Group #20285302 — subgroups

Subgroups — #20285302

Order 6784.

368 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 133 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	#16811178	C₂	1	2	yes	maximal
2	#16811003	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810975	C₂	1	2	yes	maximal, complemented
4	#16810865	C₂	1	4	yes	maximal
5	#16810717	C₂	1	4	yes	maximal, complemented
6	C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₁₂	C₂²	1	4	yes	—
9	C₂×C₈×D₅₃	C₂²	1	2	yes	—
10	#13133217	C₂²	1	2	yes	—
11	#13133208	—	2	2	no	—
12	#13133197	C₂²	1	4	yes	—
13	#13133193	—	2	2	no	—
14	Q₈×Dic₅₃	—	2	2	no	—
15	#13133182	C₂²	1	4	yes	—
16	#13133175	—	2	2	no	—
17	C₄²×D₅₃	C₂²	1	1	yes	characteristic
18	#13133140	C₂²	1	8	yes	—
19	C₄×#88322	C₂²	1	1	yes	characteristic
20	C₈×Dic₅₃	C₂²	1	2	yes	—
21	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
22	#107×C₅₃	C₂²	1	4	yes	—
23	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×Q₈×C₅₃	—	2	2	no	—
25	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
26	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
27	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
28	C₂×Dic₁₀₆	C₂³	1	4	yes	—
29	C₂×C₈×C₅₃	C₂³	1	2	yes	—
30	C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
32	C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	#11719921	—	2	4	no	—
34	C₄⋊Dic₅₃	—	2	2	no	—
35	C₄⋊Dic₅₃	—	2	2	no	—
36	C₄⋊Dic₅₃	—	2	4	no	—
37	#11719919	—	2	4	no	—
38	C₄×Dic₅₃	—	2	4	no	—
39	C₄×Dic₅₃	D₄	1	1	yes	characteristic
40	C₄×Dic₅₃	D₄	1	1	yes	characteristic
41	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
42	C₂×#88322	C₂³	1	2	yes	—
43	Dic₂₁₂	—	2	8	no	—
44	C₈×D₅₃	—	2	4	no	—
45	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₅₃	—	4	4	no	—
47	C₂×C₄×C₅₃	—	2	4	no	—
48	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₅₃	—	2	4	no	—
52	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
55	C₄×D₅₃	—	2	2	no	—
56	C₄×D₅₃	—	2	2	no	—
57	C₄×D₅₃	Q₁₆	1	4	yes	complemented
58	Dic₁₀₆	—	2	8	no	—
59	C₈×C₅₃	#41	1	4	yes	—
60	#88322	—	2	2	no	—
61	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
62	C₂×D₅₃	—	2	1	no	—
63	C₂×D₅₃	—	2	1	no	—
64	C₂×D₅₃	C₂×Q₁₆	1	2	yes	—
65	C₄×C₅₃	—	4	4	no	—
66	C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
67	C₄×C₅₃	#135	1	2	yes	—
68	C₄×C₅₃	C₂×#41	1	2	yes	—
69	Dic₅₃	—	2	1	no	—
70	Dic₅₃	—	2	1	no	—
71	Dic₅₃	—	4	4	no	—
72	Dic₅₃	C₂×Q₁₆	1	2	yes	—
73	#3170	—	53	1	no	maximal, Sylow
74	C₂×C₅₃	#532	1	1	yes	characteristic
75	C₂×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
76	C₂×C₅₃	C₂×#135	1	1	yes	characteristic
77	D₅₃	—	2	2	no	—
78	#533	—	53	2	no	—
79	#500	D₅₃	1	1	yes	characteristic, complemented
80	#494	—	53	1	no	—
81	#487	—	53	2	no	—
82	#467	—	53	4	no	—
83	C₄×Q₁₆	—	53	4	no	—
84	C₂×C₄×C₈	—	53	1	no	—
85	C₅₃	#3170	1	1	yes	characteristic, complemented, Sylow
86	C₂×Q₁₆	—	53	4	no	—
87	C₂²×C₈	—	53	2	no	—
88	C₄⋊Q₈	—	53	2	no	—
89	C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
90	#125	—	106	2	no	—
91	C₂²⋊Q₈	—	53	4	no	—
92	C₂²⋊Q₈	—	106	2	no	—
93	C₄×Q₈	—	53	4	no	—
94	C₄×Q₈	—	106	2	no	—
95	#117	—	106	2	no	—
96	C₂×C₄²	—	53	1	no	—
97	#107	C₂×D₅₃	1	4	yes	—
98	#103	—	53	8	no	—
99	C₄×C₈	—	53	2	no	—
100	C₄×C₈	—	53	1	no	—
101	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	53	4	no	—
104	C₂²×C₄	—	53	1	no	—
105	C₂²×C₄	—	53	1	no	—
106	C₂²×C₄	—	53	1	no	—
107	Q₁₆	—	106	8	no	—
108	C₂×C₈	—	53	2	no	—
109	C₂×C₈	—	106	4	no	—
110	C₂×C₈	C₂²×D₅₃	1	2	yes	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	106	4	no	—
113	C₄⋊C₄	—	106	4	no	—
114	C₄⋊C₄	—	106	2	no	—
115	C₄⋊C₄	—	106	2	no	—
116	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
117	C₂²⋊C₄	—	106	4	no	—
118	C₄²	—	53	1	no	—
119	C₄²	—	53	1	no	—
120	C₄²	—	53	1	no	—
121	C₄²	—	106	4	no	—
122	C₄²	C₂²×D₅₃	1	1	yes	characteristic
123	C₂³	—	53	1	no	—
124	Q₈	—	4	4	no	—
125	Q₈	—	106	8	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	53	1	no	—
128	C₂×C₄	—	53	4	no	—
129	C₂×C₄	—	53	1	no	—
130	C₂×C₄	—	53	1	no	—
131	C₂×C₄	—	106	2	no	—
132	C₂×C₄	—	106	2	no	—
133	C₂×C₄	—	106	4	no	—
134	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
135	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
136	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
137	C₈	—	106	2	no	—
138	C₈	#11719949	1	4	yes	—
139	C₂²	—	53	2	no	—
140	C₂²	—	106	1	no	—
141	C₂²	—	106	1	no	—
142	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
143	C₄	—	4	4	no	—
144	C₄	—	53	2	no	—
145	C₄	—	106	1	no	—
146	C₄	—	106	1	no	—
147	C₄	—	212	4	no	—
148	C₄	Q₁₆×D₅₃	1	2	yes	—
149	C₄	#13133236	1	2	yes	—
150	C₄	C₂×#11719949	1	2	yes	—
151	C₂	—	106	2	no	—
152	C₂	#16811174	1	1	yes	characteristic
153	C₂	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
154	C₂	C₂×#13133236	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811178

