Group #20273397 — subgroups

Subgroups — #20273397

Order 6784.

240 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 93 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×#13133217	C₂	1	1	yes	maximal, characteristic
2	#16810676	C₂	1	1	yes	maximal, characteristic
3	#16810371	C₂	1	4	yes	maximal, complemented
4	C₅₃×#423	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₀₆	C₄	1	2	yes	—
6	#13133217	—	2	1	no	—
7	#13133217	—	2	1	no	—
8	#13133217	C₄	1	2	yes	—
9	#13133217	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₅₃	—	2	1	no	—
11	C₂×C₄⋊Dic₅₃	—	2	1	no	—
12	C₄⋊#88322	C₂²	1	2	yes	—
13	C₂²⋊#88322	—	2	1	no	—
14	C₄×#88322	—	2	1	no	—
15	C₂×M₁₆×C₅₃	—	2	1	no	—
16	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₅₃	—	2	1	no	—
18	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
22	C₂²×Dic₅₃	—	2	2	no	—
23	C₂×Dic₁₀₆	—	2	2	no	—
24	C₂×Dic₁₀₆	—	4	4	no	—
25	C₂×Dic₁₀₆	C₂×C₄	1	4	yes	—
26	M₁₆×C₅₃	—	4	2	no	—
27	C₂×C₈×C₅₃	—	2	2	no	—
28	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₅₃	D₄	1	1	yes	characteristic
30	C₄²×C₅₃	D₄	1	1	yes	characteristic
31	C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	C₄⋊Dic₅₃	—	2	2	no	—
33	C₄⋊Dic₅₃	—	2	2	no	—
34	C₄⋊Dic₅₃	—	2	2	no	—
35	C₄⋊Dic₅₃	—	2	2	no	—
36	C₂×#88322	—	2	2	no	—
37	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	—	2	2	no	—
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
44	C₂×C₄×C₅₃	Q₁₆	1	2	yes	—
45	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₅₃	—	2	4	no	—
49	C₂×Dic₅₃	—	4	4	no	—
50	Dic₁₀₆	—	4	4	no	—
51	Dic₁₀₆	—	4	4	no	—
52	C₈×C₅₃	—	4	2	no	—
53	#88322	—	4	2	no	—
54	C₂²×C₅₃	—	2	1	no	—
55	C₂²×C₅₃	—	2	1	no	—
56	C₂²×C₅₃	#103	1	2	yes	—
57	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₄×C₅₃	—	2	1	no	—
59	C₄×C₅₃	—	2	1	no	—
60	C₄×C₅₃	#100	1	2	yes	—
61	C₄×C₅₃	#103	1	2	yes	—
62	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
63	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
64	C₄×C₅₃	#137	1	2	yes	—
65	Dic₅₃	—	4	4	no	—
66	#1548	—	53	1	no	maximal, Sylow
67	C₂×C₅₃	—	2	2	no	—
68	C₂×C₅₃	C₂×#100	1	1	yes	characteristic
69	C₂×C₅₃	C₂×#103	1	1	yes	characteristic
70	C₂×C₅₃	#419	1	1	yes	characteristic
71	C₂×C₄⋊Q₈	—	53	1	no	—
72	#431	—	53	1	no	—
73	#423	D₅₃	1	1	yes	characteristic
74	#333	—	53	4	no	—
75	C₅₃	#1548	1	1	yes	characteristic, complemented, Sylow
76	C₂²×Q₈	—	53	2	no	—
77	C₂×M₁₆	—	2	1	no	—
78	C₄⋊Q₈	—	53	2	no	—
79	C₄⋊Q₈	—	53	2	no	—
80	C₄⋊Q₈	—	106	1	no	—
81	C₄⋊Q₈	—	106	1	no	—
82	C₂×C₄⋊C₄	—	106	1	no	—
83	C₂×C₄⋊C₄	—	106	1	no	—
84	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
85	C₄⋊C₈	—	2	1	no	—
86	C₄⋊C₈	—	53	2	no	—
87	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
88	C₂²⋊C₈	—	106	1	no	—
89	C₄×C₈	—	106	1	no	—
90	C₂×Q₈	—	53	4	no	—
91	C₂×Q₈	—	106	2	no	—
92	C₂×Q₈	—	212	4	no	—
93	C₂²×C₄	—	106	2	no	—
94	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
95	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
97	M₁₆	—	4	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	106	2	no	—
100	C₄⋊C₄	—	106	2	no	—
101	C₄⋊C₄	—	106	2	no	—
102	C₄⋊C₄	—	106	2	no	—
103	C₄⋊C₄	—	106	2	no	—
104	C₄²	C₄×D₅₃	1	1	yes	characteristic
105	C₄²	D₂₁₂	1	1	yes	characteristic
106	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
107	C₄²	C₂²×D₅₃	1	1	yes	characteristic
108	C₂³	#11719921	1	1	yes	characteristic
109	Q₈	—	212	4	no	—
110	Q₈	—	212	4	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	106	4	no	—
114	C₂×C₄	—	212	4	no	—
115	C₂×C₄	#11719921	1	1	yes	characteristic
116	C₂×C₄	#11719921	1	1	yes	characteristic
117	C₂×C₄	#11719921	1	1	yes	characteristic
118	C₂×C₄	#11719923	1	2	yes	—
119	C₂×C₄	#11719925	1	2	yes	—
120	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
121	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
122	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
123	C₈	—	4	2	no	—
124	C₈	—	212	2	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	#13133140	1	2	yes	—
128	C₂²	C₂×#11719921	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	—	212	4	no	—
132	C₄	#13133133	1	2	yes	—
133	C₄	#13133140	1	2	yes	—
134	C₄	C₂×#11719923	1	1	yes	characteristic
135	C₄	C₂×#11719925	1	1	yes	characteristic
136	C₄	#13133250	1	2	yes	—
137	C₂	—	2	2	no	—
138	C₂	C₂×#13133133	1	1	yes	characteristic
139	C₂	C₂×#13133140	1	1	yes	characteristic
140	C₂	#16810633	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133217

