Group #30755328 — subgroups

Subgroups — #30755328

Order 17152.

390 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 127 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	#22560347	C₂	1	1	yes	maximal, characteristic
2	#22558894	C₂	1	2	yes	maximal
3	#22558890	C₂	1	2	yes	maximal
4	#22558501	C₂	1	1	yes	maximal, characteristic
5	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	1	yes	characteristic
7	#16863449	C₂²	1	1	yes	characteristic
8	#16863420	C₂²	1	2	yes	—
9	#16863161	—	2	2	no	—
10	#16863145	—	2	4	no	—
11	#16863142	—	2	4	no	—
12	#16863118	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
14	#500×C₆₇	—	2	2	no	—
15	#500×C₆₇	C₄	1	4	yes	—
16	C₂×#107×C₆₇	—	2	2	no	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#11761612	—	2	2	no	—
19	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
20	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
21	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
22	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
23	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
24	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
25	C₄×C₆₇⋊C₈	—	2	1	no	—
26	C₄×C₆₇⋊C₈	—	2	1	no	—
27	C₂²×Q₈×C₆₇	—	2	2	no	—
28	C₂²×C₈×C₆₇	D₄	1	2	yes	—
29	C₄⋊Q₈×C₆₇	—	2	2	no	—
30	C₄⋊Q₈×C₆₇	—	2	2	no	—
31	C₄⋊Q₈×C₆₇	—	2	2	no	—
32	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
35	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
36	#107×C₆₇	—	2	8	no	—
37	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₆₇	D₄	1	2	yes	—
39	#11761612	—	4	4	no	—
40	C₂×C₆₇⋊C₈	—	2	2	no	—
41	C₂×C₆₇⋊C₈	—	2	2	no	—
42	C₂×C₆₇⋊C₈	—	4	2	no	—
43	C₂×Q₈×C₆₇	—	2	4	no	—
44	C₂×Q₈×C₆₇	—	4	2	no	—
45	C₂×Q₈×C₆₇	—	8	4	no	—
46	C₂²×C₄×C₆₇	—	4	2	no	—
47	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
50	C₂×C₈×C₆₇	—	2	4	no	—
51	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
52	C₄⋊C₄×C₆₇	—	2	4	no	—
53	C₄⋊C₄×C₆₇	—	2	4	no	—
54	C₄⋊C₄×C₆₇	—	4	2	no	—
55	C₄⋊C₄×C₆₇	—	4	2	no	—
56	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
60	C₂³×C₆₇	#124	1	1	yes	characteristic
61	Q₈×C₆₇	—	8	4	no	—
62	Q₈×C₆₇	—	8	4	no	—
63	C₂×C₄×C₆₇	—	2	2	no	—
64	C₂×C₄×C₆₇	—	2	2	no	—
65	C₂×C₄×C₆₇	—	4	4	no	—
66	C₂×C₄×C₆₇	—	8	4	no	—
67	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
71	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
72	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
73	C₂×C₄×C₆₇	#137	1	4	yes	—
74	C₈×C₆₇	—	2	4	no	—
75	C₆₇⋊C₈	—	4	2	no	—
76	C₆₇⋊C₈	—	4	2	no	—
77	C₆₇⋊C₈	—	8	2	no	—
78	C₂²×C₆₇	—	2	1	no	—
79	C₂²×C₆₇	—	2	1	no	—
80	C₂²×C₆₇	#390	1	1	yes	characteristic
81	C₂²×C₆₇	#491	1	2	yes	—
82	C₄×C₆₇	—	2	1	no	—
83	C₄×C₆₇	—	2	1	no	—
84	C₄×C₆₇	—	8	4	no	—
85	C₄×C₆₇	#419	1	2	yes	—
86	C₄×C₆₇	#472	1	2	yes	—
87	C₄×C₆₇	#473	1	2	yes	—
88	C₄×C₆₇	#497	1	2	yes	—
89	#9778	—	67	1	no	maximal, Sylow
90	C₂×C₆₇	—	2	2	no	—
