Group #20290431 — subgroups

Subgroups — #20290431

Order 6784.

272 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 98 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	#16811582	C₂	1	1	yes	maximal, characteristic
2	C₄²×#6157	C₂	1	1	yes	maximal, characteristic
3	(C₂×#6157)⋊C₈	C₂	1	4	yes	maximal
4	#97×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×D₅₃⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄×#6157	C₄	1	4	yes	—
7	C₂×C₄×#6157	C₂²	1	2	yes	—
8	#13133272	C₄	1	4	yes	complemented
9	C₂×C₈×D₅₃	C₂²	1	2	yes	—
10	C₄²×D₅₃	C₂²	1	1	yes	characteristic
11	#13133128	C₄	1	1	yes	characteristic, complemented
12	#13133127	C₄	1	2	yes	complemented
13	#97×C₅₃	C₄	1	1	yes	characteristic, complemented
14	C₂²×#6157	C₂×C₄	1	4	yes	—
15	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
16	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
17	C₂×#88324	C₂×C₄	1	4	yes	—
18	C₄×#6157	—	2	8	no	—
19	C₄×#6157	—	2	4	no	—
20	D₅₃⋊C₈	—	2	4	no	—
21	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
25	C₂×#88322	C₂×C₄	1	2	yes	—
26	C₈×D₅₃	—	2	4	no	—
27	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
28	C₂×#6157	—	2	4	no	—
29	C₂×#6157	—	2	4	no	—
30	C₂×#6157	M₁₆	1	8	yes	—
31	C₂×C₄×C₅₃	C₄²	1	2	yes	—
32	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×Dic₅₃	C₄²	1	2	yes	—
34	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₄×D₅₃	—	2	4	no	—
36	C₄×D₅₃	#41	1	4	yes	—
37	#88324	—	2	4	no	—
38	C₈×C₅₃	—	2	2	no	—
39	#88322	—	2	2	no	—
40	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
41	C₂×D₅₃	#117	1	2	yes	—
42	C₂×D₅₃	C₂×M₁₆	1	4	yes	—
43	#6157	—	2	8	no	—
44	C₄×C₅₃	—	2	2	no	—
45	C₄×C₅₃	#117	1	2	yes	—
46	Dic₅₃	—	2	2	no	—
47	Dic₅₃	#117	1	2	yes	—
48	#1745	—	53	1	no	maximal, Sylow
49	C₂×C₅₃	#376	1	1	yes	characteristic
50	C₂×C₅₃	C₄×M₁₆	1	2	yes	—
51	D₅₃	#432	1	4	yes	—
52	C₂×#97	—	53	1	no	—
53	C₂×#97	—	53	1	no	—
54	C₄³	—	53	1	no	—
55	#336	—	53	4	no	—
56	C₅₃	#1745	1	1	yes	characteristic, complemented, Sylow
57	C₂²×C₈	—	53	2	no	—
58	C₂²×C₈	—	53	2	no	—
59	C₂×C₄²	—	53	1	no	—
60	C₂×C₄²	—	53	2	no	—
61	C₂×C₄²	—	53	4	no	—
62	#97	—	53	4	no	—
63	#97	—	53	2	no	—
64	#97	—	53	1	no	—
65	#97	#6157	1	1	yes	characteristic, complemented
66	C₂²×C₄	—	53	1	no	—
67	C₂²×C₄	—	53	4	no	—
68	C₂²×C₄	—	53	2	no	—
69	C₂×C₈	—	53	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	2	no	—
75	C₄²	—	53	1	no	—
76	C₄²	—	106	4	no	—
77	C₄²	—	106	8	no	—
78	C₄²	C₂×#6157	1	1	yes	characteristic
79	C₂³	—	53	1	no	—
80	C₂×C₄	—	53	1	no	—
81	C₂×C₄	—	53	4	no	—
82	C₂×C₄	—	53	8	no	—
83	C₂×C₄	—	53	2	no	—
84	C₂×C₄	—	106	4	no	—
85	C₂×C₄	—	106	4	no	—
86	C₂×C₄	—	106	4	no	—
87	C₂×C₄	C₄×#6157	1	2	yes	—
88	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
89	C₈	—	2	2	no	—
90	C₈	—	106	4	no	—
91	C₈	—	106	2	no	—
92	C₂²	—	53	2	no	—
93	C₂²	—	53	4	no	—
94	C₂²	C₂×C₄×#6157	1	1	yes	characteristic
95	C₄	—	2	2	no	—
96	C₄	—	53	2	no	—
97	C₄	—	106	8	no	—
98	C₄	—	106	2	no	—
99	C₄	#13133296	1	2	yes	—
100	C₂	—	53	4	no	—
101	C₂	#16811559	1	1	yes	characteristic
102	C₂	M₁₆×#6157	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811582

