Subgroups — C₅₃×#1778 #20270715

Order 6784.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#108×C₅₃	C₂	1	2	yes	maximal
2	C₄×M₁₆×C₅₃	C₂	1	2	yes	maximal
3	C₂×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#341	C₂	1	2	yes	maximal
5	C₂×M₁₆×C₅₃	C₂²	1	4	yes	—
6	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	#108×C₅₃	C₄	1	8	yes	complemented
10	#97×C₅₃	—	2	2	no	—
11	#97×C₅₃	C₄	1	4	yes	—
12	C₄×C₈×C₅₃	—	2	2	no	—
13	C₂²×C₄×C₅₃	Q₈	1	2	yes	—
14	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
15	M₁₆×C₅₃	—	2	4	no	—
16	M₁₆×C₅₃	C₂×C₄	1	8	yes	—
17	C₂×C₈×C₅₃	—	2	4	no	—
18	C₂×C₈×C₅₃	—	2	2	no	—
19	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
22	C₄²×C₅₃	C₂×C₄	1	2	yes	—
23	C₄²×C₅₃	D₄	1	2	yes	—
24	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
25	C₂×C₄×C₅₃	—	4	2	no	—
26	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
27	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
29	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₅₃	#41	1	2	yes	—
31	C₈×C₅₃	—	2	2	no	—
32	C₈×C₅₃	—	2	8	no	—
33	C₈×C₅₃	C₄²	1	4	yes	—
34	C₂²×C₅₃	—	2	2	no	—
35	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₅₃	C₄×Q₈	1	2	yes	—
37	C₄×C₅₃	—	2	4	no	—
38	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
39	C₄×C₅₃	#117	1	1	yes	characteristic
40	C₄×C₅₃	C₄×D₄	1	2	yes	—
41	#1778	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂×C₅₃	—	2	2	no	—
43	C₂×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₅₃	#430	1	2	yes	—
45	C₂×#108	C₂×C₅₃	1	2	yes	—
46	C₄×M₁₆	C₂×C₅₃	1	2	yes	—
47	C₂×#97	C₂×C₅₃	1	1	yes	characteristic
48	#341	C₂×C₅₃	1	2	yes	—
49	C₅₃	#1778	1	1	yes	characteristic, direct product factor, Sylow
50	C₂×M₁₆	C₂²×C₅₃	1	4	yes	—
51	C₂²×C₈	C₂²×C₅₃	1	1	yes	characteristic
52	C₂²×C₈	C₂²×C₅₃	1	1	yes	characteristic
53	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
54	#108	C₄×C₅₃	1	8	yes	complemented
55	#97	—	2	2	no	—
56	#97	C₄×C₅₃	1	4	yes	—
57	C₄×C₈	—	2	2	no	—
58	C₂²×C₄	Q₈×C₅₃	1	2	yes	—
59	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
60	M₁₆	—	2	4	no	—
61	M₁₆	C₂×C₄×C₅₃	1	8	yes	—
62	C₂×C₈	—	2	4	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	C₂×C₄×C₅₃	1	2	yes	—
65	C₂×C₈	C₂×C₄×C₅₃	1	2	yes	—
66	C₂×C₈	C₂×C₄×C₅₃	1	4	yes	—
67	C₄²	C₂×C₄×C₅₃	1	2	yes	—
68	C₄²	D₄×C₅₃	1	2	yes	—
69	C₂³	C₂×Q₈×C₅₃	1	1	yes	characteristic
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
72	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
73	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
74	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
75	C₂×C₄	#41×C₅₃	1	2	yes	—
76	C₈	—	2	2	no	—
77	C₈	—	2	8	no	—
78	C₈	C₄²×C₅₃	1	4	yes	—
79	C₂²	—	2	2	no	—
80	C₂²	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
81	C₂²	C₄×Q₈×C₅₃	1	2	yes	—
82	C₄	—	2	4	no	—
83	C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
84	C₄	#117×C₅₃	1	1	yes	characteristic
85	C₄	C₄×D₄×C₅₃	1	2	yes	—
86	C₂	—	2	2	no	—
87	C₂	C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
88	C₂	C₅₃×#430	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂×D₄
commutator	C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂²×C₄	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×C₄⋊C₄
prime radical	C₂³×C₅₃	C₂×Q₈
LCS	G ▷ 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
9. #108×C₅₃	C₄	semidirect
41. #1778	C₅₃	direct (N × H)
49. C₅₃	#1778	direct (N × H)
54. #108	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×#108×C₅₃	2
C₄×M₁₆×C₅₃	2
C₂×#97×C₅₃	2
C₅₃×#341	2
#1778	53

Subgroups — C53×#1778 #20270715

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#1778 #20270715