Group #27216334 — subgroups

Subgroups — C₅₃×#8822 #27216334

Order 13568.

354 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 214 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	2	yes	maximal
2	C₅₃×#1566	C₂	1	4	yes	maximal
3	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#108×C₅₃	—	2	4	no	—
5	C₅₃×#423	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#334	—	2	8	no	—
9	C₈²×C₅₃	C₄	1	1	yes	characteristic
10	C₈²×C₅₃	C₄	1	2	yes	—
11	C₈²×C₅₃	C₄	1	1	yes	characteristic
12	C₂×M₁₆×C₅₃	—	2	4	no	—
13	C₂²×C₈×C₅₃	D₄	1	2	yes	—
14	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
15	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
16	#108×C₅₃	—	4	8	no	—
17	C₄⋊C₈×C₅₃	—	2	8	no	—
18	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₅₃	D₄	1	4	yes	—
23	C₄×C₈×C₅₃	Q₈	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
26	M₁₆×C₅₃	—	4	8	no	—
27	C₂×C₈×C₅₃	—	2	8	no	—
28	C₂×C₈×C₅₃	—	4	4	no	—
29	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
30	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
32	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
33	C₂×C₈×C₅₃	SD₁₆	1	8	yes	—
34	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₄²×C₅₃	C₂×D₄	1	2	yes	—
36	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
37	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
38	C₂×C₄×C₅₃	—	2	4	no	—
39	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
41	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
43	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
44	C₈×C₅₃	—	2	4	no	—
45	C₈×C₅₃	—	8	4	no	—
46	C₈×C₅₃	#106	1	4	yes	—
47	C₈×C₅₃	#106	1	4	yes	—
48	C₈×C₅₃	#108	1	8	yes	complemented
49	#8822	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂²×C₅₃	—	2	2	no	—
51	C₂²×C₅₃	#384	1	1	yes	characteristic
52	C₂²×C₅₃	#498	1	2	yes	—
53	C₄×C₅₃	—	2	2	no	—
54	C₄×C₅₃	C₂×#106	1	2	yes	—
55	C₄×C₅₃	C₂×#108	1	4	yes	—
56	C₄×C₅₃	#492	1	2	yes	—
57	#1848	C₂×C₅₃	1	2	yes	—
58	#1566	C₂×C₅₃	1	4	yes	—
59	C₂×C₈²	C₂×C₅₃	1	1	yes	characteristic
60	C₂×C₅₃	—	2	2	no	—
61	C₂×C₅₃	#1844	1	1	yes	characteristic
62	C₂×C₅₃	#1848	1	2	yes	—
63	C₂×#108	—	2	4	no	—
64	#423	C₂²×C₅₃	1	4	yes	—
65	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
66	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
67	#334	—	2	8	no	—
68	C₈²	C₄×C₅₃	1	1	yes	characteristic
69	C₈²	C₄×C₅₃	1	2	yes	—
70	C₈²	C₄×C₅₃	1	1	yes	characteristic
71	C₅₃	#8822	1	1	yes	characteristic, direct product factor, Sylow
72	C₂×M₁₆	—	2	4	no	—
73	C₂²×C₈	D₄×C₅₃	1	2	yes	—
74	C₂²×C₈	Q₈×C₅₃	1	4	yes	—
75	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
76	#108	—	4	8	no	—
77	C₄⋊C₈	—	2	8	no	—
78	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
79	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
80	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
81	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
82	C₄×C₈	D₄×C₅₃	1	4	yes	—
83	C₄×C₈	Q₈×C₅₃	1	2	yes	—
84	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
85	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
86	M₁₆	—	4	8	no	—
87	C₂×C₈	—	2	8	no	—
88	C₂×C₈	—	4	4	no	—
89	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
90	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
91	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
92	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
93	C₂×C₈	SD₁₆×C₅₃	1	8	yes	—
94	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
95	C₄²	C₂×D₄×C₅₃	1	2	yes	—
96	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
97	C₂³	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
100	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
101	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
102	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
103	C₂×C₄	C₂×SD₁₆×C₅₃	1	4	yes	—
104	C₈	—	2	4	no	—
105	C₈	—	8	4	no	—
106	C₈	#106×C₅₃	1	4	yes	—
107	C₈	#106×C₅₃	1	4	yes	—
108	C₈	#108×C₅₃	1	8	yes	complemented
109	C₂²	—	2	2	no	—
110	C₂²	C₅₃×#384	1	1	yes	characteristic
111	C₂²	C₅₃×#498	1	2	yes	—
112	C₄	—	2	2	no	—
113	C₄	C₂×#106×C₅₃	1	2	yes	—
114	C₄	C₂×#108×C₅₃	1	4	yes	—
115	C₄	C₅₃×#492	1	2	yes	—
116	C₂	—	2	2	no	—
117	C₂	C₅₃×#1844	1	1	yes	characteristic
118	C₂	C₅₃×#1848	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#1848

