Group #20285104 — subgroups

Subgroups — #20285104

Order 6784.

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

normal subgroups only

This group has 400 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	C₂×#13133205	C₂	1	1	yes	maximal, characteristic
2	#16810985	C₂	1	4	yes	maximal, complemented
3	#16810980	C₂	1	4	yes	maximal, complemented
4	#16810631	C₂	1	2	yes	maximal, complemented
5	#16810627	C₂	1	2	yes	maximal
6	C₄×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
9	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic
10	C₂×#11719917	C₂²	1	2	yes	—
11	#13133205	—	2	2	no	—
12	#13133205	C₂²	1	2	yes	—
13	#13133205	C₂²	1	2	yes	—
14	C₂×#11719921	—	2	2	no	—
15	#13133140	C₂²	1	8	yes	complemented
16	#13133137	C₂²	1	8	yes	—
17	#13133128	C₂²	1	2	yes	complemented
18	C₄×#88322	C₂²	1	2	yes	complemented
19	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
20	#125×C₅₃	C₂²	1	2	yes	complemented
21	C₂²⋊Q₈×C₅₃	—	2	2	no	—
22	C₄×Q₈×C₅₃	—	2	2	no	—
23	#117×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
25	C₂³×D₅₃	—	2	1	no	—
26	C₂×Q₈×C₅₃	—	2	2	no	—
27	C₂²×C₄×C₅₃	D₄	1	2	yes	—
28	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	C₂²×Dic₅₃	—	2	1	no	—
30	C₂×D₂₁₂	—	2	4	no	—
31	C₂×D₂₁₂	—	2	1	no	—
32	C₂×D₂₁₂	C₂³	1	2	yes	—
33	C₂×Dic₁₀₆	—	2	4	no	—
34	C₂×Dic₁₀₆	—	2	1	no	—
35	C₂×Dic₁₀₆	C₂³	1	2	yes	—
36	C₄⋊C₄×C₅₃	—	2	4	no	—
37	C₄⋊C₄×C₅₃	—	2	2	no	—
38	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
39	C₂²⋊C₄×C₅₃	—	2	2	no	—
40	C₄²×C₅₃	—	2	2	no	—
41	C₄²×C₅₃	D₄	1	2	yes	—
42	C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	#11719921	—	2	4	no	—
45	#11719921	—	2	4	no	—
46	#11719917	—	2	4	no	—
47	C₂×#88322	C₂³	1	4	yes	—
48	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×D₅₃	—	2	2	no	—
50	C₂²×D₅₃	—	4	4	no	—
51	C₂²×D₅₃	—	4	2	no	—
52	Q₈×C₅₃	—	4	4	no	—
53	C₂×C₄×C₅₃	—	2	4	no	—
54	C₂×C₄×C₅₃	—	4	2	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	2	no	—
61	C₂×Dic₅₃	—	4	2	no	—
62	D₂₁₂	—	2	4	no	—
63	D₂₁₂	—	4	2	no	—
64	Dic₁₀₆	—	2	4	no	—
65	Dic₁₀₆	—	4	2	no	—
66	#88322	—	2	4	no	—
67	C₂²×C₅₃	—	2	2	no	—
68	C₂²×C₅₃	#124	1	2	yes	—
69	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
70	C₂×D₅₃	—	4	2	no	—
71	C₂×D₅₃	—	4	4	no	—
72	C₂×D₅₃	—	8	4	no	—
73	C₄×C₅₃	—	2	2	no	—
74	C₄×C₅₃	—	2	2	no	—
75	C₄×C₅₃	—	4	4	no	—
76	C₄×C₅₃	#124	1	2	yes	—
77	C₄×C₅₃	C₂×#41	1	2	yes	—
78	Dic₅₃	—	4	2	no	—
79	#3140	—	53	1	no	maximal, Sylow
