Group #20287389 — subgroups

Subgroups — #20287389

Order 6784.

336 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 139 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	Q₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	#16810973	C₂	1	3	yes	maximal
3	#16810918	C₂	1	3	yes	maximal
4	#16810830	C₂	1	3	yes	maximal
5	Q₈×#88322	C₂	1	1	yes	maximal, characteristic
6	C₄×#11719925	C₂	1	3	yes	maximal
7	C₅₃×#557	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719925	C₂²	1	3	yes	—
9	#13133217	C₂²	1	3	yes	—
10	#13133216	—	2	3	no	—
11	Q₈×Dic₅₃	—	2	1	no	—
12	C₄×Dic₁₀₆	C₂²	1	3	yes	—
13	#13133150	C₂²	1	3	yes	—
14	C₄⋊#88322	C₂²	1	3	yes	—
15	#13133141	C₂²	1	6	yes	—
16	#13133141	C₂²	1	3	yes	—
17	#13133140	C₂²	1	1	yes	characteristic
18	C₄×#88322	C₂²	1	3	yes	—
19	C₄⋊Q₈×C₅₃	C₂²	1	3	yes	—
20	#125×C₅₃	—	2	3	no	—
21	C₄×Q₈×C₅₃	—	2	1	no	—
22	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₄×Q₈×C₅₃	C₂²	1	3	yes	—
24	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂×Q₈×C₅₃	C₂³	1	3	yes	—
26	C₂×Dic₁₀₆	—	2	3	no	—
27	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₅₃	—	2	3	no	—
29	C₄⋊C₄×C₅₃	—	2	3	no	—
30	C₄⋊C₄×C₅₃	—	2	3	no	—
31	C₄⋊C₄×C₅₃	D₄	1	3	yes	—
32	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
33	C₄²×C₅₃	—	2	3	no	—
34	C₄²×C₅₃	C₂³	1	3	yes	—
35	#11719925	—	2	6	no	—
36	C₄⋊Dic₅₃	—	2	3	no	—
37	C₄⋊Dic₅₃	C₂³	1	3	yes	—
38	#11719919	—	2	3	no	—
39	C₄×Dic₅₃	—	2	3	no	—
40	C₂×#88322	C₂³	1	1	yes	characteristic
41	C₂×#88322	C₂³	1	3	yes	—
42	Q₈×C₅₃	—	2	3	no	—
43	Q₈×C₅₃	—	2	3	no	—
44	Q₈×C₅₃	Q₁₆	1	4	yes	complemented
45	C₂×C₄×C₅₃	—	2	3	no	—
46	C₂×C₄×C₅₃	—	2	1	no	—
47	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
48	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
49	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₅₃	—	2	1	no	—
51	C₂×Dic₅₃	—	2	3	no	—
52	Dic₁₀₆	—	2	1	no	—
53	Dic₁₀₆	—	8	3	no	—
54	Dic₁₀₆	#41	1	2	yes	—
55	#88322	—	2	3	no	—
56	#88322	—	2	1	no	—
57	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₅₃	—	2	3	no	—
59	C₄×C₅₃	—	4	3	no	—
60	C₄×C₅₃	—	4	1	no	—
61	C₄×C₅₃	C₂×Q₁₆	1	6	yes	—
62	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
63	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
64	Dic₅₃	—	2	2	no	—
65	Dic₅₃	—	4	3	no	—
66	#3363	—	53	1	no	maximal, Sylow
67	C₂×C₅₃	#548	1	1	yes	characteristic
68	C₂×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
69	C₂×C₅₃	#577	1	1	yes	characteristic
70	Q₈²	—	53	1	no	—
71	#557	D₅₃	1	1	yes	characteristic
72	#487	—	53	3	no	—
73	#477	—	53	3	no	—
74	#462	—	53	3	no	—
75	C₈×Q₈	—	53	1	no	—
76	C₄×Q₁₆	—	53	3	no	—
77	C₅₃	#3363	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₁₆	—	53	3	no	—
79	C₄⋊Q₈	—	53	3	no	—
80	C₄⋊Q₈	—	106	3	no	—
81	C₄⋊Q₈	C₂×D₅₃	1	3	yes	—
82	#125	—	2	3	no	—
83	C₄×Q₈	—	2	1	no	—
84	C₄×Q₈	—	53	3	no	—
85	C₄×Q₈	—	106	1	no	—
86	C₄×Q₈	C₂×D₅₃	1	3	yes	—
87	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
88	#107	—	53	3	no	—
89	C₄⋊C₈	—	53	3	no	—
90	#103	—	53	3	no	—
91	#103	—	53	6	no	—
92	#103	—	53	1	no	—
93	C₄×C₈	—	53	3	no	—
94	C₂×Q₈	—	53	1	no	—
95	C₂×Q₈	—	106	3	no	—
96	C₂×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
97	C₂×Q₈	C₂²×D₅₃	1	3	yes	—
98	Q₁₆	—	106	6	no	—
99	C₂×C₈	—	53	1	no	—
100	C₂×C₈	—	53	3	no	—
101	C₄⋊C₄	—	2	3	no	—
102	C₄⋊C₄	—	2	3	no	—
103	C₄⋊C₄	—	2	3	no	—
104	C₄⋊C₄	—	53	3	no	—
105	C₄⋊C₄	—	106	3	no	—
106	C₄⋊C₄	—	106	3	no	—
107	C₄⋊C₄	C₂²⋊D₅₃	1	3	yes	—
108	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
109	C₄²	—	2	3	no	—
110	C₄²	—	106	3	no	—
111	C₄²	C₂²×D₅₃	1	3	yes	—
112	Q₈	—	2	3	no	—
113	Q₈	—	2	3	no	—
114	Q₈	—	53	2	no	—
115	Q₈	—	106	1	no	—
116	Q₈	—	424	3	no	—
117	Q₈	#11719925	1	4	yes	complemented
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	3	no	—
120	C₂×C₄	—	106	1	no	—
121	C₂×C₄	—	106	3	no	—
122	C₂×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
123	C₂×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
124	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
125	C₈	—	106	3	no	—
126	C₈	—	106	1	no	—
127	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
128	C₄	—	2	3	no	—
129	C₄	—	4	3	no	—
130	C₄	—	4	1	no	—
131	C₄	—	106	2	no	—
132	C₄	—	212	3	no	—
133	C₄	C₂×#11719925	1	6	yes	—
134	C₄	#41×D₅₃	1	1	yes	characteristic
135	C₄	#13133264	1	1	yes	characteristic
136	C₂	#16811280	1	1	yes	characteristic
137	C₂	C₂²×#11719925	1	1	yes	characteristic
138	C₂	#16811431	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₀₆

