Group #20241979 — subgroups

Subgroups — #20241979

Order 6016.

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

normal subgroups only

This group has 270 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	C₄×#10625329	C₂	1	1	yes	maximal, characteristic, complemented
3	#15688862	C₂	1	4	yes	maximal, complemented
4	C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10625350	C₄	1	2	yes	—
6	C₂×#10625329	—	2	1	no	—
7	#13094024	C₄	1	1	yes	characteristic
8	#13094021	C₄	1	1	yes	characteristic
9	#13094012	C₂²	1	2	yes	—
10	#13093998	—	2	1	no	—
11	#13093998	—	2	1	no	—
12	C₄×D₁₈₈	—	2	1	no	—
13	C₄×D₁₈₈	—	2	1	no	—
14	#13093935	C₂²	1	2	yes	—
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	C₂²	1	2	yes	—
19	C₄×C₈×C₄₇	—	2	1	no	—
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	C₂×C₄	1	2	yes	—
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	—	2	1	no	—
40	#10625329	—	2	4	no	—
41	C₂×C₄₇⋊C₈	—	2	2	no	—
42	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×D₄₇	—	2	2	no	—
44	C₂×C₄×C₄₇	—	2	2	no	—
45	C₂×C₄×C₄₇	—	2	2	no	—
46	C₂×C₄×C₄₇	—	2	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂²⋊D₄₇	—	4	4	no	—
54	C₂×Dic₄₇	—	2	2	no	—
55	D₁₈₈	—	4	2	no	—
56	D₁₈₈	—	4	2	no	—
57	D₁₈₈	—	4	2	no	—
58	C₄×D₄₇	—	4	4	no	—
59	Dic₉₄	—	4	2	no	—
60	C₈×C₄₇	—	2	4	no	—
61	C₄₇⋊C₈	—	2	4	no	—
62	C₂²×C₄₇	—	2	1	no	—
63	C₂²×C₄₇	—	2	1	no	—
64	C₂²×C₄₇	—	2	1	no	—
65	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×D₄₇	—	4	2	no	—
67	C₂×D₄₇	—	4	2	no	—
68	C₂×D₄₇	—	4	2	no	—
69	C₄×C₄₇	—	2	1	no	—
70	C₄×C₄₇	—	2	1	no	—
71	C₄×C₄₇	—	2	1	no	—
72	C₄×C₄₇	#100	1	2	yes	—
73	C₄×C₄₇	#104	1	2	yes	—
74	C₄×C₄₇	#104	1	2	yes	—
75	Dic₄₇	—	4	2	no	—
76	#1525	—	47	1	no	maximal, Sylow
77	C₂×C₄₇	—	4	1	no	—
78	C₂×C₄₇	C₂×#100	1	1	yes	characteristic
79	C₂×C₄₇	C₂×#104	1	1	yes	characteristic
80	C₂×C₄₇	C₂×#104	1	1	yes	characteristic
81	D₄₇	—	8	2	no	—
82	#532	—	47	1	no	—
83	C₄×M₁₆	—	47	1	no	—
84	C₄×M₁₆	D₄₇	1	1	yes	characteristic, complemented
85	#329	—	47	4	no	—
86	C₄₇	#1525	1	1	yes	characteristic, complemented, Sylow
87	C₂×#41	—	47	2	no	—
88	C₂×M₁₆	—	2	1	no	—
89	C₂×M₁₆	—	94	1	no	—
90	C₄⋊Q₈	—	47	1	no	—
91	#127	—	47	1	no	—
92	#124	—	47	2	no	—
93	#121	—	94	1	no	—
94	#121	—	94	1	no	—
95	C₄×D₄	—	94	1	no	—
96	C₄×D₄	—	94	1	no	—
97	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
98	#97	—	47	2	no	—
99	#97	C₂×D₄₇	1	2	yes	—
100	C₄×C₈	—	2	1	no	—
101	C₄×C₈	—	94	1	no	—
102	#41	—	188	4	no	—
103	C₂×Q₈	—	47	2	no	—
104	C₂×D₄	—	47	2	no	—
105	C₂×D₄	—	94	2	no	—
106	C₂×D₄	—	94	1	no	—
107	C₂×D₄	—	94	1	no	—
108	C₂²×C₄	—	94	2	no	—
109	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
110	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
111	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
112	M₁₆	—	2	4	no	—
113	M₁₆	—	94	4	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	94	2	no	—
116	C₄⋊C₄	—	94	1	no	—
117	C₄⋊C₄	—	94	1	no	—
118	C₂²⋊C₄	—	94	2	no	—
119	C₂²⋊C₄	—	94	2	no	—
120	C₄²	C₄×D₄₇	1	1	yes	characteristic
121	C₄²	D₁₈₈	1	1	yes	characteristic
122	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
123	C₄²	C₂²×D₄₇	1	1	yes	characteristic
124	C₂³	—	94	2	no	—
125	C₂³	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
126	Q₈	—	188	2	no	—
127	D₄	—	188	2	no	—
128	D₄	—	188	4	no	—
129	D₄	—	188	2	no	—
130	D₄	—	188	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	94	2	no	—
135	C₂×C₄	—	188	4	no	—
136	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
137	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
138	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
140	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
141	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
142	C₈	—	2	4	no	—
143	C₈	—	94	4	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	—	188	2	no	—
148	C₂²	—	188	2	no	—
149	C₂²	—	188	2	no	—
150	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	188	2	no	—
155	C₄	#13093940	1	2	yes	—
156	C₄	#13093950	1	2	yes	—
157	C₄	#13093952	1	2	yes	—
158	C₂	—	4	1	no	—
159	C₂	—	376	2	no	—
160	C₂	C₂×#13093940	1	1	yes	characteristic
161	C₂	C₂×#13093950	1	1	yes	characteristic
162	C₂	C₂×#13093952	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689673

