Group #30772020 — subgroups

Subgroups — #30772020

Order 17152.

362 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 143 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	#22561993	C₂	1	2	yes	maximal, complemented
2	#22560255	C₂	1	2	yes	maximal
3	#22558311	C₂	1	2	yes	maximal, complemented
4	C₂×#504×C₆₇	C₂	1	1	yes	maximal, characteristic
5	#16863434	C₂²	1	4	yes	—
6	#16863370	—	2	2	no	—
7	#16863224	—	2	2	no	—
8	#16863222	—	2	2	no	—
9	#16863153	—	2	2	no	—
10	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
11	#504×C₆₇	—	2	2	no	—
12	#504×C₆₇	—	2	2	no	—
13	C₂×M₃₂×C₆₇	C₄	1	2	yes	complemented
14	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₆₇	C₄	1	2	yes	complemented
16	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	#14524002	—	2	4	no	—
18	#14523977	—	2	4	no	—
19	#14523975	—	2	4	no	—
20	#14523958	—	4	4	no	—
21	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
22	#131×C₆₇	—	2	2	no	—
23	#131×C₆₇	—	2	2	no	—
24	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
25	C₂×M₁₆×C₆₇	Q₈	1	2	yes	—
26	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
28	M₃₂×C₆₇	—	2	2	no	—
29	M₃₂×C₆₇	—	2	4	no	—
30	C₂×C₁₆×C₆₇	—	2	2	no	—
31	C₂×C₁₆×C₆₇	—	2	2	no	—
32	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
33	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
36	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
37	C₄⋊Dic₆₇	—	4	4	no	—
38	C₂²⋊Dic₆₇	—	4	4	no	—
39	C₄×Dic₆₇	—	4	4	no	—
40	#41×C₆₇	—	2	2	no	—
41	#41×C₆₇	—	2	2	no	—
42	C₂×Q₈×C₆₇	Q₁₆	1	1	yes	characteristic
43	C₂×D₄×C₆₇	D₈	1	2	yes	—
44	C₂×D₄×C₆₇	Q₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	M₁₆×C₆₇	—	2	2	no	—
48	M₁₆×C₆₇	—	2	4	no	—
49	C₂×C₈×C₆₇	—	2	2	no	—
50	C₂×C₈×C₆₇	—	2	2	no	—
51	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
52	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
54	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₆₇	#41	1	2	yes	—
56	C₁₆×C₆₇	—	2	2	no	—
57	C₁₆×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₁₆×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₁₆×C₆₇	C₂²⋊C₄	1	2	yes	—
60	C₂³×C₆₇	#107	1	2	yes	—
61	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
62	Q₈×C₆₇	—	2	2	no	—
63	D₄×C₆₇	—	2	4	no	—
64	D₄×C₆₇	—	2	2	no	—
65	C₂×C₄×C₆₇	—	2	2	no	—
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	#107	1	2	yes	—
68	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₆₇	#121	1	2	yes	—
70	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×C₄×C₆₇	#135	1	2	yes	—
72	C₂×Dic₆₇	—	8	4	no	—
73	C₈×C₆₇	—	2	2	no	—
74	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₈×C₆₇	#117	1	1	yes	characteristic
76	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
77	C₂²×C₆₇	—	2	2	no	—
78	C₂²×C₆₇	—	4	1	no	—
79	C₂²×C₆₇	—	4	2	no	—
80	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
81	C₂²×C₆₇	#466	1	2	yes	—
82	C₄×C₆₇	—	2	2	no	—
83	C₄×C₆₇	#380	1	1	yes	characteristic
84	C₄×C₆₇	#427	1	1	yes	characteristic
85	C₄×C₆₇	#467	1	2	yes	—
86	Dic₆₇	—	16	4	no	—
87	#12839	—	67	1	no	maximal, Sylow
88	C₂×C₆₇	—	2	2	no	—
89	C₂×C₆₇	—	2	1	no	—
90	C₂×C₆₇	—	4	2	no	—
91	C₂×C₆₇	#1808	1	1	yes	characteristic
92	C₂×#504	D₆₇	1	1	yes	characteristic
93	#2158	—	67	2	no	—
94	#1812	—	67	2	no	—
95	#1384	—	67	2	no	—
96	C₆₇	#12839	1	1	yes	characteristic, complemented, Sylow
97	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
98	#504	—	2	2	no	—
99	#504	—	2	2	no	—
100	C₂×M₃₂	Dic₆₇	1	2	yes	complemented
101	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
102	C₂²×C₁₆	Dic₆₇	1	2	yes	complemented
103	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
104	#427	—	67	4	no	—
105	#410	—	134	2	no	—
106	#367	—	134	2	no	—
107	#366	—	134	2	no	—
108	H(3,ℤ/4)	—	134	2	no	—
109	C₂×#41	D₂₆₈	1	1	yes	characteristic
110	#131	—	2	2	no	—
111	#131	—	2	2	no	—
112	C₂×M₁₆	Dic₁₃₄	1	2	yes	—
113	C₂×M₁₆	D₂₆₈	1	1	yes	characteristic
114	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
115	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
116	#117	—	134	4	no	—
117	M₃₂	—	2	2	no	—
118	M₃₂	—	2	4	no	—
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	C₂×Dic₆₇	1	2	yes	—
122	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
123	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
124	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
125	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
126	#107	—	134	4	no	—
127	#106	—	134	4	no	—
128	#99	—	268	4	no	—
129	#41	—	2	2	no	—
130	#41	—	2	2	no	—
131	C₂×Q₈	Dic₂₆₈	1	1	yes	characteristic
132	C₂×D₄	D₅₃₆	1	2	yes	—
133	C₂×D₄	Dic₂₆₈	1	1	yes	characteristic
134	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
135	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
136	M₁₆	—	2	2	no	—
137	M₁₆	—	2	4	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
141	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
142	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
143	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
144	C₂×C₈	#11761628	1	2	yes	—
145	C₄⋊C₄	—	268	4	no	—
146	C₂²⋊C₄	—	268	4	no	—
147	C₄²	—	268	4	no	—
148	C₁₆	—	2	2	no	—
149	C₁₆	C₂²⋊Dic₆₇	1	1	yes	characteristic
150	C₁₆	C₂²⋊Dic₆₇	1	1	yes	characteristic
151	C₁₆	C₂²⋊Dic₆₇	1	2	yes	—
152	C₂³	#14523977	1	2	yes	—
153	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
154	Q₈	—	2	2	no	—
155	D₄	—	2	4	no	—
156	D₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	536	4	no	—
160	C₂×C₄	#14523977	1	2	yes	—
161	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
162	C₂×C₄	#14524017	1	2	yes	—
163	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
164	C₂×C₄	#14524065	1	2	yes	—
165	C₈	—	2	2	no	—
166	C₈	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
167	C₈	#14524002	1	1	yes	characteristic
168	C₈	#14524022	1	2	yes	—
169	C₂²	—	2	2	no	—
170	C₂²	—	4	1	no	—
171	C₂²	—	4	2	no	—
172	C₂²	C₂×#14523977	1	1	yes	characteristic
173	C₂²	#16863632	1	2	yes	—
174	C₄	—	2	2	no	—
175	C₄	—	1072	4	no	—
176	C₄	#16863256	1	1	yes	characteristic
177	C₄	#16863434	1	1	yes	characteristic
178	C₄	#16863637	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	—	2	1	no	—
181	C₂	—	4	2	no	—
182	C₂	#22560238	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22561993

