Group #30748859 — subgroups

Subgroups — #30748859

Order 17152.

336 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 109 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	#22569144	C₂	1	1	yes	maximal, characteristic
2	#22558967	C₂	1	2	yes	maximal
3	#22558947	C₂	1	1	yes	maximal, characteristic
4	C₆₇⋊#1309	C₂	1	2	yes	maximal
5	C₆₇×#1561	C₂	1	1	yes	maximal, characteristic
6	#16863527	—	2	2	no	—
7	Q₈×C₆₇⋊C₈	—	2	2	no	—
8	#16863423	C₄	1	2	yes	—
9	#16863420	C₂²	1	2	yes	—
10	#16863420	C₂²	1	1	yes	characteristic
11	#16863352	C₄	1	2	yes	—
12	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	#16863148	—	2	2	no	—
14	#16863148	—	2	2	no	—
15	#16863142	—	2	2	no	—
16	#533×C₆₇	C₂²	1	1	yes	characteristic
17	#423×C₆₇	C₂²	1	2	yes	—
18	#332×C₆₇	—	2	2	no	—
19	C₂×#11761612	—	2	2	no	—
20	C₂×#11761612	C₂×C₄	1	2	yes	—
21	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
23	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
26	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
27	#14523954	—	2	2	no	—
28	C₄×C₆₇⋊C₈	—	2	2	no	—
29	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
30	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
31	C₂×M₁₆×C₆₇	—	2	2	no	—
32	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
33	#125×C₆₇	Q₈	1	2	yes	—
34	C₂²⋊Q₈×C₆₇	—	2	2	no	—
35	C₄×Q₈×C₆₇	—	2	2	no	—
36	#117×C₆₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₆₇	—	2	2	no	—
39	C₄⋊C₈×C₆₇	—	2	2	no	—
40	#11761612	—	4	2	no	—
41	#11761612	—	8	2	no	—
42	C₂×C₆₇⋊C₈	—	2	2	no	—
43	C₂×C₆₇⋊C₈	—	2	2	no	—
44	C₂×C₆₇⋊C₈	—	4	1	no	—
45	C₂×C₆₇⋊C₈	—	4	2	no	—
46	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
48	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×Q₈×C₆₇	—	2	2	no	—
50	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	#41	1	2	yes	—
52	M₁₆×C₆₇	—	8	2	no	—
53	C₂×C₈×C₆₇	—	4	2	no	—
54	C₄⋊C₄×C₆₇	—	2	2	no	—
55	C₄⋊C₄×C₆₇	—	2	4	no	—
56	C₄⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
57	C₂²⋊C₄×C₆₇	—	2	2	no	—
58	C₄²×C₆₇	—	2	2	no	—
59	C₄²×C₆₇	C₂×D₄	1	2	yes	—
60	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
62	C₂³×C₆₇	#117	1	1	yes	characteristic
63	Q₈×C₆₇	—	4	2	no	—
64	Q₈×C₆₇	—	4	2	no	—
65	C₂×C₄×C₆₇	—	2	1	no	—
66	C₂×C₄×C₆₇	—	2	1	no	—
67	C₂×C₄×C₆₇	—	2	2	no	—
68	C₂×C₄×C₆₇	—	2	2	no	—
69	C₂×C₄×C₆₇	—	2	4	no	—
70	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	#121	1	2	yes	—
73	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
74	C₈×C₆₇	—	8	2	no	—
75	C₆₇⋊C₈	—	2	1	no	—
76	C₆₇⋊C₈	—	2	2	no	—
77	C₆₇⋊C₈	—	4	2	no	—
78	C₆₇⋊C₈	—	8	2	no	—
79	C₆₇⋊C₈	#101	1	2	yes	—
80	C₂²×C₆₇	—	2	1	no	—
81	C₂²×C₆₇	—	2	2	no	—
82	C₂²×C₆₇	#380	1	1	yes	characteristic
83	C₄×C₆₇	—	2	1	no	—
84	C₄×C₆₇	—	2	2	no	—
