Group #20252100 — subgroups

Subgroups — #20252100

Order 6016.

500 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 229 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×#13094031	C₂	1	2	yes	maximal, complemented
2	#15689201	C₂	1	4	yes	maximal, complemented
3	D₄×C₄₇⋊C₈	C₂	1	4	yes	maximal, complemented
4	#15689147	C₂	1	1	yes	maximal, characteristic, complemented
5	#15689090	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#532	C₂	1	1	yes	maximal, characteristic
8	#13094031	—	2	8	no	—
9	C₂×#10625329	C₂²	1	2	yes	—
10	C₂×#10625329	C₂²	1	4	yes	—
11	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂²×C₄₇⋊C₈	C₂²	1	4	yes	—
13	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
15	C₂²⋊(C₄₇⋊C₈)	C₂²	1	8	yes	—
16	C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
17	C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
18	C₂×#41×C₄₇	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₄₇	C₄	1	1	yes	characteristic
20	#127×C₄₇	C₄	1	1	yes	characteristic
21	#124×C₄₇	C₄	1	2	yes	—
22	#121×C₄₇	C₄	1	4	yes	—
23	C₄×D₄×C₄₇	C₂²	1	4	yes	—
24	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
25	#41×C₄₇	—	2	4	no	—
26	#41×C₄₇	—	2	4	no	—
27	C₂×Q₈×C₄₇	C₂×C₄	1	2	yes	—
28	C₂×D₄×C₄₇	C₂×C₄	1	4	yes	—
29	C₂×D₄×C₄₇	C₂×C₄	1	4	yes	—
30	C₂×D₄×C₄₇	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
33	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
35	C₂²⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
36	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₄₇	C₂³	1	2	yes	—
39	#10625329	—	2	4	no	—
40	#10625329	—	2	8	no	—
41	C₂×C₄₇⋊C₈	—	2	2	no	—
42	C₂×C₄₇⋊C₈	—	2	8	no	—
43	C₂×C₄₇⋊C₈	D₄	1	4	yes	complemented
44	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
45	C₂×C₄₇⋊C₈	C₂³	1	4	yes	—
46	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
47	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
48	C₂³×C₄₇	C₂²×C₄	1	4	yes	—
49	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	Q₈×C₄₇	—	2	4	no	—
51	D₄×C₄₇	—	2	8	no	—
52	D₄×C₄₇	—	2	4	no	—
53	D₄×C₄₇	—	2	8	no	—
54	C₂×C₄×C₄₇	—	2	4	no	—
55	C₂×C₄×C₄₇	—	2	4	no	—
56	C₂×C₄×C₄₇	—	2	4	no	—
57	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
59	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₄₇	#41	1	2	yes	—
63	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
64	C₄₇⋊C₈	—	2	4	no	—
65	C₄₇⋊C₈	—	2	2	no	—
66	C₄₇⋊C₈	C₂×D₄	1	2	yes	—
67	C₄₇⋊C₈	C₂×D₄	1	2	yes	—
68	C₂²×C₄₇	—	2	2	no	—
69	C₂²×C₄₇	—	2	4	no	—
70	C₂²×C₄₇	—	2	8	no	—
71	C₂²×C₄₇	C₄×D₄	1	2	yes	—
72	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
73	C₄×C₄₇	—	2	4	no	—
74	C₄×C₄₇	—	2	2	no	—
75	C₄×C₄₇	C₄×D₄	1	2	yes	—
76	C₄×C₄₇	#131	1	4	yes	complemented
77	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
