Group #20253777 — subgroups

Subgroups — #20253777

Order 6016.

400 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 133 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×#13094012	C₂	1	1	yes	maximal, characteristic
2	#15689485	C₂	1	4	yes	maximal, complemented
3	#15689480	C₂	1	4	yes	maximal, complemented
4	#15689131	C₂	1	2	yes	maximal, complemented
5	#15689127	C₂	1	2	yes	maximal
6	C₄×#10625329	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#533	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
9	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
10	C₂×#10625329	C₂²	1	2	yes	—
11	#13094012	—	2	2	no	—
12	#13094012	C₂²	1	2	yes	—
13	#13094012	C₂²	1	2	yes	—
14	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
15	#13093947	C₂²	1	8	yes	complemented
16	#13093944	C₂²	1	8	yes	—
17	#13093935	C₂²	1	2	yes	complemented
18	C₄×C₄₇⋊C₈	C₂²	1	2	yes	complemented
19	C₄⋊Q₈×C₄₇	C₂²	1	2	yes	complemented
20	#125×C₄₇	C₂²	1	2	yes	complemented
21	C₂²⋊Q₈×C₄₇	—	2	2	no	—
22	C₄×Q₈×C₄₇	—	2	2	no	—
23	#117×C₄₇	C₂²	1	2	yes	—
24	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
25	C₂³×D₄₇	—	2	1	no	—
26	C₂×Q₈×C₄₇	—	2	2	no	—
27	C₂²×C₄×C₄₇	D₄	1	2	yes	—
28	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
29	C₂²×Dic₄₇	—	2	1	no	—
30	C₂×D₁₈₈	—	2	4	no	—
31	C₂×D₁₈₈	—	2	1	no	—
32	C₂×D₁₈₈	C₂³	1	2	yes	—
33	C₂×Dic₉₄	—	2	4	no	—
34	C₂×Dic₉₄	—	2	1	no	—
35	C₂×Dic₉₄	C₂³	1	2	yes	—
36	C₄⋊C₄×C₄₇	—	2	4	no	—
37	C₄⋊C₄×C₄₇	—	2	2	no	—
38	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
39	C₂²⋊C₄×C₄₇	—	2	2	no	—
40	C₄²×C₄₇	—	2	2	no	—
41	C₄²×C₄₇	D₄	1	2	yes	—
42	C₄²×C₄₇	C₂³	1	1	yes	characteristic
43	C₄²×C₄₇	C₂³	1	1	yes	characteristic
44	(C₂×D₄₇)⋊C₄	—	2	4	no	—
45	(C₂×D₄₇)⋊C₄	—	2	4	no	—
46	#10625329	—	2	4	no	—
47	C₂×C₄₇⋊C₈	C₂³	1	4	yes	—
48	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×D₄₇	—	2	2	no	—
50	C₂²×D₄₇	—	4	4	no	—
51	C₂²×D₄₇	—	4	2	no	—
52	Q₈×C₄₇	—	4	4	no	—
53	C₂×C₄×C₄₇	—	2	4	no	—
54	C₂×C₄×C₄₇	—	4	2	no	—
55	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₄₇	#41	1	4	yes	—
59	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
60	C₂×Dic₄₇	—	2	2	no	—
61	C₂×Dic₄₇	—	4	2	no	—
62	D₁₈₈	—	2	4	no	—
63	D₁₈₈	—	4	2	no	—
64	Dic₉₄	—	2	4	no	—
65	Dic₉₄	—	4	2	no	—
66	C₄₇⋊C₈	—	2	4	no	—
67	C₂²×C₄₇	—	2	2	no	—
68	C₂²×C₄₇	#124	1	2	yes	—
69	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
70	C₂×D₄₇	—	4	2	no	—
71	C₂×D₄₇	—	4	4	no	—
72	C₂×D₄₇	—	8	4	no	—
73	C₄×C₄₇	—	2	2	no	—
74	C₄×C₄₇	—	2	2	no	—
75	C₄×C₄₇	—	4	4	no	—
