Group #27216294 — subgroups

Subgroups — C₅₃×#8782 #27216294

Order 13568.

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

normal subgroups only

This group has 262 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×#1746	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1564	C₂	1	2	yes	maximal
3	C₅₃×#1281	C₂	1	2	yes	maximal
4	C₅₃×#1274	C₂	1	2	yes	maximal
5	C₅₃×#431	C₂²	1	4	yes	—
6	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#376	C₂²	1	1	yes	characteristic
9	C₅₃×#336	C₄	1	4	yes	—
10	C₅₃×#335	—	2	2	no	—
11	C₅₃×#334	—	2	2	no	—
12	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
13	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
16	C₄⋊C₈×C₅₃	—	2	4	no	—
17	C₂²⋊C₈×C₅₃	—	2	4	no	—
18	#97×C₅₃	—	2	2	no	—
19	C₄×C₈×C₅₃	—	2	4	no	—
20	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
21	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
22	C₄×C₈×C₅₃	Q₈	1	2	yes	—
23	#95×C₅₃	—	2	2	no	—
24	C₂²×C₄×C₅₃	—	2	2	no	—
25	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	#41	1	2	yes	—
27	C₂×C₈×C₅₃	—	2	4	no	—
28	C₂×C₈×C₅₃	—	2	2	no	—
29	C₂×C₈×C₅₃	—	2	4	no	—
30	C₂×C₈×C₅₃	—	4	4	no	—
31	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
32	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
33	C₄²×C₅₃	—	2	4	no	—
34	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
36	C₄²×C₅₃	#41	1	2	yes	—
37	C₂³×C₅₃	#117	1	1	yes	characteristic
38	C₂×C₄×C₅₃	—	4	2	no	—
39	C₂×C₄×C₅₃	—	4	4	no	—
40	C₂×C₄×C₅₃	—	4	2	no	—
41	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
43	C₂×C₄×C₅₃	#124	1	2	yes	—
44	C₂×C₄×C₅₃	#125	1	2	yes	—
45	C₂×C₄×C₅₃	#131	1	4	yes	—
46	C₈×C₅₃	—	2	2	no	—
47	C₈×C₅₃	—	2	2	no	—
48	C₈×C₅₃	—	4	2	no	—
49	C₈×C₅₃	—	8	4	no	—
50	#8782	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂²×C₅₃	—	2	2	no	—
52	C₂²×C₅₃	#382	1	1	yes	characteristic
53	C₂²×C₅₃	#433	1	2	yes	—
54	C₄×C₅₃	—	2	2	no	—
55	C₄×C₅₃	—	2	2	no	—
56	C₄×C₅₃	—	8	2	no	—
57	C₄×C₅₃	#424	1	2	yes	—
58	C₄×C₅₃	#433	1	2	yes	—
59	#1746	C₂×C₅₃	1	1	yes	characteristic
60	#1564	C₂×C₅₃	1	2	yes	—
61	#1281	C₂×C₅₃	1	2	yes	—
62	#1274	C₂×C₅₃	1	2	yes	—
63	C₂×C₅₃	—	2	2	no	—
64	C₂×C₅₃	#1832	1	1	yes	characteristic
65	C₂×C₅₃	#1833	1	2	yes	—
66	#431	C₂²×C₅₃	1	4	yes	—
67	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
68	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
69	#376	C₂²×C₅₃	1	1	yes	characteristic
70	#336	C₄×C₅₃	1	4	yes	—
71	#335	—	2	2	no	—
72	#334	—	2	2	no	—
73	C₅₃	#8782	1	1	yes	characteristic, direct product factor, Sylow
74	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
75	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
76	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
77	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
78	C₄⋊C₈	—	2	4	no	—
79	C₂²⋊C₈	—	2	4	no	—
80	#97	—	2	2	no	—
81	C₄×C₈	—	2	4	no	—
82	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
83	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
84	C₄×C₈	Q₈×C₅₃	1	2	yes	—
85	#95	—	2	2	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
88	C₂²×C₄	#41×C₅₃	1	2	yes	—
89	C₂×C₈	—	2	4	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	4	no	—
92	C₂×C₈	—	4	4	no	—
93	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
94	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
95	C₄²	—	2	4	no	—
96	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
97	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
98	C₄²	#41×C₅₃	1	2	yes	—
99	C₂³	#117×C₅₃	1	1	yes	characteristic
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	4	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
104	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
105	C₂×C₄	#124×C₅₃	1	2	yes	—
106	C₂×C₄	#125×C₅₃	1	2	yes	—
107	C₂×C₄	#131×C₅₃	1	4	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	2	2	no	—
110	C₈	—	4	2	no	—
111	C₈	—	8	4	no	—
112	C₂²	—	2	2	no	—
113	C₂²	C₅₃×#382	1	1	yes	characteristic
114	C₂²	C₅₃×#433	1	2	yes	—
115	C₄	—	2	2	no	—
116	C₄	—	2	2	no	—
117	C₄	—	8	2	no	—
118	C₄	C₅₃×#424	1	2	yes	—
119	C₄	C₅₃×#433	1	2	yes	—
120	C₂	—	2	2	no	—
121	C₂	C₅₃×#1832	1	1	yes	characteristic
122	C₂	C₅₃×#1833	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#1746

