Group #20275971 — subgroups

Subgroups — #20275971

Order 6784.

296 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 69 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×D₄₂₄	C₂	1	1	yes	maximal, characteristic
2	#16810627	C₂	1	2	yes	maximal, complemented
3	C₂×#13133133	C₂	1	2	yes	maximal, complemented
4	#16810388	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₁₂	C₂²	1	2	yes	—
7	C₂×D₄₂₄	—	2	2	no	—
8	C₂×D₄₂₄	—	2	4	no	—
9	C₂×#11719917	C₂²	1	2	yes	—
10	#13133137	—	2	4	no	—
11	#13133133	—	2	4	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	C₄⋊C₈×C₅₃	—	2	2	no	—
16	C₂³×D₅₃	—	2	2	no	—
17	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂×D₂₁₂	—	2	2	no	—
19	C₂×D₂₁₂	—	2	2	no	—
20	C₂×D₂₁₂	—	2	2	no	—
21	C₂×D₂₁₂	—	2	2	no	—
22	C₂×D₂₁₂	—	2	2	no	—
23	C₂×D₂₁₂	D₄	1	4	yes	—
24	M₁₆×C₅₃	—	2	2	no	—
25	C₂×C₈×C₅₃	—	2	2	no	—
26	C₂×C₈×C₅₃	—	2	1	no	—
27	C₂×C₈×C₅₃	D₄	1	2	yes	—
28	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
29	C₂²⋊C₄×C₅₃	—	2	1	no	—
30	C₄²×C₅₃	—	2	1	no	—
31	#11719917	—	2	2	no	—
32	#11719917	—	2	2	no	—
33	C₂×#88322	—	2	2	no	—
34	D₄₂₄	—	2	8	no	—
35	C₂³×C₅₃	#41	1	1	yes	characteristic
36	C₂²×D₅₃	—	2	4	no	—
37	C₂²×D₅₃	—	4	2	no	—
38	C₂²×D₅₃	—	4	2	no	—
39	C₂²×D₅₃	—	8	4	no	—
40	C₂×C₄×C₅₃	—	2	2	no	—
41	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	#41	1	2	yes	—
45	D₂₁₂	—	2	4	no	—
46	D₂₁₂	—	2	4	no	—
47	D₂₁₂	—	4	2	no	—
48	D₂₁₂	—	4	2	no	—
49	C₈×C₅₃	—	2	2	no	—
50	C₈×C₅₃	—	4	1	no	—
51	#88322	—	4	2	no	—
52	C₂²×C₅₃	—	2	2	no	—
53	C₂²×C₅₃	#121	1	1	yes	characteristic
54	C₂²×C₅₃	#123	1	2	yes	—
55	C₂×D₅₃	—	4	4	no	—
56	C₂×D₅₃	—	8	4	no	—
57	C₂×D₅₃	—	8	4	no	—
58	C₂×D₅₃	—	8	4	no	—
59	C₄×C₅₃	—	4	2	no	—
60	C₄×C₅₃	#120	1	1	yes	characteristic
61	C₄×C₅₃	#121	1	2	yes	—
62	C₄×C₅₃	#124	1	1	yes	characteristic
63	#2045	—	53	1	no	maximal, Sylow
64	C₂×C₅₃	—	2	2	no	—
65	C₂×C₅₃	#394	1	1	yes	characteristic
66	C₂×C₅₃	#472	1	2	yes	—
67	D₅₃	—	8	4	no	—
68	C₂²×D₈	—	53	1	no	—
69	#424	D₅₃	1	1	yes	characteristic, complemented
70	#418	—	53	2	no	—
71	C₂×#100	—	53	2	no	—
72	#341	—	53	1	no	—
73	C₅₃	#2045	1	1	yes	characteristic, complemented, Sylow
74	C₂²×D₄	—	53	2	no	—
75	C₂×D₈	—	106	4	no	—
76	C₂×D₈	—	106	2	no	—
77	C₂×M₁₆	—	53	2	no	—
78	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
79	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
80	#117	C₂×D₅₃	1	1	yes	characteristic
81	C₄⋊C₈	—	2	2	no	—
82	#102	—	106	4	no	—
83	#100	—	106	4	no	—
84	C₂⁴	—	106	2	no	—
85	C₂×D₄	—	53	4	no	—
86	C₂×D₄	—	106	2	no	—
87	C₂×D₄	—	106	2	no	—
88	C₂×D₄	—	106	2	no	—
89	C₂×D₄	—	106	2	no	—
90	C₂×D₄	—	106	2	no	—
91	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
92	D₈	—	106	8	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	106	2	no	—
95	M₁₆	—	106	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	106	2	no	—
99	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
100	C₄⋊C₄	D₂₁₂	1	2	yes	—
101	C₂²⋊C₄	—	2	1	no	—
102	C₄²	—	2	1	no	—
103	C₂³	—	106	4	no	—
104	C₂³	—	424	4	no	—
105	C₂³	—	212	2	no	—
106	C₂³	—	212	2	no	—
107	C₂³	#11719945	1	1	yes	characteristic
108	D₄	—	106	4	no	—
109	D₄	—	106	4	no	—
110	D₄	—	212	2	no	—
111	D₄	—	212	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
114	C₂×C₄	D₄×D₅₃	1	2	yes	—
115	C₂×C₄	#11719949	1	2	yes	—
116	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₈	—	4	1	no	—
119	C₈	—	212	2	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	424	4	no	—
122	C₂²	—	424	4	no	—
123	C₂²	—	424	4	no	—
124	C₂²	—	212	4	no	—
125	C₂²	#13133191	1	1	yes	characteristic
126	C₂²	#13133202	1	2	yes	—
127	C₄	—	4	2	no	—
128	C₄	#13133185	1	1	yes	characteristic
129	C₄	#13133192	1	2	yes	—
130	C₄	#13133206	1	1	yes	characteristic
131	C₂	—	2	2	no	—
132	C₂	—	424	4	no	—
133	C₂	#16810540	1	1	yes	characteristic
134	C₂	#16810893	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₄₂₄

