Group #30760611 — subgroups

Subgroups — #30760611

Order 17152.

310 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 85 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	#22561041	C₂	1	1	yes	maximal, characteristic
2	#22560329	C₂	1	1	yes	maximal, characteristic, complemented
3	#22558856	C₂	1	2	yes	maximal
4	#22558105	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1274	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863946	C₂²	1	1	yes	characteristic
7	#16863420	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×#14523971	—	2	2	no	—
10	#16863151	—	2	2	no	—
11	#16863141	—	2	2	no	—
12	#16863139	—	2	2	no	—
13	#431×C₆₇	C₂²	1	2	yes	—
14	#376×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#11761628	—	2	1	no	—
16	C₂×#11761612	—	2	1	no	—
17	C₂²×C₆₇⋊C₈	—	2	2	no	—
18	#14524043	D₄	1	1	yes	characteristic
19	#14524040	D₄	1	1	yes	characteristic
20	#14524031	—	2	1	no	—
21	#14524017	—	2	2	no	—
22	C₄×D₂₆₈	—	2	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
26	#14523971	—	4	4	no	—
27	C₂×C₄²×C₆₇	D₄	1	2	yes	—
28	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₆₇	—	2	2	no	—
30	C₂²⋊C₈×C₆₇	—	2	2	no	—
31	C₄×C₈×C₆₇	—	2	2	no	—
32	#95×C₆₇	—	2	2	no	—
33	#11761628	—	4	2	no	—
34	#11761628	—	4	2	no	—
35	C₂×Dic₁₃₄	—	2	1	no	—
36	C₂×D₂₆₈	—	2	1	no	—
37	C₂×D₂₆₈	—	2	2	no	—
38	C₂×C₂²⋊D₆₇	—	4	1	no	—
39	C₂×C₄×D₆₇	—	4	1	no	—
40	#11761612	—	4	2	no	—
41	C₂×C₆₇⋊C₈	—	4	2	no	—
42	C₂×C₆₇⋊C₈	—	4	1	no	—
43	C₂×C₆₇⋊C₈	—	4	2	no	—
44	C₂×C₆₇⋊C₈	—	4	2	no	—
45	C₄⋊Dic₆₇	—	2	2	no	—
46	(C₂×D₆₇)⋊C₄	—	4	2	no	—
47	C₂²×C₄×C₆₇	—	2	2	no	—
48	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	#41	1	2	yes	—
50	C₂×C₈×C₆₇	—	4	2	no	—
51	C₄²×C₆₇	—	2	2	no	—
52	C₄²×C₆₇	—	2	2	no	—
53	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₆₇	#41	1	2	yes	—
55	C₄²×C₆₇	#41	1	1	yes	characteristic
56	C₂³×C₆₇	#123	1	1	yes	characteristic
57	C₂²×D₆₇	—	4	1	no	—
58	C₂×C₄×C₆₇	—	4	2	no	—
59	C₂×C₄×C₆₇	—	4	4	no	—
60	C₂×C₄×C₆₇	—	4	2	no	—
61	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
62	C₂×C₄×C₆₇	#123	1	2	yes	—
63	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
64	C₂×C₄×C₆₇	#126	1	2	yes	—
65	C₂×C₄×C₆₇	#135	1	2	yes	—
66	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
67	C₂²⋊D₆₇	—	8	2	no	—
68	C₂×Dic₆₇	—	4	1	no	—
69	D₂₆₈	—	4	2	no	—
70	D₂₆₈	—	8	2	no	—
71	C₄×D₆₇	—	8	1	no	—
72	C₄×D₆₇	—	8	1	no	—
73	Dic₁₃₄	—	4	2	no	—
74	C₈×C₆₇	—	8	2	no	—
75	C₆₇⋊C₈	—	8	2	no	—
76	C₆₇⋊C₈	—	8	1	no	—
77	C₂²×C₆₇	—	2	2	no	—
78	C₂²×C₆₇	#397	1	1	yes	characteristic
79	C₂²×C₆₇	#482	1	2	yes	—
80	C₂×D₆₇	—	8	1	no	—
81	C₂×D₆₇	—	8	2	no	—
82	C₄×C₆₇	—	2	2	no	—
83	C₄×C₆₇	—	2	2	no	—
84	C₄×C₆₇	—	8	2	no	—
85	C₄×C₆₇	#463	1	2	yes	—
86	C₄×C₆₇	#488	1	1	yes	characteristic
87	C₄×C₆₇	#490	1	1	yes	characteristic
88	Dic₆₇	—	8	1	no	—
89	#10629	—	67	1	no	maximal, Sylow
90	C₂×C₆₇	—	2	2	no	—
91	C₂×C₆₇	#2073	1	1	yes	characteristic
92	C₂×C₆₇	#2080	1	2	yes	—
93	D₆₇	—	16	1	no	—
94	#1963	—	67	1	no	—
95	#1832	—	67	1	no	—
96	#1540	—	67	2	no	—
97	#1301	—	67	2	no	—
98	#1274	D₆₇	1	1	yes	characteristic, complemented
99	C₆₇	#10629	1	1	yes	characteristic, complemented, Sylow
100	#532	—	67	1	no	—
101	#431	C₂×D₆₇	1	2	yes	—
102	#423	—	67	1	no	—
103	C₂×C₄⋊C₈	—	67	2	no	—
104	C₂×#104	—	134	2	no	—
105	#376	C₂×D₆₇	1	1	yes	characteristic
106	#336	—	134	2	no	—
107	#332	—	134	2	no	—
108	#331	—	134	2	no	—
109	C₂×#41	—	134	1	no	—
110	C₂×M₁₆	—	134	1	no	—
111	C₂²×C₈	—	134	2	no	—
112	C₄⋊Q₈	—	67	1	no	—
113	#127	—	67	1	no	—
114	#124	—	134	1	no	—
115	#121	—	134	2	no	—
116	C₄×D₄	—	134	2	no	—
117	C₂×C₄²	D₂₆₈	1	2	yes	—
118	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	134	2	no	—
121	C₄⋊C₈	—	134	2	no	—
122	C₄⋊C₈	—	134	2	no	—
123	#104	—	268	4	no	—
124	C₂²⋊C₈	—	2	2	no	—
125	C₄×C₈	—	2	2	no	—
126	#95	—	2	2	no	—
127	#41	—	268	2	no	—
128	#41	—	268	2	no	—
129	C₂×Q₈	—	134	1	no	—
130	C₂×D₄	—	268	1	no	—
131	C₂×D₄	—	134	1	no	—
132	C₂×D₄	—	134	2	no	—
133	C₂²×C₄	—	2	2	no	—
134	C₂²×C₄	—	268	1	no	—
135	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
136	C₂²×C₄	#11761627	1	2	yes	—
137	M₁₆	—	268	2	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	268	2	no	—
140	C₂×C₈	—	268	1	no	—
141	C₂×C₈	—	268	2	no	—
142	C₂×C₈	—	268	2	no	—
143	C₄⋊C₄	—	134	2	no	—
144	C₂²⋊C₄	—	268	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	—	2	2	no	—
147	C₄²	D₄×D₆₇	1	1	yes	characteristic
148	C₄²	#11761628	1	2	yes	—
149	C₄²	#11761629	1	1	yes	characteristic
150	C₂³	—	268	1	no	—
151	C₂³	#14524027	1	1	yes	characteristic
152	Q₈	—	268	2	no	—
153	D₄	—	268	2	no	—
154	D₄	—	536	2	no	—
155	D₄	—	536	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	268	1	no	—
160	C₂×C₄	—	536	1	no	—
161	C₂×C₄	—	536	1	no	—
162	C₂×C₄	#14524018	1	1	yes	characteristic
163	C₂×C₄	#14524025	1	2	yes	—
164	C₂×C₄	#14524031	1	1	yes	characteristic
165	C₂×C₄	#14524038	1	2	yes	—
166	C₂×C₄	#14524065	1	2	yes	—
167	C₂×C₄	#14524071	1	2	yes	—
168	C₈	—	8	2	no	—
169	C₈	—	536	1	no	—
170	C₈	—	536	2	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	536	1	no	—
173	C₂²	—	536	2	no	—
174	C₂²	#16863323	1	1	yes	characteristic
175	C₂²	#16863721	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	8	2	no	—
179	C₄	—	536	1	no	—
180	C₄	#16863612	1	2	yes	—
181	C₄	#16863751	1	1	yes	characteristic
182	C₄	#16863761	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	—	1072	1	no	—
185	C₂	#22561624	1	1	yes	characteristic
186	C₂	#22561661	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22561041

