Group #20270962 — subgroups

Subgroups — C₅₃×#2025 #20270962

Order 6784.

294 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 100 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×#125×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#102×C₅₃	C₂	1	2	yes	maximal
3	C₅₃×#390	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#340	C₂	1	2	yes	maximal, complemented
5	C₅₃×#336	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
8	#125×C₅₃	—	2	4	no	—
9	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
10	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
12	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
13	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	#102×C₅₃	—	2	4	no	—
15	C₂⁴×C₅₃	—	2	1	no	—
16	C₂×D₄×C₅₃	—	2	2	no	—
17	C₂×D₄×C₅₃	—	2	2	no	—
18	C₂×D₄×C₅₃	—	2	1	no	—
19	C₂×D₄×C₅₃	Q₈	1	2	yes	—
20	C₂²×C₄×C₅₃	—	2	2	no	—
21	C₂²×C₄×C₅₃	—	2	1	no	—
22	C₂²×C₄×C₅₃	D₄	1	2	yes	—
23	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₅₃	—	2	2	no	—
25	C₂×C₈×C₅₃	—	2	2	no	—
26	C₂×C₈×C₅₃	—	2	2	no	—
27	C₄⋊C₄×C₅₃	—	2	8	no	—
28	C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₄⋊C₄×C₅₃	D₄	1	4	yes	—
31	C₂²⋊C₄×C₅₃	—	4	4	no	—
32	C₄²×C₅₃	—	2	2	no	—
33	C₂³×C₅₃	—	2	2	no	—
34	C₂³×C₅₃	—	4	1	no	—
35	C₂³×C₅₃	—	4	1	no	—
36	C₂³×C₅₃	—	4	2	no	—
37	C₂³×C₅₃	—	4	2	no	—
38	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
39	D₄×C₅₃	—	2	4	no	—
40	D₄×C₅₃	—	4	2	no	—
41	C₂×C₄×C₅₃	—	2	4	no	—
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	—	2	2	no	—
44	C₂×C₄×C₅₃	—	2	2	no	—
45	C₂×C₄×C₅₃	—	4	2	no	—
46	C₂×C₄×C₅₃	—	4	2	no	—
47	C₂×C₄×C₅₃	—	4	1	no	—
48	C₂×C₄×C₅₃	—	4	2	no	—
49	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	#41	1	2	yes	—
52	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
53	C₈×C₅₃	—	4	2	no	—
54	C₂²×C₅₃	—	4	2	no	—
55	C₂²×C₅₃	—	4	4	no	—
56	C₂²×C₅₃	—	8	2	no	—
57	C₂²×C₅₃	—	8	1	no	—
58	C₂²×C₅₃	—	8	1	no	—
59	C₂²×C₅₃	#120	1	1	yes	characteristic
60	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂²×C₅₃	#124	1	1	yes	characteristic
62	C₂²×C₅₃	#135	1	2	yes	—
63	C₂²×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
64	C₄×C₅₃	—	4	2	no	—
65	C₄×C₅₃	—	4	4	no	—
66	C₄×C₅₃	—	8	1	no	—
67	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
68	C₄×C₅₃	#124	1	2	yes	—
69	#2025	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂×C₅₃	—	8	2	no	—
71	C₂×C₅₃	#393	1	1	yes	characteristic
72	C₂×C₅₃	#449	1	2	yes	—
73	C₂×C₅₃	#478	1	2	yes	—
74	C₂×C₅₃	#488	1	1	yes	characteristic
75	C₂×C₅₃	#490	1	1	yes	characteristic
76	C₂×#125	C₂×C₅₃	1	1	yes	characteristic, complemented
77	C₂×#102	C₂×C₅₃	1	2	yes	—
78	#390	C₂×C₅₃	1	1	yes	characteristic
79	#340	C₂×C₅₃	1	2	yes	complemented
80	#336	C₂×C₅₃	1	1	yes	characteristic, complemented
81	C₅₃	#2025	1	1	yes	characteristic, direct product factor, Sylow
82	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
83	C₂²×C₈	C₂²×C₅₃	1	2	yes	—
84	#125	—	2	4	no	—
85	C₂×C₄⋊C₄	—	2	2	no	—
86	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
87	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
88	C₂×C₂²⋊C₄	—	2	2	no	—
89	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
90	#102	—	2	4	no	—
91	C₂⁴	—	2	1	no	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	Q₈×C₅₃	1	2	yes	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	D₄×C₅₃	1	2	yes	—
99	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₄⋊C₄	—	2	8	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	D₄×C₅₃	1	4	yes	—
107	C₂²⋊C₄	—	4	4	no	—
108	C₄²	—	2	2	no	—
109	C₂³	—	2	2	no	—
110	C₂³	—	4	1	no	—
111	C₂³	—	4	1	no	—
112	C₂³	—	4	2	no	—
113	C₂³	—	4	2	no	—
114	C₂³	C₂×D₄×C₅₃	1	1	yes	characteristic
115	D₄	—	2	4	no	—
116	D₄	—	4	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
126	C₂×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
127	C₂×C₄	#41×C₅₃	1	2	yes	—
128	C₂×C₄	#41×C₅₃	1	1	yes	characteristic
129	C₈	—	4	2	no	—
130	C₂²	—	4	2	no	—
131	C₂²	—	4	4	no	—
132	C₂²	—	8	1	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	2	no	—
135	C₂²	#120×C₅₃	1	1	yes	characteristic
136	C₂²	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
137	C₂²	#124×C₅₃	1	1	yes	characteristic
138	C₂²	#135×C₅₃	1	2	yes	—
139	C₂²	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
140	C₄	—	4	2	no	—
141	C₄	—	4	4	no	—
142	C₄	—	8	1	no	—
143	C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
144	C₄	#124×C₅₃	1	2	yes	—
145	C₂	—	8	2	no	—
146	C₂	C₅₃×#393	1	1	yes	characteristic
147	C₂	C₅₃×#449	1	2	yes	—
148	C₂	C₅₃×#478	1	2	yes	—
149	C₂	C₅₃×#488	1	1	yes	characteristic
150	C₂	C₅₃×#490	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#125×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×#102×C₅₃

