Group #20274440 — subgroups

Subgroups — #20274440

Order 6784.

372 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 137 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	C₂²×#11719912	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×D₅₃	C₂	1	2	yes	maximal
3	#16810388	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810387	C₂	1	2	yes	maximal, complemented
5	C₅₃×#341	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
7	C₂×#11719917	C₂²	1	2	yes	—
8	C₂×#11719912	—	2	2	no	—
9	C₂×#11719912	—	2	4	no	—
10	C₂×#11719912	C₄	1	4	yes	—
11	M₁₆×D₅₃	—	2	4	no	—
12	M₁₆×D₅₃	C₄	1	8	yes	—
13	C₂²×#88322	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₅₃	—	2	2	no	—
15	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
16	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂³×D₅₃	Q₈	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₅₃	Q₈	1	1	yes	characteristic
20	C₂×C₄×D₅₃	C₂×C₄	1	4	yes	—
21	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₄×D₅₃	D₄	1	2	yes	—
23	C₂×C₄×D₅₃	D₄	1	4	yes	—
24	M₁₆×C₅₃	—	2	2	no	—
25	M₁₆×C₅₃	C₂×C₄	1	4	yes	—
26	C₂×C₈×C₅₃	—	2	2	no	—
27	C₂×C₈×C₅₃	—	2	2	no	—
28	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
29	#11719917	—	2	2	no	—
30	#11719917	C₂×C₄	1	4	yes	—
31	C₂×#88322	—	2	2	no	—
32	C₂×#88322	—	2	2	no	—
33	C₂×#88322	C₂×C₄	1	2	yes	—
34	#11719912	—	2	8	no	—
35	#11719912	—	2	8	no	—
36	C₈×D₅₃	—	2	8	no	—
37	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
38	C₂²×D₅₃	—	4	2	no	—
39	C₂²×D₅₃	C₄⋊C₄	1	2	yes	—
40	C₂²×D₅₃	C₄⋊C₄	1	4	yes	—
41	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
43	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₅₃	C₄⋊C₄	1	4	yes	—
46	C₂×Dic₅₃	C₄⋊C₄	1	2	yes	—
47	C₄×D₅₃	C₄²	1	4	yes	—
48	C₄×D₅₃	C₂²⋊C₄	1	4	yes	—
49	C₄×D₅₃	C₂²⋊C₄	1	4	yes	—
50	C₄×D₅₃	C₂²⋊C₄	1	4	yes	—
51	C₈×C₅₃	—	2	4	no	—
52	C₈×C₅₃	—	2	2	no	—
53	#88322	—	2	4	no	—
54	#88322	—	2	2	no	—
55	C₂²×C₅₃	—	2	2	no	—
56	C₂²×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
57	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×D₅₃	—	2	4	no	—
59	C₂×D₅₃	—	4	4	no	—
60	C₂×D₅₃	#95	1	4	yes	—
61	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
62	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
64	Dic₅₃	#95	1	4	yes	—
65	#1738	—	53	1	no	maximal, Sylow
66	C₂×C₅₃	—	2	2	no	—
67	C₂×C₅₃	C₂×#95	1	1	yes	characteristic
68	C₂×C₅₃	#430	1	2	yes	—
69	D₅₃	—	2	4	no	—
70	C₂²×M₁₆	—	53	1	no	—
71	C₂²×M₁₆	—	53	2	no	—
72	#341	—	53	2	no	—
73	#341	—	53	1	no	—
74	#341	D₅₃	1	1	yes	characteristic, complemented
75	C₅₃	#1738	1	1	yes	characteristic, complemented, Sylow
76	C₂³×C₄	—	53	1	no	—
77	C₂×M₁₆	—	53	4	no	—
78	C₂×M₁₆	—	53	2	no	—
79	C₂×M₁₆	—	53	8	no	—
80	C₂×M₁₆	—	106	4	no	—
81	C₂×M₁₆	—	106	2	no	—
82	C₂×M₁₆	—	106	4	no	—
83	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
84	C₂²×C₈	—	53	1	no	—
85	C₂²×C₈	—	106	2	no	—
86	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
87	C₂⁴	—	53	1	no	—
88	C₂²×C₄	—	53	1	no	—
89	C₂²×C₄	—	53	4	no	—
90	C₂²×C₄	—	53	2	no	—
91	C₂²×C₄	—	53	4	no	—
92	C₂²×C₄	—	53	2	no	—
93	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
94	M₁₆	—	2	2	no	—
95	M₁₆	—	53	4	no	—
96	M₁₆	—	106	8	no	—
97	M₁₆	—	106	8	no	—
98	M₁₆	—	106	2	no	—
99	M₁₆	C₄×D₅₃	1	4	yes	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	53	2	no	—
103	C₂×C₈	—	106	8	no	—
104	C₂×C₈	—	106	2	no	—
105	C₂×C₈	—	106	2	no	—
106	C₂×C₈	C₄×D₅₃	1	2	yes	—
107	C₂³	—	53	2	no	—
108	C₂³	—	53	4	no	—
109	C₂³	—	212	2	no	—
110	C₂³	Q₈×D₅₃	1	1	yes	characteristic
111	C₂×C₄	—	53	4	no	—
112	C₂×C₄	—	53	2	no	—
113	C₂×C₄	—	53	4	no	—
114	C₂×C₄	—	53	4	no	—
115	C₂×C₄	—	53	4	no	—
116	C₂×C₄	—	53	4	no	—
117	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
118	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
119	C₂×C₄	D₄×D₅₃	1	2	yes	—
120	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	—	2	4	no	—
123	C₈	—	106	2	no	—
124	C₈	—	106	4	no	—
125	C₂²	—	2	2	no	—
126	C₂²	—	53	4	no	—
127	C₂²	—	106	4	no	—
128	C₂²	—	212	4	no	—
129	C₂²	C₄⋊C₄×D₅₃	1	2	yes	—
130	C₂²	C₄⋊C₄×D₅₃	1	1	yes	characteristic
131	C₄	—	53	4	no	—
132	C₄	C₄²×D₅₃	1	1	yes	characteristic
133	C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
134	C₄	C₂²⋊C₄×D₅₃	1	2	yes	—
135	C₂	—	2	2	no	—
136	C₂	—	106	4	no	—
137	C₂	#95×D₅₃	1	1	yes	characteristic
138	C₂	#16810670	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11719912

