Group #20253136 — subgroups

Subgroups — C₂×#15689410 #20253136

Order 6016.

480 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 173 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	C₂×C₄×D₁₈₈	C₂	1	1	yes	maximal, characteristic
2	#15689410	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13093955	C₂	1	2	yes	maximal, complemented
4	C₂×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13093944	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄×D₄₇	—	2	1	no	—
9	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
10	C₄×D₁₈₈	—	2	2	no	—
11	C₄×D₁₈₈	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
13	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
14	#13093955	C₂²	1	8	yes	complemented
15	C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	complemented
16	#13093944	C₂²	1	8	yes	—
17	C₂²×Q₈×C₄₇	—	2	1	no	—
18	C₄⋊Q₈×C₄₇	—	2	2	no	—
19	C₄⋊Q₈×C₄₇	C₂²	1	4	yes	complemented
20	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
21	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
22	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₂³×D₄₇	—	2	1	no	—
24	C₂×Q₈×C₄₇	—	2	4	no	—
25	C₂×Q₈×C₄₇	—	4	4	no	—
26	C₂²×C₄×C₄₇	—	2	2	no	—
27	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
30	C₂²×Dic₄₇	—	2	1	no	—
31	C₂×D₁₈₈	—	2	1	no	—
32	C₂×D₁₈₈	—	2	2	no	—
33	C₂×D₁₈₈	Q₈	1	4	yes	complemented
34	C₂×D₁₈₈	C₂³	1	2	yes	—
35	C₂×C₄×D₄₇	—	2	4	no	—
36	C₂×C₄×D₄₇	—	4	4	no	—
37	C₄⋊C₄×C₄₇	—	2	4	no	—
38	C₄⋊C₄×C₄₇	—	2	2	no	—
39	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
40	C₄²×C₄₇	D₄	1	2	yes	—
41	C₄²×C₄₇	C₂³	1	2	yes	—
42	(C₂×D₄₇)⋊C₄	—	2	4	no	—
43	C₄⋊Dic₄₇	—	2	1	no	—
44	C₄⋊Dic₄₇	C₂³	1	2	yes	—
45	C₂×C₄₇⋊C₈	—	2	4	no	—
46	C₂×C₄₇⋊C₈	C₂³	1	4	yes	—
47	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×D₄₇	—	2	4	no	—
49	C₂²×D₄₇	—	2	2	no	—
50	C₂²×D₄₇	—	4	2	no	—
51	C₂²×D₄₇	—	4	2	no	—
52	Q₈×C₄₇	—	4	8	no	—
53	C₂×C₄×C₄₇	—	2	2	no	—
54	C₂×C₄×C₄₇	—	2	4	no	—
55	C₂×C₄×C₄₇	—	4	4	no	—
56	C₂×C₄×C₄₇	SD₁₆	1	4	yes	—
57	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₄₇	#41	1	2	yes	—
62	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
63	C₂×Dic₄₇	—	2	2	no	—
64	C₂×Dic₄₇	—	4	2	no	—
65	D₁₈₈	—	2	4	no	—
66	D₁₈₈	—	2	2	no	—
67	D₁₈₈	C₂×Q₈	1	4	yes	complemented
68	C₄×D₄₇	—	4	8	no	—
69	C₄₇⋊C₈	—	2	4	no	—
70	C₂²×C₄₇	C₂²⋊Q₈	1	2	yes	—
71	C₂²×C₄₇	C₂×SD₁₆	1	2	yes	—
72	C₂²×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
73	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
74	C₂×D₄₇	—	2	4	no	—
75	C₂×D₄₇	—	4	2	no	—
76	C₂×D₄₇	—	4	2	no	—
77	C₂×D₄₇	—	4	8	no	—
78	C₄×C₄₇	—	2	2	no	—
79	C₄×C₄₇	—	4	4	no	—
80	C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
81	C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
82	C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
83	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
84	Dic₄₇	—	4	2	no	—
85	C₂×#476	—	47	1	no	maximal, Sylow
