Group #20276477 — subgroups

Subgroups — C₂×#16810443 #20276477

Order 6784.

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

normal subgroups only

This group has 168 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×#13133157	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#16810443	C₂	1	4	yes	maximal, direct product factor
4	C₂×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#88322	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₅₃	C₄	1	1	yes	characteristic
7	#13133157	—	2	1	no	—
8	#13133157	C₂²	1	2	yes	—
9	C₂×#11719908	—	2	1	no	—
10	C₈×Dic₅₃	C₄	1	2	yes	—
11	C₈×Dic₅₃	C₂²	1	2	yes	—
12	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	M₃₂×C₅₃	—	2	1	no	—
14	M₃₂×C₅₃	C₂²	1	2	yes	—
15	C₂×C₁₆×C₅₃	—	2	1	no	—
16	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×Dic₅₃	—	2	1	no	—
18	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	Q₈	1	1	yes	characteristic
22	C₂×C₈×C₅₃	Q₈	1	1	yes	characteristic
23	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
24	C₄×Dic₅₃	—	2	1	no	—
25	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
26	C₂×#88322	C₈	1	2	yes	—
27	C₂×#88322	C₈	1	2	yes	—
28	C₂×#88322	C₂×C₄	1	2	yes	—
29	C₁₆×C₅₃	—	2	2	no	—
30	#11719908	—	2	2	no	—
31	C₂³×C₅₃	M₁₆	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×Dic₅₃	—	2	4	no	—
39	C₂×Dic₅₃	—	2	2	no	—
40	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
44	#88322	C₂×C₈	1	4	yes	—
45	C₂²×C₅₃	—	2	2	no	—
46	C₂²×C₅₃	C₄⋊C₈	1	1	yes	characteristic
47	C₂²×C₅₃	C₄⋊C₈	1	1	yes	characteristic
48	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
49	C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
50	C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
51	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
53	Dic₅₃	—	2	4	no	—
54	C₂×#364	—	53	1	no	maximal, Sylow
55	C₂×C₅₃	—	2	1	no	—
56	C₂×C₅₃	—	2	1	no	—
57	C₂×C₅₃	#364	1	2	yes	complemented
58	C₂×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
59	C₂×M₃₂	—	53	1	no	—
60	C₂×M₃₂	D₅₃	1	1	yes	characteristic
61	C₂×C₄×C₈	—	53	1	no	—
62	#364	—	53	4	no	—
63	C₅₃	C₂×#364	1	1	yes	characteristic, complemented, Sylow
64	C₂²×C₈	—	53	1	no	—
65	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
66	C₂×C₄²	—	53	1	no	—
67	M₃₂	—	2	1	no	—
68	M₃₂	—	53	2	no	—
69	M₃₂	—	106	1	no	—
70	M₃₂	C₂×D₅₃	1	2	yes	—
71	C₂×C₁₆	—	2	1	no	—
72	C₂×C₁₆	—	106	1	no	—
73	C₄×C₈	—	53	2	no	—
74	C₄×C₈	—	53	2	no	—
75	C₂²×C₄	—	106	1	no	—
76	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
77	C₂×C₈	—	53	2	no	—
78	C₂×C₈	—	53	2	no	—
79	C₂×C₈	—	53	2	no	—
80	C₂×C₈	Dic₁₀₆	1	1	yes	characteristic
81	C₂×C₈	Dic₁₀₆	1	1	yes	characteristic
82	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
83	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
84	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
85	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
86	C₄²	—	53	2	no	—
87	C₄²	—	106	1	no	—
88	C₁₆	—	2	2	no	—
89	C₁₆	—	106	2	no	—
90	C₂³	#11719912	1	1	yes	characteristic
91	C₂×C₄	—	106	4	no	—
92	C₂×C₄	—	106	2	no	—
93	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
94	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
95	C₂×C₄	#11719912	1	1	yes	characteristic
96	C₂×C₄	#11719919	1	1	yes	characteristic
97	C₂×C₄	#11719919	1	1	yes	characteristic
98	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
99	C₈	—	53	4	no	—
100	C₈	#11719919	1	1	yes	characteristic
101	C₈	#11719919	1	1	yes	characteristic
102	C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
103	C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
104	C₂²	—	2	2	no	—
105	C₂²	#13133146	1	1	yes	characteristic
106	C₂²	#13133146	1	1	yes	characteristic
107	C₂²	C₂×#11719912	1	1	yes	characteristic
108	C₄	—	106	4	no	—
109	C₄	#13133146	1	1	yes	characteristic
110	C₄	#13133146	1	1	yes	characteristic
111	C₄	C₂×#11719919	1	1	yes	characteristic
112	C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
113	C₂	—	2	1	no	—
114	C₂	—	2	1	no	—
115	C₂	#16810443	1	2	yes	direct product factor
116	C₂	C₂×#13133146	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133157

