Group #20287357 — subgroups

Subgroups — #20287357

Order 6784.

370 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 141 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	#16811352	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810975	C₂	1	3	yes	maximal, complemented
3	#16810905	C₂	1	3	yes	maximal, complemented
4	#16810828	C₂	1	3	yes	maximal
5	Q₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#11719923	C₂	1	3	yes	maximal
7	D₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719923	C₂²	1	3	yes	—
9	#13133217	C₂²	1	3	yes	—
10	#13133208	—	2	3	no	—
11	Q₈×Dic₅₃	—	2	1	no	—
12	C₄×Dic₁₀₆	C₂²	1	3	yes	—
13	#13133148	C₂²	1	3	yes	—
14	C₄⋊#88322	C₂²	1	3	yes	—
15	#13133140	C₂²	1	6	yes	—
16	#13133140	C₂²	1	3	yes	—
17	#13133138	C₂²	1	1	yes	characteristic
18	C₄×#88322	C₂²	1	3	yes	—
19	C₂²×Q₈×C₅₃	—	2	1	no	—
20	C₄⋊Q₈×C₅₃	C₂²	1	3	yes	—
21	C₂²⋊Q₈×C₅₃	—	2	3	no	—
22	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₅₃	C₂²	1	3	yes	—
24	C₂×Q₈×C₅₃	—	2	3	no	—
25	C₂×Q₈×C₅₃	—	4	4	no	—
26	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
27	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	—	2	3	no	—
29	C₂×Dic₁₀₆	C₂³	1	3	yes	—
30	C₄⋊C₄×C₅₃	—	2	3	no	—
31	C₄⋊C₄×C₅₃	D₄	1	3	yes	—
32	C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
33	C₂²⋊C₄×C₅₃	—	2	3	no	—
34	C₄²×C₅₃	C₂³	1	3	yes	—
35	#11719923	—	2	6	no	—
36	C₄⋊Dic₅₃	—	2	3	no	—
37	C₄⋊Dic₅₃	—	2	3	no	—
38	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
39	#11719919	—	2	3	no	—
40	C₄×Dic₅₃	—	2	3	no	—
41	C₂×#88322	C₂³	1	1	yes	characteristic
42	C₂×#88322	C₂³	1	3	yes	—
43	C₂³×C₅₃	—	2	1	no	—
44	Q₈×C₅₃	—	4	6	no	—
45	Q₈×C₅₃	SD₁₆	1	4	yes	complemented
46	D₄×C₅₃	—	2	1	no	—
47	D₄×C₅₃	#41	1	2	yes	—
48	C₂×C₄×C₅₃	—	2	3	no	—
49	C₂×C₄×C₅₃	—	4	6	no	—
50	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
51	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
52	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₅₃	—	2	3	no	—
54	C₂×Dic₅₃	—	2	1	no	—
55	Dic₁₀₆	—	2	3	no	—
56	Dic₁₀₆	—	2	3	no	—
57	#88322	—	2	3	no	—
58	#88322	—	2	1	no	—
59	C₂²×C₅₃	—	2	2	no	—
60	C₂²×C₅₃	—	4	1	no	—
61	C₂²×C₅₃	—	4	1	no	—
62	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₅₃	—	2	3	no	—
64	C₄×C₅₃	—	4	3	no	—
65	C₄×C₅₃	C₂×SD₁₆	1	6	yes	—
66	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
67	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
68	Dic₅₃	—	4	3	no	—
69	Dic₅₃	—	4	1	no	—
70	#3359	—	53	1	no	maximal, Sylow
71	C₂×C₅₃	—	4	2	no	—
72	C₂×C₅₃	#548	1	1	yes	characteristic
73	C₂×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
74	C₂×C₅₃	#578	1	1	yes	characteristic
75	#557	—	53	1	no	—
76	D₄×Q₈	D₅₃	1	1	yes	characteristic
77	#487	—	53	3	no	—
78	#475	—	53	3	no	—
79	#461	—	53	3	no	—
80	C₈×Q₈	—	53	1	no	—
81	C₄×SD₁₆	—	53	3	no	—
82	C₅₃	#3359	1	1	yes	characteristic, complemented, Sylow
83	C₂²×Q₈	—	2	1	no	—
84	C₂×SD₁₆	—	53	3	no	—
85	C₄⋊Q₈	—	53	3	no	—
86	C₄⋊Q₈	C₂×D₅₃	1	3	yes	—
87	#125	—	106	3	no	—
88	C₂²⋊Q₈	—	2	3	no	—
89	C₄×Q₈	—	53	3	no	—
90	C₄×Q₈	—	106	1	no	—
91	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
92	C₄×D₄	C₂×D₅₃	1	3	yes	—
93	#106	—	53	3	no	—
94	C₄⋊C₈	—	53	3	no	—
95	#103	—	53	3	no	—
96	#103	—	53	6	no	—
97	#102	—	53	1	no	—
98	C₄×C₈	—	53	3	no	—
99	C₂×Q₈	—	2	3	no	—
100	C₂×Q₈	—	4	4	no	—
101	C₂×Q₈	—	53	3	no	—
102	C₂×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
103	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
104	C₂²×C₄	—	2	3	no	—
105	SD₁₆	—	106	6	no	—
106	C₂×C₈	—	53	1	no	—
107	C₂×C₈	—	53	3	no	—
108	C₄⋊C₄	—	2	3	no	—
109	C₄⋊C₄	—	53	1	no	—
110	C₄⋊C₄	—	106	3	no	—
111	C₄⋊C₄	—	106	3	no	—
112	C₄⋊C₄	—	106	3	no	—
113	C₄⋊C₄	C₂²⋊D₅₃	1	3	yes	—
114	C₄⋊C₄	C₂²×D₅₃	1	3	yes	—
115	C₂²⋊C₄	—	2	3	no	—
116	C₄²	—	106	3	no	—
117	C₄²	C₂²×D₅₃	1	3	yes	—
118	C₂³	—	2	1	no	—
119	Q₈	—	4	6	no	—
120	Q₈	—	106	3	no	—
121	Q₈	—	106	3	no	—
122	Q₈	#11719923	1	4	yes	complemented
123	D₄	—	2	1	no	—
124	D₄	#11719949	1	2	yes	—
125	C₂×C₄	—	2	3	no	—
126	C₂×C₄	—	4	6	no	—
127	C₂×C₄	—	106	1	no	—
128	C₂×C₄	—	106	3	no	—
129	C₂×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
130	C₂×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
131	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
132	C₈	—	106	3	no	—
133	C₈	—	106	1	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	1	no	—
137	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
138	C₄	—	2	3	no	—
139	C₄	—	4	3	no	—
140	C₄	—	212	3	no	—
141	C₄	—	212	1	no	—
142	C₄	C₂×#11719923	1	6	yes	—
143	C₄	C₂×#11719949	1	1	yes	characteristic
144	C₄	#13133265	1	1	yes	characteristic
145	C₂	—	4	2	no	—
146	C₂	#16811284	1	1	yes	characteristic
147	C₂	C₂²×#11719923	1	1	yes	characteristic
148	C₂	#16811438	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811352

