Group #30760638 — subgroups

Subgroups — #30760638

Order 17152.

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

normal subgroups only

This group has 346 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	#22561033	C₂	1	1	yes	maximal, characteristic
2	#22560350	C₂	1	1	yes	maximal, characteristic, complemented
3	#22558876	C₂	1	2	yes	maximal
4	#22558068	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1281	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14524040	C₂²	1	1	yes	characteristic
7	#16863420	C₂²	1	2	yes	—
8	C₂×#14523965	—	2	2	no	—
9	#16863262	C₂²	1	1	yes	characteristic
10	#16863161	—	2	2	no	—
11	#16863139	—	2	4	no	—
12	#431×C₆₇	C₂²	1	2	yes	—
13	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×D₂₆₈	—	2	1	no	—
15	C₂²×D₂₆₈	—	2	2	no	—
16	C₂×#11761612	—	2	2	no	—
17	#14524040	—	2	2	no	—
18	#14524040	—	2	1	no	—
19	#14524040	D₄	1	2	yes	—
20	C₂×C₄⋊Dic₆₇	—	2	1	no	—
21	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
23	#14523965	—	4	4	no	—
24	C₆₇⋊#95	—	2	2	no	—
25	C₂²×C₈×C₆₇	D₄	1	2	yes	—
26	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₆₇	—	2	2	no	—
28	C₂²⋊C₈×C₆₇	—	2	2	no	—
29	#97×C₆₇	—	2	2	no	—
30	C₄×C₈×C₆₇	—	2	2	no	—
31	C₂³×D₆₇	—	4	1	no	—
32	C₂×D₂₆₈	—	2	2	no	—
33	C₂×D₂₆₈	—	2	4	no	—
34	C₂×D₂₆₈	—	4	2	no	—
35	C₂×D₂₆₈	—	4	2	no	—
36	C₂×D₂₆₈	—	4	1	no	—
37	C₂×D₂₆₈	—	4	2	no	—
38	C₂×D₂₆₈	—	8	4	no	—
39	C₂²×Dic₆₇	—	4	1	no	—
40	#11761612	—	4	2	no	—
41	#11761612	—	4	2	no	—
42	C₂×C₆₇⋊C₈	—	4	2	no	—
43	C₄⋊Dic₆₇	—	4	2	no	—
44	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	#41	1	2	yes	—
46	C₂×C₈×C₆₇	—	2	2	no	—
47	C₂×C₈×C₆₇	—	2	2	no	—
48	C₂×C₈×C₆₇	—	2	2	no	—
49	C₂×C₈×C₆₇	—	4	2	no	—
50	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₄²×C₆₇	#41	1	2	yes	—
52	C₄²×C₆₇	#41	1	1	yes	characteristic
53	C₂³×C₆₇	#124	1	1	yes	characteristic
54	C₂²×D₆₇	—	4	2	no	—
55	C₂²×D₆₇	—	8	4	no	—
56	C₂²×D₆₇	—	8	1	no	—
57	C₂²×D₆₇	—	8	1	no	—
58	C₂×C₄×C₆₇	—	4	2	no	—
59	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
60	C₂×C₄×C₆₇	#123	1	2	yes	—
61	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
62	C₂×C₄×C₆₇	#126	1	2	yes	—
63	C₂×C₄×C₆₇	#135	1	2	yes	—
64	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
65	C₂×Dic₆₇	—	8	2	no	—
66	C₂×Dic₆₇	—	8	1	no	—
67	D₂₆₈	—	4	4	no	—
68	D₂₆₈	—	8	4	no	—
69	D₂₆₈	—	8	4	no	—
70	D₂₆₈	—	8	2	no	—
71	C₈×C₆₇	—	4	2	no	—
72	C₈×C₆₇	—	8	2	no	—
73	C₆₇⋊C₈	—	8	2	no	—
74	C₂²×C₆₇	—	2	2	no	—
75	C₂²×C₆₇	#397	1	1	yes	characteristic
76	C₂²×C₆₇	#488	1	1	yes	characteristic
77	C₂²×C₆₇	#490	1	1	yes	characteristic
78	C₂×D₆₇	—	8	2	no	—
79	C₂×D₆₇	—	8	4	no	—
80	C₂×D₆₇	—	16	2	no	—
81	C₂×D₆₇	—	16	2	no	—
82	C₄×C₆₇	—	2	2	no	—
83	C₄×C₆₇	—	2	2	no	—
