Group #27216306 — subgroups

Subgroups — C₅₃×#8794 #27216306

Order 13568.

270 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 126 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×#1848	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1835	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1452	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1286	C₂	1	4	yes	maximal
5	C₅₃×#431	C₂²	1	2	yes	—
6	C₂×#108×C₅₃	—	2	2	no	—
7	C₅₃×#423	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#382	C₂²	1	2	yes	—
10	C₅₃×#340	—	2	2	no	—
11	C₅₃×#322	C₄	1	4	yes	complemented
12	C₂×M₁₆×C₅₃	—	2	2	no	—
13	C₂²×C₈×C₅₃	D₄	1	2	yes	—
14	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
15	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
16	#108×C₅₃	—	4	4	no	—
17	C₄⋊C₈×C₅₃	—	2	2	no	—
18	C₄⋊C₈×C₅₃	—	2	4	no	—
19	C₂²⋊C₈×C₅₃	—	2	2	no	—
20	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
21	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₅₃	Q₈	1	2	yes	—
23	#95×C₅₃	—	2	4	no	—
24	C₂²×C₄×C₅₃	—	4	2	no	—
25	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	#41	1	2	yes	—
27	M₁₆×C₅₃	—	4	4	no	—
28	C₂×C₈×C₅₃	—	2	2	no	—
29	C₂×C₈×C₅₃	—	2	2	no	—
30	C₂×C₈×C₅₃	—	2	4	no	—
31	C₂×C₈×C₅₃	—	4	2	no	—
32	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
33	C₄⋊C₄×C₅₃	—	4	2	no	—
34	C₄⋊C₄×C₅₃	—	4	2	no	—
35	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
37	C₄²×C₅₃	#41	1	2	yes	—
38	C₂³×C₅₃	#124	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	4	2	no	—
40	C₂×C₄×C₅₃	—	8	4	no	—
41	C₂×C₄×C₅₃	—	8	2	no	—
42	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	#117	1	2	yes	—
44	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#125	1	2	yes	—
46	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
47	C₂×C₄×C₅₃	#137	1	2	yes	—
48	C₈×C₅₃	—	2	4	no	—
49	C₈×C₅₃	—	4	4	no	—
50	C₈×C₅₃	—	8	2	no	—
51	#8794	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂²×C₅₃	—	2	2	no	—
53	C₂²×C₅₃	#382	1	1	yes	characteristic
54	C₂²×C₅₃	#489	1	2	yes	—
55	C₄×C₅₃	—	2	2	no	—
56	C₄×C₅₃	—	2	2	no	—
57	C₄×C₅₃	—	16	2	no	—
58	C₄×C₅₃	#428	1	2	yes	—
59	C₄×C₅₃	#490	1	1	yes	characteristic
60	C₄×C₅₃	#491	1	1	yes	characteristic
61	#1848	C₂×C₅₃	1	1	yes	characteristic
62	#1835	C₂×C₅₃	1	1	yes	characteristic
63	#1452	C₂×C₅₃	1	1	yes	characteristic
64	#1286	C₂×C₅₃	1	4	yes	—
65	C₂×C₅₃	—	2	2	no	—
66	C₂×C₅₃	#1833	1	2	yes	—
67	C₂×C₅₃	#1836	1	1	yes	characteristic
68	#431	C₂²×C₅₃	1	2	yes	—
69	C₂×#108	—	2	2	no	—
70	#423	C₂²×C₅₃	1	2	yes	—
71	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
72	#382	C₂²×C₅₃	1	2	yes	—
73	#340	—	2	2	no	—
74	#322	C₄×C₅₃	1	4	yes	complemented
75	C₅₃	#8794	1	1	yes	characteristic, direct product factor, Sylow
76	C₂×M₁₆	—	2	2	no	—
77	C₂²×C₈	D₄×C₅₃	1	2	yes	—
78	C₂×C₄⋊C₄	—	2	2	no	—
79	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
80	#108	—	4	4	no	—
81	C₄⋊C₈	—	2	2	no	—
82	C₄⋊C₈	—	2	4	no	—
83	C₂²⋊C₈	—	2	2	no	—
84	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
85	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
86	C₄×C₈	Q₈×C₅₃	1	2	yes	—
87	#95	—	2	4	no	—
88	C₂²×C₄	—	4	2	no	—
89	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
90	C₂²×C₄	#41×C₅₃	1	2	yes	—
91	M₁₆	—	4	4	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	4	no	—
95	C₂×C₈	—	4	2	no	—
96	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
100	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
101	C₄²	#41×C₅₃	1	2	yes	—
102	C₂³	#124×C₅₃	1	1	yes	characteristic
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	8	4	no	—
105	C₂×C₄	—	8	2	no	—
106	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
107	C₂×C₄	#117×C₅₃	1	2	yes	—
108	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
109	C₂×C₄	#125×C₅₃	1	2	yes	—
110	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
111	C₂×C₄	#137×C₅₃	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₈	—	4	4	no	—
114	C₈	—	8	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	C₅₃×#382	1	1	yes	characteristic
117	C₂²	C₅₃×#489	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	16	2	no	—
121	C₄	C₅₃×#428	1	2	yes	—
122	C₄	C₅₃×#490	1	1	yes	characteristic
123	C₄	C₅₃×#491	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	C₅₃×#1833	1	2	yes	—
126	C₂	C₅₃×#1836	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#1848

