Group #30755297 — subgroups

Subgroups — #30755297

Order 17152.

582 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 151 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×#16863773	C₂	1	1	yes	maximal, characteristic
2	#22558878	C₂	1	4	yes	maximal, complemented
3	#22558492	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1848	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₅₃₆	—	2	2	no	—
6	C₂×#14524040	C₂²	1	2	yes	—
7	#16863773	—	2	1	no	—
8	#16863773	—	2	1	no	—
9	#16863773	C₄	1	4	yes	—
10	#16863449	C₂²	1	2	yes	—
11	#16863139	—	2	8	no	—
12	#16863117	—	2	1	no	—
13	C₈×C₆₇⋊C₈	—	2	1	no	—
14	C₂×#108×C₆₇	—	2	2	no	—
15	#423×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×D₂₆₈	—	2	2	no	—
18	C₂²×D₂₆₈	—	2	4	no	—
19	C₂×D₅₃₆	—	2	4	no	—
20	C₂×D₅₃₆	—	2	4	no	—
21	C₂×D₅₃₆	—	4	4	no	—
22	C₂×D₅₃₆	—	4	4	no	—
23	#14524040	—	2	2	no	—
24	#14524040	—	2	4	no	—
25	#14524040	C₂×C₄	1	4	yes	—
26	C₄⋊(C₆₇⋊C₈)	D₄	1	4	yes	—
27	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
28	C₄×C₆₇⋊C₈	—	2	2	no	—
29	C₂×M₁₆×C₆₇	—	2	2	no	—
30	C₂²×C₈×C₆₇	D₄	1	2	yes	—
31	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
32	#108×C₆₇	—	4	4	no	—
33	C₄⋊C₈×C₆₇	—	2	4	no	—
34	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
36	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
37	C₂³×D₆₇	—	4	2	no	—
38	C₂×D₂₆₈	—	2	4	no	—
39	C₂×D₂₆₈	—	2	8	no	—
40	C₂×D₂₆₈	—	4	4	no	—
41	C₂×D₂₆₈	—	4	4	no	—
42	C₂×D₂₆₈	—	4	4	no	—
43	C₂×D₂₆₈	—	4	2	no	—
44	C₂×D₂₆₈	—	8	8	no	—
45	D₅₃₆	—	4	8	no	—
46	D₅₃₆	—	4	8	no	—
47	C₂×C₆₇⋊C₈	—	2	4	no	—
48	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	#41	1	2	yes	—
50	M₁₆×C₆₇	—	4	4	no	—
51	C₂×C₈×C₆₇	—	4	2	no	—
52	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
53	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₆₇	D₈	1	4	yes	—
55	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
56	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₆₇	C₂×D₄	1	2	yes	—
58	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
59	C₂³×C₆₇	#124	1	1	yes	characteristic
60	C₂²×D₆₇	—	4	4	no	—
61	C₂²×D₆₇	—	8	8	no	—
62	C₂²×D₆₇	—	8	4	no	—
63	C₂×C₄×C₆₇	—	2	4	no	—
64	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
66	C₂×C₄×C₆₇	#121	1	2	yes	—
67	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
69	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
70	D₂₆₈	—	4	8	no	—
71	D₂₆₈	—	8	8	no	—
72	D₂₆₈	—	8	8	no	—
73	D₂₆₈	—	8	2	no	—
74	D₂₆₈	—	8	2	no	—
75	C₈×C₆₇	—	8	2	no	—
76	C₈×C₆₇	#102	1	4	yes	—
77	C₈×C₆₇	#102	1	4	yes	—
78	C₆₇⋊C₈	—	4	4	no	—
79	C₂²×C₆₇	—	2	2	no	—
80	C₂²×C₆₇	#390	1	1	yes	characteristic
81	C₂²×C₆₇	#486	1	2	yes	—
82	C₂×D₆₇	—	8	4	no	—
83	C₂×D₆₇	—	8	8	no	—
84	C₂×D₆₇	—	16	8	no	—
85	C₄×C₆₇	—	2	2	no	—
86	C₄×C₆₇	C₂×#102	1	2	yes	—
87	C₄×C₆₇	#472	1	4	yes	—
88	C₄×C₆₇	#492	1	1	yes	characteristic
89	C₄×C₆₇	#493	1	1	yes	characteristic
90	#9773	—	67	1	no	maximal, Sylow
91	C₂×C₆₇	—	2	2	no	—
