Group #30755267 — subgroups

Subgroups — #30755267

Order 17152.

414 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 159 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	#22560381	C₂	1	1	yes	maximal, characteristic
2	#22558894	C₂	1	2	yes	maximal
3	#22558890	C₂	1	2	yes	maximal
4	#22558492	C₂	1	1	yes	maximal, characteristic
5	C₂×#498×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	2	yes	—
7	C₂×#14523979	—	2	2	no	—
8	#16863420	C₂²	1	1	yes	characteristic
9	#16863420	C₂²	1	1	yes	characteristic
10	#16863145	—	2	4	no	—
11	#16863142	—	2	4	no	—
12	#16863117	—	2	1	no	—
13	C₈×C₆₇⋊C₈	—	2	1	no	—
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
15	#498×C₆₇	—	2	1	no	—
16	#498×C₆₇	—	2	1	no	—
17	#498×C₆₇	C₄	1	4	yes	—
18	C₂×#106×C₆₇	—	2	2	no	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#11761612	—	2	1	no	—
21	C₂×#11761612	—	2	1	no	—
22	#14523979	—	4	4	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	D₄	1	4	yes	—
26	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
27	C₄×C₆₇⋊C₈	—	2	2	no	—
28	C₂²×Q₈×C₆₇	—	2	2	no	—
29	C₂²×C₈×C₆₇	D₄	1	2	yes	—
30	C₄⋊Q₈×C₆₇	—	2	2	no	—
31	C₄⋊Q₈×C₆₇	—	2	2	no	—
32	C₄⋊Q₈×C₆₇	—	2	2	no	—
33	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
36	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
37	#106×C₆₇	—	2	4	no	—
38	#106×C₆₇	—	2	4	no	—
39	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
41	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
42	#11761612	—	4	2	no	—
43	#11761612	—	4	2	no	—
44	C₂×C₆₇⋊C₈	—	2	4	no	—
45	C₂×C₆₇⋊C₈	—	4	1	no	—
46	C₂×C₆₇⋊C₈	—	4	1	no	—
47	C₂×Q₈×C₆₇	—	2	4	no	—
48	C₂×Q₈×C₆₇	—	4	2	no	—
49	C₂×Q₈×C₆₇	—	8	4	no	—
50	C₂²×C₄×C₆₇	—	4	2	no	—
51	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	#41	1	2	yes	—
53	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
56	C₂×C₈×C₆₇	Q₁₆	1	4	yes	—
57	C₄⋊C₄×C₆₇	—	2	4	no	—
58	C₄⋊C₄×C₆₇	—	2	4	no	—
59	C₄⋊C₄×C₆₇	—	4	2	no	—
60	C₄⋊C₄×C₆₇	—	4	2	no	—
61	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₆₇	C₂×D₄	1	2	yes	—
63	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₂³×C₆₇	#124	1	1	yes	characteristic
65	Q₈×C₆₇	—	8	4	no	—
66	Q₈×C₆₇	—	8	4	no	—
67	C₂×C₄×C₆₇	—	2	4	no	—
68	C₂×C₄×C₆₇	—	4	4	no	—
69	C₂×C₄×C₆₇	—	8	4	no	—
70	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
72	C₂×C₄×C₆₇	#121	1	2	yes	—
73	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
75	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
76	C₈×C₆₇	#103	1	4	yes	—
77	C₈×C₆₇	#103	1	4	yes	—
78	C₆₇⋊C₈	—	4	4	no	—
79	C₆₇⋊C₈	—	8	1	no	—
80	C₆₇⋊C₈	—	8	1	no	—
81	C₂²×C₆₇	—	2	2	no	—
82	C₂²×C₆₇	#390	1	1	yes	characteristic
83	C₂²×C₆₇	#487	1	2	yes	—
84	C₄×C₆₇	—	2	2	no	—
85	C₄×C₆₇	—	8	4	no	—
86	C₄×C₆₇	C₂×#103	1	2	yes	—
87	C₄×C₆₇	#471	1	4	yes	—
88	C₄×C₆₇	#492	1	1	yes	characteristic
89	C₄×C₆₇	#494	1	1	yes	characteristic
90	#9767	—	67	1	no	maximal, Sylow
