Group #20287704 — subgroups

Subgroups — #20287704

Order 6784.

360 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16811346	C₂	1	2	yes	maximal, complemented
2	#16811007	C₂	1	2	yes	maximal
3	#16811003	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810846	C₂	1	4	yes	maximal, complemented
5	#16810829	C₂	1	2	yes	maximal, complemented
6	C₄×Dic₂₁₂	C₂	1	2	yes	maximal, complemented
7	C₄×D₄₂₄	C₂	1	1	yes	maximal, characteristic
8	C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₂₁₂	C₂²	1	2	yes	—
10	C₂×Dic₂₁₂	C₂²	1	2	yes	—
11	C₂×#11719913	C₂²	1	4	yes	—
12	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
13	#13133217	C₂²	1	2	yes	—
14	#13133212	—	2	4	no	—
15	#13133205	C₂²	1	4	yes	—
16	#13133193	—	2	2	no	—
17	C₄×Dic₁₀₆	C₂²	1	4	yes	—
18	C₄×D₂₁₂	C₂²	1	2	yes	—
19	#13133175	—	2	2	no	—
20	#13133151	C₂²	1	1	yes	characteristic
21	#13133142	C₂²	1	4	yes	—
22	#13133139	C₂²	1	2	yes	—
23	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
26	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₅₃	C₂²	1	2	yes	—
28	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
29	C₂×D₂₁₂	C₂³	1	2	yes	—
30	C₂×C₄×D₅₃	—	2	2	no	—
31	C₂×Dic₁₀₆	C₂³	1	4	yes	—
32	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₂³	1	2	yes	—
34	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
37	C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	C₄²×C₅₃	C₂³	1	2	yes	—
39	#11719921	—	2	4	no	—
40	#11719921	—	2	2	no	—
41	#11719921	—	2	4	no	—
42	C₄⋊Dic₅₃	—	2	2	no	—
43	C₄⋊Dic₅₃	—	2	2	no	—
44	C₄⋊Dic₅₃	C₂³	1	2	yes	—
45	#11719919	—	2	4	no	—
46	C₄×Dic₅₃	—	2	4	no	—
47	Dic₂₁₂	—	2	4	no	—
48	Dic₂₁₂	—	4	2	no	—
49	D₄₂₄	—	2	2	no	—
50	#11719913	—	4	4	no	—
51	C₂²×D₅₃	—	2	2	no	—
52	Q₈×C₅₃	—	2	2	no	—
53	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
58	C₂×Dic₅₃	—	2	4	no	—
59	C₂×Dic₅₃	—	2	2	no	—
60	D₂₁₂	—	2	2	no	—
61	D₂₁₂	—	2	2	no	—
62	C₄×D₅₃	—	4	4	no	—
63	Dic₁₀₆	—	2	4	no	—
64	Dic₁₀₆	—	2	4	no	—
65	C₈×C₅₃	—	2	2	no	—
66	C₈×C₅₃	—	2	1	no	—
67	C₈×C₅₃	#41	1	2	yes	—
68	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
69	C₂×D₅₃	—	4	2	no	—
70	C₂×D₅₃	—	4	2	no	—
71	C₂×D₅₃	—	4	2	no	—
72	C₄×C₅₃	—	2	2	no	—
73	C₄×C₅₃	—	2	2	no	—
74	C₄×C₅₃	—	2	1	no	—
75	C₄×C₅₃	#135	1	2	yes	—
76	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
77	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
78	Dic₅₃	—	4	4	no	—
79	Dic₅₃	—	4	2	no	—
80	#3393	—	53	1	no	maximal, Sylow
81	C₂×C₅₃	#550	1	1	yes	characteristic
82	C₂×C₅₃	C₂×#135	1	1	yes	characteristic
83	C₂×C₅₃	#578	1	1	yes	characteristic
84	D₅₃	—	8	2	no	—
85	#556	—	53	2	no	—
86	#495	—	53	2	no	—
87	#494	—	53	1	no	—
88	#464	—	53	4	no	—
89	#461	—	53	2	no	—
90	C₈×Q₈	D₅₃	1	1	yes	characteristic, complemented
91	C₄×Q₁₆	—	53	2	no	—
92	C₄×D₈	—	53	1	no	—
93	C₅₃	#3393	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₁₆	—	53	2	no	—
95	C₂×Q₁₆	—	53	2	no	—
96	C₂×SD₁₆	—	53	4	no	—
97	C₂×D₈	—	53	1	no	—
98	C₄⋊Q₈	—	53	2	no	—
99	#126	—	106	4	no	—
100	#124	—	53	4	no	—
101	C₂²⋊Q₈	—	106	2	no	—
102	C₄×Q₈	—	53	4	no	—
103	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
104	C₄×D₄	—	53	2	no	—
105	#117	—	106	2	no	—
106	#107	—	53	1	no	—
107	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
108	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
109	#103	—	53	4	no	—
110	#102	—	53	2	no	—
111	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
112	C₄×C₈	C₂×D₅₃	1	2	yes	—
113	C₂×Q₈	—	53	4	no	—
114	C₂×Q₈	D₂₁₂	1	1	yes	characteristic
115	C₂×D₄	—	53	2	no	—
116	C₂²×C₄	—	106	2	no	—
117	Q₁₆	—	106	4	no	—
118	Q₁₆	—	212	2	no	—
119	SD₁₆	—	212	4	no	—
120	D₈	—	106	2	no	—
121	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
122	C₂×C₈	C₂²×D₅₃	1	2	yes	—
123	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
124	C₄⋊C₄	—	53	2	no	—
125	C₄⋊C₄	—	106	2	no	—
126	C₄⋊C₄	—	106	4	no	—
127	C₄⋊C₄	—	106	2	no	—
128	C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
129	C₄⋊C₄	D₂₁₂	1	2	yes	—
130	C₂²⋊C₄	—	106	4	no	—
131	C₂²⋊C₄	—	106	2	no	—
132	C₂²⋊C₄	—	106	4	no	—
133	C₄²	—	106	4	no	—
134	C₄²	C₂²×D₅₃	1	1	yes	characteristic
135	C₄²	C₂²×D₅₃	1	2	yes	—
136	C₂³	—	106	2	no	—
137	Q₈	—	2	2	no	—
138	Q₈	—	106	4	no	—
139	Q₈	—	106	4	no	—
140	D₄	—	106	2	no	—
141	D₄	—	106	2	no	—
142	C₂×C₄	—	106	2	no	—
143	C₂×C₄	—	106	4	no	—
144	C₂×C₄	—	212	4	no	—
145	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
146	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
147	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
148	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
149	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	—	2	1	no	—
152	C₈	#11719949	1	2	yes	—
153	C₂²	—	212	2	no	—
154	C₂²	—	212	2	no	—
155	C₂²	—	212	2	no	—
156	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	2	1	no	—
160	C₄	—	212	4	no	—
161	C₄	—	212	2	no	—
162	C₄	#13133239	1	2	yes	—
163	C₄	C₂×#11719949	1	1	yes	characteristic
164	C₄	#13133263	1	1	yes	characteristic
165	C₂	—	424	2	no	—
166	C₂	#16811295	1	1	yes	characteristic
167	C₂	C₂×#13133239	1	1	yes	characteristic
168	C₂	#16811440	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811346

