Group #30775479 — subgroups

Subgroups — #30775479

Order 17152.

356 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 103 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×#16863794	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863220	C₂	1	2	yes	maximal
3	#22558314	C₂	1	2	yes	maximal
4	#22558291	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1945	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524043	—	2	1	no	—
7	C₂×#14523983	C₂²	1	2	yes	—
8	#16863794	—	2	4	no	—
9	C₂×#14523975	—	2	1	no	—
10	C₂×#14523975	C₂²	1	2	yes	—
11	#16863220	—	2	4	no	—
12	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
13	#427×C₆₇	C₂²	1	1	yes	characteristic
14	#424×C₆₇	—	2	1	no	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	#339×C₆₇	—	2	1	no	—
17	C₂²×Dic₁₃₄	—	4	1	no	—
18	#14524043	—	2	4	no	—
19	#14524043	—	4	2	no	—
20	C₂×C₄⋊Dic₆₇	—	2	2	no	—
21	C₂×C₄⋊Dic₆₇	—	4	1	no	—
22	#14523983	—	2	2	no	—
23	#14523983	Q₈	1	4	yes	complemented
24	C₂×C₆₇⋊C₁₆	—	2	2	no	—
25	#14523975	—	2	4	no	—
26	#14523975	—	2	2	no	—
27	#14523975	D₄	1	4	yes	—
28	C₂×M₁₆×C₆₇	—	2	1	no	—
29	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
31	#117×C₆₇	—	2	1	no	—
32	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
33	#108×C₆₇	—	2	2	no	—
34	#107×C₆₇	Q₈	1	2	yes	complemented
35	#106×C₆₇	—	2	1	no	—
36	C₄⋊C₈×C₆₇	—	4	2	no	—
37	C₄×C₈×C₆₇	—	2	2	no	—
38	C₂×Dic₁₃₄	—	4	4	no	—
39	C₂×Dic₁₃₄	—	8	4	no	—
40	C₂²×Dic₆₇	—	4	2	no	—
41	C₄⋊Dic₆₇	—	2	4	no	—
42	C₄⋊Dic₆₇	—	4	4	no	—
43	C₄⋊Dic₆₇	—	4	2	no	—
44	C₆₇⋊C₁₆	—	2	4	no	—
45	C₂²×C₄×C₆₇	—	2	1	no	—
46	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	M₁₆×C₆₇	—	4	1	no	—
48	M₁₆×C₆₇	—	4	1	no	—
49	C₂×C₈×C₆₇	—	2	1	no	—
50	C₂×C₈×C₆₇	—	2	1	no	—
51	C₂×C₈×C₆₇	—	4	1	no	—
52	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
53	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
54	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
57	C₄⋊C₄×C₆₇	—	2	2	no	—
58	C₂²⋊C₄×C₆₇	—	4	1	no	—
59	C₄²×C₆₇	—	2	1	no	—
60	C₄²×C₆₇	—	4	1	no	—
61	C₄²×C₆₇	SD₁₆	1	2	yes	—
62	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₂×C₄×C₆₇	—	2	4	no	—
64	C₂×C₄×C₆₇	—	2	2	no	—
65	C₂×C₄×C₆₇	—	4	2	no	—
66	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
72	C₂×Dic₆₇	—	4	4	no	—
73	C₂×Dic₆₇	—	8	4	no	—
74	Dic₁₃₄	—	8	8	no	—
75	C₈×C₆₇	—	2	2	no	—
76	C₈×C₆₇	—	8	1	no	—
77	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
78	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
79	C₂²×C₆₇	—	2	2	no	—
80	C₂²×C₆₇	#450	1	1	yes	characteristic
81	C₂²×C₆₇	#476	1	1	yes	characteristic
82	C₂²×C₆₇	#501	1	1	yes	characteristic
83	C₄×C₆₇	—	2	4	no	—
84	C₄×C₆₇	—	8	2	no	—
85	C₄×C₆₇	#395	1	1	yes	characteristic
86	C₄×C₆₇	#448	1	1	yes	characteristic
87	C₄×C₆₇	#475	1	1	yes	characteristic
88	C₄×C₆₇	#498	1	1	yes	characteristic
89	Dic₆₇	—	8	4	no	—
90	#13457	—	67	1	no	maximal, Sylow
91	C₂×C₆₇	—	2	1	no	—
92	C₂×C₆₇	—	2	1	no	—
