Group #20245793 — subgroups

Subgroups — #20245793

Order 6016.

656 conjugacy classes of subgroups; only the 261 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 261 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×#13093982	C₂	1	2	yes	maximal
2	C₂×C₄²×D₄₇	C₂	1	1	yes	maximal, characteristic
3	#15688993	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688988	C₂	1	2	yes	maximal, complemented
5	#15688980	C₂	1	4	yes	maximal
6	C₄⋊C₄×Dic₄₇	C₂	1	4	yes	maximal, complemented
7	C₄₇×#384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₄₇	C₂²	1	2	yes	—
9	C₂²×C₄×D₄₇	C₂²	1	1	yes	characteristic
10	#13093982	—	2	8	no	—
11	C₂×C₄⋊Dic₄₇	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
13	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
14	C₂×C₄×Dic₄₇	C₂²	1	4	yes	—
15	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
16	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
17	C₄²×D₄₇	C₄	1	8	yes	complemented
18	C₄₇⋊#95	C₂²	1	8	yes	—
19	C₂×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
20	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
21	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₂³×D₄₇	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
24	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
26	C₂²×Dic₄₇	C₂³	1	4	yes	—
27	C₂²×Dic₄₇	C₂³	1	2	yes	—
28	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
29	C₂×C₄×D₄₇	—	2	4	no	—
30	C₂×C₄×D₄₇	—	2	4	no	—
31	C₂×C₄×D₄₇	C₂×C₄	1	8	yes	—
32	C₂×C₄×D₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₄×D₄₇	D₄	1	4	yes	—
34	C₂×C₄×D₄₇	Q₈	1	4	yes	—
35	C₄⋊C₄×C₄₇	—	2	4	no	—
36	C₄⋊C₄×C₄₇	—	2	8	no	—
37	C₄²×C₄₇	C₂×C₄	1	4	yes	—
38	(C₂×D₄₇)⋊C₄	—	2	8	no	—
39	C₄⋊Dic₄₇	—	2	4	no	—
40	C₄⋊Dic₄₇	—	2	8	no	—
41	C₄×Dic₄₇	—	2	8	no	—
42	C₄×Dic₄₇	C₂×C₄	1	4	yes	—
43	C₄×Dic₄₇	C₂×C₄	1	8	yes	—
44	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
45	C₂²×D₄₇	—	2	2	no	—
46	C₂²×D₄₇	—	2	2	no	—
47	C₂²×D₄₇	C₂²×C₄	1	2	yes	—
48	C₂²×D₄₇	C₂×D₄	1	2	yes	—
49	C₂²×D₄₇	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₄₇	—	2	4	no	—
51	C₂×C₄×C₄₇	—	2	8	no	—
52	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
54	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
56	C₂×C₄×C₄₇	#41	1	4	yes	—
57	C₂×C₄×C₄₇	#41	1	4	yes	—
58	C₂×Dic₄₇	—	2	2	no	—
59	C₂×Dic₄₇	—	2	2	no	—
60	C₂×Dic₄₇	—	2	4	no	—
61	C₂×Dic₄₇	—	2	8	no	—
62	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
63	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
64	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
65	C₂×Dic₄₇	C₂×Q₈	1	2	yes	—
66	C₄×D₄₇	—	2	16	no	—
67	C₄×D₄₇	C₄⋊C₄	1	16	yes	complemented
68	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
69	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
70	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
71	C₂²×C₄₇	C₂×#41	1	2	yes	—
72	C₂²×C₄₇	C₂×#41	1	2	yes	—
73	C₂×D₄₇	—	2	4	no	—
74	C₂×D₄₇	—	2	2	no	—
75	C₂×D₄₇	—	2	2	no	—
76	C₂×D₄₇	—	2	4	no	—
77	C₂×D₄₇	C₂×C₄⋊C₄	1	4	yes	—
78	C₄×C₄₇	—	4	4	no	—
79	C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
