Group #20283277 — subgroups

Subgroups — #20283277

Order 6784.

484 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 213 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×M₁₆×D₅₃	C₂	1	2	yes	maximal
2	C₂×#13133174	C₂	1	1	yes	maximal, characteristic
3	#16810648	C₂	1	8	yes	maximal, complemented
4	#16810647	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810646	C₂	1	2	yes	maximal, complemented
6	C₅₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
8	C₂×#11719917	C₂²	1	2	yes	—
9	C₂×#11719912	C₂²	1	4	yes	—
10	M₁₆×D₅₃	—	2	8	no	—
11	C₂×C₈×D₅₃	C₂²	1	4	yes	—
12	#13133174	C₄	1	4	yes	—
13	#13133174	C₂²	1	4	yes	—
14	C₂×#11719919	C₄	1	2	yes	—
15	C₂×#11719921	C₄	1	2	yes	—
16	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
17	C₄⋊#88322	C₂²	1	4	yes	complemented
18	#13133146	C₂²	1	8	yes	complemented
19	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₅₃	C₂²	1	4	yes	complemented
22	C₂³×D₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
26	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₅₃	D₄	1	4	yes	—
29	C₂×C₄×D₅₃	Q₈	1	4	yes	—
30	C₂×C₄×D₅₃	C₂³	1	2	yes	—
31	M₁₆×C₅₃	—	2	4	no	—
32	C₂×C₈×C₅₃	C₂³	1	4	yes	—
33	C₄²×C₅₃	C₂×C₄	1	2	yes	—
34	C₄²×C₅₃	C₂³	1	2	yes	—
35	#11719921	C₂×C₄	1	8	yes	—
36	#11719919	C₂×C₄	1	8	yes	—
37	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
38	C₄×Dic₅₃	C₂³	1	2	yes	—
39	#11719917	—	2	4	no	—
40	C₂×#88322	C₂³	1	4	yes	—
41	#11719912	—	2	8	no	—
42	C₈×D₅₃	—	2	8	no	—
43	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×D₅₃	—	4	2	no	—
45	C₂²×D₅₃	C₄⋊C₄	1	4	yes	—
46	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₅₃	—	4	2	no	—
48	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₅₃	—	4	2	no	—
54	C₂×Dic₅₃	C₄⋊C₄	1	4	yes	—
55	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
56	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
57	C₄×D₅₃	C₄⋊C₄	1	8	yes	—
58	C₄×D₅₃	C₂×D₄	1	4	yes	—
59	C₄×D₅₃	C₂×Q₈	1	4	yes	—
60	C₈×C₅₃	—	2	4	no	—
61	#88322	—	2	4	no	—
62	C₂²×C₅₃	—	2	2	no	—
63	C₂²×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
64	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
65	C₂×D₅₃	—	2	4	no	—
66	C₂×D₅₃	—	4	4	no	—
67	C₂×D₅₃	C₂×C₄⋊C₄	1	4	yes	—
68	C₄×C₅₃	—	2	4	no	—
69	C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
70	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
71	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
72	Dic₅₃	—	2	4	no	—
73	Dic₅₃	C₂×C₄⋊C₄	1	4	yes	—
74	#2905	—	53	1	no	maximal, Sylow
75	C₂×C₅₃	—	2	2	no	—
76	C₂×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₅₃	#568	1	2	yes	—
78	D₅₃	—	2	4	no	—
79	C₂²×M₁₆	—	53	2	no	—
80	C₂×#117	—	53	1	no	—
81	#424	—	53	8	no	—
82	#423	—	53	2	no	—
83	#423	—	53	1	no	—
84	#423	D₅₃	1	1	yes	characteristic, complemented
85	C₅₃	#2905	1	1	yes	characteristic, complemented, Sylow
86	C₂³×C₄	—	53	1	no	—
87	C₂×M₁₆	—	53	2	no	—
88	C₂×M₁₆	—	53	4	no	—
89	C₂×M₁₆	—	106	8	no	—
90	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
91	C₂²×C₈	—	53	4	no	—
92	#117	—	53	4	no	—
93	#117	—	53	4	no	—
94	C₂×C₄⋊C₄	—	53	2	no	—
95	C₂×C₂²⋊C₄	—	53	2	no	—
96	C₂×C₄²	—	53	1	no	—
97	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
98	C₄⋊C₈	—	53	8	no	—
99	C₄⋊C₈	—	53	4	no	—
100	C₄⋊C₈	C₂×D₅₃	1	4	yes	complemented
101	C₂⁴	—	53	1	no	—
102	C₂²×C₄	—	53	1	no	—
103	C₂²×C₄	—	53	2	no	—
104	C₂²×C₄	—	53	4	no	—
105	C₂²×C₄	—	53	4	no	—
106	C₂²×C₄	—	53	2	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	—	2	4	no	—
111	M₁₆	—	106	8	no	—
112	M₁₆	—	106	4	no	—
113	C₂×C₈	—	53	4	no	—
114	C₂×C₈	—	106	8	no	—
115	C₂×C₈	C₂²×D₅₃	1	4	yes	—
116	C₄⋊C₄	—	53	8	no	—
117	C₂²⋊C₄	—	53	8	no	—
118	C₄²	—	53	2	no	—
119	C₄²	—	53	2	no	—
120	C₄²	C₄×D₅₃	1	2	yes	—
121	C₄²	C₂²×D₅₃	1	2	yes	—
122	C₂³	—	53	2	no	—
123	C₂³	—	53	4	no	—
124	C₂³	—	212	2	no	—
125	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	53	4	no	—
128	C₂×C₄	—	53	2	no	—
129	C₂×C₄	—	53	8	no	—
130	C₂×C₄	—	53	4	no	—
131	C₂×C₄	—	53	4	no	—
132	C₂×C₄	—	53	4	no	—
133	C₂×C₄	—	212	2	no	—
134	C₂×C₄	C₂×C₄×D₅₃	1	4	yes	—
135	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
136	C₂×C₄	D₄×D₅₃	1	2	yes	—
137	C₂×C₄	Q₈×D₅₃	1	2	yes	—
138	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₈	—	106	4	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	53	4	no	—
143	C₂²	—	106	4	no	—
144	C₂²	—	212	4	no	—
145	C₂²	C₄⋊C₄×D₅₃	1	2	yes	—
146	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
147	C₄	—	2	4	no	—
148	C₄	—	53	4	no	—
149	C₄	—	106	4	no	—
150	C₄	C₄⋊C₄×D₅₃	1	2	yes	—
151	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
152	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
153	C₂	—	2	2	no	—
154	C₂	—	106	4	no	—
155	C₂	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
156	C₂	#16811389	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×M₁₆×D₅₃

