Group #20276509 — subgroups

Subgroups — #20276509

Order 6784.

140 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 81 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×#13133153	C₂	1	1	yes	maximal, characteristic
3	#16810452	C₂	1	4	yes	maximal, complemented
4	C₂×#108×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	—	2	1	no	—
6	#13133157	C₄	1	2	yes	—
7	#13133157	C₄	1	2	yes	—
8	C₂×#11719908	C₂²	1	2	yes	—
9	#13133153	—	2	1	no	—
10	#13133153	C₂²	1	2	yes	—
11	C₂×M₁₆×C₅₃	—	2	1	no	—
12	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	#108×C₅₃	—	2	1	no	—
14	#108×C₅₃	C₂²	1	2	yes	—
15	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
16	M₁₆×C₅₃	—	2	2	no	—
17	M₁₆×C₅₃	—	4	1	no	—
18	C₂×C₈×C₅₃	—	4	1	no	—
19	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
22	C₂×C₈×C₅₃	Q₈	1	1	yes	characteristic
23	C₂×C₈×C₅₃	Q₈	1	1	yes	characteristic
24	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
25	#11719917	—	2	2	no	—
26	#11719917	—	4	1	no	—
27	C₂×#88322	—	4	1	no	—
28	#11719908	C₂×C₄	1	4	yes	—
29	C₂³×C₅₃	Q₁₆	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
33	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
34	C₂×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₈×C₅₃	—	4	2	no	—
37	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	#88322	—	4	2	no	—
42	C₂²×C₅₃	—	4	1	no	—
43	C₂²×C₅₃	#107	1	1	yes	characteristic
44	C₂²×C₅₃	#107	1	1	yes	characteristic
45	C₂²×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
46	C₄×C₅₃	#107	1	1	yes	characteristic
47	C₄×C₅₃	#107	1	1	yes	characteristic
48	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
50	#2161	—	53	1	no	maximal, Sylow
51	C₂×C₅₃	—	2	1	no	—
52	C₂×C₅₃	—	2	1	no	—
53	C₂×C₅₃	—	2	1	no	—
54	C₂×C₅₃	C₂×#107	1	1	yes	characteristic
55	C₂×M₃₂	—	53	1	no	—
56	C₂×#108	—	53	1	no	—
57	C₂×#108	D₅₃	1	1	yes	characteristic
58	#368	—	53	4	no	—
59	C₅₃	#2161	1	1	yes	characteristic, complemented, Sylow
60	C₂×M₁₆	—	2	1	no	—
61	C₂×M₁₆	—	106	1	no	—
62	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
63	M₃₂	—	53	2	no	—
64	M₃₂	—	53	2	no	—
65	C₂×C₁₆	—	53	2	no	—
66	#108	—	2	1	no	—
67	#108	—	53	2	no	—
68	#108	—	106	1	no	—
69	#108	C₂×D₅₃	1	2	yes	—
70	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	4	1	no	—
73	M₁₆	—	106	2	no	—
74	M₁₆	—	212	1	no	—
75	C₂×C₈	—	4	1	no	—
76	C₂×C₈	—	212	1	no	—
77	C₂×C₈	Dic₁₀₆	1	1	yes	characteristic
78	C₂×C₈	Dic₁₀₆	1	1	yes	characteristic
79	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
80	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
81	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
82	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
83	C₁₆	—	53	4	no	—
84	C₂³	#11719925	1	1	yes	characteristic
85	C₂×C₄	#11719919	1	1	yes	characteristic
86	C₂×C₄	#11719919	1	1	yes	characteristic
87	C₂×C₄	#11719922	1	1	yes	characteristic
88	C₂×C₄	#11719922	1	1	yes	characteristic
89	C₂×C₄	#11719925	1	1	yes	characteristic
90	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
91	C₈	—	4	2	no	—
92	C₈	—	212	2	no	—
93	C₈	#11719919	1	1	yes	characteristic
94	C₈	#11719919	1	1	yes	characteristic
95	C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
96	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
97	C₂²	—	4	1	no	—
98	C₂²	#13133150	1	1	yes	characteristic
99	C₂²	#13133150	1	1	yes	characteristic
100	C₂²	C₂×#11719925	1	1	yes	characteristic
101	C₄	#13133150	1	1	yes	characteristic
102	C₄	#13133150	1	1	yes	characteristic
103	C₄	C₂×#11719919	1	1	yes	characteristic
104	C₄	C₂×#11719922	1	1	yes	characteristic
105	C₂	—	2	1	no	—
106	C₂	—	2	1	no	—
107	C₂	—	2	1	no	—
108	C₂	C₂×#13133150	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133157

C₂

yes

maximal, characteristic

C₂×#13133153

C₂

yes

maximal, characteristic

#16810452

C₂

yes

maximal, complemented

C₂×#108×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

—

#13133157

C₄

yes

—

#13133157

C₄

yes

—

C₂×#11719908

C₂²

yes

—

#13133153

—

#13133153

C₂²

yes

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#108×C₅₃

—

#108×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

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

—

—

—

C₂×C₄

yes

—

C₂³×C₅₃

Q₁₆

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

Q₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₂²×C₄

yes

characteristic

C₈×C₅₃

C₂×Q₈

yes

characteristic

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

C₂×Q₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₂×D₈

yes

characteristic

#2161

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#107

yes

characteristic

C₂×M₃₂

—

C₂×#108

—

C₂×#108

D₅₃

yes

characteristic

#368

—

C₅₃

#2161

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

106

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#108

—

#108

—

#108

—

106

—

#108

C₂×D₅₃

yes

—

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

106

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

Dic₁₀₆

yes

characteristic

C₂×C₈

Dic₁₀₆

yes

characteristic

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₁₆

—

C₂³

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

C₈

—

C₈

—

212

—

C₈

yes

C₈

yes

C₈

C₂×Dic₁₀₆

yes

characteristic

C₈

C₂×C₄×D₅₃

yes

characteristic

C₂²

—

C₂²

yes

C₂²

yes

100

C₂²

C₂×#11719925

yes

characteristic

101

C₄

#13133150

yes

characteristic

102

C₄

#13133150

yes

characteristic

103

C₄

C₂×#11719919

yes

characteristic

104

C₄

C₂×#11719922

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

—

108

C₂

C₂×#13133150

yes

characteristic

	subgroup	G/H
center	C₄	C₂×#11719922
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₂×#108×C₅₃	C₂
socle	C₂×C₅₃	C₂×#107
hypercenter	C₂×#108	D₅₃
hypocenter	C₅₃	#2161
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#108

subgroup

G/H

center

C₄

C₂×#11719922

commutator

C₈×C₅₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₂×#108×C₅₃

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#108

N	H ≅ G/N	type
3. #16810452	C₂	semidirect
59. C₅₃	#2161	semidirect

H ≅ G/N

type

3. #16810452

C₂

semidirect

59. C₅₃

#2161

semidirect

Subgroups — #20276509

Conjugacy classes of subgroups

Complements