Group #20289755 — subgroups

Subgroups — #20289755

Order 6784.

140 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 56 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×#13133292	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133271	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133156	C₂	1	1	yes	maximal, characteristic
4	C₂×D₅₃⋊C₈	C₄	1	1	yes	characteristic
5	C₂×C₄×#6157	C₄	1	1	yes	characteristic
6	#13133292	—	2	1	no	—
7	#13133292	—	2	1	no	—
8	C₂×#11719910	C₄	1	2	yes	complemented
9	#13133271	—	2	1	no	—
10	#13133271	—	2	1	no	—
11	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	#13133156	—	2	1	no	—
13	#13133156	—	2	1	no	—
14	C₂×#11719908	C₄	1	1	yes	characteristic, complemented
15	C₂×C₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
16	C₂²×#6157	—	2	1	no	—
17	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×#88324	—	2	1	no	—
19	C₄×#6157	—	2	1	no	—
20	C₄×#6157	C₈	1	2	yes	—
21	D₅₃⋊C₈	—	2	1	no	—
22	D₅₃⋊C₈	C₈	1	2	yes	—
23	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×#88322	C₂×C₄	1	1	yes	characteristic
25	C₈×D₅₃	D₄	1	1	yes	characteristic
26	C₈×D₅₃	D₄	1	1	yes	characteristic
27	C₈×D₅₃	D₄	1	1	yes	characteristic
28	C₈×D₅₃	Q₈	1	1	yes	characteristic
29	#11719910	—	2	2	no	—
30	C₁₆×C₅₃	—	2	1	no	—
31	#11719908	—	2	1	no	—
32	C₂²×D₅₃	M₁₆	1	1	yes	characteristic
33	C₂×#6157	—	2	2	no	—
34	C₂×#6157	—	4	1	no	—
35	C₂×#6157	—	4	1	no	—
36	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
37	C₂×Dic₅₃	M₁₆	1	1	yes	characteristic
38	C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
41	C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
42	#88324	—	2	2	no	—
43	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	#88322	C₂²⋊C₄	1	1	yes	characteristic
46	#88322	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₅₃	#97	1	1	yes	characteristic
48	C₂×D₅₃	—	2	1	no	—
49	C₂×D₅₃	—	2	1	no	—
50	C₂×D₅₃	C₂²⋊C₈	1	1	yes	characteristic
51	C₂×D₅₃	C₄⋊C₈	1	1	yes	characteristic
52	#6157	—	4	2	no	—
53	C₄×C₅₃	#95	1	1	yes	characteristic
54	C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
55	Dic₅₃	C₂²⋊C₈	1	1	yes	characteristic
56	Dic₅₃	C₄⋊C₈	1	1	yes	characteristic
57	#1375	—	53	1	no	maximal, Sylow
58	C₂×C₅₃	#336	1	1	yes	characteristic
59	C₂×C₅₃	#347	1	1	yes	characteristic
60	C₂×C₅₃	#347	1	1	yes	characteristic
61	D₅₃	—	2	1	no	—
62	D₅₃	—	2	1	no	—
63	C₂×M₃₂	—	53	1	no	—
64	C₂×M₃₂	—	53	1	no	—
65	C₂×#97	—	53	1	no	—
66	C₅₃	#1375	1	1	yes	characteristic, complemented, Sylow
67	C₂²×C₈	—	53	1	no	—
68	C₂²×C₈	—	53	1	no	—
69	C₂×C₄²	—	53	1	no	—
70	M₃₂	—	106	1	no	—
71	M₃₂	—	106	1	no	—
72	M₃₂	—	106	1	no	—
73	M₃₂	—	106	1	no	—
74	C₂×C₁₆	—	53	2	no	—
75	C₂×C₁₆	—	53	1	no	—
76	C₂×C₁₆	#6157	1	1	yes	characteristic, complemented
77	#97	—	106	1	no	—
78	#97	—	106	1	no	—
79	C₂²×C₄	—	53	1	no	—
80	C₂²×C₄	—	106	1	no	—
81	C₂×C₈	—	53	1	no	—
82	C₂×C₈	—	53	1	no	—
83	C₂×C₈	—	53	1	no	—
84	C₂×C₈	—	53	1	no	—
85	C₂×C₈	—	53	1	no	—
86	C₂×C₈	—	53	2	no	—
87	C₂×C₈	—	106	1	no	—
88	C₂×C₈	—	106	1	no	—
89	C₂×C₈	C₂×#6157	1	1	yes	characteristic
90	C₄²	—	53	2	no	—
91	C₄²	—	106	1	no	—
92	C₁₆	—	2	1	no	—
93	C₁₆	—	106	2	no	—
94	C₁₆	—	106	1	no	—
95	C₂³	—	53	1	no	—
96	C₂×C₄	—	53	1	no	—
97	C₂×C₄	—	53	1	no	—
98	C₂×C₄	—	53	1	no	—
99	C₂×C₄	—	53	1	no	—
100	C₂×C₄	—	53	1	no	—
101	C₂×C₄	—	106	2	no	—
102	C₂×C₄	—	212	1	no	—
103	C₂×C₄	—	212	1	no	—
104	C₂×C₄	C₄×#6157	1	1	yes	characteristic
105	C₈	—	53	1	no	—
106	C₈	—	53	1	no	—
107	C₈	—	106	2	no	—
108	C₈	C₄⋊#6157	1	1	yes	characteristic
109	C₈	C₂²⋊#6157	1	1	yes	characteristic
110	C₂²	—	53	1	no	—
111	C₂²	—	53	1	no	—
112	C₂²	—	106	1	no	—
113	C₂²	—	106	1	no	—
114	C₂²	#13133271	1	1	yes	characteristic
115	C₄	—	53	1	no	—
116	C₄	—	53	1	no	—
117	C₄	—	212	2	no	—
118	C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
119	C₄	C₈×#6157	1	1	yes	characteristic
120	C₂	—	106	1	no	—
121	C₂	—	106	1	no	—
122	C₂	(C₂×#6157)⋊C₈	1	1	yes	characteristic
123	C₂	#16811513	1	1	yes	characteristic
124	C₂	#16811514	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133292