C₂

yes

maximal, characteristic

C₂×M₁₆×D₅₃

C₂

yes

maximal

#16810388

C₂

yes

maximal, characteristic, complemented

#16810387

C₂

yes

maximal, complemented

C₅₃×#341

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₅₃

C₂²

yes

characteristic

C₂×#11719917

C₂²

yes

—

C₂×#11719912

—

C₂×#11719912

—

C₂×#11719912

C₄

yes

—

M₁₆×D₅₃

—

M₁₆×D₅₃

C₄

yes

—

C₂²×#88322

C₂²

yes

characteristic

C₂×C₈×D₅₃

—

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂³×D₅₃

Q₈

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

Q₈

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

D₄

yes

—

C₂×C₄×D₅₃

D₄

yes

—

M₁₆×C₅₃

—

M₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

#11719917

—

#11719917

C₂×C₄

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₂×C₄

yes

—

#11719912

—

#11719912

—

C₈×D₅₃

—

C₂³×C₅₃

C₂×Q₈

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

C₄⋊C₄

yes

—

C₂²×D₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

C₄⋊C₄

yes

—

C₂×Dic₅₃

C₄⋊C₄

yes

—

C₄×D₅₃

C₄²

yes

—

C₄×D₅₃

C₂²⋊C₄

yes

—

C₄×D₅₃

C₂²⋊C₄

yes

—

C₄×D₅₃

C₂²⋊C₄

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#95

yes

—

C₄×C₅₃

C₂×C₄²

yes

characteristic

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

C₂×C₂²⋊C₄

yes

—

Dic₅₃

#95

yes

—

#1738

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×#95

yes

characteristic

C₂×C₅₃

#430

yes

—

D₅₃

—

C₂²×M₁₆

—

C₂²×M₁₆

—

#341

—

#341

—

#341

D₅₃

yes

characteristic, complemented

C₅₃

#1738

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

106

—

C₂×M₁₆

—

106

—

C₂×M₁₆

—

106

—

C₂×M₁₆

C₂×D₅₃

yes

—

C₂²×C₈

—

C₂²×C₈

—

106

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

106

—

M₁₆

—

106

—

M₁₆

—

106

—

M₁₆

C₄×D₅₃

yes

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

106

—

104

C₂×C₈

—

106

—

105

C₂×C₈

—

106

—

106

C₂×C₈

C₄×D₅₃

yes

—

107

C₂³

—

108

C₂³

—

109

C₂³

—

212

—

110

C₂³

Q₈×D₅₃

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

118

C₂×C₄

C₂×C₄×D₅₃

yes

—

119

C₂×C₄

D₄×D₅₃

yes

—

120

C₂×C₄

D₄×D₅₃

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

106

—

124

C₈

—

106

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

106

—

128

C₂²

—

212

—

129

C₂²

C₄⋊C₄×D₅₃

yes

—

130

C₂²

C₄⋊C₄×D₅₃

yes

characteristic

131

C₄

—

132

C₄

C₄²×D₅₃

yes

characteristic

133

C₄

C₂²⋊C₄×D₅₃

yes

characteristic

134

C₄

C₂²⋊C₄×D₅₃

yes

—

135

C₂

—

136

C₂

—

106

—

137

C₂

#95×D₅₃

yes

characteristic

138

C₂

#16810670

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₅₃
commutator	C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#341	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	#341	D₅₃
hypocenter	C₅₃	#1738
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₅₃

commutator

C₄×C₅₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#341

C₂

socle

C₂²×C₅₃

C₂×C₄⋊C₄

D₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
3. #16810388	C₂	semidirect
4. #16810387	C₂	semidirect
5. C₅₃×#341	C₂	semidirect
74. #341	D₅₃	semidirect
75. C₅₃	#1738	semidirect

H ≅ G/N

type

3. #16810388

C₂

semidirect

4. #16810387

C₂

semidirect

5. C₅₃×#341

C₂

semidirect

74. #341

D₅₃

semidirect

75. C₅₃

#1738

semidirect

Subgroups — #20274440

Conjugacy classes of subgroups

Complements