Subgroups — C₅₃×#1835 #20270772

Order 6784.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#382	C₂	1	2	yes	maximal
3	C₅₃×#340	C₂	1	4	yes	maximal
4	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
5	C₂×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
6	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
7	C₄×C₈×C₅₃	C₄	1	4	yes	complemented
8	#95×C₅₃	—	2	4	no	—
9	C₂²×C₄×C₅₃	—	2	4	no	—
10	C₂²×C₄×C₅₃	D₄	1	2	yes	—
11	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
12	C₂×C₈×C₅₃	—	2	4	no	—
13	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
14	C₄⋊C₄×C₅₃	—	2	8	no	—
15	C₄²×C₅₃	C₂×C₄	1	2	yes	—
16	C₄²×C₅₃	Q₈	1	2	yes	—
17	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
18	C₂×C₄×C₅₃	—	2	4	no	—
19	C₂×C₄×C₅₃	—	4	4	no	—
20	C₂×C₄×C₅₃	—	4	8	no	—
21	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
22	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
23	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
24	C₂×C₄×C₅₃	#41	1	4	yes	—
25	C₈×C₅₃	—	2	4	no	—
26	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₅₃	#124	1	2	yes	—
28	C₂²×C₅₃	#135	1	4	yes	—
29	C₄×C₅₃	—	2	4	no	—
30	C₄×C₅₃	—	8	4	no	—
31	C₄×C₅₃	#117	1	2	yes	—
32	C₄×C₅₃	#125	1	2	yes	—
33	#1835	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×C₅₃	#382	1	1	yes	characteristic
35	C₂×C₅₃	#427	1	2	yes	—
36	C₂×C₅₃	#488	1	4	yes	—
37	C₂×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic
38	#382	C₂×C₅₃	1	2	yes	—
39	#340	C₂×C₅₃	1	4	yes	—
40	C₅₃	#1835	1	1	yes	characteristic, direct product factor, Sylow
41	C₂²×C₈	C₂²×C₅₃	1	2	yes	—
42	C₂×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
43	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
44	C₄×C₈	C₄×C₅₃	1	4	yes	complemented
45	#95	—	2	4	no	—
46	C₂²×C₄	—	2	4	no	—
47	C₂²×C₄	D₄×C₅₃	1	2	yes	—
48	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
49	C₂×C₈	—	2	4	no	—
50	C₂×C₈	C₂×C₄×C₅₃	1	4	yes	—
51	C₄⋊C₄	—	2	8	no	—
52	C₄²	C₂×C₄×C₅₃	1	2	yes	—
53	C₄²	Q₈×C₅₃	1	2	yes	—
54	C₂³	C₂×D₄×C₅₃	1	1	yes	characteristic
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	4	4	no	—
57	C₂×C₄	—	4	8	no	—
58	C₂×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
59	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
60	C₂×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
61	C₂×C₄	#41×C₅₃	1	4	yes	—
62	C₈	—	2	4	no	—
63	C₂²	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
64	C₂²	#124×C₅₃	1	2	yes	—
65	C₂²	#135×C₅₃	1	4	yes	—
66	C₄	—	2	4	no	—
67	C₄	—	8	4	no	—
68	C₄	#117×C₅₃	1	2	yes	—
69	C₄	#125×C₅₃	1	2	yes	—
70	C₂	C₅₃×#382	1	1	yes	characteristic
71	C₂	C₅₃×#427	1	2	yes	—
72	C₂	C₅₃×#488	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
7. C₄×C₈×C₅₃	C₄	semidirect
33. #1835	C₅₃	direct (N × H)
40. C₅₃	#1835	direct (N × H)
44. C₄×C₈	C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×C₅₃	2
C₅₃×#382	2
C₅₃×#340	2
#1835	53

Subgroups — C53×#1835 #20270772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#1835 #20270772