86	C₂×C₄₇	#476	1	4	yes	complemented
87	C₂×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
89	C₂×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
90	D₄₇	—	4	4	no	—
91	C₂×C₄⋊Q₈	D₄₇	1	1	yes	characteristic, complemented
92	C₂×C₄×D₄	—	47	1	no	—
93	#476	—	47	8	no	—
94	C₂×#106	—	47	2	no	—
95	C₂×C₄⋊C₈	—	47	1	no	—
96	C₂×#102	—	47	2	no	—
97	C₄₇	C₂×#476	1	1	yes	characteristic, complemented, Sylow
98	C₂²×Q₈	—	2	1	no	—
99	C₂²×D₄	—	47	1	no	—
100	C₂³×C₄	—	94	1	no	—
101	C₂²×C₈	—	47	2	no	—
102	C₄⋊Q₈	—	2	2	no	—
103	C₄⋊Q₈	C₂×D₄₇	1	4	yes	complemented
104	C₄×D₄	—	47	4	no	—
105	C₄×D₄	—	94	2	no	—
106	C₂×C₄⋊C₄	—	2	1	no	—
107	C₂×C₄⋊C₄	—	47	1	no	—
108	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
109	C₂×C₂²⋊C₄	—	94	1	no	—
110	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
111	#106	—	47	8	no	—
112	C₄⋊C₈	—	47	4	no	—
113	#102	—	47	8	no	—
114	C₂⁴	—	94	1	no	—
115	C₂×Q₈	—	2	4	no	—
116	C₂×Q₈	—	4	4	no	—
117	C₂×D₄	—	47	4	no	—
118	C₂×D₄	—	47	2	no	—
119	C₂×D₄	—	94	2	no	—
120	C₂×D₄	—	94	1	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	94	1	no	—
123	C₂²×C₄	—	94	4	no	—
124	C₂²×C₄	—	188	4	no	—
125	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
126	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
128	C₂×C₈	—	47	4	no	—
129	C₂×C₈	—	94	4	no	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	47	2	no	—
133	C₄⋊C₄	—	94	1	no	—
134	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
135	C₂²⋊C₄	—	94	4	no	—
136	C₄²	C₂²⋊D₄₇	1	2	yes	—
137	C₄²	C₂²×D₄₇	1	2	yes	—
138	C₂³	—	94	4	no	—
139	C₂³	—	94	2	no	—
140	C₂³	—	188	2	no	—
141	C₂³	—	188	2	no	—
142	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
143	Q₈	—	4	8	no	—
144	D₄	—	47	4	no	—
145	D₄	—	94	4	no	—
146	D₄	—	94	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	4	4	no	—
150	C₂×C₄	—	94	2	no	—
151	C₂×C₄	—	188	2	no	—
152	C₂×C₄	—	188	8	no	—
153	C₂×C₄	#10625336	1	4	yes	—
154	C₂×C₄	Q₈×D₄₇	1	2	yes	—
155	C₂×C₄	#10625354	1	2	yes	—
156	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
157	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
158	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
159	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
160	C₈	—	94	4	no	—
161	C₂²	—	94	4	no	—
162	C₂²	—	188	2	no	—
163	C₂²	—	188	2	no	—
164	C₂²	—	188	8	no	—
165	C₂²	#13094000	1	2	yes	—
166	C₂²	C₂×#10625336	1	2	yes	—
167	C₂²	#13094050	1	2	yes	—
168	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
169	C₄	—	2	2	no	—
170	C₄	—	4	4	no	—
171	C₄	—	188	2	no	—
172	C₄	#13094000	1	2	yes	—
173	C₄	C₂×#10625336	1	4	yes	—
174	C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
175	C₄	C₂×#10625354	1	1	yes	characteristic
176	C₂	—	188	4	no	—
177	C₂	#15689410	1	4	yes	direct product factor
178	C₂	C₂×#13094000	1	1	yes	characteristic
179	C₂	C₂²×#10625336	1	1	yes	characteristic
180	C₂	C₂×#13094050	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×D₁₈₈

