Group #20270823 — subgroups

Subgroups — C₅₃×#1886 #20270823

Order 6784.

324 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 112 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×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#519	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#419	C₂	1	2	yes	maximal
4	C₂×#101×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#339	C₂	1	2	yes	maximal
6	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₅₃	—	2	2	no	—
10	C₄⋊Q₈×C₅₃	C₄	1	4	yes	complemented
11	C₄×Q₈×C₅₃	—	2	4	no	—
12	#117×C₅₃	—	2	2	no	—
13	#117×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	#103×C₅₃	—	2	4	no	—
17	#101×C₅₃	—	2	2	no	—
18	C₂×Q₈×C₅₃	—	2	2	no	—
19	C₂×Q₈×C₅₃	—	2	1	no	—
20	C₂×Q₈×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	2	yes	—
24	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₂×Q₈×C₅₃	D₄	1	2	yes	—
26	C₂²×C₄×C₅₃	—	2	1	no	—
27	C₂²×C₄×C₅₃	—	2	1	no	—
28	C₂²×C₄×C₅₃	D₄	1	2	yes	—
29	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
30	M₁₆×C₅₃	—	2	2	no	—
31	M₁₆×C₅₃	—	2	2	no	—
32	C₂×C₈×C₅₃	—	2	2	no	—
33	C₄⋊C₄×C₅₃	—	2	4	no	—
34	C₄⋊C₄×C₅₃	—	2	8	no	—
35	C₄⋊C₄×C₅₃	D₄	1	4	yes	—
36	C₂²⋊C₄×C₅₃	—	2	2	no	—
37	C₄²×C₅₃	—	2	4	no	—
38	C₄²×C₅₃	—	2	1	no	—
39	C₄²×C₅₃	—	2	2	no	—
40	C₄²×C₅₃	C₂×C₄	1	2	yes	—
41	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₅₃	—	2	4	no	—
43	Q₈×C₅₃	—	2	4	no	—
44	Q₈×C₅₃	—	2	4	no	—
45	Q₈×C₅₃	—	4	2	no	—
46	C₂×C₄×C₅₃	—	2	4	no	—
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	—	2	2	no	—
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	—	2	4	no	—
52	C₂×C₄×C₅₃	—	4	1	no	—
53	C₂×C₄×C₅₃	—	4	1	no	—
54	C₂×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
55	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#41	1	2	yes	—
59	C₈×C₅₃	—	4	2	no	—
60	C₂²×C₅₃	—	2	2	no	—
61	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₅₃	#120	1	1	yes	characteristic
63	C₂²×C₅₃	#123	1	1	yes	characteristic
64	C₄×C₅₃	—	2	4	no	—
65	C₄×C₅₃	—	2	4	no	—
66	C₄×C₅₃	—	4	2	no	—
67	C₄×C₅₃	—	4	4	no	—
68	C₄×C₅₃	C₄×D₄	1	2	yes	—
69	C₄×C₅₃	#121	1	2	yes	—
70	#1886	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂×C₅₃	—	2	1	no	—
72	C₂×C₅₃	—	2	1	no	—
73	C₂×C₅₃	#386	1	1	yes	characteristic
74	C₂×C₅₃	#456	1	2	yes	—
75	C₂×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
76	#519	C₂×C₅₃	1	1	yes	characteristic
77	#419	C₂×C₅₃	1	2	yes	—
78	C₂×#101	C₂×C₅₃	1	1	yes	characteristic
79	#339	C₂×C₅₃	1	2	yes	—
80	C₅₃	#1886	1	1	yes	characteristic, direct product factor, Sylow
81	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
82	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
83	C₂×M₁₆	C₂²×C₅₃	1	2	yes	—
84	C₄⋊Q₈	—	2	2	no	—
85	C₄⋊Q₈	C₄×C₅₃	1	4	yes	complemented
86	C₄×Q₈	—	2	4	no	—
87	#117	—	2	2	no	—
88	#117	C₂²×C₅₃	1	2	yes	—
89	C₂×C₄⋊C₄	—	2	2	no	—
90	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
91	#103	—	2	4	no	—
92	#101	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	2	1	no	—
95	C₂×Q₈	—	2	2	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	C₂×C₄×C₅₃	1	2	yes	—
99	C₂×Q₈	C₂×C₄×C₅₃	1	2	yes	—
100	C₂×Q₈	D₄×C₅₃	1	2	yes	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	D₄×C₅₃	1	2	yes	—
104	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
105	M₁₆	—	2	2	no	—
106	M₁₆	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₄⋊C₄	—	2	4	no	—
109	C₄⋊C₄	—	2	8	no	—
110	C₄⋊C₄	D₄×C₅₃	1	4	yes	—
111	C₂²⋊C₄	—	2	2	no	—
112	C₄²	—	2	4	no	—
113	C₄²	—	2	1	no	—
114	C₄²	—	2	2	no	—
115	C₄²	C₂×C₄×C₅₃	1	2	yes	—
116	C₂³	C₂×D₄×C₅₃	1	1	yes	characteristic
117	Q₈	—	2	4	no	—
118	Q₈	—	2	4	no	—
119	Q₈	—	2	4	no	—
120	Q₈	—	4	2	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
130	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
131	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
132	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
133	C₂×C₄	#41×C₅₃	1	2	yes	—
134	C₈	—	4	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
137	C₂²	#120×C₅₃	1	1	yes	characteristic
138	C₂²	#123×C₅₃	1	1	yes	characteristic
139	C₄	—	2	4	no	—
140	C₄	—	2	4	no	—
141	C₄	—	4	2	no	—
142	C₄	—	4	4	no	—
143	C₄	C₄×D₄×C₅₃	1	2	yes	—
144	C₄	#121×C₅₃	1	2	yes	—
145	C₂	—	2	1	no	—
146	C₂	—	2	1	no	—
147	C₂	C₅₃×#386	1	1	yes	characteristic
148	C₂	C₅₃×#456	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic

C₅₃×#519

C₂

yes

maximal, characteristic

C₅₃×#419

C₂

yes

maximal

C₂×#101×C₅₃

C₂

yes

maximal, characteristic

C₅₃×#339

C₂

yes

maximal

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₄

yes

complemented

C₄×Q₈×C₅₃

—

#117×C₅₃

—

#117×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#103×C₅₃

—

#101×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂×Q₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×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

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

—

C₈×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

#121

yes

—

#1886

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#386

yes

characteristic

C₂×C₅₃

#456

yes

—

C₂×C₄⋊Q₈

C₂×C₅₃

yes

characteristic

#519

C₂×C₅₃

yes

characteristic

#419

C₂×C₅₃

yes

—

C₂×#101

C₂×C₅₃

yes

characteristic

#339

C₂×C₅₃

yes

—

C₅₃

#1886

yes

characteristic, direct product factor, Sylow

C₂²×Q₈

C₂²×C₅₃

yes

characteristic

C₂²×Q₈

C₂²×C₅₃

yes

characteristic

C₂×M₁₆

C₂²×C₅₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

C₄×C₅₃

yes

complemented

C₄×Q₈

—

#117

—

#117

C₂²×C₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×C₅₃

yes

characteristic

#103

—

#101

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₄×C₅₃

yes

—

C₂×Q₈

C₂×C₄×C₅₃

yes

—

100

C₂×Q₈

D₄×C₅₃

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

D₄×C₅₃

yes

—

104

C₂²×C₄

C₂³×C₅₃

yes

characteristic

105

M₁₆

—

106

M₁₆

—

107

C₂×C₈

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

D₄×C₅₃

yes

—

111

C₂²⋊C₄

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₄²

C₂×C₄×C₅₃

yes

—

116

C₂³

C₂×D₄×C₅₃

yes

characteristic

117

Q₈

—

118

Q₈

—

119

Q₈

—

120

Q₈

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂²⋊C₄×C₅₃

yes

—

130

C₂×C₄

C₂²×C₄×C₅₃

yes

characteristic

131

C₂×C₄

C₂×D₄×C₅₃

yes

—

132

C₂×C₄

C₂×D₄×C₅₃

yes

characteristic

133

C₂×C₄

#41×C₅₃

yes

—

134

C₈

—

135

C₂²

—

136

C₂²

C₂×C₂²⋊C₄×C₅₃

yes

characteristic

137

C₂²

#120×C₅₃

yes

characteristic

138

C₂²

#123×C₅₃

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

C₄×D₄×C₅₃

yes

—

144

C₄

#121×C₅₃

yes

—

145

C₂

—

146

C₂

—

147

C₂

C₅₃×#386

yes

characteristic

148

C₂

C₅₃×#456

yes

—

	subgroup	G/H
center, socle	C₂²×C₅₃	#120
commutator	C₂×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂³×C₅₃
prime radical	C₂³×C₅₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#120

commutator

C₂×C₄

C₂²×C₄×C₅₃

Frattini, cosocle

C₂²×C₄

C₂³×C₅₃

prime radical

C₂³×C₅₃

C₂×D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ G

N	H ≅ G/N	type
10. C₄⋊Q₈×C₅₃	C₄	semidirect
70. #1886	C₅₃	direct (N × H)
80. C₅₃	#1886	direct (N × H)
85. C₄⋊Q₈	C₄×C₅₃	semidirect

H ≅ G/N

type

10. C₄⋊Q₈×C₅₃

C₄

semidirect

70. #1886

C₅₃

direct (N × H)

80. C₅₃

#1886

direct (N × H)

85. C₄⋊Q₈

C₄×C₅₃

semidirect

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

subgroup	index
C₂×C₄⋊Q₈×C₅₃	2
C₅₃×#519	2
C₅₃×#419	2
C₂×#101×C₅₃	2
C₅₃×#339	2
#1886	53

subgroup

index

C₂×C₄⋊Q₈×C₅₃

C₅₃×#519

C₅₃×#419

C₂×#101×C₅₃

C₅₃×#339

#1886

Subgroups — C₅₃×#1886 #20270823

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#1886 #20270823

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#1886 #20270823