Group #27216339 — subgroups

Subgroups — C₅₃×#8827 #27216339

Order 13568.

306 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 166 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×#1837	C₂	1	2	yes	maximal
2	C₅₃×#1567	C₂	1	2	yes	maximal
3	C₅₃×#1566	C₂	1	2	yes	maximal
4	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#423	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#341	—	2	4	no	—
10	C₅₃×#335	—	2	4	no	—
11	C₅₃×#334	—	2	4	no	—
12	C₅₃×#322	C₄	1	4	yes	—
13	C₂×M₁₆×C₅₃	—	2	4	no	—
14	C₂²×C₈×C₅₃	D₄	1	2	yes	—
15	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
16	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
17	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
18	C₄⋊C₈×C₅₃	—	2	4	no	—
19	C₄⋊C₈×C₅₃	—	2	4	no	—
20	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₅₃	D₄	1	2	yes	—
24	C₄×C₈×C₅₃	D₄	1	2	yes	—
25	C₄×C₈×C₅₃	Q₈	1	2	yes	—
26	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
29	M₁₆×C₅₃	—	4	8	no	—
30	C₂×C₈×C₅₃	—	2	4	no	—
31	C₂×C₈×C₅₃	—	4	2	no	—
32	C₂×C₈×C₅₃	—	4	2	no	—
33	C₂×C₈×C₅₃	—	4	4	no	—
34	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
35	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
36	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
37	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	—	2	2	no	—
44	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
50	C₂×C₄×C₅₃	#135	1	4	yes	—
51	C₈×C₅₃	—	2	4	no	—
52	C₈×C₅₃	—	2	4	no	—
53	C₈×C₅₃	—	2	4	no	—
54	C₈×C₅₃	—	8	4	no	—
55	#8827	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂²×C₅₃	—	2	1	no	—
57	C₂²×C₅₃	—	2	1	no	—
58	C₂²×C₅₃	#384	1	1	yes	characteristic
59	C₂²×C₅₃	#499	1	2	yes	—
60	C₄×C₅₃	—	2	1	no	—
61	C₄×C₅₃	—	2	1	no	—
62	C₄×C₅₃	#427	1	2	yes	—
63	C₄×C₅₃	#430	1	2	yes	—
64	C₄×C₅₃	#430	1	2	yes	—
65	C₄×C₅₃	#495	1	2	yes	—
66	#1837	C₂×C₅₃	1	2	yes	—
67	#1567	C₂×C₅₃	1	2	yes	—
68	#1566	C₂×C₅₃	1	2	yes	—
69	C₂×#322	C₂×C₅₃	1	1	yes	characteristic
70	C₂×C₅₃	—	2	2	no	—
71	C₂×C₅₃	#1846	1	1	yes	characteristic
72	C₂×C₅₃	#1849	1	1	yes	characteristic
73	C₂×C₅₃	#1849	1	1	yes	characteristic
74	#423	C₂²×C₅₃	1	4	yes	—
75	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
76	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
78	#341	—	2	4	no	—
79	#335	—	2	4	no	—
80	#334	—	2	4	no	—
81	#322	C₄×C₅₃	1	4	yes	—
82	C₅₃	#8827	1	1	yes	characteristic, direct product factor, Sylow
83	C₂×M₁₆	—	2	4	no	—
84	C₂²×C₈	D₄×C₅₃	1	2	yes	—
85	C₂²×C₈	Q₈×C₅₃	1	2	yes	—
86	C₂²×C₈	Q₈×C₅₃	1	2	yes	—
87	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
88	C₄⋊C₈	—	2	4	no	—
89	C₄⋊C₈	—	2	4	no	—
90	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
91	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
92	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
93	C₄×C₈	D₄×C₅₃	1	2	yes	—
94	C₄×C₈	D₄×C₅₃	1	2	yes	—
95	C₄×C₈	Q₈×C₅₃	1	2	yes	—
96	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
97	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
98	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
99	M₁₆	—	4	8	no	—
100	C₂×C₈	—	2	4	no	—
101	C₂×C₈	—	4	2	no	—
102	C₂×C₈	—	4	2	no	—
103	C₂×C₈	—	4	4	no	—
104	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
105	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
106	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
107	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
108	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
109	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
110	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
111	C₂³	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
115	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
116	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
117	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
118	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
119	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
120	C₂×C₄	#135×C₅₃	1	4	yes	—
121	C₈	—	2	4	no	—
122	C₈	—	2	4	no	—
123	C₈	—	2	4	no	—
124	C₈	—	8	4	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	C₅₃×#384	1	1	yes	characteristic
128	C₂²	C₅₃×#499	1	2	yes	—
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	C₅₃×#427	1	2	yes	—
132	C₄	C₅₃×#430	1	2	yes	—
133	C₄	C₅₃×#430	1	2	yes	—
134	C₄	C₅₃×#495	1	2	yes	—
135	C₂	—	2	2	no	—
136	C₂	C₅₃×#1846	1	1	yes	characteristic
137	C₂	C₅₃×#1849	1	1	yes	characteristic
138	C₂	C₅₃×#1849	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#1837