80	C₂×C₅₃	—	2	2	no	—
81	C₂×C₅₃	C₂×#124	1	1	yes	characteristic
82	C₂×C₅₃	#575	1	2	yes	—
83	D₅₃	—	8	2	no	—
84	#533	D₅₃	1	1	yes	characteristic, complemented
85	C₂×#124	—	53	1	no	—
86	#489	—	53	4	no	—
87	#488	—	53	4	no	—
88	#419	—	53	2	no	—
89	#418	—	53	2	no	—
90	C₄×M₁₆	—	53	1	no	—
91	C₅₃	#3140	1	1	yes	characteristic, complemented, Sylow
92	C₂²×Q₈	—	53	1	no	—
93	C₂²×D₄	—	53	1	no	—
94	C₂×M₁₆	—	53	2	no	—
95	C₄⋊Q₈	C₂×D₅₃	1	2	yes	complemented
96	#125	C₂×D₅₃	1	2	yes	complemented
97	#124	—	53	2	no	—
98	#124	—	53	2	no	—
99	#124	—	106	2	no	—
100	C₂²⋊Q₈	—	2	2	no	—
101	C₄×Q₈	—	2	2	no	—
102	#117	C₂×D₅₃	1	2	yes	—
103	C₂×C₂²⋊C₄	—	106	2	no	—
104	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
105	#103	—	53	8	no	—
106	#102	—	53	8	no	—
107	#97	—	53	2	no	—
108	C₄×C₈	—	53	2	no	—
109	C₂⁴	—	106	1	no	—
110	C₂×Q₈	—	2	2	no	—
111	C₂×Q₈	—	53	2	no	—
112	C₂×Q₈	—	106	1	no	—
113	C₂×Q₈	—	106	4	no	—
114	C₂×D₄	—	53	2	no	—
115	C₂×D₄	—	106	4	no	—
116	C₂×D₄	—	106	1	no	—
117	C₂²×C₄	—	106	1	no	—
118	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
119	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
120	M₁₆	—	106	4	no	—
121	C₂×C₈	—	53	4	no	—
122	C₄⋊C₄	—	2	4	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
125	C₂²⋊C₄	—	2	2	no	—
126	C₂²⋊C₄	—	106	4	no	—
127	C₂²⋊C₄	—	106	4	no	—
128	C₄²	—	2	2	no	—
129	C₄²	C₂²⋊D₅₃	1	2	yes	—
130	C₄²	C₂²×D₅₃	1	1	yes	characteristic
131	C₄²	C₂²×D₅₃	1	1	yes	characteristic
132	C₂³	—	106	2	no	—
133	C₂³	—	212	4	no	—
134	C₂³	—	212	2	no	—
135	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
136	Q₈	—	4	4	no	—
137	Q₈	—	106	4	no	—
138	Q₈	—	212	2	no	—
139	D₄	—	106	4	no	—
140	D₄	—	212	2	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	106	2	no	—
144	C₂×C₄	—	212	2	no	—
145	C₂×C₄	#11719949	1	4	yes	—
146	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
148	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
149	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
150	C₈	—	106	4	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	424	4	no	—
153	C₂²	—	212	2	no	—
154	C₂²	—	212	4	no	—
155	C₂²	#13133206	1	2	yes	—
156	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	4	4	no	—
160	C₄	—	212	2	no	—
161	C₄	#13133206	1	2	yes	—
162	C₄	C₂×#11719949	1	2	yes	—
163	C₂	—	2	2	no	—
164	C₂	—	424	2	no	—
165	C₂	C₂×#13133206	1	1	yes	characteristic
166	C₂	#16811421	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133205