C₂

yes

maximal, characteristic

#16810973

C₂

yes

maximal

#16810918

C₂

yes

maximal

#16810830

C₂

yes

maximal

Q₈×#88322

C₂

yes

maximal, characteristic

C₄×#11719925

C₂

yes

maximal

C₅₃×#557

C₂

yes

maximal, characteristic

C₂×#11719925

C₂²

yes

—

#13133217

C₂²

yes

—

#13133216

—

Q₈×Dic₅₃

—

C₄×Dic₁₀₆

C₂²

yes

—

#13133150

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

#13133141

C₂²

yes

—

#13133141

C₂²

yes

—

#13133140

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₂²

yes

—

#125×C₅₃

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

C₂²

yes

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×Q₈×C₅₃

C₂³

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

C₂³

yes

—

#11719925

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

C₂³

yes

characteristic

C₂×#88322

C₂³

yes

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

Q₁₆

yes

complemented

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

#41

yes

—

#88322

—

#88322

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×Q₁₆

yes

—

C₄×C₅₃

C₂×#41

yes

characteristic

C₄×C₅₃

2₋¹⁺⁴

yes

characteristic

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂²×Q₁₆

yes

characteristic

C₂×C₅₃

#577

yes

characteristic

Q₈²

—

#557

D₅₃

yes

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

C₅₃

#3363

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

C₂×D₅₃

yes

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

106

—

C₄×Q₈

C₂×D₅₃

yes

—

C₄×Q₈

C₂×D₅₃

yes

characteristic

#107

—

C₄⋊C₈

—

#103

—

#103

—

#103

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

C₂²⋊D₅₃

yes

characteristic

C₂×Q₈

C₂²×D₅₃

yes

—

Q₁₆

—

106

—

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

—

106

C₄⋊C₄

—

106

—

107

C₄⋊C₄

C₂²⋊D₅₃

yes

—

108

C₄⋊C₄

C₂²×D₅₃

yes

characteristic

109

C₄²

—

110

C₄²

—

106

—

111

C₄²

C₂²×D₅₃

yes

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

106

—

116

Q₈

—

424

—

117

Q₈

#11719925

yes

complemented

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

106

—

121

C₂×C₄

—

106

—

122

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

123

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

124

C₂×C₄

C₂³×D₅₃

yes

characteristic

125

C₈

—

106

—

126

C₈

—

106

—

127

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

106

—

132

C₄

—

212

—

133

C₄

C₂×#11719925

yes

—

134

C₄

#41×D₅₃

yes

characteristic

135

C₄

#13133264

yes

characteristic

136

C₂

#16811280

yes

characteristic

137

C₂

C₂²×#11719925

yes

characteristic

138

C₂

#16811431

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#557	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#557	D₅₃
hypocenter	C₅₃	#3363
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#557

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

N	H ≅ G/N	type
44. Q₈×C₅₃	Q₁₆	semidirect
77. C₅₃	#3363	semidirect
117. Q₈	#11719925	semidirect

H ≅ G/N

type

44. Q₈×C₅₃

Q₁₆

semidirect

77. C₅₃

#3363

semidirect

117. Q₈

#11719925

semidirect

Subgroups — #20287389

Conjugacy classes of subgroups

Complements