85	C₄×C₆₇	—	4	4	no	—
86	C₄×C₆₇	C₂×#101	1	1	yes	characteristic
87	C₄×C₆₇	#413	1	1	yes	characteristic
88	C₄×C₆₇	#472	1	2	yes	—
89	C₄×C₆₇	#473	1	2	yes	—
90	#8639	—	67	1	no	maximal, Sylow
91	C₂×C₆₇	—	4	1	no	—
92	C₂×C₆₇	#1801	1	1	yes	characteristic
93	C₂×C₆₇	#1816	1	2	yes	—
94	#2960	—	67	1	no	—
95	#1567	—	67	2	no	—
96	#1561	—	67	1	no	—
97	#1561	D₆₇	1	1	yes	characteristic
98	#1309	—	67	2	no	—
99	C₆₇	#8639	1	1	yes	characteristic, complemented, Sylow
100	#533	C₂×D₆₇	1	1	yes	characteristic
101	#446	—	134	2	no	—
102	C₈×Q₈	—	134	2	no	—
103	#424	—	67	2	no	—
104	#423	—	67	1	no	—
105	#423	—	67	2	no	—
106	#423	C₂×D₆₇	1	2	yes	—
107	#405	—	67	2	no	—
108	C₂×C₄×C₈	—	67	1	no	—
109	#335	—	134	2	no	—
110	#335	—	134	2	no	—
111	#332	—	2	2	no	—
112	#332	—	134	2	no	—
113	C₂×M₁₆	—	2	2	no	—
114	C₂×M₁₆	—	67	2	no	—
115	C₂×M₁₆	—	134	2	no	—
116	C₂²×C₈	—	67	1	no	—
117	C₂²×C₈	—	67	1	no	—
118	C₄⋊Q₈	C₂²⋊D₆₇	1	2	yes	—
119	#125	Dic₁₃₄	1	2	yes	—
120	C₂²⋊Q₈	—	2	2	no	—
121	C₄×Q₈	—	2	2	no	—
122	#117	C₄×D₆₇	1	2	yes	—
123	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	134	4	no	—
127	C₄⋊C₈	—	134	2	no	—
128	C₄⋊C₈	—	134	2	no	—
129	C₄⋊C₈	—	134	2	no	—
130	#97	—	134	2	no	—
131	C₄×C₈	—	67	2	no	—
132	C₄×C₈	—	67	2	no	—
133	C₄×C₈	—	134	2	no	—
134	C₂×Q₈	—	2	2	no	—
135	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
136	C₂²×C₄	#11761627	1	2	yes	—
137	M₁₆	—	8	2	no	—
138	M₁₆	—	268	2	no	—
139	M₁₆	—	536	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	268	2	no	—
142	C₂×C₈	—	268	1	no	—
143	C₂×C₈	—	67	1	no	—
144	C₂×C₈	—	67	1	no	—
145	C₂×C₈	—	67	2	no	—
146	C₂×C₈	—	134	2	no	—
147	C₂×C₈	—	134	2	no	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	Dic₆₇⋊C₄	1	4	yes	—
151	C₂²⋊C₄	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	D₄×D₆₇	1	2	yes	—
154	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
155	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
156	C₂³	#14523999	1	1	yes	characteristic
157	Q₈	—	4	2	no	—
158	Q₈	—	4	2	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
165	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
166	C₂×C₄	#14524015	1	2	yes	—
167	C₂×C₄	#14524020	1	2	yes	—
168	C₈	—	8	2	no	—
169	C₈	—	268	2	no	—
170	C₈	—	536	2	no	—
171	C₈	—	67	2	no	—
172	C₈	—	134	1	no	—
173	C₈	—	134	2	no	—
174	C₂²	—	2	2	no	—
175	C₂²	—	2	1	no	—
176	C₂²	#16863254	1	1	yes	characteristic
177	C₄	—	2	2	no	—
178	C₄	—	2	1	no	—
179	C₄	—	4	4	no	—
180	C₄	#101×D₆₇	1	1	yes	characteristic
181	C₄	#16863379	1	1	yes	characteristic
182	C₄	#16863664	1	2	yes	—
183	C₄	#16863669	1	2	yes	—
184	C₂	—	4	1	no	—
185	C₂	#22560203	1	1	yes	characteristic
186	C₂	#22560271	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22569144

