Group #20283313 — subgroups

Subgroups — #20283313

Order 6784.

476 conjugacy classes of subgroups; only the 253 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 253 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×#13133224	C₂	1	1	yes	maximal, characteristic
2	#16811106	C₂	1	1	yes	maximal, characteristic
3	#16810666	C₂	1	3	yes	maximal, complemented
4	#16810663	C₂	1	3	yes	maximal, complemented
5	#16810659	C₂	1	3	yes	maximal, complemented
6	C₂×#13133150	C₂	1	3	yes	maximal, complemented
7	C₅₃×#520	C₂	1	1	yes	maximal, characteristic
8	#13133224	C₄	1	8	yes	complemented
9	C₂×#11719917	C₂²	1	3	yes	—
10	C₂²×#88322	C₂²	1	3	yes	—
11	Q₈×Dic₅₃	—	2	1	no	—
12	D₄×Dic₅₃	—	2	3	no	—
13	#13133176	C₂²	1	3	yes	—
14	C₂×C₄⋊Dic₅₃	C₂²	1	3	yes	—
15	#13133150	C₂²	1	3	yes	—
16	#13133150	C₂²	1	6	yes	—
17	#13133150	C₂²	1	1	yes	characteristic
18	#13133148	C₂²	1	3	yes	—
19	#13133148	C₂²	1	3	yes	—
20	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₅₃	—	2	1	no	—
22	C₄×D₄×C₅₃	—	2	3	no	—
23	#117×C₅₃	C₂²	1	3	yes	—
24	C₂×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
25	#41×C₅₃	C₂×C₄	1	4	yes	—
26	#41×C₅₃	Q₈	1	4	yes	—
27	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
28	C₂×D₄×C₅₃	D₄	1	3	yes	—
29	C₂²×C₄×C₅₃	—	2	3	no	—
30	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
31	C₂²×Dic₅₃	—	2	3	no	—
32	C₄⋊C₄×C₅₃	—	2	3	no	—
33	C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
34	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
35	C₂²⋊C₄×C₅₃	—	2	3	no	—
36	C₄²×C₅₃	—	2	3	no	—
37	C₂²⋊Dic₅₃	—	2	3	no	—
38	C₄⋊Dic₅₃	—	2	3	no	—
39	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
40	C₄⋊Dic₅₃	C₂³	1	3	yes	—
41	C₄×Dic₅₃	—	2	3	no	—
42	#11719917	C₂×C₄	1	12	yes	complemented
43	C₂×#88322	C₂×C₄	1	12	yes	complemented
44	C₂×#88322	C₂³	1	3	yes	—
45	C₂×#88322	C₂³	1	1	yes	characteristic
46	C₂³×C₅₃	C₂×D₄	1	3	yes	—
47	Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
48	D₄×C₅₃	C₄⋊C₄	1	12	yes	—
49	C₂×C₄×C₅₃	—	2	3	no	—
50	C₂×C₄×C₅₃	—	2	1	no	—
51	C₂×C₄×C₅₃	—	8	3	no	—
52	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
53	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
54	C₂×C₄×C₅₃	C₂×Q₈	1	6	yes	—
55	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₅₃	—	2	3	no	—
57	C₂×Dic₅₃	—	2	1	no	—
58	C₂×Dic₅₃	—	8	3	no	—
59	#88322	C₂²×C₄	1	2	yes	—
60	#88322	C₂²×C₄	1	6	yes	—
61	C₂²×C₅₃	—	2	3	no	—
62	C₂²×C₅₃	—	2	3	no	—
63	C₂²×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
64	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₅₃	—	4	3	no	—
66	C₄×C₅₃	—	4	1	no	—
