Group #20275677 — subgroups

Subgroups — #20275677

Order 6784.

300 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 145 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×C₄×#88322	C₂	1	1	yes	maximal, characteristic
2	#16810493	C₂	1	1	yes	maximal, characteristic
3	#16810385	C₂	1	4	yes	maximal
4	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#88322	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₅₃	—	2	1	no	—
7	C₂×C₄⋊Dic₅₃	—	2	1	no	—
8	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
9	C₄×#88322	—	2	2	no	—
10	C₂²×Q₈×C₅₃	—	2	1	no	—
11	C₄⋊Q₈×C₅₃	—	2	2	no	—
12	C₄⋊Q₈×C₅₃	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
14	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₅₃	—	2	4	no	—
17	C₂×Q₈×C₅₃	—	4	2	no	—
18	C₂×Q₈×C₅₃	—	4	2	no	—
19	C₂²×C₄×C₅₃	—	2	2	no	—
20	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂²×Dic₅₃	—	2	2	no	—
24	C₄⋊C₄×C₅₃	—	2	4	no	—
25	C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
27	C₄²×C₅₃	C₂×C₄	1	2	yes	—
28	C₄²×C₅₃	D₄	1	2	yes	—
29	C₄⋊Dic₅₃	—	2	4	no	—
30	C₄⋊Dic₅₃	—	4	2	no	—
31	C₄⋊Dic₅₃	—	4	2	no	—
32	C₂×#88322	—	2	2	no	—
33	C₂×#88322	—	2	2	no	—
34	C₂×#88322	Q₈	1	4	yes	complemented
35	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
36	Q₈×C₅₃	—	4	8	no	—
37	C₂×C₄×C₅₃	—	2	4	no	—
38	C₂×C₄×C₅₃	—	4	2	no	—
39	C₂×C₄×C₅₃	—	4	2	no	—
40	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₅₃	D₈	1	2	yes	—
43	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
44	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
45	C₂×C₄×C₅₃	Q₁₆	1	2	yes	—
46	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₅₃	#41	1	2	yes	—
50	C₂×Dic₅₃	—	4	2	no	—
51	C₂×Dic₅₃	—	4	2	no	—
52	C₂×Dic₅₃	—	4	2	no	—
53	#88322	—	2	4	no	—
54	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₅₃	#124	1	1	yes	characteristic
56	C₂²×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
57	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
58	C₂²×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
59	C₂²×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
60	C₂²×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
61	C₄×C₅₃	—	4	4	no	—
62	C₄×C₅₃	#102	1	4	yes	—
63	C₄×C₅₃	#103	1	4	yes	—
64	C₄×C₅₃	C₄×Q₈	1	2	yes	—
65	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
66	Dic₅₃	—	8	2	no	—
67	#1990	—	53	1	no	maximal, Sylow
68	C₂×C₅₃	#391	1	1	yes	characteristic
69	C₂×C₅₃	C₂×#102	1	1	yes	characteristic
70	C₂×C₅₃	C₂×#103	1	1	yes	characteristic
71	C₂×C₅₃	#486	1	1	yes	characteristic
72	C₂×C₅₃	#487	1	1	yes	characteristic
73	C₂×C₅₃	#498	1	1	yes	characteristic
74	C₂×C₅₃	#500	1	1	yes	characteristic
75	C₂×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
76	C₂×C₄×C₈	—	53	1	no	—
77	#384	—	53	1	no	—
78	#340	—	53	4	no	—
79	C₅₃	#1990	1	1	yes	characteristic, complemented, Sylow
80	C₂²×Q₈	—	2	1	no	—
81	C₂²×C₈	—	53	2	no	—
82	C₄⋊Q₈	—	2	2	no	—
83	C₄⋊Q₈	Dic₅₃	1	4	yes	complemented
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	53	2	no	—
86	C₂×C₄⋊C₄	—	106	1	no	—
87	C₂×C₄⋊C₄	—	106	1	no	—
88	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
89	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
90	C₄×C₈	—	106	2	no	—
91	C₂×Q₈	—	2	4	no	—
92	C₂×Q₈	—	4	2	no	—
93	C₂×Q₈	—	4	2	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	106	2	no	—
96	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
99	C₂×C₈	—	53	4	no	—
100	C₂×C₈	—	106	2	no	—
101	C₂×C₈	—	106	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	2	4	no	—
104	C₄⋊C₄	—	106	4	no	—
105	C₄⋊C₄	—	212	2	no	—
106	C₄⋊C₄	—	212	2	no	—
107	C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
108	C₄²	C₂×Dic₅₃	1	2	yes	—
109	C₄²	C₂²⋊D₅₃	1	2	yes	—
110	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
111	Q₈	—	4	8	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	212	2	no	—
116	C₂×C₄	—	212	2	no	—
117	C₂×C₄	—	212	2	no	—
118	C₂×C₄	#11719922	1	2	yes	—
119	C₂×C₄	#11719923	1	2	yes	—
120	C₂×C₄	#11719924	1	2	yes	—
121	C₂×C₄	#11719925	1	2	yes	—
122	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
123	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
124	C₂×C₄	Q₈×D₅₃	1	2	yes	—
125	C₂×C₄	#11719949	1	2	yes	—
126	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
127	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
128	C₈	—	106	4	no	—
129	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
130	C₂²	#13133206	1	1	yes	characteristic
131	C₂²	#13133215	1	1	yes	characteristic
132	C₂²	C₂×#11719922	1	1	yes	characteristic
133	C₂²	C₂×#11719924	1	1	yes	characteristic
134	C₂²	C₂×#11719923	1	1	yes	characteristic
135	C₂²	C₂×#11719925	1	1	yes	characteristic
136	C₄	—	4	4	no	—
137	C₄	—	424	2	no	—
138	C₄	#13133138	1	4	yes	—
139	C₄	#13133141	1	4	yes	—
140	C₄	Q₈×Dic₅₃	1	2	yes	—
141	C₄	#13133193	1	2	yes	—
142	C₂	#16810530	1	1	yes	characteristic
143	C₂	C₂×#13133138	1	1	yes	characteristic
144	C₂	C₂×#13133141	1	1	yes	characteristic
145	C₂	#16810971	1	1	yes	characteristic
146	C₂	#16810975	1	1	yes	characteristic
147	C₂	#16811019	1	1	yes	characteristic
148	C₂	#16811026	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×#88322

