Group #30775706 — subgroups

Subgroups — #30775706

Order 17152.

284 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 87 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×#16863222	C₂	1	1	yes	maximal, characteristic, complemented
2	#22561822	C₂	1	1	yes	maximal, characteristic
3	#22560270	C₂	1	2	yes	maximal
4	#22558319	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1381	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14524049	C₂²	1	1	yes	characteristic
7	C₂×#14523983	C₂²	1	1	yes	characteristic
8	C₂×#14523977	C₂²	1	2	yes	—
9	#16863398	—	2	2	no	—
10	#16863222	—	2	2	no	—
11	(C₂×D₆₇)⋊C₁₆	—	2	2	no	—
12	#16863164	—	2	2	no	—
13	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#108×C₆₇	C₂²	1	2	yes	—
15	C₂×#11761628	D₄	1	1	yes	characteristic
16	#14524049	—	2	2	no	—
17	#14524049	—	2	2	no	—
18	C₂×#11761612	D₄	1	1	yes	characteristic
19	C₂×#11761611	—	2	1	no	—
20	C₂×C₈×D₆₇	—	2	1	no	—
21	C₂×C₄⋊Dic₆₇	—	2	2	no	—
22	#14523983	—	2	2	no	—
23	C₂×C₆₇⋊C₁₆	—	2	1	no	—
24	#14523977	—	2	2	no	—
25	#14523977	—	2	2	no	—
26	#14523968	—	4	2	no	—
27	#14523965	—	4	2	no	—
28	C₂×M₁₆×C₆₇	—	2	2	no	—
29	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₆₇	—	2	2	no	—
31	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
32	#108×C₆₇	—	2	2	no	—
33	#108×C₆₇	—	2	2	no	—
34	#11761628	—	2	2	no	—
35	#11761628	—	2	2	no	—
36	C₂×Dic₁₃₄	D₈	1	1	yes	characteristic
37	C₂×D₂₆₈	D₈	1	1	yes	characteristic
38	C₂×C₂²⋊D₆₇	—	2	1	no	—
39	C₂²×Dic₆₇	—	4	2	no	—
40	C₂×C₄×D₆₇	—	2	1	no	—
41	#11761612	—	2	2	no	—
42	#11761611	—	4	2	no	—
43	C₂×C₆₇⋊C₈	—	2	1	no	—
44	C₈×D₆₇	—	4	1	no	—
45	C₈×D₆₇	—	4	1	no	—
46	C₄⋊Dic₆₇	—	4	2	no	—
47	C₄⋊Dic₆₇	—	4	2	no	—
48	C₆₇⋊C₁₆	—	4	1	no	—
49	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
50	M₁₆×C₆₇	—	4	2	no	—
51	M₁₆×C₆₇	—	4	2	no	—
52	C₂×C₈×C₆₇	—	4	2	no	—
53	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₆₇	#41	1	2	yes	—
55	C₂×C₈×C₆₇	#41	1	2	yes	—
56	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
57	C₁₆×C₆₇	—	2	2	no	—
58	C₂³×C₆₇	#135	1	1	yes	characteristic
59	C₂²×D₆₇	—	2	1	no	—
60	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
61	C₂×C₄×C₆₇	#123	1	2	yes	—
62	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₆₇	#137	1	2	yes	—
64	C₂²⋊D₆₇	—	4	2	no	—
65	C₂×Dic₆₇	—	2	1	no	—
66	C₂×Dic₆₇	—	8	2	no	—
67	C₂×Dic₆₇	—	16	2	no	—
68	D₂₆₈	—	2	2	no	—
69	C₄×D₆₇	—	4	1	no	—
70	C₄×D₆₇	—	4	1	no	—
71	Dic₁₃₄	—	2	2	no	—
72	C₈×C₆₇	—	8	2	no	—
73	C₈×C₆₇	#123	1	1	yes	characteristic
74	C₈×C₆₇	#124	1	1	yes	characteristic
75	C₈×C₆₇	#126	1	2	yes	—
76	C₆₇⋊C₈	—	4	1	no	—
77	C₂²×C₆₇	—	2	1	no	—
78	C₂²×C₆₇	—	2	1	no	—
79	C₂²×C₆₇	#466	1	1	yes	characteristic
80	C₂²×C₆₇	#485	1	2	yes	—
81	C₂×D₆₇	—	4	1	no	—
82	C₂×D₆₇	—	4	1	no	—
83	C₂×D₆₇	—	4	1	no	—
84	C₄×C₆₇	#397	1	1	yes	characteristic
85	C₄×C₆₇	#470	1	1	yes	characteristic
86	C₄×C₆₇	#480	1	2	yes	—
87	Dic₆₇	—	4	1	no	—
88	Dic₆₇	—	16	2	no	—
