Group #20274839 — subgroups

Subgroups — #20274839

Order 6784.

172 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 89 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×#13133153	C₂	1	1	yes	maximal, characteristic
2	#16810661	C₂	1	1	yes	maximal, characteristic
3	#16810394	C₂	1	2	yes	maximal
4	#16810379	C₂	1	2	yes	maximal
5	C₅₃×#405	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	2	yes	—
7	#13133176	C₂²	1	2	yes	—
8	#13133153	—	2	1	no	—
9	#13133153	—	2	1	no	—
10	#13133151	—	2	1	no	—
11	#13133149	—	2	1	no	—
12	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	#117×C₅₃	C₂²	1	1	yes	characteristic
15	#97×C₅₃	C₄	1	2	yes	complemented
16	C₄×C₈×C₅₃	C₄	1	2	yes	complemented
17	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	M₁₆×C₅₃	—	2	1	no	—
19	M₁₆×C₅₃	—	2	1	no	—
20	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂×C₈×C₅₃	Q₈	1	1	yes	characteristic
26	C₂×C₈×C₅₃	Q₈	1	1	yes	characteristic
27	C₄⋊C₄×C₅₃	—	2	1	no	—
28	C₂²⋊C₄×C₅₃	—	2	1	no	—
29	C₄²×C₅₃	C₂×C₄	1	2	yes	—
30	C₂²⋊Dic₅₃	—	2	2	no	—
31	C₄⋊Dic₅₃	—	2	2	no	—
32	C₄×Dic₅₃	—	2	2	no	—
33	#11719917	—	2	2	no	—
34	#11719917	—	2	2	no	—
35	C₂×#88322	—	2	2	no	—
36	C₂³×C₅₃	#41	1	1	yes	characteristic
37	C₂×C₄×C₅₃	—	2	2	no	—
38	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
42	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
43	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
44	C₂×Dic₅₃	—	4	2	no	—
45	C₈×C₅₃	—	2	2	no	—
46	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
47	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	#88322	—	4	2	no	—
51	C₂²×C₅₃	—	4	1	no	—
52	C₂²×C₅₃	#117	1	1	yes	characteristic
53	C₂²×C₅₃	#124	1	1	yes	characteristic
54	C₂²×C₅₃	#125	1	1	yes	characteristic
55	C₄×C₅₃	—	4	2	no	—
56	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₄×C₅₃	#117	1	1	yes	characteristic
58	C₄×C₅₃	#124	1	1	yes	characteristic
59	C₄×C₅₃	#125	1	1	yes	characteristic
60	Dic₅₃	—	8	2	no	—
61	#1833	—	53	1	no	maximal, Sylow
62	C₂×C₅₃	—	2	1	no	—
63	C₂×C₅₃	—	2	1	no	—
64	C₂×C₅₃	—	2	1	no	—
65	C₂×C₅₃	#382	1	1	yes	characteristic
66	C₂×#108	—	53	1	no	—
67	#427	—	53	1	no	—
68	#405	D₅₃	1	1	yes	characteristic
69	#344	—	53	2	no	—
70	H(3,ℤ/4)	—	53	2	no	—
71	C₅₃	#1833	1	1	yes	characteristic, complemented, Sylow
72	C₂×M₁₆	—	53	2	no	—
73	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
75	#117	—	53	2	no	—
76	#117	C₂×D₅₃	1	1	yes	characteristic
77	#108	—	106	1	no	—
78	#108	—	106	1	no	—
79	#107	—	106	1	no	—
80	#106	—	106	1	no	—
81	#97	Dic₅₃	1	2	yes	complemented
82	C₄×C₈	Dic₅₃	1	2	yes	complemented
83	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
84	M₁₆	—	2	1	no	—
85	M₁₆	—	2	1	no	—
86	M₁₆	—	106	2	no	—
87	M₁₆	—	106	2	no	—
88	C₂×C₈	—	106	2	no	—
89	C₂×C₈	Dic₁₀₆	1	1	yes	characteristic
90	C₂×C₈	Dic₁₀₆	1	1	yes	characteristic
91	C₂×C₈	D₂₁₂	1	1	yes	characteristic
92	C₂×C₈	D₂₁₂	1	1	yes	characteristic
93	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
94	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
95	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	106	2	no	—
98	C₂²⋊C₄	—	2	1	no	—
99	C₂²⋊C₄	—	106	2	no	—
100	C₄²	—	106	2	no	—
101	C₄²	C₂×Dic₅₃	1	2	yes	—
102	C₂³	#11719945	1	1	yes	characteristic
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	212	2	no	—
105	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
106	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
107	C₂×C₄	#11719945	1	1	yes	characteristic
108	C₂×C₄	#11719945	1	1	yes	characteristic
109	C₂×C₄	#11719945	1	1	yes	characteristic
110	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	212	2	no	—
113	C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
114	C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
115	C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
116	C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
117	C₂²	—	4	1	no	—
118	C₂²	#13133176	1	1	yes	characteristic
119	C₂²	#13133205	1	1	yes	characteristic
120	C₂²	#13133209	1	1	yes	characteristic
121	C₄	—	4	2	no	—
122	C₄	—	424	2	no	—
123	C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
124	C₄	#13133176	1	1	yes	characteristic
125	C₄	#13133205	1	1	yes	characteristic
126	C₄	#13133209	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#16810489	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133153

