Group #30754450 — subgroups

Subgroups — #30754450

Order 17152.

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

normal subgroups only

This group has 294 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	C₄×#14523979	C₂	1	1	yes	maximal, characteristic
2	#22558959	C₂	1	2	yes	maximal
3	#22558501	C₂	1	1	yes	maximal, characteristic
4	#22558059	C₂	1	2	yes	maximal
5	C₂×#334×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	1	yes	characteristic
7	#16863449	C₂²	1	1	yes	characteristic
8	C₂×#14523979	C₄	1	2	yes	—
9	C₄×#11761612	C₂²	1	2	yes	—
10	#16863161	C₄	1	1	yes	characteristic
11	#16863161	C₄	1	1	yes	characteristic
12	#16863148	—	2	2	no	—
13	#16863146	—	2	2	no	—
14	#16863118	—	2	2	no	—
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#334×C₆₇	—	2	2	no	—
18	C₂×#11761612	C₂×C₄	1	2	yes	—
19	C₂×#11761612	C₂×C₄	1	2	yes	—
20	#14523979	—	2	4	no	—
21	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
22	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
23	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
24	C₂²⋊(C₆₇⋊C₈)	Q₈	1	2	yes	—
25	#14523954	—	2	2	no	—
26	C₄×C₆₇⋊C₈	—	2	2	no	—
27	C₄×C₆₇⋊C₈	—	2	1	no	—
28	C₄×C₆₇⋊C₈	—	2	1	no	—
29	C₂²×C₈×C₆₇	—	2	2	no	—
30	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₆₇	—	2	2	no	—
34	C₄⋊C₈×C₆₇	—	2	2	no	—
35	C₄×C₈×C₆₇	D₄	1	2	yes	—
36	C₄×C₈×C₆₇	Q₈	1	2	yes	—
37	#11761612	—	2	2	no	—
38	#11761612	—	2	2	no	—
39	#11761612	—	2	2	no	—
40	#11761612	M₁₆	1	4	yes	—
41	C₂×C₆₇⋊C₈	—	2	2	no	—
42	C₂×C₆₇⋊C₈	—	2	2	no	—
43	C₂×C₆₇⋊C₈	—	2	2	no	—
44	C₂×C₆₇⋊C₈	—	2	2	no	—
45	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
48	C₂×C₈×C₆₇	—	2	2	no	—
49	C₂×C₈×C₆₇	—	2	2	no	—
50	C₂×C₈×C₆₇	—	4	2	no	—
51	C₂×C₈×C₆₇	—	8	2	no	—
52	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
54	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₆₇	#41	1	1	yes	characteristic
58	C₂³×C₆₇	C₄×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₆₇	—	2	2	no	—
60	C₂×C₄×C₆₇	—	4	1	no	—
61	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
68	C₂×C₄×C₆₇	#131	1	2	yes	—
69	C₈×C₆₇	—	2	2	no	—
70	C₈×C₆₇	—	2	2	no	—
71	C₈×C₆₇	—	8	2	no	—
72	C₆₇⋊C₈	—	2	4	no	—
73	C₆₇⋊C₈	—	4	2	no	—
74	C₆₇⋊C₈	—	4	2	no	—
75	C₆₇⋊C₈	—	4	2	no	—
76	C₂²×C₆₇	—	2	1	no	—
77	C₂²×C₆₇	—	2	1	no	—
78	C₂²×C₆₇	#389	1	1	yes	characteristic
79	C₂²×C₆₇	#446	1	2	yes	—
80	C₄×C₆₇	—	2	1	no	—
81	C₄×C₆₇	—	2	1	no	—
82	C₄×C₆₇	—	2	1	no	—
83	C₄×C₆₇	#423	1	1	yes	characteristic
84	C₄×C₆₇	#424	1	1	yes	characteristic
85	C₄×C₆₇	#435	1	2	yes	—
86	C₄×C₆₇	#471	1	2	yes	—
87	#9609	—	67	1	no	maximal, Sylow
88	C₂×C₆₇	—	2	2	no	—
89	C₂×C₆₇	#1940	1	1	yes	characteristic
90	C₂×C₆₇	#1951	1	1	yes	characteristic
91	C₂×C₆₇	#1953	1	1	yes	characteristic
92	C₄×#108	—	67	1	no	—
93	#1565	—	67	2	no	—
94	C₂×#334	D₆₇	1	1	yes	characteristic
