Group #20276325 — subgroups

Subgroups — #20276325

Order 6784.

260 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 97 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×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133157	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810402	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₄×D₅₃	C₄	1	1	yes	characteristic
6	C₂²×#88322	C₄	1	1	yes	characteristic
7	C₂×C₈×D₅₃	—	2	2	no	—
8	C₂×C₈×D₅₃	—	2	2	no	—
9	C₂×C₈×D₅₃	C₄	1	2	yes	—
10	C₂×C₈×D₅₃	C₂²	1	2	yes	—
11	#13133157	—	2	2	no	—
12	C₂×#11719908	C₂²	1	2	yes	complemented
13	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	M₃₂×C₅₃	—	2	2	no	—
15	C₂×C₁₆×C₅₃	C₂²	1	2	yes	complemented
16	C₂³×D₅₃	C₈	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×Dic₅₃	C₈	1	1	yes	characteristic
19	C₂×C₄×D₅₃	—	2	2	no	—
20	C₂×C₄×D₅₃	—	2	2	no	—
21	C₂×C₄×D₅₃	C₈	1	2	yes	—
22	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₅₃	D₄	1	2	yes	—
25	C₂×C₈×C₅₃	D₄	1	2	yes	—
26	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
27	C₂×#88322	—	2	2	no	—
28	C₂×#88322	C₂×C₄	1	2	yes	—
29	C₈×D₅₃	—	2	4	no	—
30	C₈×D₅₃	—	2	2	no	—
31	C₈×D₅₃	—	2	2	no	—
32	C₁₆×C₅₃	—	2	2	no	—
33	#11719908	—	2	2	no	—
34	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂²×D₅₃	—	2	2	no	—
36	C₂²×D₅₃	—	2	4	no	—
37	C₂²×D₅₃	C₂×C₈	1	2	yes	—
38	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
40	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
41	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×Dic₅₃	—	2	2	no	—
43	C₂×Dic₅₃	C₂×C₈	1	2	yes	—
44	C₄×D₅₃	—	2	4	no	—
45	C₄×D₅₃	—	2	2	no	—
46	C₄×D₅₃	—	2	2	no	—
47	C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
48	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
50	#88322	—	2	2	no	—
51	C₂²×C₅₃	—	2	2	no	—
52	C₂²×C₅₃	C₂²⋊C₈	1	2	yes	—
53	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
54	C₂×D₅₃	—	2	8	no	—
55	C₂×D₅₃	—	2	2	no	—
56	C₂×D₅₃	—	2	2	no	—
57	C₂×D₅₃	M₃₂	1	4	yes	complemented
58	C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
59	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
61	Dic₅₃	—	2	2	no	—
62	#2112	—	53	1	no	maximal, Sylow
63	C₂×C₅₃	—	2	2	no	—
64	C₂×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
65	C₂×C₅₃	C₂×M₃₂	1	2	yes	—
66	D₅₃	—	2	4	no	—
67	C₂³×C₈	—	53	1	no	—
68	C₂×M₃₂	—	53	1	no	—
69	C₂×M₃₂	D₅₃	1	1	yes	characteristic, complemented
70	C₂²⋊C₁₆	—	53	4	no	—
71	C₅₃	#2112	1	1	yes	characteristic, complemented, Sylow
72	C₂³×C₄	—	53	1	no	—
73	C₂²×C₈	—	53	1	no	—
74	C₂²×C₈	—	53	2	no	—
75	C₂²×C₈	—	53	2	no	—
76	C₂²×C₈	—	106	2	no	—
77	C₂²×C₈	—	106	2	no	—
78	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
79	M₃₂	—	2	2	no	—
80	M₃₂	—	106	2	no	—
81	C₂×C₁₆	—	53	2	no	—
82	C₂×C₁₆	C₂×D₅₃	1	2	yes	complemented
83	C₂⁴	—	53	1	no	—
84	C₂²×C₄	—	53	1	no	—
85	C₂²×C₄	—	53	2	no	—
86	C₂²×C₄	—	53	2	no	—
87	C₂²×C₄	—	106	2	no	—
88	C₂²×C₄	—	106	2	no	—
89	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
90	C₂×C₈	—	53	2	no	—
91	C₂×C₈	—	106	2	no	—
92	C₂×C₈	—	106	2	no	—
93	C₂×C₈	—	106	2	no	—
94	C₂×C₈	—	106	4	no	—
95	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
96	C₂×C₈	D₂₁₂	1	2	yes	—
97	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
98	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
99	C₁₆	—	2	2	no	—
100	C₁₆	—	106	2	no	—
101	C₂³	—	53	2	no	—
102	C₂³	—	106	4	no	—
103	C₂³	—	106	2	no	—
104	C₂³	C₈×D₅₃	1	1	yes	characteristic
105	C₂×C₄	—	53	2	no	—
106	C₂×C₄	—	106	2	no	—
107	C₂×C₄	—	106	4	no	—
108	C₂×C₄	—	106	2	no	—
109	C₂×C₄	—	106	2	no	—
110	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
111	C₂×C₄	#11719912	1	2	yes	—
112	C₂×C₄	#11719921	1	2	yes	—
113	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
114	C₈	—	106	2	no	—
115	C₈	#11719921	1	2	yes	—
116	C₈	C₂×D₂₁₂	1	1	yes	characteristic
117	C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
118	C₂²	—	2	2	no	—
119	C₂²	—	53	4	no	—
120	C₂²	—	106	8	no	—
121	C₂²	—	106	2	no	—
122	C₂²	—	106	2	no	—
123	C₂²	#13133129	1	2	yes	—
124	C₂²	C₂×C₈×D₅₃	1	1	yes	characteristic
125	C₄	—	106	2	no	—
126	C₄	#13133129	1	2	yes	—
127	C₄	C₂×#11719921	1	1	yes	characteristic
128	C₄	C₂×#11719912	1	1	yes	characteristic
129	C₂	—	2	2	no	—
130	C₂	—	106	4	no	—
131	C₂	C₂×#13133129	1	1	yes	characteristic
132	C₂	M₃₂×D₅₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×D₅₃

