Group #20274756 — subgroups

Subgroups — #20274756

Order 6784.

272 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂²×#11719912	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133152	C₂	1	2	yes	maximal, complemented
3	#16810586	C₂	1	2	yes	maximal
4	#16810388	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#341	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
7	C₂×#11719917	C₂²	1	2	yes	—
8	C₂×#11719912	—	2	4	no	—
9	C₂×#11719912	C₄	1	4	yes	—
10	C₂²×#88322	C₂²	1	1	yes	characteristic
11	#13133152	—	2	4	no	—
12	#13133129	—	2	4	no	—
13	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
14	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂³×D₅₃	Q₈	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
17	C₂²×Dic₅₃	Q₈	1	1	yes	characteristic
18	C₂×C₄×D₅₃	—	2	2	no	—
19	C₂×C₄×D₅₃	—	2	2	no	—
20	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₄×D₅₃	D₄	1	2	yes	—
22	M₁₆×C₅₃	—	2	2	no	—
23	M₁₆×C₅₃	—	2	2	no	—
24	C₂×C₈×C₅₃	—	2	2	no	—
25	C₂×C₈×C₅₃	—	2	2	no	—
26	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
27	#11719917	—	2	2	no	—
28	#11719917	—	2	2	no	—
29	C₂×#88322	—	2	2	no	—
30	C₂×#88322	C₂×C₄	1	2	yes	—
31	C₂×#88322	D₄	1	4	yes	—
32	#11719912	—	2	8	no	—
33	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
34	C₂²×D₅₃	—	2	2	no	—
35	C₂²×D₅₃	—	4	2	no	—
36	C₂²×D₅₃	C₄⋊C₄	1	2	yes	—
37	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	#41	1	2	yes	—
41	C₂×Dic₅₃	—	2	2	no	—
42	C₂×Dic₅₃	C₄⋊C₄	1	2	yes	—
43	C₄×D₅₃	—	2	2	no	—
44	C₄×D₅₃	—	2	2	no	—
45	C₄×D₅₃	—	2	4	no	—
46	C₈×C₅₃	—	2	2	no	—
47	C₈×C₅₃	—	4	2	no	—
48	#88322	—	4	2	no	—
49	#88322	C₂²⋊C₄	1	4	yes	—
50	C₂²×C₅₃	—	2	2	no	—
51	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₅₃	C₂²⋊Q₈	1	2	yes	—
53	C₂×D₅₃	—	2	2	no	—
54	C₂×D₅₃	—	2	4	no	—
55	C₂×D₅₃	—	4	4	no	—
56	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₄×C₅₃	#117	1	1	yes	characteristic
58	C₄×C₅₃	#121	1	2	yes	—
59	Dic₅₃	—	2	2	no	—
60	#1811	—	53	1	no	maximal, Sylow
61	C₂×C₅₃	—	2	2	no	—
62	C₂×C₅₃	#380	1	1	yes	characteristic
63	C₂×C₅₃	#430	1	2	yes	—
64	D₅₃	—	4	2	no	—
65	C₂²×M₁₆	—	53	1	no	—
66	C₂×#108	—	53	2	no	—
67	#407	—	53	2	no	—
68	#341	—	53	1	no	—
69	#341	D₅₃	1	1	yes	characteristic, complemented
70	C₅₃	#1811	1	1	yes	characteristic, complemented, Sylow
71	C₂³×C₄	—	53	1	no	—
72	C₂×M₁₆	—	53	4	no	—
73	C₂×M₁₆	—	53	2	no	—
74	C₂×M₁₆	—	106	4	no	—
75	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
76	C₂²×C₈	—	53	1	no	—
77	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
78	#108	—	106	4	no	—
79	C₂²⋊C₈	—	106	4	no	—
80	C₂⁴	—	53	1	no	—
81	C₂²×C₄	—	53	1	no	—
82	C₂²×C₄	—	53	2	no	—
83	C₂²×C₄	—	53	2	no	—
84	C₂²×C₄	—	106	2	no	—
85	C₂²×C₄	—	106	2	no	—
86	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
87	M₁₆	—	2	2	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	106	8	no	—
90	M₁₆	—	106	2	no	—
91	M₁₆	—	106	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	53	2	no	—
95	C₂×C₈	—	53	4	no	—
96	C₂×C₈	—	106	2	no	—
97	C₂×C₈	C₄×D₅₃	1	2	yes	—
98	C₂³	—	53	2	no	—
99	C₂³	—	106	2	no	—
100	C₂³	—	212	2	no	—
101	C₂³	Q₈×D₅₃	1	1	yes	characteristic
102	C₂×C₄	—	53	2	no	—
103	C₂×C₄	—	106	2	no	—
104	C₂×C₄	—	106	2	no	—
105	C₂×C₄	—	106	2	no	—
106	C₂×C₄	—	106	4	no	—
107	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
108	C₂×C₄	#11719945	1	2	yes	—
109	C₂×C₄	D₄×D₅₃	1	2	yes	—
110	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	4	2	no	—
113	C₈	—	53	4	no	—
114	C₈	—	212	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	—	106	4	no	—
117	C₂²	—	106	2	no	—
118	C₂²	—	212	4	no	—
119	C₂²	C₄⋊C₄×D₅₃	1	1	yes	characteristic
120	C₂²	#13133195	1	2	yes	—
121	C₄	—	106	2	no	—
122	C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
123	C₄	#13133174	1	1	yes	characteristic
124	C₄	#13133190	1	2	yes	—
125	C₂	—	2	2	no	—
126	C₂	—	212	2	no	—
127	C₂	#16810482	1	1	yes	characteristic
128	C₂	#16810670	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11719912