95	#1452	—	67	1	no	—
96	#1283	—	67	2	no	—
97	C₆₇	#9609	1	1	yes	characteristic, complemented, Sylow
98	#431	—	67	1	no	—
99	#431	—	67	1	no	—
100	C₂×#108	—	67	2	no	—
101	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
102	C₄×M₁₆	—	67	2	no	—
103	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
104	#341	—	67	1	no	—
105	#341	—	67	1	no	—
106	#335	—	134	2	no	—
107	#334	—	2	2	no	—
108	#334	—	134	2	no	—
109	#322	—	134	2	no	—
110	C₂×M₁₆	—	67	2	no	—
111	C₂×M₁₆	—	67	2	no	—
112	C₂²×C₈	—	2	2	no	—
113	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
114	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
115	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
116	#108	—	134	4	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	67	2	no	—
120	C₄⋊C₈	—	134	1	no	—
121	C₂²⋊C₈	—	67	2	no	—
122	C₂²⋊C₈	—	134	1	no	—
123	#97	—	134	2	no	—
124	C₄×C₈	—	134	2	no	—
125	C₄×C₈	—	134	1	no	—
126	C₄×C₈	—	134	1	no	—
127	C₄×C₈	Dic₁₃₄	1	2	yes	—
128	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
129	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
130	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
131	C₂²×C₄	#11761627	1	1	yes	characteristic
132	M₁₆	—	67	4	no	—
133	M₁₆	—	134	2	no	—
134	M₁₆	—	134	2	no	—
135	M₁₆	—	134	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	8	2	no	—
140	C₂×C₈	—	134	2	no	—
141	C₂×C₈	—	134	2	no	—
142	C₂×C₈	—	134	2	no	—
143	C₂×C₈	—	134	2	no	—
144	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
145	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
146	C₄²	D₄×D₆₇	1	1	yes	characteristic
147	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
148	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
149	C₄²	#11761629	1	1	yes	characteristic
150	C₂³	#14524008	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
154	C₂×C₄	#14524005	1	1	yes	characteristic
155	C₂×C₄	#14524014	1	1	yes	characteristic
156	C₂×C₄	#14524019	1	1	yes	characteristic
157	C₂×C₄	#14524034	1	1	yes	characteristic
158	C₂×C₄	#14524042	1	1	yes	characteristic
159	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
160	C₂×C₄	#14524049	1	2	yes	—
161	C₈	—	2	2	no	—
162	C₈	—	2	2	no	—
163	C₈	—	8	2	no	—
164	C₈	—	268	2	no	—
165	C₈	—	268	2	no	—
166	C₈	—	268	2	no	—
167	C₈	—	134	4	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	#16863292	1	1	yes	characteristic
171	C₂²	#16863525	1	2	yes	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	#16863419	1	1	yes	characteristic
176	C₄	#16863422	1	1	yes	characteristic
177	C₄	#16863466	1	2	yes	—
178	C₄	#16863656	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	#22560930	1	1	yes	characteristic
181	C₂	#22560987	1	1	yes	characteristic
182	C₂	#22560997	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14523979

