Group #27215638 — subgroups

Subgroups — C₅₃×#8126 #27215638

Order 13568.

520 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 230 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	C₅₃×#2920	C₂	1	3	yes	maximal
2	C₅₃×#1308	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
4	C₅₃×#432	—	2	3	no	—
5	C₅₃×#432	C₂²	1	6	yes	—
6	C₅₃×#431	—	2	3	no	—
7	C₅₃×#431	C₄	1	6	yes	—
8	C₅₃×#423	—	2	3	no	—
9	C₅₃×#407	C₄	1	6	yes	—
10	C₄×M₁₆×C₅₃	—	2	3	no	—
11	C₂³×C₄×C₅₃	C₂×C₄	1	3	yes	—
12	C₂×M₁₆×C₅₃	—	2	6	no	—
13	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
14	C₂×C₄²×C₅₃	D₄	1	3	yes	—
15	C₂×C₄²×C₅₃	D₄	1	3	yes	—
16	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
18	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
19	C₄⋊C₈×C₅₃	—	2	6	no	—
20	C₄⋊C₈×C₅₃	—	2	6	no	—
21	C₂²⋊C₈×C₅₃	—	2	6	no	—
22	C₂²⋊C₈×C₅₃	C₂×C₄	1	12	yes	—
23	#97×C₅₃	—	2	6	no	—
24	C₄×C₈×C₅₃	—	2	6	no	—
25	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	—	4	6	no	—
27	C₂²×C₄×C₅₃	C₄²	1	3	yes	—
28	C₂²×C₄×C₅₃	C₂²⋊C₄	1	3	yes	—
29	C₂²×C₄×C₅₃	C₂²⋊C₄	1	3	yes	—
30	C₂²×C₄×C₅₃	C₄⋊C₄	1	3	yes	—
31	C₂²×C₄×C₅₃	C₄⋊C₄	1	3	yes	—
32	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
33	M₁₆×C₅₃	—	4	12	no	—
34	C₂×C₈×C₅₃	—	2	12	no	—
35	C₄²×C₅₃	C₂²⋊C₄	1	3	yes	—
36	C₄²×C₅₃	C₂²⋊C₄	1	3	yes	—
37	C₄²×C₅₃	C₄⋊C₄	1	3	yes	—
38	C₄²×C₅₃	C₄⋊C₄	1	3	yes	—
39	C₄²×C₅₃	C₂×D₄	1	3	yes	—
40	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₂³×C₅₃	—	4	3	no	—
42	C₂³×C₅₃	#95	1	1	yes	characteristic
43	C₂³×C₅₃	#95	1	1	yes	characteristic
44	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
45	C₂×C₄×C₅₃	—	2	6	no	—
46	C₂×C₄×C₅₃	—	2	6	no	—
47	C₂×C₄×C₅₃	—	2	6	no	—
48	C₂×C₄×C₅₃	—	4	3	no	—
49	C₂×C₄×C₅₃	—	4	3	no	—
50	C₂×C₄×C₅₃	—	4	3	no	—
51	C₂×C₄×C₅₃	#95	1	3	yes	—
52	C₂×C₄×C₅₃	#95	1	3	yes	—
53	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
54	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
55	C₈×C₅₃	—	4	12	no	—
56	#8126	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂²×C₅₃	—	2	3	no	—
58	C₂²×C₅₃	—	2	3	no	—
59	C₂²×C₅₃	—	2	3	no	—
60	C₂²×C₅₃	—	4	4	no	—
61	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
62	C₄×C₅₃	—	2	3	no	—
63	C₄×C₅₃	—	2	3	no	—
64	C₄×C₅₃	—	2	3	no	—