84	C₄×C₆₇	#463	1	2	yes	—
85	C₄×C₆₇	#482	1	2	yes	—
86	Dic₆₇	—	16	1	no	—
87	#10634	—	67	1	no	maximal, Sylow
88	C₂×C₆₇	—	2	1	no	—
89	C₂×C₆₇	—	2	1	no	—
90	C₂×C₆₇	#2073	1	1	yes	characteristic
91	C₂×C₆₇	#2081	1	2	yes	—
92	D₆₇	—	16	2	no	—
93	#1961	—	67	1	no	—
94	#1838	—	67	1	no	—
95	#1545	—	67	2	no	—
96	#1286	—	67	2	no	—
97	#1281	D₆₇	1	1	yes	characteristic, complemented
98	C₆₇	#10634	1	1	yes	characteristic, complemented, Sylow
99	C₂×#127	—	67	1	no	—
100	#431	C₂×D₆₇	1	2	yes	—
101	#423	—	67	2	no	—
102	C₂×#102	—	134	2	no	—
103	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
104	#382	—	67	1	no	—
105	#341	—	134	2	no	—
106	#331	—	134	4	no	—
107	C₂²×D₄	—	134	1	no	—
108	C₂²×D₄	—	134	2	no	—
109	C₂×M₁₆	—	134	2	no	—
110	C₂²×C₈	D₂₆₈	1	2	yes	—
111	#127	—	67	2	no	—
112	#127	—	134	2	no	—
113	#127	—	134	1	no	—
114	C₂×C₄⋊C₄	—	134	1	no	—
115	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	134	2	no	—
118	C₄⋊C₈	—	134	2	no	—
119	#102	—	268	4	no	—
120	C₂²⋊C₈	—	2	2	no	—
121	#97	—	2	2	no	—
122	C₄×C₈	—	2	2	no	—
123	#95	—	134	2	no	—
124	C₂⁴	—	268	1	no	—
125	C₂×D₄	—	268	1	no	—
126	C₂×D₄	—	268	2	no	—
127	C₂×D₄	—	268	2	no	—
128	C₂×D₄	—	268	2	no	—
129	C₂×D₄	—	536	4	no	—
130	C₂×D₄	—	134	2	no	—
131	C₂×D₄	—	134	4	no	—
132	C₂²×C₄	—	268	1	no	—
133	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
134	C₂²×C₄	#11761628	1	2	yes	—
135	M₁₆	—	268	2	no	—
136	M₁₆	—	268	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	268	2	no	—
142	C₄⋊C₄	—	268	2	no	—
143	C₄²	D₄×D₆₇	1	1	yes	characteristic
144	C₄²	#11761627	1	2	yes	—
145	C₄²	#11761629	1	1	yes	characteristic
146	C₂³	—	268	2	no	—
147	C₂³	—	536	4	no	—
148	C₂³	—	536	1	no	—
149	C₂³	—	536	1	no	—
150	C₂³	#14524031	1	1	yes	characteristic
151	D₄	—	268	4	no	—
152	D₄	—	536	4	no	—
153	D₄	—	536	4	no	—
154	D₄	—	536	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	536	2	no	—
157	C₂×C₄	—	536	1	no	—
158	C₂×C₄	#14524018	1	1	yes	characteristic
159	C₂×C₄	#14524025	1	2	yes	—
160	C₂×C₄	#14524027	1	1	yes	characteristic
161	C₂×C₄	#14524038	1	2	yes	—
162	C₂×C₄	#14524065	1	2	yes	—
163	C₂×C₄	#14524071	1	2	yes	—
164	C₈	—	4	2	no	—
165	C₈	—	8	2	no	—
166	C₈	—	536	2	no	—
167	C₂²	—	2	2	no	—
168	C₂²	—	536	2	no	—
169	C₂²	—	536	4	no	—
170	C₂²	—	1072	2	no	—
171	C₂²	—	1072	2	no	—
172	C₂²	#16863323	1	1	yes	characteristic
173	C₂²	#16863751	1	1	yes	characteristic
174	C₂²	#16863761	1	1	yes	characteristic
175	C₄	—	2	2	no	—
176	C₄	—	2	2	no	—
177	C₄	—	1072	1	no	—
178	C₄	#16863612	1	2	yes	—
179	C₄	#16863721	1	2	yes	—
180	C₂	—	2	1	no	—
181	C₂	—	2	1	no	—
182	C₂	—	1072	2	no	—
183	C₂	#22561624	1	1	yes	characteristic
184	C₂	#22561668	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22561033

