Group #27221168 — subgroups

Subgroups — C₅₃×#13656 #27221168

Order 13568.

184 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 86 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×#2262	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2178	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1425	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1417	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1397	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#508	C₂²	1	2	yes	—
7	C₅₃×#443	—	2	2	no	—
8	C₅₃×Q₆₄	C₄	1	1	yes	characteristic, complemented
9	C₅₃×SD₆₄	C₄	1	2	yes	complemented
10	C₅₃×D₃₂	C₄	1	1	yes	characteristic, complemented
11	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#368	C₂²	1	2	yes	—
14	C₅₃×#359	—	2	2	no	—
15	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	#135×C₅₃	—	2	2	no	—
17	#131×C₅₃	—	4	2	no	—
18	Q₃₂×C₅₃	C₂×C₄	1	2	yes	—
19	SD₃₂×C₅₃	—	2	2	no	—
20	D₁₆×C₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
23	#108×C₅₃	—	2	2	no	—
24	#104×C₅₃	—	4	2	no	—
25	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
26	C₃₂×C₅₃	—	2	1	no	—
27	C₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
29	#41×C₅₃	—	4	2	no	—
30	Q₁₆×C₅₃	—	2	2	no	—
31	SD₁₆×C₅₃	—	4	2	no	—
32	D₈×C₅₃	—	2	2	no	—
33	M₁₆×C₅₃	—	4	2	no	—
34	M₁₆×C₅₃	—	8	2	no	—
35	C₂×C₈×C₅₃	—	8	2	no	—
36	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₄²×C₅₃	D₈	1	1	yes	characteristic
39	C₁₆×C₅₃	—	2	1	no	—
40	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₁₆×C₅₃	#41	1	1	yes	characteristic
42	Q₈×C₅₃	—	4	2	no	—
43	D₄×C₅₃	—	4	2	no	—
44	D₄×C₅₃	—	8	2	no	—
45	C₂×C₄×C₅₃	—	2	1	no	—
46	C₂×C₄×C₅₃	—	8	2	no	—
47	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
48	C₈×C₅₃	—	8	2	no	—
49	C₈×C₅₃	D₁₆	1	2	yes	—
50	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
51	C₈×C₅₃	#135	1	1	yes	characteristic
52	#13656	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂²×C₅₃	—	8	2	no	—
54	C₂²×C₅₃	#508	1	1	yes	characteristic
55	C₄×C₅₃	—	4	1	no	—
56	C₄×C₅₃	—	8	2	no	—
57	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
58	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
59	#2262	C₂×C₅₃	1	1	yes	characteristic
60	#2178	C₂×C₅₃	1	2	yes	complemented
61	#1425	C₂×C₅₃	1	1	yes	characteristic, complemented
62	#1417	C₂×C₅₃	1	2	yes	complemented
63	#1397	C₂×C₅₃	1	1	yes	characteristic, complemented
64	C₂×C₅₃	—	2	1	no	—
65	C₂×C₅₃	—	16	2	no	—
66	C₂×C₅₃	C₄×D₁₆	1	1	yes	characteristic
67	#508	C₂²×C₅₃	1	2	yes	—
68	#443	—	2	2	no	—
69	Q₆₄	C₄×C₅₃	1	1	yes	characteristic, complemented
70	SD₆₄	C₄×C₅₃	1	2	yes	complemented
71	D₃₂	C₄×C₅₃	1	1	yes	characteristic, complemented
72	C₂×C₃₂	C₂²×C₅₃	1	1	yes	characteristic
73	C₂×C₃₂	C₂²×C₅₃	1	1	yes	characteristic
74	#368	C₂²×C₅₃	1	2	yes	—
75	#359	—	2	2	no	—
76	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
77	C₅₃	#13656	1	1	yes	characteristic, direct product factor, Sylow
78	#135	—	2	2	no	—
79	#131	—	4	2	no	—
80	Q₃₂	C₂×C₄×C₅₃	1	2	yes	—
81	SD₃₂	—	2	2	no	—
82	D₁₆	C₂×C₄×C₅₃	1	2	yes	—
83	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
84	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
85	#108	—	2	2	no	—
86	#104	—	4	2	no	—
87	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
88	C₃₂	—	2	1	no	—
89	C₃₂	C₂×C₄×C₅₃	1	1	yes	characteristic
90	C₃₂	C₂×C₄×C₅₃	1	1	yes	characteristic
91	#41	—	4	2	no	—
92	Q₁₆	—	2	2	no	—
93	SD₁₆	—	4	2	no	—
94	D₈	—	2	2	no	—
95	M₁₆	—	4	2	no	—
96	M₁₆	—	8	2	no	—
97	C₂×C₈	—	8	2	no	—
98	C₂×C₈	D₈×C₅₃	1	1	yes	characteristic
99	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
100	C₄²	D₈×C₅₃	1	1	yes	characteristic
101	C₁₆	—	2	1	no	—
102	C₁₆	C₂²×C₄×C₅₃	1	1	yes	characteristic
103	C₁₆	#41×C₅₃	1	1	yes	characteristic
104	Q₈	—	4	2	no	—
105	D₄	—	4	2	no	—
106	D₄	—	8	2	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	8	2	no	—
109	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
110	C₈	—	8	2	no	—
111	C₈	D₁₆×C₅₃	1	2	yes	—
112	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
113	C₈	#135×C₅₃	1	1	yes	characteristic
114	C₂²	—	8	2	no	—
115	C₂²	C₅₃×#508	1	1	yes	characteristic
116	C₄	—	4	1	no	—
117	C₄	—	8	2	no	—
118	C₄	C₄×D₈×C₅₃	1	1	yes	characteristic
119	C₄	C₂×D₁₆×C₅₃	1	1	yes	characteristic
120	C₂	—	2	1	no	—
121	C₂	—	16	2	no	—
122	C₂	C₄×D₁₆×C₅₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#2262

