Group #20290274 — subgroups

Subgroups — #20290274

Order 6784.

184 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 78 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	#16811627	C₂	1	2	yes	maximal
2	#16811487	C₂	1	4	yes	maximal, complemented
3	C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719936	—	2	2	no	—
5	C₄⋊#88324	—	2	2	no	—
6	C₄⋊#88324	C₂²	1	4	yes	—
7	C₂×C₈×D₅₃	C₄	1	2	yes	—
8	C₄²×D₅₃	C₂²	1	1	yes	characteristic
9	C₄×#88322	C₄	1	1	yes	characteristic
10	C₈×Dic₅₃	C₂²	1	2	yes	—
11	C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
12	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
14	C₂×C₄×D₅₃	Q₈	1	1	yes	characteristic
15	C₂×#88324	—	2	4	no	—
16	#11719936	—	4	4	no	—
17	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₄×Dic₅₃	D₄	1	1	yes	characteristic
20	C₄×Dic₅₃	Q₈	1	1	yes	characteristic
21	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
22	C₂×#88322	C₂×C₄	1	2	yes	—
23	C₈×D₅₃	—	2	4	no	—
24	C₂²×D₅₃	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
28	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
30	C₂×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
31	C₄×D₅₃	—	2	2	no	—
32	C₄×D₅₃	—	2	2	no	—
33	C₄×D₅₃	D₈	1	2	yes	—
34	C₄×D₅₃	Q₁₆	1	2	yes	—
35	#88324	—	4	4	no	—
36	C₈×C₅₃	—	2	2	no	—
37	#88322	M₁₆	1	4	yes	complemented
38	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂×D₅₃	—	2	1	no	—
40	C₂×D₅₃	—	2	1	no	—
41	C₂×D₅₃	#107	1	2	yes	—
42	C₄×C₅₃	#107	1	2	yes	—
43	C₄×C₅₃	#108	1	2	yes	—
44	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
45	Dic₅₃	—	2	1	no	—
46	Dic₅₃	—	2	1	no	—
47	Dic₅₃	C₂×D₈	1	1	yes	characteristic
48	Dic₅₃	C₂×Q₁₆	1	1	yes	characteristic
49	#1567	—	53	1	no	maximal, Sylow
50	C₂×C₅₃	#423	1	1	yes	characteristic
51	C₂×C₅₃	C₂×#107	1	1	yes	characteristic
52	C₂×C₅₃	C₂×#108	1	1	yes	characteristic
53	D₅₃	—	2	2	no	—
54	#423	—	53	2	no	—
55	C₂×C₄×C₈	—	53	1	no	—
56	#335	—	53	4	no	—
57	C₅₃	#1567	1	1	yes	characteristic, complemented, Sylow
58	C₂×M₁₆	—	106	2	no	—
59	C₂²×C₈	—	53	2	no	—
60	C₂×C₄²	—	53	1	no	—
61	C₄⋊C₈	—	53	4	no	—
62	C₄⋊C₈	—	106	2	no	—
63	C₄×C₈	—	53	2	no	—
64	C₄×C₈	—	53	1	no	—
65	C₄×C₈	#6157	1	1	yes	characteristic
66	C₂²×C₄	—	53	1	no	—
67	C₂²×C₄	—	53	1	no	—
68	C₂²×C₄	—	53	1	no	—
69	M₁₆	—	212	4	no	—
70	C₂×C₈	—	53	2	no	—
71	C₂×C₈	—	106	4	no	—
72	C₂×C₈	—	106	4	no	—
73	C₂×C₈	C₂×#6157	1	2	yes	—
74	C₄²	—	53	1	no	—
75	C₄²	—	53	1	no	—
76	C₄²	—	53	1	no	—
77	C₄²	C₂×#6157	1	1	yes	characteristic
78	C₂³	—	53	1	no	—
79	C₂×C₄	—	53	1	no	—
80	C₂×C₄	—	53	2	no	—
81	C₂×C₄	—	53	2	no	—
82	C₂×C₄	—	53	1	no	—
83	C₂×C₄	—	53	1	no	—
84	C₂×C₄	—	106	2	no	—
85	C₂×C₄	—	106	2	no	—
86	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
87	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
88	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
89	C₈	—	2	2	no	—
90	C₈	—	53	4	no	—
91	C₈	—	212	4	no	—
92	C₂²	—	53	2	no	—
93	C₂²	—	106	1	no	—
94	C₂²	—	106	1	no	—
95	C₂²	C₂×C₄⋊#6157	1	1	yes	characteristic
96	C₄	—	53	1	no	—
97	C₄	—	53	1	no	—
98	C₄	—	106	1	no	—
99	C₄	—	106	1	no	—
100	C₄	#13133286	1	2	yes	—
101	C₄	#13133287	1	2	yes	—
102	C₄	D₅₃⋊M₁₆	1	2	yes	—
103	C₂	—	106	2	no	—
104	C₂	#16811628	1	1	yes	characteristic
105	C₂	C₂×#13133286	1	1	yes	characteristic
106	C₂	C₂×#13133287	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811627

C₂

yes

maximal

#16811487

C₂

yes

maximal, complemented

C₄×C₈×D₅₃

C₂

yes

maximal, characteristic

C₂×#11719936

—

C₄⋊#88324

—

C₄⋊#88324

C₂²

yes

—

C₂×C₈×D₅₃

C₄

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

C₄×#88322

C₄

yes

characteristic

C₈×Dic₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

Q₈

yes

characteristic

C₂×#88324

—

#11719936

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

Q₈

yes

characteristic

C₄×Dic₅₃

C₂³

yes

characteristic

C₂×#88322

C₂×C₄

yes

—

C₈×D₅₃

—

C₂²×D₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

C₂×Q₈

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

D₈

yes

—

C₄×D₅₃

Q₁₆

yes

—

#88324

—

C₈×C₅₃

—

#88322

M₁₆

yes

complemented

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#107

yes

—

C₄×C₅₃

#107

yes

—

C₄×C₅₃

#108

yes

—

C₄×C₅₃

C₂×M₁₆

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂×D₈

yes

characteristic

Dic₅₃

C₂×Q₁₆

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#423

—

C₂×C₄×C₈

—

#335

—

C₅₃

#1567

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#6157

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₂×#6157

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#6157

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²×#6157

yes

characteristic

C₈

—

C₈

—

C₈

—

212

—

C₂²

—

C₂²

—

106

—

C₂²

—

106

—

C₂²

C₂×C₄⋊#6157

yes

characteristic

C₄

—

C₄

—

C₄

—

106

—

C₄

—

106

—

100

C₄

#13133286

yes

—

101

C₄

#13133287

yes

—

102

C₄

D₅₃⋊M₁₆

yes

—

103

C₂

—

106

—

104

C₂

#16811628

yes

characteristic

105

C₂

C₂×#13133286

yes

characteristic

106

C₂

C₂×#13133287

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	C₄×C₈	#6157
hypocenter	C₅₃	#1567
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₄×C₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂×C₄⋊C₄

C₄×C₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
2. #16811487	C₂	semidirect
37. #88322	M₁₆	semidirect
57. C₅₃	#1567	semidirect

H ≅ G/N

type

2. #16811487

C₂

semidirect

37. #88322

M₁₆

semidirect

57. C₅₃

#1567

semidirect

Subgroups — #20290274

Conjugacy classes of subgroups

Complements