C₂

yes

maximal

C₂×#13133174

C₂

yes

maximal, characteristic

#16810648

C₂

yes

maximal, complemented

#16810647

C₂

yes

maximal, characteristic, complemented

#16810646

C₂

yes

maximal, complemented

C₅₃×#423

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₅₃

C₂²

yes

characteristic

C₂×#11719917

C₂²

yes

—

C₂×#11719912

C₂²

yes

—

M₁₆×D₅₃

—

C₂×C₈×D₅₃

C₂²

yes

—

#13133174

C₄

yes

—

#13133174

C₂²

yes

—

C₂×#11719919

C₄

yes

—

C₂×#11719921

C₄

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

complemented

#13133146

C₂²

yes

complemented

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

complemented

C₂³×D₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

D₄

yes

—

C₂×C₄×D₅₃

Q₈

yes

—

C₂×C₄×D₅₃

C₂³

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂³

yes

—

#11719921

C₂×C₄

yes

—

#11719919

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂³

yes

—

#11719917

—

C₂×#88322

C₂³

yes

—

#11719912

—

C₈×D₅₃

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

C₄⋊C₄

yes

—

C₂²×D₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₄⋊C₄

yes

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₄×D₅₃

C₄⋊C₄

yes

—

C₄×D₅₃

C₂×D₄

yes

—

C₄×D₅₃

C₂×Q₈

yes

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₂×C₄⋊C₄

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₄×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

C₂²×Q₈

yes

characteristic

Dic₅₃

—

Dic₅₃

C₂×C₄⋊C₄

yes

—

#2905

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₅₃

#568

yes

—

D₅₃

—

C₂²×M₁₆

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#2905

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

106

—

C₂×M₁₆

C₂×D₅₃

yes

—

C₂²×C₈

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

100

C₄⋊C₈

C₂×D₅₃

yes

complemented

101

C₂⁴

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₄×D₅₃

yes

—

109

C₂²×C₄

C₂²×D₅₃

yes

characteristic

110

M₁₆

—

111

M₁₆

—

106

—

112

M₁₆

—

106

—

113

C₂×C₈

—

114

C₂×C₈

—

106

—

115

C₂×C₈

C₂²×D₅₃

yes

—

116

C₄⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₄×D₅₃

yes

—

121

C₄²

C₂²×D₅₃

yes

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

212

—

125

C₂³

C₂×C₄×D₅₃

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

212

—

134

C₂×C₄

C₂×C₄×D₅₃

yes

—

135

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

136

C₂×C₄

D₄×D₅₃

yes

—

137

C₂×C₄

Q₈×D₅₃

yes

—

138

C₂×C₄

C₂³×D₅₃

yes

characteristic

139

C₈

—

140

C₈

—

106

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

106

—

144

C₂²

—

212

—

145

C₂²

C₄⋊C₄×D₅₃

yes

—

146

C₂²

C₂²×C₄×D₅₃

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

106

—

150

C₄

C₄⋊C₄×D₅₃

yes

—

151

C₄

C₂×D₄×D₅₃

yes

characteristic

152

C₄

C₂×Q₈×D₅₃

yes

characteristic

153

C₂

—

154

C₂

—

106

—

155

C₂

C₂×C₄⋊C₄×D₅₃

yes

characteristic

156

C₂

#16811389

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#423	C₂
hypercenter	#423	D₅₃
hypocenter	C₅₃	#2905
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₃

commutator, socle

C₂²×C₅₃

C₂³×C₄

Fitting

C₅₃×#423

C₂

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
3. #16810648	C₂	semidirect
4. #16810647	C₂	semidirect
5. #16810646	C₂	semidirect
6. C₅₃×#423	C₂	semidirect
17. C₄⋊#88322	C₂²	semidirect
18. #13133146	C₂²	semidirect
21. C₄⋊C₈×C₅₃	C₂²	semidirect
84. #423	D₅₃	semidirect
85. C₅₃	#2905	semidirect
100. C₄⋊C₈	C₂×D₅₃	semidirect

H ≅ G/N

type

3. #16810648

C₂

semidirect

4. #16810647

C₂

semidirect

5. #16810646

C₂

semidirect

6. C₅₃×#423

C₂

semidirect

17. C₄⋊#88322

C₂²

semidirect

18. #13133146

C₂²

semidirect

21. C₄⋊C₈×C₅₃

C₂²

semidirect

84. #423

D₅₃

semidirect

85. C₅₃

#2905

semidirect

100. C₄⋊C₈

C₂×D₅₃

semidirect

Subgroups — #20283277

Conjugacy classes of subgroups

Complements