C₂

yes

maximal, characteristic

#16810627

C₂

yes

maximal, complemented

C₂×#13133133

C₂

yes

maximal, complemented

#16810388

C₂

yes

maximal, characteristic, complemented

C₅₃×#424

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

C₂²

yes

—

C₂×D₄₂₄

—

C₂×D₄₂₄

—

C₂×#11719917

C₂²

yes

—

#13133137

—

#13133133

—

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂³×D₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

D₄

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

#11719917

—

#11719917

—

C₂×#88322

—

D₄₂₄

—

C₂³×C₅₃

#41

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

#121

yes

characteristic

C₂²×C₅₃

#123

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#120

yes

characteristic

C₄×C₅₃

#121

yes

—

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

#394

yes

characteristic

C₂×C₅₃

#472

yes

—

D₅₃

—

C₂²×D₈

—

#424

D₅₃

yes

characteristic, complemented

#418

—

C₂×#100

—

#341

—

C₅₃

#2045

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂×D₈

—

106

—

C₂×D₈

—

106

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

characteristic

#117

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

#102

—

106

—

#100

—

106

—

C₂⁴

—

106

—

C₂×D₄

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

D₈

—

106

—

M₁₆

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₂²⋊D₅₃

yes

—

100

C₄⋊C₄

D₂₁₂

yes

—

101

C₂²⋊C₄

—

102

C₄²

—

103

C₂³

—

106

—

104

C₂³

—

424

—

105

C₂³

—

212

—

106

C₂³

—

212

—

107

C₂³

#11719945

yes

characteristic

108

D₄

—

106

—

109

D₄

—

106

—

110

D₄

—

212

—

111

D₄

—

212

—

112

C₂×C₄

—

113

C₂×C₄

C₂×D₂₁₂

yes

characteristic

114

C₂×C₄

D₄×D₅₃

yes

—

115

C₂×C₄

#11719949

yes

—

116

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

—

212

—

120

C₂²

—

121

C₂²

—

424

—

122

C₂²

—

424

—

123

C₂²

—

424

—

124

C₂²

—

212

—

125

C₂²

#13133191

yes

characteristic

126

C₂²

#13133202

yes

—

127

C₄

—

128

C₄

#13133185

yes

characteristic

129

C₄

#13133192

yes

—

130

C₄

#13133206

yes

characteristic

131

C₂

—

132

C₂

—

424

—

133

C₂

#16810540

yes

characteristic

134

C₂

#16810893

yes

—

	subgroup	G/H
center	C₂²	#13133191
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#424	C₂
socle	C₂²×C₅₃	#121
hypercenter	#424	D₅₃
hypocenter	C₅₃	#2045
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#13133191

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#424

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
2. #16810627	C₂	semidirect
3. C₂×#13133133	C₂	semidirect
4. #16810388	C₂	semidirect
5. C₅₃×#424	C₂	semidirect
69. #424	D₅₃	semidirect
73. C₅₃	#2045	semidirect

H ≅ G/N

type

2. #16810627

C₂

semidirect

3. C₂×#13133133

C₂

semidirect

4. #16810388

C₂

semidirect

5. C₅₃×#424

C₂

semidirect

69. #424

D₅₃

semidirect

73. C₅₃

#2045

semidirect

Subgroups — #20275971

Conjugacy classes of subgroups

Complements