92	C₂×C₆₇	#1957	1	1	yes	characteristic
93	C₂×C₆₇	#1978	1	2	yes	—
94	D₆₇	—	16	4	no	—
95	C₂×#493	—	67	1	no	—
96	#1848	D₆₇	1	1	yes	characteristic, complemented
97	#1545	—	67	4	no	—
98	#1450	—	67	1	no	—
99	C₆₇	#9773	1	1	yes	characteristic, complemented, Sylow
100	C₂²×D₈	—	134	2	no	—
101	C₂×#127	—	67	2	no	—
102	#493	—	67	4	no	—
103	#493	—	134	1	no	—
104	#493	—	134	1	no	—
105	#431	—	67	2	no	—
106	C₂×#108	—	2	2	no	—
107	#423	C₂×D₆₇	1	2	yes	—
108	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
109	#331	—	134	8	no	—
110	#322	—	134	1	no	—
111	C₈²	—	134	1	no	—
112	C₂²×D₄	—	134	2	no	—
113	C₂²×D₄	—	134	4	no	—
114	C₂×D₈	—	268	4	no	—
115	C₂×D₈	—	268	4	no	—
116	C₂×D₈	—	134	4	no	—
117	C₂×D₈	—	134	4	no	—
118	C₂×M₁₆	—	2	2	no	—
119	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
120	#127	—	67	4	no	—
121	#127	—	134	4	no	—
122	#127	—	134	2	no	—
123	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
124	#108	—	4	4	no	—
125	C₄⋊C₈	—	2	4	no	—
126	C₄⋊C₈	—	67	4	no	—
127	C₂²⋊C₈	—	134	2	no	—
128	C₄×C₈	—	134	2	no	—
129	C₄×C₈	C₄×D₆₇	1	2	yes	—
130	C₄×C₈	D₂₆₈	1	1	yes	characteristic
131	C₄×C₈	D₂₆₈	1	1	yes	characteristic
132	C₂⁴	—	268	2	no	—
133	C₂×D₄	—	268	4	no	—
134	C₂×D₄	—	268	2	no	—
135	C₂×D₄	—	268	4	no	—
136	C₂×D₄	—	268	4	no	—
137	C₂×D₄	—	536	8	no	—
138	C₂×D₄	—	134	4	no	—
139	C₂×D₄	—	134	8	no	—
140	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
141	C₂²×C₄	#11761629	1	2	yes	—
142	D₈	—	268	8	no	—
143	D₈	—	268	8	no	—
144	M₁₆	—	4	4	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	—	134	4	no	—
147	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
148	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
149	C₂×C₈	#11761613	1	4	yes	—
150	C₂×C₈	#11761615	1	4	yes	—
151	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
152	C₄²	D₄×D₆₇	1	2	yes	—
153	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
154	C₂³	—	268	4	no	—
155	C₂³	—	536	8	no	—
156	C₂³	—	536	4	no	—
157	C₂³	#14524032	1	1	yes	characteristic
158	D₄	—	268	8	no	—
159	D₄	—	536	8	no	—
160	D₄	—	536	2	no	—
161	D₄	—	536	2	no	—
162	D₄	—	536	8	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
165	C₂×C₄	#14524005	1	2	yes	—
166	C₂×C₄	#14524018	1	2	yes	—
167	C₂×C₄	#14524039	1	1	yes	characteristic
168	C₂×C₄	C₂×#11761613	1	2	yes	—
169	C₂×C₄	C₂×#11761615	1	2	yes	—
170	C₈	—	8	2	no	—
171	C₈	—	268	4	no	—
172	C₈	#14523963	1	4	yes	—
173	C₈	#14523963	1	4	yes	—
174	C₂²	—	2	2	no	—
175	C₂²	—	536	8	no	—
176	C₂²	—	536	4	no	—
177	C₂²	—	1072	8	no	—
178	C₂²	#16863297	1	1	yes	characteristic
179	C₂²	#16863744	1	2	yes	—
180	C₄	—	2	2	no	—
181	C₄	C₂×#14523963	1	2	yes	—
182	C₄	#16863667	1	4	yes	—
183	C₄	#16863770	1	1	yes	characteristic
184	C₄	#16863772	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	—	1072	4	no	—
187	C₂	#22561014	1	1	yes	characteristic
188	C₂	#22561117	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863773