C₂

yes

maximal, characteristic

#22558967

C₂

yes

maximal

#22558947

C₂

yes

maximal, characteristic

C₆₇⋊#1309

C₂

yes

maximal

C₆₇×#1561

C₂

yes

maximal, characteristic

#16863527

—

Q₈×C₆₇⋊C₈

—

#16863423

C₄

yes

—

#16863420

C₂²

yes

—

#16863420

C₂²

yes

characteristic

#16863352

C₄

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

—

—

#533×C₆₇

C₂²

yes

characteristic

#423×C₆₇

C₂²

yes

—

#332×C₆₇

—

C₂×#11761612

—

C₂×#11761612

C₂×C₄

yes

—

C₂²×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

#14523954

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₂×M₁₆×C₆₇

—

C₄⋊Q₈×C₆₇

D₄

yes

—

#125×C₆₇

Q₈

yes

—

C₂²⋊Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

#117×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₆₇⋊C₈

C₂²⋊C₄

yes

—

C₂×C₆₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₄⋊C₄

yes

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₂³×C₆₇

#117

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

#101

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#380

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₄×C₆₇

#473

yes

—

#8639

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1801

yes

characteristic

C₂×C₆₇

#1816

yes

—

—

—

—

D₆₇

yes

characteristic

#1309

—

C₆₇

#8639

yes

characteristic, complemented, Sylow

100

#533

C₂×D₆₇

yes

characteristic

101

#446

—

134

—

102

C₈×Q₈

—

134

—

103

#424

—

104

#423

—

105

#423

—

106

#423

C₂×D₆₇

yes

—

107

#405

—

108

C₂×C₄×C₈

—

109

#335

—

134

—

110

#335

—

134

—

111

#332

—

112

#332

—

134

—

113

C₂×M₁₆

—

114

C₂×M₁₆

—

115

C₂×M₁₆

—

134

—

116

C₂²×C₈

—

117

C₂²×C₈

—

118

C₄⋊Q₈

C₂²⋊D₆₇

yes

—

119

#125

Dic₁₃₄

yes

—

120

C₂²⋊Q₈

—

121

C₄×Q₈

—

122

#117

C₄×D₆₇

yes

—

123

C₂×C₄²

C₂²×D₆₇

yes

characteristic

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

134

—

127

C₄⋊C₈

—

134

—

128

C₄⋊C₈

—

134

—

129

C₄⋊C₈

—

134

—

130

#97

—

134

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

—

134

—

134

C₂×Q₈

—

135

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

136

C₂²×C₄

#11761627

yes

—

137

M₁₆

—

138

M₁₆

—

268

—

139

M₁₆

—

536

—

140

C₂×C₈

—

141

C₂×C₈

—

268

—

142

C₂×C₈

—

268

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

134

—

147

C₂×C₈

—

134

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

Dic₆₇⋊C₄

yes

—

151

C₂²⋊C₄

—

152

C₄²

—

153

C₄²

D₄×D₆₇

yes

—

154

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

155

C₄²

C₂×Dic₁₃₄

yes

characteristic

156

C₂³

#14523999

yes

characteristic

157

Q₈

—

158

Q₈

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

C₂²⋊C₄×D₆₇

yes

characteristic

165

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

166

C₂×C₄

#14524015

yes

—

167

C₂×C₄

#14524020

yes

—

168

C₈

—

169

C₈

—

268

—

170

C₈

—

536

—

171

C₈

—

172

C₈

—

134

—

173

C₈

—

134

—

174

C₂²

—

175

C₂²

—

176

C₂²

#16863254

yes

characteristic

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

#101×D₆₇

yes

characteristic

181

C₄

#16863379

yes

characteristic

182

C₄

#16863664

yes

—

183

C₄

#16863669

yes

—

184

C₂

—

185

C₂

#22560203

yes

characteristic

186

C₂

#22560271

yes

—

	subgroup	G/H
center	C₂²	#16863254
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1561	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂³×C₆₇	#117
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1561	D₆₇
hypocenter	C₆₇	#8639
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1561

subgroup

G/H

center

C₂²

#16863254

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1561

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1561

N	H ≅ G/N	type
99. C₆₇	#8639	semidirect

H ≅ G/N

type

99. C₆₇

#8639

semidirect

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

subgroup	index
#22569144	2
#22558967	2
#22558947	2
C₆₇⋊#1309	2
C₆₇×#1561	2
#8639	67

subgroup

index

#22569144

#22558967

#22558947

C₆₇⋊#1309

C₆₇×#1561

#8639

Subgroups — #30748859

Conjugacy classes of subgroups

Complements

Maximal subgroups