Group #27221172 — subgroups

Subgroups — C₅₃×#13660 #27221172

Order 13568.

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

normal subgroups only

This group has 158 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	Q₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	C₅₃×C₄⋊C₃₂	C₂	1	2	yes	maximal
4	C₅₃×#1397	C₂	1	2	yes	maximal
5	C₄×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₈×Q₈×C₅₃	C₄	1	1	yes	characteristic
7	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₃₂×C₅₃	C₂²	1	2	yes	—
9	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×C₄⋊C₁₆	C₄	1	1	yes	characteristic
11	C₅₃×C₄⋊C₁₆	C₄	1	2	yes	—
12	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
14	C₄×Q₈×C₅₃	C₈	1	1	yes	characteristic
15	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
17	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
18	C₄⋊C₈×C₅₃	C₈	1	2	yes	—
19	C₄⋊C₈×C₅₃	C₈	1	1	yes	characteristic
20	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₃₂×C₅₃	—	2	1	no	—
23	C₃₂×C₅₃	—	2	2	no	—
24	C₃₂×C₅₃	Q₈	1	2	yes	complemented
25	C₂×Q₈×C₅₃	C₁₆	1	1	yes	characteristic
26	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
27	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
29	C₄⋊C₄×C₅₃	C₁₆	1	2	yes	—
30	C₄⋊C₄×C₅₃	C₁₆	1	1	yes	characteristic
31	C₄²×C₅₃	C₂×C₈	1	2	yes	—
32	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
33	C₁₆×C₅₃	—	2	1	no	—
34	C₁₆×C₅₃	—	2	2	no	—
35	C₁₆×C₅₃	C₂×Q₈	1	1	yes	characteristic
36	C₁₆×C₅₃	#41	1	1	yes	characteristic
37	Q₈×C₅₃	—	2	2	no	—
38	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
39	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
40	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
43	C₈×C₅₃	—	2	1	no	—
44	C₈×C₅₃	—	2	2	no	—
45	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
46	C₈×C₅₃	#131	1	1	yes	characteristic
47	#13660	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
49	C₄×C₅₃	—	2	2	no	—
50	C₄×C₅₃	—	2	2	no	—
51	C₄×C₅₃	—	2	1	no	—
52	C₄×C₅₃	M₆₄	1	2	yes	—
53	C₄×C₅₃	C₈×Q₈	1	1	yes	characteristic
54	C₄×C₅₃	#504	1	1	yes	characteristic
55	Q₈×C₁₆	C₂×C₅₃	1	1	yes	characteristic
56	C₄⋊C₃₂	C₂×C₅₃	1	1	yes	characteristic
57	C₄⋊C₃₂	C₂×C₅₃	1	2	yes	—
58	#1397	C₂×C₅₃	1	2	yes	—
59	C₄×C₃₂	C₂×C₅₃	1	1	yes	characteristic
60	C₂×C₅₃	Q₈×C₁₆	1	1	yes	characteristic
61	C₂×C₅₃	C₂×M₆₄	1	1	yes	characteristic
62	C₂×C₅₃	#2258	1	1	yes	characteristic
63	C₈×Q₈	C₄×C₅₃	1	1	yes	characteristic
64	C₂×C₃₂	C₂²×C₅₃	1	1	yes	characteristic
65	C₂×C₃₂	C₂²×C₅₃	1	2	yes	—
66	C₂×C₃₂	C₂²×C₅₃	1	1	yes	characteristic
67	C₄⋊C₁₆	C₄×C₅₃	1	1	yes	characteristic
68	C₄⋊C₁₆	C₄×C₅₃	1	2	yes	—
69	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
70	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
71	C₅₃	#13660	1	1	yes	characteristic, direct product factor, Sylow
72	C₄×Q₈	C₈×C₅₃	1	1	yes	characteristic
73	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
74	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
75	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
76	C₄⋊C₈	C₈×C₅₃	1	2	yes	—
77	C₄⋊C₈	C₈×C₅₃	1	1	yes	characteristic
78	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
79	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
80	C₃₂	—	2	1	no	—
81	C₃₂	—	2	2	no	—
82	C₃₂	Q₈×C₅₃	1	2	yes	complemented
83	C₂×Q₈	C₁₆×C₅₃	1	1	yes	characteristic
84	C₂×C₈	C₂×C₈×C₅₃	1	1	yes	characteristic
85	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
86	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
87	C₄⋊C₄	C₁₆×C₅₃	1	2	yes	—
88	C₄⋊C₄	C₁₆×C₅₃	1	1	yes	characteristic
89	C₄²	C₂×C₈×C₅₃	1	2	yes	—
90	C₄²	C₂×C₈×C₅₃	1	1	yes	characteristic
91	C₁₆	—	2	1	no	—
92	C₁₆	—	2	2	no	—
93	C₁₆	C₂×Q₈×C₅₃	1	1	yes	characteristic
94	C₁₆	#41×C₅₃	1	1	yes	characteristic
95	Q₈	—	2	2	no	—
96	C₂×C₄	C₂×C₁₆×C₅₃	1	2	yes	—
97	C₂×C₄	C₂×C₁₆×C₅₃	1	1	yes	characteristic
98	C₂×C₄	C₂×C₁₆×C₅₃	1	2	yes	—
99	C₂×C₄	C₂×C₁₆×C₅₃	1	1	yes	characteristic
100	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	2	2	no	—
103	C₈	C₄×Q₈×C₅₃	1	1	yes	characteristic
104	C₈	#131×C₅₃	1	1	yes	characteristic
105	C₂²	C₂²×C₁₆×C₅₃	1	1	yes	characteristic
106	C₄	—	2	2	no	—
107	C₄	—	2	2	no	—
108	C₄	—	2	1	no	—
109	C₄	C₅₃×M₆₄	1	2	yes	—
110	C₄	C₈×Q₈×C₅₃	1	1	yes	characteristic
111	C₄	C₅₃×#504	1	1	yes	characteristic
112	C₂	Q₈×C₁₆×C₅₃	1	1	yes	characteristic
113	C₂	C₂×C₅₃×M₆₄	1	1	yes	characteristic
114	C₂	C₅₃×#2258	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×C₁₆×C₅₃