93	C₂×C₆₇	#2058	1	1	yes	characteristic
94	C₂×C₆₇	#2230	1	2	yes	—
95	C₂×#498	—	67	1	no	—
96	C₂×#365	—	67	2	no	—
97	#1945	D₆₇	1	1	yes	characteristic
98	#1386	—	67	2	no	—
99	#1376	—	67	1	no	—
100	C₆₇	#13457	1	1	yes	characteristic, complemented, Sylow
101	C₂×C₄⋊Q₈	—	134	1	no	—
102	C₂×M₃₂	—	67	2	no	—
103	#498	—	134	4	no	—
104	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
105	#427	C₂×D₆₇	1	1	yes	characteristic
106	C₂×#106	—	67	2	no	—
107	C₂×#106	—	134	1	no	—
108	#424	—	2	1	no	—
109	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
110	#365	—	134	4	no	—
111	#339	—	2	1	no	—
112	C₂²×Q₈	—	268	1	no	—
113	C₂×M₁₆	—	2	1	no	—
114	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
116	C₄⋊Q₈	—	268	2	no	—
117	C₄⋊Q₈	—	134	4	no	—
118	#117	—	2	1	no	—
119	C₂×C₄⋊C₄	—	268	1	no	—
120	C₂×C₄⋊C₄	—	134	2	no	—
121	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
122	M₃₂	—	67	4	no	—
123	M₃₂	—	134	2	no	—
124	C₂×C₁₆	—	134	2	no	—
125	#108	—	2	2	no	—
126	#107	Dic₁₃₄	1	2	yes	complemented
127	#106	—	2	1	no	—
128	#106	—	67	4	no	—
129	#106	—	134	4	no	—
130	#106	—	134	2	no	—
131	C₄⋊C₈	—	4	2	no	—
132	C₄×C₈	—	2	2	no	—
133	C₂×Q₈	—	268	4	no	—
134	C₂×Q₈	—	536	4	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	268	2	no	—
137	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
138	M₁₆	—	4	1	no	—
139	M₁₆	—	4	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	4	1	no	—
143	C₂×C₈	#11761616	1	2	yes	—
144	C₂×C₈	D₄×D₆₇	1	2	yes	—
145	C₂×C₈	Q₈×D₆₇	1	2	yes	—
146	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
147	C₂×C₈	#11761628	1	1	yes	characteristic
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	268	4	no	—
150	C₄⋊C₄	—	268	2	no	—
151	C₄⋊C₄	—	134	4	no	—
152	C₂²⋊C₄	—	4	1	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	4	1	no	—
155	C₄²	#11761615	1	2	yes	—
156	C₁₆	—	134	4	no	—
157	C₂³	#14524068	1	1	yes	characteristic
158	Q₈	—	536	8	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	268	4	no	—
163	C₂×C₄	—	536	4	no	—
164	C₂×C₄	#14524011	1	1	yes	characteristic
165	C₂×C₄	#14524022	1	1	yes	characteristic
166	C₂×C₄	#14524041	1	1	yes	characteristic
167	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
168	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
169	C₂×C₄	#14524073	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	8	1	no	—
172	C₈	#14524021	1	2	yes	—
173	C₈	#14524042	1	2	yes	—
174	C₂²	—	2	2	no	—
175	C₂²	#16863546	1	1	yes	characteristic
176	C₂²	#16863686	1	1	yes	characteristic
177	C₂²	#16863806	1	1	yes	characteristic
178	C₄	—	2	4	no	—
179	C₄	—	8	2	no	—
180	C₄	—	536	4	no	—
181	C₄	#16863316	1	1	yes	characteristic
182	C₄	#16863534	1	1	yes	characteristic
183	C₄	#16863681	1	1	yes	characteristic
184	C₄	#16863792	1	1	yes	characteristic
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	#22561542	1	1	yes	characteristic
188	C₂	#22562380	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863794