C₂

yes

maximal, characteristic

C₄²×#6157

C₂

yes

maximal, characteristic

(C₂×#6157)⋊C₈

C₂

yes

maximal

#97×D₅₃

C₂

yes

maximal, characteristic

C₂×D₅₃⋊C₈

C₂²

yes

—

C₂×C₄×#6157

C₄

yes

—

C₂×C₄×#6157

C₂²

yes

—

#13133272

C₄

yes

complemented

C₂×C₈×D₅₃

C₂²

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

#13133128

C₄

yes

characteristic, complemented

#13133127

C₄

yes

complemented

#97×C₅₃

C₄

yes

characteristic, complemented

C₂²×#6157

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×#88324

C₂×C₄

yes

—

C₄×#6157

—

C₄×#6157

—

D₅₃⋊C₈

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

C₈×D₅₃

—

C₂²×D₅₃

C₂²×C₄

yes

—

—

M₁₆

yes

—

C₂×C₄×C₅₃

C₄²

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₄²

yes

—

C₂×Dic₅₃

C₂²×C₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

#41

yes

—

#88324

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₂×C₄²

yes

characteristic

C₂×D₅₃

#117

yes

—

C₂×D₅₃

C₂×M₁₆

yes

—

#6157

—

C₄×C₅₃

—

C₄×C₅₃

#117

yes

—

Dic₅₃

—

Dic₅₃

#117

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₄×M₁₆

yes

—

D₅₃

#432

yes

—

C₂×#97

—

C₂×#97

—

C₄³

—

#336

—

C₅₃

#1745

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

—

—

—

yes

characteristic, complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₂×#6157

yes

—

C₄²

—

C₄²

—

C₄²

—

106

—

C₄²

—

106

—

C₄²

C₂×#6157

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

C₄×#6157

yes

—

C₂×C₄

C₂²×#6157

yes

characteristic

C₈

—

C₈

—

106

—

C₈

—

106

—

C₂²

—

C₂²

—

C₂²

C₂×C₄×#6157

yes

characteristic

C₄

—

C₄

—

C₄

—

106

—

C₄

—

106

—

C₄

#13133296

yes

—

100

C₂

—

101

C₂

#16811559

yes

characteristic

102

C₂

M₁₆×#6157

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#6157
commutator, socle	C₂²×C₅₃	C₂×C₄²
Fitting	#97×C₅₃	C₄
hypercenter	#97	#6157
hypocenter	C₅₃	#1745
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #97

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#6157

commutator, socle

C₂²×C₅₃

C₂×C₄²

Fitting

#97×C₅₃

C₄

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ #97

N	H ≅ G/N	type
8. #13133272	C₄	semidirect
11. #13133128	C₄	semidirect
12. #13133127	C₄	semidirect
13. #97×C₅₃	C₄	semidirect
56. C₅₃	#1745	semidirect
65. #97	#6157	semidirect

H ≅ G/N

type

8. #13133272

C₄

semidirect

11. #13133128

C₄

semidirect

12. #13133127

C₄

semidirect

13. #97×C₅₃

C₄

semidirect

56. C₅₃

#1745

semidirect

65. #97

#6157

semidirect

Subgroups — #20290431

Conjugacy classes of subgroups

Complements