C₂

yes

maximal, complemented

#22560255

C₂

yes

maximal

#22558311

C₂

yes

maximal, complemented

C₂×#504×C₆₇

C₂

yes

maximal, characteristic

#16863434

C₂²

yes

—

—

—

—

—

C₂×#131×C₆₇

C₂²

yes

characteristic

#504×C₆₇

—

#504×C₆₇

—

C₂×M₃₂×C₆₇

C₄

yes

complemented

C₂×M₃₂×C₆₇

C₂²

yes

characteristic

C₂²×C₁₆×C₆₇

C₄

yes

complemented

C₂²×C₁₆×C₆₇

C₂²

yes

—

—

—

—

C₂×#41×C₆₇

D₄

yes

characteristic

#131×C₆₇

—

#131×C₆₇

—

C₂×M₁₆×C₆₇

D₄

yes

characteristic

C₂×M₁₆×C₆₇

Q₈

yes

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

M₃₂×C₆₇

—

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

C₂×C₄

yes

—

C₂×C₁₆×C₆₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₇

D₄

yes

—

C₂×C₁₆×C₆₇

D₄

yes

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

Q₁₆

yes

characteristic

C₂×D₄×C₆₇

D₈

yes

—

C₂×D₄×C₆₇

Q₁₆

yes

characteristic

C₂²×C₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₁₆×C₆₇

—

C₁₆×C₆₇

C₂²⋊C₄

yes

characteristic

C₁₆×C₆₇

C₂²⋊C₄

yes

characteristic

C₁₆×C₆₇

C₂²⋊C₄

yes

—

C₂³×C₆₇

#107

yes

—

C₂³×C₆₇

C₂×D₈

yes

characteristic

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#107

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₇

#135

yes

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₆₇

#117

yes

characteristic

C₈×C₆₇

C₂²⋊Q₈

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×#107

yes

characteristic

C₂²×C₆₇

#466

yes

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

#12839

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#1808

yes

characteristic

C₂×#504

D₆₇

yes

—

—

—

C₆₇

#12839

yes

characteristic, complemented, Sylow

C₂×#131

C₂×D₆₇

yes

characteristic

#504

—

#504

—

100

C₂×M₃₂

Dic₆₇

yes

complemented

101

C₂×M₃₂

C₂×D₆₇

yes

characteristic

102

C₂²×C₁₆

Dic₆₇

yes

complemented

103

C₂²×C₁₆

C₂×D₆₇

yes

characteristic

104

#427

—

105

#410

—

134

—

106

#367

—

134

—

107

#366

—

134

—

108

H(3,ℤ/4)

