Group #20274116 — subgroups

Subgroups — #20274116

Order 6784.

144 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16810915	C₂	1	2	yes	maximal
2	C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
3	#16810376	C₂	1	1	yes	maximal, characteristic
4	#16810372	C₂	1	2	yes	maximal
5	C₅₃×#500	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
7	#13133150	—	2	2	no	—
8	C₄⋊#88322	C₂²	1	1	yes	characteristic
9	C₄⋊#88322	C₂²	1	2	yes	—
10	#13133146	—	2	1	no	—
11	#13133140	—	2	2	no	—
12	C₈×Dic₅₃	—	2	1	no	—
13	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
14	#107×C₅₃	—	2	1	no	—
15	#107×C₅₃	—	2	1	no	—
16	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₅₃	—	2	2	no	—
18	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
19	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
21	C₄⋊C₄×C₅₃	—	2	2	no	—
22	C₄⋊C₄×C₅₃	—	2	2	no	—
23	C₄²×C₅₃	C₂³	1	1	yes	characteristic
24	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
25	#11719919	—	2	1	no	—
26	C₄×Dic₅₃	—	2	1	no	—
27	C₂×#88322	—	2	1	no	—
28	C₂×#88322	—	2	2	no	—
29	Q₈×C₅₃	—	8	2	no	—
30	C₂×C₄×C₅₃	—	4	2	no	—
31	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
34	C₂×Dic₅₃	—	2	1	no	—
35	C₂×Dic₅₃	—	2	1	no	—
36	Dic₁₀₆	—	2	1	no	—
37	Dic₁₀₆	Q₁₆	1	2	yes	—
38	C₈×C₅₃	—	2	1	no	—
39	C₈×C₅₃	Q₁₆	1	2	yes	—
40	#88322	—	4	2	no	—
41	#88322	—	4	1	no	—
42	C₂²×C₅₃	#121	1	1	yes	characteristic
43	C₄×C₅₃	—	2	1	no	—
44	C₄×C₅₃	—	8	2	no	—
45	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
46	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
47	C₄×C₅₃	#135	1	1	yes	characteristic
48	C₄×C₅₃	#137	1	1	yes	characteristic
49	Dic₅₃	—	2	2	no	—
50	Dic₅₃	—	4	1	no	—
51	#1676	—	53	1	no	maximal, Sylow
52	C₂×C₅₃	#462	1	1	yes	characteristic
53	C₂×C₅₃	#467	1	1	yes	characteristic
54	C₂×C₅₃	#472	1	1	yes	characteristic
55	#500	D₅₃	1	1	yes	characteristic
56	#477	—	53	2	no	—
57	C₈×Q₈	—	53	1	no	—
58	#335	—	53	1	no	—
59	#333	—	53	2	no	—
60	C₅₃	#1676	1	1	yes	characteristic, complemented, Sylow
61	C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
62	C₄×Q₈	—	53	1	no	—
63	#107	—	2	1	no	—
64	#107	—	2	1	no	—
65	#107	—	106	2	no	—
66	C₄⋊C₈	—	53	1	no	—
67	C₄⋊C₈	—	53	2	no	—
68	C₄⋊C₈	—	106	1	no	—
69	#103	—	106	2	no	—
70	C₄×C₈	—	106	1	no	—
71	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
72	C₂×Q₈	—	2	2	no	—
73	C₂×Q₈	—	53	1	no	—
74	C₂×C₈	—	106	2	no	—
75	C₂×C₈	—	106	1	no	—
76	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
77	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
78	C₄⋊C₄	—	2	2	no	—
79	C₄⋊C₄	—	2	2	no	—
80	C₄⋊C₄	—	53	1	no	—
81	C₄⋊C₄	—	106	1	no	—
82	C₄²	—	106	1	no	—
83	C₄²	C₂²×D₅₃	1	1	yes	characteristic
84	Q₈	—	8	2	no	—
85	Q₈	—	53	2	no	—
86	Q₈	—	106	1	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	—	106	1	no	—
89	C₂×C₄	—	106	1	no	—
90	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
91	C₂×C₄	#11719947	1	1	yes	characteristic
92	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	212	2	no	—
95	C₈	—	212	1	no	—
96	C₈	#11719925	1	2	yes	—
97	C₂²	#13133188	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	8	2	no	—
100	C₄	—	106	2	no	—
101	C₄	—	212	1	no	—
102	C₄	Q₁₆×D₅₃	1	1	yes	characteristic
103	C₄	C₂×#11719925	1	1	yes	characteristic
104	C₄	#13133240	1	1	yes	characteristic
105	C₄	#13133248	1	1	yes	characteristic
106	C₂	#16810830	1	1	yes	characteristic
107	C₂	#16810861	1	1	yes	characteristic
108	C₂	#16810890	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810915

C₂

yes

maximal

C₈×Dic₁₀₆

C₂

yes

maximal, characteristic

#16810376

C₂

yes

maximal, characteristic

#16810372

C₂

yes

maximal

C₅₃×#500

C₂

yes

maximal, characteristic

C₄×Dic₁₀₆

C₂²

yes

characteristic

#13133150

—

C₄⋊#88322

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

#13133146

—

#13133140

—

C₈×Dic₅₃

—

C₄⋊Q₈×C₅₃

C₂²

yes

—

#107×C₅₃

—

#107×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Dic₁₀₆

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

D₄

yes

characteristic

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

Q₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

Q₁₆

yes

—

#88322

—

#88322

—

C₂²×C₅₃

#121

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×Q₁₆

yes

characteristic

C₄×C₅₃

C₂×Q₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

characteristic

#477

—

C₈×Q₈

—

#335

—

#333

—

C₅₃

#1676

yes

characteristic, complemented, Sylow

C₄⋊Q₈

C₂×D₅₃

yes

—

C₄×Q₈

—

#107

—

#107

—

#107

—

106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

#103

—

106

—

C₄×C₈

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄²

—

106

—

C₄²

C₂²×D₅₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

106

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

D₄×D₅₃

yes

characteristic

C₂×C₄

#11719947

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

C₈

—

C₈

—

212

—

C₈

—

212

—

C₈

#11719925

yes

—

C₂²

#13133188

yes

characteristic

C₄

—

C₄

—

100

C₄

—

106

—

101

C₄

—

212

—

102

C₄

Q₁₆×D₅₃

yes

characteristic

103

C₄

C₂×#11719925

yes

characteristic

104

C₄

#13133240

yes

characteristic

105

C₄

#13133248

yes

characteristic

106

C₂

#16810830

yes

characteristic

107

C₂

#16810861

yes

characteristic

108

C₂

#16810890

yes

characteristic

	subgroup	G/H
center	C₂²	#13133188
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#500	C₂
socle	C₂²×C₅₃	#121
hypercenter	#500	D₅₃
hypocenter	C₅₃	#1676
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#13133188

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#500

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
60. C₅₃	#1676	semidirect

H ≅ G/N

type

60. C₅₃

#1676

semidirect

Subgroups — #20274116

Conjugacy classes of subgroups

Complements