Group #20251893 — subgroups

Subgroups — C₂×#15689131 #20251893

Order 6016.

752 conjugacy classes of subgroups; only the 269 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 269 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	C₂³×Dic₉₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×#10625329	C₂	1	1	yes	maximal, characteristic
3	#15689131	C₂	1	8	yes	maximal, direct product factor
4	C₂×#13093947	C₂	1	4	yes	maximal, complemented
5	C₂×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₉₄	—	2	1	no	—
7	C₂²×Dic₉₄	—	2	2	no	—
8	C₂²×Dic₉₄	—	2	4	no	—
9	C₂²×Dic₉₄	C₄	1	8	yes	complemented
10	C₂²×Dic₉₄	C₂²	1	2	yes	—
11	C₂²×Dic₉₄	C₂²	1	4	yes	—
12	C₂³×Dic₄₇	—	2	1	no	—
13	C₂×#10625329	—	2	4	no	—
14	C₂×#10625329	C₂²	1	4	yes	—
15	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
16	#13093947	C₂²	1	16	yes	complemented
17	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
18	#117×C₄₇	—	2	2	no	—
19	#117×C₄₇	C₂²	1	4	yes	—
20	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
22	C₂×C₄²×C₄₇	—	2	1	no	—
23	C₂⁴×C₄₇	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	—	2	2	no	—
25	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₄₇	D₄	1	2	yes	—
28	C₂²×C₄×C₄₇	D₄	1	4	yes	—
29	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
31	C₂²×Dic₄₇	—	2	8	no	—
32	C₂²×Dic₄₇	—	2	2	no	—
33	C₂²×Dic₄₇	—	2	4	no	—
34	C₂×Dic₉₄	—	2	16	no	—
35	C₂×Dic₉₄	—	2	8	no	—
36	C₂×Dic₉₄	—	2	2	no	—
37	C₂×Dic₉₄	—	2	4	no	—
38	C₂×Dic₉₄	—	2	4	no	—
39	C₂×Dic₉₄	—	2	8	no	—
40	C₂×Dic₉₄	C₂×C₄	1	8	yes	—
41	C₂×Dic₉₄	C₂×C₄	1	16	yes	complemented
42	C₂×Dic₉₄	C₂³	1	4	yes	—
43	C₄⋊C₄×C₄₇	—	2	2	no	—
44	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
45	C₂²⋊C₄×C₄₇	—	2	4	no	—
46	C₄²×C₄₇	—	2	4	no	—
47	#10625329	—	2	8	no	—
48	C₂×C₄₇⋊C₈	—	2	4	no	—
49	C₂×C₄₇⋊C₈	C₂³	1	4	yes	—
50	C₂³×C₄₇	—	2	4	no	—
51	C₂³×C₄₇	C₂²⋊C₄	1	4	yes	—
52	C₂³×C₄₇	C₂×D₄	1	2	yes	—
53	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₄₇	—	2	4	no	—
55	C₂×C₄×C₄₇	—	4	4	no	—
56	C₂×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
57	C₂×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
58	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
62	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
65	C₂×Dic₄₇	—	2	8	no	—
66	C₂×Dic₄₇	—	2	16	no	—
67	C₂×Dic₄₇	—	2	4	no	—
68	Dic₉₄	—	2	8	no	—
69	Dic₉₄	—	2	16	no	—
70	Dic₉₄	—	2	4	no	—
71	Dic₉₄	C₂²×C₄	1	8	yes	—
72	C₄₇⋊C₈	—	2	4	no	—
73	C₂²×C₄₇	—	2	4	no	—
74	C₂²×C₄₇	—	2	8	no	—
75	C₂²×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
76	C₂²×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
77	C₂²×C₄₇	#137	1	4	yes	—
78	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
79	C₄×C₄₇	—	4	4	no	—
80	C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
81	C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
82	C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
83	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
84	Dic₄₇	—	2	8	no	—
85	C₂×#419	—	47	1	no	maximal, Sylow
86	C₂×C₄₇	—	2	4	no	—
87	C₂×C₄₇	#419	1	4	yes	complemented
