Group #27215652 — subgroups

Subgroups — C₅₃×#8140 #27215652

Order 13568.

330 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 174 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×#1726	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1452	C₂	1	2	yes	maximal
3	C₅₃×#1275	C₂	1	2	yes	maximal
4	C₅₃×#1274	C₂	1	2	yes	maximal
5	C₅₃×#431	C₂²	1	4	yes	—
6	C₂×#97×C₅₃	C₄	1	2	yes	—
7	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#376	C₂²	1	1	yes	characteristic
11	C₅₃×#322	C₄	1	8	yes	complemented
12	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
17	C₄⋊C₈×C₅₃	—	2	4	no	—
18	C₂²⋊C₈×C₅₃	—	2	4	no	—
19	#97×C₅₃	—	2	4	no	—
20	#97×C₅₃	—	2	2	no	—
21	C₄×C₈×C₅₃	—	2	4	no	—
22	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
23	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
24	#95×C₅₃	—	2	2	no	—
25	C₂²×C₄×C₅₃	—	2	2	no	—
26	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	#41	1	2	yes	—
28	C₂×C₈×C₅₃	—	2	2	no	—
29	C₂×C₈×C₅₃	—	2	2	no	—
30	C₂×C₈×C₅₃	—	2	2	no	—
31	C₂×C₈×C₅₃	—	2	2	no	—
32	C₂×C₈×C₅₃	—	2	8	no	—
33	C₂×C₈×C₅₃	C₄²	1	2	yes	—
34	C₂×C₈×C₅₃	C₄²	1	2	yes	—
35	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
36	C₄²×C₅₃	—	2	2	no	—
37	C₄²×C₅₃	M₁₆	1	4	yes	—
38	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
39	C₄²×C₅₃	#41	1	2	yes	—
40	C₂³×C₅₃	#117	1	1	yes	characteristic
41	C₂×C₄×C₅₃	—	2	4	no	—
42	C₂×C₄×C₅₃	—	4	2	no	—
43	C₂×C₄×C₅₃	—	4	4	no	—
44	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#117	1	4	yes	—
46	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
48	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
49	C₈×C₅₃	—	2	2	no	—
50	C₈×C₅₃	—	2	2	no	—
51	C₈×C₅₃	—	2	4	no	—
52	C₈×C₅₃	—	4	8	no	—
53	#8140	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂²×C₅₃	—	2	2	no	—
55	C₂²×C₅₃	#376	1	1	yes	characteristic
56	C₂²×C₅₃	#432	1	2	yes	—
57	C₄×C₅₃	—	2	4	no	—
58	C₄×C₅₃	—	4	4	no	—
59	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
60	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
61	C₄×C₅₃	#432	1	2	yes	—
62	#1726	C₂×C₅₃	1	1	yes	characteristic
63	#1452	C₂×C₅₃	1	2	yes	—
64	#1275	C₂×C₅₃	1	2	yes	—
65	#1274	C₂×C₅₃	1	2	yes	—
66	C₂×C₅₃	—	2	2	no	—
67	C₂×C₅₃	#1745	1	1	yes	characteristic
68	C₂×C₅₃	#1747	1	2	yes	—
69	#431	C₂²×C₅₃	1	4	yes	—
70	C₂×#97	C₄×C₅₃	1	2	yes	—
71	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
72	C₂×C₄×C₈	C₄×C₅₃	1	2	yes	—
73	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
74	#376	C₂²×C₅₃	1	1	yes	characteristic
75	#322	C₄×C₅₃	1	8	yes	complemented
76	C₅₃	#8140	1	1	yes	characteristic, direct product factor, Sylow
77	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
78	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
79	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
80	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
81	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
82	C₄⋊C₈	—	2	4	no	—
83	C₂²⋊C₈	—	2	4	no	—
84	#97	—	2	2	no	—
85	#97	—	2	4	no	—
86	C₄×C₈	—	2	4	no	—
87	C₄×C₈	C₂×C₄×C₅₃	1	8	yes	—
88	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
89	#95	—	2	2	no	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
92	C₂²×C₄	#41×C₅₃	1	2	yes	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	8	no	—
98	C₂×C₈	C₄²×C₅₃	1	2	yes	—
99	C₂×C₈	C₄²×C₅₃	1	2	yes	—
100	C₂×C₈	M₁₆×C₅₃	1	4	yes	—
101	C₄²	—	2	2	no	—
102	C₄²	M₁₆×C₅₃	1	4	yes	—
103	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
104	C₄²	#41×C₅₃	1	2	yes	—
105	C₂³	#117×C₅₃	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	4	no	—
109	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
110	C₂×C₄	#117×C₅₃	1	4	yes	—
111	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
112	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
113	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
114	C₈	—	2	2	no	—
115	C₈	—	2	2	no	—
116	C₈	—	2	4	no	—
117	C₈	—	4	8	no	—
118	C₂²	—	2	2	no	—
119	C₂²	C₅₃×#376	1	1	yes	characteristic
120	C₂²	C₅₃×#432	1	2	yes	—
121	C₄	—	2	4	no	—
122	C₄	—	4	4	no	—
123	C₄	C₄×M₁₆×C₅₃	1	1	yes	characteristic
124	C₄	C₄×M₁₆×C₅₃	1	1	yes	characteristic
125	C₄	C₅₃×#432	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	C₅₃×#1745	1	1	yes	characteristic
128	C₂	C₅₃×#1747	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#1726

