Group #20274430 — subgroups

Subgroups — #20274430

Order 6784.

424 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 153 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	C₂²×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133175	C₂	1	2	yes	maximal
3	#16810386	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810385	C₂	1	2	yes	maximal, complemented
5	C₅₃×#340	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
7	C₂²×#88322	C₂²	1	1	yes	characteristic
8	C₂×C₈×D₅₃	—	2	4	no	—
9	C₂×C₈×D₅₃	C₄	1	4	yes	complemented
10	#13133175	—	2	4	no	—
11	#13133175	C₄	1	8	yes	complemented
12	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
13	C₂×#11719921	—	2	2	no	—
14	C₂×C₄×Dic₅₃	—	2	2	no	—
15	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
17	C₂³×D₅₃	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	—	2	2	no	—
19	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₅₃	—	2	2	no	—
21	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
22	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
23	C₂×C₄×D₅₃	C₂×C₄	1	4	yes	—
24	C₂×C₄×D₅₃	D₄	1	4	yes	—
25	C₂×C₄×D₅₃	Q₈	1	2	yes	—
26	C₂×C₈×C₅₃	—	2	2	no	—
27	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
28	C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
30	#11719921	—	2	8	no	—
31	C₄⋊Dic₅₃	—	2	2	no	—
32	C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
33	C₄×Dic₅₃	—	2	8	no	—
34	C₂×#88322	—	2	2	no	—
35	C₂×#88322	C₂×C₄	1	2	yes	—
36	C₈×D₅₃	—	2	8	no	—
37	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×D₅₃	—	2	1	no	—
39	C₂²×D₅₃	—	2	2	no	—
40	C₂²×D₅₃	—	2	1	no	—
41	C₂²×D₅₃	C₂²⋊C₄	1	4	yes	—
42	C₂²×D₅₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₅₃	—	2	4	no	—
44	C₂×C₄×C₅₃	—	4	2	no	—
45	C₂×C₄×C₅₃	—	4	2	no	—
46	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×Dic₅₃	—	2	4	no	—
51	C₂×Dic₅₃	—	4	2	no	—
52	C₂×Dic₅₃	—	4	2	no	—
53	C₂×Dic₅₃	C₂²⋊C₄	1	4	yes	—
54	C₂×Dic₅₃	C₄⋊C₄	1	2	yes	—
55	C₄×D₅₃	C₄²	1	4	yes	—
56	C₄×D₅₃	C₂²⋊C₄	1	4	yes	—
57	C₄×D₅₃	C₄⋊C₄	1	8	yes	—
58	C₈×C₅₃	—	2	2	no	—
59	#88322	—	2	2	no	—
60	C₂²×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
61	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₅₃	#135	1	1	yes	characteristic
63	C₂²×C₅₃	#135	1	2	yes	—
64	C₂²×C₅₃	#135	1	1	yes	characteristic
65	C₂×D₅₃	—	2	4	no	—
66	C₂×D₅₃	—	2	4	no	—
67	C₂×D₅₃	—	2	2	no	—
68	C₂×D₅₃	—	2	2	no	—
69	C₂×D₅₃	#95	1	4	yes	—
70	C₄×C₅₃	—	4	4	no	—
71	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
72	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
74	Dic₅₃	—	4	4	no	—
75	Dic₅₃	#95	1	4	yes	—
76	#1735	—	53	1	no	maximal, Sylow
77	C₂×C₅₃	C₂×#95	1	1	yes	characteristic
78	C₂×C₅₃	#416	1	2	yes	—
79	C₂×C₅₃	#416	1	2	yes	—
80	C₂×C₅₃	#427	1	1	yes	characteristic
81	C₂×C₅₃	#427	1	1	yes	characteristic
82	D₅₃	—	2	4	no	—
83	C₂³×C₈	—	53	1	no	—
84	C₂×#117	—	53	2	no	—
85	#340	—	53	1	no	—
86	#340	—	53	2	no	—
87	#340	D₅₃	1	1	yes	characteristic, complemented
88	C₅₃	#1735	1	1	yes	characteristic, complemented, Sylow
89	C₂³×C₄	—	53	1	no	—
90	C₂²×C₈	—	53	1	no	—
91	C₂²×C₈	—	53	4	no	—
92	C₂²×C₈	—	106	4	no	—
93	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
94	#117	—	53	8	no	—
95	#117	—	106	4	no	—
96	C₂×C₄⋊C₄	—	53	2	no	—
97	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
98	C₂×C₂²⋊C₄	—	106	2	no	—
99	C₂×C₄²	—	106	2	no	—
100	C₂⁴	—	53	1	no	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	—	53	1	no	—
103	C₂²×C₄	—	53	4	no	—
104	C₂²×C₄	—	53	2	no	—
105	C₂²×C₄	—	53	4	no	—
106	C₂²×C₄	—	53	2	no	—
107	C₂²×C₄	—	106	2	no	—
108	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	53	2	no	—
111	C₂×C₈	—	106	8	no	—
112	C₂×C₈	—	106	2	no	—
113	C₂×C₈	C₄×D₅₃	1	2	yes	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	53	4	no	—
116	C₄⋊C₄	—	106	2	no	—
117	C₄⋊C₄	C₄×D₅₃	1	4	yes	—
118	C₂²⋊C₄	—	106	8	no	—
119	C₄²	—	106	8	no	—
120	C₂³	—	53	4	no	—
121	C₂³	—	53	2	no	—
122	C₂³	—	106	1	no	—
123	C₂³	—	106	2	no	—
124	C₂³	—	106	1	no	—
125	C₂³	D₄×D₅₃	1	1	yes	characteristic
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	53	4	no	—
130	C₂×C₄	—	53	2	no	—
131	C₂×C₄	—	53	4	no	—
132	C₂×C₄	—	53	4	no	—
133	C₂×C₄	—	53	8	no	—
134	C₂×C₄	—	106	4	no	—
135	C₂×C₄	—	212	2	no	—
136	C₂×C₄	—	212	2	no	—
137	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
138	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
139	C₂×C₄	D₄×D₅₃	1	2	yes	—
140	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	106	2	no	—
143	C₂²	—	53	4	no	—
144	C₂²	—	106	4	no	—
145	C₂²	—	106	4	no	—
146	C₂²	—	106	2	no	—
147	C₂²	—	106	2	no	—
148	C₂²	C₂²⋊C₄×D₅₃	1	2	yes	—
149	C₂²	C₄⋊C₄×D₅₃	1	1	yes	characteristic
150	C₂²	#13133236	1	1	yes	characteristic
151	C₂²	#13133238	1	2	yes	—
152	C₂²	#13133240	1	1	yes	characteristic
153	C₄	—	4	4	no	—
154	C₄	—	53	4	no	—
155	C₄	—	212	4	no	—
156	C₄	C₄²×D₅₃	1	1	yes	characteristic
157	C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
158	C₄	C₄⋊C₄×D₅₃	1	2	yes	—
159	C₂	—	106	4	no	—
160	C₂	#95×D₅₃	1	1	yes	characteristic
161	C₂	#16810616	1	2	yes	—
162	C₂	#16810618	1	2	yes	—
163	C₂	#16810658	1	1	yes	characteristic
164	C₂	#16810660	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×D₅₃

