Group #20274470 — subgroups

Subgroups — #20274470

Order 6784.

200 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 105 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	#16810578	C₂	1	2	yes	maximal
2	#16810381	C₂	1	1	yes	maximal, characteristic
3	#16810380	C₂	1	1	yes	maximal, characteristic
4	#16810379	C₂	1	2	yes	maximal
5	C₅₃×#405	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	2	yes	—
7	#13133176	C₂²	1	2	yes	—
8	#13133127	C₄	1	4	yes	complemented
9	C₈×Dic₅₃	C₄	1	4	yes	complemented
10	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	#117×C₅₃	C₂²	1	1	yes	characteristic
13	#97×C₅₃	C₄	1	2	yes	complemented
14	C₄×C₈×C₅₃	C₄	1	2	yes	complemented
15	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
16	M₁₆×C₅₃	—	2	2	no	—
17	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄⋊C₄×C₅₃	—	2	1	no	—
23	C₂²⋊C₄×C₅₃	—	2	1	no	—
24	C₄²×C₅₃	C₂×C₄	1	2	yes	—
25	C₂²⋊Dic₅₃	—	2	2	no	—
26	C₄⋊Dic₅₃	—	2	2	no	—
27	C₄×Dic₅₃	C₂×C₄	1	4	yes	—
28	#11719917	—	2	2	no	—
29	#11719917	—	2	2	no	—
30	C₂×#88322	C₂×C₄	1	4	yes	—
31	C₂³×C₅₃	#41	1	1	yes	characteristic
32	C₂×C₄×C₅₃	—	2	2	no	—
33	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
34	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
36	C₂×C₄×C₅₃	#41	1	2	yes	—
37	C₂×Dic₅₃	—	2	4	no	—
38	C₈×C₅₃	—	2	2	no	—
39	C₈×C₅₃	C₄²	1	2	yes	—
40	C₈×C₅₃	C₄²	1	1	yes	characteristic
41	C₈×C₅₃	C₄²	1	1	yes	characteristic
42	#88322	—	2	4	no	—
43	C₂²×C₅₃	—	4	1	no	—
44	C₂²×C₅₃	#117	1	2	yes	—
45	C₂²×C₅₃	#117	1	1	yes	characteristic
46	C₄×C₅₃	—	4	2	no	—
47	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₅₃	#117	1	2	yes	—
49	C₄×C₅₃	#117	1	1	yes	characteristic
50	Dic₅₃	—	4	4	no	—
51	#1747	—	53	1	no	maximal, Sylow
52	C₂×C₅₃	—	2	2	no	—
53	C₂×C₅₃	—	2	1	no	—
54	C₂×C₅₃	#376	1	1	yes	characteristic
55	#405	—	53	2	no	—
56	#405	D₅₃	1	1	yes	characteristic
57	#338	—	53	1	no	—
58	H(3,ℤ/4)	—	53	1	no	—
59	H(3,ℤ/4)	—	53	2	no	—
60	C₅₃	#1747	1	1	yes	characteristic, complemented, Sylow
61	C₂×M₁₆	—	53	2	no	—
62	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
63	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
64	#117	—	53	2	no	—
65	#117	C₂×D₅₃	1	1	yes	characteristic
66	#97	—	53	4	no	—
67	#97	Dic₅₃	1	2	yes	complemented
68	C₄×C₈	—	53	4	no	—
69	C₄×C₈	Dic₅₃	1	2	yes	complemented
70	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	106	2	no	—
73	M₁₆	—	106	2	no	—
74	C₂×C₈	—	53	4	no	—
75	C₂×C₈	C₄×D₅₃	1	2	yes	—
76	C₂×C₈	C₄×D₅₃	1	2	yes	—
77	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
78	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
79	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
80	C₄⋊C₄	—	2	1	no	—
81	C₄⋊C₄	—	106	2	no	—
82	C₂²⋊C₄	—	2	1	no	—
83	C₂²⋊C₄	—	106	2	no	—
84	C₄²	—	53	4	no	—
85	C₄²	C₂×Dic₅₃	1	2	yes	—
86	C₂³	#11719945	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	106	4	no	—
89	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
90	C₂×C₄	#11719945	1	1	yes	characteristic
91	C₂×C₄	#11719945	1	2	yes	—
92	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
93	C₈	—	2	2	no	—
94	C₈	—	106	4	no	—
95	C₈	C₄×Dic₅₃	1	1	yes	characteristic
96	C₈	C₄×Dic₅₃	1	2	yes	—
97	C₈	C₄×Dic₅₃	1	1	yes	characteristic
98	C₂²	—	4	1	no	—
99	C₂²	#13133174	1	2	yes	—
100	C₂²	#13133176	1	1	yes	characteristic
101	C₄	—	4	2	no	—
102	C₄	—	212	4	no	—
103	C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
104	C₄	#13133174	1	2	yes	—
105	C₄	#13133176	1	1	yes	characteristic
106	C₂	—	2	1	no	—
107	C₂	—	2	2	no	—
108	C₂	#16810469	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810578