67	C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
68	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
70	Dic₅₃	—	4	3	no	—
71	Dic₅₃	—	4	1	no	—
72	#2913	—	53	1	no	maximal, Sylow
73	C₂×C₅₃	—	2	6	no	—
74	C₂×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₅₃	#576	1	1	yes	characteristic
76	C₂×C₅₃	#578	1	1	yes	characteristic
77	C₂×#131	—	53	1	no	—
78	#520	—	53	1	no	—
79	#520	D₅₃	1	1	yes	characteristic
80	#428	—	53	3	no	—
81	#428	—	53	3	no	—
82	#427	—	53	3	no	—
83	C₂×#107	—	53	3	no	—
84	C₅₃	#2913	1	1	yes	characteristic, complemented, Sylow
85	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
86	#131	—	53	8	no	—
87	C₂×M₁₆	—	53	3	no	—
88	C₂²×C₈	—	53	3	no	—
89	C₄×Q₈	—	2	1	no	—
90	C₄×Q₈	—	106	1	no	—
91	C₄×D₄	—	2	3	no	—
92	C₄×D₄	—	106	3	no	—
93	#117	—	53	3	no	—
94	#117	C₂×D₅₃	1	3	yes	—
95	C₂×C₄⋊C₄	—	53	3	no	—
96	C₂×C₄⋊C₄	C₂×D₅₃	1	3	yes	—
97	#107	—	53	3	no	—
98	#107	—	53	6	no	—
99	#107	—	53	1	no	—
100	#106	—	53	3	no	—
101	#106	—	53	3	no	—
102	#41	Dic₁₀₆	1	4	yes	—
103	#41	C₄×D₅₃	1	4	yes	—
104	C₂×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
105	C₂×D₄	C₂²⋊D₅₃	1	3	yes	—
106	C₂²×C₄	—	2	3	no	—
107	C₂²×C₄	—	106	3	no	—
108	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
109	M₁₆	—	53	12	no	—
110	C₂×C₈	—	53	12	no	—
111	C₂×C₈	—	53	3	no	—
112	C₂×C₈	—	53	1	no	—
113	C₄⋊C₄	—	2	3	no	—
114	C₄⋊C₄	—	53	1	no	—
115	C₄⋊C₄	—	53	3	no	—
116	C₄⋊C₄	—	106	3	no	—
117	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
118	C₄⋊C₄	C₂²×D₅₃	1	3	yes	—
119	C₂²⋊C₄	—	2	3	no	—
120	C₂²⋊C₄	—	106	3	no	—
121	C₄²	—	2	3	no	—
122	C₄²	—	106	3	no	—
123	C₂³	C₂×C₂²⋊D₅₃	1	3	yes	—
124	Q₈	#11719919	1	4	yes	—
125	D₄	#11719919	1	12	yes	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	3	no	—
128	C₂×C₄	—	8	3	no	—
129	C₂×C₄	—	106	1	no	—
130	C₂×C₄	—	106	3	no	—
131	C₂×C₄	—	424	3	no	—
132	C₂×C₄	C₂×Dic₁₀₆	1	6	yes	—
133	C₂×C₄	C₂×C₄×D₅₃	1	6	yes	—
134	C₂×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
135	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
136	C₈	—	53	2	no	—
137	C₈	—	53	6	no	—
138	C₂²	—	2	3	no	—
139	C₂²	—	2	3	no	—
140	C₂²	C₂×#11719919	1	6	yes	—
141	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
142	C₄	—	4	3	no	—
143	C₄	—	4	1	no	—
144	C₄	—	212	3	no	—
145	C₄	—	212	1	no	—
146	C₄	C₂×#11719919	1	6	yes	—
147	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
148	C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
149	C₂	—	2	6	no	—
150	C₂	C₂²×#11719919	1	1	yes	characteristic
151	C₂	#16811426	1	1	yes	characteristic
152	C₂	#16811438	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133224