C₂

yes

maximal, characteristic

C₂×C₈×Dic₅₃

C₂

yes

maximal, characteristic

#16810443

C₂

yes

maximal, direct product factor

C₂×M₃₂×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₄

yes

characteristic

#13133157

—

#13133157

C₂²

yes

—

C₂×#11719908

—

C₈×Dic₅₃

C₄

yes

—

C₈×Dic₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

M₃₂×C₅₃

—

M₃₂×C₅₃

C₂²

yes

—

C₂×C₁₆×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

Q₈

yes

characteristic

C₂×C₈×C₅₃

Q₈

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₂×#88322

C₈

yes

—

C₂×#88322

C₈

yes

—

C₂×#88322

C₂×C₄

yes

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

M₁₆

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

M₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₂×D₄

yes

characteristic

C₈×C₅₃

C₂×Q₈

yes

characteristic

#88322

C₂×C₈

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄⋊C₈

yes

characteristic

C₂²×C₅₃

C₄⋊C₈

yes

characteristic

C₂²×C₅₃

C₂×M₁₆

yes

characteristic

C₄×C₅₃

C₄⋊C₈

yes

characteristic

C₄×C₅₃

C₄⋊C₈

yes

characteristic

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₂²×C₈

yes

characteristic

Dic₅₃

—

C₂×#364

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#364

yes

complemented

C₂×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

D₅₃

yes

characteristic

C₂×C₄×C₈

—

#364

—

C₅₃

C₂×#364

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄²

—

M₃₂

—

M₃₂

—

M₃₂

—

106

—

M₃₂

C₂×D₅₃

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

106

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₀₆

yes

characteristic

C₂×C₈

Dic₁₀₆

yes

characteristic

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄²

—

C₄²

—

106

—

C₁₆

—

C₁₆

—

106

—

C₂³

#11719912

yes

characteristic

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

C₈×D₅₃

yes

characteristic

C₂×C₄

C₈×D₅₃

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₈

—

100

C₈

#11719919

yes

characteristic

101

C₈

#11719919

yes

characteristic

102

C₈

C₂×Dic₁₀₆

yes

characteristic

103

C₈

C₂×C₂²⋊D₅₃

yes

characteristic

104

C₂²

—

105

C₂²

#13133146

yes

characteristic

106

C₂²

#13133146

yes

characteristic

107

C₂²

C₂×#11719912

yes

characteristic

108

C₄

—

106

—

109

C₄

#13133146

yes

characteristic

110

C₄

#13133146

yes

characteristic

111

C₄

C₂×#11719919

yes

characteristic

112

C₄

C₂×C₈×D₅₃

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

#16810443

yes

direct product factor

116

C₂

C₂×#13133146

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₂×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₄⋊C₈
hypercenter	C₂×M₃₂	D₅₃
hypocenter	C₅₃	C₂×#364
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₂×C₈

C₂²⋊D₅₃

commutator

C₄×C₅₃

C₂²×C₈

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₂×M₃₂×C₅₃

C₂

socle

C₂²×C₅₃

C₄⋊C₈

hypercenter

C₂×M₃₂

D₅₃

hypocenter

C₅₃

C₂×#364

LCS

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. #16810443	C₂	direct (N × H)
57. C₂×C₅₃	#364	semidirect
63. C₅₃	C₂×#364	semidirect
115. C₂	#16810443	direct (N × H)

H ≅ G/N

type

3. #16810443

C₂

direct (N × H)

57. C₂×C₅₃

#364

semidirect

63. C₅₃

C₂×#364

semidirect

115. C₂

#16810443

direct (N × H)

Subgroups — C₂×#16810443 #20276477

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810443 #20276477

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810443 #20276477