C₂

yes

maximal, characteristic

#16810493

C₂

yes

maximal, characteristic

#16810385

C₂

yes

maximal

C₂×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₄×#88322

—

C₂²×Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

Q₈

yes

complemented

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

D₈

yes

—

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

Q₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

#88322

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

#124

yes

characteristic

C₂²×C₅₃

C₄⋊Q₈

yes

characteristic

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₂²×C₅₃

C₂×SD₁₆

yes

characteristic

C₂²×C₅₃

C₂×SD₁₆

yes

characteristic

C₂²×C₅₃

C₂×Q₁₆

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#102

yes

—

C₄×C₅₃

#103

yes

—

C₄×C₅₃

C₄×Q₈

yes

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊Q₈

D₅₃

yes

characteristic

C₂×C₄×C₈

—

#384

—

#340

—

C₅₃

#1990

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₅₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄×C₈

—

106

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

106

—

101

C₂×C₈

—

106

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

106

—

105

C₄⋊C₄

—

212

—

106

C₄⋊C₄

—

212

—

107

C₄⋊C₄

C₂×Dic₅₃

yes

—

108

C₄²

C₂×Dic₅₃

yes

—

109

C₄²

C₂²⋊D₅₃

yes

—

110

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

111

Q₈

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

212

—

116

C₂×C₄

—

212

—

117

C₂×C₄

—

212

—

118

C₂×C₄

#11719922

yes

—

119

C₂×C₄

#11719923

yes

—

120

C₂×C₄

#11719924

yes

—

121

C₂×C₄

#11719925

yes

—

122

C₂×C₄

C₂²⋊Dic₅₃

yes

—

123

C₂×C₄

C₂²⋊Dic₅₃

yes

—

124

C₂×C₄

Q₈×D₅₃

yes

—

125

C₂×C₄

#11719949

yes

—

126

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

127

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

128

C₈

—

106

—

129

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

130

C₂²

#13133206

yes

characteristic

131

C₂²

#13133215

yes

characteristic

132

C₂²

C₂×#11719922

yes

characteristic

133

C₂²

C₂×#11719924

yes

characteristic

134

C₂²

C₂×#11719923

yes

characteristic

135

C₂²

C₂×#11719925

yes

characteristic

136

C₄

—

137

C₄

—

424

—

138

C₄

#13133138

yes

—

139

C₄

#13133141

yes

—

140

C₄

Q₈×Dic₅₃

yes

—

141

C₄

#13133193

yes

—

142

C₂

#16810530

yes

characteristic

143

C₂

C₂×#13133138

yes

characteristic

144

C₂

C₂×#13133141

yes

characteristic

145

C₂

#16810971

yes

characteristic

146

C₂

#16810975

yes

characteristic

147

C₂

#16811019

yes

characteristic

148

C₂

#16811026

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×C₄⋊Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#1990
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×C₄⋊Q₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₅₃

hypocenter

C₅₃

#1990

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₅₃	C₄	semidirect
34. C₂×#88322	Q₈	semidirect
79. C₅₃	#1990	semidirect
83. C₄⋊Q₈	Dic₅₃	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₅₃

C₄

semidirect

34. C₂×#88322

Q₈

semidirect

79. C₅₃

#1990

semidirect

83. C₄⋊Q₈

Dic₅₃

semidirect

Subgroups — #20275677

Conjugacy classes of subgroups

Complements