Group #20285037 — subgroups

Subgroups — #20285037

Order 6784.

400 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 141 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	#16811180	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810977	C₂	1	4	yes	maximal, complemented
3	#16810973	C₂	1	2	yes	maximal, complemented
4	#16810969	C₂	1	2	yes	maximal, complemented
5	#16810619	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#532	C₂	1	1	yes	maximal, characteristic
8	C₂²×#88322	C₂²	1	2	yes	—
9	#13133217	C₂²	1	2	yes	—
10	#13133209	C₂²	1	2	yes	—
11	#13133196	—	2	2	no	—
12	C₄×Dic₁₀₆	—	2	2	no	—
13	#13133176	C₂²	1	2	yes	—
14	#13133141	C₂²	1	8	yes	—
15	#13133138	C₂²	1	8	yes	—
16	C₄×#88322	C₂²	1	2	yes	—
17	C₄×#88322	C₂²	1	2	yes	—
18	C₂×#41×C₅₃	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
20	#127×C₅₃	C₂²	1	1	yes	characteristic
21	#124×C₅₃	C₂²	1	2	yes	—
22	#121×C₅₃	—	2	2	no	—
23	C₄×D₄×C₅₃	—	2	2	no	—
24	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
25	#41×C₅₃	—	2	8	no	—
26	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
27	C₂×D₄×C₅₃	—	2	2	no	—
28	C₂×D₄×C₅₃	—	2	2	no	—
29	C₂×D₄×C₅₃	C₂³	1	2	yes	—
30	C₂²×C₄×C₅₃	—	2	2	no	—
31	C₂²×C₄×C₅₃	D₄	1	2	yes	—
32	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₀₆	—	2	2	no	—
34	C₄⋊C₄×C₅₃	—	2	2	no	—
35	C₂²⋊C₄×C₅₃	—	2	4	no	—
36	C₄²×C₅₃	D₄	1	2	yes	—
37	C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	C₂²⋊Dic₅₃	—	2	2	no	—
40	C₄⋊Dic₅₃	—	2	2	no	—
41	C₄⋊Dic₅₃	C₂³	1	4	yes	—
42	#11719919	—	2	4	no	—
43	C₄×Dic₅₃	—	2	2	no	—
44	C₂×#88322	—	2	4	no	—
45	C₂×#88322	C₂³	1	4	yes	—
46	C₂³×C₅₃	—	2	2	no	—
47	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
48	Q₈×C₅₃	—	2	4	no	—
49	D₄×C₅₃	—	2	4	no	—
50	D₄×C₅₃	—	4	4	no	—
51	D₄×C₅₃	—	4	4	no	—
52	C₂×C₄×C₅₃	—	2	4	no	—
53	C₂×C₄×C₅₃	—	2	2	no	—
54	C₂×C₄×C₅₃	—	4	4	no	—
55	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#41	1	4	yes	—
59	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
60	C₂×Dic₅₃	—	2	4	no	—
61	Dic₁₀₆	—	4	4	no	—
62	#88322	—	2	4	no	—
63	C₂²×C₅₃	—	2	2	no	—
64	C₂²×C₅₃	—	4	2	no	—
65	C₂²×C₅₃	—	4	4	no	—
66	C₂²×C₅₃	#124	1	2	yes	—
67	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₅₃	—	2	2	no	—
69	C₄×C₅₃	—	4	2	no	—
70	C₄×C₅₃	#124	1	2	yes	—
71	C₄×C₅₃	#135	1	4	yes	—
72	C₄×C₅₃	C₂×#41	1	2	yes	—
73	Dic₅₃	—	4	4	no	—
74	#3131	—	53	1	no	maximal, Sylow
75	C₂×C₅₃	—	2	2	no	—
76	C₂×C₅₃	—	8	2	no	—
77	C₂×C₅₃	C₂×#124	1	1	yes	characteristic
78	C₂×C₅₃	C₂×#135	1	2	yes	—
79	#533	—	53	1	no	—
80	#532	D₅₃	1	1	yes	characteristic
81	#488	—	53	4	no	—
82	#487	—	53	2	no	—
83	#486	—	53	2	no	—
84	#416	—	53	4	no	—
85	C₂×C₄×C₈	—	53	1	no	—
86	C₅₃	#3131	1	1	yes	characteristic, complemented, Sylow
87	C₂×#41	C₂×D₅₃	1	2	yes	—
88	C₂²×C₈	—	53	2	no	—
89	C₄⋊Q₈	—	53	2	no	—
90	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
91	#127	C₂×D₅₃	1	1	yes	characteristic
92	#125	—	53	2	no	—
93	#124	C₂×D₅₃	1	2	yes	—
94	C₂²⋊Q₈	—	106	2	no	—
95	#121	—	2	2	no	—
96	C₄×Q₈	—	106	2	no	—
97	C₄×D₄	—	2	2	no	—
98	#117	—	53	2	no	—
99	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
100	#103	—	53	8	no	—
101	#102	—	53	8	no	—
102	C₄×C₈	—	53	2	no	—
103	C₄×C₈	—	53	2	no	—
104	#41	—	2	8	no	—
105	C₂×Q₈	—	106	2	no	—
106	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
107	C₂×D₄	—	2	2	no	—
108	C₂×D₄	—	2	2	no	—
109	C₂×D₄	C₂²×D₅₃	1	2	yes	—
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
112	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
113	C₂×C₈	—	53	4	no	—
114	C₂×C₈	—	106	4	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	53	4	no	—
117	C₄⋊C₄	—	106	4	no	—
118	C₄⋊C₄	—	106	2	no	—
119	C₂²⋊C₄	—	2	4	no	—
120	C₂²⋊C₄	—	106	2	no	—
121	C₄²	—	106	2	no	—
122	C₄²	C₂²⋊D₅₃	1	2	yes	—
123	C₄²	C₂²×D₅₃	1	1	yes	characteristic
124	C₄²	C₂²×D₅₃	1	1	yes	characteristic
125	C₂³	—	2	2	no	—
126	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
127	Q₈	—	2	4	no	—
128	Q₈	—	212	4	no	—
129	D₄	—	2	4	no	—
130	D₄	—	4	4	no	—
131	D₄	—	4	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	106	4	no	—
136	C₂×C₄	#11719947	1	4	yes	—
137	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
138	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
139	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
140	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
141	C₈	—	106	4	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	4	4	no	—
145	C₂²	#13133204	1	2	yes	—
146	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	4	2	no	—
149	C₄	—	212	4	no	—
150	C₄	#13133204	1	2	yes	—
151	C₄	#13133237	1	4	yes	—
152	C₄	C₂×#11719947	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	—	8	2	no	—
155	C₂	C₂×#13133204	1	1	yes	characteristic
156	C₂	C₂×#13133237	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811180