C₂

yes

maximal, characteristic

C₅₃×#1835

C₂

yes

maximal, characteristic

C₅₃×#1452

C₂

yes

maximal, characteristic

C₅₃×#1286

C₂

yes

maximal

C₅₃×#431

C₂²

yes

—

C₂×#108×C₅₃

—

C₅₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#382

C₂²

yes

—

C₅₃×#340

—

C₅₃×#322

C₄

yes

complemented

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#108×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

#41

yes

—

C₂³×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

#125

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#8794

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#382

yes

characteristic

C₂²×C₅₃

#489

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#428

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₂×C₅₃

yes

characteristic

#1835

C₂×C₅₃

yes

characteristic

#1452

C₂×C₅₃

yes

characteristic

#1286

C₂×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

#1833

yes

—

C₂×C₅₃

#1836

yes

characteristic

#431

C₂²×C₅₃

yes

—

C₂×#108

—

#423

C₂²×C₅₃

yes

—

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

#382

C₂²×C₅₃

yes

—

#340

—

#322

C₄×C₅₃

yes

complemented

C₅₃

#8794

yes

characteristic, direct product factor, Sylow

C₂×M₁₆

—

C₂²×C₈

D₄×C₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³×C₅₃

yes

characteristic

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

C₂×C₄×C₅₃

yes

—

C₄×C₈

C₂×C₄×C₅₃

yes

—

C₄×C₈

Q₈×C₅₃

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

C₂²×C₄

#41×C₅₃

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×C₄×C₅₃

yes

characteristic

100

C₄²

C₂×Q₈×C₅₃

yes

characteristic

101

C₄²

#41×C₅₃

yes

—

102

C₂³

#124×C₅₃

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×C₄⋊C₄×C₅₃

yes

characteristic

107

C₂×C₄

#117×C₅₃

yes

—

108

C₂×C₄

#124×C₅₃

yes

characteristic

109

C₂×C₄

#125×C₅₃

yes

—

110

C₂×C₄

AGL(1,ℤ/8)×C₅₃

yes

—

111

C₂×C₄

#137×C₅₃

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₂²

—

116

C₂²

C₅₃×#382

yes

characteristic

117

C₂²

C₅₃×#489

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

C₅₃×#428

yes

—

122

C₄

C₅₃×#490

yes

characteristic

123

C₄

C₅₃×#491

yes

characteristic

124

C₂

—

125

C₂

C₅₃×#1833

yes

—

126

C₂

C₅₃×#1836

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#382
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂³×C₅₃	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#382

commutator

C₄²

C₂²×C₄×C₅₃

Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

C₂³×C₅₃

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
11. C₅₃×#322	C₄	semidirect
51. #8794	C₅₃	direct (N × H)
74. #322	C₄×C₅₃	semidirect
75. C₅₃	#8794	direct (N × H)

H ≅ G/N

type

11. C₅₃×#322

C₄

semidirect

51. #8794

C₅₃

direct (N × H)

74. #322

C₄×C₅₃

semidirect

75. C₅₃

#8794

direct (N × H)

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

subgroup	index
C₅₃×#1848	2
C₅₃×#1835	2
C₅₃×#1452	2
C₅₃×#1286	2
#8794	53

subgroup

index

C₅₃×#1848

C₅₃×#1835

C₅₃×#1452

C₅₃×#1286

#8794

Subgroups — C₅₃×#8794 #27216306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#8794 #27216306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8794 #27216306