C₂

yes

maximal, characteristic

#16810985

C₂

yes

maximal, complemented

#16810980

C₂

yes

maximal, complemented

#16810631

C₂

yes

maximal, complemented

#16810627

C₂

yes

maximal

C₄×#11719917

C₂

yes

maximal, characteristic, complemented

C₅₃×#533

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₁₀₆

C₂²

yes

characteristic

C₂²×D₂₁₂

C₂²

yes

characteristic

C₂×#11719917

C₂²

yes

—

#13133205

—

#13133205

C₂²

yes

—

#13133205

C₂²

yes

—

C₂×#11719921

—

#13133140

C₂²

yes

complemented

#13133137

C₂²

yes

—

#13133128

C₂²

yes

complemented

C₄×#88322

C₂²

yes

complemented

C₄⋊Q₈×C₅₃

C₂²

yes

complemented

#125×C₅₃

C₂²

yes

complemented

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂³×D₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

C₂³

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂³

yes

—

—

—

C₂³

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

Q₈×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

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

Dic₁₀₆

—

Dic₁₀₆

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

#124

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#124

yes

—

C₄×C₅₃

C₂×#41

yes

—

Dic₅₃

—

#3140

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×#124

yes

characteristic

C₂×C₅₃

#575

yes

—

D₅₃

—

#533

D₅₃

yes

characteristic, complemented

—

—

—

—

—

C₄×M₁₆

—

C₅₃

#3140

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×D₄

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₅₃

yes

complemented

#125

C₂×D₅₃

yes

—

—

—

106

—

100

C₂²⋊Q₈

—

101

C₄×Q₈

—

102

#117

C₂×D₅₃

yes

—

103

C₂×C₂²⋊C₄

—

106

—

104

C₂×C₄²

C₂×D₅₃

yes

characteristic

105

#103

—

106

#102

—

107

#97

—

108

C₄×C₈

—

109

C₂⁴

—

106

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

106

—

113

C₂×Q₈

—

106

—

114

C₂×D₄

—

115

C₂×D₄

—

106

—

116

C₂×D₄

—

106

—

117

C₂²×C₄

—

106

—

118

C₂²×C₄

C₂²⋊D₅₃

yes

—

119

C₂²×C₄

C₂²×D₅₃

yes

characteristic

120

M₁₆

—

106

—

121

C₂×C₈

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

C₂²×D₅₃

yes

—

125

C₂²⋊C₄

—

126

C₂²⋊C₄

—

106

—

127

C₂²⋊C₄

—

106

—

128

C₄²

—

129

C₄²

C₂²⋊D₅₃

yes

—

130

C₄²

C₂²×D₅₃

yes

characteristic

131

C₄²

C₂²×D₅₃

yes

characteristic

132

C₂³

—

106

—

133

C₂³

—

212

—

134

C₂³

—

212

—

135

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

136

Q₈

—

137

Q₈

—

106

—

138

Q₈

—

212

—

139

D₄

—

106

—

140

D₄

—

212

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

106

—

144

C₂×C₄

—

212

—

145

C₂×C₄

#11719949

yes

—

146

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

147

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

148

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

149

C₂×C₄

C₂³×D₅₃

yes

characteristic

150

C₈

—

106

—

151

C₂²

—

152

C₂²

—

424

—

153

C₂²

—

212

—

154

C₂²

—

212

—

155

C₂²

#13133206

yes

—

156

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

212

—

161

C₄

#13133206

yes

—

162

C₄

C₂×#11719949

yes

—

163

C₂

—

164

C₂

—

424

—

165

C₂

C₂×#13133206

yes

characteristic

166

C₂

#16811421

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#533	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#533	D₅₃
hypocenter	C₅₃	#3140
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#533

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #16810985	C₂	semidirect
3. #16810980	C₂	semidirect
4. #16810631	C₂	semidirect
6. C₄×#11719917	C₂	semidirect
7. C₅₃×#533	C₂	semidirect
15. #13133140	C₂²	semidirect
17. #13133128	C₂²	semidirect
18. C₄×#88322	C₂²	semidirect
19. C₄⋊Q₈×C₅₃	C₂²	semidirect
20. #125×C₅₃	C₂²	semidirect
84. #533	D₅₃	semidirect
91. C₅₃	#3140	semidirect
95. C₄⋊Q₈	C₂×D₅₃	semidirect
96. #125	C₂×D₅₃	semidirect

H ≅ G/N

type

2. #16810985

C₂

semidirect

3. #16810980

C₂

semidirect

4. #16810631

C₂

semidirect

6. C₄×#11719917

C₂

semidirect

7. C₅₃×#533

C₂

semidirect

15. #13133140

C₂²

semidirect

17. #13133128

C₂²

semidirect

18. C₄×#88322

C₂²

semidirect

19. C₄⋊Q₈×C₅₃

C₂²

semidirect

20. #125×C₅₃

C₂²

semidirect

84. #533

D₅₃

semidirect

91. C₅₃

#3140

semidirect

95. C₄⋊Q₈

C₂×D₅₃

semidirect

96. #125

C₂×D₅₃

semidirect

Subgroups — #20285104

Conjugacy classes of subgroups

Complements