89	#13495	—	67	1	no	maximal, Sylow
90	C₂×C₆₇	—	2	2	no	—
91	C₂×C₆₇	#2075	1	1	yes	characteristic
92	C₂×C₆₇	#2221	1	1	yes	characteristic
93	C₂×C₆₇	#2222	1	1	yes	characteristic
94	D₆₇	—	8	1	no	—
95	C₂×#366	—	67	1	no	—
96	#2113	—	67	1	no	—
97	#1816	—	67	2	no	—
98	#1387	—	67	2	no	—
99	#1381	D₆₇	1	1	yes	characteristic, complemented
100	C₆₇	#13495	1	1	yes	characteristic, complemented, Sylow
101	C₂×#131	—	67	1	no	—
102	C₂×M₃₂	—	67	1	no	—
103	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
104	C₂×#108	C₂×D₆₇	1	2	yes	—
105	C₂×#107	—	67	2	no	—
106	#417	—	134	2	no	—
107	#366	—	134	2	no	—
108	C₂²⋊C₁₆	—	134	2	no	—
109	#343	—	134	2	no	—
110	C₂×#41	—	67	1	no	—
111	#131	—	134	2	no	—
112	#131	—	134	2	no	—
113	C₂×M₁₆	—	2	2	no	—
114	C₂×M₁₆	—	67	1	no	—
115	C₂×M₁₆	—	134	1	no	—
116	C₂²×C₈	—	134	1	no	—
117	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
118	C₂×C₄⋊C₄	—	134	2	no	—
119	M₃₂	—	134	2	no	—
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	—	134	1	no	—
122	C₂×C₁₆	D₂₆₈	1	2	yes	—
123	#108	—	2	2	no	—
124	#108	—	2	2	no	—
125	#107	—	134	2	no	—
126	#107	—	134	2	no	—
127	#103	—	268	2	no	—
128	#102	—	268	2	no	—
129	#41	—	134	2	no	—
130	#41	—	134	2	no	—
131	C₂×Q₈	—	67	1	no	—
132	C₂×D₄	—	67	1	no	—
133	C₂×D₄	—	134	1	no	—
134	C₂²×C₄	—	268	2	no	—
135	C₂²×C₄	—	134	1	no	—
136	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
137	M₁₆	—	4	2	no	—
138	M₁₆	—	4	2	no	—
139	M₁₆	—	268	2	no	—
140	M₁₆	—	134	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	268	1	no	—
143	C₂×C₈	—	268	1	no	—
144	C₂×C₈	—	134	1	no	—
145	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
146	C₂×C₈	#11761627	1	2	yes	—
147	C₂×C₈	#11761628	1	2	yes	—
148	C₂×C₈	#11761629	1	1	yes	characteristic
149	C₄⋊C₄	—	268	2	no	—
150	C₄⋊C₄	—	268	2	no	—
151	C₁₆	—	2	2	no	—
152	C₁₆	—	268	1	no	—
153	C₂³	—	134	1	no	—
154	C₂³	#14524066	1	1	yes	characteristic
155	Q₈	—	134	2	no	—
156	D₄	—	268	2	no	—
157	D₄	—	134	2	no	—
158	C₂×C₄	—	268	1	no	—
159	C₂×C₄	—	268	1	no	—
160	C₂×C₄	—	536	2	no	—
161	C₂×C₄	—	1072	2	no	—
162	C₂×C₄	—	134	1	no	—
163	C₂×C₄	#14524018	1	1	yes	characteristic
164	C₂×C₄	#14524025	1	2	yes	—
165	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
166	C₂×C₄	#14524072	1	2	yes	—
167	C₈	—	8	2	no	—
168	C₈	—	268	1	no	—
169	C₈	#14524027	1	1	yes	characteristic
170	C₈	#14524031	1	1	yes	characteristic
171	C₈	#14524038	1	2	yes	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	—	268	1	no	—
175	C₂²	—	268	1	no	—
176	C₂²	—	268	1	no	—
177	C₂²	#16863633	1	1	yes	characteristic
178	C₂²	#16863736	1	2	yes	—
179	C₄	—	268	1	no	—
180	C₄	—	1072	2	no	—
181	C₄	#16863323	1	1	yes	characteristic
182	C₄	#16863655	1	1	yes	characteristic
183	C₄	#16863708	1	2	yes	—
184	C₂	—	2	2	no	—
185	C₂	—	536	1	no	—
186	C₂	#22561634	1	1	yes	characteristic
187	C₂	#22562330	1	1	yes	characteristic
188	C₂	#22562335	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863222

