Group #20241806 — subgroups

Subgroups — #20241806

Order 6016.

262 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 77 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	#15689673	C₂	1	1	yes	maximal, characteristic
2	#15689176	C₂	1	1	yes	maximal, characteristic, complemented
3	#15688852	C₂	1	4	yes	maximal, complemented
4	C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10625350	—	2	1	no	—
6	#13094024	C₄	1	1	yes	characteristic
7	#13094021	C₄	1	1	yes	characteristic
8	#13094012	—	2	1	no	—
9	#13093998	—	2	1	no	—
10	#13093998	C₄	1	2	yes	—
11	C₄×D₁₈₈	—	2	1	no	—
12	C₄×D₁₈₈	C₂²	1	2	yes	—
13	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
14	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
15	C₄×C₄₇⋊C₈	—	2	1	no	—
16	C₂×M₁₆×C₄₇	—	2	1	no	—
17	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	#97×C₄₇	—	2	1	no	—
19	C₄×C₈×C₄₇	C₂²	1	2	yes	—
20	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
23	C₂×C₂²⋊D₄₇	—	2	2	no	—
24	#10625350	—	4	4	no	—
25	C₂×D₁₈₈	—	2	1	no	—
26	C₂×D₁₈₈	—	2	1	no	—
27	C₂×D₁₈₈	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₄₇	—	2	2	no	—
29	C₂×Dic₉₄	—	2	1	no	—
30	M₁₆×C₄₇	—	2	4	no	—
31	C₂×C₈×C₄₇	—	2	2	no	—
32	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₄₇	D₄	1	1	yes	characteristic
34	C₄²×C₄₇	D₄	1	1	yes	characteristic
35	C₄²×C₄₇	C₂³	1	1	yes	characteristic
36	(C₂×D₄₇)⋊C₄	—	2	2	no	—
37	(C₂×D₄₇)⋊C₄	—	2	2	no	—
38	C₄⋊Dic₄₇	—	2	1	no	—
39	C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₄₇⋊C₈	—	2	2	no	—
41	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×D₄₇	—	2	2	no	—
43	C₂×C₄×C₄₇	—	2	2	no	—
44	C₂×C₄×C₄₇	—	2	2	no	—
45	C₂×C₄×C₄₇	—	2	2	no	—
46	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂²⋊D₄₇	—	4	4	no	—
53	C₂×Dic₄₇	—	2	2	no	—
54	D₁₈₈	—	2	4	no	—
55	D₁₈₈	—	4	2	no	—
56	D₁₈₈	—	4	2	no	—
57	C₄×D₄₇	—	4	4	no	—
58	Dic₉₄	—	4	2	no	—
59	C₈×C₄₇	—	2	4	no	—
60	C₄₇⋊C₈	—	4	2	no	—
61	C₂²×C₄₇	—	2	1	no	—
62	C₂²×C₄₇	—	2	1	no	—
63	C₂²×C₄₇	—	2	1	no	—
64	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×D₄₇	—	4	2	no	—
66	C₂×D₄₇	—	4	2	no	—
67	C₂×D₄₇	—	4	2	no	—
68	C₄×C₄₇	—	2	1	no	—
69	C₄×C₄₇	—	2	1	no	—
70	C₄×C₄₇	—	2	1	no	—
71	C₄×C₄₇	#104	1	2	yes	—
72	C₄×C₄₇	#131	1	2	yes	—
73	C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
74	Dic₄₇	—	4	2	no	—
75	#1484	—	47	1	no	maximal, Sylow
76	C₂×C₄₇	—	4	1	no	—
77	C₂×C₄₇	#408	1	1	yes	characteristic
78	C₂×C₄₇	#418	1	1	yes	characteristic
79	C₂×C₄₇	C₂×#104	1	1	yes	characteristic
80	D₄₇	—	8	2	no	—
81	#532	—	47	1	no	—
82	#431	—	47	1	no	—
83	C₄×M₁₆	D₄₇	1	1	yes	characteristic, complemented
84	D₄⋊C₈	—	47	4	no	—
85	C₄₇	#1484	1	1	yes	characteristic, complemented, Sylow
86	C₂×#41	—	94	1	no	—
87	C₂×M₁₆	—	2	1	no	—
88	C₄⋊Q₈	—	47	1	no	—
89	#127	—	47	1	no	—
90	#124	—	94	1	no	—
91	#121	—	47	2	no	—
92	#121	—	94	1	no	—
93	C₄×D₄	—	47	2	no	—
94	C₄×D₄	—	94	1	no	—
95	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
96	C₄⋊C₈	—	47	2	no	—
97	C₂²⋊C₈	—	94	1	no	—
98	#97	—	2	1	no	—
99	C₄×C₈	—	94	1	no	—
100	C₄×C₈	C₂×D₄₇	1	2	yes	—
101	#41	—	188	4	no	—
102	C₂×Q₈	—	94	1	no	—
103	C₂×D₄	—	47	2	no	—
104	C₂×D₄	—	94	2	no	—
105	C₂×D₄	—	94	1	no	—
106	C₂×D₄	—	94	1	no	—
107	C₂²×C₄	—	94	2	no	—
108	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
109	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
110	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
111	M₁₆	—	2	4	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	94	2	no	—
114	C₄⋊C₄	—	47	2	no	—
115	C₄⋊C₄	—	94	1	no	—
116	C₂²⋊C₄	—	94	2	no	—
117	C₂²⋊C₄	—	94	2	no	—
118	C₄²	C₄×D₄₇	1	1	yes	characteristic
119	C₄²	D₁₈₈	1	1	yes	characteristic
120	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
121	C₄²	C₂²×D₄₇	1	1	yes	characteristic
122	C₂³	—	94	2	no	—
123	C₂³	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
124	Q₈	—	188	2	no	—
125	D₄	—	94	4	no	—
126	D₄	—	188	2	no	—
127	D₄	—	188	4	no	—
128	D₄	—	188	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	94	2	no	—
133	C₂×C₄	—	188	4	no	—
134	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
135	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
136	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
138	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
139	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
140	C₈	—	2	4	no	—
141	C₈	—	188	2	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	188	2	no	—
146	C₂²	—	188	2	no	—
147	C₂²	—	188	2	no	—
148	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	188	2	no	—
153	C₄	#13093952	1	2	yes	—
154	C₄	#13094032	1	2	yes	—
155	C₄	#13094052	1	2	yes	—
156	C₂	—	4	1	no	—
157	C₂	—	376	2	no	—
158	C₂	#15689091	1	1	yes	characteristic
159	C₂	#15689129	1	1	yes	characteristic
160	C₂	C₂×#13093952	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689673

