Group #20243466 — subgroups

Subgroups — #20243466

Order 6016.

316 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂²×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	#15689087	C₂	1	1	yes	maximal, characteristic
3	#15688891	C₂	1	4	yes	maximal, complemented
4	C₄₇×#407	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₄₇	C₄	1	2	yes	—
6	C₂×#10625329	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₄₇	—	2	1	no	—
8	C₂×C₄⋊Dic₄₇	—	2	1	no	—
9	C₂×C₄⋊Dic₄₇	—	2	1	no	—
10	C₂×C₄⋊Dic₄₇	—	2	1	no	—
11	C₂×C₄⋊Dic₄₇	C₄	1	2	yes	—
12	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
13	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
14	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
15	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₄₇	C₂²	1	2	yes	—
17	C₂²⋊C₈×C₄₇	—	2	1	no	—
18	C₂²⋊C₈×C₄₇	—	2	1	no	—
19	C₂⁴×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	—	2	4	no	—
21	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
27	C₂²×Dic₄₇	—	2	2	no	—
28	C₂²×Dic₄₇	—	2	2	no	—
29	C₂²×Dic₄₇	—	4	4	no	—
30	C₂²×Dic₄₇	C₂×C₄	1	4	yes	—
31	M₁₆×C₄₇	—	2	4	no	—
32	C₂×C₈×C₄₇	—	2	2	no	—
33	C₄⋊Dic₄₇	—	2	2	no	—
34	C₄⋊Dic₄₇	—	2	2	no	—
35	C₄⋊Dic₄₇	—	2	2	no	—
36	C₄⋊Dic₄₇	—	2	2	no	—
37	#10625329	—	2	4	no	—
38	C₂×C₄₇⋊C₈	—	2	2	no	—
39	C₂³×C₄₇	—	2	2	no	—
40	C₂³×C₄₇	—	2	2	no	—
41	C₂³×C₄₇	—	2	2	no	—
42	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₄₇	—	2	2	no	—
46	C₂×C₄×C₄₇	—	2	2	no	—
47	C₂×C₄×C₄₇	—	2	2	no	—
48	C₂×C₄×C₄₇	—	2	2	no	—
49	C₂×C₄×C₄₇	—	2	2	no	—
50	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₄₇	#41	1	4	yes	—
55	C₂×Dic₄₇	—	2	4	no	—
56	C₂×Dic₄₇	—	4	4	no	—
57	C₂×Dic₄₇	—	4	4	no	—
58	C₂×Dic₄₇	—	4	4	no	—
59	C₈×C₄₇	—	4	2	no	—
60	C₄₇⋊C₈	—	4	2	no	—
61	C₂²×C₄₇	—	2	8	no	—
62	C₂²×C₄₇	—	2	1	no	—
63	C₂²×C₄₇	—	2	1	no	—
64	C₂²×C₄₇	—	2	1	no	—
65	C₂²×C₄₇	—	2	1	no	—
66	C₂²×C₄₇	—	2	1	no	—
67	C₂²×C₄₇	—	2	1	no	—
68	C₂²×C₄₇	#100	1	2	yes	—
69	C₂²×C₄₇	#101	1	2	yes	—
70	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄₇	#117	1	2	yes	—
72	C₄×C₄₇	—	2	1	no	—
73	C₄×C₄₇	—	2	1	no	—
74	C₄×C₄₇	#123	1	2	yes	—
75	C₄×C₄₇	#123	1	2	yes	—
76	Dic₄₇	—	4	4	no	—
77	#1821	—	47	1	no	maximal, Sylow
78	C₂×C₄₇	—	2	2	no	—
79	C₂×C₄₇	—	2	2	no	—
80	C₂×C₄₇	—	2	2	no	—
81	C₂×C₄₇	#381	1	1	yes	characteristic
82	C₂×C₄₇	C₂×#100	1	1	yes	characteristic
83	C₂×C₄₇	C₂×#101	1	1	yes	characteristic
84	C₂²×C₄⋊C₄	—	47	1	no	—
85	#407	—	47	1	no	—
86	#407	D₄₇	1	1	yes	characteristic
87	#343	—	47	4	no	—
88	C₄₇	#1821	1	1	yes	characteristic, complemented, Sylow
89	C₂³×C₄	—	47	2	no	—
90	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
91	C₂×M₁₆	—	47	2	no	—
92	C₂×M₁₆	C₂×D₄₇	1	2	yes	—
93	C₂×C₄⋊C₄	—	47	2	no	—
94	C₂×C₄⋊C₄	—	47	2	no	—
95	C₂×C₄⋊C₄	—	94	1	no	—
96	C₂×C₄⋊C₄	—	94	1	no	—
97	C₂×C₄⋊C₄	—	94	1	no	—
98	C₂×C₄⋊C₄	—	94	1	no	—
99	C₂²⋊C₈	—	2	1	no	—
100	C₂²⋊C₈	—	2	1	no	—
101	C₂²⋊C₈	—	94	1	no	—
102	C₂²⋊C₈	—	94	1	no	—
103	C₂⁴	C₄×D₄₇	1	1	yes	characteristic
104	C₂²×C₄	—	2	4	no	—
105	C₂²×C₄	—	47	4	no	—
106	C₂²×C₄	—	94	2	no	—
107	C₂²×C₄	—	94	2	no	—
108	C₂²×C₄	—	188	4	no	—
109	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
110	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
111	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
112	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
113	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
115	M₁₆	—	2	4	no	—
116	M₁₆	—	94	4	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	94	2	no	—
119	C₄⋊C₄	—	94	2	no	—
120	C₄⋊C₄	—	94	2	no	—
121	C₄⋊C₄	—	94	2	no	—
122	C₄⋊C₄	—	94	2	no	—
123	C₂³	—	2	2	no	—
124	C₂³	—	2	2	no	—
125	C₂³	—	2	2	no	—
126	C₂³	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
127	C₂³	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
128	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	94	4	no	—
135	C₂×C₄	—	188	4	no	—
136	C₂×C₄	—	188	4	no	—
137	C₂×C₄	—	188	4	no	—
138	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
139	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
140	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
141	C₂×C₄	#10625352	1	4	yes	—
142	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
143	C₈	—	4	2	no	—
144	C₈	—	188	2	no	—
145	C₂²	—	2	8	no	—
146	C₂²	—	2	1	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#13093940	1	2	yes	—
153	C₂²	#13093942	1	2	yes	—
154	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
155	C₂²	#13093980	1	2	yes	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	188	4	no	—
159	C₄	#13094005	1	2	yes	—
160	C₄	#13094008	1	2	yes	—
161	C₂	—	2	2	no	—
162	C₂	—	2	2	no	—
163	C₂	—	2	2	no	—
164	C₂	#15688984	1	1	yes	characteristic
165	C₂	C₂×#13093940	1	1	yes	characteristic
166	C₂	C₂×#13093942	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Dic₄₇