C₂

yes

maximal, characteristic

#22558878

C₂

yes

maximal, complemented

#22558492

C₂

yes

maximal, characteristic, complemented

C₆₇×#1848

C₂

yes

maximal, characteristic, complemented

C₂²×D₅₃₆

—

C₂×#14524040

C₂²

yes

—

#16863773

—

#16863773

—

#16863773

C₄

yes

—

#16863449

C₂²

yes

—

#16863139

—

#16863117

—

C₈×C₆₇⋊C₈

—

C₂×#108×C₆₇

—

#423×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂²×D₂₆₈

—

C₂²×D₂₆₈

—

C₂×D₅₃₆

—

C₂×D₅₃₆

—

C₂×D₅₃₆

—

C₂×D₅₃₆

—

#14524040

—

#14524040

—

#14524040

C₂×C₄

yes

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

—

C₂²⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#108×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

characteristic

C₂³×D₆₇

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

D₅₃₆

—

D₅₃₆

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₂³×C₆₇

#124

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

—

C₂×C₄×C₆₇

C₂×SD₁₆

yes

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₈×C₆₇

—

C₈×C₆₇

#102

yes

—

C₈×C₆₇

#102

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#390

yes

characteristic

C₂²×C₆₇

#486

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×#102

yes

—

C₄×C₆₇

#472

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₂×C₆₇

—

C₂×C₆₇

#1957

yes

characteristic

C₂×C₆₇

#1978

yes

—

D₆₇

—

C₂×#493

—

#1848

D₆₇

yes

characteristic, complemented

#1545

—

#1450

—

C₆₇

#9773

yes

characteristic, complemented, Sylow

100

C₂²×D₈

—

134

—

101

C₂×#127

—

102

#493

—

103

#493

—

134

—

104

#493

—

134

—

105

#431

—

106

C₂×#108

—

107

#423

C₂×D₆₇

yes

—

108

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

109

#331

—

134

—

110

#322

—

134

—

111

C₈²

—

134

—

112

C₂²×D₄

—

134

—

113

C₂²×D₄

—

134

—

114

C₂×D₈

—

268

—

115

C₂×D₈

—

268

—

116

C₂×D₈

—

134

—

117

C₂×D₈

—

134

—

118

C₂×M₁₆

—

119

C₂²×C₈

C₂²⋊D₆₇

yes

—

120

#127

—

121

#127

—

134

—

122

#127

—

134

—

123

C₂×C₄²

C₂²×D₆₇

yes

characteristic

124

#108

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₂²⋊C₈

—

134

—

128

C₄×C₈

—

134

—

129

C₄×C₈

C₄×D₆₇

yes

—

130

C₄×C₈

D₂₆₈

yes

characteristic

131

C₄×C₈

D₂₆₈

yes

characteristic

132

C₂⁴

—

268

—

133

C₂×D₄

—

268

—

134

C₂×D₄

—

268

—

135

C₂×D₄

—

268

—

136

C₂×D₄

—

268

—

137

C₂×D₄

—

536

—

138

C₂×D₄

—

134

—

139

C₂×D₄

—

134

—

140

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

141

C₂²×C₄

#11761629

yes

—

142

D₈

—

268

—

143

D₈

—

268

—

144

M₁₆

—

145

C₂×C₈

—

146

C₂×C₈

—

134

—

147

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

148

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

149

C₂×C₈

#11761613

yes

—

150

C₂×C₈

#11761615

yes

—

151

C₄²

C₂×C₄×D₆₇

yes

characteristic

152

C₄²

D₄×D₆₇

yes

—

153

C₄²

C₂×D₂₆₈

yes

characteristic

154

C₂³

—

268

—

155

C₂³

—

536

—

156

C₂³

—

536

—

157

C₂³

#14524032

yes

characteristic

158

D₄

—

268

—

159

D₄

—

536

—

160

D₄

—

536

—

161

D₄

—

536

—

162

D₄

—

536

—

163

C₂×C₄

—

164

C₂×C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

165

C₂×C₄

#14524005

yes

—

166

C₂×C₄

#14524018

yes

—

167

C₂×C₄

#14524039

yes

characteristic

168

C₂×C₄

C₂×#11761613

yes

—

169

C₂×C₄

C₂×#11761615

yes

—

170

C₈

—

171

C₈

—

268

—

172

C₈

#14523963

yes

—

173

C₈

#14523963

yes

—

174

C₂²

—

175

C₂²

—

536

—

176

C₂²

—

536

—

177

C₂²

—

1072

—

178

C₂²

#16863297

yes

characteristic

179

C₂²

#16863744

yes

—

180

C₄

—

181

C₄

C₂×#14523963

yes

—

182

C₄

#16863667

yes

—

183

C₄

#16863770

yes

characteristic

184

C₄

#16863772

yes

characteristic

185

C₂

—

186

C₂

—

1072

—

187

C₂

#22561014

yes

characteristic

188

C₂

#22561117

yes

—

	subgroup	G/H
center	C₂²	#16863297
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1848	C₂
socle	C₂²×C₆₇	#390
prime radical	C₂×#16863773	C₂
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1848	D₆₇
hypocenter	C₆₇	#9773
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1848

subgroup

G/H

center

C₂²

#16863297

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1848

C₂

C₂²×C₆₇

C₂

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1848

N	H ≅ G/N	type
2. #22558878	C₂	semidirect
3. #22558492	C₂	semidirect
4. C₆₇×#1848	C₂	semidirect
96. #1848	D₆₇	semidirect
99. C₆₇	#9773	semidirect

H ≅ G/N

type

2. #22558878

C₂

semidirect

3. #22558492

C₂

semidirect

4. C₆₇×#1848

C₂

semidirect

96. #1848

D₆₇

semidirect

99. C₆₇

#9773

semidirect

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

subgroup	index
C₂×#16863773	2
#22558878	2
#22558492	2
C₆₇×#1848	2
#9773	67

subgroup

index

C₆₇×#1848

Subgroups — #30755297

Conjugacy classes of subgroups

Complements

Maximal subgroups