Group #20291683 — subgroups

Subgroups — #20291683

Order 6784.

360 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 198 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	#16811682	C₂	1	6	yes	maximal, complemented
2	#16811681	C₂	1	2	yes	maximal, complemented
3	C₂×#13133292	C₂	1	3	yes	maximal, complemented
4	D₅₃⋊M₃₂	C₂	1	3	yes	maximal, complemented
5	#131×D₅₃	C₂	1	1	yes	maximal, characteristic
6	#13133292	C₂²	1	3	yes	—
7	#13133292	C₂²	1	1	yes	characteristic
8	#13133292	C₂²	1	6	yes	complemented
9	#13133291	C₂²	1	6	yes	complemented
10	D₅₃⋊C₁₆	C₂²	1	6	yes	complemented
11	C₂×#11719910	C₂²	1	6	yes	complemented
12	#41×D₅₃	C₄	1	1	yes	characteristic
13	#13133225	C₂²	1	3	yes	—
14	#13133224	C₂²	1	1	yes	characteristic
15	#13133223	C₄	1	3	yes	—
16	M₁₆×D₅₃	C₄	1	3	yes	—
17	C₂×C₈×D₅₃	C₂²	1	3	yes	—
18	#131×C₅₃	C₄	1	1	yes	characteristic
19	#41×C₅₃	C₂×C₄	1	1	yes	characteristic
20	#11719949	C₈	1	1	yes	characteristic
21	Q₈×D₅₃	C₈	1	1	yes	characteristic
22	#11719947	C₈	1	3	yes	—
23	D₄×D₅₃	C₈	1	3	yes	—
24	#11719945	C₂×C₄	1	3	yes	—
25	C₂×C₄×D₅₃	C₂×C₄	1	3	yes	—
26	M₁₆×C₅₃	C₂×C₄	1	3	yes	—
27	C₂×C₈×C₅₃	C₂×C₄	1	3	yes	—
28	#11719917	C₂×C₄	1	3	yes	—
29	C₂×#88322	C₂³	1	3	yes	—
30	#11719912	C₂×C₄	1	6	yes	—
31	C₈×D₅₃	C₂×C₄	1	3	yes	—
32	C₈×D₅₃	C₂×C₄	1	3	yes	—
33	C₈×D₅₃	C₂³	1	1	yes	characteristic
34	C₈×D₅₃	C₂³	1	3	yes	—
35	#11719910	C₂³	1	2	yes	—
36	#11719910	C₂³	1	6	yes	—
37	C₂²×D₅₃	C₂×C₈	1	3	yes	—
38	Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
39	D₄×C₅₃	C₂×C₈	1	3	yes	—
40	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
41	C₂²⋊D₅₃	C₂×C₈	1	6	yes	—
42	C₂×Dic₅₃	C₂×C₈	1	3	yes	—
43	D₂₁₂	C₂×C₈	1	3	yes	—
44	C₄×D₅₃	C₂×C₈	1	3	yes	—
45	C₄×D₅₃	C₂×C₈	1	3	yes	—
46	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₄×D₅₃	C₂²×C₄	1	3	yes	—
48	Dic₁₀₆	C₂×C₈	1	3	yes	—
49	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₈×C₅₃	C₂²×C₄	1	3	yes	—
51	#88322	C₂²×C₄	1	3	yes	—
52	#88322	C₂⁴	1	1	yes	characteristic
53	C₂²×C₅₃	C₂²×C₈	1	3	yes	—
