Group #20274103 — subgroups

Subgroups — #20274103

Order 6784.

172 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16811026	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810829	C₂	1	1	yes	maximal, characteristic
3	#16810825	C₂	1	1	yes	maximal, characteristic
4	#16810688	C₂	1	1	yes	maximal, characteristic
5	#16810375	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810368	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719913	—	2	1	no	—
9	C₂×#11719924	—	2	1	no	—
10	C₂×C₈×D₅₃	—	2	1	no	—
11	#13133217	C₂²	1	1	yes	characteristic
12	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
13	#13133150	—	2	1	no	—
14	#13133150	—	2	1	no	—
15	C₄⋊#88322	C₂²	1	1	yes	characteristic
16	#13133142	—	2	1	no	—
17	#13133141	—	2	1	no	—
18	#13133129	—	2	1	no	—
19	C₄×#88322	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₃	—	2	1	no	—
24	C₂×D₂₁₂	D₄	1	1	yes	characteristic
25	C₂×C₄×D₅₃	—	2	1	no	—
26	C₂×Dic₁₀₆	—	2	1	no	—
27	C₂×C₈×C₅₃	—	2	1	no	—
28	C₂×C₈×C₅₃	—	2	1	no	—
29	C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	#11719924	—	4	2	no	—
33	#11719921	—	2	1	no	—
34	C₄⋊Dic₅₃	—	2	1	no	—
35	C₄⋊Dic₅₃	—	2	1	no	—
36	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
37	C₂×#88322	—	2	1	no	—
38	C₂×#88322	D₄	1	1	yes	characteristic
39	C₂×#88322	D₄	1	1	yes	characteristic
40	#11719913	—	4	2	no	—
41	C₈×D₅₃	—	4	2	no	—
42	C₂²×D₅₃	—	2	1	no	—
43	Q₈×C₅₃	—	4	1	no	—
44	Q₈×C₅₃	—	4	1	no	—
45	C₂×C₄×C₅₃	—	4	1	no	—
46	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
49	C₂×Dic₅₃	—	2	1	no	—
50	C₂×Dic₅₃	—	4	1	no	—
51	D₂₁₂	—	2	1	no	—
52	D₂₁₂	D₈	1	2	yes	—
53	C₄×D₅₃	—	4	1	no	—
54	C₄×D₅₃	—	4	1	no	—
55	Dic₁₀₆	—	4	1	no	—
56	Dic₁₀₆	—	4	1	no	—
57	C₈×C₅₃	—	4	1	no	—
58	C₈×C₅₃	—	4	1	no	—
59	#88322	—	2	1	no	—
60	#88322	—	4	1	no	—
61	#88322	SD₁₆	1	2	yes	complemented
62	C₂²×C₅₃	#121	1	1	yes	characteristic
63	C₂×D₅₃	—	2	2	no	—
64	C₂×D₅₃	—	4	1	no	—
65	C₂×D₅₃	—	4	1	no	—
66	C₄×C₅₃	—	2	1	no	—
67	C₄×C₅₃	—	8	1	no	—
68	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
69	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
70	C₄×C₅₃	#135	1	1	yes	characteristic
71	C₄×C₅₃	#137	1	1	yes	characteristic
72	Dic₅₃	—	4	1	no	—
73	Dic₅₃	—	8	1	no	—
74	#1673	—	53	1	no	maximal, Sylow
75	C₂×C₅₃	#461	1	1	yes	characteristic
76	C₂×C₅₃	#466	1	1	yes	characteristic
77	C₂×C₅₃	#473	1	1	yes	characteristic
78	D₅₃	—	4	2	no	—
79	#500	—	53	1	no	—
80	#461	—	53	1	no	—
81	#461	—	53	1	no	—
82	C₈×D₄	—	53	1	no	—
83	#335	—	53	1	no	—
84	#332	—	53	1	no	—
85	#332	D₅₃	1	1	yes	characteristic, complemented
86	C₅₃	#1673	1	1	yes	characteristic, complemented, Sylow
87	C₂×SD₁₆	—	106	1	no	—
88	C₂×SD₁₆	—	106	1	no	—
89	C₂²×C₈	—	106	1	no	—
90	C₄⋊Q₈	—	53	1	no	—
91	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
92	C₄×D₄	—	53	1	no	—
93	#107	—	106	1	no	—
94	#107	—	106	1	no	—
95	C₄⋊C₈	—	53	1	no	—
96	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
97	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
98	#103	—	106	1	no	—
99	#103	—	106	1	no	—
100	C₂²⋊C₈	—	106	1	no	—
101	C₄×C₈	—	53	1	no	—
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	106	1	no	—
104	C₂×D₄	—	53	1	no	—
105	C₂²×C₄	—	106	1	no	—
106	SD₁₆	—	212	2	no	—
107	SD₁₆	—	212	2	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	53	1	no	—
111	C₂×C₈	—	53	1	no	—
112	C₂×C₈	—	106	1	no	—
113	C₂×C₈	—	212	2	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	53	1	no	—
117	C₄⋊C₄	—	106	1	no	—
118	C₄⋊C₄	—	106	1	no	—
119	C₂²⋊C₄	—	106	1	no	—
120	C₄²	C₂²×D₅₃	1	1	yes	characteristic
121	C₂³	—	106	1	no	—
122	Q₈	—	4	1	no	—
123	Q₈	—	4	1	no	—
124	Q₈	—	212	1	no	—
125	Q₈	—	212	1	no	—
126	D₄	—	53	2	no	—
127	D₄	—	106	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	106	1	no	—
130	C₂×C₄	—	212	1	no	—
131	C₂×C₄	—	212	1	no	—
132	C₂×C₄	—	212	1	no	—
133	C₂×C₄	#11719945	1	1	yes	characteristic
134	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
135	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
136	C₈	—	4	1	no	—
137	C₈	—	4	1	no	—
138	C₈	—	53	2	no	—
139	C₈	—	106	1	no	—
140	C₈	—	212	1	no	—
141	C₂²	—	106	2	no	—
142	C₂²	—	212	1	no	—
143	C₂²	—	212	1	no	—
144	C₂²	#13133190	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	8	1	no	—
147	C₄	—	424	1	no	—
148	C₄	—	212	1	no	—
149	C₄	D₈×D₅₃	1	1	yes	characteristic
150	C₄	SD₁₆×D₅₃	1	1	yes	characteristic
151	C₄	#13133238	1	1	yes	characteristic
152	C₄	#13133249	1	1	yes	characteristic
153	C₂	—	212	2	no	—
154	C₂	#16810827	1	1	yes	characteristic
155	C₂	#16810858	1	1	yes	characteristic
156	C₂	#16810897	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811026

