Group #20276820 — subgroups

Subgroups — #20276820

Order 6784.

150 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 49 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	C₂×#13133156	C₂	1	1	yes	maximal, characteristic
2	#16810861	C₂	1	2	yes	maximal
3	#16810451	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810427	C₂	1	2	yes	maximal, complemented
5	C₂×Q₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
7	#13133193	—	2	2	no	—
8	#13133156	—	2	2	no	—
9	C₂×#11719908	C₂²	1	1	yes	characteristic
10	#13133151	C₂²	1	2	yes	—
11	#13133141	—	2	2	no	—
12	C₂×Q₁₆×C₅₃	C₂²	1	2	yes	—
13	Q₃₂×C₅₃	—	2	2	no	—
14	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₅₃	—	2	2	no	—
16	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
17	Q₁₆×C₅₃	—	2	2	no	—
18	Q₁₆×C₅₃	—	2	2	no	—
19	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
20	#11719921	—	4	2	no	—
21	C₄⋊Dic₅₃	—	2	2	no	—
22	#11719919	—	4	2	no	—
23	C₂×#88322	D₄	1	1	yes	characteristic
24	C₈×D₅₃	—	2	1	no	—
25	C₈×D₅₃	—	2	1	no	—
26	C₁₆×C₅₃	D₄	1	2	yes	—
27	#11719908	—	2	1	no	—
28	C₂²×D₅₃	D₈	1	1	yes	characteristic
29	Q₈×C₅₃	—	4	2	no	—
30	Q₈×C₅₃	—	4	2	no	—
31	C₂×C₄×C₅₃	—	4	2	no	—
32	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₂×Dic₅₃	—	4	2	no	—
34	C₂×Dic₅₃	D₈	1	1	yes	characteristic
35	C₄×D₅₃	—	2	1	no	—
36	C₄×D₅₃	—	2	1	no	—
37	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₈×C₅₃	#41	1	1	yes	characteristic
39	#88322	—	2	1	no	—
40	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
41	C₂×D₅₃	—	2	1	no	—
42	C₂×D₅₃	—	2	2	no	—
43	C₄×C₅₃	—	8	2	no	—
44	C₄×C₅₃	#121	1	1	yes	characteristic
45	C₄×C₅₃	#135	1	1	yes	characteristic
46	Dic₅₃	—	2	1	no	—
47	Dic₅₃	—	8	2	no	—
48	#2219	—	53	1	no	maximal, Sylow
49	C₂×C₅₃	#466	1	1	yes	characteristic
50	C₂×C₅₃	#510	1	2	yes	—
51	D₅₃	—	4	1	no	—
52	C₂×Q₃₂	D₅₃	1	1	yes	characteristic, complemented
53	C₂×M₃₂	—	53	1	no	—
54	#467	—	53	2	no	—
55	#367	—	53	1	no	—
56	#358	—	53	2	no	—
57	C₅₃	#2219	1	1	yes	characteristic, complemented, Sylow
58	C₂×Q₁₆	C₂×D₅₃	1	2	yes	—
59	C₂²×C₈	—	53	1	no	—
60	C₂²⋊Q₈	—	106	2	no	—
61	Q₃₂	—	2	2	no	—
62	M₃₂	—	106	2	no	—
63	C₂×C₁₆	—	53	1	no	—
64	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
65	#107	—	53	2	no	—
66	#103	—	106	2	no	—
67	C₂×Q₈	—	2	2	no	—
68	C₂²×C₄	—	53	1	no	—
69	Q₁₆	—	2	2	no	—
70	Q₁₆	—	2	2	no	—
71	C₂×C₈	—	53	1	no	—
72	C₂×C₈	—	106	1	no	—
73	C₂×C₈	—	106	1	no	—
74	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
75	C₄⋊C₄	—	106	2	no	—
76	C₄⋊C₄	—	212	2	no	—
77	C₂²⋊C₄	—	212	2	no	—
78	C₁₆	—	106	1	no	—
79	C₁₆	C₂²⋊D₅₃	1	2	yes	—
80	C₂³	—	53	1	no	—
81	Q₈	—	4	2	no	—
82	Q₈	—	4	2	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	—	53	1	no	—
85	C₂×C₄	—	106	1	no	—
86	C₂×C₄	—	106	1	no	—
87	C₂×C₄	—	212	2	no	—
88	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
89	C₈	—	106	1	no	—
90	C₈	#11719947	1	1	yes	characteristic
91	C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
92	C₂²	—	106	2	no	—
93	C₂²	—	106	1	no	—
94	C₂²	D₈×D₅₃	1	1	yes	characteristic
95	C₄	—	8	2	no	—
96	C₄	—	106	1	no	—
97	C₄	—	424	2	no	—
98	C₄	#13133188	1	1	yes	characteristic
99	C₄	#13133236	1	1	yes	characteristic
100	C₂	—	212	1	no	—
101	C₂	#16810856	1	1	yes	characteristic
102	C₂	#16811065	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133156

