Group #20290362 — subgroups

Subgroups — #20290362

Order 6784.

148 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 56 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	#16811674	C₂	1	2	yes	maximal
2	#16811484	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#88324	C₂	1	2	yes	maximal
4	#16811464	C₂	1	1	yes	maximal, characteristic
5	Q₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	#13133284	—	2	2	no	—
7	C₄⋊#88324	C₂²	1	2	yes	—
8	#13133272	—	2	2	no	—
9	C₄×#88324	C₂²	1	1	yes	characteristic
10	C₄×#88324	C₂²	1	1	yes	characteristic
11	Q₈×Dic₅₃	C₂²	1	2	yes	—
12	#13133151	C₄	1	1	yes	characteristic
13	#13133141	C₄	1	2	yes	—
14	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
16	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
17	C₂×#88324	—	2	2	no	—
18	C₂×#88324	—	2	1	no	—
19	C₂×#88324	D₄	1	2	yes	—
20	Q₁₆×C₅₃	—	2	2	no	—
21	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄⋊Dic₅₃	—	2	2	no	—
23	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
24	C₄×Dic₅₃	—	2	2	no	—
25	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
26	C₂×#88322	C₂×C₄	1	1	yes	characteristic
27	Q₈×C₅₃	—	2	2	no	—
28	Q₈×C₅₃	—	2	2	no	—
29	C₂×C₄×C₅₃	—	2	2	no	—
30	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₅₃	—	2	2	no	—
32	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
33	C₂×Dic₅₃	#41	1	1	yes	characteristic
34	#88324	—	2	2	no	—
35	#88324	—	4	1	no	—
36	#88324	—	4	2	no	—
37	C₈×C₅₃	M₁₆	1	2	yes	—
38	#88322	—	2	1	no	—
39	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
40	C₄×C₅₃	—	4	2	no	—
41	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
42	C₄×C₅₃	#131	1	1	yes	characteristic
43	Dic₅₃	—	2	1	no	—
44	Dic₅₃	—	4	2	no	—
45	Dic₅₃	#137	1	2	yes	—
46	#1593	—	53	1	no	maximal, Sylow
47	C₂×C₅₃	#436	1	1	yes	characteristic
48	C₂×C₅₃	#441	1	1	yes	characteristic
49	C₂×C₅₃	#444	1	1	yes	characteristic
50	#446	—	53	2	no	—
51	C₄×Q₁₆	—	53	1	no	—
52	#334	—	53	1	no	—
53	Q₈⋊C₈	—	53	2	no	—
54	#322	—	53	1	no	—
55	C₅₃	#1593	1	1	yes	characteristic, complemented, Sylow
56	C₂×Q₁₆	#6157	1	1	yes	characteristic
57	C₄×Q₈	—	53	2	no	—
58	#107	—	53	1	no	—
59	C₄⋊C₈	—	53	2	no	—
60	C₄⋊C₈	—	106	2	no	—
61	#103	—	53	2	no	—
62	#97	—	106	2	no	—
63	C₄×C₈	—	53	1	no	—
64	C₄×C₈	—	53	1	no	—
65	C₄×C₈	—	53	1	no	—
66	C₂×Q₈	C₂×#6157	1	2	yes	—
67	Q₁₆	—	2	2	no	—
68	C₂×C₈	—	53	2	no	—
69	C₂×C₈	—	53	1	no	—
70	C₂×C₈	—	106	1	no	—
71	C₂×C₈	—	106	2	no	—
72	C₂×C₈	C₂×#6157	1	1	yes	characteristic
73	C₄⋊C₄	—	53	2	no	—
74	C₄⋊C₄	—	106	2	no	—
75	C₄²	—	53	1	no	—
76	C₄²	—	106	2	no	—
77	Q₈	—	2	2	no	—
78	Q₈	—	2	2	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	53	1	no	—
81	C₂×C₄	—	53	1	no	—
82	C₂×C₄	—	106	2	no	—
83	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
84	C₈	—	106	2	no	—
85	C₈	—	106	1	no	—
86	C₈	—	212	2	no	—
87	C₈	—	212	1	no	—
88	C₈	#11719940	1	2	yes	—
89	C₂²	D₄×#6157	1	1	yes	characteristic
90	C₄	—	4	2	no	—
91	C₄	—	53	2	no	—
92	C₄	—	106	1	no	—
93	C₄	—	212	2	no	—
94	C₄	C₂×#11719940	1	1	yes	characteristic
95	C₄	#13133304	1	1	yes	characteristic
96	C₂	#16811658	1	1	yes	characteristic
97	C₂	#16811663	1	1	yes	characteristic
98	C₂	#16811669	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811674

C₂

yes

maximal

#16811484

C₂

yes

maximal, characteristic

Q₈⋊#88324

C₂

yes

maximal

#16811464

C₂

yes

maximal, characteristic

Q₁₆×Dic₅₃

C₂

yes

maximal, characteristic

#13133284

—

C₄⋊#88324

C₂²

yes

—

#13133272

—

C₄×#88324

C₂²

yes

characteristic

C₄×#88324

C₂²

yes

characteristic

Q₈×Dic₅₃

C₂²

yes

—

#13133151

C₄

yes

characteristic

#13133141

C₄

yes

—

C₈×Dic₅₃

C₂²

yes

characteristic

C₂×Q₁₆×C₅₃

C₄

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂×#88324

—

C₂×#88324

—

C₂×#88324

D₄

yes

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂³

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

C₂×Dic₅₃

yes

—

—

—

C₈×C₅₃

M₁₆

yes

—

#88322

—

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₄×C₅₃

#131

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#137

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₄×Q₁₆

—

#334

—

Q₈⋊C₈

—

#322

—

C₅₃

#1593

yes

characteristic, complemented, Sylow

C₂×Q₁₆

#6157

yes

characteristic

C₄×Q₈

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

#103

—

#97

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

C₂×#6157

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₂×#6157

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄²

—

C₄²

—

106

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

C₂²×#6157

yes

characteristic

C₈

—

106

—

C₈

—

106

—

C₈

—

212

—

C₈

—

212

—

C₈

#11719940

yes

—

C₂²

D₄×#6157

yes

characteristic

C₄

—

C₄

—

C₄

—

106

—

C₄

—

212

—

C₄

yes

C₄

yes

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²	D₄×#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₂×Q₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₄×D₄
hypercenter	C₂×Q₁₆	#6157
hypocenter	C₅₃	#1593
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂²

D₄×#6157

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₂×Q₁₆×C₅₃

C₄

socle

C₂²×C₅₃

C₄×D₄

C₂×Q₁₆

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

N	H ≅ G/N	type
55. C₅₃	#1593	semidirect

H ≅ G/N

type

55. C₅₃

#1593

semidirect

Subgroups — #20290362

Conjugacy classes of subgroups

Complements