76	C₄×C₄₇	#124	1	2	yes	—
77	C₄×C₄₇	C₂×#41	1	2	yes	—
78	Dic₄₇	—	4	2	no	—
79	#3140	—	47	1	no	maximal, Sylow
80	C₂×C₄₇	—	2	2	no	—
81	C₂×C₄₇	C₂×#124	1	1	yes	characteristic
82	C₂×C₄₇	#575	1	2	yes	—
83	D₄₇	—	8	2	no	—
84	#533	D₄₇	1	1	yes	characteristic, complemented
85	C₂×#124	—	47	1	no	—
86	#489	—	47	4	no	—
87	#488	—	47	4	no	—
88	#419	—	47	2	no	—
89	#418	—	47	2	no	—
90	C₄×M₁₆	—	47	1	no	—
91	C₄₇	#3140	1	1	yes	characteristic, complemented, Sylow
92	C₂²×Q₈	—	47	1	no	—
93	C₂²×D₄	—	47	1	no	—
94	C₂×M₁₆	—	47	2	no	—
95	C₄⋊Q₈	C₂×D₄₇	1	2	yes	complemented
96	#125	C₂×D₄₇	1	2	yes	complemented
97	#124	—	47	2	no	—
98	#124	—	47	2	no	—
99	#124	—	94	2	no	—
100	C₂²⋊Q₈	—	2	2	no	—
101	C₄×Q₈	—	2	2	no	—
102	#117	C₂×D₄₇	1	2	yes	—
103	C₂×C₂²⋊C₄	—	94	2	no	—
104	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
105	#103	—	47	8	no	—
106	#102	—	47	8	no	—
107	#97	—	47	2	no	—
108	C₄×C₈	—	47	2	no	—
109	C₂⁴	—	94	1	no	—
110	C₂×Q₈	—	2	2	no	—
111	C₂×Q₈	—	47	2	no	—
112	C₂×Q₈	—	94	1	no	—
113	C₂×Q₈	—	94	4	no	—
114	C₂×D₄	—	47	2	no	—
115	C₂×D₄	—	94	4	no	—
116	C₂×D₄	—	94	1	no	—
117	C₂²×C₄	—	94	1	no	—
118	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
119	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
120	M₁₆	—	94	4	no	—
121	C₂×C₈	—	47	4	no	—
122	C₄⋊C₄	—	2	4	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
125	C₂²⋊C₄	—	2	2	no	—
126	C₂²⋊C₄	—	94	4	no	—
127	C₂²⋊C₄	—	94	4	no	—
128	C₄²	—	2	2	no	—
129	C₄²	C₂²⋊D₄₇	1	2	yes	—
130	C₄²	C₂²×D₄₇	1	1	yes	characteristic
131	C₄²	C₂²×D₄₇	1	1	yes	characteristic
132	C₂³	—	94	2	no	—
133	C₂³	—	188	4	no	—
134	C₂³	—	188	2	no	—
135	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
136	Q₈	—	4	4	no	—
137	Q₈	—	94	4	no	—
138	Q₈	—	188	2	no	—
139	D₄	—	94	4	no	—
140	D₄	—	188	2	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	94	2	no	—
144	C₂×C₄	—	188	2	no	—
145	C₂×C₄	#10625354	1	4	yes	—
146	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
148	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
149	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
150	C₈	—	94	4	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	376	4	no	—
153	C₂²	—	188	2	no	—
154	C₂²	—	188	4	no	—
155	C₂²	#13094013	1	2	yes	—
156	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	4	4	no	—
160	C₄	—	188	2	no	—
161	C₄	#13094013	1	2	yes	—
162	C₄	C₂×#10625354	1	2	yes	—
163	C₂	—	2	2	no	—
164	C₂	—	376	2	no	—
165	C₂	C₂×#13094013	1	1	yes	characteristic
166	C₂	#15689921	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094012