C₂

yes

maximal, characteristic

C₂×#16863220

C₂

yes

maximal

#22558314

C₂

yes

maximal

#22558291

C₂

yes

maximal, characteristic

C₆₇×#1945

C₂

yes

maximal, characteristic

C₂×#14524043

—

C₂×#14523983

C₂²

yes

—

#16863794

—

C₂×#14523975

—

C₂×#14523975

C₂²

yes

—

#16863220

—

C₂×#108×C₆₇

C₂²

yes

characteristic

#427×C₆₇

C₂²

yes

characteristic

#424×C₆₇

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#339×C₆₇

—

C₂²×Dic₁₃₄

—

#14524043

—

#14524043

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

#14523983

—

#14523983

Q₈

yes

complemented

C₂×C₆₇⋊C₁₆

—

#14523975

—

#14523975

—

#14523975

D₄

yes

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

#117×C₆₇

—

C₂×C₄²×C₆₇

D₄

yes

characteristic

#108×C₆₇

—

#107×C₆₇

Q₈

yes

complemented

#106×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×Q₈

yes

—

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

#41

yes

characteristic

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

SD₁₆

yes

—

C₂³×C₆₇

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#120

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

#137

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₂²⋊Q₈

yes

—

C₈×C₆₇

C₄⋊Q₈

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

#13457

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#2058

yes

characteristic

C₂×C₆₇

#2230

yes

—

C₂×#498

—

C₂×#365

—

#1945

D₆₇

yes

characteristic

#1386

—

#1376

—

100

C₆₇

#13457

yes

characteristic, complemented, Sylow

101

C₂×C₄⋊Q₈

—

134

—

102

C₂×M₃₂

—

103

#498

—

134

—

104

C₂×#108

C₂×D₆₇

yes

characteristic

105

#427

C₂×D₆₇

yes

characteristic

106

C₂×#106

—

107

C₂×#106

—

134

—

108

#424

—

109

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

110

#365

—

134

—

111

#339

—

112

C₂²×Q₈

—

268

—

113

C₂×M₁₆

—

114

C₂²×C₈

C₂²⋊D₆₇

yes

characteristic

115

C₂²×C₈

C₂²×D₆₇

yes

characteristic

116

C₄⋊Q₈

—

268

—

117

C₄⋊Q₈

—

134

—

118

#117

—

119

C₂×C₄⋊C₄

—

268

—

120

C₂×C₄⋊C₄

—

134

—

121

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

122

M₃₂

—

123

M₃₂

—

134

—

124

C₂×C₁₆

—

134

—

125

#108

—

126

#107

Dic₁₃₄

yes

complemented

127

#106

—

128

#106

—

129

#106

—

134

—

130

#106

—

134

—

131

C₄⋊C₈

—

132

C₄×C₈

—

133

C₂×Q₈

—

268

—

134

C₂×Q₈

—

536

—

135

C₂²×C₄

—

136

C₂²×C₄

—

268

—

137

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

138

M₁₆

—

139

M₁₆

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

#11761616

yes

—

144

C₂×C₈

D₄×D₆₇

yes

—

145

C₂×C₈

Q₈×D₆₇

yes

—

146

C₂×C₈

C₂×Dic₁₃₄

yes

characteristic

147

C₂×C₈

#11761628

yes

characteristic

148

C₄⋊C₄

—

149

C₄⋊C₄

—

268

—

150

C₄⋊C₄

—

268

—

151

C₄⋊C₄

—

134

—

152

C₂²⋊C₄

—

153

C₄²

—

154

C₄²

—

155

C₄²

#11761615

yes

—

156

C₁₆

—

134

—

157

C₂³

#14524068

yes

characteristic

158

Q₈

—

536

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

268

—

163

C₂×C₄

—

536

—

164

C₂×C₄

#14524011

yes

characteristic

165

C₂×C₄

#14524022

yes

characteristic

166

C₂×C₄

#14524041

yes

characteristic

167

C₂×C₄

C₂×#11761615

yes

characteristic

168

C₂×C₄

C₂×#11761616

yes

characteristic

169

C₂×C₄

#14524073

yes

characteristic

170

C₈

—

171

C₈

—

172

C₈

#14524021

yes

—

173

C₈

#14524042

yes

—

174

C₂²

—

175

C₂²

#16863546

yes

characteristic

176

C₂²

#16863686

yes

characteristic

177

C₂²

#16863806

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

536

—

181

C₄

#16863316

yes

characteristic

182

C₄

#16863534

yes

characteristic

183

C₄

#16863681

yes

characteristic

184

C₄

#16863792

yes

characteristic

185

C₂

—

186

C₂

—

187

C₂

#22561542

yes

characteristic

188

C₂

#22562380

yes

—

	subgroup	G/H
center	C₂²	#16863686
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1945	C₂
socle	C₂²×C₆₇	#476
prime radical	C₂³×C₆₇	AGL(1,ℤ/8)
hypercenter	#1945	D₆₇
hypocenter	C₆₇	#13457
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1945

subgroup

G/H

center

C₂²

#16863686

commutator, cosocle

C₂²×C₈×C₆₇

C₂³

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1945

C₂

C₂²×C₆₇

C₂³×C₆₇

D₆₇

C₆₇

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1945

N	H ≅ G/N	type
23. #14523983	Q₈	semidirect
34. #107×C₆₇	Q₈	semidirect
100. C₆₇	#13457	semidirect
126. #107	Dic₁₃₄	semidirect

H ≅ G/N

type

23. #14523983

Q₈

semidirect

34. #107×C₆₇

Q₈

semidirect

100. C₆₇

#13457

semidirect

126. #107

Dic₁₃₄

semidirect

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

subgroup	index
C₂×#16863794	2
C₂×#16863220	2
#22558314	2
#22558291	2
C₆₇×#1945	2
#13457	67

subgroup

index

C₆₇×#1945

Subgroups — #30775479

Conjugacy classes of subgroups

Complements

Maximal subgroups