C₂

yes

maximal, characteristic, complemented

#16810975

C₂

yes

maximal, complemented

#16810905

C₂

yes

maximal, complemented

#16810828

C₂

yes

maximal

Q₈×#88322

C₂

yes

maximal, characteristic, complemented

C₄×#11719923

C₂

yes

maximal

D₄×Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719923

C₂²

yes

—

#13133217

C₂²

yes

—

#13133208

—

Q₈×Dic₅₃

—

C₄×Dic₁₀₆

C₂²

yes

—

#13133148

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

#13133140

C₂²

yes

—

#13133140

C₂²

yes

—

#13133138

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

—

C₂²×Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₂²

yes

—

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄×D₄×C₅₃

C₂²

yes

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×D₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

—

C₂×Dic₁₀₆

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

—

#11719923

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

characteristic

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

C₂³

yes

characteristic

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

SD₁₆

yes

complemented

D₄×C₅₃

—

D₄×C₅₃

#41

yes

—

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

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×SD₁₆

yes

—

C₄×C₅₃

C₂×#41

yes

characteristic

C₄×C₅₃

2₋¹⁺⁴

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#3359

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#548

yes

characteristic

C₂×C₅₃

C₂²×SD₁₆

yes

C₂×C₅₃

yes

—

D₄×Q₈

D₅₃

yes

—

—

—

C₈×Q₈

—

C₄×SD₁₆

—

C₅₃

#3359

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

—

#125

—

106

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

106

—

C₄×Q₈

C₂×D₅₃

yes

characteristic

C₄×D₄

C₂×D₅₃

yes

—

#106

—

C₄⋊C₈

—

#103

—

#103

—

#102

—

C₄×C₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

C₂²⋊D₅₃

yes

characteristic

103

C₂×D₄

C₂²×D₅₃

yes

characteristic

104

C₂²×C₄

—

105

SD₁₆

—

106

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

106

—

111

C₄⋊C₄

—

106

—

112

C₄⋊C₄

—

106

—

113

C₄⋊C₄

C₂²⋊D₅₃

yes

—

114

C₄⋊C₄

C₂²×D₅₃

yes

—

115

C₂²⋊C₄

—

116

C₄²

—

106

—

117

C₄²

C₂²×D₅₃

yes

—

118

C₂³

—

119

Q₈

—

120

Q₈

—

106

—

121

Q₈

—

106

—

122

Q₈

#11719923

yes

complemented

123

D₄

—

124

D₄

#11719949

yes

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

106

—

128

C₂×C₄

—

106

—

129

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

130

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

131

C₂×C₄

C₂³×D₅₃

yes

characteristic

132

C₈

—

106

—

133

C₈

—

106

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

212

—

141

C₄

—

212

—

142

C₄

C₂×#11719923

yes

—

143

C₄

C₂×#11719949

yes

characteristic

144

C₄

#13133265

yes

characteristic

145

C₂

—

146

C₂

#16811284

yes

characteristic

147

C₂

C₂²×#11719923

yes

characteristic

148

C₂

#16811438

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	D₄×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	D₄×Q₈	D₅₃
hypocenter	C₅₃	#3359
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

D₄×Q₈×C₅₃

C₂

socle

C₂²×C₅₃

C₂²×D₄

hypercenter

D₄×Q₈

D₅₃

hypocenter

C₅₃

#3359

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

N	H ≅ G/N	type
1. #16811352	C₂	semidirect
2. #16810975	C₂	semidirect
3. #16810905	C₂	semidirect
5. Q₈×#88322	C₂	semidirect
45. Q₈×C₅₃	SD₁₆	semidirect
82. C₅₃	#3359	semidirect
122. Q₈	#11719923	semidirect

H ≅ G/N

type

1. #16811352

C₂

semidirect

2. #16810975

C₂

semidirect

3. #16810905

C₂

semidirect

5. Q₈×#88322

C₂

semidirect

45. Q₈×C₅₃

SD₁₆

semidirect

82. C₅₃

#3359

semidirect

122. Q₈

#11719923

semidirect

Subgroups — #20287357

Conjugacy classes of subgroups

Complements