54	C₂×D₅₃	—	4	3	no	—
55	C₂×D₅₃	C₂²×C₈	1	1	yes	characteristic
56	C₂×D₅₃	C₂²×C₈	1	3	yes	—
57	C₄×C₅₃	C₂²×C₈	1	3	yes	—
58	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
59	Dic₅₃	C₂²×C₈	1	3	yes	—
60	Dic₅₃	C₂²×C₈	1	1	yes	characteristic
61	#3407	—	53	1	no	maximal, Sylow
62	C₂×C₅₃	—	2	3	no	—
63	C₂×C₅₃	C₂³×C₈	1	1	yes	characteristic
64	D₅₃	—	2	1	no	—
65	D₅₃	—	2	3	no	—
66	C₂×#131	—	53	1	no	—
67	#504	—	53	2	no	—
68	#504	—	53	6	no	—
69	C₂×M₃₂	—	53	3	no	—
70	C₂×M₃₂	—	53	3	no	—
71	C₅₃	#3407	1	1	yes	characteristic, complemented, Sylow
72	C₂×#41	—	53	1	no	—
73	#131	—	53	1	no	—
74	#131	—	53	3	no	—
75	#131	—	53	3	no	—
76	#131	#6157	1	1	yes	characteristic
77	C₂×M₁₆	—	53	3	no	—
78	C₂²×C₈	—	53	3	no	—
79	M₃₂	—	53	6	no	—
80	M₃₂	—	53	3	no	—
81	M₃₂	—	53	1	no	—
82	M₃₂	—	53	6	no	—
83	C₂×C₁₆	—	53	6	no	—
84	C₂×C₁₆	—	53	6	no	—
85	#41	—	53	1	no	—
86	#41	—	53	3	no	—
87	#41	—	53	3	no	—
88	#41	C₂×#6157	1	1	yes	characteristic
89	C₂×Q₈	—	53	1	no	—
90	C₂×D₄	—	53	3	no	—
91	C₂²×C₄	—	53	3	no	—
92	M₁₆	—	53	3	no	—
93	M₁₆	—	53	6	no	—
94	M₁₆	C₂×#6157	1	3	yes	—
95	C₂×C₈	—	53	3	no	—
96	C₂×C₈	—	53	3	no	—
97	C₂×C₈	—	53	1	no	—
98	C₂×C₈	—	53	3	no	—
99	C₂×C₈	—	53	3	no	—
100	C₂×C₈	C₂×#6157	1	3	yes	—
101	C₁₆	—	53	2	no	—
102	C₁₆	—	53	6	no	—
103	C₂³	—	53	3	no	—
104	Q₈	—	53	3	no	—
105	Q₈	D₅₃⋊C₈	1	1	yes	characteristic
106	D₄	—	53	6	no	—
107	D₄	—	53	3	no	—
108	D₄	D₅₃⋊C₈	1	3	yes	—
109	C₂×C₄	—	53	3	no	—
110	C₂×C₄	—	53	3	no	—
111	C₂×C₄	—	53	3	no	—
112	C₂×C₄	—	53	1	no	—
113	C₂×C₄	—	53	3	no	—
114	C₂×C₄	C₂²×#6157	1	3	yes	—
115	C₈	—	53	3	no	—
116	C₈	—	53	1	no	—
117	C₈	C₂²×#6157	1	1	yes	characteristic
118	C₈	C₂²×#6157	1	3	yes	—
119	C₂²	—	53	1	no	—
120	C₂²	—	53	3	no	—
121	C₂²	—	212	3	no	—
122	C₂²	C₂×D₅₃⋊C₈	1	3	yes	—
123	C₄	—	53	1	no	—
124	C₄	—	53	3	no	—
125	C₄	C₂×D₅₃⋊C₈	1	3	yes	—
126	C₄	C₂³×#6157	1	1	yes	characteristic
127	C₂	—	2	3	no	—
128	C₂	—	106	1	no	—
129	C₂	—	106	3	no	—
130	C₂	C₂²×D₅₃⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811682