C₂

yes

maximal, characteristic, complemented

#22561822

C₂

yes

maximal, characteristic

#22560270

C₂

yes

maximal

#22558319

C₂

yes

maximal, complemented

C₆₇×#1381

C₂

yes

maximal, characteristic, complemented

C₂×#14524049

C₂²

yes

characteristic

C₂×#14523983

C₂²

yes

characteristic

C₂×#14523977

C₂²

yes

—

#16863398

—

#16863222

—

(C₂×D₆₇)⋊C₁₆

—

#16863164

—

C₂²×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×#108×C₆₇

C₂²

yes

—

C₂×#11761628

D₄

yes

—

—

D₄

yes

characteristic

C₂×#11761611

—

C₂×C₈×D₆₇

—

C₂×C₄⋊Dic₆₇

—

#14523983

—

C₂×C₆₇⋊C₁₆

—

—

—

—

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

—

#108×C₆₇

—

#108×C₆₇

—

#11761628

—

#11761628

—

C₂×Dic₁₃₄

D₈

yes

characteristic

C₂×D₂₆₈

D₈

yes

characteristic

C₂×C₂²⋊D₆₇

—

C₂²×Dic₆₇

—

C₂×C₄×D₆₇

—

#11761612

—

#11761611

—

C₂×C₆₇⋊C₈

—

C₈×D₆₇

—

C₈×D₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₂×C₈×C₆₇

#41

yes

—

C₂×C₈×C₆₇

#41

yes

characteristic

C₁₆×C₆₇

—

C₂³×C₆₇

#135

yes

characteristic

C₂²×D₆₇

—

C₂×C₄×C₆₇

#121

yes

characteristic

C₂×C₄×C₆₇

#123

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

#137

yes

—

C₂²⋊D₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

C₄×D₆₇

—

C₄×D₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

yes

C₈×C₆₇

yes

C₈×C₆₇

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#466

yes

characteristic

C₂²×C₆₇

#485

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

Dic₆₇

—

#13495

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

—

—

—

—

—

D₆₇

yes

characteristic, complemented

100

C₆₇

#13495

yes

characteristic, complemented, Sylow

101

C₂×#131

—

102

C₂×M₃₂

—

103

C₂²×C₁₆

C₂×D₆₇

yes

characteristic

104

C₂×#108

C₂×D₆₇

yes

—

105

C₂×#107

—

106

#417

—

134

—

107

#366

—

134

—

108

C₂²⋊C₁₆

—

134

—

109

#343

—

134

—

110

C₂×#41

—

111

#131

—

134

—

112

#131

—

134

—

113

C₂×M₁₆

—

114

C₂×M₁₆

—

115

C₂×M₁₆

—

134

—

116

C₂²×C₈

—

134

—

117

C₂²×C₈

C₂²×D₆₇

yes

characteristic

118

C₂×C₄⋊C₄

—

134

—

119

M₃₂

—

134

—

120

C₂×C₁₆

—

121

C₂×C₁₆

—

134

—

122

C₂×C₁₆

D₂₆₈

yes

—

123

#108

—

124

#108

—

125

#107

—

134

—

126

#107

—

134

—

127

#103

—

268

—

128

#102

—

268

—

129

#41

—

134

—

130

#41

—

134

—

131

C₂×Q₈

—

132

C₂×D₄

—

133

C₂×D₄

—

134

—

134

C₂²×C₄

—

268

—

135

