Group #30775710 — subgroups

Subgroups — #30775710

Order 17152.

284 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 87 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	#22561822	C₂	1	1	yes	maximal, characteristic
2	#22560274	C₂	1	2	yes	maximal
3	#22558317	C₂	1	2	yes	maximal, complemented
4	#22558303	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#367×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14524049	C₂²	1	1	yes	characteristic
7	C₂×#14523983	C₂²	1	1	yes	characteristic
8	C₂×#14523979	C₂²	1	2	yes	—
9	#16863395	—	2	2	no	—
10	(C₂×D₆₇)⋊C₁₆	—	2	2	no	—
11	#16863165	—	2	2	no	—
12	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#107×C₆₇	C₂²	1	2	yes	—
14	#367×C₆₇	—	2	2	no	—
15	C₂×#11761628	D₄	1	1	yes	characteristic
16	#14524049	—	2	2	no	—
17	#14524049	—	2	2	no	—
18	C₂×#11761612	—	2	2	no	—
19	C₂×#11761612	D₄	1	1	yes	characteristic
20	C₂×#11761611	—	2	1	no	—
21	C₂×C₈×D₆₇	—	2	1	no	—
22	#14523983	—	2	2	no	—
23	C₂×C₆₇⋊C₁₆	—	2	1	no	—
24	#14523979	—	2	2	no	—
25	#14523979	—	2	2	no	—
26	#14523966	—	4	2	no	—
27	#14523963	—	4	2	no	—
28	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
30	C₂×C₁₆×C₆₇	—	2	2	no	—
31	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
32	#107×C₆₇	—	2	2	no	—
33	#107×C₆₇	—	2	2	no	—
34	#11761628	—	2	2	no	—
35	#11761628	—	2	2	no	—
36	C₂×Dic₁₃₄	D₈	1	1	yes	characteristic
37	C₂×D₂₆₈	D₈	1	1	yes	characteristic
38	C₂×C₂²⋊D₆₇	—	2	1	no	—
39	C₂×C₄×D₆₇	—	2	1	no	—
40	#11761612	—	2	2	no	—
41	#11761612	—	4	2	no	—
42	#11761612	—	4	2	no	—
43	#11761611	—	4	2	no	—
44	C₂×C₆₇⋊C₈	—	2	1	no	—
45	C₂×C₆₇⋊C₈	—	4	2	no	—
46	C₈×D₆₇	—	4	1	no	—
47	C₈×D₆₇	—	4	1	no	—
48	C₆₇⋊C₁₆	—	4	1	no	—
49	C₂²×C₄×C₆₇	—	4	2	no	—
50	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
53	C₂×C₈×C₆₇	#41	1	2	yes	—
54	C₂×C₈×C₆₇	#41	1	2	yes	—
55	C₄⋊C₄×C₆₇	—	4	2	no	—
56	C₄⋊C₄×C₆₇	—	4	2	no	—
57	C₁₆×C₆₇	—	2	2	no	—
58	C₂³×C₆₇	#135	1	1	yes	characteristic
59	C₂²×D₆₇	—	2	1	no	—
60	C₂×C₄×C₆₇	—	8	2	no	—
61	C₂×C₄×C₆₇	—	16	2	no	—
62	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
63	C₂×C₄×C₆₇	#123	1	2	yes	—
64	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₆₇	#137	1	2	yes	—
66	C₂²⋊D₆₇	—	4	2	no	—
67	C₂×Dic₆₇	—	2	1	no	—
68	D₂₆₈	—	2	2	no	—
69	C₄×D₆₇	—	4	1	no	—
70	C₄×D₆₇	—	4	1	no	—
71	Dic₁₃₄	—	2	2	no	—
72	C₈×C₆₇	#123	1	1	yes	characteristic
73	C₈×C₆₇	#124	1	1	yes	characteristic
74	C₈×C₆₇	#126	1	2	yes	—
75	C₆₇⋊C₈	—	4	1	no	—
76	C₆₇⋊C₈	—	8	2	no	—
77	C₂²×C₆₇	—	2	2	no	—
78	C₂²×C₆₇	#466	1	1	yes	characteristic
79	C₂²×C₆₇	#485	1	2	yes	—
80	C₂×D₆₇	—	4	1	no	—
81	C₂×D₆₇	—	4	2	no	—
82	C₄×C₆₇	—	16	2	no	—
83	C₄×C₆₇	#397	1	1	yes	characteristic
84	C₄×C₆₇	#470	1	1	yes	characteristic
85	C₄×C₆₇	#480	1	2	yes	—
86	Dic₆₇	—	4	1	no	—
87	#13496	—	67	1	no	maximal, Sylow