91	C₂×C₆₇	—	2	2	no	—
92	C₂×C₆₇	#1956	1	1	yes	characteristic
93	C₂×C₆₇	#1977	1	2	yes	—
94	C₂×#498	D₆₇	1	1	yes	characteristic
95	#1848	—	67	1	no	—
96	#1549	—	67	2	no	—
97	#1548	—	67	2	no	—
98	#1450	—	67	1	no	—
99	C₆₇	#9767	1	1	yes	characteristic, complemented, Sylow
100	C₂×C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
101	#498	—	2	1	no	—
102	#498	—	2	1	no	—
103	#498	Dic₆₇	1	4	yes	—
104	#431	—	67	2	no	—
105	C₂×#108	—	134	2	no	—
106	C₂×#106	—	2	2	no	—
107	#423	—	67	1	no	—
108	#423	—	67	1	no	—
109	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
110	#333	—	134	4	no	—
111	#332	—	134	4	no	—
112	#322	—	134	1	no	—
113	C₈²	—	134	1	no	—
114	C₂²×Q₈	—	2	2	no	—
115	C₂×M₁₆	—	134	1	no	—
116	C₂×M₁₆	—	134	1	no	—
117	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
118	C₄⋊Q₈	—	2	2	no	—
119	C₄⋊Q₈	—	2	2	no	—
120	C₄⋊Q₈	—	2	2	no	—
121	C₄⋊Q₈	C₂×Dic₆₇	1	4	yes	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
125	#108	—	268	4	no	—
126	#106	—	2	4	no	—
127	#106	—	2	4	no	—
128	C₄⋊C₈	—	67	4	no	—
129	C₄⋊C₈	—	134	2	no	—
130	C₄⋊C₈	—	134	2	no	—
131	C₂²⋊C₈	—	134	2	no	—
132	C₄×C₈	—	134	2	no	—
133	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
134	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
135	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
136	C₂×Q₈	—	2	4	no	—
137	C₂×Q₈	—	4	2	no	—
138	C₂×Q₈	—	8	4	no	—
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
141	C₂²×C₄	#11761627	1	2	yes	—
142	M₁₆	—	268	2	no	—
143	M₁₆	—	268	2	no	—
144	C₂×C₈	—	268	1	no	—
145	C₂×C₈	—	268	1	no	—
146	C₂×C₈	—	134	4	no	—
147	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
148	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
149	C₂×C₈	#11761615	1	4	yes	—
150	C₂×C₈	#11761618	1	4	yes	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
156	C₄²	D₄×D₆₇	1	2	yes	—
157	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
158	C₂³	#14524030	1	1	yes	characteristic
159	Q₈	—	8	4	no	—
160	Q₈	—	8	4	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	8	4	no	—
164	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
165	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
166	C₂×C₄	#14524014	1	2	yes	—
167	C₂×C₄	#14524039	1	1	yes	characteristic
168	C₂×C₄	C₂×#11761615	1	2	yes	—
169	C₂×C₄	C₂×#11761618	1	2	yes	—
170	C₈	—	268	4	no	—
171	C₈	—	536	1	no	—
172	C₈	—	536	1	no	—
173	C₈	#14523967	1	4	yes	—
174	C₈	#14523967	1	4	yes	—
175	C₂²	—	2	2	no	—
176	C₂²	#16863299	1	1	yes	characteristic
177	C₂²	#16863746	1	2	yes	—
178	C₄	—	2	2	no	—
179	C₄	—	8	4	no	—
180	C₄	C₂×#14523967	1	2	yes	—
181	C₄	#16863656	1	4	yes	—
182	C₄	#16863770	1	1	yes	characteristic
183	C₄	#16863775	1	1	yes	characteristic
184	C₂	—	2	2	no	—
185	C₂	#22561012	1	1	yes	characteristic
186	C₂	#22561114	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22560381