78	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
79	#2931	—	47	1	no	maximal, Sylow
80	C₂×C₄₇	—	2	2	no	—
81	C₂×C₄₇	—	4	4	no	—
82	C₂×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
83	C₂×C₄₇	C₂×#131	1	2	yes	—
84	C₂×#131	—	47	2	no	—
85	#532	D₄₇	1	1	yes	characteristic
86	#436	—	47	4	no	—
87	C₈×D₄	—	47	4	no	—
88	#423	—	47	1	no	—
89	#408	—	47	2	no	—
90	C₂×C₄×C₈	—	47	1	no	—
91	C₄₇	#2931	1	1	yes	characteristic, complemented, Sylow
92	C₂×#41	C₂×D₄₇	1	2	yes	—
93	#131	—	94	8	no	—
94	C₂×M₁₆	—	47	2	no	—
95	C₂×M₁₆	—	47	4	no	—
96	C₂²×C₈	—	47	1	no	—
97	C₂²×C₈	—	47	4	no	—
98	C₂²×C₈	—	47	1	no	—
99	C₄⋊Q₈	Dic₄₇	1	1	yes	characteristic
100	#127	Dic₄₇	1	1	yes	characteristic
101	#124	Dic₄₇	1	2	yes	—
102	#121	Dic₄₇	1	4	yes	—
103	C₄×D₄	C₂×D₄₇	1	4	yes	—
104	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
105	C₄⋊C₈	—	47	4	no	—
106	C₂²⋊C₈	—	47	8	no	—
107	C₄×C₈	—	47	2	no	—
108	C₄×C₈	—	47	2	no	—
109	#41	—	2	4	no	—
110	#41	—	2	4	no	—
111	C₂×Q₈	C₂×Dic₄₇	1	2	yes	—
112	C₂×D₄	C₂×Dic₄₇	1	4	yes	—
113	C₂×D₄	C₂×Dic₄₇	1	4	yes	—
114	C₂×D₄	C₂×Dic₄₇	1	2	yes	—
115	C₂²×C₄	C₂×Dic₄₇	1	2	yes	—
116	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
117	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
118	M₁₆	—	94	8	no	—
119	M₁₆	—	94	4	no	—
120	C₂×C₈	—	47	4	no	—
121	C₂×C₈	—	47	1	no	—
122	C₂×C₈	—	47	1	no	—
123	C₂×C₈	—	47	2	no	—
124	C₂×C₈	—	47	4	no	—
125	C₂×C₈	—	94	8	no	—
126	C₂×C₈	—	94	2	no	—
127	C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
128	C₂²⋊C₄	C₂×Dic₄₇	1	8	yes	—
129	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
130	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
131	C₄²	C₂²×D₄₇	1	2	yes	—
132	C₂³	C₂²×Dic₄₇	1	4	yes	—
133	C₂³	C₂²×Dic₄₇	1	1	yes	characteristic
134	Q₈	—	2	4	no	—
135	D₄	—	2	8	no	—
136	D₄	—	2	4	no	—
137	D₄	—	2	8	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	D₄×D₄₇	1	2	yes	—
142	C₂×C₄	#10625352	1	2	yes	—
143	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
144	C₂×C₄	C₂²×Dic₄₇	1	4	yes	—
145	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
146	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
147	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
148	C₈	—	47	2	no	—
149	C₈	—	47	2	no	—
150	C₈	—	94	4	no	—
151	C₈	—	94	2	no	—
152	C₂²	—	2	2	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	2	8	no	—
155	C₂²	D₄×Dic₄₇	1	2	yes	—
156	C₂²	C₂³×Dic₄₇	1	1	yes	characteristic
157	C₄	—	2	4	no	—
158	C₄	—	2	2	no	—
159	C₄	D₄×Dic₄₇	1	2	yes	—
160	C₄	#13094031	1	4	yes	complemented
161	C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
162	C₄	C₂×#10625352	1	1	yes	characteristic
163	C₂	—	2	2	no	—
164	C₂	—	4	4	no	—
165	C₂	C₂×D₄×Dic₄₇	1	1	yes	characteristic
166	C₂	C₂×#13094031	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094031