C₂

yes

maximal, complemented

#16811007

C₂

yes

maximal

#16811003

C₂

yes

maximal, characteristic, complemented

#16810846

C₂

yes

maximal, complemented

#16810829

C₂

yes

maximal, complemented

C₄×Dic₂₁₂

C₂

yes

maximal, complemented

C₄×D₄₂₄

C₂

yes

maximal, characteristic

C₈×Q₈×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₂

C₂²

yes

—

C₂×Dic₂₁₂

C₂²

yes

—

C₂×#11719913

C₂²

yes

—

C₂×D₄₂₄

C₂²

yes

characteristic

#13133217

C₂²

yes

—

#13133212

—

#13133205

C₂²

yes

—

#13133193

—

C₄×Dic₁₀₆

C₂²

yes

—

C₄×D₂₁₂

C₂²

yes

—

#13133175

—

#13133151

C₂²

yes

characteristic

#13133142

C₂²

yes

—

#13133139

C₂²

yes

—

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×D₂₁₂

C₂³

yes

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

C₂³

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂³

yes

—

#11719921

—

#11719921

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

C₄×Dic₅₃

—

Dic₂₁₂

—

Dic₂₁₂

—

D₄₂₄

—

#11719913

—

C₂²×D₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#41

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#135

yes

—

C₄×C₅₃

C₂×#41

yes

characteristic

C₄×C₅₃

H(5,𝔽₂)