C₂

yes

maximal, characteristic

#15689087

C₂

yes

maximal, characteristic

#15688891

C₂

yes

maximal, complemented

C₄₇×#407

C₂

yes

maximal, characteristic

C₂³×Dic₄₇

C₄

yes

—

C₂×#10625329

C₂²

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₄

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂×M₁₆×C₄₇

C₂²

yes

—

C₂²⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₂⁴×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

C₂×C₄

yes

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#100

yes

—

C₂²×C₄₇

#101

yes

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

#117

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#123

yes

—

C₄×C₄₇

#123

yes

—

Dic₄₇

—

#1821

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂²×C₄⋊C₄

—

#407

—

#407

D₄₇

yes

characteristic

#343

—

C₄₇

#1821

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

C₂×D₄₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₄₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂²⋊C₈

—

100

C₂²⋊C₈

—

101

C₂²⋊C₈

—

102

C₂²⋊C₈

—

103

C₂⁴

C₄×D₄₇

yes

characteristic

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

188

—

109

C₂²×C₄

C₄×D₄₇

yes

characteristic

110

C₂²×C₄

D₁₈₈

yes

characteristic

111

C₂²×C₄

D₁₈₈

yes

characteristic

112

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

113

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

114

C₂²×C₄

C₂²×D₄₇

yes

characteristic

115

M₁₆

—

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

(C₂×D₄₇)⋊C₄

yes

characteristic

127

C₂³

(C₂×D₄₇)⋊C₄

yes

characteristic

128

C₂³

C₂×C₄×D₄₇

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

188

—

136

C₂×C₄

—

188

—

137

C₂×C₄

—

188

—

138

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

139

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

140

C₂×C₄

C₂×D₁₈₈

yes

characteristic

141

C₂×C₄

#10625352

yes

—

142

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

143

C₈

—

144

C₈

—

188

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

#13093940

yes

—

153

C₂²

#13093942

yes

—

154

C₂²

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

155

C₂²

#13093980

yes

—

156

C₄

—

157

C₄

—

158

C₄

—

188

—

159

C₄

#13094005

yes

—

160

C₄

#13094008

yes

—

161

C₂

—

162

C₂

—

163

C₂

—

164

C₂

#15688984

yes

characteristic

165

C₂

C₂×#13093940

yes

characteristic

166

C₂

C₂×#13093942

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂³×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#407	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	#407	D₄₇
hypocenter	C₄₇	#1821
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂²

C₂×(C₂×D₄₇)⋊C₄

commutator

C₂³×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#407

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
3. #15688891	C₂	semidirect
88. C₄₇	#1821	semidirect

H ≅ G/N

type

3. #15688891

C₂

semidirect

88. C₄₇

#1821

semidirect

Subgroups — #20243466

Conjugacy classes of subgroups

Complements