Group #20244104 — subgroups

Subgroups — #20244104

Order 6016.

260 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 129 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×Dic₄₇⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
4	#15688877	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₄₇	C₄	1	1	yes	characteristic
9	C₂×C₄⋊Dic₄₇	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
11	Dic₄₇⋊C₈	—	2	4	no	—
12	C₄×C₄₇⋊C₈	—	2	2	no	—
13	C₄₇⋊#95	C₄	1	2	yes	—
14	C₂²×C₈×C₄₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₄₇	—	2	2	no	—
17	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
20	C₂²×Dic₄₇	C₂×C₄	1	2	yes	—
21	C₂²×Dic₄₇	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₄₇	—	2	2	no	—
23	C₂×C₈×C₄₇	—	2	2	no	—
24	C₂×C₈×C₄₇	—	2	2	no	—
25	C₄²×C₄₇	D₄	1	2	yes	—
26	C₄²×C₄₇	Q₈	1	2	yes	—
27	C₄⋊Dic₄₇	—	2	2	no	—
28	C₄⋊Dic₄₇	C₈	1	4	yes	complemented
29	C₄×Dic₄₇	—	2	2	no	—
30	C₄×Dic₄₇	—	2	2	no	—
31	C₂×C₄₇⋊C₈	—	2	1	no	—
32	C₂×C₄₇⋊C₈	—	2	1	no	—
33	C₂×C₄₇⋊C₈	—	2	1	no	—
34	C₂×C₄₇⋊C₈	—	2	1	no	—
35	C₂×C₄₇⋊C₈	D₄	1	2	yes	—
36	C₂×C₄₇⋊C₈	Q₈	1	2	yes	—
37	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₄₇	—	2	1	no	—
39	C₂×C₄×C₄₇	—	2	1	no	—
40	C₂×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₄₇	C₂×C₈	1	2	yes	—
43	C₂×C₄×C₄₇	M₁₆	1	2	yes	—
44	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
49	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
50	C₂×Dic₄₇	—	2	2	no	—
51	C₂×Dic₄₇	—	2	2	no	—
52	C₂×Dic₄₇	—	2	2	no	—
53	C₂×Dic₄₇	—	2	2	no	—
54	C₂×Dic₄₇	—	2	2	no	—
55	C₂×Dic₄₇	C₂×C₈	1	4	yes	—
56	C₈×C₄₇	—	4	2	no	—
57	C₄₇⋊C₈	—	2	2	no	—
58	C₄₇⋊C₈	—	2	2	no	—
59	C₂²×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄₇	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₄₇	C₄×Q₈	1	1	yes	characteristic
62	C₂²×C₄₇	C₂²×C₈	1	1	yes	characteristic
63	C₂²×C₄₇	C₂×M₁₆	1	1	yes	characteristic
64	C₂²×C₄₇	#131	1	1	yes	characteristic
65	C₂²×C₄₇	#131	1	1	yes	characteristic
66	C₄×C₄₇	—	2	2	no	—
67	C₄×C₄₇	C₄⋊C₈	1	4	yes	complemented
68	C₄×C₄₇	#121	1	1	yes	characteristic
69	C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
70	C₄×C₄₇	#125	1	1	yes	characteristic
71	C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
72	Dic₄₇	—	2	4	no	—
73	Dic₄₇	—	4	2	no	—
74	#1939	—	47	1	no	maximal, Sylow
75	C₂×C₄₇	#389	1	1	yes	characteristic
76	C₂×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
77	C₂×C₄₇	#424	1	1	yes	characteristic
78	C₂×C₄₇	C₈×D₄	1	1	yes	characteristic
79	C₂×C₄₇	#436	1	1	yes	characteristic
