Group #20285215 — subgroups

Subgroups — #499×D₅₃ #20285215

Order 6784.

360 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 145 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	#125×D₅₃	C₂	1	2	yes	maximal
2	#16811025	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811024	C₂	1	2	yes	maximal, complemented
4	#16811023	C₂	1	2	yes	maximal, complemented
5	#16811022	C₂	1	2	yes	maximal, complemented
6	#107×D₅₃	C₂	1	2	yes	maximal
7	#106×D₅₃	C₂	1	2	yes	maximal
8	C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	#13133209	C₂²	1	2	yes	—
13	#13133208	C₂²	1	4	yes	—
14	#13133207	—	2	4	no	—
15	C₄⋊C₄×D₅₃	—	2	4	no	—
16	C₄⋊C₄×D₅₃	C₂²	1	4	yes	—
17	C₄²×D₅₃	C₂²	1	1	yes	characteristic
18	#13133151	C₂²	1	2	yes	—
19	#13133150	C₂²	1	4	yes	—
20	#13133149	C₂²	1	2	yes	—
21	#13133148	C₂²	1	4	yes	—
22	C₄×#88322	C₂²	1	1	yes	characteristic
23	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
24	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
25	#125×C₅₃	C₂²	1	2	yes	—
26	#107×C₅₃	C₂²	1	2	yes	—
27	#106×C₅₃	C₂²	1	2	yes	—
28	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₅₃	—	2	4	no	—
30	C₂×C₄×D₅₃	D₄	1	2	yes	—
31	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₅₃	—	2	4	no	—
35	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
36	C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	C₄⋊Dic₅₃	—	2	4	no	—
38	C₄⋊Dic₅₃	C₂³	1	4	yes	—
39	#11719919	—	2	4	no	—
40	#11719919	—	2	4	no	—
41	#11719919	—	2	4	no	—
42	C₄×Dic₅₃	D₄	1	2	yes	—
43	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
44	C₂×#88322	C₂³	1	1	yes	characteristic
45	C₂×#88322	C₂³	1	1	yes	characteristic
46	C₈×D₅₃	Q₈	1	4	yes	—
47	C₈×D₅₃	Q₈	1	4	yes	—
48	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	—	2	4	no	—
50	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
52	C₂×Dic₅₃	—	2	4	no	—
53	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
54	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
55	C₄×D₅₃	—	2	4	no	—
56	C₄×D₅₃	—	4	8	no	—
57	C₄×D₅₃	C₂×Q₈	1	2	yes	—
58	C₄×D₅₃	C₂×Q₈	1	2	yes	—
59	C₈×C₅₃	C₂×Q₈	1	2	yes	—
60	C₈×C₅₃	C₂×Q₈	1	2	yes	—
61	#88322	C₂×Q₈	1	2	yes	—
62	#88322	C₂×Q₈	1	2	yes	—
63	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
64	C₂×D₅₃	C₄⋊Q₈	1	2	yes	—
65	C₂×D₅₃	#135	1	4	yes	—
66	C₄×C₅₃	—	2	2	no	—
67	C₄×C₅₃	—	4	4	no	—
68	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
