Group #20283582 — subgroups

Subgroups — #20283582

Order 6784.

456 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 181 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×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	#16811100	C₂	1	2	yes	maximal
3	#16810729	C₂	1	4	yes	maximal, complemented
4	#16810717	C₂	1	4	yes	maximal, complemented
5	#16810631	C₂	1	2	yes	maximal
6	#16810578	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#427	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₀₆	C₂²	1	2	yes	—
9	C₂×Dic₂₁₂	C₄	1	8	yes	complemented
10	C₂×Dic₂₁₂	C₂²	1	2	yes	—
11	C₂×Dic₂₁₂	C₂²	1	2	yes	—
12	C₂×#11719917	C₂²	1	1	yes	characteristic
13	C₄×Dic₁₀₆	—	2	4	no	—
14	#13133182	C₂²	1	8	yes	—
15	#13133176	C₂²	1	1	yes	characteristic
16	#13133173	—	2	4	no	—
17	#13133140	C₂²	1	8	yes	—
18	#13133127	C₂²	1	2	yes	—
19	C₈×Dic₅₃	C₂²	1	2	yes	—
20	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	#117×C₅₃	C₂²	1	2	yes	—
22	#107×C₅₃	C₂²	1	2	yes	—
23	#106×C₅₃	C₂²	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₅₃	—	2	2	no	—
26	C₂×Dic₁₀₆	—	2	2	no	—
27	C₂×Dic₁₀₆	—	2	4	no	—
28	C₂×Dic₁₀₆	C₂×C₄	1	8	yes	—
29	C₂×Dic₁₀₆	C₂³	1	4	yes	—
30	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
31	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
34	C₂²⋊C₄×C₅₃	—	2	2	no	—
35	C₄²×C₅₃	—	2	2	no	—
36	C₂²⋊Dic₅₃	D₄	1	2	yes	—
37	C₄⋊Dic₅₃	D₄	1	2	yes	—
38	#11719919	—	2	8	no	—
39	C₄×Dic₅₃	—	2	8	no	—
40	C₄×Dic₅₃	C₂³	1	2	yes	—
41	#11719917	—	2	2	no	—
42	C₂×#88322	C₂³	1	2	yes	—
43	Dic₂₁₂	C₂×C₄	1	8	yes	—
44	Dic₂₁₂	C₂×C₄	1	8	yes	—
45	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	—	2	4	no	—
47	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	#41	1	2	yes	—
50	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₅₃	—	2	8	no	—
52	C₂×Dic₅₃	—	2	4	no	—
53	C₂×Dic₅₃	C₂×D₄	1	4	yes	—
54	Dic₁₀₆	—	2	8	no	—
55	Dic₁₀₆	—	2	4	no	—
56	Dic₁₀₆	C₂²×C₄	1	8	yes	—
57	C₈×C₅₃	C₂²×C₄	1	2	yes	—
58	C₈×C₅₃	C₂²×C₄	1	2	yes	—
59	#88322	—	2	2	no	—
60	C₂²×C₅₃	—	2	2	no	—
61	C₂²×C₅₃	C₄×D₄	1	2	yes	—
62	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₅₃	—	4	4	no	—
64	C₄×C₅₃	C₄×D₄	1	2	yes	—
65	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
67	Dic₅₃	—	2	8	no	—
68	Dic₅₃	—	2	4	no	—
69	#2950	—	53	1	no	maximal, Sylow
70	C₂×C₅₃	—	2	2	no	—
71	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
72	C₂×C₅₃	#578	1	2	yes	—
73	C₂²×Q₁₆	—	53	1	no	—
74	#519	—	53	2	no	—
75	#441	—	53	4	no	—
76	C₄×Q₁₆	—	53	4	no	—
77	#427	D₅₃	1	1	yes	characteristic
78	#419	—	53	2	no	—
79	#405	—	53	1	no	—
80	C₅₃	#2950	1	1	yes	characteristic, complemented, Sylow
81	C₂²×Q₈	—	53	2	no	—
82	C₂×Q₁₆	—	53	8	no	—
83	C₂×Q₁₆	—	53	2	no	—
84	C₂×Q₁₆	—	53	2	no	—
85	C₂×M₁₆	—	53	1	no	—
86	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
87	C₄×Q₈	—	53	8	no	—
88	C₄×Q₈	—	106	4	no	—
89	#117	—	53	1	no	—
90	#117	—	106	4	no	—
91	#117	C₂×D₅₃	1	2	yes	—
92	#107	C₂×D₅₃	1	2	yes	—
93	#106	C₂×D₅₃	1	2	yes	—
94	#103	—	53	8	no	—
95	#97	—	53	2	no	—
96	C₄×C₈	—	53	2	no	—
97	C₂×Q₈	—	53	8	no	—
98	C₂×Q₈	—	53	4	no	—
99	C₂×Q₈	—	106	2	no	—
100	C₂×Q₈	—	106	4	no	—
101	C₂²×C₄	—	106	2	no	—
102	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
103	Q₁₆	—	53	8	no	—
104	Q₁₆	—	53	8	no	—
105	M₁₆	—	106	2	no	—
106	C₂×C₈	—	53	2	no	—
107	C₂×C₈	C₄×D₅₃	1	4	yes	—
108	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
109	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
110	C₄⋊C₄	—	53	2	no	—
111	C₄⋊C₄	—	106	8	no	—
112	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
113	C₂²⋊C₄	—	2	2	no	—
114	C₂²⋊C₄	—	53	2	no	—
115	C₄²	—	2	2	no	—
116	C₄²	—	53	2	no	—
117	C₄²	—	106	8	no	—
118	C₂³	D₄×D₅₃	1	1	yes	characteristic
119	Q₈	—	53	8	no	—
120	Q₈	—	106	8	no	—
121	Q₈	—	106	4	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	53	4	no	—
124	C₂×C₄	—	106	8	no	—
125	C₂×C₄	—	106	4	no	—
126	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
127	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
128	C₂×C₄	#11719949	1	2	yes	—
129	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
130	C₈	—	106	2	no	—
131	C₈	C₂×C₄×D₅₃	1	2	yes	—
132	C₈	C₂×C₄×D₅₃	1	2	yes	—
133	C₂²	—	2	2	no	—
134	C₂²	#13133179	1	2	yes	—
135	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
136	C₄	—	4	4	no	—
137	C₄	—	106	8	no	—
138	C₄	—	106	4	no	—
139	C₄	#13133179	1	2	yes	—
140	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
141	C₄	C₂×#11719949	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	C₂×#13133179	1	1	yes	characteristic
144	C₂	#16811437	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₁₂

