Group #20275625 — subgroups

Subgroups — #20275625

Order 6784.

316 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂×#13133153	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810627	C₂	1	2	yes	maximal, complemented
4	C₂×#13133133	C₂	1	2	yes	maximal, complemented
5	C₅₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₁₂	C₂²	1	2	yes	—
7	C₂×D₄₂₄	—	2	2	no	—
8	C₂×D₄₂₄	—	2	1	no	—
9	C₂×D₄₂₄	—	2	1	no	—
10	C₂×D₄₂₄	C₄	1	4	yes	complemented
11	C₂×#11719917	C₂²	1	2	yes	—
12	#13133153	—	2	2	no	—
13	#13133137	—	2	4	no	—
14	#13133133	—	2	4	no	—
15	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	#117×C₅₃	C₂²	1	1	yes	characteristic
18	#97×C₅₃	—	2	1	no	—
19	C₄×C₈×C₅₃	—	2	1	no	—
20	C₂³×D₅₃	—	2	2	no	—
21	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂×D₂₁₂	—	2	2	no	—
23	C₂×D₂₁₂	—	2	2	no	—
24	C₂×D₂₁₂	—	2	2	no	—
25	C₂×D₂₁₂	—	2	2	no	—
26	C₂×D₂₁₂	—	2	2	no	—
27	C₂×D₂₁₂	C₂×C₄	1	4	yes	—
28	M₁₆×C₅₃	—	2	1	no	—
29	M₁₆×C₅₃	D₄	1	2	yes	—
30	C₂×C₈×C₅₃	—	2	1	no	—
31	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₅₃	D₄	1	2	yes	—
33	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
34	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
35	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
36	C₂²⋊C₄×C₅₃	—	2	1	no	—
37	C₄²×C₅₃	—	2	1	no	—
38	#11719917	—	2	2	no	—
39	#11719917	—	2	2	no	—
40	C₂×#88322	—	2	2	no	—
41	D₄₂₄	—	2	4	no	—
42	D₄₂₄	—	2	4	no	—
43	C₂³×C₅₃	#41	1	1	yes	characteristic
44	C₂²×D₅₃	—	2	4	no	—
45	C₂²×D₅₃	—	4	2	no	—
46	C₂²×D₅₃	—	4	2	no	—
47	C₂²×D₅₃	—	8	4	no	—
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
55	D₂₁₂	—	2	4	no	—
56	D₂₁₂	—	2	4	no	—
57	D₂₁₂	—	4	2	no	—
58	D₂₁₂	—	4	2	no	—
59	C₈×C₅₃	—	2	2	no	—
60	C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
62	#88322	—	4	2	no	—
63	C₂²×C₅₃	—	2	2	no	—
64	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
65	C₂²×C₅₃	#121	1	1	yes	characteristic
66	C₂²×C₅₃	#124	1	1	yes	characteristic
67	C₂×D₅₃	—	4	4	no	—
68	C₂×D₅₃	—	8	4	no	—
69	C₂×D₅₃	—	8	4	no	—
70	C₂×D₅₃	—	8	4	no	—
71	C₄×C₅₃	—	4	2	no	—
72	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
74	C₄×C₅₃	#121	1	1	yes	characteristic
75	C₄×C₅₃	#127	1	1	yes	characteristic
76	#1978	—	53	1	no	maximal, Sylow
77	C₂×C₅₃	—	2	1	no	—
78	C₂×C₅₃	—	2	1	no	—
79	C₂×C₅₃	#390	1	1	yes	characteristic
80	C₂×C₅₃	#472	1	2	yes	—
81	D₅₃	—	8	4	no	—
82	C₂²×D₈	—	53	1	no	—
83	C₂×#108	—	53	1	no	—
84	#418	—	53	2	no	—