C₂

yes

maximal, characteristic

C₅₃×C₄⋊C₃₂

C₂

yes

maximal, characteristic

C₅₃×C₄⋊C₃₂

C₂

yes

maximal

C₅₃×#1397

C₂

yes

maximal

C₄×C₃₂×C₅₃

C₂

yes

maximal, characteristic

C₈×Q₈×C₅₃

C₄

yes

characteristic

C₂×C₃₂×C₅₃

C₂²

yes

characteristic

C₂×C₃₂×C₅₃

C₂²

yes

—

C₂×C₃₂×C₅₃

C₂²

yes

characteristic

C₅₃×C₄⋊C₁₆

C₄

yes

characteristic

C₅₃×C₄⋊C₁₆

C₄

yes

—

C₄×C₁₆×C₅₃

C₂²

yes

characteristic

C₄×C₁₆×C₅₃

C₂²

yes

—

C₄×Q₈×C₅₃

C₈

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₁₆×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

C₈

yes

—

C₄⋊C₈×C₅₃

C₈

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₃₂×C₅₃

—

C₃₂×C₅₃

—

C₃₂×C₅₃

Q₈

yes

complemented

C₂×Q₈×C₅₃

C₁₆

yes

characteristic

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₅₃

C₁₆

yes

—

C₄⋊C₄×C₅₃

C₁₆

yes

characteristic

C₄²×C₅₃

C₂×C₈

yes

—

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₁₆×C₅₃

—

C₁₆×C₅₃

—

C₁₆×C₅₃

C₂×Q₈

yes

characteristic

C₁₆×C₅₃

#41

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

C₂×C₁₆

yes

—

C₂×C₄×C₅₃

C₂×C₁₆

yes

—

C₂×C₄×C₅₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₄×Q₈

yes

C₈×C₅₃

yes

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

C₂²×C₁₆

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

M₆₄

yes

—

C₄×C₅₃

C₈×Q₈

yes

characteristic

C₄×C₅₃

#504

yes

characteristic

Q₈×C₁₆

C₂×C₅₃

yes

characteristic

C₄⋊C₃₂

C₂×C₅₃

yes

characteristic

C₄⋊C₃₂

C₂×C₅₃

yes

—

#1397

C₂×C₅₃

yes

—

C₄×C₃₂

C₂×C₅₃

yes

characteristic

C₂×C₅₃

Q₈×C₁₆

yes

characteristic

C₂×C₅₃

C₂×M₆₄

yes

characteristic

C₂×C₅₃

#2258

yes

characteristic

C₈×Q₈

C₄×C₅₃

yes

characteristic

C₂×C₃₂

C₂²×C₅₃

yes

characteristic

C₂×C₃₂

C₂²×C₅₃

yes

—

C₂×C₃₂

C₂²×C₅₃

yes

characteristic

C₄⋊C₁₆

C₄×C₅₃

yes

characteristic

C₄⋊C₁₆

C₄×C₅₃

yes

—

C₄×C₁₆

C₂²×C₅₃

yes

characteristic

C₄×C₁₆

C₂²×C₅₃

yes

—

C₅₃

#13660

yes

characteristic, direct product factor, Sylow

C₄×Q₈

C₈×C₅₃

yes

characteristic

C₂×C₁₆

C₂×C₄×C₅₃

yes

characteristic

C₂×C₁₆

C₂×C₄×C₅₃

