Group #20244497 — subgroups

Subgroups — #20244497

Order 6016.

382 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 73 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×#13094049	C₂	1	2	yes	maximal, complemented
2	C₂²×D₃₇₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13093940	C₂	1	2	yes	maximal, complemented
4	#15688888	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#408	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₈₈	C₂²	1	2	yes	—
7	#13094049	—	2	4	no	—
8	#13094049	—	2	4	no	—
9	C₂×#10625336	—	2	2	no	—
10	C₂×D₃₇₆	—	2	4	no	—
11	C₂×D₃₇₆	—	2	2	no	—
12	C₂×#10625334	—	2	2	no	—
13	C₂×#10625329	C₂²	1	2	yes	—
14	#13093940	—	2	4	no	—
15	C₂×#41×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²⋊C₈×C₄₇	—	2	2	no	—
19	C₂³×D₄₇	—	2	2	no	—
20	#41×C₄₇	—	2	2	no	—
21	#41×C₄₇	—	2	2	no	—
22	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
23	C₂×D₄×C₄₇	—	2	1	no	—
24	C₂×D₄×C₄₇	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	—	2	1	no	—
26	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
27	C₂×D₁₈₈	—	2	2	no	—
28	C₂×D₁₈₈	—	2	2	no	—
29	C₂×D₁₈₈	—	2	2	no	—
30	C₂×D₁₈₈	—	2	2	no	—
31	C₂×D₁₈₈	—	2	2	no	—
32	C₂×D₁₈₈	D₄	1	4	yes	—
33	M₁₆×C₄₇	—	2	2	no	—
34	C₂×C₈×C₄₇	—	2	2	no	—
35	C₂×C₈×C₄₇	—	2	1	no	—
36	C₂×C₈×C₄₇	D₄	1	2	yes	—
37	#10625336	—	2	8	no	—
38	#10625334	—	2	8	no	—
39	#10625329	—	2	2	no	—
40	#10625329	D₄	1	4	yes	—
41	C₂×C₄₇⋊C₈	—	2	2	no	—
42	D₃₇₆	—	2	8	no	—
43	C₂³×C₄₇	—	2	1	no	—
44	C₂³×C₄₇	#41	1	1	yes	characteristic
45	C₂²×D₄₇	—	2	4	no	—
46	C₂²×D₄₇	—	4	2	no	—
47	C₂²×D₄₇	—	4	2	no	—
48	C₂²×D₄₇	—	8	4	no	—
49	Q₈×C₄₇	—	2	2	no	—
50	D₄×C₄₇	—	2	2	no	—
51	D₄×C₄₇	—	4	2	no	—
52	C₂×C₄×C₄₇	—	2	1	no	—
53	C₂×C₄×C₄₇	—	4	1	no	—
54	C₂×C₄×C₄₇	—	4	1	no	—
55	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	D₁₈₈	—	2	4	no	—
60	D₁₈₈	—	2	4	no	—
61	D₁₈₈	—	4	2	no	—
62	D₁₈₈	—	4	2	no	—
63	C₈×C₄₇	—	2	2	no	—
64	C₈×C₄₇	—	4	1	no	—
65	C₄₇⋊C₈	—	2	4	no	—
66	C₂²×C₄₇	—	2	2	no	—
67	C₂²×C₄₇	—	4	1	no	—
68	C₂²×C₄₇	—	4	2	no	—
69	C₂²×C₄₇	#121	1	2	yes	—
70	C₂²×C₄₇	#121	1	1	yes	characteristic
71	C₂×D₄₇	—	4	4	no	—
72	C₂×D₄₇	—	8	4	no	—
73	C₂×D₄₇	—	8	4	no	—
74	C₂×D₄₇	—	8	4	no	—
75	C₄×C₄₇	—	4	1	no	—
76	C₄×C₄₇	#120	1	1	yes	characteristic
77	C₄×C₄₇	#120	1	2	yes	—
78	C₄×C₄₇	#127	1	1	yes	characteristic
79	#2018	—	47	1	no	maximal, Sylow
80	C₂×C₄₇	—	2	2	no	—
