Group #30744149 — subgroups

Subgroups — #30744149

Order 17152.

384 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 119 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	#22560352	C₂	1	2	yes	maximal
2	#22558032	C₂	1	4	yes	maximal
3	C₄×#125×C₆₇	C₂	1	1	yes	maximal, characteristic
4	#16863151	—	2	2	no	—
5	#16863151	—	2	2	no	—
6	#16863151	C₂²	1	4	yes	—
7	C₂×#125×C₆₇	C₄	1	1	yes	characteristic
8	#389×C₆₇	C₂²	1	2	yes	—
9	#387×C₆₇	—	2	2	no	—
10	#382×C₆₇	C₄	1	1	yes	characteristic
11	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
12	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
13	C₄³×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₆₇⋊C₈	—	2	4	no	—
15	#125×C₆₇	—	2	4	no	—
16	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
17	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
18	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
19	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₆₇	—	2	2	no	—
21	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₆₇	D₄	1	2	yes	—
24	C₂×C₄²×C₆₇	D₄	1	2	yes	—
25	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
26	#95×C₆₇	—	2	2	no	—
27	#95×C₆₇	—	2	2	no	—
28	#95×C₆₇	—	2	2	no	—
29	C₂×C₆₇⋊C₈	—	4	4	no	—
30	C₂×C₆₇⋊C₈	—	4	4	no	—
31	C₂×C₆₇⋊C₈	—	4	4	no	—
32	C₂²×C₄×C₆₇	—	2	2	no	—
33	C₂²×C₄×C₆₇	—	2	2	no	—
34	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
39	C₄⋊C₄×C₆₇	—	2	8	no	—
40	C₄⋊C₄×C₆₇	—	2	2	no	—
41	C₄⋊C₄×C₆₇	—	4	2	no	—
42	C₄⋊C₄×C₆₇	—	4	4	no	—
43	C₄⋊C₄×C₆₇	M₁₆	1	4	yes	—
44	C₄²×C₆₇	—	2	2	no	—
45	C₄²×C₆₇	—	2	2	no	—
46	C₄²×C₆₇	—	2	4	no	—
47	C₄²×C₆₇	—	4	2	no	—
48	C₄²×C₆₇	—	4	1	no	—
49	C₄²×C₆₇	—	4	4	no	—
50	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₆₇	—	2	1	no	—
52	C₂×C₄×C₆₇	—	2	2	no	—
53	C₂×C₄×C₆₇	—	2	1	no	—
54	C₂×C₄×C₆₇	—	2	2	no	—
55	C₂×C₄×C₆₇	—	2	2	no	—
56	C₂×C₄×C₆₇	—	2	2	no	—
57	C₂×C₄×C₆₇	—	2	4	no	—
58	C₂×C₄×C₆₇	—	2	1	no	—
59	C₂×C₄×C₆₇	—	2	2	no	—
60	C₂×C₄×C₆₇	—	2	2	no	—
61	C₂×C₄×C₆₇	—	2	2	no	—
62	C₂×C₄×C₆₇	—	4	2	no	—
63	C₂×C₄×C₆₇	—	4	2	no	—
64	C₂×C₄×C₆₇	—	4	2	no	—
65	C₂×C₄×C₆₇	—	4	2	no	—
66	C₂×C₄×C₆₇	—	4	2	no	—
67	C₂×C₄×C₆₇	#101	1	2	yes	—
68	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
69	C₂×C₄×C₆₇	#135	1	4	yes	—
70	C₆₇⋊C₈	—	8	4	no	—
71	C₂²×C₆₇	#407	1	1	yes	characteristic
72	C₂²×C₆₇	#410	1	1	yes	characteristic
73	C₂²×C₆₇	C₂×#101	1	1	yes	characteristic
74	C₂²×C₆₇	#416	1	2	yes	—
75	C₂²×C₆₇	#421	1	2	yes	—
76	C₄×C₆₇	—	2	2	no	—
77	C₄×C₆₇	—	2	4	no	—
78	C₄×C₆₇	—	2	2	no	—
79	C₄×C₆₇	—	4	1	no	—
80	C₄×C₆₇	—	4	2	no	—
81	C₄×C₆₇	—	4	4	no	—
82	C₄×C₆₇	—	8	2	no	—
83	#7650	—	67	1	no	maximal, Sylow
84	C₂×C₆₇	#1467	1	1	yes	characteristic
85	C₂×C₆₇	#1488	1	2	yes	—
86	C₂×C₆₇	#1521	1	2	yes	—
87	C₂×C₆₇	#1534	1	1	yes	characteristic
88	C₂×C₆₇	#1544	1	1	yes	characteristic
89	C₄×#125	D₆₇	1	1	yes	characteristic
90	#1839	—	67	2	no	—
91	#1271	—	67	4	no	—
92	C₆₇	#7650	1	1	yes	characteristic, complemented, Sylow
93	C₂×#125	Dic₆₇	1	1	yes	characteristic
94	#389	C₂×D₆₇	1	2	yes	—
95	#387	—	2	2	no	—
96	#382	Dic₆₇	1	1	yes	characteristic
97	C₄×C₄⋊C₄	—	2	2	no	—
98	C₄×C₄⋊C₄	Dic₆₇	1	2	yes	—
99	C₄³	C₂×D₆₇	1	1	yes	characteristic
100	#336	—	67	4	no	—
101	#336	—	134	2	no	—
102	#336	—	134	2	no	—
103	C₂²×C₈	—	134	4	no	—
104	#125	—	2	4	no	—
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
108	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
109	C₂×C₄²	—	2	2	no	—
110	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
111	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
112	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
113	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
114	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
115	#95	—	2	2	no	—
116	#95	—	2	2	no	—
117	#95	—	2	2	no	—
118	C₂²×C₄	—	2	2	no	—
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
121	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
122	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
123	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
124	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
125	C₂×C₈	—	268	4	no	—
126	C₂×C₈	—	268	4	no	—
127	C₂×C₈	—	268	4	no	—
128	C₄⋊C₄	—	2	8	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	—	4	4	no	—
132	C₄⋊C₄	#11761612	1	4	yes	—
133	C₄²	—	2	2	no	—
134	C₄²	—	2	2	no	—
135	C₄²	—	2	4	no	—
136	C₄²	—	4	2	no	—
137	C₄²	—	4	1	no	—
138	C₄²	—	4	4	no	—
139	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	#14523962	1	2	yes	—
157	C₂×C₄	C₂×#11761612	1	2	yes	—
158	C₂×C₄	#14524063	1	4	yes	—
159	C₈	—	536	4	no	—
160	C₂²	#16863360	1	1	yes	characteristic
161	C₂²	#16863370	1	1	yes	characteristic
162	C₂²	C₂×#14523962	1	1	yes	characteristic
163	C₂²	#16863392	1	2	yes	—
164	C₂²	#16863413	1	2	yes	—
165	C₄	—	2	4	no	—
166	C₄	—	2	2	no	—
167	C₄	—	2	2	no	—
168	C₄	—	4	1	no	—
169	C₄	—	4	2	no	—
170	C₄	—	4	4	no	—
171	C₄	—	8	2	no	—
172	C₂	#22558553	1	1	yes	characteristic
173	C₂	#22558643	1	2	yes	—
174	C₂	#22558786	1	2	yes	—
175	C₂	#22558836	1	1	yes	characteristic
176	C₂	#22558874	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560352

