Subgroups — C₅₃×#1838 #20270775

Order 6784.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#423	C₂	1	2	yes	maximal
2	C₂×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#341	C₂	1	4	yes	maximal
4	C₂×M₁₆×C₅₃	C₂²	1	4	yes	—
5	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
6	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
7	C₄⋊C₈×C₅₃	—	2	4	no	—
8	#97×C₅₃	C₄	1	4	yes	—
9	C₂²×C₄×C₅₃	Q₈	1	2	yes	—
10	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
11	M₁₆×C₅₃	—	2	8	no	—
12	C₂×C₈×C₅₃	—	2	4	no	—
13	C₂×C₈×C₅₃	—	2	4	no	—
14	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
15	C₄²×C₅₃	C₂×C₄	1	2	yes	—
16	C₄²×C₅₃	D₄	1	2	yes	—
17	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
18	C₂×C₄×C₅₃	—	4	2	no	—
19	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
20	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
21	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
22	C₂×C₄×C₅₃	#41	1	4	yes	—
23	C₈×C₅₃	—	2	4	no	—
24	C₈×C₅₃	—	4	4	no	—
25	C₂²×C₅₃	—	2	2	no	—
26	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₅₃	#125	1	2	yes	—
28	C₄×C₅₃	—	2	4	no	—
29	C₄×C₅₃	#117	1	2	yes	—
30	C₄×C₅₃	#124	1	2	yes	—
31	#1838	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂×C₅₃	—	2	2	no	—
33	C₂×C₅₃	#382	1	1	yes	characteristic
34	C₂×C₅₃	#430	1	2	yes	—
35	#423	C₂×C₅₃	1	2	yes	—
36	C₂×#97	C₂×C₅₃	1	1	yes	characteristic
37	#341	C₂×C₅₃	1	4	yes	—
38	C₅₃	#1838	1	1	yes	characteristic, direct product factor, Sylow
39	C₂×M₁₆	C₂²×C₅₃	1	4	yes	—
40	C₂²×C₈	C₂²×C₅₃	1	2	yes	—
41	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
42	C₄⋊C₈	—	2	4	no	—
43	#97	C₄×C₅₃	1	4	yes	—
44	C₂²×C₄	Q₈×C₅₃	1	2	yes	—
45	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
46	M₁₆	—	2	8	no	—
47	C₂×C₈	—	2	4	no	—
48	C₂×C₈	—	2	4	no	—
49	C₂×C₈	C₂×C₄×C₅₃	1	4	yes	—
50	C₄²	C₂×C₄×C₅₃	1	2	yes	—
51	C₄²	D₄×C₅₃	1	2	yes	—
52	C₂³	C₂×Q₈×C₅₃	1	1	yes	characteristic
53	C₂×C₄	—	4	2	no	—
54	C₂×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
55	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
56	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
57	C₂×C₄	#41×C₅₃	1	4	yes	—
58	C₈	—	2	4	no	—
59	C₈	—	4	4	no	—
60	C₂²	—	2	2	no	—
61	C₂²	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
62	C₂²	#125×C₅₃	1	2	yes	—
63	C₄	—	2	4	no	—
64	C₄	#117×C₅₃	1	2	yes	—
65	C₄	#124×C₅₃	1	2	yes	—
66	C₂	—	2	2	no	—
67	C₂	C₅₃×#382	1	1	yes	characteristic
68	C₂	C₅₃×#430	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×C₄⋊C₄
prime radical	C₂³×C₅₃	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
31. #1838	C₅₃	direct (N × H)
38. C₅₃	#1838	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#423	2
C₂×#97×C₅₃	2
C₅₃×#341	2
#1838	53

Subgroups — C53×#1838 #20270775

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#1838 #20270775