C₂

yes

maximal, characteristic

C₄×#10625329

C₂

yes

maximal, characteristic, complemented

#15688862

C₂

yes

maximal, complemented

C₄×M₁₆×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×#10625350

C₄

yes

—

C₂×#10625329

—

#13094024

C₄

yes

characteristic

#13094021

C₄

yes

characteristic

#13094012

C₂²

yes

—

#13093998

—

#13093998

—

C₄×D₁₈₈

—

C₄×D₁₈₈

—

#13093935

C₂²

yes

—

C₄×C₄₇⋊C₈

—

C₂×M₁₆×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#97×C₄₇

C₂²

yes

—

C₄×C₈×C₄₇

—

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

C₂×C₄

yes

—

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

—

#10625329

—

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

#100

yes

—

C₄×C₄₇

#104

yes

—

C₄×C₄₇

#104

yes

—

Dic₄₇

—

#1525

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

#532

—

C₄×M₁₆

—

C₄×M₁₆

D₄₇

yes

characteristic, complemented

#329

—

C₄₇

#1525

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

—

—

—

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

#97

—

#97

C₂×D₄₇

yes

—

100

C₄×C₈

—

101

C₄×C₈

—

102

#41

—

188

—

103

C₂×Q₈

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₄×D₄₇

yes

characteristic

110

C₂²×C₄

D₁₈₈

yes

characteristic

111

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₂²⋊C₄

—

120

C₄²

C₄×D₄₇

yes

characteristic

121

C₄²

D₁₈₈

yes

characteristic

122

C₄²

C₂²⋊D₄₇

yes

characteristic

123

C₄²

C₂²×D₄₇

yes

characteristic

124

C₂³

—

125

C₂³

(C₂×D₄₇)⋊C₄

yes

characteristic

126

Q₈

—

188

—

127

D₄

—

188

—

128

D₄

—

188

—

129

D₄

—

188

—

130

D₄

—

188

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

188

—

136

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

137

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

138

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

139

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

140

C₂×C₄

C₂×D₁₈₈

yes

characteristic

141

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

142

C₈

—

143

C₈

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

188

—

148

C₂²

—

188

—

149

C₂²

—

188

—

150

C₂²

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

188

—

155

C₄

#13093940

yes

—

156

C₄

#13093950

yes

—

157

C₄

#13093952

yes

—

158

C₂

—

159

C₂

—

376

—

160

C₂

C₂×#13093940

yes

characteristic

161

C₂

C₂×#13093950

yes

characteristic

162

C₂

C₂×#13093952

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₄₇)⋊C₄
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	#1525
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

(C₂×D₄₇)⋊C₄

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

#1525

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
2. C₄×#10625329	C₂	semidirect
3. #15688862	C₂	semidirect
4. C₄×M₁₆×C₄₇	C₂	semidirect
84. C₄×M₁₆	D₄₇	semidirect
86. C₄₇	#1525	semidirect

H ≅ G/N

type

2. C₄×#10625329

C₂

semidirect

3. #15688862

C₂

semidirect

4. C₄×M₁₆×C₄₇

C₂

semidirect

84. C₄×M₁₆

D₄₇

semidirect

86. C₄₇

#1525

semidirect

Subgroups — #20241979

Conjugacy classes of subgroups

Complements