Group #20273564 — subgroups

Subgroups — #20273564

Order 6784.

152 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 63 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	#16810750	C₂	1	1	yes	maximal, characteristic
2	#16810747	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
4	#16810353	C₂	1	1	yes	maximal, characteristic
5	#16810353	C₂	1	1	yes	maximal, characteristic
6	#16810345	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#335	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₁₂	C₄	1	1	yes	characteristic
9	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
10	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
11	#13133151	C₄	1	1	yes	characteristic
12	#13133146	—	2	1	no	—
13	#13133146	—	2	1	no	—
14	#13133142	C₄	1	1	yes	characteristic
15	#13133142	C₄	1	1	yes	characteristic
16	#13133127	—	2	1	no	—
17	C₄×#88322	C₂²	1	1	yes	characteristic
18	C₄×#88322	C₂²	1	1	yes	characteristic
19	C₈×Dic₅₃	—	2	1	no	—
20	C₄⋊C₈×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×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
24	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₅₃	—	2	1	no	—
26	C₂×C₈×C₅₃	—	2	1	no	—
27	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₅₃	C₂³	1	1	yes	characteristic
30	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
32	#11719919	—	2	1	no	—
33	#11719919	—	2	1	no	—
34	C₄×Dic₅₃	—	2	1	no	—
35	C₄×Dic₅₃	—	2	1	no	—
36	C₂×#88322	—	2	1	no	—
37	C₂×#88322	D₄	1	1	yes	characteristic
38	C₂×#88322	D₄	1	1	yes	characteristic
39	Dic₂₁₂	—	2	2	no	—
40	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
43	C₂×Dic₅₃	—	2	1	no	—
44	C₂×Dic₅₃	—	2	1	no	—
45	C₂×Dic₅₃	—	2	1	no	—
46	C₂×Dic₅₃	—	2	1	no	—
47	Dic₁₀₆	—	2	1	no	—
48	Dic₁₀₆	—	2	1	no	—
49	Dic₁₀₆	—	2	1	no	—
50	Dic₁₀₆	M₁₆	1	2	yes	—
51	C₈×C₅₃	—	2	1	no	—
52	C₈×C₅₃	—	2	1	no	—
53	C₈×C₅₃	—	4	1	no	—
54	C₈×C₅₃	—	4	1	no	—
55	#88322	—	2	1	no	—
56	#88322	—	4	1	no	—
57	#88322	Q₁₆	1	2	yes	complemented
58	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
59	C₄×C₅₃	—	2	1	no	—
60	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
61	C₄×C₅₃	#131	1	1	yes	characteristic
62	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
63	C₄×C₅₃	#135	1	1	yes	characteristic
64	Dic₅₃	—	2	2	no	—
65	Dic₅₃	—	4	1	no	—
66	Dic₅₃	—	4	1	no	—
67	Dic₅₃	—	4	1	no	—
68	#1584	—	53	1	no	maximal, Sylow
69	C₂×C₅₃	#435	1	1	yes	characteristic
70	C₂×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
71	C₂×C₅₃	#444	1	1	yes	characteristic
72	#446	—	53	1	no	—
73	C₈×Q₈	—	53	1	no	—
74	C₄×Q₁₆	—	53	1	no	—
75	#335	D₅₃	1	1	yes	characteristic
76	Q₈⋊C₈	—	53	1	no	—
77	Q₈⋊C₈	—	53	1	no	—
78	#322	—	53	1	no	—
79	C₅₃	#1584	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₁₆	—	53	1	no	—
81	C₄×Q₈	—	53	1	no	—
82	C₄×Q₈	—	53	1	no	—
83	#107	—	53	1	no	—
84	C₄⋊C₈	—	106	1	no	—
85	C₄⋊C₈	—	106	1	no	—
86	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
87	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
88	#103	—	53	1	no	—
89	#103	—	53	1	no	—
90	#97	—	106	1	no	—
91	C₄×C₈	—	53	1	no	—
92	C₄×C₈	—	53	1	no	—
93	C₄×C₈	—	106	1	no	—
94	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
95	C₂×Q₈	—	53	1	no	—
96	C₂×Q₈	—	53	1	no	—
97	Q₁₆	—	106	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	53	1	no	—
101	C₂×C₈	—	53	1	no	—
102	C₂×C₈	—	106	1	no	—
103	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
104	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
105	C₄⋊C₄	—	53	1	no	—
106	C₄⋊C₄	—	53	1	no	—
107	C₄⋊C₄	—	106	1	no	—
108	C₄⋊C₄	—	106	1	no	—
109	C₄²	—	106	1	no	—
110	C₄²	—	106	1	no	—
111	C₄²	C₂²×D₅₃	1	1	yes	characteristic
112	Q₈	—	53	2	no	—
113	Q₈	—	106	1	no	—
114	Q₈	—	106	1	no	—
115	Q₈	—	106	1	no	—
116	C₂×C₄	—	106	1	no	—
117	C₂×C₄	—	106	1	no	—
118	C₂×C₄	—	106	1	no	—
119	C₂×C₄	—	106	1	no	—
120	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
121	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
122	C₂×C₄	#11719949	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	2	1	no	—
125	C₈	—	4	1	no	—
126	C₈	—	4	1	no	—
127	C₈	—	53	2	no	—
128	C₈	—	106	1	no	—
129	C₈	—	212	1	no	—
130	C₂²	#13133179	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	106	2	no	—
133	C₄	—	212	1	no	—
134	C₄	—	212	1	no	—
135	C₄	—	212	1	no	—
136	C₄	M₁₆×D₅₃	1	1	yes	characteristic
137	C₄	#13133223	1	1	yes	characteristic
138	C₄	Q₁₆×D₅₃	1	1	yes	characteristic
139	C₄	#13133236	1	1	yes	characteristic
140	C₂	#16810693	1	1	yes	characteristic
141	C₂	#16810717	1	1	yes	characteristic
142	C₂	#16810744	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810750