C₂

yes

maximal, characteristic

#15689485

C₂

yes

maximal, complemented

#15689480

C₂

yes

maximal, complemented

#15689131

C₂

yes

maximal, complemented

#15689127

C₂

yes

maximal

C₄×#10625329

C₂

yes

maximal, characteristic, complemented

C₄₇×#533

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₉₄

C₂²

yes

characteristic

C₂²×D₁₈₈

C₂²

yes

characteristic

C₂×#10625329

C₂²

yes

—

#13094012

—

#13094012

C₂²

yes

—

#13094012

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

—

#13093947

C₂²

yes

complemented

#13093944

C₂²

yes

—

#13093935

C₂²

yes

complemented

C₄×C₄₇⋊C₈

C₂²

yes

complemented

C₄⋊Q₈×C₄₇

C₂²

yes

complemented

#125×C₄₇

C₂²

yes

complemented

C₂²⋊Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

#117×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂³×D₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

C₂³

yes

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

C₂³

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

D₄

yes

—

C₄²×C₄₇

C₂³

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

#10625329

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

Dic₉₄

—

Dic₉₄

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

#124

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#124

yes

—

C₄×C₄₇

C₂×#41

yes

—

Dic₄₇

—

#3140

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂×#124

yes

characteristic

C₂×C₄₇

#575

yes

—

D₄₇

—

#533

D₄₇

yes

characteristic, complemented

—

—

—

—

—

C₄×M₁₆

—

C₄₇

#3140

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×D₄

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₄₇

yes

complemented

#125

C₂×D₄₇

yes

—

—

—

100

C₂²⋊Q₈

—

101

C₄×Q₈

—

102

#117

C₂×D₄₇

yes

—

103

C₂×C₂²⋊C₄

—

104

C₂×C₄²

C₂×D₄₇

yes

characteristic

105

#103

—

106

#102

—

107

#97

—

108

C₄×C₈

—

109

C₂⁴

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂²×C₄

—

118

C₂²×C₄

C₂²⋊D₄₇

yes

—

119

C₂²×C₄

C₂²×D₄₇

yes

characteristic

120

M₁₆

—

121

C₂×C₈

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

C₂²×D₄₇

yes

—

125

C₂²⋊C₄

—

126

C₂²⋊C₄

—

127

C₂²⋊C₄

—

128

C₄²

—

129

C₄²

C₂²⋊D₄₇

yes

—

130

C₄²

C₂²×D₄₇

yes

characteristic

131

C₄²

C₂²×D₄₇

yes

characteristic

132

C₂³

—

133

C₂³

—

188

—

134

C₂³

—

188

—

135

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

136

Q₈

—

137

Q₈

—

138

Q₈

—

188

—

139

D₄

—

140

D₄

—

188

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

188

—

145

C₂×C₄

#10625354

yes

—

146

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

147

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

148

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

149

C₂×C₄

C₂³×D₄₇

yes

characteristic

150

C₈

—

151

C₂²

—

152

C₂²

—

376

—

153

C₂²

—

188

—

154

C₂²

—

188

—

155

C₂²

#13094013

yes

—

156

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

188

—

161

C₄

#13094013

yes

—

162

C₄

C₂×#10625354

yes

—

163

C₂

—

164

C₂

—

376

—

165

C₂

C₂×#13094013

yes

characteristic

166

C₂

#15689921

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#533	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#533	D₄₇
hypocenter	C₄₇	#3140
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#533

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #15689485	C₂	semidirect
3. #15689480	C₂	semidirect
4. #15689131	C₂	semidirect
6. C₄×#10625329	C₂	semidirect
7. C₄₇×#533	C₂	semidirect
15. #13093947	C₂²	semidirect
17. #13093935	C₂²	semidirect
18. C₄×C₄₇⋊C₈	C₂²	semidirect
19. C₄⋊Q₈×C₄₇	C₂²	semidirect
20. #125×C₄₇	C₂²	semidirect
84. #533	D₄₇	semidirect
91. C₄₇	#3140	semidirect
95. C₄⋊Q₈	C₂×D₄₇	semidirect
96. #125	C₂×D₄₇	semidirect

H ≅ G/N

type

2. #15689485

C₂

semidirect

3. #15689480

C₂

semidirect

4. #15689131

C₂

semidirect

6. C₄×#10625329

C₂

semidirect

7. C₄₇×#533

C₂

semidirect

15. #13093947

C₂²

semidirect

17. #13093935

C₂²

semidirect

18. C₄×C₄₇⋊C₈

C₂²

semidirect

19. C₄⋊Q₈×C₄₇

C₂²

semidirect

20. #125×C₄₇

C₂²

semidirect

84. #533

D₄₇

semidirect

91. C₄₇

#3140

semidirect

95. C₄⋊Q₈

C₂×D₄₇

semidirect

96. #125

C₂×D₄₇

semidirect

Subgroups — #20253777

Conjugacy classes of subgroups

Complements