Group #20275253 — subgroups

Subgroups — #20275253

Order 6784.

314 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 93 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	#16811101	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810640	C₂	1	2	yes	maximal
3	C₂×#13133136	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810379	C₂	1	2	yes	maximal, complemented
5	C₅₃×#536	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	2	yes	—
7	Q₈×Dic₅₃	—	2	4	no	—
8	#13133176	—	2	2	no	—
9	#13133176	C₂²	1	2	yes	—
10	#13133144	—	2	4	no	—
11	#13133136	—	2	2	no	—
12	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₄⋊Q₈×C₅₃	C₄	1	2	yes	complemented
15	#124×C₅₃	C₄	1	2	yes	complemented
16	#123×C₅₃	—	2	2	no	—
17	C₂²⋊Q₈×C₅₃	—	2	2	no	—
18	#117×C₅₃	C₂²	1	1	yes	characteristic
19	#41×C₅₃	—	2	2	no	—
20	#41×C₅₃	—	2	2	no	—
21	C₂×Q₈×C₅₃	—	2	2	no	—
22	C₂×Q₈×C₅₃	—	2	2	no	—
23	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₂×Q₈×C₅₃	D₄	1	2	yes	—
26	C₂×D₄×C₅₃	—	2	1	no	—
27	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	—	2	1	no	—
29	C₂²×C₄×C₅₃	D₄	1	2	yes	—
30	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₅₃	—	2	4	no	—
32	C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₂²⋊C₄×C₅₃	—	2	1	no	—
34	C₂²⋊C₄×C₅₃	—	2	4	no	—
35	C₄²×C₅₃	C₂×C₄	1	2	yes	—
36	C₂²⋊Dic₅₃	—	2	2	no	—
37	C₄⋊Dic₅₃	—	2	4	no	—
38	C₄⋊Dic₅₃	—	2	2	no	—
39	C₄×Dic₅₃	—	2	4	no	—
40	C₄×Dic₅₃	D₄	1	4	yes	—
41	#11719917	—	2	2	no	—
42	#11719917	—	2	2	no	—
43	C₂×#88322	—	2	2	no	—
44	C₂³×C₅₃	—	2	1	no	—
45	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₅₃	—	2	4	no	—
47	Q₈×C₅₃	—	2	4	no	—
48	Q₈×C₅₃	—	2	2	no	—
49	D₄×C₅₃	—	2	2	no	—
50	D₄×C₅₃	—	4	1	no	—
51	D₄×C₅₃	—	4	1	no	—
52	C₂×C₄×C₅₃	—	2	2	no	—
53	C₂×C₄×C₅₃	—	2	2	no	—
54	C₂×C₄×C₅₃	—	2	1	no	—
55	C₂×C₄×C₅₃	—	2	2	no	—
56	C₂×C₄×C₅₃	—	4	2	no	—
57	C₂×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
58	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#41	1	2	yes	—
62	C₂×Dic₅₃	—	2	4	no	—
63	#88322	—	4	2	no	—
64	C₂²×C₅₃	—	4	1	no	—
65	C₂²×C₅₃	—	4	2	no	—
66	C₂²×C₅₃	—	4	1	no	—
67	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₅₃	#120	1	1	yes	characteristic
69	C₂²×C₅₃	#123	1	1	yes	characteristic
70	C₄×C₅₃	—	2	4	no	—
71	C₄×C₅₃	—	4	2	no	—
72	C₄×C₅₃	—	4	1	no	—
73	C₄×C₅₃	C₄×D₄	1	2	yes	—
74	C₄×C₅₃	#121	1	2	yes	—
75	Dic₅₃	—	4	4	no	—
76	#1912	—	53	1	no	maximal, Sylow
77	C₂×C₅₃	—	2	1	no	—
78	C₂×C₅₃	—	2	1	no	—
79	C₂×C₅₃	—	2	1	no	—
80	C₂×C₅₃	—	8	1	no	—
81	C₂×C₅₃	#386	1	1	yes	characteristic
82	#536	D₅₃	1	1	yes	characteristic
83	#519	—	53	1	no	—
84	#421	—	53	2	no	—
85	C₂×#101	—	53	1	no	—
86	H(3,ℤ/4)	—	53	2	no	—
87	C₅₃	#1912	1	1	yes	characteristic, complemented, Sylow
88	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
89	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
90	C₂×M₁₆	—	53	2	no	—
91	C₄⋊Q₈	Dic₅₃	1	2	yes	complemented
92	#124	Dic₅₃	1	2	yes	complemented
93	#123	—	2	2	no	—
94	C₂²⋊Q₈	—	2	2	no	—
95	C₄×Q₈	—	106	4	no	—
96	#117	—	53	2	no	—
97	#117	—	106	2	no	—
98	#117	C₂×D₅₃	1	1	yes	characteristic
99	#104	—	106	4	no	—
100	#101	—	106	2	no	—
101	#41	—	2	2	no	—
102	#41	—	2	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
106	C₂×Q₈	C₂×Dic₅₃	1	2	yes	—
107	C₂×Q₈	C₂²⋊D₅₃	1	2	yes	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	C₂×Dic₅₃	1	1	yes	characteristic
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
112	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
113	M₁₆	—	106	2	no	—
114	M₁₆	—	106	2	no	—
115	C₂×C₈	—	106	2	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	2	4	no	—
118	C₄⋊C₄	—	106	4	no	—
119	C₄⋊C₄	—	106	2	no	—
120	C₂²⋊C₄	—	2	1	no	—
121	C₂²⋊C₄	—	2	4	no	—
122	C₂²⋊C₄	—	106	2	no	—
123	C₄²	—	53	4	no	—
124	C₄²	—	106	4	no	—
125	C₄²	C₂×Dic₅₃	1	2	yes	—
126	C₂³	—	2	1	no	—
127	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
128	Q₈	—	2	4	no	—
129	Q₈	—	2	4	no	—
130	Q₈	—	2	2	no	—
131	D₄	—	2	2	no	—
132	D₄	—	4	1	no	—
133	D₄	—	4	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	106	4	no	—
140	C₂×C₄	C₂²⋊Dic₅₃	1	4	yes	—
141	C₂×C₄	D₄×D₅₃	1	2	yes	—
142	C₂×C₄	#11719947	1	2	yes	—
143	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
145	C₈	—	212	2	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	4	2	no	—
148	C₂²	—	4	1	no	—
149	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
150	C₂²	#13133185	1	1	yes	characteristic
151	C₂²	#13133198	1	1	yes	characteristic
152	C₄	—	2	4	no	—
153	C₄	—	4	2	no	—
154	C₄	—	4	1	no	—
155	C₄	—	212	4	no	—
156	C₄	D₄×Dic₅₃	1	2	yes	—
157	C₄	#13133189	1	2	yes	—
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	—	2	1	no	—
161	C₂	—	8	1	no	—
162	C₂	#16810503	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811101

