Group #20275639 — subgroups

Subgroups — #20275639

Order 6784.

324 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 129 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	C₂×#13133141	C₂	1	4	yes	maximal
3	C₄×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#88322	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
7	C₂×C₄×Dic₅₃	—	2	1	no	—
8	C₄⋊#88322	—	2	2	no	—
9	#13133141	—	2	8	no	—
10	#13133124	—	2	1	no	—
11	C₂²×Q₈×C₅₃	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₅₃	—	2	2	no	—
13	C₄⋊Q₈×C₅₃	C₄	1	4	yes	complemented
14	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
15	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
16	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₅₃	—	2	2	no	—
18	C₂×Q₈×C₅₃	—	2	4	no	—
19	C₂×Q₈×C₅₃	—	2	4	no	—
20	C₂×Q₈×C₅₃	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₅₃	—	2	2	no	—
22	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₅₃	—	2	2	no	—
26	C₄⋊C₄×C₅₃	—	2	4	no	—
27	C₄⋊C₄×C₅₃	—	2	4	no	—
28	C₄²×C₅₃	C₂×C₄	1	2	yes	—
29	C₄²×C₅₃	D₄	1	2	yes	—
30	C₄⋊Dic₅₃	—	2	2	no	—
31	C₄⋊Dic₅₃	D₄	1	4	yes	—
32	C₄×Dic₅₃	—	4	2	no	—
33	C₂×#88322	—	2	4	no	—
34	C₂×#88322	—	2	2	no	—
35	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
36	Q₈×C₅₃	—	2	8	no	—
37	Q₈×C₅₃	—	4	2	no	—
38	Q₈×C₅₃	—	4	2	no	—
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	—	2	4	no	—
41	C₂×C₄×C₅₃	—	4	4	no	—
42	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₅₃	C₂²⋊C₄	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	2	yes	—
48	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	#41	1	2	yes	—
50	C₂×Dic₅₃	—	2	4	no	—
51	C₂×Dic₅₃	—	4	2	no	—
52	C₂×Dic₅₃	—	4	2	no	—
53	#88322	—	4	2	no	—
54	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₅₃	#121	1	2	yes	—
56	C₂²×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
57	C₂²×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
58	C₂²×C₅₃	#137	1	2	yes	—
59	C₄×C₅₃	—	2	2	no	—
60	C₄×C₅₃	—	4	4	no	—
61	C₄×C₅₃	#103	1	4	yes	complemented
62	C₄×C₅₃	C₄×D₄	1	2	yes	—
63	C₄×C₅₃	#124	1	1	yes	characteristic
64	C₄×C₅₃	#127	1	1	yes	characteristic
65	Dic₅₃	—	4	4	no	—
66	#1981	—	53	1	no	maximal, Sylow
67	C₂×C₅₃	#390	1	1	yes	characteristic
68	C₂×C₅₃	C₂×#103	1	1	yes	characteristic
69	C₂×C₅₃	#419	1	1	yes	characteristic
70	C₂×C₅₃	#461	1	2	yes	—
71	C₂×C₅₃	#462	1	2	yes	—
72	C₂×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
73	C₂×C₄⋊C₈	—	53	1	no	—
74	C₂×#103	—	53	4	no	—
75	C₄×C₄⋊C₄	—	53	1	no	—
76	C₅₃	#1981	1	1	yes	characteristic, complemented, Sylow
77	C₂²×Q₈	C₂×D₅₃	1	2	yes	—
78	C₂²×C₈	—	53	2	no	—
79	C₄⋊Q₈	—	2	2	no	—
80	C₄⋊Q₈	Dic₅₃	1	4	yes	complemented
81	C₂×C₄⋊C₄	—	2	1	no	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	53	2	no	—
84	C₂×C₄²	—	106	1	no	—
85	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
86	C₄⋊C₈	—	106	2	no	—
87	#103	—	106	8	no	—
88	#95	—	106	1	no	—
89	C₂×Q₈	—	2	2	no	—
90	C₂×Q₈	—	2	4	no	—
91	C₂×Q₈	—	2	4	no	—
92	C₂×Q₈	C₂×Dic₅₃	1	4	yes	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	106	2	no	—
95	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
98	C₂×C₈	—	106	4	no	—
99	C₂×C₈	—	106	2	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	2	4	no	—
102	C₄⋊C₄	—	53	4	no	—
103	C₄⋊C₄	—	106	2	no	—
104	C₄²	—	212	2	no	—
105	C₄²	C₂×Dic₅₃	1	2	yes	—
106	C₄²	C₂²⋊D₅₃	1	2	yes	—
107	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
108	Q₈	—	2	8	no	—
109	Q₈	—	4	2	no	—
110	Q₈	—	4	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	4	no	—
114	C₂×C₄	—	106	4	no	—
115	C₂×C₄	—	212	2	no	—
116	C₂×C₄	—	212	2	no	—
117	C₂×C₄	#11719924	1	2	yes	—
118	C₂×C₄	#11719925	1	2	yes	—
119	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
120	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
121	C₂×C₄	D₄×D₅₃	1	2	yes	—
122	C₂×C₄	#11719947	1	2	yes	—
123	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
124	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
125	C₈	—	212	2	no	—
126	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
127	C₂²	#13133188	1	2	yes	—
128	C₂²	C₂×#11719924	1	1	yes	characteristic
129	C₂²	C₂×#11719925	1	1	yes	characteristic
130	C₂²	#13133248	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	—	4	4	no	—
133	C₄	—	212	4	no	—
134	C₄	#13133141	1	4	yes	complemented
135	C₄	D₄×Dic₅₃	1	2	yes	—
136	C₄	#13133204	1	1	yes	characteristic
137	C₄	#13133213	1	1	yes	characteristic
138	C₂	#16810526	1	1	yes	characteristic
139	C₂	C₂×#13133141	1	1	yes	characteristic
140	C₂	#16810632	1	1	yes	characteristic
141	C₂	#16810825	1	2	yes	—
142	C₂	#16810830	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#88322