—

134

—

109

C₂×#41

D₂₆₈

yes

characteristic

110

#131

—

111

#131

—

112

C₂×M₁₆

Dic₁₃₄

yes

—

113

C₂×M₁₆

D₂₆₈

yes

characteristic

114

C₂²×C₈

C₂×Dic₆₇

yes

—

115

C₂²×C₈

C₂²×D₆₇

yes

characteristic

116

#117

—

134

—

117

M₃₂

—

118

M₃₂

—

119

C₂×C₁₆

—

120

C₂×C₁₆

—

121

C₂×C₁₆

C₂×Dic₆₇

yes

—

122

C₂×C₁₆

C₂×Dic₆₇

yes

characteristic

123

C₂×C₁₆

C₂×Dic₆₇

yes

characteristic

124

C₂×C₁₆

C₂²⋊D₆₇

yes

—

125

C₂×C₁₆

C₂²⋊D₆₇

yes

—

126

#107

—

134

—

127

#106

—

134

—

128

#99

—

268

—

129

#41

—

130

#41

—

131

C₂×Q₈

Dic₂₆₈

yes

characteristic

132

C₂×D₄

D₅₃₆

yes

—

133

C₂×D₄

Dic₂₆₈

yes

characteristic

134

C₂²×C₄

C₄⋊Dic₆₇

yes

—

135

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

136

M₁₆

—

137

M₁₆

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

C₄⋊Dic₆₇

yes

—

141

C₂×C₈

C₂²×Dic₆₇

yes

characteristic

142

C₂×C₈

C₂×C₂²⋊D₆₇

yes

—

143

C₂×C₈

C₂×Dic₁₃₄

yes

characteristic

144

C₂×C₈

#11761628

yes

—

145

C₄⋊C₄

—

268

—

146

C₂²⋊C₄

—

268

—

147

C₄²

—

268

—

148

C₁₆

—

149

C₁₆

C₂²⋊Dic₆₇

yes

characteristic

150

C₁₆

C₂²⋊Dic₆₇

yes

characteristic

151

C₁₆

C₂²⋊Dic₆₇

yes

—

152

C₂³

#14523977

yes

—

153

C₂³

C₂×D₅₃₆

yes

characteristic

154

Q₈

—

155

D₄

—

156

D₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

536

—

160

C₂×C₄

#14523977

yes

—

161

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

162

C₂×C₄

#14524017

yes

—

163

C₂×C₄

C₂×Dic₂₆₈

yes

characteristic

164

C₂×C₄

#14524065

yes

—

165

C₈

—

166

C₈

C₂×C₂²⋊Dic₆₇

yes

characteristic

167

C₈

#14524002

yes

characteristic

168

C₈

#14524022

yes

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

C₂×#14523977

yes

characteristic

173

C₂²

#16863632

yes

—

174

C₄

—

175

C₄

—

1072

—

176

C₄

#16863256

yes

characteristic

177

C₄

#16863434

yes

characteristic

178

C₄

#16863637

yes

—

179

C₂

—

180

C₂

—

181

C₂

—

182

C₂

#22560238

yes

characteristic

	subgroup	G/H
center	C₄	#16863434
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#504×C₆₇	C₂
socle	C₂×C₆₇	#1808
prime radical	C₂×#41×C₆₇	D₄
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#504	D₆₇
hypocenter	C₆₇	#12839
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ C₂×#504

subgroup

G/H

center

C₄

#16863434

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₂×#504×C₆₇

C₂

socle

C₂×C₆₇

#1808

prime radical

C₂×#41×C₆₇

D₄

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ C₂×#504

N	H ≅ G/N	type
1. #22561993	C₂	semidirect
3. #22558311	C₂	semidirect
13. C₂×M₃₂×C₆₇	C₄	semidirect
15. C₂²×C₁₆×C₆₇	C₄	semidirect
96. C₆₇	#12839	semidirect
100. C₂×M₃₂	Dic₆₇	semidirect
102. C₂²×C₁₆	Dic₆₇	semidirect

H ≅ G/N

type

1. #22561993

C₂

semidirect

3. #22558311

C₂

semidirect

13. C₂×M₃₂×C₆₇

C₄

semidirect

15. C₂²×C₁₆×C₆₇

C₄

semidirect

96. C₆₇

#12839

semidirect

100. C₂×M₃₂

Dic₆₇

semidirect

102. C₂²×C₁₆

Dic₆₇

semidirect

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

subgroup	index
#22561993	2
#22560255	2
#22558311	2
C₂×#504×C₆₇	2
#12839	67

subgroup

index

C₂×#504×C₆₇

Subgroups — #30772020

Conjugacy classes of subgroups

Complements

Maximal subgroups