Group #20270918 — subgroups

Subgroups — C₅₃×#1981 #20270918

Order 6784.

324 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 144 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	C₂×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#103×C₅₃	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₅₃	C₂²	1	2	yes	—
6	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
7	C₄⋊Q₈×C₅₃	—	2	2	no	—
8	C₄⋊Q₈×C₅₃	C₄	1	4	yes	complemented
9	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
10	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
11	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄²×C₅₃	—	2	1	no	—
13	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₅₃	—	2	2	no	—
15	#103×C₅₃	—	2	8	no	—
16	#95×C₅₃	—	2	1	no	—
17	C₂×Q₈×C₅₃	—	2	2	no	—
18	C₂×Q₈×C₅₃	—	2	4	no	—
19	C₂×Q₈×C₅₃	—	2	4	no	—
20	C₂×Q₈×C₅₃	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₅₃	—	2	2	no	—
22	C₂²×C₄×C₅₃	—	2	2	no	—
23	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₅₃	—	2	4	no	—
27	C₂×C₈×C₅₃	—	2	2	no	—
28	C₄⋊C₄×C₅₃	—	2	4	no	—
29	C₄⋊C₄×C₅₃	—	2	4	no	—
30	C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₄⋊C₄×C₅₃	D₄	1	4	yes	—
32	C₄²×C₅₃	—	4	2	no	—
33	C₄²×C₅₃	C₂×C₄	1	2	yes	—
34	C₄²×C₅₃	D₄	1	2	yes	—
35	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
36	Q₈×C₅₃	—	2	8	no	—
37	Q₈×C₅₃	—	4	2	no	—
38	Q₈×C₅₃	—	4	2	no	—
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	—	2	4	no	—
41	C₂×C₄×C₅₃	—	2	4	no	—
42	C₂×C₄×C₅₃	—	4	4	no	—
43	C₂×C₄×C₅₃	—	4	2	no	—
44	C₂×C₄×C₅₃	—	4	2	no	—
45	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
48	C₂×C₄×C₅₃	Q₁₆	1	2	yes	—
49	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#41	1	2	yes	—
53	C₈×C₅₃	—	4	2	no	—
54	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₅₃	#121	1	2	yes	—
56	C₂²×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
57	C₂²×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
58	C₂²×C₅₃	#137	1	2	yes	—
59	C₄×C₅₃	—	2	2	no	—
60	C₄×C₅₃	—	4	4	no	—
61	C₄×C₅₃	—	4	4	no	—
62	C₄×C₅₃	#103	1	4	yes	complemented
63	C₄×C₅₃	C₄×D₄	1	2	yes	—
64	C₄×C₅₃	#124	1	1	yes	characteristic
65	C₄×C₅₃	#127	1	1	yes	characteristic
66	#1981	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂×C₅₃	#390	1	1	yes	characteristic
68	C₂×C₅₃	C₂×#103	1	1	yes	characteristic
69	C₂×C₅₃	#419	1	1	yes	characteristic
70	C₂×C₅₃	#461	1	2	yes	—
71	C₂×C₅₃	#462	1	2	yes	—
72	C₂×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
73	C₂×C₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic
74	C₂×#103	C₂×C₅₃	1	4	yes	—
75	C₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic
76	C₅₃	#1981	1	1	yes	characteristic, direct product factor, Sylow
77	C₂²×Q₈	C₂²×C₅₃	1	2	yes	—
78	C₂²×C₈	C₂²×C₅₃	1	2	yes	—
79	C₄⋊Q₈	—	2	2	no	—
80	C₄⋊Q₈	C₄×C₅₃	1	4	yes	complemented
81	C₂×C₄⋊C₄	—	2	1	no	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
84	C₂×C₄²	—	2	1	no	—
85	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
86	C₄⋊C₈	—	2	2	no	—
87	#103	—	2	8	no	—
88	#95	—	2	1	no	—
89	C₂×Q₈	—	2	2	no	—
90	C₂×Q₈	—	2	4	no	—
91	C₂×Q₈	—	2	4	no	—
92	C₂×Q₈	C₂×C₄×C₅₃	1	4	yes	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	D₄×C₅₃	1	1	yes	characteristic
96	C₂²×C₄	D₄×C₅₃	1	1	yes	characteristic
97	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
98	C₂×C₈	—	2	4	no	—
99	C₂×C₈	—	2	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	2	4	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	D₄×C₅₃	1	4	yes	—
104	C₄²	—	4	2	no	—
105	C₄²	C₂×C₄×C₅₃	1	2	yes	—
106	C₄²	D₄×C₅₃	1	2	yes	—
107	C₂³	C₂×D₄×C₅₃	1	1	yes	characteristic
108	Q₈	—	2	8	no	—
109	Q₈	—	4	2	no	—
110	Q₈	—	4	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	4	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
118	C₂×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
119	C₂×C₄	SD₁₆×C₅₃	1	2	yes	—
120	C₂×C₄	Q₁₆×C₅₃	1	2	yes	—
121	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
122	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
123	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
124	C₂×C₄	#41×C₅₃	1	2	yes	—
125	C₈	—	4	2	no	—
126	C₂²	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
127	C₂²	#121×C₅₃	1	2	yes	—
128	C₂²	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
129	C₂²	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
130	C₂²	#137×C₅₃	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	—	4	4	no	—
133	C₄	—	4	4	no	—
134	C₄	#103×C₅₃	1	4	yes	complemented
135	C₄	C₄×D₄×C₅₃	1	2	yes	—
136	C₄	#124×C₅₃	1	1	yes	characteristic
137	C₄	#127×C₅₃	1	1	yes	characteristic
138	C₂	C₅₃×#390	1	1	yes	characteristic
139	C₂	C₂×#103×C₅₃	1	1	yes	characteristic
140	C₂	C₅₃×#419	1	1	yes	characteristic
141	C₂	C₅₃×#461	1	2	yes	—
142	C₂	C₅₃×#462	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₅₃

