Group #27220188 — subgroups

Subgroups — C₅₃×#12676 #27220188

Order 13568.

306 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 190 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	C₂×C₅₃×#368	C₂	1	2	yes	maximal
2	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
3	C₄×#108×C₅₃	C₂	1	2	yes	maximal
4	C₅₃×#1381	C₂	1	2	yes	maximal
5	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
6	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×#108×C₅₃	C₂²	1	4	yes	—
8	C₄×M₁₆×C₅₃	—	2	2	no	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#368	C₄	1	8	yes	complemented
11	C₅₃×#346	C₄	1	4	yes	—
12	C₅₃×#341	—	2	2	no	—
13	C₂×M₁₆×C₅₃	—	2	4	no	—
14	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
17	C₂×C₁₆×C₅₃	—	2	2	no	—
18	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
21	#108×C₅₃	—	2	4	no	—
22	#108×C₅₃	C₂×C₄	1	8	yes	—
23	#97×C₅₃	—	4	2	no	—
24	C₄×C₈×C₅₃	—	4	2	no	—
25	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₅₃	Q₈	1	2	yes	—
27	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
28	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
29	M₁₆×C₅₃	—	2	8	no	—
30	M₁₆×C₅₃	—	4	4	no	—
31	C₂×C₈×C₅₃	—	2	2	no	—
32	C₂×C₈×C₅₃	—	4	4	no	—
33	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
34	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
36	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₈×C₅₃	#41	1	2	yes	—
38	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
39	C₄²×C₅₃	D₈	1	2	yes	—
40	C₁₆×C₅₃	—	2	2	no	—
41	C₁₆×C₅₃	C₄²	1	4	yes	—
42	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
43	C₂×C₄×C₅₃	—	4	2	no	—
44	C₂×C₄×C₅₃	#107	1	4	yes	—
45	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
47	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#135	1	2	yes	—
49	C₈×C₅₃	—	2	2	no	—
50	C₈×C₅₃	—	4	8	no	—
51	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
52	C₈×C₅₃	#117	1	1	yes	characteristic
53	C₈×C₅₃	C₄×Q₈	1	2	yes	—
54	#12676	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂²×C₅₃	—	2	2	no	—
56	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
57	C₂²×C₅₃	C₄×Q₁₆	1	2	yes	—
58	C₄×C₅₃	—	2	4	no	—
59	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
60	C₄×C₅₃	#427	1	1	yes	characteristic
61	C₄×C₅₃	C₄×D₈	1	2	yes	—
62	C₂×#368	C₂×C₅₃	1	2	yes	—
63	C₂×#346	C₂×C₅₃	1	1	yes	characteristic
64	C₄×#108	C₂×C₅₃	1	2	yes	—
65	#1381	C₂×C₅₃	1	2	yes	—
66	C₂×C₅₃	—	2	2	no	—
67	C₂×C₅₃	C₄×#107	1	1	yes	characteristic
68	C₂×C₅₃	#2161	1	2	yes	—
69	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
70	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
71	C₂×#108	C₂²×C₅₃	1	4	yes	—
72	C₄×M₁₆	—	2	2	no	—
73	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
74	#368	C₄×C₅₃	1	8	yes	complemented
75	#346	C₄×C₅₃	1	4	yes	—
76	#341	—	2	2	no	—
77	C₅₃	#12676	1	1	yes	characteristic, direct product factor, Sylow
78	C₂×M₁₆	—	2	4	no	—
79	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
80	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
81	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
82	C₂×C₁₆	—	2	2	no	—
83	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
84	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
85	C₂×C₁₆	C₂×C₄×C₅₃	1	4	yes	—
86	#108	—	2	4	no	—
87	#108	C₂×C₄×C₅₃	1	8	yes	—
88	#97	—	4	2	no	—
89	C₄×C₈	—	4	2	no	—
90	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
91	C₄×C₈	Q₈×C₅₃	1	2	yes	—
92	C₂²×C₄	Q₁₆×C₅₃	1	2	yes	—
93	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
94	M₁₆	—	2	8	no	—
95	M₁₆	—	4	4	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	4	4	no	—
98	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
99	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
100	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
101	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
102	C₂×C₈	#41×C₅₃	1	2	yes	—
103	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
104	C₄²	D₈×C₅₃	1	2	yes	—
105	C₁₆	—	2	2	no	—
106	C₁₆	C₄²×C₅₃	1	4	yes	—
107	C₂³	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	#107×C₅₃	1	4	yes	—
110	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
111	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
112	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
113	C₂×C₄	#135×C₅₃	1	2	yes	—
114	C₈	—	2	2	no	—
115	C₈	—	4	8	no	—
116	C₈	C₂×C₄²×C₅₃	1	1	yes	characteristic
117	C₈	#117×C₅₃	1	1	yes	characteristic
118	C₈	C₄×Q₈×C₅₃	1	2	yes	—
119	C₂²	—	2	2	no	—
120	C₂²	C₂×#107×C₅₃	1	1	yes	characteristic
121	C₂²	C₄×Q₁₆×C₅₃	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
124	C₄	C₅₃×#427	1	1	yes	characteristic
125	C₄	C₄×D₈×C₅₃	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	C₄×#107×C₅₃	1	1	yes	characteristic
128	C₂	C₅₃×#2161	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅₃×#368

