Group #20290355 — subgroups

Subgroups — #20290355

Order 6784.

152 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 58 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	#16811486	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#88324	C₂	1	2	yes	maximal
4	C₈×#88324	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	1	no	—
18	C₂×#88324	—	2	2	no	—
19	C₂×#88324	D₄	1	1	yes	characteristic
20	C₂×#88324	D₄	1	1	yes	characteristic
21	Q₁₆×C₅₃	—	2	2	no	—
22	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄⋊Dic₅₃	—	2	2	no	—
24	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
25	C₄×Dic₅₃	—	2	2	no	—
26	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
27	C₂×#88322	C₂×C₄	1	1	yes	characteristic
28	Q₈×C₅₃	—	2	2	no	—
29	Q₈×C₅₃	—	2	2	no	—
30	C₂×C₄×C₅₃	—	2	2	no	—
31	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₅₃	—	2	2	no	—
33	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂×Dic₅₃	#41	1	1	yes	characteristic
35	#88324	—	2	2	no	—
36	#88324	—	2	1	no	—
37	#88324	—	4	2	no	—
38	#88324	Q₁₆	1	2	yes	complemented
39	C₈×C₅₃	M₁₆	1	2	yes	—
40	#88322	—	2	1	no	—
41	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
42	C₄×C₅₃	—	4	2	no	—
43	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
44	C₄×C₅₃	#131	1	1	yes	characteristic
45	Dic₅₃	—	2	1	no	—
46	Dic₅₃	—	4	2	no	—
47	Dic₅₃	C₂×Q₁₆	1	1	yes	characteristic
48	Dic₅₃	#135	1	1	yes	characteristic
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	—	53	1	no	—
58	C₅₃	#1591	1	1	yes	characteristic, complemented, Sylow
59	C₂×Q₁₆	#6157	1	1	yes	characteristic
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	—	53	1	no	—
67	C₄×C₈	—	53	1	no	—
68	C₄×C₈	—	53	1	no	—
69	C₂×Q₈	C₂×#6157	1	2	yes	—
70	Q₁₆	—	2	2	no	—
71	C₂×C₈	—	53	1	no	—
72	C₂×C₈	—	53	1	no	—
73	C₂×C₈	—	53	1	no	—
74	C₂×C₈	—	106	1	no	—
75	C₂×C₈	—	106	2	no	—
76	C₂×C₈	C₂×#6157	1	1	yes	characteristic
77	C₄⋊C₄	—	53	2	no	—
78	C₄⋊C₄	—	106	2	no	—
79	C₄²	—	53	1	no	—
80	C₄²	—	106	2	no	—
81	Q₈	—	2	2	no	—
82	Q₈	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	53	1	no	—
85	C₂×C₄	—	53	1	no	—
86	C₂×C₄	—	106	2	no	—
87	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
88	C₈	—	53	2	no	—
89	C₈	—	106	1	no	—
90	C₈	—	106	2	no	—
91	C₈	—	106	1	no	—
92	C₈	—	212	2	no	—
93	C₈	#11719940	1	2	yes	—
94	C₂²	D₄×#6157	1	1	yes	characteristic
95	C₄	—	4	2	no	—
96	C₄	—	53	1	no	—
97	C₄	—	53	1	no	—
98	C₄	—	106	1	no	—
99	C₄	—	212	2	no	—
100	C₄	C₂×#11719940	1	1	yes	characteristic
101	C₄	#13133304	1	1	yes	characteristic
102	C₂	#16811658	1	1	yes	characteristic
103	C₂	Q₁₆×#6157	1	1	yes	characteristic
104	C₂	#16811667	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811674

C₂

yes

maximal

#16811486

C₂

yes

maximal, characteristic

Q₈⋊#88324

C₂

yes

maximal

C₈×#88324

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

characteristic

C₂×#88324

D₄

yes

characteristic

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

—

—

—

Q₁₆

yes

complemented

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

C₂×Q₁₆

yes

Dic₅₃

yes

—

C₂×C₅₃

yes

C₂×C₅₃

C₄×Q₁₆

yes

C₂×C₅₃

yes

—

C₄×Q₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

—

C₅₃

#1591

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

—

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

—

C₈

—

106

—

C₈

—

106

—

C₈

—

106

—

C₈

—

212

—

C₈

#11719940

yes

—

C₂²

D₄×#6157

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

106

—

C₄

—

212

—

100

C₄

C₂×#11719940

yes

characteristic

101

C₄

#13133304

yes

characteristic

102

C₂

#16811658

yes

characteristic

103

C₂

Q₁₆×#6157

yes

characteristic

104

C₂

#16811667

yes

characteristic

	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	#1591
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
38. #88324	Q₁₆	semidirect
58. C₅₃	#1591	semidirect

H ≅ G/N

type

38. #88324

Q₁₆

semidirect

58. C₅₃

#1591

semidirect

Subgroups — #20290355

Conjugacy classes of subgroups

Complements