C₂

yes

maximal

C₅₃×#390

C₂

yes

maximal, characteristic

C₅₃×#340

C₂

yes

maximal, complemented

C₅₃×#336

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

—

#125×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#102×C₅₃

—

C₂⁴×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

#41

yes

characteristic

C₈×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#120

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

AGL(1,ℤ/8)

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

#124

yes

—

#2025

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₅₃

—

C₂×C₅₃

#393

yes

characteristic

C₂×C₅₃

#449

yes

—

C₂×C₅₃

#478

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

characteristic, complemented

C₂×#102

C₂×C₅₃

yes

—

#390

C₂×C₅₃

yes

characteristic

#340

C₂×C₅₃

yes

complemented

#336

C₂×C₅₃

yes

characteristic, complemented

C₅₃

#2025

yes

characteristic, direct product factor, Sylow

C₂²×D₄

C₂²×C₅₃

yes

characteristic

C₂²×C₈

C₂²×C₅₃

yes

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₅₃

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₅₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂²×C₅₃

yes

characteristic

#102

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

Q₈×C₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₅₃

yes

—

C₂²×C₄

C₂³×C₅₃

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

D₄×C₅₃

yes

—

107

C₂²⋊C₄

—

108

C₄²

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

—

114

C₂³

C₂×D₄×C₅₃

yes

characteristic

115

D₄

—

116

D₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₂×D₄×C₅₃

yes

—

126

C₂×C₄

C₂×Q₈×C₅₃

yes

characteristic

127

C₂×C₄

#41×C₅₃

yes

—

128

C₂×C₄

#41×C₅₃

yes

characteristic

129

C₈

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

#120×C₅₃

yes

characteristic

136

C₂²

C₂²⋊Q₈×C₅₃

yes

characteristic

137

C₂²

#124×C₅₃

yes

characteristic

138

C₂²

#135×C₅₃

yes

—

139

C₂²

AGL(1,ℤ/8)×C₅₃

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

C₂²⋊Q₈×C₅₃

yes

—

144

C₄

#124×C₅₃

yes

—

145

C₂

—

146

C₂

C₅₃×#393

yes

characteristic

147

C₂

C₅₃×#449

yes

—

148

C₂

C₅₃×#478

yes

—

149

C₂

C₅₃×#488

yes

characteristic

150

C₂

C₅₃×#490

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₅₃
prime radical	C₂²×D₄×C₅₃	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂³×C₅₃

C₂×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₅₃

prime radical

C₂²×D₄×C₅₃

C₂²

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
1. C₂×#125×C₅₃	C₂	semidirect
4. C₅₃×#340	C₂	semidirect
5. C₅₃×#336	C₂	semidirect
69. #2025	C₅₃	direct (N × H)
76. C₂×#125	C₂×C₅₃	semidirect
79. #340	C₂×C₅₃	semidirect
80. #336	C₂×C₅₃	semidirect
81. C₅₃	#2025	direct (N × H)

H ≅ G/N

type

1. C₂×#125×C₅₃

C₂

semidirect

4. C₅₃×#340

C₂

semidirect

5. C₅₃×#336

C₂

semidirect

69. #2025

C₅₃

direct (N × H)

76. C₂×#125

C₂×C₅₃

semidirect

79. #340

C₂×C₅₃

semidirect

80. #336

C₂×C₅₃

semidirect

81. C₅₃

#2025

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#125×C₅₃	2
C₂×#102×C₅₃	2
C₅₃×#390	2
C₅₃×#340	2
C₅₃×#336	2
#2025	53

subgroup

index

C₂×#125×C₅₃

C₂×#102×C₅₃

C₅₃×#390

C₅₃×#340

C₅₃×#336

#2025

Subgroups — C₅₃×#2025 #20270962

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#2025 #20270962

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2025 #20270962