Group #20274840 — subgroups

Subgroups — #20274840

Order 6784.

172 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 81 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	#16810661	C₂	1	1	yes	maximal, characteristic
2	#16810578	C₂	1	1	yes	maximal, characteristic
3	#16810395	C₂	1	2	yes	maximal
4	#16810380	C₂	1	2	yes	maximal
5	C₂×#108×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	1	yes	characteristic
7	#13133176	C₂²	1	2	yes	—
8	#13133176	C₂²	1	1	yes	characteristic
9	#13133151	—	2	1	no	—
10	#13133149	—	2	1	no	—
11	#13133127	C₄	1	2	yes	complemented
12	C₈×Dic₅₃	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
14	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	#108×C₅₃	—	2	1	no	—
16	#108×C₅₃	—	2	1	no	—
17	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	M₁₆×C₅₃	—	2	2	no	—
19	M₁₆×C₅₃	—	2	2	no	—
20	C₂×C₈×C₅₃	—	2	2	no	—
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	C₂²⋊Dic₅₃	—	2	1	no	—
28	C₂²⋊Dic₅₃	—	2	2	no	—
29	C₄⋊Dic₅₃	—	2	1	no	—
30	C₄⋊Dic₅₃	—	2	2	no	—
31	C₄×Dic₅₃	—	2	2	no	—
32	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
33	#11719917	—	2	1	no	—
34	#11719917	—	2	1	no	—
35	C₂×#88322	C₂×C₄	1	2	yes	—
36	C₂³×C₅₃	#41	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
41	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
42	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
43	C₂×Dic₅₃	—	2	2	no	—
44	C₂×Dic₅₃	—	4	2	no	—
45	C₈×C₅₃	—	4	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	—	2	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	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₄×C₅₃	#117	1	1	yes	characteristic
57	C₄×C₅₃	#124	1	1	yes	characteristic
58	C₄×C₅₃	#125	1	1	yes	characteristic
59	Dic₅₃	—	4	2	no	—
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	D₅₃	1	1	yes	characteristic
67	#427	—	53	1	no	—
68	#405	—	53	1	no	—
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	1	no	—
73	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
74	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
75	#117	—	53	1	no	—
76	#117	—	53	2	no	—
77	#108	—	2	1	no	—
78	#108	—	2	1	no	—
79	#107	—	106	1	no	—
80	#106	—	106	1	no	—
81	#97	—	53	2	no	—
82	C₄×C₈	—	53	2	no	—
83	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	—	2	2	no	—
86	M₁₆	—	106	1	no	—
87	M₁₆	—	106	1	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	53	2	no	—
90	C₂×C₈	Dic₁₀₆	1	1	yes	characteristic
91	C₂×C₈	Dic₁₀₆	1	1	yes	characteristic
92	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
93	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
94	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
95	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
96	C₄⋊C₄	—	106	1	no	—
97	C₄⋊C₄	—	106	2	no	—
98	C₂²⋊C₄	—	106	1	no	—
99	C₂²⋊C₄	—	106	2	no	—
100	C₄²	—	53	2	no	—
101	C₄²	—	106	2	no	—
102	C₂³	#11719949	1	1	yes	characteristic
103	C₂×C₄	—	106	2	no	—
104	C₂×C₄	—	212	2	no	—
105	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
106	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
107	C₂×C₄	#11719947	1	1	yes	characteristic
108	C₂×C₄	#11719947	1	1	yes	characteristic
109	C₂×C₄	#11719949	1	1	yes	characteristic
110	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
111	C₈	—	4	2	no	—
112	C₈	—	106	2	no	—
113	C₈	#11719919	1	1	yes	characteristic
114	C₈	#11719919	1	1	yes	characteristic
115	C₈	#11719919	1	1	yes	characteristic
116	C₈	#11719919	1	1	yes	characteristic
117	C₂²	—	4	1	no	—
118	C₂²	#13133175	1	1	yes	characteristic
119	C₂²	#13133206	1	1	yes	characteristic
120	C₂²	#13133208	1	1	yes	characteristic
121	C₄	—	424	2	no	—
122	C₄	—	212	2	no	—
123	C₄	C₂×#11719919	1	1	yes	characteristic
124	C₄	#13133175	1	1	yes	characteristic
125	C₄	#13133204	1	1	yes	characteristic
126	C₄	#13133208	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#16810488	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810661

C₂

yes

maximal, characteristic

#16810578

C₂

yes

maximal, characteristic

#16810395

C₂

yes

maximal

#16810380

C₂

yes

maximal

C₂×#108×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

characteristic

#13133176

C₂²

yes

—

#13133176

C₂²

yes

—

—

C₄

yes

complemented

C₈×Dic₅₃

C₄

yes

complemented

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#108×C₅₃

—

#108×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

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

C₂²⋊Dic₅₃

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₄

yes

—

#11719917

—

#11719917

—

C₂×#88322

C₂×C₄

yes

—

C₂³×C₅₃

#41

yes

characteristic

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

—

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

C₂×C₄⋊C₄

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

#1833

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#382

yes

characteristic

C₂×#108

D₅₃

yes

—

—

—

—

C₅₃

#1833

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

—

C₂²×C₈

C₂×D₅₃

yes

—

—

—

—

—

106

—

#106

—

106

—

#97

—

C₄×C₈

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₀₆

yes

characteristic

C₂×C₈

Dic₁₀₆

yes

characteristic

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

100

C₄²

—

101

C₄²

—

106

—

102

C₂³

#11719949

yes

characteristic

103

C₂×C₄

—

106

—

104

C₂×C₄

—

212

—

105

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

106

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

107

C₂×C₄

#11719947

yes

characteristic

108

C₂×C₄

#11719947

yes

characteristic

109

C₂×C₄

#11719949

yes

characteristic

110

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

111

C₈

—

112

C₈

—

106

—

113

C₈

#11719919

yes

characteristic

114

C₈

#11719919

yes

characteristic

115

C₈

#11719919

yes

characteristic

116

C₈

#11719919

yes

characteristic

117

C₂²

—

118

C₂²

#13133175

yes

characteristic

119

C₂²

#13133206

yes

characteristic

120

C₂²

#13133208

yes

characteristic

121

C₄

—

424

—

122

C₄

—

212

—

123

C₄

C₂×#11719919

yes

characteristic

124

C₄

#13133175

yes

characteristic

125

C₄

#13133204

yes

characteristic

126

C₄

#13133208

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#16810488

yes

characteristic

	subgroup	G/H
center	C₄	#13133175
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#108×C₅₃	C₂
socle	C₂×C₅₃	#382
hypercenter	C₂×#108	D₅₃
hypocenter	C₅₃	#1833
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₄

#13133175

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#108×C₅₃

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
11. #13133127	C₄	semidirect
12. C₈×Dic₅₃	C₄	semidirect
71. C₅₃	#1833	semidirect

H ≅ G/N

type

11. #13133127

C₄

semidirect

12. C₈×Dic₅₃

C₄

semidirect

71. C₅₃

#1833

semidirect

Subgroups — #20274840

Conjugacy classes of subgroups

Complements