80	C₂×C₄₇	C₈×Q₈	1	1	yes	characteristic
81	C₂×C₄₇	#446	1	1	yes	characteristic
82	C₂×C₄⋊C₈	—	47	2	no	—
83	C₂×C₄⋊C₈	D₄₇	1	1	yes	characteristic
84	C₂×C₄×C₈	—	47	1	no	—
85	C₄×C₄⋊C₄	—	47	1	no	—
86	#336	—	47	2	no	—
87	C₄₇	#1939	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₈	—	47	1	no	—
89	C₂²×C₈	—	47	1	no	—
90	C₂²×C₈	C₂×D₄₇	1	2	yes	—
91	C₂×C₄⋊C₄	—	47	1	no	—
92	C₂×C₄⋊C₄	—	47	1	no	—
93	C₂×C₄²	—	47	2	no	—
94	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
95	C₄⋊C₈	—	2	2	no	—
96	C₄⋊C₈	—	94	4	no	—
97	C₄×C₈	—	94	2	no	—
98	#95	—	47	2	no	—
99	C₂²×C₄	—	47	2	no	—
100	C₂²×C₄	—	47	2	no	—
101	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
102	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	47	2	no	—
108	C₂×C₈	—	47	2	no	—
109	C₂×C₈	—	94	1	no	—
110	C₂×C₈	—	94	1	no	—
111	C₂×C₈	—	94	1	no	—
112	C₂×C₈	—	94	1	no	—
113	C₄⋊C₄	—	47	4	no	—
114	C₄⋊C₄	—	94	2	no	—
115	C₄²	—	94	2	no	—
116	C₄²	—	94	2	no	—
117	C₄²	Dic₉₄	1	2	yes	—
118	C₄²	C₂²⋊D₄₇	1	2	yes	—
119	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	47	4	no	—
123	C₂×C₄	—	94	2	no	—
124	C₂×C₄	—	94	2	no	—
125	C₂×C₄	—	94	2	no	—
126	C₂×C₄	—	94	2	no	—
127	C₂×C₄	—	94	2	no	—
128	C₂×C₄	C₈×D₄₇	1	2	yes	—
129	C₂×C₄	#10625324	1	2	yes	—
130	C₂×C₄	Dic₄₇⋊C₄	1	2	yes	—
131	C₂×C₄	Dic₄₇⋊C₄	1	2	yes	—
132	C₂×C₄	C₂×Dic₉₄	1	1	yes	characteristic
133	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
134	C₂×C₄	#10625352	1	1	yes	characteristic
135	C₂×C₄	Q₈×D₄₇	1	1	yes	characteristic
136	C₂×C₄	#10625354	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
138	C₈	—	4	2	no	—
139	C₈	—	94	2	no	—
140	C₈	—	94	2	no	—
141	C₂²	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
142	C₂²	#13093986	1	1	yes	characteristic
143	C₂²	#13093989	1	1	yes	characteristic
144	C₂²	C₂×C₈×D₄₇	1	1	yes	characteristic
145	C₂²	C₂×#10625324	1	1	yes	characteristic
146	C₂²	#13094032	1	1	yes	characteristic
147	C₂²	#13094032	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	94	4	no	—
150	C₄	—	188	2	no	—
151	C₄	Dic₄₇⋊C₈	1	4	yes	complemented
152	C₄	#13093995	1	1	yes	characteristic
153	C₄	#13094000	1	1	yes	characteristic
154	C₄	#13094015	1	1	yes	characteristic
155	C₄	#13094023	1	1	yes	characteristic
156	C₂	#15689019	1	1	yes	characteristic
157	C₂	C₂×Dic₄₇⋊C₈	1	1	yes	characteristic
158	C₂	#15689151	1	1	yes	characteristic
159	C₂	#15689188	1	1	yes	characteristic
160	C₂	#15689200	1	1	yes	characteristic
161	C₂	#15689247	1	1	yes	characteristic
162	C₂	#15689250	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₄₇⋊C₈