C₂

yes

maximal, characteristic

#22558894

C₂

yes

maximal

#22558890

C₂

yes

maximal

#22558492

C₂

yes

maximal, characteristic

C₂×#498×C₆₇

C₂

yes

maximal, characteristic

#16863449

C₂²

yes

—

C₂×#14523979

—

#16863420

C₂²

yes

characteristic

#16863420

C₂²

yes

—

—

—

C₈×C₆₇⋊C₈

—

C₂×C₄⋊Q₈×C₆₇

C₂²

yes

—

#498×C₆₇

—

#498×C₆₇

—

#498×C₆₇

C₄

yes

—

C₂×#106×C₆₇

—

C₂×C₄×C₈×C₆₇

C₂²

yes

—

—

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

—

C₂²⋊(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

#106×C₆₇

—

#106×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

characteristic

#11761612

—

#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

characteristic

C₂²×C₄×C₆₇

#41

yes

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₂×C₈×C₆₇

Q₁₆

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

—

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×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×SD₁₆

yes

—

C₂×C₄×C₆₇

C₂×Q₁₆

yes

—

C₈×C₆₇

#103

yes

—

C₈×C₆₇

#103

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#390

yes

characteristic

C₂²×C₆₇

#487

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×#103

yes

—

C₄×C₆₇

#471

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₂×C₆₇

—

C₂×C₆₇

#1956

yes

characteristic

C₂×C₆₇

#1977

yes

—

C₂×#498

D₆₇

yes

—

—

—

—

C₆₇

#9767

yes

characteristic, complemented, Sylow

100

C₂×C₄⋊Q₈

C₂×D₆₇

yes

—

101

#498

—

102

#498

—

103

#498

Dic₆₇

yes

—

104

#431

—

105

C₂×#108

—

134

—

106

C₂×#106

—

107

#423

—

108

#423

—

109

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

110

#333

—

134

—

111

#332

—

134

—

112

#322

—

134

—

113

C₈²

—

134

—

114

C₂²×Q₈

—

115

C₂×M₁₆

—

134

—

116

C₂×M₁₆

—

134

—

117

C₂²×C₈

C₂²⋊D₆₇

yes

—

118

C₄⋊Q₈

—

119

C₄⋊Q₈

—

120

C₄⋊Q₈

—

121

C₄⋊Q₈

C₂×Dic₆₇

yes

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄²

C₂²×D₆₇

yes

characteristic

125

#108

—

268

—

126

#106

—

127

#106

—

128

C₄⋊C₈

—

129

C₄⋊C₈

—

134

—

130

C₄⋊C₈

—

134

—

131

C₂²⋊C₈

—

134

—

132

C₄×C₈

—

134

—

133

C₄×C₈

C₂×Dic₆₇

yes

—

134

C₄×C₈

C₂²⋊D₆₇

yes

characteristic

135

C₄×C₈

C₂²⋊D₆₇

yes

characteristic

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂²×C₄

—

140

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

141

C₂²×C₄

#11761627

yes

—

142

M₁₆

—

268

—

143

M₁₆

—

268

—

144

C₂×C₈

—

268

—

145

C₂×C₈

—

268

—

146

C₂×C₈

—

134

—

147

C₂×C₈

C₂²⋊Dic₆₇

yes

—

148

C₂×C₈

C₂²⋊Dic₆₇

yes

—

149

C₂×C₈

#11761615

yes

—

150

C₂×C₈

#11761618

yes

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄²

C₂²×Dic₆₇

yes

characteristic

156

C₄²

D₄×D₆₇

yes

—

157

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

158

C₂³

#14524030

yes

characteristic

159

Q₈

—

160

Q₈

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

C₂×C₂²⋊Dic₆₇

yes

characteristic

165

C₂×C₄

D₄×Dic₆₇

yes

—

166

C₂×C₄

#14524014

yes

—

167

C₂×C₄

#14524039

yes

characteristic

168

C₂×C₄

C₂×#11761615

yes

—

169

C₂×C₄

C₂×#11761618

yes

—

170

C₈

—

268

—

171

C₈

—

536

—

172

C₈

—

536

—

173

C₈

#14523967

yes

—

174

C₈

#14523967

yes

—

175

C₂²

—

176

C₂²

#16863299

yes

characteristic

177

C₂²

#16863746

yes

—

178

C₄

—

179

C₄

—

180

C₄

C₂×#14523967

yes

—

181

C₄

#16863656

yes

—

182

C₄

#16863770

yes

characteristic

183

C₄

#16863775

yes

characteristic

184

C₂

—

185

C₂

#22561012

yes

characteristic

186

C₂

#22561114

yes

—

	subgroup	G/H
center	C₂²	#16863299
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#498×C₆₇	C₂
socle	C₂²×C₆₇	#390
prime radical	C₂³×C₆₇	#124
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#498	D₆₇
hypocenter	C₆₇	#9767
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#498

subgroup

G/H

center

C₂²

#16863299

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₂×#498×C₆₇

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#498

N	H ≅ G/N	type
99. C₆₇	#9767	semidirect

H ≅ G/N

type

99. C₆₇

#9767

semidirect

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

subgroup	index
#22560381	2
#22558894	2
#22558890	2
#22558492	2
C₂×#498×C₆₇	2
#9767	67

subgroup

index

C₂×#498×C₆₇

Subgroups — #30755267

Conjugacy classes of subgroups

Complements

Maximal subgroups