C₂

yes

maximal, characteristic

#22560329

C₂

yes

maximal, characteristic, complemented

#22558856

C₂

yes

maximal

#22558105

C₂

yes

maximal, complemented

C₆₇×#1274

C₂

yes

maximal, characteristic, complemented

#16863946

C₂²

yes

characteristic

#16863420

C₂²

yes

characteristic

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

—

—

—

—

#431×C₆₇

C₂²

yes

—

#376×C₆₇

C₂²

yes

characteristic

C₂×#11761628

—

C₂×#11761612

—

C₂²×C₆₇⋊C₈

—

#14524043

D₄

yes

characteristic

#14524040

D₄

yes

characteristic

#14524031

—

#14524017

—

C₄×D₂₆₈

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

#14523971

—

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

C₄×C₈×C₆₇

—

#95×C₆₇

—

#11761628

—

#11761628

—

C₂×Dic₁₃₄

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×C₂²⋊D₆₇

—

C₂×C₄×D₆₇

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

(C₂×D₆₇)⋊C₄

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

#41

yes

—

C₄²×C₆₇

yes

C₂³×C₆₇

yes

C₂²×D₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#121

yes

characteristic

C₂×C₄×C₆₇

#123

yes

—

C₂×C₄×C₆₇

#124

yes

characteristic

C₂×C₄×C₆₇

#126

yes

—

C₂×C₄×C₆₇

#135

yes

—

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂²⋊D₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