C₂

yes

maximal, characteristic

#15689176

C₂

yes

maximal, characteristic, complemented

#15688852

C₂

yes

maximal, complemented

C₄×M₁₆×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×#10625350

—

#13094024

C₄

yes

characteristic

#13094021

C₄

yes

—

—

C₄

yes

—

C₄×D₁₈₈

—

C₄×D₁₈₈

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₂×M₁₆×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#97×C₄₇

—

C₄×C₈×C₄₇

C₂²

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂×C₂²⋊D₄₇

—

#10625350

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

C₂×C₄

yes

—

C₂×C₄×D₄₇

—

C₂×Dic₉₄

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

C₂²×D₄₇

—

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

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

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#104

yes

—

C₄×C₄₇

#131

yes

—

C₄×C₄₇

AGL(1,ℤ/8)

yes

—

Dic₄₇

—

#1484

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

#532

—

#431

—

C₄×M₁₆

D₄₇

yes

characteristic, complemented

D₄⋊C₈

—

C₄₇

#1484

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

—

—

—

—

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

100

C₄×C₈

C₂×D₄₇

yes

—

101

#41

—

188

—

102

C₂×Q₈

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₄×D₄₇

yes

characteristic

109

C₂²×C₄

D₁₈₈

yes

characteristic

110

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

111

M₁₆

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

C₄×D₄₇

yes

characteristic

119

C₄²

D₁₈₈

yes

characteristic

120

C₄²

C₂²⋊D₄₇

yes

characteristic

121

C₄²

C₂²×D₄₇

yes

characteristic

122

C₂³

—

123

C₂³

(C₂×D₄₇)⋊C₄

yes

characteristic

124

Q₈

—

188

—

125

D₄

—

126

D₄

—

188

—

127

D₄

—

188

—

128

D₄

—

188

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

188

—

134

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

135

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

136

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

137

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

138

C₂×C₄

C₂×D₁₈₈

yes

characteristic

139

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

140

C₈

—

141

C₈

—

188

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

188

—

146

C₂²

—

188

—

147

C₂²

—

188

—

148

C₂²

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

188

—

153

C₄

#13093952

yes

—

154

C₄

#13094032

yes

—

155

C₄

#13094052

yes

—

156

C₂

—

157

C₂

—

376

—

158

C₂

#15689091

yes

characteristic

159

C₂

#15689129

yes

characteristic

160

C₂

C₂×#13093952

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄×M₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	C₄×M₁₆	D₄₇
hypocenter	C₄₇	#1484
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₁₈₈

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄×M₁₆×C₄₇

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

hypercenter

C₄×M₁₆

D₄₇

hypocenter

C₄₇

#1484

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #15689176	C₂	semidirect
3. #15688852	C₂	semidirect
4. C₄×M₁₆×C₄₇	C₂	semidirect
83. C₄×M₁₆	D₄₇	semidirect
85. C₄₇	#1484	semidirect

H ≅ G/N

type

2. #15689176

C₂

semidirect

3. #15688852

C₂

semidirect

4. C₄×M₁₆×C₄₇

C₂

semidirect

83. C₄×M₁₆

D₄₇

semidirect

85. C₄₇

#1484

semidirect

Subgroups — #20241806

Conjugacy classes of subgroups

Complements