C₂

yes

maximal, characteristic

C₅₃×#2178

C₂

yes

maximal, complemented

C₅₃×#1425

C₂

yes

maximal, characteristic, complemented

C₅₃×#1417

C₂

yes

maximal, complemented

C₅₃×#1397

C₂

yes

maximal, characteristic, complemented

C₅₃×#508

C₂²

yes

—

C₅₃×#443

—

C₅₃×Q₆₄

C₄

yes

characteristic, complemented

C₅₃×SD₆₄

C₄

yes

complemented

C₅₃×D₃₂

C₄

yes

characteristic, complemented

C₂×C₃₂×C₅₃

C₂²

yes

characteristic

C₂×C₃₂×C₅₃

C₂²

yes

characteristic

C₅₃×#368

C₂²

yes

—

C₅₃×#359

—

C₄×C₁₆×C₅₃

C₂²

yes

characteristic

#135×C₅₃

—

#131×C₅₃

—

Q₃₂×C₅₃

C₂×C₄

yes

—

SD₃₂×C₅₃

—

D₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₁₆×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂³

yes

characteristic

#108×C₅₃

—

#104×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

characteristic

C₃₂×C₅₃

—

C₃₂×C₅₃

C₂×C₄

yes

characteristic

C₃₂×C₅₃

C₂×C₄

yes

characteristic

#41×C₅₃

—

Q₁₆×C₅₃

—

SD₁₆×C₅₃

—

D₈×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

D₈

yes

characteristic

C₁₆×C₅₃

—

C₁₆×C₅₃

C₂²×C₄

yes

characteristic

C₁₆×C₅₃

#41

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

D₁₆

yes

—

C₈×C₅₃

C₄×D₄

yes

C₈×C₅₃

yes

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#508

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₈

yes

characteristic

C₄×C₅₃

C₂×D₁₆

yes

characteristic

#2262

C₂×C₅₃

yes

characteristic

#2178

C₂×C₅₃

yes

complemented

#1425

C₂×C₅₃

yes

characteristic, complemented

#1417

C₂×C₅₃

yes

complemented

#1397

C₂×C₅₃

yes

characteristic, complemented

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₄×D₁₆

yes

characteristic

#508

C₂²×C₅₃

yes

—

#443

—

Q₆₄

C₄×C₅₃

yes

characteristic, complemented

SD₆₄

C₄×C₅₃

yes

complemented

D₃₂

C₄×C₅₃

yes

characteristic, complemented

C₂×C₃₂

C₂²×C₅₃

yes

characteristic

C₂×C₃₂

C₂²×C₅₃

yes

characteristic

#368

C₂²×C₅₃

yes

—

#359

—

C₄×C₁₆

C₂²×C₅₃

yes

characteristic

C₅₃

#13656

yes

characteristic, direct product factor, Sylow

#135

—

#131

—

Q₃₂

C₂×C₄×C₅₃

yes

—

SD₃₂

—

D₁₆

C₂×C₄×C₅₃

yes

—

C₂×C₁₆

D₄×C₅₃

yes

characteristic

C₂×C₁₆

C₂³×C₅₃

yes

characteristic

#108

—

#104

—

C₄×C₈

D₄×C₅₃

yes

characteristic

C₃₂

—

C₃₂

C₂×C₄×C₅₃

yes

characteristic

C₃₂

C₂×C₄×C₅₃

yes

characteristic

#41

—

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

D₈×C₅₃

yes

characteristic

C₂×C₈