C₂

yes

maximal, characteristic

#22558959

C₂

yes

maximal

#22558501

C₂

yes

maximal, characteristic

#22558059

C₂

yes

maximal

C₂×#334×C₆₇

C₂

yes

maximal, characteristic

#16863449

C₂²

yes

characteristic

#16863449

C₂²

yes

characteristic

C₂×#14523979

C₄

yes

—

C₄×#11761612

C₂²

yes

—

#16863161

C₄

yes

characteristic

#16863161

C₄

yes

—

—

—

C₂×C₄⋊C₈×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#334×C₆₇

—

C₂×#11761612

C₂×C₄

yes

—

C₂×#11761612

C₂×C₄

yes

—

#14523979

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

—

C₂²⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

Q₈

yes

—

#14523954

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₂²×C₈×C₆₇

—

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

Q₈

yes

—

—

—

—

M₁₆

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₄²×C₆₇

#41

yes

characteristic

C₂³×C₆₇

C₄×Q₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

characteristic

C₂×C₄×C₆₇

#121

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

#125

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#389

yes

characteristic

C₂²×C₆₇

#446

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₄×C₆₇

#471

yes

—

#9609

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

—

D₆₇

yes

characteristic

#1452

—

#1283

—

C₆₇

#9609

yes

characteristic, complemented, Sylow

#431

—

#431

—

100

C₂×#108

—

101

C₂×C₄⋊C₈

C₂×D₆₇

yes

—

102

C₄×M₁₆

—

103

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

104

#341

—

105

#341

—

106

#335

—

134

—

107

#334

—

108

#334

—

134

—

109

#322

—

134

—

110

C₂×M₁₆

—

111

C₂×M₁₆

—

112

C₂²×C₈

—

113

C₂²×C₈

C₄×D₆₇

yes

characteristic

114

C₂²×C₈

C₄×D₆₇

yes

characteristic

115

C₂×C₄²

C₂²×D₆₇

yes

characteristic

116

#108

—

134

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

134

—

121

C₂²⋊C₈

—

122

C₂²⋊C₈

—

134

—

123

#97

—

134

—

124

C₄×C₈

—

134

—

125

C₄×C₈

—

134

—

126

C₄×C₈

—

134

—

127

C₄×C₈

Dic₁₃₄

yes

—

128

C₄×C₈

C₂²⋊D₆₇

yes

—

129

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

130

C₂²×C₄

Q₈×D₆₇

yes

characteristic

131

C₂²×C₄

#11761627

yes

characteristic

132

M₁₆

—

133

M₁₆

—

134

—

134

M₁₆

—

134

—

135

M₁₆

—

134

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

134

—

141

C₂×C₈

—

134

—

142

C₂×C₈

—

134

—

143

C₂×C₈

—

134

—

144

C₂×C₈

Dic₆₇⋊C₄

yes

—

145

C₂×C₈

Dic₆₇⋊C₄

yes

—

146

C₄²

D₄×D₆₇

yes

characteristic

147

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

148

C₄²

C₂×Dic₁₃₄

yes

characteristic

149

C₄²

#11761629

yes

characteristic

150

C₂³

#14524008

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

154

C₂×C₄

#14524005

yes

characteristic

155

C₂×C₄

#14524014

yes

characteristic

156

C₂×C₄

#14524019

yes

characteristic

157

C₂×C₄

#14524034

yes

characteristic

158

C₂×C₄

#14524042

yes

characteristic

159

C₂×C₄

M₁₆×D₆₇

yes

—

160

C₂×C₄

#14524049

yes

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₈

—

268

—

165

C₈

—

268

—

166

C₈

—

268

—

167

C₈

—

134

—

168

C₂²

—

169

C₂²

—

170

C₂²

#16863292

yes

characteristic

171

C₂²

#16863525

yes

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

#16863419

yes

characteristic

176

C₄

#16863422

yes

characteristic

177

C₄

#16863466

yes

—

178

C₄

#16863656

yes

—

179

C₂

—

180

C₂

#22560930

yes

characteristic

181

C₂

#22560987

yes

characteristic

182

C₂

#22560997

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14524014
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#334×C₆₇	C₂
socle	C₂²×C₆₇	#389
prime radical	C₂³×C₆₇	C₄×Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#334	D₆₇
hypocenter	C₆₇	#9609
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#334

subgroup

G/H

center

C₂×C₄

#14524014

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₂×#334×C₆₇

C₂

socle

C₂²×C₆₇

#389

prime radical

C₂³×C₆₇

C₄×Q₈

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#334

N	H ≅ G/N	type
97. C₆₇	#9609	semidirect

H ≅ G/N

type

97. C₆₇

#9609

semidirect

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

subgroup	index
C₄×#14523979	2
#22558959	2
#22558501	2
#22558059	2
C₂×#334×C₆₇	2
#9609	67

subgroup

index

C₂×#334×C₆₇

Subgroups — #30754450

Conjugacy classes of subgroups

Complements

Maximal subgroups