C₂

yes

maximal, characteristic

C₂×#103×C₅₃

C₂

yes

maximal

C₄×C₄⋊C₄×C₅₃

C₂

yes

maximal, characteristic

C₂²×Q₈×C₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

#103×C₅₃

—

#95×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

D₄

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

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

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

Q₁₆

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

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

#121

yes

—

C₂²×C₅₃

C₂×SD₁₆

yes

characteristic

C₂²×C₅₃

C₂×Q₁₆

yes

characteristic

C₂²×C₅₃

#137

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#103

yes

complemented

C₄×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#462

yes

—

C₂×C₄⋊Q₈

C₂×C₅₃

yes

characteristic

C₂×C₄⋊C₈

C₂×C₅₃

yes

characteristic

C₂×#103

C₂×C₅₃

yes

—

C₄×C₄⋊C₄

C₂×C₅₃

yes

characteristic

C₅₃

#1981

yes

characteristic, direct product factor, Sylow

C₂²×Q₈

C₂²×C₅₃

yes

—

C₂²×C₈

C₂²×C₅₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

C₄×C₅₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×C₅₃

yes

characteristic

C₄⋊C₈

—

#103

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₄×C₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₅₃

yes

characteristic

C₂²×C₄

D₄×C₅₃

yes

characteristic

C₂²×C₄

C₂³×C₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

D₄×C₅₃

yes

—

104

C₄²

—

105

C₄²

C₂×C₄×C₅₃

yes

—

106

C₄²

D₄×C₅₃

yes

—

107

C₂³

C₂×D₄×C₅₃

yes

characteristic

108

Q₈

—

109

Q₈

—

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂²⋊C₄×C₅₃

yes

—

118

C₂×C₄

C₂²⋊C₄×C₅₃

yes

—

119

C₂×C₄

SD₁₆×C₅₃

yes

—

120

C₂×C₄

Q₁₆×C₅₃

yes

—

121

C₂×C₄

C₂²×C₄×C₅₃

yes

characteristic

122

C₂×C₄

C₂×D₄×C₅₃

yes

—

123

C₂×C₄

C₂×D₄×C₅₃

yes

characteristic

124

C₂×C₄

#41×C₅₃

yes

—

125

C₈

—

126

C₂²

C₂×C₂²⋊C₄×C₅₃

yes

characteristic

127

C₂²

#121×C₅₃

yes

—

128

C₂²

C₂×SD₁₆×C₅₃

yes

characteristic

129

C₂²

C₂×Q₁₆×C₅₃

yes

characteristic

130

C₂²

#137×C₅₃

yes

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

#103×C₅₃

yes

complemented

135

C₄

C₄×D₄×C₅₃

yes

—

136

C₄

#124×C₅₃

yes

characteristic

137

C₄

#127×C₅₃

yes

characteristic

138

C₂

C₅₃×#390

yes

characteristic

139

C₂

C₂×#103×C₅₃

yes

characteristic

140

C₂

C₅₃×#419

yes

characteristic

141

C₂

C₅₃×#461

yes

—

142

C₂

C₅₃×#462

yes

—

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

subgroup

G/H

center, socle, prime radical

C₂³×C₅₃

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₅₃

Frattini, cosocle

C₂²×C₄

C₂³×C₅₃

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
8. C₄⋊Q₈×C₅₃	C₄	semidirect
62. C₄×C₅₃	#103	semidirect
66. #1981	C₅₃	direct (N × H)
76. C₅₃	#1981	direct (N × H)
80. C₄⋊Q₈	C₄×C₅₃	semidirect
134. C₄	#103×C₅₃	semidirect

H ≅ G/N

type

8. C₄⋊Q₈×C₅₃

C₄

semidirect

62. C₄×C₅₃

#103

semidirect

66. #1981

C₅₃

direct (N × H)

76. C₅₃

#1981

direct (N × H)

80. C₄⋊Q₈

C₄×C₅₃

semidirect

134. C₄

#103×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
C₂×C₄⋊C₈×C₅₃	2
C₂×#103×C₅₃	2
C₄×C₄⋊C₄×C₅₃	2
#1981	53

subgroup

index

C₂×C₄⋊Q₈×C₅₃

C₂×C₄⋊C₈×C₅₃

C₂×#103×C₅₃

C₄×C₄⋊C₄×C₅₃

#1981

Subgroups — C₅₃×#1981 #20270918

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#1981 #20270918

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#1981 #20270918