C₂

yes

maximal, characteristic

C₂×#13133175

C₂

yes

maximal

#16810386

C₂

yes

maximal, characteristic, complemented

#16810385

C₂

yes

maximal, complemented

C₅₃×#340

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₅₃

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

C₄

yes

complemented

#13133175

—

#13133175

C₄

yes

complemented

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719921

—

C₂×C₄×Dic₅₃

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂³×D₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

D₄

yes

—

C₂×C₄×D₅₃

Q₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

C₂×C₄

yes

—

C₈×D₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₂²⋊C₄

yes

—

C₂²×D₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²⋊C₄

yes

—

C₂×Dic₅₃

C₄⋊C₄

yes

—

C₄×D₅₃

C₄²

yes

—

C₄×D₅₃

C₂²⋊C₄

yes

—

C₄×D₅₃

C₄⋊C₄

yes

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#135

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#95

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄²

yes

characteristic

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

C₂×C₄⋊C₄

yes

—

Dic₅₃

—

Dic₅₃

#95

yes

—

#1735

—

maximal, Sylow

C₂×C₅₃

C₂×#95

yes

characteristic

C₂×C₅₃

#416

yes

—

C₂×C₅₃

#416

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₂³×C₈

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#1735

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

106

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

#117

—

#117

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₂²⋊C₄

—

106

—

C₂×C₄²

—

106

—

100

C₂⁴

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

106

—

108

C₂²×C₄

C₂²×D₅₃

yes

characteristic

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

106

—

112

C₂×C₈

—

106

—

113

C₂×C₈

C₄×D₅₃

yes

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

106

—

117

C₄⋊C₄

C₄×D₅₃

yes

—

118

C₂²⋊C₄

—

106

—

119

C₄²

—

106

—

120

C₂³

—

121

C₂³

—

122

C₂³

—

106

—

123

C₂³

—

106

—

124

C₂³

—

106

—

125

C₂³

D₄×D₅₃

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

106

—

135

C₂×C₄

—

212

—

136

C₂×C₄

—

212

—

137

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

138

C₂×C₄

C₂×C₄×D₅₃

yes

—

139

C₂×C₄

D₄×D₅₃

yes

—

140

C₂×C₄

Q₈×D₅₃

yes

characteristic

141

C₈

—

142

C₈

—

106

—

143

C₂²

—

144

C₂²

—

106

—

145

C₂²

—

106

—

146

C₂²

—

106

—

147

C₂²

—

106

—

148

C₂²

C₂²⋊C₄×D₅₃

yes

—

149

C₂²

C₄⋊C₄×D₅₃

yes

characteristic

150

C₂²

#13133236

yes

characteristic

151

C₂²

#13133238

yes

—

152

C₂²

#13133240

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

212

—

156

C₄

C₄²×D₅₃

yes

characteristic

157

C₄

C₂²⋊C₄×D₅₃

yes

characteristic

158

C₄

C₄⋊C₄×D₅₃

yes

—

159

C₂

—

106

—

160

C₂

#95×D₅₃

yes

characteristic

161

C₂

#16810616

yes

—

162

C₂

#16810618

yes

—

163

C₂

#16810658

yes

characteristic

164

C₂

#16810660

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₅₃
commutator	C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#340	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	#340	D₅₃
hypocenter	C₅₃	#1735
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₅₃

commutator

C₄×C₅₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#340

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
3. #16810386	C₂	semidirect
4. #16810385	C₂	semidirect
5. C₅₃×#340	C₂	semidirect
9. C₂×C₈×D₅₃	C₄	semidirect
11. #13133175	C₄	semidirect
87. #340	D₅₃	semidirect
88. C₅₃	#1735	semidirect

H ≅ G/N

type

3. #16810386

C₂

semidirect

4. #16810385

C₂

semidirect

5. C₅₃×#340

C₂

semidirect

9. C₂×C₈×D₅₃

C₄

semidirect

11. #13133175

C₄

semidirect

87. #340

D₅₃

semidirect

88. C₅₃

#1735

semidirect

Subgroups — #20274430

Conjugacy classes of subgroups

Complements