C₂²×C₄

—

134

—

136

C₂²×C₄

D₄×D₆₇

yes

characteristic

137

M₁₆

—

138

M₁₆

—

139

M₁₆

—

268

—

140

M₁₆

—

134

—

141

C₂×C₈

—

142

C₂×C₈

—

268

—

143

C₂×C₈

—

268

—

144

C₂×C₈

—

134

—

145

C₂×C₈

C₂×D₂₆₈

yes

characteristic

146

C₂×C₈

#11761627

yes

—

147

C₂×C₈

#11761628

yes

—

148

C₂×C₈

#11761629

yes

characteristic

149

C₄⋊C₄

—

268

—

150

C₄⋊C₄

—

268

—

151

C₁₆

—

152

C₁₆

—

268

—

153

C₂³

—

134

—

154

C₂³

#14524066

yes

characteristic

155

Q₈

—

134

—

156

D₄

—

268

—

157

D₄

—

134

—

158

C₂×C₄

—

268

—

159

C₂×C₄

—

268

—

160

C₂×C₄

—

536

—

161

C₂×C₄

—

1072

—

162

C₂×C₄

—

134

—

163

C₂×C₄

#14524018

yes

characteristic

164

C₂×C₄

#14524025

yes

—

165

C₂×C₄

D₈×D₆₇

yes

characteristic

166

C₂×C₄

#14524072

yes

—

167

C₈

—

168

C₈

—

268

—

169

C₈

#14524027

yes

characteristic

170

C₈

#14524031

yes

characteristic

171

C₈

#14524038

yes

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

268

—

175

C₂²

—

268

—

176

C₂²

—

268

—

177

C₂²

#16863633

yes

characteristic

178

C₂²

#16863736

yes

—

179

C₄

—

268

—

180

C₄

—

1072

—

181

C₄

#16863323

yes

characteristic

182

C₄

#16863655

yes

characteristic

183

C₄

#16863708

yes

—

184

C₂

—

185

C₂

—

536

—

186

C₂

#22561634

yes

characteristic

187

C₂

#22562330

yes

characteristic

188

C₂

#22562335

yes

characteristic

	subgroup	G/H
center	C₂²	#16863633
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1381	C₂
socle	C₂²×C₆₇	#466
prime radical	C₂×#11761628	D₄
hypercenter	#1381	D₆₇
hypocenter	C₆₇	#13495
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂²

#16863633

commutator, cosocle

C₂²×C₈×C₆₇

C₂³

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1381

C₂

C₂²×C₆₇

D₄

D₆₇

C₆₇

G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
1. C₂×#16863222	C₂	semidirect
4. #22558319	C₂	semidirect
5. C₆₇×#1381	C₂	semidirect
99. #1381	D₆₇	semidirect
100. C₆₇	#13495	semidirect

H ≅ G/N

type

1. C₂×#16863222

C₂

semidirect

4. #22558319

C₂

semidirect

5. C₆₇×#1381

C₂

semidirect

99. #1381

D₆₇

semidirect

100. C₆₇

#13495

semidirect

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

subgroup	index
C₂×#16863222	2
#22561822	2
#22560270	2
#22558319	2
C₆₇×#1381	2
#13495	67

subgroup

index

C₆₇×#1381

Subgroups — #30775706

Conjugacy classes of subgroups

Complements

Maximal subgroups