C₂

yes

maximal

C₂×C₄×C₄₇⋊C₈

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₄₇

C₂

yes

maximal, characteristic

#15688877

C₂

yes

maximal

C₂×C₄⋊C₈×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₄

yes

characteristic

C₂×C₄⋊Dic₄₇

C₄

yes

characteristic

C₂×C₄×Dic₄₇

C₂²

yes

—

Dic₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₄₇⋊#95

C₄

yes

—

C₂²×C₈×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

—

C₂²×Dic₄₇

C₂×C₄

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

D₄

yes

—

C₄²×C₄₇

Q₈

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₈

yes

complemented

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₄

yes

—

C₂×C₄₇⋊C₈

Q₈

yes

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₂×C₈

yes

—

C₂×C₄×C₄₇

M₁₆

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×C₈

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄₇

C₄×D₄

yes

characteristic

C₂²×C₄₇

C₄×Q₈

yes

characteristic

C₂²×C₄₇

C₂²×C₈

yes

characteristic

C₂²×C₄₇

C₂×M₁₆

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

C₄⋊C₈

yes

complemented

C₄×C₄₇

#121

yes

characteristic

C₄×C₄₇

C₂²⋊Q₈

yes

characteristic

C₄×C₄₇

#125

yes

characteristic

C₄×C₄₇

C₄⋊Q₈

yes

characteristic

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

C₂×C₄⋊C₈

yes

characteristic

C₂×C₄₇

#424

yes

characteristic

C₂×C₄₇

C₈×D₄

yes

characteristic

C₂×C₄₇

#436

yes

characteristic

C₂×C₄₇

C₈×Q₈

yes

characteristic

C₂×C₄₇

#446

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₄₇

yes

characteristic

C₂×C₄×C₈

—

C₄×C₄⋊C₄

—

#336

—

C₄₇

#1939

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

#95

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₄×D₄₇

yes

characteristic

102

C₂²×C₄

C₄×D₄₇

yes

characteristic

103

C₂²×C₄

C₂²×D₄₇

yes

characteristic

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

—

116

C₄²

—

117

C₄²

Dic₉₄

yes

—

118

C₄²

C₂²⋊D₄₇

yes

—

119

C₂³

C₂×C₄×D₄₇

yes

characteristic

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₈×D₄₇

yes

—

129

C₂×C₄

#10625324

yes

—

130

C₂×C₄

Dic₄₇⋊C₄

yes

—

131

C₂×C₄

Dic₄₇⋊C₄

yes

—

132

C₂×C₄

C₂×Dic₉₄

yes

characteristic

133

C₂×C₄

D₄×D₄₇

yes

characteristic

134

C₂×C₄

#10625352

yes

characteristic

135

C₂×C₄

Q₈×D₄₇

yes

characteristic

136

C₂×C₄

#10625354

yes

characteristic

137

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

141

C₂²

C₂×Dic₄₇⋊C₄

yes

characteristic

142

C₂²

#13093986

yes

characteristic

143

C₂²

#13093989

yes

characteristic

144

C₂²

C₂×C₈×D₄₇

yes

characteristic

145

C₂²

C₂×#10625324

yes

characteristic

146

C₂²

#13094032

yes

characteristic

147

C₂²

#13094032

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

188

—

151

C₄

Dic₄₇⋊C₈

yes

complemented

152

C₄

#13093995

yes

characteristic

153

C₄

#13094000

yes

characteristic

154

C₄

#13094015

yes

characteristic

155

C₄

#13094023

yes

characteristic

156

C₂

#15689019

yes

characteristic

157

C₂

C₂×Dic₄₇⋊C₈

yes

characteristic

158

C₂

#15689151

yes

characteristic

159

C₂

#15689188

yes

characteristic

160

C₂

#15689200

yes

characteristic

161

C₂

#15689247

yes

characteristic

162

C₂

#15689250

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂²×C₈
Fitting	C₂×C₄⋊C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×C₄
hypercenter	C₂×C₄⋊C₈	D₄₇
hypocenter	C₄₇	#1939
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₄₇

commutator

C₂²×C₄₇

C₂²×C₈

Fitting

C₂×C₄⋊C₈×C₄₇

C₂

socle

C₂³×C₄₇

C₂²×C₄

hypercenter

C₂×C₄⋊C₈

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
28. C₄⋊Dic₄₇	C₈	semidirect
67. C₄×C₄₇	C₄⋊C₈	semidirect
87. C₄₇	#1939	semidirect
151. C₄	Dic₄₇⋊C₈	semidirect

H ≅ G/N

type

28. C₄⋊Dic₄₇

C₈

semidirect

67. C₄×C₄₇

C₄⋊C₈

semidirect

87. C₄₇

#1939

semidirect

151. C₄

Dic₄₇⋊C₈

semidirect

Subgroups — #20244104

Conjugacy classes of subgroups

Complements