C₂

yes

maximal

C₅₃×#1566

C₂

yes

maximal

C₂×C₈²×C₅₃

C₂

yes

maximal, characteristic

C₂×#108×C₅₃

—

C₅₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#334

—

C₈²×C₅₃

C₄

yes

characteristic

C₈²×C₅₃

C₄

yes

—

C₈²×C₅₃

C₄

yes

characteristic

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

Q₈

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#108×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₂³×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#106

yes

—

C₈×C₅₃

#106

yes

—

C₈×C₅₃

#108

yes

complemented

#8822

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#384

yes

characteristic

C₂²×C₅₃

#498

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#106

yes

—

C₄×C₅₃

C₂×#108

yes

—

C₄×C₅₃

#492

yes

—

#1848

C₂×C₅₃

yes

—

#1566

C₂×C₅₃

yes

—

C₂×C₈²

C₂×C₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

#1844

yes

characteristic

C₂×C₅₃

#1848

yes

—

C₂×#108

—

#423

C₂²×C₅₃

yes

—

C₂×C₄×C₈

C₂²×C₅₃

yes

—

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

#334

—

C₈²

C₄×C₅₃

yes

characteristic

C₈²

C₄×C₅₃

yes

—

C₈²

C₄×C₅₃

yes

characteristic

C₅₃

#8822

yes

characteristic, direct product factor, Sylow

C₂×M₁₆

—

C₂²×C₈

D₄×C₅₃

yes

—

C₂²×C₈

Q₈×C₅₃

yes

—

C₂×C₄²

C₂³×C₅₃

yes

characteristic

#108

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄×C₅₃

yes

—

C₄×C₈

C₂×C₄×C₅₃

yes

characteristic

C₄×C₈

C₂×C₄×C₅₃

yes

characteristic

C₄×C₈

C₂×C₄×C₅₃

yes

—

C₄×C₈

D₄×C₅₃

yes

—

C₄×C₈

Q₈×C₅₃

yes

—

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

C₂²×C₄

C₂×Q₈×C₅₃

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₅₃

yes

—

C₂×C₈

C₄⋊C₄×C₅₃

yes

—

C₂×C₈

C₄⋊C₄×C₅₃

yes

—

C₂×C₈

C₄⋊C₄×C₅₃

yes

—

C₂×C₈

SD₁₆×C₅₃

yes

—

C₄²

C₂²×C₄×C₅₃

yes

characteristic

C₄²

C₂×D₄×C₅₃

yes

—

C₄²

C₂×Q₈×C₅₃

yes

characteristic

C₂³

C₄⋊Q₈×C₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₄×C₅₃

yes

characteristic

100

C₂×C₄

C₂×C₄⋊C₄×C₅₃

yes

—

101

C₂×C₄

#127×C₅₃

yes

characteristic

102

C₂×C₄

C₄⋊Q₈×C₅₃

yes

—

103

C₂×C₄

C₂×SD₁₆×C₅₃

yes

—

104

C₈

—

105

C₈

—

106

C₈

#106×C₅₃

yes

—

107

C₈

#106×C₅₃

yes

—

108

C₈

#108×C₅₃

yes

complemented

109

C₂²

—

110

C₂²

C₅₃×#384

yes

characteristic

111

C₂²

C₅₃×#498

yes

—

112

C₄

—

113

C₄

C₂×#106×C₅₃

yes

—

114

C₄

C₂×#108×C₅₃

yes

—

115

C₄

C₅₃×#492

yes

—

116

C₂

—

117

C₂

C₅₃×#1844

yes

characteristic

118

C₂

C₅₃×#1848

yes

—

	subgroup	G/H
center	C₂×C₄×C₅₃	#127
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#384
prime radical	C₂³×C₅₃	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center

C₂×C₄×C₅₃

#127

commutator

C₄²

C₂²×C₄×C₅₃

Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

socle

C₂²×C₅₃

#384

prime radical

C₂³×C₅₃

C₄⋊Q₈

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
48. C₈×C₅₃	#108	semidirect
49. #8822	C₅₃	direct (N × H)
71. C₅₃	#8822	direct (N × H)
108. C₈	#108×C₅₃	semidirect

H ≅ G/N

type

48. C₈×C₅₃

#108

semidirect

49. #8822

C₅₃

direct (N × H)

71. C₅₃

#8822

direct (N × H)

108. C₈

#108×C₅₃

semidirect

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

subgroup	index
C₅₃×#1848	2
C₅₃×#1566	2
C₂×C₈²×C₅₃	2
#8822	53

subgroup

index

C₅₃×#1848

C₅₃×#1566

C₂×C₈²×C₅₃

#8822

Subgroups — C₅₃×#8822 #27216334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#8822 #27216334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8822 #27216334