91	C₂×C₆₇	#1960	1	1	yes	characteristic
92	C₂×C₆₇	#1978	1	1	yes	characteristic
93	C₂×C₆₇	#1979	1	1	yes	characteristic
94	C₂×#500	D₆₇	1	1	yes	characteristic
95	#1837	—	67	1	no	—
96	#1549	—	67	2	no	—
97	#1548	—	67	2	no	—
98	#1452	—	67	1	no	—
99	C₆₇	#9778	1	1	yes	characteristic, complemented, Sylow
100	C₂×C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
101	#500	—	2	2	no	—
102	#500	Dic₆₇	1	4	yes	—
103	#431	—	67	1	no	—
104	#431	—	67	1	no	—
105	C₂×#107	—	2	2	no	—
106	#423	—	67	2	no	—
107	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
108	#341	—	134	2	no	—
109	#333	—	134	4	no	—
110	#332	—	134	4	no	—
111	#322	—	134	2	no	—
112	C₂²×Q₈	—	2	2	no	—
113	C₂×M₁₆	—	134	2	no	—
114	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
115	C₄⋊Q₈	—	2	2	no	—
116	C₄⋊Q₈	—	2	2	no	—
117	C₄⋊Q₈	—	2	2	no	—
118	C₄⋊Q₈	C₂×Dic₆₇	1	4	yes	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
122	#107	—	2	8	no	—
123	C₄⋊C₈	—	67	2	no	—
124	C₄⋊C₈	—	67	2	no	—
125	C₄⋊C₈	—	134	2	no	—
126	C₄⋊C₈	—	134	2	no	—
127	C₂²⋊C₈	—	134	1	no	—
128	C₂²⋊C₈	—	134	1	no	—
129	C₄×C₈	—	134	1	no	—
130	C₄×C₈	—	134	1	no	—
131	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
132	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
133	C₂×Q₈	—	2	4	no	—
134	C₂×Q₈	—	4	2	no	—
135	C₂×Q₈	—	8	4	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
138	C₂²×C₄	#11761627	1	1	yes	characteristic
139	C₂²×C₄	#11761627	1	1	yes	characteristic
140	M₁₆	—	268	4	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	268	2	no	—
143	C₂×C₈	—	134	2	no	—
144	C₂×C₈	—	134	2	no	—
145	C₂×C₈	C₂²⋊Dic₆₇	1	4	yes	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
151	C₄²	D₄×D₆₇	1	1	yes	characteristic
152	C₄²	D₄×D₆₇	1	1	yes	characteristic
153	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
154	C₂³	#14524030	1	1	yes	characteristic
155	Q₈	—	8	4	no	—
156	Q₈	—	8	4	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	—	8	4	no	—
161	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
162	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
163	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
164	C₂×C₄	#14524014	1	1	yes	characteristic
165	C₂×C₄	#14524014	1	1	yes	characteristic
166	C₂×C₄	#14524039	1	1	yes	characteristic
167	C₂×C₄	#14524074	1	4	yes	—
168	C₈	—	2	4	no	—
169	C₈	—	268	2	no	—
170	C₈	—	268	2	no	—
171	C₈	—	536	2	no	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	#16863299	1	1	yes	characteristic
175	C₂²	#16863764	1	2	yes	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	8	4	no	—
179	C₄	#16863405	1	2	yes	—
180	C₄	#16863663	1	2	yes	—
181	C₄	#16863668	1	2	yes	—
182	C₄	#16863788	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	#22561030	1	1	yes	characteristic
185	C₂	#22561120	1	1	yes	characteristic
186	C₂	#22561125	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560347

