Group #20283357 — subgroups

Subgroups — #20283357

Order 6784.

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

normal subgroups only

This group has 400 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	C₄×#11719945	C₂	1	1	yes	maximal, characteristic
2	#16810682	C₂	1	4	yes	maximal, complemented
3	#16810674	C₂	1	4	yes	maximal, complemented
4	#16810649	C₂	1	2	yes	maximal, complemented
5	#16810589	C₂	1	2	yes	maximal
6	C₄×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719945	C₂²	1	1	yes	characteristic
9	C₂×#11719917	C₂²	1	2	yes	—
10	C₄×Dic₁₀₆	C₄	1	2	yes	—
11	C₄×D₂₁₂	C₄	1	2	yes	—
12	C₄×C₂²⋊D₅₃	C₄	1	4	yes	—
13	#13133176	C₂²	1	1	yes	characteristic
14	#13133174	C₂²	1	2	yes	—
15	C₄²×D₅₃	C₂²	1	2	yes	—
16	#13133146	C₂²	1	8	yes	—
17	#13133129	C₂²	1	8	yes	—
18	#13133128	C₂²	1	2	yes	—
19	C₄×#88322	C₂²	1	2	yes	—
20	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₅₃	C₂²	1	2	yes	—
23	C₄×C₈×C₅₃	C₂²	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
27	#11719945	—	2	4	no	—
28	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₅₃	C₂×C₄	1	4	yes	—
30	C₂×C₄×D₅₃	C₂³	1	2	yes	—
31	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	—	2	4	no	—
33	C₂×C₈×C₅₃	C₂³	1	4	yes	—
34	C₄²×C₅₃	C₂×C₄	1	2	yes	—
35	C₄²×C₅₃	C₂³	1	1	yes	characteristic
36	C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
38	#11719921	C₂×C₄	1	4	yes	—
39	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
40	#11719919	C₂×C₄	1	4	yes	—
41	C₄×Dic₅₃	C₂×C₄	1	4	yes	—
42	C₄×Dic₅₃	C₂³	1	2	yes	—
43	#11719917	—	2	4	no	—
44	C₂×#88322	C₂³	1	4	yes	—
45	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₅₃	—	2	4	no	—
48	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₅₃	#41	1	4	yes	—
52	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
53	C₂²⋊D₅₃	—	2	4	no	—
54	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
55	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
56	D₂₁₂	—	2	2	no	—
57	C₄×D₅₃	—	2	8	no	—
58	C₄×D₅₃	—	2	4	no	—
59	Dic₁₀₆	—	2	2	no	—
60	C₈×C₅₃	—	2	4	no	—
61	#88322	—	2	4	no	—
62	C₂²×C₅₃	—	2	2	no	—
63	C₂²×C₅₃	#117	1	2	yes	—
64	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
65	C₂×D₅₃	—	2	2	no	—
66	C₂×D₅₃	—	2	4	no	—
67	C₄×C₅₃	—	2	2	no	—
68	C₄×C₅₃	#117	1	2	yes	—
69	C₄×C₅₃	#131	1	4	yes	—
70	C₄×C₅₃	C₂×#41	1	2	yes	—
71	Dic₅₃	—	2	2	no	—
72	Dic₅₃	—	2	4	no	—
73	#2922	—	53	1	no	maximal, Sylow
74	C₂×C₅₃	—	2	2	no	—
75	C₂×C₅₃	C₂×#117	1	1	yes	characteristic
76	C₂×C₅₃	C₂×#131	1	2	yes	—
77	D₅₃	—	4	2	no	—
78	C₄×#41	—	53	1	no	—
79	#433	—	53	4	no	—
80	#431	—	53	4	no	—
81	#424	—	53	2	no	—
82	#408	—	53	2	no	—
83	C₄×M₁₆	—	53	1	no	—
84	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic, complemented
85	C₅₃	#2922	1	1	yes	characteristic, complemented, Sylow
86	C₂×#41	—	53	1	no	—
87	C₂×M₁₆	—	53	2	no	—
88	C₂²×C₈	C₂×D₅₃	1	2	yes	—
89	C₄×Q₈	—	53	2	no	—
90	C₄×D₄	—	53	4	no	—
91	C₄×D₄	—	53	2	no	—
92	#117	—	53	1	no	—
93	#117	—	53	2	no	—
94	C₂×C₄²	—	53	2	no	—
95	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
96	C₄⋊C₈	—	53	8	no	—
97	C₂²⋊C₈	—	53	8	no	—
98	#97	—	53	2	no	—
99	C₄×C₈	—	53	2	no	—
100	C₄×C₈	C₂×D₅₃	1	2	yes	—
101	C₄×C₈	C₂×D₅₃	1	2	yes	—
102	#41	—	106	4	no	—
103	C₂×Q₈	—	53	1	no	—
104	C₂×D₄	—	53	2	no	—
105	C₂×D₄	—	53	1	no	—
106	C₂²×C₄	—	53	4	no	—
107	C₂²×C₄	—	53	2	no	—
108	C₂²×C₄	C₄×D₅₃	1	2	yes	—
109	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
110	M₁₆	—	106	4	no	—
111	C₂×C₈	—	2	4	no	—
112	C₂×C₈	—	53	4	no	—
113	C₂×C₈	C₂²×D₅₃	1	4	yes	—
114	C₄⋊C₄	—	53	4	no	—
115	C₄⋊C₄	—	53	2	no	—
116	C₂²⋊C₄	—	53	2	no	—
117	C₂²⋊C₄	—	53	4	no	—
118	C₄²	—	53	4	no	—
119	C₄²	—	53	2	no	—
120	C₄²	C₄×D₅₃	1	2	yes	—
121	C₄²	C₂²×D₅₃	1	1	yes	characteristic
122	C₄²	C₂²×D₅₃	1	1	yes	characteristic
123	C₂³	—	53	2	no	—
124	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
125	Q₈	—	106	2	no	—
126	D₄	—	106	4	no	—
127	D₄	—	106	2	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	53	4	no	—
130	C₂×C₄	—	53	2	no	—
131	C₂×C₄	—	106	8	no	—
132	C₂×C₄	—	106	4	no	—
133	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
134	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
135	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
136	C₂×C₄	#11719945	1	4	yes	—
137	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
138	C₈	—	2	4	no	—
139	C₈	—	106	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	106	2	no	—
142	C₂²	—	106	4	no	—
143	C₂²	#13133174	1	2	yes	—
144	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	106	2	no	—
147	C₄	—	106	4	no	—
148	C₄	#13133174	1	2	yes	—
149	C₄	#13133223	1	4	yes	—
150	C₄	C₂×#11719945	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	212	2	no	—
153	C₂	C₂×#13133174	1	1	yes	characteristic
154	C₂	C₂×#13133223	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#11719945