88	C₂×C₆₇	—	2	2	no	—
89	C₂×C₆₇	#2075	1	1	yes	characteristic
90	C₂×C₆₇	#2223	1	2	yes	—
91	D₆₇	—	8	1	no	—
92	C₂×#367	D₆₇	1	1	yes	characteristic, complemented
93	#2113	—	67	1	no	—
94	#1816	—	67	2	no	—
95	#1387	—	67	2	no	—
96	#1381	—	67	1	no	—
97	C₆₇	#13496	1	1	yes	characteristic, complemented, Sylow
98	C₂×#131	—	67	1	no	—
99	C₂×M₃₂	—	67	1	no	—
100	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
101	C₂×#108	—	67	2	no	—
102	C₂×#107	C₂×D₆₇	1	2	yes	—
103	#417	—	134	2	no	—
104	#367	—	2	2	no	—
105	C₂²⋊C₁₆	—	134	2	no	—
106	#343	—	134	2	no	—
107	C₂×#41	—	67	1	no	—
108	#131	—	134	2	no	—
109	#131	—	134	2	no	—
110	C₂×M₁₆	—	67	1	no	—
111	C₂×M₁₆	—	134	2	no	—
112	C₂×M₁₆	—	134	1	no	—
113	C₂²×C₈	—	134	1	no	—
114	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
115	C₂×C₄⋊C₄	—	2	2	no	—
116	M₃₂	—	134	2	no	—
117	C₂×C₁₆	—	2	2	no	—
118	C₂×C₁₆	—	134	1	no	—
119	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
120	#108	—	134	2	no	—
121	#108	—	134	2	no	—
122	#107	—	2	2	no	—
123	#107	—	2	2	no	—
124	#103	—	268	2	no	—
125	#102	—	268	2	no	—
126	#41	—	134	2	no	—
127	#41	—	134	2	no	—
128	C₂×Q₈	—	67	1	no	—
129	C₂×D₄	—	67	1	no	—
130	C₂×D₄	—	134	1	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	134	1	no	—
133	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
134	M₁₆	—	268	2	no	—
135	M₁₆	—	268	2	no	—
136	M₁₆	—	268	2	no	—
137	M₁₆	—	134	2	no	—
138	C₂×C₈	—	268	1	no	—
139	C₂×C₈	—	268	1	no	—
140	C₂×C₈	—	268	2	no	—
141	C₂×C₈	—	134	1	no	—
142	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
143	C₂×C₈	#11761627	1	1	yes	characteristic
144	C₂×C₈	#11761628	1	2	yes	—
145	C₂×C₈	#11761629	1	2	yes	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₁₆	—	2	2	no	—
149	C₁₆	—	268	1	no	—
150	C₂³	—	134	1	no	—
151	C₂³	#14524062	1	1	yes	characteristic
152	Q₈	—	134	2	no	—
153	D₄	—	268	2	no	—
154	D₄	—	134	2	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	268	1	no	—
157	C₂×C₄	—	268	1	no	—
158	C₂×C₄	—	16	2	no	—
159	C₂×C₄	—	134	1	no	—
160	C₂×C₄	#14524014	1	1	yes	characteristic
161	C₂×C₄	#14524028	1	2	yes	—
162	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
163	C₂×C₄	#14524075	1	2	yes	—
164	C₈	—	268	1	no	—
165	C₈	—	536	2	no	—
166	C₈	#14524026	1	1	yes	characteristic
167	C₈	#14524032	1	1	yes	characteristic
168	C₈	#14524038	1	2	yes	—
169	C₂²	—	2	2	no	—
170	C₂²	—	268	2	no	—
171	C₂²	—	268	1	no	—
172	C₂²	#16863629	1	1	yes	characteristic
173	C₂²	#16863739	1	2	yes	—
174	C₄	—	268	1	no	—
175	C₄	—	16	2	no	—
176	C₄	#16863322	1	1	yes	characteristic
177	C₄	#16863651	1	1	yes	characteristic
178	C₄	#16863711	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	—	536	1	no	—
181	C₂	#22561633	1	1	yes	characteristic
182	C₂	#22562336	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22561822

