Group #20276488 — subgroups

Subgroups — #20276488

Order 6784.

144 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 89 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×#13133153	C₂	1	1	yes	maximal, characteristic
2	#16810661	C₂	1	1	yes	maximal, characteristic
3	#16810446	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	—	2	1	no	—
6	#13133176	—	2	1	no	—
7	#13133153	—	2	1	no	—
8	#13133153	C₂²	1	2	yes	—
9	#13133151	—	2	1	no	—
10	#13133149	C₂²	1	2	yes	—
11	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	M₃₂×C₅₃	C₄	1	2	yes	complemented
13	M₃₂×C₅₃	C₂²	1	2	yes	—
14	C₂×C₁₆×C₅₃	C₄	1	2	yes	complemented
15	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
16	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
19	C₂×C₈×C₅₃	Q₈	1	1	yes	characteristic
20	C₂×C₈×C₅₃	Q₈	1	1	yes	characteristic
21	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
22	C₂²⋊Dic₅₃	—	4	1	no	—
23	C₄⋊Dic₅₃	—	2	2	no	—
24	C₄×Dic₅₃	—	4	1	no	—
25	#11719917	—	2	2	no	—
26	#11719917	—	4	1	no	—
27	C₂×#88322	—	4	1	no	—
28	C₁₆×C₅₃	C₂×C₄	1	4	yes	—
29	C₂³×C₅₃	SD₁₆	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	SD₁₆	1	1	yes	characteristic
33	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
34	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂×Dic₅₃	—	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	#106	1	1	yes	characteristic
44	C₂²×C₅₃	#106	1	1	yes	characteristic
45	C₂²×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
46	C₄×C₅₃	#106	1	1	yes	characteristic
47	C₄×C₅₃	#106	1	1	yes	characteristic
48	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
50	Dic₅₃	—	8	2	no	—
51	#2155	—	53	1	no	maximal, Sylow
52	C₂×C₅₃	—	2	1	no	—
53	C₂×C₅₃	—	2	1	no	—
54	C₂×C₅₃	—	2	1	no	—
55	C₂×C₅₃	C₂×#106	1	1	yes	characteristic
56	C₂×M₃₂	D₅₃	1	1	yes	characteristic
57	C₂×#108	—	53	1	no	—
58	#427	—	53	1	no	—
59	#365	—	53	4	no	—
60	C₅₃	#2155	1	1	yes	characteristic, complemented, Sylow
61	C₂×M₁₆	—	106	1	no	—
62	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
63	#117	—	106	1	no	—
64	M₃₂	Dic₅₃	1	2	yes	complemented
65	M₃₂	C₂×D₅₃	1	2	yes	—
66	C₂×C₁₆	Dic₅₃	1	2	yes	complemented
67	#108	—	53	2	no	—
68	#108	—	106	1	no	—
69	#107	—	106	1	no	—
70	#106	—	53	2	no	—
71	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
72	M₁₆	—	106	2	no	—
73	M₁₆	—	212	1	no	—
74	C₂×C₈	—	212	1	no	—
75	C₂×C₈	Dic₁₀₆	1	1	yes	characteristic
76	C₂×C₈	Dic₁₀₆	1	1	yes	characteristic
77	C₂×C₈	D₂₁₂	1	1	yes	characteristic
78	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
79	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
80	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
81	C₄⋊C₄	—	106	2	no	—
82	C₂²⋊C₄	—	212	1	no	—
83	C₄²	—	212	1	no	—
84	C₁₆	C₂×Dic₅₃	1	4	yes	—
85	C₂³	#11719913	1	1	yes	characteristic
86	C₂×C₄	—	212	2	no	—
87	C₂×C₄	#11719913	1	1	yes	characteristic
88	C₂×C₄	#11719913	1	1	yes	characteristic
89	C₂×C₄	#11719913	1	1	yes	characteristic
90	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
91	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
92	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
93	C₈	—	212	2	no	—
94	C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
95	C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
96	C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
97	C₈	C₂²×Dic₅₃	1	1	yes	characteristic
98	C₂²	—	4	1	no	—
99	C₂²	#13133149	1	1	yes	characteristic
100	C₂²	#13133149	1	1	yes	characteristic
101	C₂²	C₂×#11719913	1	1	yes	characteristic
102	C₄	—	424	2	no	—
103	C₄	#13133149	1	1	yes	characteristic
104	C₄	#13133149	1	1	yes	characteristic
105	C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
106	C₄	C₂×#11719913	1	1	yes	characteristic
107	C₂	—	2	1	no	—
108	C₂	—	2	1	no	—
109	C₂	—	2	1	no	—
110	C₂	C₂×#13133149	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133153