C₂

yes

maximal, characteristic

#16810661

C₂

yes

maximal, characteristic

#16810394

C₂

yes

maximal

#16810379

C₂

yes

maximal

C₅₃×#405

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

#13133176

C₂²

yes

—

—

—

—

—

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₂²

yes

characteristic

#97×C₅₃

C₄

yes

complemented

C₄×C₈×C₅₃

C₄

yes

complemented

C₂²×C₄×C₅₃

C₂³

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

Q₈

yes

characteristic

C₂×C₈×C₅₃

Q₈

yes

characteristic

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

#11719917

—

#11719917

—

C₂×#88322

—

C₂³×C₅₃

#41

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₄⋊C₄

yes

characteristic

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#1833

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

—

—

D₅₃

yes

characteristic

#344

—

H(3,ℤ/4)

—

C₅₃

#1833

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

characteristic

#117

—

#117

C₂×D₅₃

yes

characteristic

#108

—

106

—

#108

—

106

—

#107

—

106

—

#106

—

106

—

#97

Dic₅₃

yes

complemented

C₄×C₈

Dic₅₃

yes

complemented

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

106

—

C₂×C₈

Dic₁₀₆

yes

characteristic

C₂×C₈

Dic₁₀₆

yes

characteristic

C₂×C₈

D₂₁₂

yes

characteristic

C₂×C₈

D₂₁₂

yes

characteristic

C₂×C₈

C₂×Dic₅₃

yes

—

C₂×C₈

C₂×Dic₅₃

yes

characteristic

C₂×C₈

C₂×Dic₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

C₂²⋊C₄

—

106

—

100

C₄²

—

106

—

101

C₄²

C₂×Dic₅₃

yes

—

102

C₂³

#11719945

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

212

—

105

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

106

C₂×C₄

C₂×D₂₁₂

yes

characteristic

107

C₂×C₄

#11719945

yes

characteristic

108

C₂×C₄

#11719945

yes

characteristic

109

C₂×C₄

#11719945

yes

characteristic

110

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

111

C₈

—

112

C₈

—

212

—

113

C₈

C₄⋊Dic₅₃

yes

characteristic

114

C₈

C₄⋊Dic₅₃

yes

characteristic

115

C₈

C₄⋊Dic₅₃

yes

characteristic

116

C₈

C₄⋊Dic₅₃

yes

characteristic

117

C₂²

—

118

C₂²

#13133176

yes

characteristic

119

C₂²

#13133205

yes

characteristic

120

C₂²

#13133209

yes

characteristic

121

C₄

—

122

C₄

—

424

—

123

C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

124

C₄

#13133176

yes

characteristic

125

C₄

#13133205

yes

characteristic

126

C₄

#13133209

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#16810489

yes

characteristic

	subgroup	G/H
center	C₄	#13133176
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#405	C₂
socle	C₂×C₅₃	#382
hypercenter	#405	D₅₃
hypocenter	C₅₃	#1833
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ #117 ◁ #405

subgroup

G/H

center

C₄

#13133176

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#405

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ #117 ◁ #405

N	H ≅ G/N	type
15. #97×C₅₃	C₄	semidirect
16. C₄×C₈×C₅₃	C₄	semidirect
71. C₅₃	#1833	semidirect
81. #97	Dic₅₃	semidirect
82. C₄×C₈	Dic₅₃	semidirect

H ≅ G/N

type

15. #97×C₅₃

C₄

semidirect

16. C₄×C₈×C₅₃

C₄

semidirect

71. C₅₃

#1833

semidirect

81. #97

Dic₅₃

semidirect

82. C₄×C₈

Dic₅₃

semidirect

Subgroups — #20274839

Conjugacy classes of subgroups

Complements