80	C₄×C₄₇	#117	1	8	yes	complemented
81	Dic₄₇	—	2	4	no	—
82	Dic₄₇	—	4	4	no	—
83	Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
84	#2285	—	47	1	no	maximal, Sylow
85	C₂×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄₇	C₂×#117	1	2	yes	—
87	C₂×C₄₇	#532	1	1	yes	characteristic
88	C₂×C₄₇	#532	1	1	yes	characteristic
89	C₂×C₄₇	#533	1	2	yes	—
90	D₄₇	—	2	4	no	—
91	C₂×#117	—	47	2	no	—
92	C₂²×C₄²	—	47	1	no	—
93	#384	—	47	1	no	—
94	#384	D₄₇	1	1	yes	characteristic, complemented
95	#382	—	47	2	no	—
96	#380	—	47	4	no	—
97	C₄×C₄⋊C₄	—	47	4	no	—
98	C₄₇	#2285	1	1	yes	characteristic, complemented, Sylow
99	C₂³×C₄	—	47	1	no	—
100	C₂³×C₄	—	47	2	no	—
101	#117	—	94	8	no	—
102	C₂×C₄⋊C₄	—	47	2	no	—
103	C₂×C₄⋊C₄	—	47	4	no	—
104	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
105	C₂×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
106	C₂×C₂²⋊C₄	—	47	4	no	—
107	C₂×C₄²	—	47	2	no	—
108	C₂×C₄²	—	47	1	no	—
109	C₂×C₄²	—	47	8	no	—
110	C₂×C₄²	—	47	4	no	—
111	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
112	#95	—	47	8	no	—
113	C₂⁴	—	47	1	no	—
114	C₂²×C₄	—	47	1	no	—
115	C₂²×C₄	—	47	4	no	—
116	C₂²×C₄	—	47	4	no	—
117	C₂²×C₄	—	47	4	no	—
118	C₂²×C₄	—	47	2	no	—
119	C₂²×C₄	—	47	8	no	—
120	C₂²×C₄	—	47	4	no	—
121	C₂²×C₄	—	94	4	no	—
122	C₂²×C₄	—	94	4	no	—
123	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
124	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
126	C₄⋊C₄	—	2	4	no	—
127	C₄⋊C₄	—	2	8	no	—
128	C₄⋊C₄	—	94	4	no	—
129	C₄⋊C₄	—	94	8	no	—
130	C₂²⋊C₄	—	94	8	no	—
131	C₄²	—	47	8	no	—
132	C₄²	—	47	4	no	—
133	C₄²	—	94	8	no	—
134	C₄²	C₄×D₄₇	1	4	yes	—
135	C₂³	—	47	2	no	—
136	C₂³	—	47	2	no	—
137	C₂³	—	47	2	no	—
138	C₂³	—	94	2	no	—
139	C₂³	—	94	2	no	—
140	C₂³	C₂³×D₄₇	1	1	yes	characteristic
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	8	no	—
143	C₂×C₄	—	47	2	no	—
144	C₂×C₄	—	47	2	no	—
145	C₂×C₄	—	47	2	no	—
146	C₂×C₄	—	47	16	no	—
147	C₂×C₄	—	47	4	no	—
148	C₂×C₄	—	94	2	no	—
149	C₂×C₄	—	94	2	no	—
150	C₂×C₄	—	94	16	no	—
151	C₂×C₄	—	94	4	no	—
152	C₂×C₄	—	94	8	no	—
153	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
154	C₂×C₄	C₂×C₄×D₄₇	1	4	yes	—
155	C₂×C₄	D₄×D₄₇	1	2	yes	—
156	C₂×C₄	#10625352	1	4	yes	—
157	C₂×C₄	Q₈×D₄₇	1	2	yes	—
158	C₂×C₄	#10625354	1	4	yes	—
159	C₂²	—	47	4	no	—
160	C₂²	—	94	4	no	—
161	C₂²	—	94	2	no	—
162	C₂²	—	94	2	no	—
163	C₂²	—	94	4	no	—
164	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
165	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
166	C₂²	C₂×Q₈×D₄₇	1	1	yes	characteristic
167	C₂²	C₂×#10625352	1	2	yes	—
168	C₂²	C₂×#10625354	1	2	yes	—
169	C₄	—	4	4	no	—
170	C₄	—	47	4	no	—
171	C₄	—	94	4	no	—
172	C₄	—	188	4	no	—
173	C₄	C₄⋊C₄×D₄₇	1	4	yes	—
174	C₄	#13093982	1	8	yes	complemented
175	C₂	—	94	4	no	—
176	C₂	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
177	C₂	C₂×#13093982	1	2	yes	—
178	C₂	#15689671	1	1	yes	characteristic
179	C₂	#15689674	1	1	yes	characteristic
180	C₂	#15689678	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093982

