—	4	1	no	—
179	C₂²	—	268	2	no	—
180	C₂²	—	134	2	no	—
181	C₂²	#107×D₆₇	1	1	yes	characteristic
182	C₂²	#16863636	1	2	yes	—
183	C₄	—	134	2	no	—
184	C₄	#16863255	1	1	yes	characteristic
185	C₄	#16863431	1	1	yes	characteristic
186	C₄	#16863631	1	2	yes	—
187	C₂	—	2	1	no	—
188	C₂	—	2	2	no	—
189	C₂	—	268	2	no	—
190	C₂	#22560237	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863814

C₂

yes

maximal, characteristic

C₂×#16863225

C₂

yes

maximal, complemented

#22560256

C₂

yes

maximal

#22558313

C₂

yes

maximal, characteristic, complemented

C₆₇×#1385

C₂

yes

maximal, characteristic, complemented

C₂×#14524049

C₂²

yes

characteristic

#16863814

—

C₂×#14523982

C₄

yes

—

C₂²×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₂×C₁₆×D₆₇

C₄

yes

—

C₂×#14523979

C₂²

yes

—

#16863378

—

#16863225

—

C₆₇⋊#338

—

C₂×M₃₂×C₆₇

C₂²

yes

characteristic

C₂×#108×C₆₇

C₂²

yes

—

C₂×#11761628

D₄

yes

—

—

—

D₄

yes

characteristic

C₂×#11761611

Q₈

yes

—

C₂×C₈×D₆₇

C₂×C₄

yes

—

#14523982

—

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₁₆

D₄

yes

—

C₁₆×D₆₇

—

—

—

—

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

C₂×C₄

yes

—

#108×C₆₇

—

#108×C₆₇

—

#11761628

—

#11761628

—

C₂×Dic₁₃₄

D₈

yes

characteristic

C₂×D₂₆₈

D₈

yes

characteristic

C₂×C₂²⋊D₆₇

Q₁₆

yes

—

C₂×C₄×D₆₇

C₄⋊C₄

yes

—

—

—

—

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

—

C₈×D₆₇

—

C₈×D₆₇

—

C₆₇⋊C₁₆

C₂²⋊C₄

yes

—

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

#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

—

C₂²⋊D₆₇

—

C₂×Dic₆₇

#107

yes

—

D₂₆₈

—

C₄×D₆₇

—

C₄×D₆₇

—

Dic₁₃₄

—

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

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

#12840

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#1808

yes

characteristic

D₆₇

—

—

—

—

—

100

#1385

D₆₇

yes

characteristic, complemented

101

C₆₇

#12840

yes

characteristic, complemented, Sylow

102

C₂×#131

—

103

#504

—

134

—

104

C₂×M₃₂

—

105

C₂×M₃₂

C₂×D₆₇

yes

characteristic

106

C₂²×C₁₆

—

107

C₂²×C₁₆

—

108

C₂×#108

—

109

C₂×#108

C₂×D₆₇

yes

—

110

#413

—

134

—

111

#368

—

134

—

112

#338

—

134

—

113

C₂×#41

—

114

#131

—

134

—

115

#131

—

134

—

116

C₂×M₁₆

—

117

C₂×M₁₆

—

118

C₂×M₁₆

—

119

C₂×M₁₆

—

134

—

120

C₂²×C₈

—

121

C₂²×C₈

C₂²×D₆₇

yes

characteristic

122

M₃₂

—

123

M₃₂

—

134

—

124

C₂×C₁₆

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

127

C₂×C₁₆

—

134

—

128

C₂×C₁₆

C₄×D₆₇

yes

—

129

#108

—

130

#108

—

131

#108

—

134

—

132

#108

—

134

—

133

#101

—

268

—

134

#100

—

268

—

135

#41

—

134

—

136

#41

—

134

—

137

C₂×Q₈

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂²×C₄

—

141

C₂²×C₄

D₄×D₆₇

yes

characteristic

142

M₁₆

—

143

M₁₆

—

144

M₁₆

—

268

—

145

M₁₆

—

268

—

146

M₁₆

—

134

—

147

M₁₆

—

134

—

148

C₂×C₈

—

149

C₂×C₈

—

268

—

150

C₂×C₈

—

151

C₂×C₈

—

134

—

152

C₂×C₈

—

134

—

153

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

154

C₂×C₈

D₄×D₆₇

yes

—

155

C₂×C₈

Q₈×D₆₇

yes

characteristic

156

C₂×C₈

#11761628

yes

—

157

C₁₆

—

158

C₁₆

—

159

C₁₆

—

160

C₂³

—

161

C₂³

Q₁₆×D₆₇

yes

characteristic

162

Q₈

—

134

—

163

D₄

—

134

—

164

D₄

—

134

—

165

C₂×C₄

—

166

C₂×C₄

—

134

—

167

C₂×C₄

—

134

—

168

C₂×C₄

C₄⋊C₄×D₆₇

yes

characteristic

169

C₂×C₄

#14524016

yes

—

170

C₂×C₄

D₈×D₆₇

yes

characteristic

171

C₂×C₄

#14524064

yes

—

172

C₈

—

173

C₈

—

536

—

174

C₈

—

134

—

175

C₈

C₂²⋊C₄×D₆₇

yes

characteristic

176

C₈

#14524000

yes

characteristic

177

C₈

#14524021

yes

—

178

C₂²

—

179

C₂²

—

268

—

180

C₂²

—

134

—

181

C₂²

#107×D₆₇

yes

characteristic

182

C₂²

#16863636

yes

—

183

C₄

—

134

—

184

C₄

#16863255

yes

characteristic

185

C₄

#16863431

yes

characteristic

186

C₄

#16863631

yes

—

187

C₂

—

188

C₂

—

189

C₂

—

268

—

190

C₂

#22560237

yes

characteristic

	subgroup	G/H
center	C₄	#16863431
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1385	C₂
socle	C₂×C₆₇	#1808
prime radical	C₂×#11761628	D₄
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1385	D₆₇
hypocenter	C₆₇	#12840
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

subgroup

G/H

center

C₄

#16863431

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1385

C₂

C₂×C₆₇

D₄

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

N	H ≅ G/N	type
2. C₂×#16863225	C₂	semidirect
4. #22558313	C₂	semidirect
5. C₆₇×#1385	C₂	semidirect
100. #1385	D₆₇	semidirect
101. C₆₇	#12840	semidirect

H ≅ G/N

type

2. C₂×#16863225

C₂

semidirect

4. #22558313

C₂

semidirect

5. C₆₇×#1385

C₂

semidirect

100. #1385

D₆₇

semidirect

101. C₆₇

#12840

semidirect

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

subgroup	index
C₂×#16863814	2
C₂×#16863225	2
#22560256	2
#22558313	2
C₆₇×#1385	2
#12840	67

subgroup

index

C₆₇×#1385

Subgroups — #30772023

Conjugacy classes of subgroups

Complements

Maximal subgroups