C₂

yes

maximal, characteristic

C₂×#13133152

C₂

yes

maximal, complemented

#16810586

C₂

yes

maximal

#16810388

C₂

yes

maximal, characteristic, complemented

C₅₃×#341

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₅₃

C₂²

yes

characteristic

C₂×#11719917

C₂²

yes

—

C₂×#11719912

—

C₂×#11719912

C₄

yes

—

C₂²×#88322

C₂²

yes

characteristic

#13133152

—

#13133129

—

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂³×D₅₃

Q₈

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

Q₈

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

D₄

yes

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂×#88322

D₄

yes

—

#11719912

—

C₂³×C₅₃

C₂×Q₈

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₄⋊C₄

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

C₂²⋊C₄

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

#1811

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#380

yes

characteristic

C₂×C₅₃

#430

yes

—

D₅₃

—

C₂²×M₁₆

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#1811

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

106

—

C₂×M₁₆

C₂×D₅₃

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

#108

—

106

—

C₂²⋊C₈

—

106

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

106

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₄×D₅₃

yes

—

C₂³

—

C₂³

—

106

—

100

C₂³

—

212

—

101

C₂³

Q₈×D₅₃

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

106

—

104

C₂×C₄

—

106

—

105

C₂×C₄

—

106

—

106

C₂×C₄

—

106

—

107

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

108

C₂×C₄

#11719945

yes

—

109

C₂×C₄

D₄×D₅₃

yes

—

110

C₂×C₄

D₄×D₅₃

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

212

—

115

C₂²

—

116

C₂²

—

106

—

117

C₂²

—

106

—

118

C₂²

—

212

—

119

C₂²

C₄⋊C₄×D₅₃

yes

characteristic

120

C₂²

#13133195

yes

—

121

C₄

—

106

—

122

C₄

C₂²⋊C₄×D₅₃

yes

characteristic

123

C₄

#13133174

yes

characteristic

124

C₄

#13133190

yes

—

125

C₂

—

126

C₂

—

212

—

127

C₂

#16810482

yes

characteristic

128

C₂

#16810670

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#341	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	#341	D₅₃
hypocenter	C₅₃	#1811
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#341

C₂

socle

C₂²×C₅₃

C₂×C₄⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
2. C₂×#13133152	C₂	semidirect
4. #16810388	C₂	semidirect
5. C₅₃×#341	C₂	semidirect
69. #341	D₅₃	semidirect
70. C₅₃	#1811	semidirect

H ≅ G/N

type

2. C₂×#13133152

C₂

semidirect

4. #16810388

C₂

semidirect

5. C₅₃×#341

C₂

semidirect

69. #341

D₅₃

semidirect

70. C₅₃

#1811

semidirect

Subgroups — #20274756

Conjugacy classes of subgroups

Complements