Group #20273613 — subgroups

Subgroups — #20273613

Order 6784.

152 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 69 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	#16810753	C₂	1	2	yes	maximal
2	C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
3	#16810376	C₂	1	1	yes	maximal, characteristic
4	#16810355	C₂	1	2	yes	maximal
5	C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₁₂	C₄	1	1	yes	characteristic
7	C₄×Dic₁₀₆	C₂²	1	2	yes	—
8	#13133151	C₄	1	1	yes	characteristic
9	C₄⋊#88322	C₂²	1	2	yes	—
10	#13133146	—	2	2	no	—
11	#13133142	C₄	1	2	yes	—
12	#13133127	—	2	2	no	—
13	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₅₃	—	2	1	no	—
18	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
22	C₄²×C₅₃	C₂³	1	1	yes	characteristic
23	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
24	#11719919	—	2	2	no	—
25	C₄×Dic₅₃	—	2	2	no	—
26	C₂×#88322	—	2	2	no	—
27	Dic₂₁₂	—	2	2	no	—
28	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
31	C₂×Dic₅₃	—	2	2	no	—
32	C₂×Dic₅₃	—	2	2	no	—
33	Dic₁₀₆	—	2	2	no	—
34	Dic₁₀₆	—	2	2	no	—
35	C₈×C₅₃	—	2	2	no	—
36	C₈×C₅₃	—	2	1	no	—
37	C₈×C₅₃	—	2	1	no	—
38	C₈×C₅₃	M₁₆	1	2	yes	—
39	C₈×C₅₃	Q₁₆	1	2	yes	complemented
40	#88322	—	4	2	no	—
41	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
42	C₄×C₅₃	—	2	1	no	—
43	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
44	C₄×C₅₃	#131	1	1	yes	characteristic
45	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
46	C₄×C₅₃	#135	1	1	yes	characteristic
47	Dic₅₃	—	4	2	no	—
48	Dic₅₃	—	4	2	no	—
49	#1591	—	53	1	no	maximal, Sylow
50	C₂×C₅₃	#436	1	1	yes	characteristic
51	C₂×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
52	C₂×C₅₃	#443	1	1	yes	characteristic
53	#446	—	53	2	no	—
54	C₄×Q₁₆	—	53	1	no	—
55	#335	—	53	1	no	—
56	Q₈⋊C₈	—	53	2	no	—
57	C₈²	D₅₃	1	1	yes	characteristic
58	C₅₃	#1591	1	1	yes	characteristic, complemented, Sylow
59	C₂×Q₁₆	—	53	1	no	—
60	C₄×Q₈	—	53	2	no	—
61	#107	—	53	1	no	—
62	C₄⋊C₈	—	53	2	no	—
63	C₄⋊C₈	—	106	2	no	—
64	#103	—	53	2	no	—
65	#97	—	106	2	no	—
66	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
67	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
68	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
69	C₂×Q₈	—	53	2	no	—
70	Q₁₆	—	106	2	no	—
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	106	2	no	—
73	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
74	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
75	C₂×C₈	D₂₁₂	1	1	yes	characteristic
76	C₂×C₈	D₂₁₂	1	1	yes	characteristic
77	C₄⋊C₄	—	53	2	no	—
78	C₄⋊C₄	—	106	2	no	—
79	C₄²	—	106	2	no	—
80	C₄²	C₂²×D₅₃	1	1	yes	characteristic
81	Q₈	—	106	2	no	—
82	Q₈	—	106	2	no	—
83	C₂×C₄	—	106	2	no	—
84	C₂×C₄	—	106	2	no	—
85	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
86	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
87	C₂×C₄	#11719945	1	1	yes	characteristic
88	C₈	—	2	2	no	—
89	C₈	—	2	1	no	—
90	C₈	—	2	1	no	—
91	C₈	—	212	2	no	—
92	C₈	#11719912	1	2	yes	—
93	C₈	Dic₂₁₂	1	2	yes	complemented
94	C₂²	C₄×D₂₁₂	1	1	yes	characteristic
95	C₄	—	2	1	no	—
96	C₄	—	212	2	no	—
97	C₄	—	212	2	no	—
98	C₄	C₂×#11719912	1	1	yes	characteristic
99	C₄	#13133223	1	1	yes	characteristic
100	C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
101	C₄	#13133239	1	1	yes	characteristic
102	C₂	#16810702	1	1	yes	characteristic
103	C₂	C₄×Dic₂₁₂	1	1	yes	characteristic
104	C₂	#16810741	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810753

C₂

yes

maximal

C₄×Dic₂₁₂

C₂

yes

maximal, characteristic

#16810376

C₂

yes

maximal, characteristic

#16810355

C₂

yes

maximal

C₈²×C₅₃

C₂

yes

maximal, characteristic

C₂×Dic₂₁₂

C₄

yes

characteristic

C₄×Dic₁₀₆

C₂²

yes

—

#13133151

C₄

yes

characteristic

C₄⋊#88322

C₂²

yes

—

#13133146

—

#13133142

C₄

yes

—

#13133127

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Dic₁₀₆

C₂×C₄

yes

—

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

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

Dic₂₁₂

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

M₁₆

yes

—

C₈×C₅₃

Q₁₆

yes

complemented

#88322

—

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₄×C₅₃

#131

yes

characteristic

C₄×C₅₃

C₂×Q₁₆

yes

characteristic

C₄×C₅₃

#135

yes

characteristic

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₄×Q₁₆

yes

C₂×C₅₃

yes

—

C₄×Q₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

D₅₃

yes

characteristic

C₅₃

#1591

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

#103

—

#97

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₄×C₈

C₂×D₅₃

yes

characteristic

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

Q₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

D₂₁₂

yes

characteristic

C₂×C₈

D₂₁₂

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄²

—

106

—

C₄²

C₂²×D₅₃

yes

characteristic

Q₈

—

106

—

Q₈

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₂×C₄

#11719945

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

212

—

C₈

#11719912

yes

—

C₈

Dic₂₁₂

yes

complemented

C₂²

C₄×D₂₁₂

yes

characteristic

C₄

—

C₄

—

212

—

C₄

—

212

—

C₄

yes

C₄

yes

100

C₄

C₂×Dic₂₁₂

yes

characteristic

101

C₄

#13133239

yes

characteristic

102

C₂

#16810702

yes

characteristic

103

C₂

C₄×Dic₂₁₂

yes

characteristic

104

C₂

#16810741

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₈²×C₅₃	C₂
socle	C₂²×C₅₃	C₄×D₄
hypercenter	C₈²	D₅₃
hypocenter	C₅₃	#1591
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈²

subgroup

G/H

center

C₂×C₄

C₂×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₈²×C₅₃

C₂

socle

C₂²×C₅₃

C₄×D₄

hypercenter

C₈²

D₅₃

hypocenter

C₅₃

#1591

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈²

N	H ≅ G/N	type
39. C₈×C₅₃	Q₁₆	semidirect
58. C₅₃	#1591	semidirect
93. C₈	Dic₂₁₂	semidirect

H ≅ G/N

type

39. C₈×C₅₃

Q₁₆

semidirect

58. C₅₃

#1591

semidirect

93. C₈

Dic₂₁₂

semidirect

Subgroups — #20273613

Conjugacy classes of subgroups

Complements