C₂

yes

maximal, characteristic

#16810861

C₂

yes

maximal

#16810451

C₂

yes

maximal, characteristic, complemented

#16810427

C₂

yes

maximal, complemented

C₂×Q₃₂×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₅₃

C₂²

yes

—

—

C₂²

yes

characteristic

#13133151

C₂²

yes

—

#13133141

—

C₂×Q₁₆×C₅₃

C₂²

yes

—

Q₃₂×C₅₃

—

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×C₄×D₅₃

D₄

yes

characteristic

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

—

#11719919

—

C₂×#88322

D₄

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

—

C₁₆×C₅₃

D₄

yes

—

#11719908

—

C₂²×D₅₃

D₈

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

D₈

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

C₂×D₄

yes

C₈×C₅₃

yes

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

—

C₂×Q₃₂

D₅₃

yes

characteristic, complemented

C₂×M₃₂

—

#467

—

#367

—

#358

—

C₅₃

#2219

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×D₅₃

yes

—

C₂²×C₈

—

C₂²⋊Q₈

—

106

—

Q₃₂

—

M₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₃

yes

characteristic

#107

—

#103

—

106

—

C₂×Q₈

—

C₂²×C₄

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₂²⋊C₄

—

212

—

C₁₆

—

106

—

C₁₆

C₂²⋊D₅₃

yes

—

C₂³

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

D₄×D₅₃

yes

characteristic

C₈

—

106

—

C₈

#11719947

yes

characteristic

C₈

C₂×C₂²⋊D₅₃

yes

characteristic

C₂²

—

106

—

C₂²

—

106

—

C₂²

D₈×D₅₃

yes

characteristic

C₄

—

C₄

—

106

—

C₄

—

424

—

C₄

yes

C₄

yes

100

C₂

—

212

—

101

C₂

#16810856

yes

characteristic

102

C₂

#16811065

yes

—

	subgroup	G/H
center	C₂²	D₈×D₅₃
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₂×Q₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	C₂×Q₃₂	D₅₃
hypocenter	C₅₃	#2219
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂²

D₈×D₅₃

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₂×Q₃₂×C₅₃

C₂

socle

C₂²×C₅₃

C₂×D₈

hypercenter

C₂×Q₃₂

D₅₃

hypocenter

C₅₃

#2219

LCS

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

N	H ≅ G/N	type
3. #16810451	C₂	semidirect
4. #16810427	C₂	semidirect
5. C₂×Q₃₂×C₅₃	C₂	semidirect
52. C₂×Q₃₂	D₅₃	semidirect
57. C₅₃	#2219	semidirect

H ≅ G/N

type

3. #16810451

C₂

semidirect

4. #16810427

C₂

semidirect

5. C₂×Q₃₂×C₅₃

C₂

semidirect

52. C₂×Q₃₂

D₅₃

semidirect

57. C₅₃

#2219

semidirect

Subgroups — #20276820

Conjugacy classes of subgroups

Complements