C₂

yes

maximal, characteristic

#22560274

C₂

yes

maximal

#22558317

C₂

yes

maximal, complemented

#22558303

C₂

yes

maximal, characteristic, complemented

C₂×#367×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#14524049

C₂²

yes

characteristic

C₂×#14523983

C₂²

yes

characteristic

C₂×#14523979

C₂²

yes

—

#16863395

—

(C₂×D₆₇)⋊C₁₆

—

#16863165

—

C₂²×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×#107×C₆₇

C₂²

yes

—

#367×C₆₇

—

C₂×#11761628

D₄

yes

—

—

—

D₄

yes

characteristic

C₂×#11761611

—

C₂×C₈×D₆₇

—

#14523983

—

C₂×C₆₇⋊C₁₆

—

—

—

—

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₆₇

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

—

#107×C₆₇

—

#107×C₆₇

—

#11761628

—

#11761628

—

C₂×Dic₁₃₄

D₈

yes

characteristic

C₂×D₂₆₈

D₈

yes

characteristic

C₂×C₂²⋊D₆₇

—

C₂×C₄×D₆₇

—

—

—

—

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₈×D₆₇

—

C₈×D₆₇

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

C₂×C₈×C₆₇

yes

C₂×C₈×C₆₇

yes

—

C₂×C₈×C₆₇

#41

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₁₆×C₆₇

—

C₂³×C₆₇

#135

yes

characteristic

C₂²×D₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#121

yes

characteristic

C₂×C₄×C₆₇

#123

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

#137

yes

—

C₂²⋊D₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

C₄×D₆₇

—

C₄×D₆₇

—

Dic₁₃₄

—

C₈×C₆₇

yes

C₈×C₆₇

yes

C₈×C₆₇

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#466

yes

characteristic

C₂²×C₆₇

#485

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

#13496

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#2075

yes

characteristic

C₂×C₆₇

#2223

yes

—

D₆₇

—

C₂×#367

D₆₇

yes

characteristic, complemented

—

—

—

—

C₆₇

#13496

yes

characteristic, complemented, Sylow

C₂×#131

—

C₂×M₃₂

—

100

C₂²×C₁₆

C₂×D₆₇

yes

characteristic

101

C₂×#108

—

102

C₂×#107

C₂×D₆₇

yes

—

103

#417

—

134

—

104

#367

—

105

C₂²⋊C₁₆

—

134

—

106

#343

—

134

—

107

C₂×#41

—

108

#131

—

134

—

109

#131

—

134

—

110

C₂×M₁₆

—

111

C₂×M₁₆

—

134

—

112

C₂×M₁₆

—

134

—

113

C₂²×C₈

—

134

—

114

C₂²×C₈

C₂²×D₆₇

yes

characteristic

115

C₂×C₄⋊C₄

—

116

M₃₂

—

134

—

117

C₂×C₁₆

—

118

C₂×C₁₆

—

134

—

119

C₂×C₁₆

C₂²⋊D₆₇

yes

—

120

#108

—

134

—

121

#108

—

134

—

122

#107

—

123

#107

—

124

#103

—

268

—

125

#102

—

268

—

126

#41

—

134

—

127

#41

—

134

—

128

C₂×Q₈

—

129

C₂×D₄

—

130

C₂×D₄

—

134

—

131

C₂²×C₄

—

132

C₂²×C₄

—

134

—

133

C₂²×C₄

D₄×D₆₇

yes

characteristic

134

M₁₆

—

268

—

135

M₁₆

—

268

—

136

M₁₆

—

268

—

137

M₁₆

—

134

—

138

C₂×C₈

—

268

—

139

C₂×C₈

—

268

—

140

C₂×C₈

—

268

—

141

C₂×C₈

—

134

—

142

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

143

C₂×C₈

#11761627

yes

characteristic

144

C₂×C₈

#11761628

yes

—

145

C₂×C₈

#11761629

yes

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₁₆

—

149

C₁₆

—

268

—

150

C₂³

—

134

—

151

C₂³

#14524062

yes

characteristic

152

Q₈

—

134

—

153

D₄

—

268

—

154

D₄

—

134

—

155

C₂×C₄

—

156

C₂×C₄

—

268

—

157

C₂×C₄

—

268

—

158

C₂×C₄

—

159

C₂×C₄

—

134

—

160

C₂×C₄

#14524014

yes

characteristic

161

C₂×C₄

#14524028

yes

—

162

C₂×C₄

D₈×D₆₇

yes

characteristic

163

C₂×C₄

#14524075

yes

—

164

C₈

—

268

—

165

C₈

—

536

—

166

C₈

#14524026

yes

characteristic

167

C₈

#14524032

yes

characteristic

168

C₈

#14524038

yes

—

169

C₂²

—

170

C₂²

—

268

—

171

C₂²

—

268

—

172

C₂²

#16863629

yes

characteristic

173

C₂²

#16863739

yes

—

174

C₄

—

268

—

175

C₄

—

176

C₄

#16863322

yes

characteristic

177

C₄

#16863651

yes

characteristic

178

C₄

#16863711

yes

—

179

C₂

—

180

C₂

—

536

—

181

C₂

#22561633

yes

characteristic

182

C₂

#22562336

yes

—

	subgroup	G/H
center	C₂²	#16863629
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#367×C₆₇	C₂
socle	C₂²×C₆₇	#466
prime radical	C₂×#11761628	D₄
hypercenter	C₂×#367	D₆₇
hypocenter	C₆₇	#13496
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

subgroup

G/H

center

C₂²

#16863629

commutator, cosocle

C₂²×C₈×C₆₇

C₂³

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₂×#367×C₆₇

C₂

C₂²×C₆₇

D₄

D₆₇

C₆₇

G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

N	H ≅ G/N	type
3. #22558317	C₂	semidirect
4. #22558303	C₂	semidirect
5. C₂×#367×C₆₇	C₂	semidirect
92. C₂×#367	D₆₇	semidirect
97. C₆₇	#13496	semidirect

H ≅ G/N

type

3. #22558317

C₂

semidirect

4. #22558303

C₂

semidirect

5. C₂×#367×C₆₇

C₂

semidirect

92. C₂×#367

D₆₇

semidirect

97. C₆₇

#13496

semidirect

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

subgroup	index
#22561822	2
#22560274	2
#22558317	2
#22558303	2
C₂×#367×C₆₇	2
#13496	67

subgroup

index

C₂×#367×C₆₇

Subgroups — #30775710

Conjugacy classes of subgroups

Complements

Maximal subgroups