88	C₂×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄₇	C₂×#137	1	2	yes	—
90	C₂³×Q₈	—	47	1	no	—
91	C₂²×M₁₆	—	47	1	no	—
92	C₂×#117	D₄₇	1	1	yes	characteristic
93	#419	—	47	8	no	—
94	C₂×#103	—	47	4	no	—
95	C₄₇	C₂×#419	1	1	yes	characteristic, complemented, Sylow
96	C₂²×Q₈	—	47	8	no	—
97	C₂²×Q₈	—	47	2	no	—
98	C₂²×Q₈	—	47	4	no	—
99	C₂²×Q₈	—	94	4	no	—
100	C₂²×Q₈	—	94	2	no	—
101	C₂²×Q₈	—	94	1	no	—
102	C₂³×C₄	—	94	1	no	—
103	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
104	C₂×M₁₆	—	47	4	no	—
105	C₂×M₁₆	—	94	4	no	—
106	C₂²×C₈	—	47	2	no	—
107	#117	—	2	2	no	—
108	#117	C₂×D₄₇	1	4	yes	—
109	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
110	C₂×C₂²⋊C₄	—	2	1	no	—
111	C₂×C₄²	—	2	1	no	—
112	#103	—	47	16	no	—
113	C₂⁴	C₂²⋊D₄₇	1	1	yes	characteristic
114	C₂×Q₈	—	47	8	no	—
115	C₂×Q₈	—	47	16	no	—
116	C₂×Q₈	—	47	4	no	—
117	C₂×Q₈	—	94	16	no	—
118	C₂×Q₈	—	94	8	no	—
119	C₂×Q₈	—	94	4	no	—
120	C₂×Q₈	—	94	8	no	—
121	C₂×Q₈	—	94	4	no	—
122	C₂×Q₈	—	94	2	no	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	—	94	8	no	—
125	C₂²×C₄	—	94	4	no	—
126	C₂²×C₄	—	94	2	no	—
127	C₂²×C₄	C₄×D₄₇	1	4	yes	—
128	C₂²×C₄	D₁₈₈	1	4	yes	—
129	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
130	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
131	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
132	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
133	M₁₆	—	94	8	no	—
134	C₂×C₈	—	47	4	no	—
135	C₂×C₈	—	94	4	no	—
136	C₄⋊C₄	—	2	2	no	—
137	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
138	C₂²⋊C₄	—	2	4	no	—
139	C₄²	—	2	4	no	—
140	C₂³	—	2	4	no	—
141	C₂³	(C₂×D₄₇)⋊C₄	1	4	yes	—
142	C₂³	C₂×C₂²⋊D₄₇	1	2	yes	—
143	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
144	Q₈	—	47	8	no	—
145	Q₈	—	94	8	no	—
146	Q₈	—	94	16	no	—
147	Q₈	—	94	4	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	4	4	no	—
150	C₂×C₄	—	94	16	no	—
151	C₂×C₄	—	94	8	no	—
152	C₂×C₄	—	94	4	no	—
153	C₂×C₄	(C₂×D₄₇)⋊C₄	1	8	yes	—
154	C₂×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
155	C₂×C₄	C₂×C₄×D₄₇	1	4	yes	—
156	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
157	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
158	C₂×C₄	C₂×D₁₈₈	1	4	yes	—
159	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
160	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
161	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
162	C₈	—	94	4	no	—
163	C₂²	—	2	8	no	—
164	C₂²	—	2	4	no	—
165	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	4	yes	—
166	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
167	C₂²	#13094054	1	4	yes	—
168	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
169	C₄	—	4	4	no	—
170	C₄	—	94	8	no	—
171	C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
172	C₄	C₂×(C₂×D₄₇)⋊C₄	1	4	yes	—
173	C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
174	C₄	C₂²×D₁₈₈	1	1	yes	characteristic
175	C₂	—	2	4	no	—
176	C₂	#15689131	1	4	yes	direct product factor
177	C₂	C₂²×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
178	C₂	C₂×#13094054	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Dic₉₄

