Subgroups — C₅₃×#1400 #20270337

Order 6784.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×M₆₄	C₂	1	2	yes	maximal, complemented
3	C₂²×C₈×C₅₃	C₄	1	1	yes	characteristic
4	M₃₂×C₅₃	—	2	2	no	—
5	C₂×C₁₆×C₅₃	C₄	1	1	yes	characteristic
6	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
7	C₃₂×C₅₃	—	2	2	no	—
8	C₂²×C₄×C₅₃	C₈	1	1	yes	characteristic
9	C₂×C₈×C₅₃	—	2	1	no	—
10	C₂×C₈×C₅₃	—	2	1	no	—
11	C₂×C₈×C₅₃	C₈	1	1	yes	characteristic
12	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₁₆×C₅₃	—	2	1	no	—
14	C₁₆×C₅₃	D₄	1	2	yes	—
15	C₂³×C₅₃	C₁₆	1	1	yes	characteristic
16	C₂×C₄×C₅₃	—	2	1	no	—
17	C₂×C₄×C₅₃	—	2	1	no	—
18	C₂×C₄×C₅₃	C₁₆	1	1	yes	characteristic
19	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
20	C₈×C₅₃	—	2	1	no	—
21	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₈×C₅₃	M₁₆	1	1	yes	characteristic
23	C₂²×C₅₃	—	2	1	no	—
24	C₂²×C₅₃	—	4	1	no	—
25	C₂²×C₅₃	C₂×C₁₆	1	1	yes	characteristic
26	C₄×C₅₃	—	2	1	no	—
27	C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
28	C₄×C₅₃	M₃₂	1	1	yes	characteristic
29	#1400	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂×C₅₃	—	2	1	no	—
31	C₂×C₅₃	—	4	1	no	—
32	C₂×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
33	C₂×M₃₂	C₂×C₅₃	1	1	yes	characteristic
34	M₆₄	C₂×C₅₃	1	2	yes	complemented
35	C₅₃	#1400	1	1	yes	characteristic, direct product factor, Sylow
36	C₂²×C₈	C₄×C₅₃	1	1	yes	characteristic
37	M₃₂	—	2	2	no	—
38	C₂×C₁₆	C₄×C₅₃	1	1	yes	characteristic
39	C₂×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
40	C₃₂	—	2	2	no	—
41	C₂²×C₄	C₈×C₅₃	1	1	yes	characteristic
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	—	2	1	no	—
44	C₂×C₈	C₈×C₅₃	1	1	yes	characteristic
45	C₂×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
46	C₁₆	—	2	1	no	—
47	C₁₆	D₄×C₅₃	1	2	yes	—
48	C₂³	C₁₆×C₅₃	1	1	yes	characteristic
49	C₂×C₄	—	2	1	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	C₁₆×C₅₃	1	1	yes	characteristic
52	C₂×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
53	C₈	—	2	1	no	—
54	C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
55	C₈	M₁₆×C₅₃	1	1	yes	characteristic
56	C₂²	—	2	1	no	—
57	C₂²	—	4	1	no	—
58	C₂²	C₂×C₁₆×C₅₃	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	C₂²⋊C₈×C₅₃	1	1	yes	characteristic
61	C₄	M₃₂×C₅₃	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	4	1	no	—
64	C₂	C₅₃×C₂²⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₅₃	C₂²⋊C₄
commutator	C₂²	C₂×C₁₆×C₅₃
Frattini, cosocle	C₂×C₁₆	C₂²×C₅₃
socle	C₂×C₅₃	C₂²⋊C₁₆
prime radical	C₂³×C₅₃	C₁₆
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₅₃ ◁ C₂×C₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×M₆₄	C₂	semidirect
29. #1400	C₅₃	direct (N × H)
34. M₆₄	C₂×C₅₃	semidirect
35. C₅₃	#1400	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×M₃₂×C₅₃	2
C₅₃×M₆₄	2
#1400	53

Subgroups — C53×#1400 #20270337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#1400 #20270337