yes

characteristic

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

—

—

—

—

—

C₈×Q₈

D₅₃

yes

characteristic, complemented

C₄×Q₁₆

—

C₄×D₈

—

C₅₃

#3393

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

#126

—

106

—

100

#124

—

101

C₂²⋊Q₈

—

106

—

102

C₄×Q₈

—

103

C₄×Q₈

C₂×D₅₃

yes

characteristic

104

C₄×D₄

—

105

#117

—

106

—

106

#107

—

107

C₄⋊C₈

C₂×D₅₃

yes

characteristic

108

C₄⋊C₈

C₂×D₅₃

yes

—

109

#103

—

110

#102

—

111

C₄×C₈

C₂×D₅₃

yes

characteristic

112

C₄×C₈

C₂×D₅₃

yes

—

113

C₂×Q₈

—

114

C₂×Q₈

D₂₁₂

yes

characteristic

115

C₂×D₄

—

116

C₂²×C₄

—

106

—

117

Q₁₆

—

106

—

118

Q₁₆

—

212

—

119

SD₁₆

—

212

—

120

D₈

—

106

—

121

C₂×C₈

C₂²×D₅₃

yes

characteristic

122

C₂×C₈

C₂²×D₅₃

yes

—

123

C₂×C₈

C₂²×D₅₃

yes

characteristic

124

C₄⋊C₄

—

125

C₄⋊C₄

—

106

—

126

C₄⋊C₄

—

106

—

127

C₄⋊C₄

—

106

—

128

C₄⋊C₄

D₂₁₂

yes

characteristic

129

C₄⋊C₄

D₂₁₂

yes

—

130

C₂²⋊C₄

—

106

—

131

C₂²⋊C₄

—

106

—

132

C₂²⋊C₄

—

106

—

133

C₄²

—

106

—

134

C₄²

C₂²×D₅₃

yes

characteristic

135

C₄²

C₂²×D₅₃

yes

—

136

C₂³

—

106

—

137

Q₈

—

138

Q₈

—

106

—

139

Q₈

—

106

—

140

D₄

—

106

—

141

D₄

—

106

—

142

C₂×C₄

—

106

—

143

C₂×C₄

—

106

—

144

C₂×C₄

—

212

—

145

C₂×C₄

C₂×D₂₁₂

yes

—

146

C₂×C₄

C₂×D₂₁₂

yes

characteristic

147

C₂×C₄

C₂×D₂₁₂

yes

—

148

C₂×C₄

C₂×D₂₁₂

yes

characteristic

149

C₂×C₄

C₂³×D₅₃

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

#11719949

yes

—

153

C₂²

—

212

—

154

C₂²

—

212

—

155

C₂²

—

212

—

156

C₂²

C₂²×D₂₁₂

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

212

—

161

C₄

—

212

—

162

C₄

#13133239

yes

—

163

C₄

C₂×#11719949

yes

characteristic

164

C₄

#13133263

yes

characteristic

165

C₂

—

424

—

166

C₂

#16811295

yes

characteristic

167

C₂

C₂×#13133239

yes

characteristic

168

C₂

#16811440

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₈×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	C₈×Q₈	D₅₃
hypocenter	C₅₃	#3393
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₈×Q₈×C₅₃

C₂

socle

C₂²×C₅₃

C₂²×D₄

hypercenter

C₈×Q₈

D₅₃

hypocenter

C₅₃

#3393

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
1. #16811346	C₂	semidirect
3. #16811003	C₂	semidirect
4. #16810846	C₂	semidirect
5. #16810829	C₂	semidirect
6. C₄×Dic₂₁₂	C₂	semidirect
8. C₈×Q₈×C₅₃	C₂	semidirect
90. C₈×Q₈	D₅₃	semidirect
93. C₅₃	#3393	semidirect

H ≅ G/N

type

1. #16811346

C₂

semidirect

3. #16811003

C₂

semidirect

4. #16810846

C₂

semidirect

5. #16810829

C₂

semidirect

6. C₄×Dic₂₁₂

C₂

semidirect

8. C₈×Q₈×C₅₃

C₂

semidirect

90. C₈×Q₈

D₅₃

semidirect

93. C₅₃

#3393

semidirect

Subgroups — #20287704

Conjugacy classes of subgroups

Complements