C₂

yes

maximal

#16810381

C₂

yes

maximal, characteristic

#16810380

C₂

yes

maximal, characteristic

#16810379

C₂

yes

maximal

C₅₃×#405

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

#13133176

C₂²

yes

—

#13133127

C₄

yes

complemented

C₈×Dic₅₃

C₄

yes

complemented

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₂²

yes

characteristic

#97×C₅₃

C₄

yes

complemented

C₄×C₈×C₅₃

C₄

yes

complemented

C₂²×C₄×C₅₃

C₂³

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

C₂×C₄

yes

—

#11719917

—

#11719917

—

C₂×#88322

C₂×C₄

yes

—

C₂³×C₅₃

#41

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₄²

yes

—

C₈×C₅₃

C₄²

yes

characteristic

C₈×C₅₃

C₄²

yes

characteristic

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

#117

yes

—

C₂²×C₅₃

#117

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄²

yes

characteristic

C₄×C₅₃

#117

yes

—

C₄×C₅₃

#117

yes

characteristic

Dic₅₃

—

#1747

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

—

D₅₃

yes

—

—

—

C₅₃

#1747

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

characteristic

#117

—

#117

C₂×D₅₃

yes

characteristic

#97

—

#97

Dic₅₃

yes

complemented

C₄×C₈

—

C₄×C₈

Dic₅₃

yes

complemented

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

C₄×D₅₃

yes

—

C₂×C₈

C₄×D₅₃

yes

—

C₂×C₈

C₂×Dic₅₃

yes

—

C₂×C₈

C₂×Dic₅₃

yes

characteristic

C₂×C₈

C₂×Dic₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

C₂²⋊C₄

—

106

—

C₄²

—

C₄²

C₂×Dic₅₃

yes

—

C₂³

#11719945

yes

characteristic

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

C₂×C₄×D₅₃

yes

—

C₂×C₄

#11719945

yes

characteristic

C₂×C₄

#11719945

yes

—

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

C₈

—

C₈

—

106

—

C₈

C₄×Dic₅₃

yes

characteristic

C₈

C₄×Dic₅₃

yes

—

C₈

C₄×Dic₅₃

yes

characteristic

C₂²

—

C₂²

#13133174

yes

—

100

C₂²

#13133176

yes

characteristic

101

C₄

—

102

C₄

—

212

—

103

C₄

C₂×C₄×Dic₅₃

yes

characteristic

104

C₄

#13133174

yes

—

105

C₄

#13133176

yes

characteristic

106

C₂

—

107

C₂

—

108

C₂

#16810469

yes

characteristic

	subgroup	G/H
center	C₈	C₄×Dic₅₃
commutator	C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#405	C₂
socle	C₂×C₅₃	#376
hypercenter	#405	D₅₃
hypocenter	C₅₃	#1747
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ #405

subgroup

G/H

center

C₈

C₄×Dic₅₃

commutator

C₄×C₅₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#405

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₅₃

C₁ ◁ C₈ ◁ #405

N	H ≅ G/N	type
8. #13133127	C₄	semidirect
9. C₈×Dic₅₃	C₄	semidirect
13. #97×C₅₃	C₄	semidirect
14. C₄×C₈×C₅₃	C₄	semidirect
60. C₅₃	#1747	semidirect
67. #97	Dic₅₃	semidirect
69. C₄×C₈	Dic₅₃	semidirect

H ≅ G/N

type

8. #13133127

C₄

semidirect

9. C₈×Dic₅₃

C₄

semidirect

13. #97×C₅₃

C₄

semidirect

14. C₄×C₈×C₅₃

C₄

semidirect

60. C₅₃

#1747

semidirect

67. #97

Dic₅₃

semidirect

69. C₄×C₈

Dic₅₃

semidirect

Subgroups — #20274470

Conjugacy classes of subgroups

Complements