C₂

yes

maximal, characteristic

#16811100

C₂

yes

maximal

#16810729

C₂

yes

maximal, complemented

#16810717

C₂

yes

maximal, complemented

#16810631

C₂

yes

maximal

#16810578

C₂

yes

maximal, characteristic

C₅₃×#427

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

—

C₂×Dic₂₁₂

C₄

yes

complemented

C₂×Dic₂₁₂

C₂²

yes

—

C₂×Dic₂₁₂

C₂²

yes

—

C₂×#11719917

C₂²

yes

characteristic

C₄×Dic₁₀₆

—

#13133182

C₂²

yes

—

#13133176

C₂²

yes

characteristic

#13133173

—

#13133140

C₂²

yes

—

#13133127

C₂²

yes

—

C₈×Dic₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₂²

yes

—

#107×C₅₃

C₂²

yes

—

#106×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×Dic₁₀₆

C₂³

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

C₂³

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₂²⋊Dic₅₃

D₄

yes

—

C₄⋊Dic₅₃

D₄

yes

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂³

yes

—

#11719917

—

C₂×#88322

C₂³

yes

—

Dic₂₁₂

C₂×C₄

yes

—

Dic₂₁₂

C₂×C₄

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

C₂²×C₄

yes

—

C₈×C₅₃

C₂²×C₄

yes

—

C₈×C₅₃

C₂²×C₄

yes

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄×D₄

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

C₂³×C₄

yes

characteristic

C₄×C₅₃

C₂×#41

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#2950

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×C₄×D₄

yes

characteristic

C₂×C₅₃

#578

yes

—

C₂²×Q₁₆

—

#519

—

#441

—

C₄×Q₁₆

—

#427

D₅₃

yes

characteristic

#419

—

#405

—

C₅₃

#2950

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

106

—

#117

—

#117

—

106

—

#117

C₂×D₅₃

yes

—

#107

C₂×D₅₃

yes

—

#106

C₂×D₅₃

yes

—

#103

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

106

—

100

C₂×Q₈

—

106

—

101

C₂²×C₄

—

106

—

102

C₂²×C₄

C₂²×D₅₃

yes

characteristic

103

Q₁₆

—

104

Q₁₆

—

105

M₁₆

—

106

—

106

C₂×C₈

—

107

C₂×C₈

C₄×D₅₃

yes

—

108

C₂×C₈

C₂²×D₅₃

yes

characteristic

109

C₂×C₈

C₂²×D₅₃

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

106

—

112

C₄⋊C₄

C₂²×D₅₃

yes

—

113

C₂²⋊C₄

—

114

C₂²⋊C₄

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

106

—

118

C₂³

D₄×D₅₃

yes

characteristic

119

Q₈

—

120

Q₈

—

106

—

121

Q₈

—

106

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

106

—

125

C₂×C₄

—

106

—

126

C₂×C₄

C₂×C₄×D₅₃

yes

—

127

C₂×C₄

D₄×D₅₃

yes

characteristic

128

C₂×C₄

#11719949

yes

—

129

C₂×C₄

C₂³×D₅₃

yes

characteristic

130

C₈

—

106

—

131

C₈

C₂×C₄×D₅₃

yes

—

132

C₈

C₂×C₄×D₅₃

yes

—

133

C₂²

—

134

C₂²

#13133179

yes

—

135

C₂²

C₂×D₄×D₅₃

yes

characteristic

136

C₄

—

137

C₄

—

106

—

138

C₄

—

106

—

139

C₄

#13133179

yes

—

140

C₄

C₂²×C₄×D₅₃

yes

characteristic

141

C₄

C₂×#11719949

yes

characteristic

142

C₂

—

143

C₂

C₂×#13133179

yes

characteristic

144

C₂

#16811437

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#427	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#427	D₅₃
hypocenter	C₅₃	#2950
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂²

C₂×D₄×D₅₃

commutator

C₄×C₅₃

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#427

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
3. #16810729	C₂	semidirect
4. #16810717	C₂	semidirect
9. C₂×Dic₂₁₂	C₄	semidirect
80. C₅₃	#2950	semidirect

H ≅ G/N

type

3. #16810729

C₂

semidirect

4. #16810717

C₂

semidirect

9. C₂×Dic₂₁₂

C₄

semidirect

80. C₅₃

#2950

semidirect

Subgroups — #20283582

Conjugacy classes of subgroups

Complements