yes

—

C₂×C₁₆

C₂³×C₅₃

yes

characteristic

C₄⋊C₈

C₈×C₅₃

yes

—

C₄⋊C₈

C₈×C₅₃

yes

characteristic

C₄×C₈

C₂×C₄×C₅₃

yes

characteristic

C₄×C₈

C₂×C₄×C₅₃

yes

—

C₃₂

—

C₃₂

—

C₃₂

Q₈×C₅₃

yes

complemented

C₂×Q₈

C₁₆×C₅₃

yes

characteristic

C₂×C₈

C₂×C₈×C₅₃

yes

characteristic

C₂×C₈

C₂×C₈×C₅₃

yes

—

C₂×C₈

C₂²×C₄×C₅₃

yes

characteristic

C₄⋊C₄

C₁₆×C₅₃

yes

—

C₄⋊C₄

C₁₆×C₅₃

yes

characteristic

C₄²

C₂×C₈×C₅₃

yes

—

C₄²

C₂×C₈×C₅₃

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

C₂×Q₈×C₅₃

yes

characteristic

C₁₆

#41×C₅₃

yes

characteristic

Q₈

—

C₂×C₄

C₂×C₁₆×C₅₃

yes

—

C₂×C₄

C₂×C₁₆×C₅₃

yes

characteristic

C₂×C₄

C₂×C₁₆×C₅₃

yes

—

C₂×C₄

C₂×C₁₆×C₅₃

yes

characteristic

100

C₂×C₄

C₂²×C₈×C₅₃

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

C₄×Q₈×C₅₃

yes

characteristic

104

C₈

#131×C₅₃

yes

characteristic

105

C₂²

C₂²×C₁₆×C₅₃

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

C₅₃×M₆₄

yes

—

110

C₄

C₈×Q₈×C₅₃

yes

characteristic

111

C₄

C₅₃×#504

yes

characteristic

112

C₂

Q₈×C₁₆×C₅₃

yes

characteristic

113

C₂

C₂×C₅₃×M₆₄

yes

characteristic

114

C₂

C₅₃×#2258

yes

characteristic

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

subgroup

G/H

center

C₂×C₁₆×C₅₃

C₂³

commutator

C₂²

C₂²×C₁₆×C₅₃

Frattini, cosocle

C₂×C₁₆

C₂³×C₅₃

socle, prime radical

C₂²×C₅₃

C₂²×C₁₆

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₁₆×C₅₃ ◁ G

N	H ≅ G/N	type
24. C₃₂×C₅₃	Q₈	semidirect
47. #13660	C₅₃	direct (N × H)
71. C₅₃	#13660	direct (N × H)
82. C₃₂	Q₈×C₅₃	semidirect

H ≅ G/N

type

24. C₃₂×C₅₃

Q₈

semidirect

47. #13660

C₅₃

direct (N × H)

71. C₅₃

#13660

direct (N × H)

82. C₃₂

Q₈×C₅₃

semidirect

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

subgroup	index
Q₈×C₁₆×C₅₃	2
C₅₃×C₄⋊C₃₂	2
C₅₃×#1397	2
C₄×C₃₂×C₅₃	2
#13660	53

subgroup

index

Q₈×C₁₆×C₅₃

C₅₃×C₄⋊C₃₂

C₅₃×#1397

C₄×C₃₂×C₅₃

#13660

Subgroups — C₅₃×#13660 #27221172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#13660 #27221172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13660 #27221172