C₂×D₄×C₅₃

yes

characteristic

100

C₄²

D₈×C₅₃

yes

characteristic

101

C₁₆

—

102

C₁₆

C₂²×C₄×C₅₃

yes

characteristic

103

C₁₆

#41×C₅₃

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂×D₈×C₅₃

yes

characteristic

110

C₈

—

111

C₈

D₁₆×C₅₃

yes

—

112

C₈

C₄×D₄×C₅₃

yes

characteristic

113

C₈

#135×C₅₃

yes

characteristic

114

C₂²

—

115

C₂²

C₅₃×#508

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

C₄×D₈×C₅₃

yes

characteristic

119

C₄

C₂×D₁₆×C₅₃

yes

characteristic

120

C₂

—

121

C₂

—

122

C₂

C₄×D₁₆×C₅₃

yes

characteristic

	subgroup	G/H
center	C₄×C₅₃	C₂×D₁₆
commutator	C₁₆	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₅₃
socle	C₂×C₅₃	C₄×D₁₆
prime radical	C₅₃×#2262	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂×C₈×C₅₃ ◁ C₄×C₈×C₅₃ ◁ C₄×C₁₆×C₅₃ ◁ G

subgroup

G/H

center

C₄×C₅₃

C₂×D₁₆

commutator

C₁₆

C₂²×C₄×C₅₃

Frattini, cosocle

C₂×C₁₆

C₂³×C₅₃

socle

C₂×C₅₃

C₄×D₁₆

prime radical

C₅₃×#2262

C₂

LCS

G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₅₃ ◁ C₂×C₈×C₅₃ ◁ C₄×C₈×C₅₃ ◁ C₄×C₁₆×C₅₃ ◁ G

N	H ≅ G/N	type
2. C₅₃×#2178	C₂	semidirect
3. C₅₃×#1425	C₂	semidirect
4. C₅₃×#1417	C₂	semidirect
5. C₅₃×#1397	C₂	semidirect
8. C₅₃×Q₆₄	C₄	semidirect
9. C₅₃×SD₆₄	C₄	semidirect
10. C₅₃×D₃₂	C₄	semidirect
52. #13656	C₅₃	direct (N × H)
60. #2178	C₂×C₅₃	semidirect
61. #1425	C₂×C₅₃	semidirect
62. #1417	C₂×C₅₃	semidirect
63. #1397	C₂×C₅₃	semidirect
69. Q₆₄	C₄×C₅₃	semidirect
70. SD₆₄	C₄×C₅₃	semidirect
71. D₃₂	C₄×C₅₃	semidirect
77. C₅₃	#13656	direct (N × H)

H ≅ G/N

type

2. C₅₃×#2178

C₂

semidirect

3. C₅₃×#1425

C₂

semidirect

4. C₅₃×#1417

C₂

semidirect

5. C₅₃×#1397

C₂

semidirect

8. C₅₃×Q₆₄

C₄

semidirect

9. C₅₃×SD₆₄

C₄

semidirect

10. C₅₃×D₃₂

C₄

semidirect

52. #13656

C₅₃

direct (N × H)

60. #2178

C₂×C₅₃

semidirect

61. #1425

C₂×C₅₃

semidirect

62. #1417

C₂×C₅₃

semidirect

63. #1397

C₂×C₅₃

semidirect

69. Q₆₄

C₄×C₅₃

semidirect

70. SD₆₄

C₄×C₅₃

semidirect

71. D₃₂

C₄×C₅₃

semidirect

77. C₅₃

#13656

direct (N × H)

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

subgroup	index
C₅₃×#2262	2
C₅₃×#2178	2
C₅₃×#1425	2
C₅₃×#1417	2
C₅₃×#1397	2
#13656	53

subgroup

index

C₅₃×#2262

C₅₃×#2178

C₅₃×#1425

C₅₃×#1417

C₅₃×#1397

#13656

Subgroups — C₅₃×#13656 #27221168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#13656 #27221168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13656 #27221168