C₂

yes

maximal, characteristic

#16810676

C₂

yes

maximal, characteristic

#16810371

C₂

yes

maximal, complemented

C₅₃×#423

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₄

yes

—

#13133217

—

#13133217

—

#13133217

C₄

yes

—

#13133217

C₂²

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

C₄⋊#88322

C₂²

yes

—

C₂²⋊#88322

—

C₄×#88322

—

C₂×M₁₆×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

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×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#103

yes

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#100

yes

—

C₄×C₅₃

#103

yes

—

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₄×C₅₃

C₂×Q₁₆

yes

characteristic

C₄×C₅₃

#137

yes

—

Dic₅₃

—

#1548

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊Q₈

—

#431

—

#423

D₅₃

yes

characteristic

#333

—

C₅₃

#1548

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

—

106

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₃

yes

—

C₂²⋊C₈

—

106

—

C₄×C₈

—

106

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

212

—

C₂²×C₄

—

106

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

106

—

100

C₄⋊C₄

—

106

—

101

C₄⋊C₄

—

106

—

102

C₄⋊C₄

—

106

—

103

C₄⋊C₄

—

106

—

104

C₄²

C₄×D₅₃

yes

characteristic

105

C₄²

D₂₁₂

yes

characteristic

106

C₄²

C₂²⋊D₅₃

yes

characteristic

107

C₄²

C₂²×D₅₃

yes

characteristic

108

C₂³

#11719921

yes

characteristic

109

Q₈

—

212

—

110

Q₈

—

212

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

106

—

114

C₂×C₄

—

212

—

115

C₂×C₄

#11719921

yes

characteristic

116

C₂×C₄

#11719921

yes

characteristic

117

C₂×C₄

#11719921

yes

characteristic

118

C₂×C₄

#11719923

yes

—

119

C₂×C₄

#11719925

yes

—

120

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

121

C₂×C₄

C₂×D₂₁₂

yes

characteristic

122

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

123

C₈

—

124

C₈

—

212

—

125

C₂²

—

126

C₂²

—

127

C₂²

#13133140

yes

—

128

C₂²

C₂×#11719921

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

212

—

132

C₄

#13133133

yes

—

133

C₄

#13133140

yes

—

134

C₄

C₂×#11719923

yes

characteristic

135

C₄

C₂×#11719925

yes

characteristic

136

C₄

#13133250

yes

—

137

C₂

—

138

C₂

C₂×#13133133

yes

characteristic

139

C₂

C₂×#13133140

yes

characteristic

140

C₂

#16810633

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11719921
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#423	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#423	D₅₃
hypocenter	C₅₃	#1548
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×#11719921

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#423

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
3. #16810371	C₂	semidirect
75. C₅₃	#1548	semidirect

H ≅ G/N

type

3. #16810371

C₂

semidirect

75. C₅₃

#1548

semidirect

Subgroups — #20273397

Conjugacy classes of subgroups

Complements