C₂

yes

maximal, complemented

#16811681

C₂

yes

maximal, complemented

C₂×#13133292

C₂

yes

maximal, complemented

D₅₃⋊M₃₂

C₂

yes

maximal, complemented

#131×D₅₃

C₂

yes

maximal, characteristic

#13133292

C₂²

yes

—

#13133292

C₂²

yes

characteristic

#13133292

C₂²

yes

complemented

#13133291

C₂²

yes

complemented

D₅₃⋊C₁₆

C₂²

yes

complemented

C₂×#11719910

C₂²

yes

complemented

#41×D₅₃

C₄

yes

characteristic

#13133225

C₂²

yes

—

#13133224

C₂²

yes

characteristic

#13133223

C₄

yes

—

M₁₆×D₅₃

C₄

yes

—

C₂×C₈×D₅₃

C₂²

yes

—

#131×C₅₃

C₄

yes

characteristic

#41×C₅₃

C₂×C₄

yes

characteristic

#11719949

C₈

yes

characteristic

Q₈×D₅₃

C₈

yes

characteristic

#11719947

C₈

yes

—

D₄×D₅₃

C₈

yes

—

#11719945

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

M₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

#11719917

C₂×C₄

yes

—

C₂×#88322

C₂³

yes

—

#11719912

C₂×C₄

yes

—

C₈×D₅₃

C₂×C₄

yes

—

C₈×D₅₃

C₂×C₄

yes

—

C₈×D₅₃

C₂³

yes

characteristic

C₈×D₅₃

C₂³

yes

—

#11719910

C₂³

yes

—

#11719910

C₂³

yes

—

C₂²×D₅₃

C₂×C₈

yes

—

Q₈×C₅₃

C₂×C₈

yes

characteristic

D₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂²⋊D₅₃

C₂×C₈

yes

—

C₂×Dic₅₃

C₂×C₈

yes

—

D₂₁₂

C₂×C₈

yes

—

C₄×D₅₃

C₂×C₈

yes

—

C₄×D₅₃

C₂×C₈

yes

—

C₄×D₅₃

C₂²×C₄

yes

characteristic

C₄×D₅₃

C₂²×C₄

yes

—

Dic₁₀₆

C₂×C₈

yes

—

C₈×C₅₃

C₂²×C₄

yes

characteristic

C₈×C₅₃

C₂²×C₄

yes

—

#88322

C₂²×C₄

yes

—

#88322

C₂⁴

yes

characteristic

C₂²×C₅₃

C₂²×C₈

yes

—

C₂×D₅₃

—

C₂×D₅₃

C₂²×C₈

yes

characteristic

C₂×D₅₃

C₂²×C₈

yes

—

C₄×C₅₃

C₂²×C₈

yes

—

C₄×C₅₃

C₂³×C₄

yes

characteristic

Dic₅₃

C₂²×C₈

yes

—

Dic₅₃

C₂²×C₈

yes

characteristic

#3407

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂³×C₈

yes

characteristic

D₅₃

—

D₅₃

—

C₂×#131

—

#504

—

#504

—

C₂×M₃₂

—

C₂×M₃₂

—

C₅₃

#3407

yes

characteristic, complemented, Sylow

C₂×#41

—

—

—

—

yes

C₂×M₁₆

—

C₂²×C₈

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

yes

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

M₁₆

C₂×#6157

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

C₂×#6157

yes

—

101

C₁₆

—

102

C₁₆

—

103

C₂³

—

104

Q₈

—

105

Q₈

D₅₃⋊C₈

yes

characteristic

106

D₄

—

107

D₄

—

108

D₄

D₅₃⋊C₈

yes

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂²×#6157

yes

—

115

C₈

—

116

C₈

—

117

C₈

C₂²×#6157

yes

characteristic

118

C₈

C₂²×#6157

yes

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

212

—

122

C₂²

C₂×D₅₃⋊C₈

yes

—

123

C₄

—

124

C₄

—

125

C₄

C₂×D₅₃⋊C₈

yes

—

126

C₄

C₂³×#6157

yes

characteristic

127

C₂

—

128

C₂

—

106

—

129

C₂

—

106

—

130

C₂

C₂²×D₅₃⋊C₈

yes

characteristic

	subgroup	G/H
center, Frattini	C₄	C₂³×#6157
commutator, socle	C₂×C₅₃	C₂³×C₈
Fitting	#131×C₅₃	C₄
hypercenter	#131	#6157
hypocenter	C₅₃	#3407
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ #131

subgroup

G/H

center, Frattini

C₄

C₂³×#6157

commutator, socle

C₂×C₅₃

C₂³×C₈

Fitting

#131×C₅₃

C₄

C₅₃

G ▷ C₂×C₅₃ ▷ C₅₃

C₁ ◁ C₄ ◁ #131

N	H ≅ G/N	type
1. #16811682	C₂	semidirect
2. #16811681	C₂	semidirect
3. C₂×#13133292	C₂	semidirect
4. D₅₃⋊M₃₂	C₂	semidirect
8. #13133292	C₂²	semidirect
9. #13133291	C₂²	semidirect
10. D₅₃⋊C₁₆	C₂²	semidirect
11. C₂×#11719910	C₂²	semidirect
71. C₅₃	#3407	semidirect

H ≅ G/N

type

1. #16811682

C₂

semidirect

2. #16811681

C₂

semidirect

3. C₂×#13133292

C₂

semidirect

4. D₅₃⋊M₃₂

C₂

semidirect

8. #13133292

C₂²

semidirect

9. #13133291

C₂²

semidirect

10. D₅₃⋊C₁₆

C₂²

semidirect

11. C₂×#11719910

C₂²

semidirect

71. C₅₃

#3407

semidirect

Subgroups — #20291683

Conjugacy classes of subgroups

Complements