C₂

yes

maximal, characteristic

C₅₃×#1564

C₂

yes

maximal

C₅₃×#1281

C₂

yes

maximal

C₅₃×#1274

C₂

yes

maximal

C₅₃×#431

C₂²

yes

—

C₂×#97×C₅₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#376

C₂²

yes

characteristic

C₅₃×#336

C₄

yes

—

C₅₃×#335

—

C₅₃×#334

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

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

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

Q₈

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

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

#41

yes

—

C₂³×C₅₃

#117

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#125

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#8782

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#382

yes

characteristic

C₂²×C₅₃

#433

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#424

yes

—

C₄×C₅₃

#433

yes

—

#1746

C₂×C₅₃

yes

characteristic

#1564

C₂×C₅₃

yes

—

#1281

C₂×C₅₃

yes

—

#1274

C₂×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

#1832

yes

characteristic

C₂×C₅₃

#1833

yes

—

#431

C₂²×C₅₃

yes

—

C₂×#97

C₂²×C₅₃

yes

characteristic

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

#376

C₂²×C₅₃

yes

characteristic

#336

C₄×C₅₃

yes

—

#335

—

#334

—

C₅₃

#8782

yes

characteristic, direct product factor, Sylow

C₂²×C₈

C₂×C₄×C₅₃

yes

—

C₂²×C₈

C₂×C₄×C₅₃

yes

—

C₂×C₄²

C₂×C₄×C₅₃

yes

—

C₂×C₄²

C₂³×C₅₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

D₄×C₅₃

yes

characteristic

C₄×C₈

D₄×C₅₃

yes

characteristic

C₄×C₈

Q₈×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

C₄⋊C₄×C₅₃

yes

—

C₂×C₈

C₄⋊C₄×C₅₃

yes

—

C₄²

—

C₄²

C₂×D₄×C₅₃

yes

characteristic

C₄²

C₂×Q₈×C₅₃

yes

characteristic

C₄²

#41×C₅₃

yes

—

C₂³

#117×C₅₃

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂×C₄⋊C₄×C₅₃

yes

characteristic

104

C₂×C₄

#117×C₅₃

yes

characteristic

105

C₂×C₄

#124×C₅₃

yes

—

106

C₂×C₄

#125×C₅₃

yes

—

107

C₂×C₄

#131×C₅₃

yes

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₂²

—

113

C₂²

C₅₃×#382

yes

characteristic

114

C₂²

C₅₃×#433

yes

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

C₅₃×#424

yes

—

119

C₄

C₅₃×#433

yes

—

120

C₂

—

121

C₂

C₅₃×#1832

yes

characteristic

122

C₂

C₅₃×#1833

yes

—

	subgroup	G/H
center	C₂×C₄×C₅₃	#117
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#382
prime radical	C₂³×C₅₃	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₅₃×#376 ◁ G

subgroup

G/H

center

C₂×C₄×C₅₃

#117

commutator

C₂²×C₄

C₂²×C₄×C₅₃

Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

C₂²×C₅₃

C₂³×C₅₃

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂×C₄×C₅₃ ◁ C₅₃×#376 ◁ G

N	H ≅ G/N	type
50. #8782	C₅₃	direct (N × H)
73. C₅₃	#8782	direct (N × H)

H ≅ G/N

type

50. #8782

C₅₃

direct (N × H)

73. C₅₃

#8782

direct (N × H)

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

subgroup	index
C₅₃×#1746	2
C₅₃×#1564	2
C₅₃×#1281	2
C₅₃×#1274	2
#8782	53

subgroup

index

C₅₃×#1746

C₅₃×#1564

C₅₃×#1281

C₅₃×#1274

#8782

Subgroups — C₅₃×#8782 #27216294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#8782 #27216294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8782 #27216294