C₂

yes

maximal, characteristic

#16810747

C₂

yes

maximal, characteristic

C₄×Dic₂₁₂

C₂

yes

maximal, characteristic

#16810353

C₂

yes

maximal, characteristic

#16810353

C₂

yes

maximal, characteristic

#16810345

C₂

yes

maximal, characteristic

C₅₃×#335

C₂

yes

maximal, characteristic

C₂×Dic₂₁₂

C₄

yes

characteristic

C₄×Dic₁₀₆

C₂²

yes

characteristic

C₄×Dic₁₀₆

C₂²

yes

characteristic

#13133151

C₄

yes

—

—

C₄

yes

characteristic

#13133142

C₄

yes

characteristic

#13133127

—

C₄×#88322

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₈×Dic₅₃

—

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Dic₁₀₆

C₂×C₄

yes

characteristic

C₂×Dic₁₀₆

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

C₂×C₄

yes

characteristic

C₄⋊Dic₅₃

C₂×C₄

yes

characteristic

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

Dic₂₁₂

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

M₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

#88322

Q₁₆

yes

complemented

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₄×C₅₃

#131

yes

characteristic

C₄×C₅₃

C₂×Q₁₆

yes

characteristic

C₄×C₅₃

#135

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₄×Q₁₆

yes

C₂×C₅₃

yes

—

C₈×Q₈

—

C₄×Q₁₆

—

#335

D₅₃

yes

characteristic

Q₈⋊C₈

—

Q₈⋊C₈

—

#322

—

C₅₃

#1584

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

C₄×Q₈

—

#107

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

C₄⋊C₈

C₂×D₅₃

yes

characteristic

C₄⋊C₈

C₂×D₅₃

yes

—

—

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

106

—

103

C₂×C₈

C₄×D₅₃

yes

characteristic

104

C₂×C₈

C₄×D₅₃

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

106

—

108

C₄⋊C₄

—

106

—

109

C₄²

—

106

—

110

C₄²

—

106

—

111

C₄²

C₂²×D₅₃

yes

characteristic

112

Q₈

—

113

Q₈

—

106

—

114

Q₈

—

106

—

115

Q₈

—

106

—

116

C₂×C₄

—

106

—

117

C₂×C₄

—

106

—

118

C₂×C₄

—

106

—

119

C₂×C₄

—

106

—

120

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

121

C₂×C₄

D₄×D₅₃

yes

characteristic

122

C₂×C₄

#11719949

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

106

—

129

C₈

—

212

—

130

C₂²

#13133179

yes

characteristic

131

C₄

—

132

C₄

—

106

—

133

C₄

—

212

—

134

C₄

—

212

—

135

C₄

—

212

—

136

C₄

M₁₆×D₅₃

yes

characteristic

137

C₄

#13133223

yes

characteristic

138

C₄

Q₁₆×D₅₃

yes

characteristic

139

C₄

#13133236

yes

characteristic

140

C₂

#16810693

yes

characteristic

141

C₂

#16810717

yes

characteristic

142

C₂

#16810744

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#335	C₂
socle	C₂²×C₅₃	C₄×D₄
hypercenter	#335	D₅₃
hypocenter	C₅₃	#1584
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

subgroup

G/H

center

C₂×C₄

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#335

C₂

socle

C₂²×C₅₃

C₄×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

N	H ≅ G/N	type
57. #88322	Q₁₆	semidirect
79. C₅₃	#1584	semidirect

H ≅ G/N

type

57. #88322

Q₁₆

semidirect

79. C₅₃

#1584

semidirect

Subgroups — #20273564

Conjugacy classes of subgroups

Complements