C₂

yes

maximal

#16811003

C₂

yes

maximal, characteristic, complemented

#16810975

C₂

yes

maximal, complemented

#16810865

C₂

yes

maximal

#16810717

C₂

yes

maximal, complemented

C₄×C₈×D₅₃

C₂

yes

maximal, characteristic

C₅₃×#500

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₂

C₂²

yes

—

C₂×C₈×D₅₃

C₂²

yes

—

#13133217

C₂²

yes

—

#13133208

—

#13133197

C₂²

yes

—

#13133193

—

Q₈×Dic₅₃

—

#13133182

C₂²

yes

—

#13133175

—

C₄²×D₅₃

C₂²

yes

characteristic

#13133140

C₂²

yes

—

C₄×#88322

C₂²

yes

characteristic

C₈×Dic₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₂²

yes

—

#107×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

C₂³

yes

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

C₂³

yes

characteristic

C₂×#88322

C₂³

yes

—

Dic₂₁₂

—

C₈×D₅₃

—

C₂²×D₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

Q₁₆

yes

complemented

Dic₁₀₆

—

C₈×C₅₃

#41

yes

—

#88322

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₂×Q₁₆

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×Q₁₆

yes

—

C₄×C₅₃

#135

yes

—

C₄×C₅₃

C₂×#41

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂×Q₁₆

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂²×Q₁₆

yes

characteristic

C₂×C₅₃

C₂×#135

yes

characteristic

D₅₃

—

#533

—

#500

D₅₃

yes

characteristic, complemented

#494

—

#487

—

#467

—

C₄×Q₁₆

—

C₂×C₄×C₈

—

C₅₃

#3170

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

—

#125

—

106

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

106

—

C₄×Q₈

—

C₄×Q₈

—

106

—

#117

—

106

—

C₂×C₄²

—

#107

C₂×D₅₃

yes

—

#103

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

C₂×D₅₃

yes

characteristic

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

Q₁₆

—

106

—

108

C₂×C₈

—

109

C₂×C₈

—

106

—

110

C₂×C₈

C₂²×D₅₃

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

106

—

113

C₄⋊C₄

—

106

—

114

C₄⋊C₄

—

106

—

115

C₄⋊C₄

—

106

—

116

C₄⋊C₄

C₂²×D₅₃

yes

—

117

C₂²⋊C₄

—

106

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

106

—

122

C₄²

C₂²×D₅₃

yes

characteristic

123

C₂³

—

124

Q₈

—

125

Q₈

—

106

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

106

—

132

C₂×C₄

—

106

—

133

C₂×C₄

—

106

—

134

C₂×C₄

D₄×D₅₃

yes

characteristic

135

C₂×C₄

D₄×D₅₃

yes

characteristic

136

C₂×C₄

C₂³×D₅₃

yes

characteristic

137

C₈

—

106

—

138

C₈

#11719949

yes

—

139

C₂²

—

140

C₂²

—

106

—

141

C₂²

—

106

—

142

C₂²

C₂×D₄×D₅₃

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

106

—

146

C₄

—

106

—

147

C₄

—

212

—

148

C₄

Q₁₆×D₅₃

yes

—

149

C₄

#13133236

yes

—

150

C₄

C₂×#11719949

yes

—

151

C₂

—

106

—

152

C₂

#16811174

yes

characteristic

153

C₂

C₂×Q₁₆×D₅₃

yes

characteristic

154

C₂

C₂×#13133236

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#500	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#500	D₅₃
hypocenter	C₅₃	#3170
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

C₂×D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#500

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
2. #16811003	C₂	semidirect
3. #16810975	C₂	semidirect
5. #16810717	C₂	semidirect
7. C₅₃×#500	C₂	semidirect
57. C₄×D₅₃	Q₁₆	semidirect
79. #500	D₅₃	semidirect
85. C₅₃	#3170	semidirect

H ≅ G/N

type

2. #16811003

C₂

semidirect

3. #16810975

C₂

semidirect

5. #16810717

C₂

semidirect

7. C₅₃×#500

C₂

semidirect

57. C₄×D₅₃

Q₁₆

semidirect

79. #500

D₅₃

semidirect

85. C₅₃

#3170

semidirect

Subgroups — #20285302

Conjugacy classes of subgroups

Complements