65	C₄×C₅₃	#344	1	6	yes	—
66	#2920	C₂×C₅₃	1	3	yes	—
67	#1308	C₂×C₅₃	1	4	yes	complemented
68	C₂×C₅₃	—	4	1	no	—
69	C₂×C₅₃	—	4	1	no	—
70	C₂×C₅₃	—	4	1	no	—
71	C₂×C₅₃	C₂×#344	1	3	yes	—
72	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
73	#432	—	2	3	no	—
74	#432	C₂²×C₅₃	1	6	yes	—
75	#431	—	2	3	no	—
76	#431	C₄×C₅₃	1	6	yes	—
77	#423	—	2	3	no	—
78	#407	C₄×C₅₃	1	6	yes	—
79	C₄×M₁₆	—	2	3	no	—
80	C₅₃	#8126	1	1	yes	characteristic, direct product factor, Sylow
81	C₂³×C₄	C₂×C₄×C₅₃	1	3	yes	—
82	C₂×M₁₆	—	2	6	no	—
83	C₂×C₄²	C₂×C₄×C₅₃	1	3	yes	—
84	C₂×C₄²	D₄×C₅₃	1	3	yes	—
85	C₂×C₄²	D₄×C₅₃	1	3	yes	—
86	C₂×C₄²	Q₈×C₅₃	1	1	yes	characteristic
87	C₂×C₄²	Q₈×C₅₃	1	1	yes	characteristic
88	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
89	C₄⋊C₈	—	2	6	no	—
90	C₄⋊C₈	—	2	6	no	—
91	C₂²⋊C₈	—	2	6	no	—
92	C₂²⋊C₈	C₂×C₄×C₅₃	1	12	yes	—
93	#97	—	2	6	no	—
94	C₄×C₈	—	2	6	no	—
95	C₂⁴	C₄²×C₅₃	1	1	yes	characteristic
96	C₂²×C₄	—	4	6	no	—
97	C₂²×C₄	C₄²×C₅₃	1	3	yes	—
98	C₂²×C₄	C₂²⋊C₄×C₅₃	1	3	yes	—
99	C₂²×C₄	C₂²⋊C₄×C₅₃	1	3	yes	—
100	C₂²×C₄	C₄⋊C₄×C₅₃	1	3	yes	—
101	C₂²×C₄	C₄⋊C₄×C₅₃	1	3	yes	—
102	C₂²×C₄	C₂²×C₄×C₅₃	1	3	yes	—
103	M₁₆	—	4	12	no	—
104	C₂×C₈	—	2	12	no	—
105	C₄²	C₂²⋊C₄×C₅₃	1	3	yes	—
106	C₄²	C₂²⋊C₄×C₅₃	1	3	yes	—
107	C₄²	C₄⋊C₄×C₅₃	1	3	yes	—
108	C₄²	C₄⋊C₄×C₅₃	1	3	yes	—
109	C₄²	C₂×D₄×C₅₃	1	3	yes	—
110	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
111	C₂³	—	4	3	no	—
112	C₂³	#95×C₅₃	1	1	yes	characteristic
113	C₂³	#95×C₅₃	1	1	yes	characteristic
114	C₂³	C₂×C₄²×C₅₃	1	1	yes	characteristic
115	C₂×C₄	—	2	6	no	—
116	C₂×C₄	—	2	6	no	—
117	C₂×C₄	—	2	6	no	—
118	C₂×C₄	—	4	3	no	—
119	C₂×C₄	—	4	3	no	—
120	C₂×C₄	—	4	3	no	—
121	C₂×C₄	#95×C₅₃	1	3	yes	—
122	C₂×C₄	#95×C₅₃	1	3	yes	—
123	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	3	yes	—
124	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	3	yes	—
125	C₈	—	4	12	no	—
126	C₂²	—	2	3	no	—
127	C₂²	—	2	3	no	—
128	C₂²	—	2	3	no	—
129	C₂²	—	4	4	no	—
130	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
131	C₄	—	2	3	no	—
132	C₄	—	2	3	no	—
133	C₄	—	2	3	no	—
134	C₄	C₅₃×#344	1	6	yes	—
135	C₂	—	4	1	no	—
136	C₂	—	4	1	no	—
137	C₂	—	4	1	no	—
138	C₂	C₂×C₅₃×#344	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#2920