81	C₂×C₄₇	—	8	1	no	—
82	C₂×C₄₇	#392	1	1	yes	characteristic
83	C₂×C₄₇	#472	1	2	yes	—
84	D₄₇	—	8	4	no	—
85	C₂×AGL(1,ℤ/8)	—	47	2	no	—
86	C₂²×D₈	—	47	1	no	—
87	C₂×#100	—	47	2	no	—
88	#408	D₄₇	1	1	yes	characteristic, complemented
89	#341	—	47	1	no	—
90	C₄₇	#2018	1	1	yes	characteristic, complemented, Sylow
91	C₂×#41	C₂×D₄₇	1	1	yes	characteristic
92	C₂²×D₄	—	47	2	no	—
93	AGL(1,ℤ/8)	—	94	4	no	—
94	AGL(1,ℤ/8)	—	94	4	no	—
95	C₂×SD₁₆	—	94	2	no	—
96	C₂×D₈	—	94	4	no	—
97	C₂×D₈	—	94	2	no	—
98	C₂×D₈	—	94	2	no	—
99	C₂×M₁₆	—	47	2	no	—
100	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic
101	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
102	#100	—	94	4	no	—
103	C₂²⋊C₈	—	2	2	no	—
104	C₂⁴	—	94	2	no	—
105	#41	—	2	2	no	—
106	#41	—	2	2	no	—
107	C₂×Q₈	D₁₈₈	1	1	yes	characteristic
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	47	4	no	—
110	C₂×D₄	—	94	2	no	—
111	C₂×D₄	—	94	2	no	—
112	C₂×D₄	—	94	2	no	—
113	C₂×D₄	—	94	2	no	—
114	C₂×D₄	—	94	2	no	—
115	C₂×D₄	D₁₈₈	1	1	yes	characteristic
116	C₂²×C₄	—	2	1	no	—
117	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
118	SD₁₆	—	94	8	no	—
119	D₈	—	94	8	no	—
120	D₈	—	94	8	no	—
121	M₁₆	—	2	2	no	—
122	M₁₆	—	47	4	no	—
123	M₁₆	—	94	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	94	2	no	—
127	C₂×C₈	C₂²⋊D₄₇	1	2	yes	—
128	C₂³	—	2	1	no	—
129	C₂³	—	94	4	no	—
130	C₂³	—	376	4	no	—
131	C₂³	—	188	2	no	—
132	C₂³	—	188	2	no	—
133	C₂³	#10625350	1	1	yes	characteristic
134	Q₈	—	2	2	no	—
135	D₄	—	2	2	no	—
136	D₄	—	4	2	no	—
137	D₄	—	94	4	no	—
138	D₄	—	94	4	no	—
139	D₄	—	188	2	no	—
140	D₄	—	188	2	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
145	C₂×C₄	D₄×D₄₇	1	2	yes	—
146	C₂×C₄	D₄×D₄₇	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	—	4	1	no	—
150	C₈	—	94	4	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	4	1	no	—
153	C₂²	—	4	2	no	—
154	C₂²	—	376	4	no	—
155	C₂²	—	376	4	no	—
156	C₂²	—	376	4	no	—
157	C₂²	—	188	4	no	—
158	C₂²	#13093997	1	2	yes	—
159	C₂²	#13093998	1	1	yes	characteristic
160	C₄	—	4	1	no	—
161	C₄	#13093992	1	1	yes	characteristic
162	C₄	#13093994	1	2	yes	—
163	C₄	#13094020	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	—	8	1	no	—
166	C₂	—	376	4	no	—
167	C₂	#15689032	1	1	yes	characteristic
168	C₂	#15689393	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094049