C₂

yes

maximal

#22558032

C₂

yes

maximal

C₄×#125×C₆₇

C₂

yes

maximal, characteristic

#16863151

—

#16863151

—

#16863151

C₂²

yes

—

C₂×#125×C₆₇

C₄

yes

characteristic

#389×C₆₇

C₂²

yes

—

#387×C₆₇

—

#382×C₆₇

C₄

yes

characteristic

C₄×C₄⋊C₄×C₆₇

—

C₄×C₄⋊C₄×C₆₇

C₄

yes

—

C₄³×C₆₇

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

—

#125×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#95×C₆₇

—

#95×C₆₇

—

#95×C₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

M₁₆

yes

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#101

yes

—

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₂×C₄×C₆₇

#135

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₂²×C₆₇

#421

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×C₆₇

#1521

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

yes

characteristic

#1839

—

#1271

—

C₆₇

#7650

yes

characteristic, complemented, Sylow

C₂×#125

Dic₆₇

yes

characteristic

#389

C₂×D₆₇

yes

—

#387

—

#382

Dic₆₇

yes

characteristic

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

Dic₆₇

yes

—

C₄³

C₂×D₆₇

yes

characteristic

100

#336

—

101

#336

—

134

—

102

#336

—

134

—

103

C₂²×C₈

—

134

—

104

#125

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

C₂×Dic₆₇

yes

—

108

C₂×C₄⋊C₄

C₂×Dic₆₇

yes

—

109

C₂×C₄²

—

110

C₂×C₄²

C₂×Dic₆₇

yes

characteristic

111

C₂×C₄²

C₂×Dic₆₇

yes

characteristic

112

C₂×C₄²

C₂²⋊D₆₇

yes

—

113

C₂×C₄²

C₂²⋊D₆₇

yes

—

114

C₂×C₄²

C₂²×D₆₇

yes

characteristic

115

#95

—

116

#95

—

117

#95

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

C₂²⋊Dic₆₇

yes

characteristic

121

C₂²×C₄

C₂²⋊Dic₆₇

yes

—

122

C₂²×C₄

C₂²⋊Dic₆₇

yes

characteristic

123

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

124

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

—

125

C₂×C₈

—

268

—

126

C₂×C₈

—

268

—

127

C₂×C₈

—

268

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

#11761612

yes

—

133

C₄²

—

134

C₄²

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₂³

C₂×C₂²⋊Dic₆₇

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

#14523962

yes

—

157

C₂×C₄

C₂×#11761612

yes

—

158

C₂×C₄

#14524063

yes

—

159

C₈

—

536

—

160

C₂²

#16863360

yes

characteristic

161

C₂²

#16863370

yes

characteristic

162

C₂²

C₂×#14523962

yes

characteristic

163

C₂²

#16863392

yes

—

164

C₂²

#16863413

yes

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₂

#22558553

yes

characteristic

173

C₂

#22558643

yes

—

174

C₂

#22558786

yes

—

175

C₂

#22558836

yes

characteristic

176

C₂

#22558874

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×#125×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×#125	D₆₇
hypocenter	C₆₇	#7650
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₆₇

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₄×#125×C₆₇

C₂

socle, prime radical

C₂³×C₆₇

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
92. C₆₇	#7650	semidirect

H ≅ G/N

type

92. C₆₇

#7650

semidirect

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

subgroup	index
#22560352	2
#22558032	2
C₄×#125×C₆₇	2
#7650	67

subgroup

index

#22560352

#22558032

C₄×#125×C₆₇

#7650

Subgroups — #30744149

Conjugacy classes of subgroups

Complements

Maximal subgroups