C₂

yes

maximal, characteristic

#16810661

C₂

yes

maximal, characteristic

#16810446

C₂

yes

maximal, complemented

C₂×M₃₂×C₅₃

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

#13133151

—

#13133149

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

M₃₂×C₅₃

C₄

yes

complemented

M₃₂×C₅₃

C₂²

yes

—

C₂×C₁₆×C₅₃

C₄

yes

complemented

C₂²×C₄×C₅₃

D₄

yes

characteristic

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

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

#11719917

—

#11719917

—

C₂×#88322

—

C₁₆×C₅₃

C₂×C₄

yes

—

C₂³×C₅₃

SD₁₆

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

SD₁₆

yes

characteristic

C₂×C₄×C₅₃

SD₁₆

yes

characteristic

C₂×C₄×C₅₃

SD₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

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×SD₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

Dic₅₃

—

#2155

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#106

yes

characteristic

C₂×M₃₂

D₅₃

yes

—

—

—

C₅₃

#2155

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

#117

—

106

—

M₃₂

Dic₅₃

yes

complemented

M₃₂

C₂×D₅₃

yes

—

C₂×C₁₆

Dic₅₃

yes

complemented

#108

—

#108

—

106

—

#107

—

106

—

#106

—

C₂²×C₄

D₂₁₂

yes

characteristic

M₁₆

—

106

—

M₁₆

—

212

—

C₂×C₈

—

212

—

C₂×C₈

Dic₁₀₆

yes

characteristic

C₂×C₈

Dic₁₀₆

yes

characteristic

C₂×C₈

D₂₁₂

yes

characteristic

C₂×C₈

C₂×Dic₅₃

yes

characteristic

C₂×C₈

C₂×Dic₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

212

—

C₄²

—

212

—

C₁₆

C₂×Dic₅₃

yes

—

C₂³

#11719913

yes

characteristic

C₂×C₄

—

212

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₈

—

212

—

C₈

C₄⋊Dic₅₃

yes

characteristic

C₈

C₄⋊Dic₅₃

yes

characteristic

C₈

C₂×Dic₁₀₆

yes

characteristic

C₈

C₂²×Dic₅₃

yes

characteristic

C₂²

—

C₂²

#13133149

yes

characteristic

100

C₂²

#13133149

yes

characteristic

101

C₂²

C₂×#11719913

yes

characteristic

102

C₄

—

424

—

103

C₄

#13133149

yes

characteristic

104

C₄

#13133149

yes

characteristic

105

C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

106

C₄

C₂×#11719913

yes

characteristic

107

C₂

—

108

C₂

—

109

C₂

—

110

C₂

C₂×#13133149

yes

characteristic

	subgroup	G/H
center	C₄	#13133149
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₂×M₃₂×C₅₃	C₂
socle	C₂×C₅₃	C₂×#106
hypercenter	C₂×M₃₂	D₅₃
hypocenter	C₅₃	#2155
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₄

#13133149

commutator

C₈×C₅₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₂×M₃₂×C₅₃

C₂

socle

C₂×C₅₃

C₂×#106

hypercenter

C₂×M₃₂

D₅₃

hypocenter

C₅₃

#2155

LCS

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. #16810446	C₂	semidirect
12. M₃₂×C₅₃	C₄	semidirect
14. C₂×C₁₆×C₅₃	C₄	semidirect
60. C₅₃	#2155	semidirect
64. M₃₂	Dic₅₃	semidirect
66. C₂×C₁₆	Dic₅₃	semidirect

H ≅ G/N

type

3. #16810446

C₂

semidirect

12. M₃₂×C₅₃

C₄

semidirect

14. C₂×C₁₆×C₅₃

C₄

semidirect

60. C₅₃

#2155

semidirect

64. M₃₂

Dic₅₃

semidirect

66. C₂×C₁₆

Dic₅₃

semidirect

Subgroups — #20276488

Conjugacy classes of subgroups

Complements