C₂

yes

maximal, characteristic

C₂×#13133271

C₂

yes

maximal, characteristic

C₂×#13133156

C₂

yes

maximal, characteristic

C₂×D₅₃⋊C₈

C₄

yes

characteristic

C₂×C₄×#6157

C₄

yes

—

—

C₄

yes

complemented

#13133271

—

#13133271

—

C₂×C₈×D₅₃

C₂²

yes

—

—

C₄

yes

characteristic, complemented

C₂×C₁₆×C₅₃

C₄

yes

characteristic, complemented

C₂²×#6157

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

—

C₈

yes

—

D₅₃⋊C₈

—

D₅₃⋊C₈

C₈

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₈×D₅₃

D₄

yes

characteristic

C₈×D₅₃

D₄

yes

characteristic

C₈×D₅₃

D₄

yes

characteristic

C₈×D₅₃

Q₈

yes

characteristic

#11719910

—

C₁₆×C₅₃

—

#11719908

—

C₂²×D₅₃

M₁₆

yes

—

—

—

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×Dic₅₃

M₁₆

yes

characteristic

C₄×D₅₃

C₂²⋊C₄

yes

characteristic

C₄×D₅₃

C₄⋊C₄

yes

characteristic

C₄×D₅₃

C₂×C₈

yes

characteristic

C₄×D₅₃

C₂×C₈

yes

characteristic

#88324

—

C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₈×C₅₃

C₄⋊C₄

yes

characteristic

#88322

C₂²⋊C₄

yes

characteristic

#88322

C₄⋊C₄

yes

characteristic

C₂²×C₅₃

#97

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₂²⋊C₈

yes

characteristic

C₂×D₅₃

C₄⋊C₈

yes

characteristic

#6157

—

C₄×C₅₃

#95

yes

characteristic

C₄×C₅₃

C₄×C₈

yes

characteristic

Dic₅₃

C₂²⋊C₈

yes

characteristic

Dic₅₃

C₄⋊C₈

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

D₅₃

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×#97

—

C₅₃

#1375

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

106

—

M₃₂

—

106

—

M₃₂

—

106

—

M₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

#6157

yes

characteristic, complemented

#97

—

106

—

#97

—

106

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₂×#6157

yes

characteristic

C₄²

—

C₄²

—

106

—

C₁₆

—

C₁₆

—

106

—

C₁₆

—

106

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

106

—

102

C₂×C₄

—

212

—

103

C₂×C₄

—

212

—

104

C₂×C₄

C₄×#6157

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

106

—

108

C₈

C₄⋊#6157

yes

characteristic

109

C₈

C₂²⋊#6157

yes

characteristic

110

C₂²

—

111

C₂²

—

112

C₂²

—

106

—

113

C₂²

—

106

—

114

C₂²

#13133271

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

212

—

118

C₄

(C₂×#6157)⋊C₄

yes

characteristic

119

C₄

C₈×#6157

yes

characteristic

120

C₂

—

106

—

121

C₂

—

106

—

122

C₂

(C₂×#6157)⋊C₈

yes

characteristic

123

C₂

#16811513

yes

characteristic

124

C₂

#16811514

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄×#6157
commutator	C₄×C₅₃	C₄×C₈
Frattini	C₂×C₈	C₂×#6157
Fitting	C₂×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	#97
hypercenter	C₂×C₁₆	#6157
hypocenter	C₅₃	#1375
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₁₆

subgroup

G/H

center

C₂×C₄

C₄×#6157

commutator

C₄×C₅₃

C₄×C₈

Frattini

C₂×C₈

C₂×#6157

Fitting

C₂×C₁₆×C₅₃

C₄

C₂²×C₅₃

C₂×C₁₆

C₅₃

G ▷ C₄×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ C₂×C₁₆

N	H ≅ G/N	type
8. C₂×#11719910	C₄	semidirect
14. C₂×#11719908	C₄	semidirect
15. C₂×C₁₆×C₅₃	C₄	semidirect
66. C₅₃	#1375	semidirect
76. C₂×C₁₆	#6157	semidirect

H ≅ G/N

type

8. C₂×#11719910

C₄

semidirect

14. C₂×#11719908

C₄

semidirect

15. C₂×C₁₆×C₅₃

C₄

semidirect

66. C₅₃

#1375

semidirect

76. C₂×C₁₆

#6157

semidirect

Subgroups — #20289755

Conjugacy classes of subgroups

Complements