Subgroups — C₅₃×#1412 #20270349

Order 6784.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#490	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#353	C₂	1	2	yes	maximal
3	#127×C₅₃	C₂²	1	1	yes	characteristic
4	#125×C₅₃	C₄	1	1	yes	characteristic, complemented
5	#102×C₅₃	—	2	2	no	—
6	#99×C₅₃	—	2	2	no	—
7	#97×C₅₃	C₄	1	1	yes	characteristic, complemented
8	C₂×D₄×C₅₃	—	4	1	no	—
9	C₂×D₄×C₅₃	D₄	1	2	yes	—
10	C₂×C₈×C₅₃	—	2	1	no	—
11	C₄⋊C₄×C₅₃	—	2	1	no	—
12	C₄⋊C₄×C₅₃	—	4	1	no	—
13	C₂²⋊C₄×C₅₃	—	4	2	no	—
14	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₅₃	—	2	2	no	—
16	D₄×C₅₃	—	4	2	no	—
17	D₄×C₅₃	—	8	2	no	—
18	C₂×C₄×C₅₃	—	2	1	no	—
19	C₂×C₄×C₅₃	—	4	1	no	—
20	C₂×C₄×C₅₃	—	8	2	no	—
21	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₈×C₅₃	—	4	1	no	—
23	C₂²×C₅₃	—	4	2	no	—
24	C₂²×C₅₃	—	8	2	no	—
25	C₂²×C₅₃	#99	1	1	yes	characteristic
26	C₄×C₅₃	—	2	1	no	—
27	C₄×C₅₃	—	4	1	no	—
28	C₄×C₅₃	—	8	1	no	—
29	C₄×C₅₃	—	16	2	no	—
30	#1412	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂×C₅₃	—	2	1	no	—
32	C₂×C₅₃	—	8	2	no	—
33	C₂×C₅₃	#354	1	1	yes	characteristic
34	#490	C₂×C₅₃	1	1	yes	characteristic
35	#353	C₂×C₅₃	1	2	yes	—
36	C₅₃	#1412	1	1	yes	characteristic, direct product factor, Sylow
37	#127	C₂²×C₅₃	1	1	yes	characteristic
38	#125	C₄×C₅₃	1	1	yes	characteristic, complemented
39	#102	—	2	2	no	—
40	#99	—	2	2	no	—
41	#97	C₄×C₅₃	1	1	yes	characteristic, complemented
42	C₂×D₄	—	4	1	no	—
43	C₂×D₄	D₄×C₅₃	1	2	yes	—
44	C₂×C₈	—	2	1	no	—
45	C₄⋊C₄	—	2	1	no	—
46	C₄⋊C₄	—	4	1	no	—
47	C₂²⋊C₄	—	4	2	no	—
48	C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
49	C₂³	—	2	2	no	—
50	D₄	—	4	2	no	—
51	D₄	—	8	2	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	—	8	2	no	—
55	C₂×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
56	C₈	—	4	1	no	—
57	C₂²	—	4	2	no	—
58	C₂²	—	8	2	no	—
59	C₂²	#99×C₅₃	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	—	4	1	no	—
62	C₄	—	8	1	no	—
63	C₄	—	16	2	no	—
64	C₂	—	2	1	no	—
65	C₂	—	8	2	no	—
66	C₂	C₅₃×#354	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅₃	#354
commutator	C₄²	C₂×C₄×C₅₃
Frattini, cosocle	#127	C₂²×C₅₃
prime radical	#127×C₅₃	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂²×C₅₃ ◁ C₂×C₄×C₅₃ ◁ #127×C₅₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. #125×C₅₃	C₄	semidirect
7. #97×C₅₃	C₄	semidirect
30. #1412	C₅₃	direct (N × H)
36. C₅₃	#1412	direct (N × H)
38. #125	C₄×C₅₃	semidirect
41. #97	C₄×C₅₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅₃×#490	2
C₅₃×#353	2
#1412	53

Subgroups — C53×#1412 #20270349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#1412 #20270349