C₂

yes

maximal

C₅₃×#1567

C₂

yes

maximal

C₅₃×#1566

C₂

yes

maximal

C₂×C₅₃×#322

C₂

yes

maximal, characteristic

C₅₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#341

—

C₅₃×#335

—

C₅₃×#334

—

C₅₃×#322

C₄

yes

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

Q₈

yes

—

C₂²×C₈×C₅₃

Q₈

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₂³×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#8827

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#384

yes

characteristic

C₂²×C₅₃

#499

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#427

yes

—

C₄×C₅₃

#430

yes

—

C₄×C₅₃

#430

yes

—

C₄×C₅₃

#495

yes

—

#1837

C₂×C₅₃

yes

—

#1567

C₂×C₅₃

yes

—

#1566

C₂×C₅₃

yes

—

C₂×#322

C₂×C₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂²×C₅₃

yes

—

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

C₂×C₄×C₈

C₂²×C₅₃

yes

—

—

—

C₄×C₅₃

yes

—

C₅₃

#8827

yes

characteristic, direct product factor, Sylow

C₂×M₁₆

—

C₂²×C₈

D₄×C₅₃

yes

—

C₂²×C₈

Q₈×C₅₃

yes

—

C₂²×C₈

Q₈×C₅₃

yes

—

C₂×C₄²

C₂³×C₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄×C₅₃

yes

—

C₄×C₈

C₂×C₄×C₅₃

yes

—

C₄×C₈

C₂×C₄×C₅₃

yes

—

C₄×C₈

D₄×C₅₃

yes

—

C₄×C₈

D₄×C₅₃

yes

—

C₄×C₈

Q₈×C₅₃

yes

—

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

C₂²×C₄

C₂×Q₈×C₅₃

yes

characteristic

C₂²×C₄

C₂×Q₈×C₅₃

yes

characteristic

M₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

C₄⋊C₄×C₅₃

yes

—

105

C₂×C₈

C₄⋊C₄×C₅₃

yes

—

106

C₂×C₈

C₄⋊C₄×C₅₃

yes

—

107

C₄²

C₂²×C₄×C₅₃

yes

characteristic

108

C₄²

C₂×D₄×C₅₃

yes

characteristic

109

C₄²

C₂×D₄×C₅₃

yes

characteristic

110

C₄²

C₂×Q₈×C₅₃

yes

characteristic

111

C₂³

C₄⋊Q₈×C₅₃

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂×C₄⋊C₄×C₅₃

yes

characteristic

115

C₂×C₄

C₂×C₄⋊C₄×C₅₃

yes

characteristic

116

C₂×C₄

C₂×C₄⋊C₄×C₅₃

yes

characteristic

117

C₂×C₄

#127×C₅₃

yes

characteristic

118

C₂×C₄

C₄⋊Q₈×C₅₃

yes

characteristic

119

C₂×C₄

C₄⋊Q₈×C₅₃

yes

characteristic

120

C₂×C₄

#135×C₅₃

yes

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₂²

—

126

C₂²

—

127

C₂²

C₅₃×#384

yes

characteristic

128

C₂²

C₅₃×#499

yes

—

129

C₄

—

130

C₄

—

131

C₄

C₅₃×#427

yes

—

132

C₄

C₅₃×#430

yes

—

133

C₄

C₅₃×#430

yes

—

134

C₄

C₅₃×#495

yes

—

135

C₂

—

136

C₂

C₅₃×#1846

yes

characteristic

137

C₂

C₅₃×#1849

yes

characteristic

138

C₂

C₅₃×#1849

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#127
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#384
prime radical	C₂³×C₅₃	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center

C₂×C₄×C₅₃

#127

commutator

C₄²

C₂²×C₄×C₅₃

Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

socle

C₂²×C₅₃

#384

prime radical

C₂³×C₅₃

C₄⋊Q₈

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
55. #8827	C₅₃	direct (N × H)
82. C₅₃	#8827	direct (N × H)

H ≅ G/N

type

55. #8827

C₅₃

direct (N × H)

82. C₅₃

#8827

direct (N × H)

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

subgroup	index
C₅₃×#1837	2
C₅₃×#1567	2
C₅₃×#1566	2
C₂×C₅₃×#322	2
#8827	53

subgroup

index

C₅₃×#1837

C₅₃×#1567

C₅₃×#1566

C₂×C₅₃×#322

#8827

Subgroups — C₅₃×#8827 #27216339

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#8827 #27216339

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8827 #27216339