C₂

yes

maximal, characteristic

C₂×#13133141

C₂

yes

maximal

C₄×C₄⋊Dic₅₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

—

C₄⋊#88322

—

#13133141

—

#13133124

—

C₂²×Q₈×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

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

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

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

SD₁₆

yes

—

C₂×C₄×C₅₃

Q₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

#88322

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

#121

yes

—

C₂²×C₅₃

C₂×SD₁₆

yes

characteristic

C₂²×C₅₃

C₂×Q₁₆

yes

characteristic

C₂²×C₅₃

#137

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#103

yes

complemented

C₄×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#462

yes

—

C₂×C₄⋊Q₈

D₅₃

yes

characteristic

C₂×C₄⋊C₈

—

C₂×#103

—

C₄×C₄⋊C₄

—

C₅₃

#1981

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₃

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₅₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

#103

—

106

—

#95

—

106

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₅₃

yes

—

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

—

106

—

C₂×C₈

—

106

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

106

—

104

C₄²

—

212

—

105

C₄²

C₂×Dic₅₃

yes

—

106

C₄²

C₂²⋊D₅₃

yes

—

107

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

108

Q₈

—

109

Q₈

—

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

106

—

115

C₂×C₄

—

212

—

116

C₂×C₄

—

212

—

117

C₂×C₄

#11719924

yes

—

118

C₂×C₄

#11719925

yes

—

119

C₂×C₄

C₂²⋊Dic₅₃

yes

—

120

C₂×C₄

C₂²⋊Dic₅₃

yes

—

121

C₂×C₄

D₄×D₅₃

yes

—

122

C₂×C₄

#11719947

yes

—

123

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

124

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

125

C₈

—

212

—

126

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

127

C₂²

#13133188

yes

—

128

C₂²

C₂×#11719924

yes

characteristic

129

C₂²

C₂×#11719925

yes

characteristic

130

C₂²

#13133248

yes

—

131

C₄

—

132

C₄

—

133

C₄

—

212

—

134

C₄

#13133141

yes

complemented

135

C₄

D₄×Dic₅₃

yes

—

136

C₄

#13133204

yes

characteristic

137

C₄

#13133213

yes

characteristic

138

C₂

#16810526

yes

characteristic

139

C₂

C₂×#13133141

yes

characteristic

140

C₂

#16810632

yes

characteristic

141

C₂

#16810825

yes

—

142

C₂

#16810830

yes

—

	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	#1981
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

#1981

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
13. C₄⋊Q₈×C₅₃	C₄	semidirect
61. C₄×C₅₃	#103	semidirect
76. C₅₃	#1981	semidirect
80. C₄⋊Q₈	Dic₅₃	semidirect
134. C₄	#13133141	semidirect

H ≅ G/N

type

13. C₄⋊Q₈×C₅₃

C₄

semidirect

61. C₄×C₅₃

#103

semidirect

76. C₅₃

#1981

semidirect

80. C₄⋊Q₈

Dic₅₃

semidirect

134. C₄

#13133141

semidirect

Subgroups — #20275639

Conjugacy classes of subgroups

Complements