C₂

yes

maximal, characteristic

#16811106

C₂

yes

maximal, characteristic

#16810666

C₂

yes

maximal, complemented

#16810663

C₂

yes

maximal, complemented

#16810659

C₂

yes

maximal, complemented

C₂×#13133150

C₂

yes

maximal, complemented

C₅₃×#520

C₂

yes

maximal, characteristic

#13133224

C₄

yes

complemented

C₂×#11719917

C₂²

yes

—

C₂²×#88322

C₂²

yes

—

Q₈×Dic₅₃

—

D₄×Dic₅₃

—

#13133176

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

#13133150

C₂²

yes

—

#13133150

C₂²

yes

—

#13133150

C₂²

yes

characteristic

#13133148

C₂²

yes

—

#13133148

C₂²

yes

—

C₂×#41×C₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

—

C₄×D₄×C₅₃

—

#117×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

#41×C₅₃

C₂×C₄

yes

—

#41×C₅₃

Q₈

yes

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×D₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂³

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

C₂³

yes

—

C₄×Dic₅₃

—

#11719917

C₂×C₄

yes

complemented

C₂×#88322

C₂×C₄

yes

complemented

C₂×#88322

C₂³

yes

—

C₂×#88322

C₂³

yes

characteristic

C₂³×C₅₃

C₂×D₄

yes

—

Q₈×C₅₃

C₄⋊C₄

yes

—

D₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

#88322

C₂²×C₄

yes

—

#88322

C₂²×C₄

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₄×C₅₃

C₂³×C₄

yes

characteristic

C₄×C₅₃

C₂²×Q₈

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#2913

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂²×C₄⋊C₄

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

D₅₃

yes

—

—

—

—

C₅₃

#2913

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₃

yes

characteristic

#131

—

C₂×M₁₆

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

106

—

C₄×D₄

—

C₄×D₄

—

106

—

#117

—

#117

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

#107

—

#107

—

#107

—

100

#106

—

101

#106

—

102

#41

Dic₁₀₆

yes

—

103

#41

C₄×D₅₃

yes

—

104

C₂×Q₈

C₂²⋊D₅₃

yes

characteristic

105

C₂×D₄

C₂²⋊D₅₃

yes

—

106

C₂²×C₄

—

107

C₂²×C₄

—

106

—

108

C₂²×C₄

C₂²×D₅₃

yes

—

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

106

—

117

C₄⋊C₄

C₂²×D₅₃

yes

characteristic

118

C₄⋊C₄

C₂²×D₅₃

yes

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

106

—

121

C₄²

—

122

C₄²

—

106

—

123

C₂³

C₂×C₂²⋊D₅₃

yes

—

124

Q₈

#11719919

yes

—

125

D₄

#11719919

yes

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

106

—

130

C₂×C₄

—

106

—

131

C₂×C₄

—

424

—

132

C₂×C₄

C₂×Dic₁₀₆

yes

—

133

C₂×C₄

C₂×C₄×D₅₃

yes

—

134

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

135

C₂×C₄

C₂³×D₅₃

yes

characteristic

136

C₈

—

137

C₈

—

138

C₂²

—

139

C₂²

—

140

C₂²

C₂×#11719919

yes

—

141

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

212

—

145

C₄

—

212

—

146

C₄

C₂×#11719919

yes

—

147

C₄

C₂²×C₄×D₅₃

yes

characteristic

148

C₄

C₂²×Dic₁₀₆

yes

characteristic

149

C₂

—

150

C₂

C₂²×#11719919

yes

characteristic

151

C₂

#16811426

yes

characteristic

152

C₂

#16811438

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#520	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#520	D₅₃
hypocenter	C₅₃	#2913
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ #520

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₅₃

commutator

C₄×C₅₃

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#520

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×#41 ◁ #520

N	H ≅ G/N	type
3. #16810666	C₂	semidirect
4. #16810663	C₂	semidirect
5. #16810659	C₂	semidirect
6. C₂×#13133150	C₂	semidirect
8. #13133224	C₄	semidirect
42. #11719917	C₂×C₄	semidirect
43. C₂×#88322	C₂×C₄	semidirect
84. C₅₃	#2913	semidirect

H ≅ G/N

type

3. #16810666

C₂

semidirect

4. #16810663

C₂

semidirect

5. #16810659

C₂

semidirect

6. C₂×#13133150

C₂

semidirect

8. #13133224

C₄

semidirect

42. #11719917

C₂×C₄

semidirect

43. C₂×#88322

C₂×C₄

semidirect

84. C₅₃

#2913

semidirect

Subgroups — #20283313

Conjugacy classes of subgroups

Complements