C₂

yes

maximal, complemented

#15689201

C₂

yes

maximal, complemented

D₄×C₄₇⋊C₈

C₂

yes

maximal, complemented

#15689147

C₂

yes

maximal, characteristic, complemented

#15689090

C₂

yes

maximal, complemented

C₂×C₄×C₄₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄₇×#532

C₂

yes

maximal, characteristic

#13094031

—

C₂×#10625329

C₂²

yes

—

C₂×#10625329

C₂²

yes

—

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂²⋊(C₄₇⋊C₈)

C₂²

yes

—

C₄×C₄₇⋊C₈

C₂²

yes

—

C₄×C₄₇⋊C₈

C₂²

yes

—

C₂×#41×C₄₇

C₂²

yes

—

C₄⋊Q₈×C₄₇

C₄

yes

characteristic

#127×C₄₇

C₄

yes

characteristic

#124×C₄₇

C₄

yes

—

#121×C₄₇

C₄

yes

—

C₄×D₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

C₂×C₄

yes

—

C₂×D₄×C₄₇

C₂×C₄

yes

—

C₂×D₄×C₄₇

C₂×C₄

yes

—

C₂×D₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂²⋊C₄×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂³

yes

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₄

yes

complemented

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂×C₄₇⋊C₈

C₂³

yes

characteristic

C₂×C₄₇⋊C₈

C₂³

yes

characteristic

C₂³×C₄₇

C₂²×C₄

yes

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

C₂×D₄

yes

—

C₄₇⋊C₈

C₂×D₄

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₄×D₄

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

#131

yes

complemented

C₄×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

C₂×#41

yes

characteristic

#2931

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂×C₄×D₄

yes

characteristic

C₂×C₄₇

C₂×#131

yes

—

C₂×#131

—

#532

D₄₇

yes

characteristic

#436

—

C₈×D₄

—

#423

—

#408

—

C₂×C₄×C₈

—

C₄₇

#2931

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₄₇

yes

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

Dic₄₇

yes

characteristic

100

#127

Dic₄₇

yes

characteristic

101

#124

Dic₄₇

yes

—

102

#121

Dic₄₇

yes

—

103

C₄×D₄

C₂×D₄₇

yes

—

104

C₂×C₄²

C₂×D₄₇

yes

characteristic

105

C₄⋊C₈

—

106

C₂²⋊C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

#41

—

110

#41

—

111

C₂×Q₈

C₂×Dic₄₇

yes

—

112

C₂×D₄

C₂×Dic₄₇

yes

—

113

C₂×D₄

C₂×Dic₄₇

yes

—

114

C₂×D₄

C₂×Dic₄₇

yes

—

115

C₂²×C₄

C₂×Dic₄₇

yes

—

116

C₂²×C₄

C₂²×D₄₇

yes

—

117

C₂²×C₄

C₂²×D₄₇

yes

characteristic

118

M₁₆

—

119

M₁₆

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₄⋊C₄

C₂×Dic₄₇

yes

—

128

C₂²⋊C₄

C₂×Dic₄₇

yes

—

129

C₄²

C₂×Dic₄₇

yes

characteristic

130

C₄²

C₂×Dic₄₇

yes

characteristic

131

C₄²

C₂²×D₄₇

yes

—

132

C₂³

C₂²×Dic₄₇

yes

—

133

C₂³

C₂²×Dic₄₇

yes

characteristic

134

Q₈

—

135

D₄

—

136

D₄

—

137

D₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

D₄×D₄₇

yes

—

142

C₂×C₄

#10625352

yes

—

143

C₂×C₄

C₂²×Dic₄₇

yes

—

144

C₂×C₄

C₂²×Dic₄₇

yes

—

145

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

146

C₂×C₄

C₂²×Dic₄₇

yes

—

147

C₂×C₄

C₂³×D₄₇

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

D₄×Dic₄₇

yes

—

156

C₂²

C₂³×Dic₄₇

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

D₄×Dic₄₇

yes

—

160

C₄

#13094031

yes

complemented

161

C₄

C₂×D₄×D₄₇

yes

characteristic

162

C₄

C₂×#10625352

yes

characteristic

163

C₂

—

164

C₂

—

165

C₂

C₂×D₄×Dic₄₇

yes

characteristic

166

C₂

C₂×#13094031

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₄
Fitting	C₄₇×#532	C₂
hypercenter	#532	D₄₇
hypocenter	C₄₇	#2931
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #532

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₄₇

commutator, socle

C₂²×C₄₇

C₂³×C₄

Fitting

C₄₇×#532

C₂

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₄ ◁ #532

N	H ≅ G/N	type
1. C₂×#13094031	C₂	semidirect
2. #15689201	C₂	semidirect
3. D₄×C₄₇⋊C₈	C₂	semidirect
4. #15689147	C₂	semidirect
5. #15689090	C₂	semidirect
6. C₂×C₄×C₄₇⋊C₈	C₂	semidirect
43. C₂×C₄₇⋊C₈	D₄	semidirect
76. C₄×C₄₇	#131	semidirect
91. C₄₇	#2931	semidirect
160. C₄	#13094031	semidirect

H ≅ G/N

type

1. C₂×#13094031

C₂

semidirect

2. #15689201

C₂

semidirect

3. D₄×C₄₇⋊C₈

C₂

semidirect

4. #15689147

C₂

semidirect

5. #15689090

C₂

semidirect

6. C₂×C₄×C₄₇⋊C₈

C₂

semidirect

43. C₂×C₄₇⋊C₈

D₄

semidirect

76. C₄×C₄₇

#131

semidirect

91. C₄₇

#2931

semidirect

160. C₄

#13094031

semidirect

Subgroups — #20252100

Conjugacy classes of subgroups

Complements