C₂

yes

maximal

C₂×C₄²×D₄₇

C₂

yes

maximal, characteristic

#15688993

C₂

yes

maximal, characteristic, complemented

#15688988

C₂

yes

maximal, complemented

#15688980

C₂

yes

maximal

C₄⋊C₄×Dic₄₇

C₂

yes

maximal, complemented

C₄₇×#384

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₄₇

C₂²

yes

—

C₂²×C₄×D₄₇

C₂²

yes

characteristic

#13093982

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

C₄²×D₄₇

C₄

yes

complemented

C₄₇⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂³×D₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

characteristic

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

C₂×C₄

yes

—

C₂×C₄×D₄₇

C₂×C₄

yes

—

C₂×C₄×D₄₇

D₄

yes

—

C₂×C₄×D₄₇

Q₈

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

C₂×C₄

yes

—

C₄×Dic₄₇

C₂×C₄

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

C₂²×C₄

yes

—

C₂²×D₄₇

C₂×D₄

yes

—

C₂²×D₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂×Dic₄₇

C₂×D₄

yes

—

C₂×Dic₄₇

C₂×Q₈

yes

—

C₄×D₄₇

—

C₄×D₄₇

C₄⋊C₄

yes

complemented

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂²×Q₈

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

C₂×#41

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

C₂×C₄⋊C₄

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₄×C₄₇

#117

yes

complemented

Dic₄₇

—

Dic₄₇

—

Dic₄₇

C₂×C₄⋊C₄

yes

—

#2285

—

maximal, Sylow

C₂×C₄₇

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₄₇

C₂×#117

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

D₄₇

—

C₂×#117

—

C₂²×C₄²

—

#384

—

#384

D₄₇

yes

characteristic, complemented

#382

—

#380

—

C₄×C₄⋊C₄

—

C₄₇

#2285

yes

characteristic, complemented, Sylow

C₂³×C₄

—

100

C₂³×C₄

—

101

#117

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

105

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

106

C₂×C₂²⋊C₄

—

107

C₂×C₄²

—

108

C₂×C₄²

—

109

C₂×C₄²

—

110

C₂×C₄²

—

111

C₂×C₄²

C₂×D₄₇

yes

characteristic

112

#95

—

113

C₂⁴

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

C₂²×D₄₇

yes

—

124

C₂²×C₄

C₂²×D₄₇

yes

characteristic

125

C₂²×C₄

C₂²×D₄₇

yes

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₂²⋊C₄

—

131

C₄²

—

132

C₄²

—

133

C₄²

—

134

C₄²

C₄×D₄₇

yes

—

135

C₂³

—

136

C₂³

—

137

C₂³

—

138

C₂³

—

139

C₂³

—

140

C₂³

C₂³×D₄₇

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

C₂×C₄×D₄₇

yes

—

154

C₂×C₄

C₂×C₄×D₄₇

yes

—

155

C₂×C₄

D₄×D₄₇

yes

—

156

C₂×C₄

#10625352

yes

—

157

C₂×C₄

Q₈×D₄₇

yes

—

158

C₂×C₄

#10625354

yes

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

C₂²×C₄×D₄₇

yes

characteristic

165

C₂²

C₂×D₄×D₄₇

yes

characteristic

166

C₂²

C₂×Q₈×D₄₇

yes

characteristic

167

C₂²

C₂×#10625352

yes

—

168

C₂²

C₂×#10625354

yes

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

188

—

173

C₄

C₄⋊C₄×D₄₇

yes

—

174

C₄

#13093982

yes

complemented

175

C₂

—

176

C₂

C₂×C₄⋊C₄×D₄₇

yes

characteristic

177

C₂

C₂×#13093982

yes

—

178

C₂

#15689671

yes

characteristic

179

C₂

#15689674

yes

characteristic

180

C₂

#15689678

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Fitting	C₄₇×#384	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	#384	D₄₇
hypocenter	C₄₇	#2285
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator

C₂²×C₄₇

C₂³×C₄

Fitting

C₄₇×#384

C₂

socle

C₂³×C₄₇

C₂⁴

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
3. #15688993	C₂	semidirect
4. #15688988	C₂	semidirect
6. C₄⋊C₄×Dic₄₇	C₂	semidirect
7. C₄₇×#384	C₂	semidirect
17. C₄²×D₄₇	C₄	semidirect
67. C₄×D₄₇	C₄⋊C₄	semidirect
80. C₄×C₄₇	#117	semidirect
94. #384	D₄₇	semidirect
98. C₄₇	#2285	semidirect
174. C₄	#13093982	semidirect

H ≅ G/N

type

3. #15688993

C₂

semidirect

4. #15688988

C₂

semidirect

6. C₄⋊C₄×Dic₄₇

C₂

semidirect

7. C₄₇×#384

C₂

semidirect

17. C₄²×D₄₇

C₄

semidirect

67. C₄×D₄₇

C₄⋊C₄

semidirect

80. C₄×C₄₇

#117

semidirect

94. #384

D₄₇

semidirect

98. C₄₇

#2285

semidirect

174. C₄

#13093982

semidirect

Subgroups — #20245793

Conjugacy classes of subgroups

Complements