Group #27216335 — subgroups

Subgroups — C₅₃×#8823 #27216335

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×#1567	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×#335	—	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	D₈	1	4	yes	—
34	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
35	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₅₃	C₂×D₄	1	2	yes	—
37	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
38	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	2	4	no	—
40	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
44	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
45	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
46	C₈×C₅₃	—	2	4	no	—
47	C₈×C₅₃	—	8	4	no	—
48	C₈×C₅₃	#107	1	4	yes	—
49	C₈×C₅₃	#107	1	4	yes	—
50	C₈×C₅₃	#108	1	8	yes	complemented
51	#8823	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂²×C₅₃	—	2	2	no	—
53	C₂²×C₅₃	#384	1	1	yes	characteristic
54	C₂²×C₅₃	#500	1	2	yes	—
55	C₄×C₅₃	—	2	2	no	—
56	C₄×C₅₃	C₂×#107	1	2	yes	—
57	C₄×C₅₃	C₂×#108	1	4	yes	—
58	C₄×C₅₃	#493	1	1	yes	characteristic
59	C₄×C₅₃	#494	1	1	yes	characteristic
60	#1848	C₂×C₅₃	1	2	yes	—
61	#1567	C₂×C₅₃	1	4	yes	—
62	C₂×C₈²	C₂×C₅₃	1	1	yes	characteristic
63	C₂×C₅₃	—	2	2	no	—
64	C₂×C₅₃	#1845	1	1	yes	characteristic
65	C₂×C₅₃	#1848	1	2	yes	—
66	C₂×#108	—	2	4	no	—
67	#423	C₂²×C₅₃	1	4	yes	—
68	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
69	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
70	#335	—	2	8	no	—
71	C₈²	C₄×C₅₃	1	1	yes	characteristic
72	C₈²	C₄×C₅₃	1	2	yes	—
73	C₈²	C₄×C₅₃	1	1	yes	characteristic
74	C₅₃	#8823	1	1	yes	characteristic, direct product factor, Sylow
75	C₂×M₁₆	—	2	4	no	—
76	C₂²×C₈	D₄×C₅₃	1	2	yes	—
77	C₂²×C₈	Q₈×C₅₃	1	4	yes	—
78	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
79	#108	—	4	8	no	—
80	C₄⋊C₈	—	2	8	no	—
81	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
82	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
83	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
84	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
85	C₄×C₈	D₄×C₅₃	1	4	yes	—
86	C₄×C₈	Q₈×C₅₃	1	2	yes	—
87	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
88	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
89	M₁₆	—	4	8	no	—
90	C₂×C₈	—	2	8	no	—
91	C₂×C₈	—	4	4	no	—
92	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
93	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
94	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
95	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
96	C₂×C₈	D₈×C₅₃	1	4	yes	—
97	C₂×C₈	Q₁₆×C₅₃	1	4	yes	—
98	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
99	C₄²	C₂×D₄×C₅₃	1	2	yes	—
100	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
101	C₂³	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
104	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
105	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
106	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
107	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
108	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
109	C₈	—	2	4	no	—
110	C₈	—	8	4	no	—
111	C₈	#107×C₅₃	1	4	yes	—
112	C₈	#107×C₅₃	1	4	yes	—
113	C₈	#108×C₅₃	1	8	yes	complemented
114	C₂²	—	2	2	no	—
115	C₂²	C₅₃×#384	1	1	yes	characteristic
116	C₂²	C₅₃×#500	1	2	yes	—
117	C₄	—	2	2	no	—
118	C₄	C₂×#107×C₅₃	1	2	yes	—
119	C₄	C₂×#108×C₅₃	1	4	yes	—
120	C₄	C₅₃×#493	1	1	yes	characteristic
121	C₄	C₅₃×#494	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	C₅₃×#1845	1	1	yes	characteristic
124	C₂	C₅₃×#1848	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#1848

C₂

yes

maximal

C₅₃×#1567

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×#335

—

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

D₈

yes

—

C₂×C₈×C₅₃

Q₁₆

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×D₈

yes

—

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#107

yes

—

C₈×C₅₃

#107

yes

—

C₈×C₅₃

#108

yes

complemented

#8823

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#384

yes

characteristic

C₂²×C₅₃

#500

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#107

yes

—

C₄×C₅₃

C₂×#108

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₂×C₅₃

yes

—

#1567

C₂×C₅₃

yes

—

C₂×C₈²

C₂×C₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

#1845

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

#335

—

C₈²

C₄×C₅₃

yes

characteristic

C₈²

C₄×C₅₃

yes

—

C₈²

C₄×C₅₃

yes

characteristic

C₅₃

#8823

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

D₈×C₅₃

yes

—

C₂×C₈

Q₁₆×C₅₃

yes

—

C₄²

C₂²×C₄×C₅₃

yes

characteristic

C₄²

C₂×D₄×C₅₃

yes

—

100

C₄²

C₂×Q₈×C₅₃

yes

characteristic

101

C₂³

C₄⋊Q₈×C₅₃

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

C₂×C₄⋊C₄×C₅₃

yes

characteristic

104

C₂×C₄

C₂×C₄⋊C₄×C₅₃

yes

—

105

C₂×C₄

#127×C₅₃

yes

characteristic

106

C₂×C₄

C₄⋊Q₈×C₅₃

yes

—

107

C₂×C₄

C₂×D₈×C₅₃

yes

—

108

C₂×C₄

C₂×Q₁₆×C₅₃

yes

—

109

C₈

—

110

C₈

—

111

C₈

#107×C₅₃

yes

—

112

C₈

#107×C₅₃

yes

—

113

C₈

#108×C₅₃

yes

complemented

114

C₂²

—

115

C₂²

C₅₃×#384

yes

characteristic

116

C₂²

C₅₃×#500

yes

—

117

C₄

—

118

C₄

C₂×#107×C₅₃

yes

—

119

C₄

C₂×#108×C₅₃

yes

—

120

C₄

C₅₃×#493

yes

characteristic

121

C₄

C₅₃×#494

yes

characteristic

122

C₂

—

123

C₂

C₅₃×#1845

yes

characteristic

124

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
50. C₈×C₅₃	#108	semidirect
51. #8823	C₅₃	direct (N × H)
74. C₅₃	#8823	direct (N × H)
113. C₈	#108×C₅₃	semidirect

H ≅ G/N

type

50. C₈×C₅₃

#108

semidirect

51. #8823

C₅₃

direct (N × H)

74. C₅₃

#8823

direct (N × H)

113. 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×#1567	2
C₂×C₈²×C₅₃	2
#8823	53

subgroup

index

C₅₃×#1848

C₅₃×#1567

C₂×C₈²×C₅₃

#8823

Subgroups — C₅₃×#8823 #27216335

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#8823 #27216335

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8823 #27216335