C₂

yes

maximal

C₂×C₅₃×#346

C₂

yes

maximal, characteristic

C₄×#108×C₅₃

C₂

yes

maximal

C₅₃×#1381

C₂

yes

maximal

C₂²×C₁₆×C₅₃

C₂²

yes

characteristic

C₂²×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×#108×C₅₃

C₂²

yes

—

C₄×M₁₆×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#368

C₄

yes

complemented

C₅₃×#346

C₄

yes

—

C₅₃×#341

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

#108×C₅₃

—

#108×C₅₃

C₂×C₄

yes

—

#97×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

Q₁₆

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²×C₄

yes

—

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₄²×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

D₈

yes

—

C₁₆×C₅₃

—

C₁₆×C₅₃

C₄²

yes

—

C₂³×C₅₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#107

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂×C₄²

yes

characteristic

C₈×C₅₃

#117

yes

characteristic

C₈×C₅₃

C₄×Q₈

yes

—

#12676

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

C₂×#107

yes

characteristic

C₂²×C₅₃

C₄×Q₁₆

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₄×C₄⋊C₄

yes

characteristic

C₄×C₅₃

#427

yes

characteristic

C₄×C₅₃

C₄×D₈

yes

—

C₂×#368

C₂×C₅₃

yes

—

C₂×#346

C₂×C₅₃

yes

characteristic

C₄×#108

C₂×C₅₃

yes

—

#1381

C₂×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

C₄×#107

yes

characteristic

C₂×C₅₃

#2161

yes

—

C₂²×C₁₆

C₂²×C₅₃

yes

characteristic

C₂²×C₁₆

C₂²×C₅₃

yes

characteristic

C₂×#108

C₂²×C₅₃

yes

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

#368

C₄×C₅₃

yes

complemented

#346

C₄×C₅₃

yes

—

#341

—

C₅₃

#12676

yes

characteristic, direct product factor, Sylow

C₂×M₁₆

—

C₂²×C₈

D₄×C₅₃

yes

characteristic

C₂²×C₈

C₂³×C₅₃

yes

characteristic

C₂×C₄²

D₄×C₅₃

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄×C₅₃

yes

—

C₂×C₁₆

C₂×C₄×C₅₃

yes

—

C₂×C₁₆

C₂×C₄×C₅₃

yes

—

#108

—

#108

C₂×C₄×C₅₃

yes

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×C₄×C₅₃

yes

—

C₄×C₈

Q₈×C₅₃

yes

—

C₂²×C₄

Q₁₆×C₅₃

yes

—

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₅₃

yes

—

C₂×C₈

C₂²×C₄×C₅₃

yes

characteristic

100

C₂×C₈

C₂²×C₄×C₅₃

yes

—

101

C₂×C₈

C₂×Q₈×C₅₃

yes

characteristic

102

C₂×C₈

#41×C₅₃

yes

—

103

C₄²

C₄⋊C₄×C₅₃

yes

—

104

C₄²

D₈×C₅₃

yes

—

105

C₁₆

—

106

C₁₆

C₄²×C₅₃

yes

—

107

C₂³

C₂×Q₁₆×C₅₃

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

#107×C₅₃

yes

—

110

C₂×C₄

C₂×C₄⋊C₄×C₅₃

yes

characteristic

111

C₂×C₄

C₄×D₄×C₅₃

yes

—

112

C₂×C₄

C₂×D₈×C₅₃

yes

characteristic

113

C₂×C₄

#135×C₅₃

yes

—

114

C₈

—

115

C₈

—

116

C₈

C₂×C₄²×C₅₃

yes

characteristic

117

C₈

#117×C₅₃

yes

characteristic

118

C₈

C₄×Q₈×C₅₃

yes

—

119

C₂²

—

120

C₂²

C₂×#107×C₅₃

yes

characteristic

121

C₂²

C₄×Q₁₆×C₅₃

yes

—

122

C₄

—

123

C₄

C₄×C₄⋊C₄×C₅₃

yes

characteristic

124

C₄

C₅₃×#427

yes

characteristic

125

C₄

C₄×D₈×C₅₃

yes

—

126

C₂

—

127

C₂

C₄×#107×C₅₃

yes

characteristic

128

C₂

C₅₃×#2161

yes

—

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂×D₈
commutator	C₈	C₂×C₄²×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂³×C₅₃	C₂×Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

subgroup

G/H

center

C₂×C₄×C₅₃

C₂×D₈

commutator

C₈

C₂×C₄²×C₅₃

Frattini, cosocle

C₂²×C₈

C₂³×C₅₃

socle

C₂²×C₅₃

C₂×#107

prime radical

C₂³×C₅₃

C₂×Q₁₆

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

N	H ≅ G/N	type
10. C₅₃×#368	C₄	semidirect
54. #12676	C₅₃	direct (N × H)
74. #368	C₄×C₅₃	semidirect
77. C₅₃	#12676	direct (N × H)

H ≅ G/N

type

10. C₅₃×#368

C₄

semidirect

54. #12676

C₅₃

direct (N × H)

74. #368

C₄×C₅₃

semidirect

77. C₅₃

#12676

direct (N × H)

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

subgroup	index
C₂×C₅₃×#368	2
C₂×C₅₃×#346	2
C₄×#108×C₅₃	2
C₅₃×#1381	2
#12676	53

subgroup

index

C₂×C₅₃×#368

C₂×C₅₃×#346

C₄×#108×C₅₃

C₅₃×#1381

#12676

Subgroups — C₅₃×#12676 #27220188

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#12676 #27220188

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12676 #27220188