Group #20272649 — subgroups

Subgroups — #20272649

Order 6784.

134 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 41 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	#16810349	C₂	1	2	yes	maximal
2	C₅₃×#391	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#88322	—	2	2	no	—
4	C₂²×Q₈×C₅₃	C₄	1	1	yes	characteristic
5	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
6	C₂×C₄²×C₅₃	C₄	1	1	yes	characteristic
7	#95×C₅₃	—	2	2	no	—
8	C₂×Q₈×C₅₃	—	2	1	no	—
9	C₂×Q₈×C₅₃	—	4	1	no	—
10	C₂×Q₈×C₅₃	—	4	1	no	—
11	C₂×Q₈×C₅₃	C₈	1	2	yes	complemented
12	C₂²×C₄×C₅₃	—	2	1	no	—
13	C₂²×C₄×C₅₃	—	2	1	no	—
14	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	D₄	1	2	yes	—
16	C₄⋊C₄×C₅₃	—	4	1	no	—
17	C₄²×C₅₃	—	4	1	no	—
18	C₂×#88322	—	4	2	no	—
19	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
20	Q₈×C₅₃	—	4	2	no	—
21	Q₈×C₅₃	—	4	2	no	—
22	C₂×C₄×C₅₃	—	2	1	no	—
23	C₂×C₄×C₅₃	—	2	1	no	—
24	C₂×C₄×C₅₃	—	2	2	no	—
25	C₂×C₄×C₅₃	—	2	2	no	—
26	C₂×C₄×C₅₃	—	2	2	no	—
27	C₂×C₄×C₅₃	—	4	1	no	—
28	C₂×C₄×C₅₃	—	4	1	no	—
29	C₂×C₄×C₅₃	—	4	1	no	—
30	C₂×C₄×C₅₃	—	4	1	no	—
31	C₂×C₄×C₅₃	—	4	2	no	—
32	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
34	#88322	—	8	2	no	—
35	C₂²×C₅₃	—	2	1	no	—
36	C₂²×C₅₃	—	2	1	no	—
37	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
38	C₂²×C₅₃	#99	1	1	yes	characteristic
39	C₂²×C₅₃	#100	1	1	yes	characteristic
40	C₄×C₅₃	—	2	1	no	—
41	C₄×C₅₃	—	2	1	no	—
42	C₄×C₅₃	—	4	2	no	—
43	C₄×C₅₃	—	4	2	no	—
44	C₄×C₅₃	—	4	2	no	—
45	#1319	—	53	1	no	maximal, Sylow
46	C₂×C₅₃	—	2	1	no	—
47	C₂×C₅₃	—	2	1	no	—
48	C₂×C₅₃	#323	1	1	yes	characteristic
49	C₂×C₅₃	#354	1	1	yes	characteristic
50	C₂×C₅₃	#356	1	1	yes	characteristic
51	#391	D₅₃	1	1	yes	characteristic
52	#324	—	53	2	no	—
53	C₅₃	#1319	1	1	yes	characteristic, complemented, Sylow
54	C₂²×Q₈	Dic₅₃	1	1	yes	characteristic
55	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
56	C₂×C₄²	Dic₅₃	1	1	yes	characteristic
57	C₂²⋊C₈	—	106	2	no	—
58	#95	—	2	2	no	—
59	C₂×Q₈	—	2	1	no	—
60	C₂×Q₈	—	4	1	no	—
61	C₂×Q₈	—	4	1	no	—
62	C₂×Q₈	#88322	1	2	yes	complemented
63	C₂²×C₄	—	2	1	no	—
64	C₂²×C₄	—	2	1	no	—
65	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
66	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
67	C₂×C₈	—	212	2	no	—
68	C₄⋊C₄	—	4	1	no	—
69	C₄²	—	4	1	no	—
70	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
71	Q₈	—	4	2	no	—
72	Q₈	—	4	2	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	4	1	no	—
79	C₂×C₄	—	4	1	no	—
80	C₂×C₄	—	4	1	no	—
81	C₂×C₄	—	4	1	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	C₂×#88322	1	1	yes	characteristic
84	C₂×C₄	#11719917	1	1	yes	characteristic
85	C₈	—	424	2	no	—
86	C₂²	—	2	1	no	—
87	C₂²	—	2	1	no	—
88	C₂²	C₂²⋊#88322	1	1	yes	characteristic
89	C₂²	#13133132	1	1	yes	characteristic
90	C₂²	#13133134	1	1	yes	characteristic
91	C₄	—	2	1	no	—
92	C₄	—	2	1	no	—
93	C₄	—	4	2	no	—
94	C₄	—	4	2	no	—
95	C₄	—	4	2	no	—
96	C₂	—	2	1	no	—
97	C₂	—	2	1	no	—
98	C₂	#16810347	1	1	yes	characteristic
99	C₂	#16810418	1	1	yes	characteristic
100	C₂	#16810422	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810349

C₂

yes

maximal

C₅₃×#391

C₂

yes

maximal, characteristic

C₂²⋊#88322

—

C₂²×Q₈×C₅₃

C₄

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₄

yes

characteristic

#95×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₈

yes

complemented

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₂×#88322

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

M₁₆

yes

characteristic

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²⋊C₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

#1319

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

characteristic

#324

—

C₅₃

#1319

yes

characteristic, complemented, Sylow

C₂²×Q₈

Dic₅₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

C₂×C₄²

Dic₅₃

yes

characteristic

C₂²⋊C₈

—

106

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

#88322

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂×C₈

—

212

—

C₄⋊C₄

—

C₄²

—

C₂³

C₂²⋊Dic₅₃

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

424

—

C₂²

—

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₂

—

C₂

—

C₂

yes

C₂

yes

100

C₂

#16810422

yes

characteristic

	subgroup	G/H
center	C₂²	#13133132
commutator	C₂×C₄×C₅₃	C₂×C₈
Frattini	C₂×C₄⋊C₄	C₂×D₅₃
Fitting	C₅₃×#391	C₂
socle	C₂²×C₅₃	#99
hypercenter	#391	D₅₃
hypocenter	C₅₃	#1319
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #391

subgroup

G/H

center

C₂²

#13133132

commutator

C₂×C₄×C₅₃

C₂×C₈

Frattini

C₂×C₄⋊C₄

C₂×D₅₃

Fitting

C₅₃×#391

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #391

N	H ≅ G/N	type
11. C₂×Q₈×C₅₃	C₈	semidirect
53. C₅₃	#1319	semidirect
62. C₂×Q₈	#88322	semidirect

H ≅ G/N

type

11. C₂×Q₈×C₅₃

C₈

semidirect

53. C₅₃

#1319

semidirect

62. C₂×Q₈

#88322

semidirect

Subgroups — #20272649

Conjugacy classes of subgroups

Complements