85	C₂×#100	—	53	2	no	—
86	#405	D₅₃	1	1	yes	characteristic, complemented
87	C₅₃	#1978	1	1	yes	characteristic, complemented, Sylow
88	C₂²×D₄	—	53	2	no	—
89	C₂×D₈	—	53	4	no	—
90	C₂×D₈	—	106	2	no	—
91	C₂×D₈	—	106	1	no	—
92	C₂×D₈	—	106	1	no	—
93	C₂×M₁₆	—	53	2	no	—
94	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
95	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
96	#117	C₂×D₅₃	1	1	yes	characteristic
97	#108	—	106	2	no	—
98	#102	—	106	4	no	—
99	#100	—	106	4	no	—
100	#97	—	2	1	no	—
101	C₄×C₈	—	2	1	no	—
102	C₂⁴	—	106	2	no	—
103	C₂×D₄	—	53	4	no	—
104	C₂×D₄	—	106	2	no	—
105	C₂×D₄	—	106	2	no	—
106	C₂×D₄	—	106	2	no	—
107	C₂×D₄	—	106	2	no	—
108	C₂×D₄	—	106	2	no	—
109	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
110	D₈	—	106	4	no	—
111	D₈	—	106	4	no	—
112	M₁₆	—	2	1	no	—
113	M₁₆	—	106	2	no	—
114	M₁₆	—	106	2	no	—
115	M₁₆	D₂₁₂	1	2	yes	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	106	2	no	—
118	C₂×C₈	C₄×D₅₃	1	2	yes	—
119	C₂×C₈	D₂₁₂	1	2	yes	—
120	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
121	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
122	C₄⋊C₄	D₂₁₂	1	2	yes	—
123	C₂²⋊C₄	—	2	1	no	—
124	C₄²	—	2	1	no	—
125	C₂³	—	106	4	no	—
126	C₂³	—	424	4	no	—
127	C₂³	—	212	2	no	—
128	C₂³	—	212	2	no	—
129	C₂³	#11719945	1	1	yes	characteristic
130	D₄	—	106	4	no	—
131	D₄	—	106	4	no	—
132	D₄	—	212	2	no	—
133	D₄	—	212	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
136	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
137	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
138	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
139	C₂×C₄	#11719945	1	1	yes	characteristic
140	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	212	2	no	—
143	C₈	#11719921	1	2	yes	—
144	C₈	#11719921	1	2	yes	—
145	C₂²	—	2	2	no	—
146	C₂²	—	424	4	no	—
147	C₂²	—	424	4	no	—
148	C₂²	—	424	4	no	—
149	C₂²	—	212	4	no	—
150	C₂²	C₄×D₂₁₂	1	1	yes	characteristic
151	C₂²	#13133191	1	1	yes	characteristic
152	C₂²	#13133205	1	1	yes	characteristic
153	C₄	—	4	2	no	—
154	C₄	C₂×#11719921	1	1	yes	characteristic
155	C₄	C₄×D₂₁₂	1	1	yes	characteristic
156	C₄	#13133191	1	1	yes	characteristic
157	C₄	#13133214	1	1	yes	characteristic
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	—	424	4	no	—
161	C₂	#16810525	1	1	yes	characteristic
162	C₂	#16810893	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₄₂₄