69	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
70	Dic₅₃	—	2	2	no	—
71	Dic₅₃	—	4	4	no	—
72	Dic₅₃	C₄⋊Q₈	1	2	yes	—
73	C₂×#499	—	53	1	no	maximal, Sylow
74	C₂×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₅₃	C₂×#135	1	2	yes	—
76	D₅₃	#499	1	4	yes	direct product factor
77	C₂×#125	—	53	2	no	—
78	#499	—	53	2	no	—
79	#499	—	53	2	no	—
80	#499	—	53	2	no	—
81	#499	—	53	1	no	—
82	#499	D₅₃	1	1	yes	characteristic, direct product factor
83	C₂×#107	—	53	2	no	—
84	C₂×#106	—	53	2	no	—
85	C₂×C₄×C₈	—	53	1	no	—
86	C₅₃	C₂×#499	1	1	yes	characteristic, complemented, Sylow
87	C₂²×C₈	—	53	1	no	—
88	C₂²×C₈	—	53	1	no	—
89	#125	—	53	4	no	—
90	#125	—	53	2	no	—
91	#125	—	106	4	no	—
92	#125	C₂×D₅₃	1	2	yes	—
93	C₂×C₄⋊C₄	—	53	4	no	—
94	C₂×C₄⋊C₄	—	106	4	no	—
95	C₂×C₄²	—	53	1	no	—
96	#107	—	53	2	no	—
97	#107	—	53	4	no	—
98	#107	C₂×D₅₃	1	2	yes	—
99	#106	—	53	2	no	—
100	#106	—	53	4	no	—
101	#106	C₂×D₅₃	1	2	yes	—
102	C₄×C₈	—	53	1	no	—
103	C₄×C₈	—	53	1	no	—
104	C₄×C₈	—	53	1	no	—
105	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
106	C₂²×C₄	—	53	1	no	—
107	C₂²×C₄	—	53	2	no	—
108	C₂²×C₄	—	106	4	no	—
109	C₂×C₈	—	53	1	no	—
110	C₂×C₈	—	53	1	no	—
111	C₂×C₈	—	53	4	no	—
112	C₂×C₈	—	53	4	no	—
113	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
114	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	53	4	no	—
117	C₄⋊C₄	—	106	4	no	—
118	C₄⋊C₄	—	106	4	no	—
119	C₄⋊C₄	—	106	4	no	—
120	C₄⋊C₄	—	106	4	no	—
121	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
122	C₄²	—	53	2	no	—
123	C₄²	—	53	1	no	—
124	C₄²	C₂²×D₅₃	1	1	yes	characteristic
125	C₂³	—	53	1	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	53	1	no	—
128	C₂×C₄	—	53	2	no	—
129	C₂×C₄	—	53	2	no	—
130	C₂×C₄	—	53	2	no	—
131	C₂×C₄	—	106	4	no	—
132	C₂×C₄	—	106	4	no	—
133	C₂×C₄	—	212	8	no	—
134	C₂×C₄	D₄×D₅₃	1	2	yes	—
135	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
136	C₈	—	53	2	no	—
137	C₈	—	53	2	no	—
138	C₈	Q₈×D₅₃	1	2	yes	—
139	C₈	Q₈×D₅₃	1	2	yes	—
140	C₂²	—	53	4	no	—
141	C₂²	—	53	2	no	—
142	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	4	4	no	—
145	C₄	—	53	2	no	—
146	C₄	—	106	2	no	—
147	C₄	—	212	4	no	—
148	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
149	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
150	C₂	—	53	4	no	—
151	C₂	C₄⋊Q₈×D₅₃	1	1	yes	characteristic
152	C₂	#135×D₅₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#125×D₅₃