C₂

yes

maximal, characteristic

C₂²×#10625329

C₂

yes

maximal, characteristic

#15689131

C₂

yes

maximal, direct product factor

C₂×#13093947

C₂

yes

maximal, complemented

C₂×#117×C₄₇

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

—

C₂²×Dic₉₄

—

C₂²×Dic₉₄

—

C₂²×Dic₉₄

C₄

yes

complemented

C₂²×Dic₉₄

C₂²

yes

—

C₂²×Dic₉₄

C₂²

yes

—

C₂³×Dic₄₇

—

C₂×#10625329

—

C₂×#10625329

C₂²

yes

—

C₂²×C₄₇⋊C₈

C₂²

yes

—

#13093947

C₂²

yes

complemented

C₂³×C₄×C₄₇

C₂²

yes

characteristic

#117×C₄₇

—

#117×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

—

C₂⁴×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

C₂×C₄

yes

—

C₂×Dic₉₄

C₂×C₄

yes

complemented

C₂×Dic₉₄

C₂³

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²⋊C₄

yes

—

C₂³×C₄₇

C₂×D₄

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

C₂²×C₄

yes

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

—

C₂²×C₄₇

#137

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

C₂×C₂²⋊C₄

yes

—

C₄×C₄₇

C₂×C₂²⋊C₄

yes

—

C₄×C₄₇

C₂³×C₄

yes

characteristic

C₄×C₄₇

C₂²×D₄

yes

characteristic

Dic₄₇

—

C₂×#419

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#419

yes

complemented

C₂×C₄₇

C₂²×C₂²⋊C₄

yes

characteristic

C₂×C₄₇

C₂×#137

yes

—

C₂³×Q₈

—

C₂²×M₁₆

—

C₂×#117

D₄₇

yes

characteristic

#419

—

C₂×#103

—

C₄₇

C₂×#419

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

—

100

C₂²×Q₈

—

101

C₂²×Q₈

—

102

C₂³×C₄

—

103

C₂³×C₄

C₂×D₄₇

yes

characteristic

104

C₂×M₁₆

—

105

C₂×M₁₆

—

106

C₂²×C₈

—

107

#117

—

108

#117

C₂×D₄₇

yes

—

109

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

110

C₂×C₂²⋊C₄

—

111

C₂×C₄²

—

112

#103

—

113

C₂⁴

C₂²⋊D₄₇

yes

characteristic

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂×Q₈

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

C₄×D₄₇

yes

—

128

C₂²×C₄

D₁₈₈

yes

—

129

C₂²×C₄

C₂²⋊D₄₇

yes

—

130

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

131

C₂²×C₄

C₂²×D₄₇

yes

—

132

C₂²×C₄

C₂²×D₄₇

yes

characteristic

133

M₁₆

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₄⋊C₄

—

137

C₄⋊C₄

C₂²×D₄₇

yes

—

138

C₂²⋊C₄

—

139

C₄²

—

140

C₂³

—

141

C₂³

(C₂×D₄₇)⋊C₄

yes

—

142

C₂³

C₂×C₂²⋊D₄₇

yes

—

143

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

144

Q₈

—

145

Q₈

—

146

Q₈

—

147

Q₈

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

—

154

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

—

155

C₂×C₄

C₂×C₄×D₄₇

yes

—

156

C₂×C₄

C₂×C₄×D₄₇

yes

—

157

C₂×C₄

C₂×D₁₈₈

yes

—

158

C₂×C₄

C₂×D₁₈₈

yes

—

159

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

160

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

161

C₂×C₄

C₂³×D₄₇

yes

characteristic

162

C₈

—

163

C₂²

—

164

C₂²

—

165

C₂²

C₂×(C₂×D₄₇)⋊C₄

yes

—

166

C₂²

C₂×(C₂×D₄₇)⋊C₄

yes

—

167

C₂²

#13094054

yes

—

168

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

C₂×(C₂×D₄₇)⋊C₄

yes

—

172

C₄

C₂×(C₂×D₄₇)⋊C₄

yes

—

173

C₄

C₂²×C₄×D₄₇

yes

characteristic

174

C₄

C₂²×D₁₈₈

yes

characteristic

175

C₂

—

176

C₂

#15689131

yes

direct product factor

177

C₂

C₂²×(C₂×D₄₇)⋊C₄

yes

characteristic

178

C₂

C₂×#13094054

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₂×#117×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂×#117	D₄₇
hypocenter	C₄₇	C₂×#419
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#117

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₇

commutator

C₄×C₄₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₂×#117×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

D₄₇

C₄₇

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#117

N	H ≅ G/N	type
3. #15689131	C₂	direct (N × H)
4. C₂×#13093947	C₂	semidirect
9. C₂²×Dic₉₄	C₄	semidirect
16. #13093947	C₂²	semidirect
41. C₂×Dic₉₄	C₂×C₄	semidirect
87. C₂×C₄₇	#419	semidirect
95. C₄₇	C₂×#419	semidirect
176. C₂	#15689131	direct (N × H)

H ≅ G/N

type

3. #15689131

C₂

direct (N × H)

4. C₂×#13093947

C₂

semidirect

9. C₂²×Dic₉₄

C₄

semidirect

16. #13093947

C₂²

semidirect

41. C₂×Dic₉₄

C₂×C₄

semidirect

87. C₂×C₄₇

#419

semidirect

95. C₄₇

C₂×#419

semidirect

176. C₂

#15689131

direct (N × H)

Subgroups — C₂×#15689131 #20251893

Conjugacy classes of subgroups

Complements

Subgroups — C2×#15689131 #20251893

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15689131 #20251893