C₂

yes

maximal, characteristic

C₅₃×#1452

C₂

yes

maximal

C₅₃×#1275

C₂

yes

maximal

C₅₃×#1274

C₂

yes

maximal

C₅₃×#431

C₂²

yes

—

C₂×#97×C₅₃

C₄

yes

—

C₂×#97×C₅₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₃

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#376

C₂²

yes

characteristic

C₅₃×#322

C₄

yes

complemented

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

—

#97×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

—

C₂×C₈×C₅₃

C₄²

yes

—

C₂×C₈×C₅₃

M₁₆

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

M₁₆

yes

—

C₄²×C₅₃

C₂²×C₄

yes

—

C₄²×C₅₃

#41

yes

—

C₂³×C₅₃

#117

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

#117

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#8140

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#376

yes

characteristic

C₂²×C₅₃

#432

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×M₁₆

yes

characteristic

C₄×C₅₃

C₄×M₁₆

yes

characteristic

C₄×C₅₃

#432

yes

—

#1726

C₂×C₅₃

yes

characteristic

#1452

C₂×C₅₃

yes

—

#1275

C₂×C₅₃

yes

—

#1274

C₂×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

#1745

yes

characteristic

C₂×C₅₃

#1747

yes

—

#431

C₂²×C₅₃

yes

—

C₂×#97

C₄×C₅₃

yes

—

C₂×#97

C₂²×C₅₃

yes

characteristic

C₂×C₄×C₈

C₄×C₅₃

yes

—

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

#376

C₂²×C₅₃

yes

characteristic

#322

C₄×C₅₃

yes

complemented

C₅₃

#8140

yes

characteristic, direct product factor, Sylow

C₂²×C₈

C₂×C₄×C₅₃

yes

characteristic

C₂²×C₈

C₂×C₄×C₅₃

yes

characteristic

C₂²×C₈

C₂×C₄×C₅₃

yes

—

C₂×C₄²

C₂×C₄×C₅₃

yes

—

C₂×C₄²

C₂³×C₅₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×C₄×C₅₃

yes

—

C₄×C₈

C₂×C₄×C₅₃

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₂²×C₄×C₅₃

yes

characteristic

C₂²×C₄

#41×C₅₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²×C₅₃

yes

—

C₂×C₈

C₄²×C₅₃

yes

—

100

C₂×C₈

M₁₆×C₅₃

yes

—

101

C₄²

—

102

C₄²

M₁₆×C₅₃

yes

—

103

C₄²

C₂²×C₄×C₅₃

yes

—

104

C₄²

#41×C₅₃

yes

—

105

C₂³

#117×C₅₃

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂×C₄²×C₅₃

yes

characteristic

110

C₂×C₄

#117×C₅₃

yes

—

111

C₂×C₄

#117×C₅₃

yes

characteristic

112

C₂×C₄

C₂×M₁₆×C₅₃

yes

—

113

C₂×C₄

C₂×M₁₆×C₅₃

yes

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₂²

—

119

C₂²

C₅₃×#376

yes

characteristic

120

C₂²

C₅₃×#432

yes

—

121

C₄

—

122

C₄

—

123

C₄

C₄×M₁₆×C₅₃

yes

characteristic

124

C₄

C₄×M₁₆×C₅₃

yes

characteristic

125

C₄

C₅₃×#432

yes

—

126

C₂

—

127

C₂

C₅₃×#1745

yes

characteristic

128

C₂

C₅₃×#1747

yes

—

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#376
prime radical	C₂³×C₅₃	#117
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ G

subgroup

G/H

center

C₂×C₄×C₅₃

C₂×C₄²

commutator

C₂×C₄

C₂×C₄²×C₅₃

Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

C₂²×C₅₃

C₂³×C₅₃

G ▷ C₂×C₄ ▷ C₁

C₁ ◁ C₂×C₄×C₅₃ ◁ G

N	H ≅ G/N	type
11. C₅₃×#322	C₄	semidirect
53. #8140	C₅₃	direct (N × H)
75. #322	C₄×C₅₃	semidirect
76. C₅₃	#8140	direct (N × H)

H ≅ G/N

type

11. C₅₃×#322

C₄

semidirect

53. #8140

C₅₃

direct (N × H)

75. #322

C₄×C₅₃

semidirect

76. C₅₃

#8140

direct (N × H)

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

subgroup	index
C₅₃×#1726	2
C₅₃×#1452	2
C₅₃×#1275	2
C₅₃×#1274	2
#8140	53

subgroup

index

C₅₃×#1726

C₅₃×#1452

C₅₃×#1275

C₅₃×#1274

#8140

Subgroups — C₅₃×#8140 #27215652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#8140 #27215652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8140 #27215652