C₂

yes

maximal, characteristic

C₂×#13133157

C₂

yes

maximal, characteristic, complemented

#16810402

C₂

yes

maximal, complemented

C₂×M₃₂×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₅₃

C₄

yes

characteristic

C₂²×#88322

C₄

yes

characteristic

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

C₄

yes

—

C₂×C₈×D₅₃

C₂²

yes

—

#13133157

—

C₂×#11719908

C₂²

yes

complemented

C₂²×C₈×C₅₃

C₂²

yes

characteristic

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

C₂²

yes

complemented

C₂³×D₅₃

C₈

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₈

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₈

yes

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×#88322

—

C₂×#88322

C₂×C₄

yes

—

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

C₂×C₈

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×C₈

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

C₂²⋊C₄

yes

—

C₈×C₅₃

C₂×D₄

yes

characteristic

C₈×C₅₃

C₂×D₄

yes

characteristic

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²⋊C₈

yes

—

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

M₃₂

yes

complemented

C₄×C₅₃

C₂²⋊C₈

yes

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

C₂×M₁₆

yes

characteristic

Dic₅₃

—

#2112

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₅₃

C₂×M₃₂

yes

—

D₅₃

—

C₂³×C₈

—

C₂×M₃₂

—

C₂×M₃₂

D₅₃

yes

characteristic, complemented

C₂²⋊C₁₆

—

C₅₃

#2112

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

106

—

C₂²×C₈

—

106

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

M₃₂

—

M₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₃

yes

complemented

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

D₂₁₂

yes

—

C₂×C₈

C₂²⋊D₅₃

yes

—

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₁₆

—

100

C₁₆

—

106

—

101

C₂³

—

102

C₂³

—

106

—

103

C₂³

—

106

—

104

C₂³

C₈×D₅₃

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

106

—

107

C₂×C₄

—

106

—

108

C₂×C₄

—

106

—

109

C₂×C₄

—

106

—

110

C₂×C₄

C₈×D₅₃

yes

characteristic

111

C₂×C₄

#11719912

yes

—

112

C₂×C₄

#11719921

yes

—

113

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

114

C₈

—

106

—

115

C₈

#11719921

yes

—

116

C₈

C₂×D₂₁₂

yes

characteristic

117

C₈

C₂×C₂²⋊D₅₃

yes

characteristic

118

C₂²

—

119

C₂²

—

120

C₂²

—

106

—

121

C₂²

—

106

—

122

C₂²

—

106

—

123

C₂²

#13133129

yes

—

124

C₂²

C₂×C₈×D₅₃

yes

characteristic

125

C₄

—

106

—

126

C₄

#13133129

yes

—

127

C₄

C₂×#11719921

yes

characteristic

128

C₄

C₂×#11719912

yes

characteristic

129

C₂

—

130

C₂

—

106

—

131

C₂

C₂×#13133129

yes

characteristic

132

C₂

M₃₂×D₅₃

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₈
Fitting	C₂×M₃₂×C₅₃	C₂
hypercenter	C₂×M₃₂	D₅₃
hypocenter	C₅₃	#2112
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₅₃

commutator, socle

C₂²×C₅₃

C₂²×C₈

Fitting

C₂×M₃₂×C₅₃

C₂

hypercenter

C₂×M₃₂

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
2. C₂×#13133157	C₂	semidirect
3. #16810402	C₂	semidirect
4. C₂×M₃₂×C₅₃	C₂	semidirect
12. C₂×#11719908	C₂²	semidirect
15. C₂×C₁₆×C₅₃	C₂²	semidirect
57. C₂×D₅₃	M₃₂	semidirect
69. C₂×M₃₂	D₅₃	semidirect
71. C₅₃	#2112	semidirect
82. C₂×C₁₆	C₂×D₅₃	semidirect

H ≅ G/N

type

2. C₂×#13133157

C₂

semidirect

3. #16810402

C₂

semidirect

4. C₂×M₃₂×C₅₃

C₂

semidirect

12. C₂×#11719908

C₂²

semidirect

15. C₂×C₁₆×C₅₃

C₂²

semidirect

57. C₂×D₅₃

M₃₂

semidirect

69. C₂×M₃₂

D₅₃

semidirect

71. C₅₃

#2112

semidirect

82. C₂×C₁₆

C₂×D₅₃

semidirect

Subgroups — #20276325

Conjugacy classes of subgroups

Complements