C₂

yes

maximal, characteristic, complemented

#16810829

C₂

yes

maximal, characteristic

#16810825

C₂

yes

maximal, characteristic

#16810688

C₂

yes

maximal, characteristic

#16810375

C₂

yes

maximal, characteristic, complemented

#16810368

C₂

yes

maximal, characteristic, complemented

C₅₃×#332

C₂

yes

maximal, characteristic, complemented

C₂×#11719913

—

C₂×#11719924

—

C₂×C₈×D₅₃

—

#13133217

C₂²

yes

characteristic

C₄×D₂₁₂

C₂²

yes

—

—

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×D₂₁₂

D₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719924

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

characteristic

C₂×#88322

—

C₂×#88322

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

#11719913

—

C₈×D₅₃

—

C₂²×D₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

D₈

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

#88322

SD₁₆

yes

complemented

C₂²×C₅₃

#121

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

C₂×SD₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#500

—

#461

—

#461

—

C₈×D₄

—

#335

—

#332

—

#332

D₅₃

yes

characteristic, complemented

C₅₃

#1673

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

106

—

C₂×SD₁₆

—

106

—

C₂²×C₈

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

C₄×D₄

—

#107

—

106

—

#107

—

106

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₃

yes

characteristic

C₄⋊C₈

C₂×D₅₃

yes

characteristic

#103

—

106

—

#103

—

106

—

100

C₂²⋊C₈

—

106

—

101

C₄×C₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

106

—

104

C₂×D₄

—

105

C₂²×C₄

—

106

—

106

SD₁₆

—

212

—

107

SD₁₆

—

212

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

106

—

113

C₂×C₈

—

212

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

106

—

118

C₄⋊C₄

—

106

—

119

C₂²⋊C₄

—

106

—

120

C₄²

C₂²×D₅₃

yes

characteristic

121

C₂³

—

106

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

212

—

125

Q₈

—

212

—

126

D₄

—

127

D₄

—

106

—

128

C₂×C₄

—

129

C₂×C₄

—

106

—

130

C₂×C₄

—

212

—

131

C₂×C₄

—

212

—

132

C₂×C₄

—

212

—

133

C₂×C₄

#11719945

yes

characteristic

134

C₂×C₄

D₄×D₅₃

yes

characteristic

135

C₂×C₄

D₄×D₅₃

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

106

—

140

C₈

—

212

—

141

C₂²

—

106

—

142

C₂²

—

212

—

143

C₂²

—

212

—

144

C₂²

#13133190

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

424

—

148

C₄

—

212

—

149

C₄

D₈×D₅₃

yes

characteristic

150

C₄

SD₁₆×D₅₃

yes

characteristic

151

C₄

#13133238

yes

characteristic

152

C₄

#13133249

yes

characteristic

153

C₂

—

212

—

154

C₂

#16810827

yes

characteristic

155

C₂

#16810858

yes

characteristic

156

C₂

#16810897

yes

characteristic

	subgroup	G/H
center	C₂²	#13133190
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#332	C₂
socle	C₂²×C₅₃	#121
hypercenter	#332	D₅₃
hypocenter	C₅₃	#1673
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#13133190

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#332

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
1. #16811026	C₂	semidirect
5. #16810375	C₂	semidirect
6. #16810368	C₂	semidirect
7. C₅₃×#332	C₂	semidirect
61. #88322	SD₁₆	semidirect
85. #332	D₅₃	semidirect
86. C₅₃	#1673	semidirect

H ≅ G/N

type

1. #16811026

C₂

semidirect

5. #16810375

C₂

semidirect

6. #16810368

C₂

semidirect

7. C₅₃×#332

C₂

semidirect

61. #88322

SD₁₆

semidirect

85. #332

D₅₃

semidirect

86. C₅₃

#1673

semidirect

Subgroups — #20274103

Conjugacy classes of subgroups

Complements