C₂

yes

maximal, characteristic, complemented

#16810977

C₂

yes

maximal, complemented

#16810973

C₂

yes

maximal, complemented

#16810969

C₂

yes

maximal, complemented

#16810619

C₂

yes

maximal, complemented

C₂×C₄×#88322

C₂

yes

maximal, characteristic, complemented

C₅₃×#532

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

#13133217

C₂²

yes

—

#13133209

C₂²

yes

—

#13133196

—

C₄×Dic₁₀₆

—

#13133176

C₂²

yes

—

#13133141

C₂²

yes

—

#13133138

C₂²

yes

—

C₄×#88322

C₂²

yes

—

C₄×#88322

C₂²

yes

—

C₂×#41×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

#127×C₅₃

C₂²

yes

characteristic

#124×C₅₃

C₂²

yes

—

#121×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#41×C₅₃

—

C₂×Q₈×C₅₃

C₂³

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

Dic₁₀₆

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#124

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#124

yes

—

C₄×C₅₃

#135

yes

—

C₄×C₅₃

C₂×#41

yes

—

Dic₅₃

—

#3131

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#124

yes

characteristic

C₂×C₅₃

C₂×#135

yes

—

#533

—

#532

D₅₃

yes

—

—

—

—

C₂×C₄×C₈

—

C₅₃

#3131

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₃

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

#127

C₂×D₅₃

yes

characteristic

#125

—

#124

C₂×D₅₃

yes

—

C₂²⋊Q₈

—

106

—

#121

—

C₄×Q₈

—

106

—

C₄×D₄

—

#117

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

100

#103

—

101

#102

—

102

C₄×C₈

—

103

C₄×C₈

—

104

#41

—

105

C₂×Q₈

—

106

—

106

C₂×Q₈

C₂²×D₅₃

yes

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

C₂²×D₅₃

yes

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²⋊D₅₃

yes

—

112

C₂²×C₄

C₂²×D₅₃

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

106

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

106

—

118

C₄⋊C₄

—

106

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

106

—

121

C₄²

—

106

—

122

C₄²

C₂²⋊D₅₃

yes

—

123

C₄²

C₂²×D₅₃

yes

characteristic

124

C₄²

C₂²×D₅₃

yes

characteristic

125

C₂³

—

126

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

127

Q₈

—

128

Q₈

—

212

—

129

D₄

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

106

—

136

C₂×C₄

#11719947

yes

—

137

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

138

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

139

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

140

C₂×C₄

C₂³×D₅₃

yes

characteristic

141

C₈

—

106

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

#13133204

yes

—

146

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

212

—

150

C₄

#13133204

yes

—

151

C₄

#13133237

yes

—

152

C₄

C₂×#11719947

yes

—

153

C₂

—

154

C₂

—

155

C₂

C₂×#13133204

yes

characteristic

156

C₂

C₂×#13133237

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#532	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#532	D₅₃
hypocenter	C₅₃	#3131
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#532

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #16811180	C₂	semidirect
2. #16810977	C₂	semidirect
3. #16810973	C₂	semidirect
4. #16810969	C₂	semidirect
5. #16810619	C₂	semidirect
6. C₂×C₄×#88322	C₂	semidirect
86. C₅₃	#3131	semidirect

H ≅ G/N

type

1. #16811180

C₂

semidirect

2. #16810977

C₂

semidirect

3. #16810973

C₂

semidirect

4. #16810969

C₂

semidirect

5. #16810619

C₂

semidirect

6. C₂×C₄×#88322

C₂

semidirect

86. C₅₃

#3131

semidirect

Subgroups — #20285037

Conjugacy classes of subgroups

Complements