Subgroups — C₅₃×#1414 #20270351

Order 6784.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#488	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#355	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#354	C₂	1	1	yes	maximal, characteristic
4	#125×C₅₃	C₄	1	1	yes	characteristic, complemented
5	#124×C₅₃	C₂²	1	1	yes	characteristic
6	#103×C₅₃	—	2	1	no	—
7	#102×C₅₃	—	2	1	no	—
8	#101×C₅₃	—	2	1	no	—
9	#99×C₅₃	—	2	1	no	—
10	C₄×C₈×C₅₃	C₄	1	1	yes	characteristic, complemented
11	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
12	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
13	M₁₆×C₅₃	—	4	1	no	—
14	C₂×C₈×C₅₃	—	2	1	no	—
15	C₄⋊C₄×C₅₃	—	2	1	no	—
16	C₄⋊C₄×C₅₃	—	4	1	no	—
17	C₂²⋊C₄×C₅₃	—	4	1	no	—
18	C₂²⋊C₄×C₅₃	—	4	1	no	—
19	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₅₃	—	2	1	no	—
21	Q₈×C₅₃	—	4	1	no	—
22	D₄×C₅₃	—	4	1	no	—
23	C₂×C₄×C₅₃	—	2	1	no	—
24	C₂×C₄×C₅₃	—	2	1	no	—
25	C₂×C₄×C₅₃	—	4	1	no	—
26	C₂×C₄×C₅₃	—	8	1	no	—
27	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₈×C₅₃	—	4	1	no	—
29	C₈×C₅₃	—	8	1	no	—
30	C₂²×C₅₃	—	4	1	no	—
31	C₂²×C₅₃	—	8	1	no	—
32	C₂²×C₅₃	#99	1	1	yes	characteristic
33	C₄×C₅₃	—	2	1	no	—
34	C₄×C₅₃	—	4	1	no	—
35	C₄×C₅₃	—	4	1	no	—
36	C₄×C₅₃	—	8	1	no	—
37	C₄×C₅₃	—	16	1	no	—
38	#1414	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂×C₅₃	—	2	1	no	—
40	C₂×C₅₃	—	8	1	no	—
41	C₂×C₅₃	#354	1	1	yes	characteristic
42	#488	C₂×C₅₃	1	1	yes	characteristic
43	#355	C₂×C₅₃	1	1	yes	characteristic
44	#354	C₂×C₅₃	1	1	yes	characteristic
45	C₅₃	#1414	1	1	yes	characteristic, direct product factor, Sylow
46	#125	C₄×C₅₃	1	1	yes	characteristic, complemented
47	#124	C₂²×C₅₃	1	1	yes	characteristic
48	#103	—	2	1	no	—
49	#102	—	2	1	no	—
50	#101	—	2	1	no	—
51	#99	—	2	1	no	—
52	C₄×C₈	C₄×C₅₃	1	1	yes	characteristic, complemented
53	C₂×Q₈	D₄×C₅₃	1	1	yes	characteristic
54	C₂×D₄	D₄×C₅₃	1	1	yes	characteristic
55	M₁₆	—	4	1	no	—
56	C₂×C₈	—	2	1	no	—
57	C₄⋊C₄	—	2	1	no	—
58	C₄⋊C₄	—	4	1	no	—
59	C₂²⋊C₄	—	4	1	no	—
60	C₂²⋊C₄	—	4	1	no	—
61	C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
62	C₂³	—	2	1	no	—
63	Q₈	—	4	1	no	—
64	D₄	—	4	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	4	1	no	—
68	C₂×C₄	—	8	1	no	—
69	C₂×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
70	C₈	—	4	1	no	—
71	C₈	—	8	1	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	8	1	no	—
74	C₂²	#99×C₅₃	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	8	1	no	—
79	C₄	—	16	1	no	—
80	C₂	—	2	1	no	—
81	C₂	—	8	1	no	—
82	C₂	C₅₃×#354	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅₃	#354
commutator	C₄²	C₂×C₄×C₅₃
Frattini, cosocle	#124	C₂²×C₅₃
prime radical	C₂×D₄×C₅₃	D₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂²×C₅₃ ◁ C₂×C₄×C₅₃ ◁ #124×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. #125×C₅₃	C₄	semidirect
10. C₄×C₈×C₅₃	C₄	semidirect
38. #1414	C₅₃	direct (N × H)
45. C₅₃	#1414	direct (N × H)
46. #125	C₄×C₅₃	semidirect
52. 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
C₅₃×#488	2
C₅₃×#355	2
C₅₃×#354	2
#1414	53

Subgroups — C53×#1414 #20270351

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#1414 #20270351