C₂

yes

maximal, characteristic

#22560350

C₂

yes

maximal, characteristic, complemented

#22558876

C₂

yes

maximal

#22558068

C₂

yes

maximal, complemented

C₆₇×#1281

C₂

yes

maximal, characteristic, complemented

C₂×#14524040

C₂²

yes

characteristic

#16863420

C₂²

yes

—

C₂×#14523965

—

#16863262

C₂²

yes

characteristic

#16863161

—

#16863139

—

#431×C₆₇

C₂²

yes

—

C₂×#97×C₆₇

C₂²

yes

characteristic

C₂²×D₂₆₈

—

C₂²×D₂₆₈

—

—

—

—

D₄

yes

—

C₂×C₄⋊Dic₆₇

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

#14523965

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

#97×C₆₇

—

C₄×C₈×C₆₇

—

C₂³×D₆₇

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂²×Dic₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

#41

yes

—

C₄²×C₆₇

yes

C₂³×C₆₇

yes

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#121

yes

characteristic

C₂×C₄×C₆₇

#123

yes

—

C₂×C₄×C₆₇

#123

yes

characteristic

C₂×C₄×C₆₇

#126

yes

—

C₂×C₄×C₆₇

#135

yes

—

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#463

yes

—

C₄×C₆₇

#482

yes

—

Dic₆₇

—

#10634

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#2073

yes

characteristic

C₂×C₆₇

#2081

yes

—

D₆₇

—

—

—

—

—

D₆₇

yes

characteristic, complemented

C₆₇

#10634

yes

characteristic, complemented, Sylow

C₂×#127

—

100

#431

C₂×D₆₇

yes

—

101

#423

—

102

C₂×#102

—

134

—

103

C₂×#97

C₂×D₆₇

yes

characteristic

104

#382

—

105

#341

—

134

—

106

#331

—

134

—

107

C₂²×D₄

—

134

—

108

C₂²×D₄

—

134

—

109

C₂×M₁₆

—

134

—

110

C₂²×C₈

D₂₆₈

yes

—

111

#127

—

112

#127

—

134

—

113

#127

—

134

—

114

C₂×C₄⋊C₄

—

134

—

115

C₂×C₄²

C₂²×D₆₇

yes

characteristic

116

C₄⋊C₈

—

117

C₄⋊C₈

—

134

—

118

C₄⋊C₈

—

134

—

119

#102

—

268

—

120

C₂²⋊C₈

—

121

#97

—

122

C₄×C₈

—

123

#95

—

134

—

124

C₂⁴

—

268

—

125

C₂×D₄

—

268

—

126

C₂×D₄

—

268

—

127

C₂×D₄

—

268

—

128

C₂×D₄

—

268

—

129

C₂×D₄

—

536

—

130

C₂×D₄

—

134

—

131

C₂×D₄

—

134

—

132

C₂²×C₄

—

268

—

133

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

134

C₂²×C₄

#11761628

yes

—

135

M₁₆

—

268

—

136

M₁₆

—

268

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

268

—

142

C₄⋊C₄

—

268

—

143

C₄²

D₄×D₆₇

yes

characteristic

144

C₄²

#11761627

yes

—

145

C₄²

#11761629

yes

characteristic

146

C₂³

—

268

—

147

C₂³

—

536

—

148

C₂³

—

536

—

149

C₂³

—

536

—

150

C₂³

#14524031

yes

characteristic

151

D₄

—

268

—

152

D₄

—

536

—

153

D₄

—

536

—

154

D₄

—

536

—

155

C₂×C₄

—

156

C₂×C₄

—

536

—

157

C₂×C₄

—

536

—

158

C₂×C₄

#14524018

yes

characteristic

159

C₂×C₄

#14524025

yes

—

160

C₂×C₄

#14524027

yes

characteristic

161

C₂×C₄

#14524038

yes

—

162

C₂×C₄

#14524065

yes

—

163

C₂×C₄

#14524071

yes

—

164

C₈

—

165

C₈

—

166

C₈

—

536

—

167

C₂²

—

168

C₂²

—

536

—

169

C₂²

—

536

—

170

C₂²

—

1072

—

171

C₂²

—

1072

—

172

C₂²

#16863323

yes

characteristic

173

C₂²

#16863751

yes

characteristic

174

C₂²

#16863761

yes

characteristic

175

C₄

—

176

C₄

—

177

C₄

—

1072

—

178

C₄

#16863612

yes

—

179

C₄

#16863721

yes

—

180

C₂

—

181

C₂

—

182

C₂

—

1072

—

183

C₂

#22561624

yes

characteristic

184

C₂

#22561668

yes

—

	subgroup	G/H
center	C₂²	#16863323
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1281	C₂
socle	C₂²×C₆₇	#397
prime radical	C₂×#14524040	C₂²
hypercenter	#1281	D₆₇
hypocenter	C₆₇	#10634
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1281

subgroup

G/H

center

C₂²

#16863323

commutator, cosocle

C₂×C₄²×C₆₇

C₂³

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1281

C₂

C₂²×C₆₇

C₂²

D₆₇

C₆₇

G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1281

N	H ≅ G/N	type
2. #22560350	C₂	semidirect
4. #22558068	C₂	semidirect
5. C₆₇×#1281	C₂	semidirect
97. #1281	D₆₇	semidirect
98. C₆₇	#10634	semidirect

H ≅ G/N

type

2. #22560350

C₂

semidirect

4. #22558068

C₂

semidirect

5. C₆₇×#1281

C₂

semidirect

97. #1281

D₆₇

semidirect

98. C₆₇

#10634

semidirect

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

subgroup	index
#22561033	2
#22560350	2
#22558876	2
#22558068	2
C₆₇×#1281	2
#10634	67

subgroup

index

C₆₇×#1281

Subgroups — #30760638

Conjugacy classes of subgroups

Complements

Maximal subgroups