C₂

yes

maximal, complemented

C₂²×D₃₇₆

C₂

yes

maximal, characteristic, complemented

C₂×#13093940

C₂

yes

maximal, complemented

#15688888

C₂

yes

maximal, characteristic, complemented

C₄₇×#408

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₈₈

C₂²

yes

—

#13094049

—

#13094049

—

C₂×#10625336

—

C₂×D₃₇₆

—

C₂×D₃₇₆

—

C₂×#10625334

—

C₂×#10625329

C₂²

yes

—

#13093940

—

C₂×#41×C₄₇

C₂²

yes

characteristic

C₂×M₁₆×C₄₇

C₂²

yes

characteristic

C₂²×C₈×C₄₇

C₂²

yes

characteristic

C₂²⋊C₈×C₄₇

—

C₂³×D₄₇

—

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

D₄

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

D₄

yes

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₄

yes

—

—

—

—

D₄

yes

—

C₂×C₄₇⋊C₈

—

D₃₇₆

—

C₂³×C₄₇

—

C₂³×C₄₇

#41

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#121

yes

—

C₂²×C₄₇

#121

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#120

yes

characteristic

C₄×C₄₇

#120

yes

—

C₄×C₄₇

yes

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#392

yes

characteristic

C₂×C₄₇

#472

yes

—

D₄₇

—

C₂×AGL(1,ℤ/8)

—

C₂²×D₈

—

C₂×#100

—

#408

D₄₇

yes

characteristic, complemented

#341

—

C₄₇

#2018

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₄₇

yes

characteristic

C₂²×D₄

—

AGL(1,ℤ/8)

—

AGL(1,ℤ/8)

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×M₁₆

—

100

C₂×M₁₆

C₂×D₄₇

yes

characteristic

101

C₂²×C₈

C₂×D₄₇

yes

characteristic

102

#100

—

103

C₂²⋊C₈

—

104

C₂⁴

—

105

#41

—

106

#41

—

107

C₂×Q₈

D₁₈₈

yes

characteristic

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

D₁₈₈

yes

characteristic

116

C₂²×C₄

—

117

C₂²×C₄

C₂²×D₄₇

yes

characteristic

118

SD₁₆

—

119

D₈

—

120

D₈

—

121

M₁₆

—

122

M₁₆

—

123

M₁₆

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

C₂²⋊D₄₇

yes

—

128

C₂³

—

129

C₂³

—

130

C₂³

—

376

—

131

C₂³

—

188

—

132

C₂³

—

188

—

133

C₂³

#10625350

yes

characteristic

134

Q₈

—

135

D₄

—

136

D₄

—

137

D₄

—

138

D₄

—

139

D₄

—

188

—

140

D₄

—

188

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

C₂×D₁₈₈

yes

characteristic

145

C₂×C₄

D₄×D₄₇

yes

—

146

C₂×C₄

D₄×D₄₇

yes

—

147

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

376

—

155

C₂²

—

376

—

156

C₂²

—

376

—

157

C₂²

—

188

—

158

C₂²

#13093997

yes

—

159

C₂²

#13093998

yes

characteristic

160

C₄

—

161

C₄

#13093992

yes

characteristic

162

C₄

#13093994

yes

—

163

C₄

#13094020

yes

characteristic

164

C₂

—

165

C₂

—

166

C₂

—

376

—

167

C₂

#15689032

yes

characteristic

168

C₂

#15689393

yes

—

	subgroup	G/H
center	C₂²	#13093998
commutator	C₂²×C₄×C₄₇	C₂³
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#408	C₂
socle	C₂²×C₄₇	#121
hypercenter	#408	D₄₇
hypocenter	C₄₇	#2018
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₂²

#13093998

commutator

C₂²×C₄×C₄₇

C₂³

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#408

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
1. C₂×#13094049	C₂	semidirect
2. C₂²×D₃₇₆	C₂	semidirect
3. C₂×#13093940	C₂	semidirect
4. #15688888	C₂	semidirect
5. C₄₇×#408	C₂	semidirect
88. #408	D₄₇	semidirect
90. C₄₇	#2018	semidirect

H ≅ G/N

type

1. C₂×#13094049

C₂

semidirect

2. C₂²×D₃₇₆

C₂

semidirect

3. C₂×#13093940

C₂

semidirect

4. #15688888

C₂

semidirect

5. C₄₇×#408

C₂

semidirect

88. #408

D₄₇

semidirect

90. C₄₇

#2018

semidirect

Subgroups — #20244497

Conjugacy classes of subgroups

Complements