C₂

yes

maximal, characteristic

#16810682

C₂

yes

maximal, complemented

#16810674

C₂

yes

maximal, complemented

#16810649

C₂

yes

maximal, complemented

#16810589

C₂

yes

maximal

C₄×#11719917

C₂

yes

maximal, characteristic, complemented

C₂×C₄×C₈×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×#11719945

C₂²

yes

characteristic

C₂×#11719917

C₂²

yes

—

C₄×Dic₁₀₆

C₄

yes

—

C₄×D₂₁₂

C₄

yes

—

C₄×C₂²⋊D₅₃

C₄

yes

—

#13133176

C₂²

yes

characteristic

#13133174

C₂²

yes

—

C₄²×D₅₃

C₂²

yes

—

#13133146

C₂²

yes

—

#13133129

C₂²

yes

—

#13133128

C₂²

yes

—

C₄×#88322

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₂²⋊D₅₃

C₂×C₄

yes

—

#11719945

—

C₂×D₂₁₂

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

—

C₂×Dic₁₀₆

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

C₂×C₄

yes

—

#11719921

C₂×C₄

yes

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂³

yes

—

#11719917

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂²×D₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂²×C₄

yes

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

#117

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#117

yes

—

C₄×C₅₃

#131

yes

—

C₄×C₅₃

C₂×#41

yes

—

Dic₅₃

—

Dic₅₃

—

#2922

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×#117

yes

characteristic

C₂×C₅₃

C₂×#131

yes

—

D₅₃

—

C₄×#41

—

—

—

—

—

C₄×M₁₆

—

C₂×C₄×C₈

D₅₃

yes

characteristic, complemented

C₅₃

#2922

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₅₃

yes

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

#117

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

100

C₄×C₈

C₂×D₅₃

yes

—

101

C₄×C₈

C₂×D₅₃

yes

—

102

#41

—

106

—

103

C₂×Q₈

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₄×D₅₃

yes

—

109

C₂²×C₄

C₂²×D₅₃

yes

characteristic

110

M₁₆

—

106

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₂²×D₅₃

yes

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₄×D₅₃

yes

—

121

C₄²

C₂²×D₅₃

yes

characteristic

122

C₄²

C₂²×D₅₃

yes

characteristic

123

C₂³

—

124

C₂³

C₂×C₄×D₅₃

yes

characteristic

125

Q₈

—

106

—

126

D₄

—

106

—

127

D₄

—

106

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

106

—

132

C₂×C₄

—

106

—

133

C₂×C₄

C₂×C₄×D₅₃

yes

—

134

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

135

C₂×C₄

C₂×C₄×D₅₃

yes

—

136

C₂×C₄

#11719945

yes

—

137

C₂×C₄

C₂³×D₅₃

yes

characteristic

138

C₈

—

139

C₈

—

106

—

140

C₂²

—

141

C₂²

—

106

—

142

C₂²

—

106

—

143

C₂²

#13133174

yes

—

144

C₂²

C₂²×C₄×D₅₃

yes

characteristic

145

C₄

—

146

C₄

—

106

—

147

C₄

—

106

—

148

C₄

#13133174

yes

—

149

C₄

#13133223

yes

—

150

C₄

C₂×#11719945

yes

—

151

C₂

—

152

C₂

—

212

—

153

C₂

C₂×#13133174

yes

characteristic

154

C₂

C₂×#13133223

yes

—

	subgroup	G/H
center	C₄²	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×C₄×C₈×C₅₃	C₂
hypercenter	C₂×C₄×C₈	D₅₃
hypocenter	C₅₃	#2922
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₄²

C₂²×D₅₃

commutator, socle

C₂²×C₅₃

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂×C₄×C₈×C₅₃

C₂

hypercenter

C₂×C₄×C₈

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #16810682	C₂	semidirect
3. #16810674	C₂	semidirect
4. #16810649	C₂	semidirect
6. C₄×#11719917	C₂	semidirect
7. C₂×C₄×C₈×C₅₃	C₂	semidirect
84. C₂×C₄×C₈	D₅₃	semidirect
85. C₅₃	#2922	semidirect

H ≅ G/N

type

2. #16810682

C₂

semidirect

3. #16810674

C₂

semidirect

4. #16810649

C₂

semidirect

6. C₄×#11719917

C₂

semidirect

7. C₂×C₄×C₈×C₅₃

C₂

semidirect

84. C₂×C₄×C₈

D₅₃

semidirect

85. C₅₃

#2922

semidirect

Subgroups — #20283357

Conjugacy classes of subgroups

Complements