—

C₄×D₆₇

—

C₄×D₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#397

yes

characteristic

C₂²×C₆₇

#482

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#463

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

#10629

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#2073

yes

characteristic

C₂×C₆₇

#2080

yes

—

D₆₇

—

—

—

—

—

D₆₇

yes

characteristic, complemented

C₆₇

#10629

yes

characteristic, complemented, Sylow

100

#532

—

101

#431

C₂×D₆₇

yes

—

102

#423

—

103

C₂×C₄⋊C₈

—

104

C₂×#104

—

134

—

105

#376

C₂×D₆₇

yes

characteristic

106

#336

—

134

—

107

#332

—

134

—

108

#331

—

134

—

109

C₂×#41

—

134

—

110

C₂×M₁₆

—

134

—

111

C₂²×C₈

—

134

—

112

C₄⋊Q₈

—

113

#127

—

114

#124

—

134

—

115

#121

—

134

—

116

C₄×D₄

—

134

—

117

C₂×C₄²

D₂₆₈

yes

—

118

C₂×C₄²

C₂²×D₆₇

yes

characteristic

119

C₄⋊C₈

—

120

C₄⋊C₈

—

134

—

121

C₄⋊C₈

—

134

—

122

C₄⋊C₈

—

134

—

123

#104

—

268

—

124

C₂²⋊C₈

—

125

C₄×C₈

—

126

#95

—

127

#41

—

268

—

128

#41

—

268

—

129

C₂×Q₈

—

134

—

130

C₂×D₄

—

268

—

131

C₂×D₄

—

134

—

132

C₂×D₄

—

134

—

133

C₂²×C₄

—

134

C₂²×C₄

—

268

—

135

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

136

C₂²×C₄

#11761627

yes

—

137

M₁₆

—

268

—

138

C₂×C₈

—

139

C₂×C₈

—

268

—

140

C₂×C₈

—

268

—

141

C₂×C₈

—

268

—

142

C₂×C₈

—

268

—

143

C₄⋊C₄

—

134

—

144

C₂²⋊C₄

—

268

—

145

C₄²

—

146

C₄²

—

147

C₄²

D₄×D₆₇

yes

characteristic

148

C₄²

#11761628

yes

—

149

C₄²

#11761629

yes

characteristic

150

C₂³

—

268

—

151

C₂³

#14524027

yes

characteristic

152

Q₈

—

268

—

153

D₄

—

268

—

154

D₄

—

536

—

155

D₄

—

536

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

268

—

160

C₂×C₄

—

536

—

161

C₂×C₄

—

536

—

162

C₂×C₄

#14524018

yes

characteristic

163

C₂×C₄

#14524025

yes

—

164

C₂×C₄

#14524031

yes

characteristic

165

C₂×C₄

#14524038

yes

—

166

C₂×C₄

#14524065

yes

—

167

C₂×C₄

#14524071

yes

—

168

C₈

—

169

C₈

—

536

—

170

C₈

—

536

—

171

C₂²

—

172

C₂²

—

536

—

173

C₂²

—

536

—

174

C₂²

#16863323

yes

characteristic

175

C₂²

#16863721

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

536

—

180

C₄

#16863612

yes

—

181

C₄

#16863751

yes

characteristic

182

C₄

#16863761

yes

characteristic

183

C₂

—

184

C₂

—

1072

—

185

C₂

#22561624

yes

characteristic

186

C₂

#22561661

yes

—

	subgroup	G/H
center	C₂×C₄	#14524031
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1274	C₂
socle	C₂²×C₆₇	#397
prime radical	#16863946	C₂²
hypercenter	#1274	D₆₇
hypocenter	C₆₇	#10629
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

subgroup

G/H

center

C₂×C₄

#14524031

commutator, cosocle

C₂×C₄²×C₆₇

C₂³

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1274

C₂

C₂²×C₆₇

C₂²

D₆₇

C₆₇

G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

N	H ≅ G/N	type
2. #22560329	C₂	semidirect
4. #22558105	C₂	semidirect
5. C₆₇×#1274	C₂	semidirect
98. #1274	D₆₇	semidirect
99. C₆₇	#10629	semidirect

H ≅ G/N

type

2. #22560329

C₂

semidirect

4. #22558105

C₂

semidirect

5. C₆₇×#1274

C₂

semidirect

98. #1274

D₆₇

semidirect

99. C₆₇

#10629

semidirect

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

subgroup	index
#22561041	2
#22560329	2
#22558856	2
#22558105	2
C₆₇×#1274	2
#10629	67

subgroup

index

C₆₇×#1274

Subgroups — #30760611

Conjugacy classes of subgroups

Complements

Maximal subgroups