C₂

yes

maximal

#16811025

C₂

yes

maximal, characteristic, complemented

#16811024

C₂

yes

maximal, complemented

#16811023

C₂

yes

maximal, complemented

#16811022

C₂

yes

maximal, complemented

#107×D₅₃

C₂

yes

maximal

#106×D₅₃

C₂

yes

maximal

C₄×C₈×D₅₃

C₂

yes

maximal, characteristic

C₅₃×#499

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₅₃

C₂²

yes

characteristic

C₂×C₈×D₅₃

C₂²

yes

characteristic

#13133209

C₂²

yes

—

#13133208

C₂²

yes

—

#13133207

—

C₄⋊C₄×D₅₃

—

C₄⋊C₄×D₅₃

C₂²

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

#13133151

C₂²

yes

—

#13133150

C₂²

yes

—

#13133149

C₂²

yes

—

#13133148

C₂²

yes

—

C₄×#88322

C₂²

yes

characteristic

C₈×Dic₅₃

C₂²

yes

characteristic

C₈×Dic₅₃

C₂²

yes

characteristic

#125×C₅₃

C₂²

yes

—

#107×C₅₃

C₂²

yes

—

#106×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

D₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

#11719919

—

#11719919

—

C₄×Dic₅₃

D₄

yes

—

C₄×Dic₅₃

C₂³

yes

characteristic

C₂×#88322

C₂³

yes

characteristic

C₂×#88322

C₂³

yes

characteristic

C₈×D₅₃

Q₈

yes

—

C₈×D₅₃

Q₈

yes

—

C₂²×D₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂×Q₈

yes

—

C₄×D₅₃

C₂×Q₈

yes

—

C₈×C₅₃

C₂×Q₈

yes

—

C₈×C₅₃

C₂×Q₈

yes

—

#88322

C₂×Q₈

yes

—

#88322

C₂×Q₈

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

C₄⋊Q₈

yes

—

C₂×D₅₃

#135

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²×Q₈

yes

characteristic

C₄×C₅₃

C₂²×Q₈

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₄⋊Q₈

yes

—

C₂×#499

—

maximal, Sylow

C₂×C₅₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₅₃

C₂×#135

yes

—

D₅₃

#499

yes

direct product factor

—

—

—

—

—

D₅₃

yes

characteristic, direct product factor

C₂×#107

—

C₂×#106

—

C₂×C₄×C₈

—

C₅₃

C₂×#499

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

#125

—

#125

—

#125

—

106

—

#125

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

—

#107

—

#107

—

#107

C₂×D₅₃

yes

—

#106

—

100

#106

—

101

#106

C₂×D₅₃

yes

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

C₂×D₅₃

yes

characteristic

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

106

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₂²×D₅₃

yes

characteristic

114

C₂×C₈

C₂²×D₅₃

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

106

—

118

C₄⋊C₄

—

106

—

119

C₄⋊C₄

—

106

—

120

C₄⋊C₄

—

106

—

121

C₄⋊C₄

C₂²×D₅₃

yes

—

122

C₄²

—

123

C₄²

—

124

C₄²

C₂²×D₅₃

yes

characteristic

125

C₂³

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

106

—

132

C₂×C₄

—

106

—

133

C₂×C₄

—

212

—

134

C₂×C₄

D₄×D₅₃

yes

—

135

C₂×C₄

C₂³×D₅₃

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

Q₈×D₅₃

yes

—

139

C₈

Q₈×D₅₃

yes

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₂×D₄×D₅₃

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

106

—

147

C₄

—

212

—

148

C₄

C₂×Q₈×D₅₃

yes

characteristic

149

C₄

C₂×Q₈×D₅₃

yes

characteristic

150

C₂

—

151

C₂

C₄⋊Q₈×D₅₃

yes

characteristic

152

C₂

#135×D₅₃

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#499	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#499	D₅₃
hypocenter	C₅₃	C₂×#499
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

subgroup

G/H

center

C₂²

C₂×D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#499

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #499

N	H ≅ G/N	type
2. #16811025	C₂	semidirect
3. #16811024	C₂	semidirect
4. #16811023	C₂	semidirect
5. #16811022	C₂	semidirect
9. C₅₃×#499	C₂	semidirect
76. D₅₃	#499	direct (N × H)
82. #499	D₅₃	direct (N × H)
86. C₅₃	C₂×#499	semidirect

H ≅ G/N

type

2. #16811025

C₂

semidirect

3. #16811024

C₂

semidirect

4. #16811023

C₂

semidirect

5. #16811022

C₂

semidirect

9. C₅₃×#499

C₂

semidirect

76. D₅₃

#499

direct (N × H)

82. #499

D₅₃

direct (N × H)

86. C₅₃

C₂×#499

semidirect

Subgroups — #499×D₅₃ #20285215

Conjugacy classes of subgroups

Complements

Subgroups — #499×D53 #20285215

Conjugacy classes of subgroups

Complements

Subgroups — #499×D₅₃ #20285215