C₂

yes

maximal, characteristic, complemented

#16810640

C₂

yes

maximal

C₂×#13133136

C₂

yes

maximal, characteristic, complemented

#16810379

C₂

yes

maximal, complemented

C₅₃×#536

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

Q₈×Dic₅₃

—

#13133176

—

#13133176

C₂²

yes

—

#13133144

—

#13133136

—

C₂×#41×C₅₃

C₂²

yes

characteristic

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₄

yes

complemented

#124×C₅₃

C₄

yes

complemented

#123×C₅₃

—

C₂²⋊Q₈×C₅₃

—

#117×C₅₃

C₂²

yes

characteristic

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂×Q₈×C₅₃

D₄

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₄

yes

—

#11719917

—

#11719917

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

#121

yes

—

Dic₅₃

—

#1912

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#386

yes

characteristic

#536

D₅₃

yes

—

—

—

—

C₅₃

#1912

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₃

yes

characteristic

C₂²×Q₈

C₂×D₅₃

yes

characteristic

C₂×M₁₆

—

C₄⋊Q₈

Dic₅₃

yes

complemented

#124

Dic₅₃

yes

complemented

#123

—

C₂²⋊Q₈

—

C₄×Q₈

—

106

—

#117

—

#117

—

106

—

#117

C₂×D₅₃

yes

characteristic

#104

—

106

—

100

#101

—

106

—

101

#41

—

102

#41

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

C₂×Dic₅₃

yes

characteristic

106

C₂×Q₈

C₂×Dic₅₃

yes

—

107

C₂×Q₈

C₂²⋊D₅₃

yes

—

108

C₂×D₄

—

109

C₂×D₄

C₂×Dic₅₃

yes

characteristic

110

C₂²×C₄

—

111

C₂²×C₄

C₂²⋊D₅₃

yes

—

112

C₂²×C₄

C₂²×D₅₃

yes

characteristic

113

M₁₆

—

106

—

114

M₁₆

—

106

—

115

C₂×C₈

—

106

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

106

—

119

C₄⋊C₄

—

106

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

122

C₂²⋊C₄

—

106

—

123

C₄²

—

124

C₄²

—

106

—

125

C₄²

C₂×Dic₅₃

yes

—

126

C₂³

—

127

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

D₄

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

106

—

140

C₂×C₄

C₂²⋊Dic₅₃

yes

—

141

C₂×C₄

D₄×D₅₃

yes

—

142

C₂×C₄

#11719947

yes

—

143

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

145

C₈

—

212

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

150

C₂²

#13133185

yes

characteristic

151

C₂²

#13133198

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

212

—

156

C₄

D₄×Dic₅₃

yes

—

157

C₄

#13133189

yes

—

158

C₂

—

159

C₂

—

160

C₂

—

161

C₂

—

162

C₂

#16810503

yes

characteristic

	subgroup	G/H
center	C₂	#16810503
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#536	C₂
socle	C₂×C₅₃	#386
hypercenter	#536	D₅₃
hypocenter	C₅₃	#1912
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #536

subgroup

G/H

center

C₂

#16810503

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#536

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #536

N	H ≅ G/N	type
1. #16811101	C₂	semidirect
3. C₂×#13133136	C₂	semidirect
4. #16810379	C₂	semidirect
14. C₄⋊Q₈×C₅₃	C₄	semidirect
15. #124×C₅₃	C₄	semidirect
87. C₅₃	#1912	semidirect
91. C₄⋊Q₈	Dic₅₃	semidirect
92. #124	Dic₅₃	semidirect

H ≅ G/N

type

1. #16811101

C₂

semidirect

3. C₂×#13133136

C₂

semidirect

4. #16810379

C₂

semidirect

14. C₄⋊Q₈×C₅₃

C₄

semidirect

15. #124×C₅₃

C₄

semidirect

87. C₅₃

#1912

semidirect

91. C₄⋊Q₈

Dic₅₃

semidirect

92. #124

Dic₅₃

semidirect

Subgroups — #20275253

Conjugacy classes of subgroups

Complements