C₂

yes

maximal, characteristic

#22558894

C₂

yes

maximal

#22558890

C₂

yes

maximal

#22558501

C₂

yes

maximal, characteristic

C₂×#500×C₆₇

C₂

yes

maximal, characteristic

#16863449

C₂²

yes

characteristic

#16863449

C₂²

yes

characteristic

#16863420

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Q₈×C₆₇

C₂²

yes

—

#500×C₆₇

—

#500×C₆₇

C₄

yes

—

C₂×#107×C₆₇

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×#11761612

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

—

C₂²⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₂²×Q₈×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#107×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

C₂²×C₄×C₆₇

yes

C₂²×C₄×C₆₇

yes

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₂³×C₆₇

#124

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#390

yes

characteristic

C₂²×C₆₇

#491

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#419

yes

—

C₄×C₆₇

#472

yes

—

C₄×C₆₇

#473

yes

—

C₄×C₆₇

#497

yes

—

#9778

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

yes

—

—

—

—

C₆₇

#9778

yes

characteristic, complemented, Sylow

100

C₂×C₄⋊Q₈

C₂×D₆₇

yes

—

101

#500

—

102

#500

Dic₆₇

yes

—

103

#431

—

104

#431

—

105

C₂×#107

—

106

#423

—

107

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

108

#341

—

134

—

109

#333

—

134

—

110

#332

—

134

—

111

#322

—

134

—

112

C₂²×Q₈

—

113

C₂×M₁₆

—

134

—

114

C₂²×C₈

C₂²⋊D₆₇

yes

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

C₂×Dic₆₇

yes

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄²

C₂²×D₆₇

yes

characteristic

122

#107

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄⋊C₈

—

134

—

126

C₄⋊C₈

—

134

—

127

C₂²⋊C₈

—

134

—

128

C₂²⋊C₈

—

134

—

129

C₄×C₈

—

134

—

130

C₄×C₈

—

134

—

131

C₄×C₈

C₂×Dic₆₇

yes

—

132

C₄×C₈

C₂²⋊D₆₇

yes

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂²×C₄

—

137

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

138

C₂²×C₄

#11761627

yes

characteristic

139

C₂²×C₄

#11761627

yes

characteristic

140

M₁₆

—

268

—

141

C₂×C₈

—

142

C₂×C₈

—

268

—

143

C₂×C₈

—

134

—

144

C₂×C₈

—

134

—

145

C₂×C₈

C₂²⋊Dic₆₇

yes

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄²

C₂²×Dic₆₇

yes

characteristic

151

C₄²

D₄×D₆₇

yes

characteristic

152

C₄²

D₄×D₆₇

yes

characteristic

153

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

154

C₂³

#14524030

yes

characteristic

155

Q₈

—

156

Q₈

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

C₂×C₂²⋊Dic₆₇

yes

characteristic

162

C₂×C₄

D₄×Dic₆₇

yes

characteristic

163

C₂×C₄

D₄×Dic₆₇

yes

characteristic

164

C₂×C₄

#14524014

yes

characteristic

165

C₂×C₄

#14524014

yes

characteristic

166

C₂×C₄

#14524039

yes

characteristic

167

C₂×C₄

#14524074

yes

—

168

C₈

—

169

C₈

—

268

—

170

C₈

—

268

—

171

C₈

—

536

—

172

C₂²

—

173

C₂²

—

174

C₂²

#16863299

yes

characteristic

175

C₂²

#16863764

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

#16863405

yes

—

180

C₄

#16863663

yes

—

181

C₄

#16863668

yes

—

182

C₄

#16863788

yes

—

183

C₂

—

184

C₂

#22561030

yes

characteristic

185

C₂

#22561120

yes

characteristic

186

C₂

#22561125

yes

characteristic

	subgroup	G/H
center	C₂²	#16863299
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#500×C₆₇	C₂
socle	C₂²×C₆₇	#390
prime radical	C₂³×C₆₇	#124
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#500	D₆₇
hypocenter	C₆₇	#9778
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

subgroup

G/H

center

C₂²

#16863299

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₂×#500×C₆₇

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

N	H ≅ G/N	type
99. C₆₇	#9778	semidirect

H ≅ G/N

type

99. C₆₇

#9778

semidirect

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

subgroup	index
#22560347	2
#22558894	2
#22558890	2
#22558501	2
C₂×#500×C₆₇	2
#9778	67

subgroup

index

C₂×#500×C₆₇

Subgroups — #30755328

Conjugacy classes of subgroups

Complements

Maximal subgroups