C₂

yes

maximal, characteristic

C₂×#13133153

C₂

yes

maximal, characteristic, complemented

#16810627

C₂

yes

maximal, complemented

C₂×#13133133

C₂

yes

maximal, complemented

C₅₃×#405

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

C₂²

yes

—

C₂×D₄₂₄

—

C₂×D₄₂₄

—

C₂×D₄₂₄

—

C₂×D₄₂₄

C₄

yes

complemented

C₂×#11719917

C₂²

yes

—

#13133153

—

#13133137

—

#13133133

—

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₂²

yes

characteristic

#97×C₅₃

—

C₄×C₈×C₅₃

—

C₂³×D₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

C₂×C₄

yes

—

M₁₆×C₅₃

—

M₁₆×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

D₄

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

#11719917

—

#11719917

—

C₂×#88322

—

D₄₂₄

—

D₄₂₄

—

C₂³×C₅₃

#41

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

—

C₈×C₅₃

C₂²⋊C₄

yes

—

C₈×C₅₃

C₂²⋊C₄

yes

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄×D₄

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

C₄×D₄

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#390

yes

characteristic

C₂×C₅₃

#472

yes

—

D₅₃

—

C₂²×D₈

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#1978

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

106

—

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

#108

—

106

—

#102

—

106

—

#100

—

106

—

100

#97

—

101

C₄×C₈

—

102

C₂⁴

—

106

—

103

C₂×D₄

—

104

C₂×D₄

—

106

—

105

C₂×D₄

—

106

—

106

C₂×D₄

—

106

—

107

C₂×D₄

—

106

—

108

C₂×D₄

—

106

—

109

C₂²×C₄

C₂²×D₅₃

yes

characteristic

110

D₈

—

106

—

111

D₈

—

106

—

112

M₁₆

—

113

M₁₆

—

106

—

114

M₁₆

—

106

—

115

M₁₆

D₂₁₂

yes

—

116

C₂×C₈

—

117

C₂×C₈

—

106

—

118

C₂×C₈

C₄×D₅₃

yes

—

119

C₂×C₈

D₂₁₂

yes

—

120

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

121

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

122

C₄⋊C₄

D₂₁₂

yes

—

123

C₂²⋊C₄

—

124

C₄²

—

125

C₂³

—

106

—

126

C₂³

—

424

—

127

C₂³

—

212

—

128

C₂³

—

212

—

129

C₂³

#11719945

yes

characteristic

130

D₄

—

106

—

131

D₄

—

106

—

132

D₄

—

212

—

133

D₄

—

212

—

134

C₂×C₄

—

135

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

136

C₂×C₄

C₂×D₂₁₂

yes

characteristic

137

C₂×C₄

C₂×D₂₁₂

yes

characteristic

138

C₂×C₄

C₂×D₂₁₂

yes

characteristic

139

C₂×C₄

#11719945

yes

characteristic

140

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

141

C₈

—

142

C₈

—

212

—

143

C₈

#11719921

yes

—

144

C₈

#11719921

yes

—

145

C₂²

—

146

C₂²

—

424

—

147

C₂²

—

424

—

148

C₂²

—

424

—

149

C₂²

—

212

—

150

C₂²

C₄×D₂₁₂

yes

characteristic

151

C₂²

#13133191

yes

characteristic

152

C₂²

#13133205

yes

characteristic

153

C₄

—

154

C₄

C₂×#11719921

yes

characteristic

155

C₄

C₄×D₂₁₂

yes

characteristic

156

C₄

#13133191

yes

characteristic

157

C₄

#13133214

yes

characteristic

158

C₂

—

159

C₂

—

160

C₂

—

424

—

161

C₂

#16810525

yes

characteristic

162

C₂

#16810893

yes

—

	subgroup	G/H
center	C₂²	#13133191
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#405	C₂
socle	C₂²×C₅₃	#121
hypercenter	#405	D₅₃
hypocenter	C₅₃	#1978
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

subgroup

G/H

center

C₂²

#13133191

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#405

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

N	H ≅ G/N	type
2. C₂×#13133153	C₂	semidirect
3. #16810627	C₂	semidirect
4. C₂×#13133133	C₂	semidirect
5. C₅₃×#405	C₂	semidirect
10. C₂×D₄₂₄	C₄	semidirect
86. #405	D₅₃	semidirect
87. C₅₃	#1978	semidirect

H ≅ G/N

type

2. C₂×#13133153

C₂

semidirect

3. #16810627

C₂

semidirect

4. C₂×#13133133

C₂

semidirect

5. C₅₃×#405

C₂

semidirect

10. C₂×D₄₂₄

C₄

semidirect

86. #405

D₅₃

semidirect

87. C₅₃

#1978

semidirect

Subgroups — #20275625

Conjugacy classes of subgroups

Complements