C₂

yes

maximal, characteristic

#15689410

C₂

yes

maximal, direct product factor

C₂×#13093955

C₂

yes

maximal, complemented

C₂×C₄⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

C₂×#13093944

C₂

yes

maximal

C₂×C₄⋊Q₈×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₈₈

C₂²

yes

characteristic

C₂²×C₄×D₄₇

—

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₄×D₁₈₈

—

C₄×D₁₈₈

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×(C₂×D₄₇)⋊C₄

—

#13093955

C₂²

yes

complemented

C₄⋊(C₄₇⋊C₈)

C₂²

yes

complemented

#13093944

C₂²

yes

—

C₂²×Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

C₂²

yes

complemented

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂³×D₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

Q₈

yes

complemented

C₂×D₁₈₈

C₂³

yes

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₄²×C₄₇

D₄

yes

—

C₄²×C₄₇

C₂³

yes

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂³

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

C₂×Q₈

yes

complemented

C₄×D₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₂²⋊Q₈

yes

—

C₂²×C₄₇

C₂×SD₁₆

yes

—

C₂²×C₄₇

AGL(1,ℤ/8)

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂²⋊Q₈

yes

—

C₄×C₄₇

C₂×SD₁₆

yes

—

C₄×C₄₇

C₂²×Q₈

yes

characteristic

C₄×C₄₇

C₂×#41

yes

characteristic

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₄₇

C₂²×SD₁₆

yes

characteristic

C₂×C₄₇

C₂×AGL(1,ℤ/8)

yes

characteristic

D₄₇

—

C₂×C₄⋊Q₈

D₄₇

yes

characteristic, complemented

C₂×C₄×D₄

—

#476

—

C₂×#106

—

C₂×C₄⋊C₈

—

C₂×#102

—

C₄₇

C₂×#476

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×D₄

—

100

C₂³×C₄

—

101

C₂²×C₈

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

C₂×D₄₇

yes

complemented

104

C₄×D₄

—

105

C₄×D₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

109

C₂×C₂²⋊C₄

—

110

C₂×C₄²

C₂×D₄₇

yes

characteristic

111

#106

—

112

C₄⋊C₈

—

113

#102

—

114

C₂⁴

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

188

—

125

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

126

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

127

C₂²×C₄

C₂²×D₄₇

yes

characteristic

128

C₂×C₈

—

129

C₂×C₈

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

C₂²×D₄₇

yes

—

135

C₂²⋊C₄

—

136

C₄²

C₂²⋊D₄₇

yes

—

137

C₄²

C₂²×D₄₇

yes

—

138

C₂³

—

139

C₂³

—

140

C₂³

—

188

—

141

C₂³

—

188

—

142

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

143

Q₈

—

144

D₄

—

145

D₄

—

146

D₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

188

—

152

C₂×C₄

—

188

—

153

C₂×C₄

#10625336

yes

—

154

C₂×C₄

Q₈×D₄₇

yes

—

155

C₂×C₄

#10625354

yes

—

156

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

157

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

158

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

159

C₂×C₄

C₂³×D₄₇

yes

characteristic

160

C₈

—

161

C₂²

—

162

C₂²

—

188

—

163

C₂²

—

188

—

164

C₂²

—

188

—

165

C₂²

#13094000

yes

—

166

C₂²

C₂×#10625336

yes

—

167

C₂²

#13094050

yes

—

168

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

188

—

172

C₄

#13094000

yes

—

173

C₄

C₂×#10625336

yes

—

174

C₄

C₂×Q₈×D₄₇

yes

characteristic

175

C₄

C₂×#10625354

yes

characteristic

176

C₂

—

188

—

177

C₂

#15689410

yes

direct product factor

178

C₂

C₂×#13094000

yes

characteristic

179

C₂

C₂²×#10625336

yes

characteristic

180

C₂

C₂×#13094050

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₂×C₄⋊Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	C₂×#476
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₂×C₄⋊Q₈×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₄₇

hypocenter

C₄₇

C₂×#476

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #15689410	C₂	direct (N × H)
3. C₂×#13093955	C₂	semidirect
4. C₂×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
6. C₂×C₄⋊Q₈×C₄₇	C₂	semidirect
14. #13093955	C₂²	semidirect
15. C₄⋊(C₄₇⋊C₈)	C₂²	semidirect
19. C₄⋊Q₈×C₄₇	C₂²	semidirect
33. C₂×D₁₈₈	Q₈	semidirect
67. D₁₈₈	C₂×Q₈	semidirect
86. C₂×C₄₇	#476	semidirect
91. C₂×C₄⋊Q₈	D₄₇	semidirect
97. C₄₇	C₂×#476	semidirect
103. C₄⋊Q₈	C₂×D₄₇	semidirect
177. C₂	#15689410	direct (N × H)

H ≅ G/N

type

2. #15689410

C₂

direct (N × H)

3. C₂×#13093955

C₂

semidirect

4. C₂×C₄⋊(C₄₇⋊C₈)

C₂

semidirect

6. C₂×C₄⋊Q₈×C₄₇

C₂

semidirect

14. #13093955

C₂²

semidirect

15. C₄⋊(C₄₇⋊C₈)

C₂²

semidirect

19. C₄⋊Q₈×C₄₇

C₂²

semidirect

33. C₂×D₁₈₈

Q₈

semidirect

67. D₁₈₈

C₂×Q₈

semidirect

86. C₂×C₄₇

#476

semidirect

91. C₂×C₄⋊Q₈

D₄₇

semidirect

97. C₄₇

C₂×#476

semidirect

103. C₄⋊Q₈

C₂×D₄₇

semidirect

177. C₂

#15689410

direct (N × H)

Subgroups — C₂×#15689410 #20253136

Conjugacy classes of subgroups

Complements

Subgroups — C2×#15689410 #20253136

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15689410 #20253136