C₂

yes

maximal

C₅₃×#1308

C₂

yes

maximal, complemented

C₂²×C₄²×C₅₃

C₂²

yes

characteristic

C₅₃×#432

—

C₅₃×#432

C₂²

yes

—

C₅₃×#431

—

C₅₃×#431

C₄

yes

—

C₅₃×#423

—

C₅₃×#407

C₄

yes

—

C₄×M₁₆×C₅₃

—

C₂³×C₄×C₅₃

C₂×C₄

yes

—

C₂×M₁₆×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

Q₈

yes

characteristic

C₂×C₄²×C₅₃

Q₈

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

C₂×C₄

yes

—

#97×C₅₃

—

C₄×C₈×C₅₃

—

C₂⁴×C₅₃

C₄²

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₄²

yes

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂²⋊C₄

yes

—

C₄²×C₅₃

C₂²⋊C₄

yes

—

C₄²×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₂³×C₅₃

—

C₂³×C₅₃

yes

C₂³×C₅₃

yes

C₂³×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#95

yes

—

C₂×C₄×C₅₃

#95

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₈×C₅₃

—

#8126

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×#95

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#344

yes

—

#2920

C₂×C₅₃

yes

—

#1308

C₂×C₅₃

yes

complemented

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#344

yes

—

C₂²×C₄²

C₂²×C₅₃

yes

characteristic

#432

—

#432

C₂²×C₅₃

yes

—

#431

—

#431

C₄×C₅₃

yes

—

#423

—

#407

C₄×C₅₃

yes

—

C₄×M₁₆

—

C₅₃

#8126

yes

characteristic, direct product factor, Sylow

C₂³×C₄

C₂×C₄×C₅₃

yes

—

C₂×M₁₆

—

C₂×C₄²

C₂×C₄×C₅₃

yes

—

C₂×C₄²

D₄×C₅₃

yes

—

C₂×C₄²

D₄×C₅₃

yes

—

C₂×C₄²

Q₈×C₅₃

yes

characteristic

C₂×C₄²

Q₈×C₅₃

yes

characteristic

C₂×C₄²

C₂³×C₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

C₂×C₄×C₅₃

yes

—

#97

—

C₄×C₈

—

C₂⁴

C₄²×C₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄²×C₅₃

yes

—

C₂²×C₄

C₂²⋊C₄×C₅₃

yes

—

C₂²×C₄

C₂²⋊C₄×C₅₃

yes

—

100

C₂²×C₄

C₄⋊C₄×C₅₃

yes

—

101

C₂²×C₄

C₄⋊C₄×C₅₃

yes

—

102

C₂²×C₄

C₂²×C₄×C₅₃

yes

—

103

M₁₆

—

104

C₂×C₈

—

105

C₄²

C₂²⋊C₄×C₅₃

yes

—

106

C₄²

C₂²⋊C₄×C₅₃

yes

—

107

C₄²

C₄⋊C₄×C₅₃

yes

—

108

C₄²

C₄⋊C₄×C₅₃

yes

—

109

C₄²

C₂×D₄×C₅₃

yes

—

110

C₄²

C₂×Q₈×C₅₃

yes

characteristic

111

C₂³

—

112

C₂³

#95×C₅₃

yes

characteristic

113

C₂³

#95×C₅₃

yes

characteristic

114

C₂³

C₂×C₄²×C₅₃

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

#95×C₅₃

yes

—

122

C₂×C₄

#95×C₅₃

yes

—

123

C₂×C₄

C₂×C₂²⋊C₄×C₅₃

yes

—

124

C₂×C₄

C₂×C₄⋊C₄×C₅₃

yes

—

125

C₈

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

C₂×#95×C₅₃

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

C₅₃×#344

yes

—

135

C₂

—

136

C₂

—

137

C₂

—

138

C₂

C₂×C₅₃×#344

yes

—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#95
commutator	C₂³	C₂×C₄²×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂⁴×C₅₃	C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

C₂×#95

commutator

C₂³

C₂×C₄²×C₅₃

Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

prime radical

C₂⁴×C₅₃

C₄²

LCS

G ▷ C₂³ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
2. C₅₃×#1308	C₂	semidirect
56. #8126	C₅₃	direct (N × H)
67. #1308	C₂×C₅₃	semidirect
80. C₅₃	#8126	direct (N × H)

H ≅ G/N

type

2. C₅₃×#1308

C₂

semidirect

56. #8126

C₅₃

direct (N × H)

67. #1308

C₂×C₅₃

semidirect

80. C₅₃

#8126

direct (N × H)

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

subgroup	index
C₅₃×#2920	2
C₅₃×#1308	2
#8126	53

subgroup

index

C₅₃×#2920

C₅₃×#1308

#8126

Subgroups — C₅₃×#8126 #27215638

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#8126 #27215638

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8126 #27215638