Subgroups — C₅₃×#2110 #20271047

Order 6784.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×C₅₃	C₂	1	2	yes	maximal
2	C₅₃×#405	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#347	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₅₃	C₄	1	1	yes	characteristic
5	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
6	#117×C₅₃	C₄	1	1	yes	characteristic
7	M₃₂×C₅₃	C₄	1	4	yes	complemented
8	M₃₂×C₅₃	C₂²	1	4	yes	—
9	C₂×C₁₆×C₅₃	C₄	1	4	yes	complemented
10	#97×C₅₃	C₂²	1	2	yes	—
11	C₄×C₈×C₅₃	C₄	1	2	yes	—
12	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
13	M₁₆×C₅₃	—	2	2	no	—
14	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
15	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
19	C₄⋊C₄×C₅₃	—	2	1	no	—
20	C₂²⋊C₄×C₅₃	—	2	1	no	—
21	C₄²×C₅₃	C₂×C₄	1	2	yes	—
22	C₁₆×C₅₃	C₂×C₄	1	8	yes	—
23	C₂³×C₅₃	M₁₆	1	1	yes	characteristic
24	C₂×C₄×C₅₃	—	2	2	no	—
25	C₂×C₄×C₅₃	C₄²	1	2	yes	—
26	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
27	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
28	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
29	C₈×C₅₃	—	2	2	no	—
30	C₈×C₅₃	C₄²	1	1	yes	characteristic
31	C₈×C₅₃	C₄²	1	1	yes	characteristic
32	C₈×C₅₃	C₂²×C₄	1	2	yes	—
33	C₂²×C₅₃	—	4	1	no	—
34	C₂²×C₅₃	#97	1	2	yes	—
35	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
36	C₄×C₅₃	—	4	2	no	—
37	C₄×C₅₃	#97	1	2	yes	—
38	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
39	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
40	#2110	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂×C₅₃	—	2	1	no	—
42	C₂×C₅₃	—	2	2	no	—
43	C₂×C₅₃	C₂×#97	1	1	yes	characteristic
44	C₂×M₃₂	C₂×C₅₃	1	2	yes	—
45	#405	C₂×C₅₃	1	1	yes	characteristic
46	#347	C₂×C₅₃	1	4	yes	complemented
47	C₅₃	#2110	1	1	yes	characteristic, direct product factor, Sylow
48	C₂×M₁₆	C₄×C₅₃	1	1	yes	characteristic
49	C₂²×C₈	C₂²×C₅₃	1	1	yes	characteristic
50	#117	C₄×C₅₃	1	1	yes	characteristic
51	M₃₂	C₄×C₅₃	1	4	yes	complemented
52	M₃₂	C₂²×C₅₃	1	4	yes	—
53	C₂×C₁₆	C₄×C₅₃	1	4	yes	complemented
54	#97	C₂²×C₅₃	1	2	yes	—
55	C₄×C₈	C₄×C₅₃	1	2	yes	—
56	C₂²×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
57	M₁₆	—	2	2	no	—
58	C₂×C₈	C₂×C₄×C₅₃	1	2	yes	—
59	C₂×C₈	C₂×C₄×C₅₃	1	2	yes	—
60	C₂×C₈	C₂×C₄×C₅₃	1	2	yes	—
61	C₂×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
62	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
63	C₄⋊C₄	—	2	1	no	—
64	C₂²⋊C₄	—	2	1	no	—
65	C₄²	C₂×C₄×C₅₃	1	2	yes	—
66	C₁₆	C₂×C₄×C₅₃	1	8	yes	—
67	C₂³	M₁₆×C₅₃	1	1	yes	characteristic
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	C₄²×C₅₃	1	2	yes	—
70	C₂×C₄	M₁₆×C₅₃	1	1	yes	characteristic
71	C₂×C₄	M₁₆×C₅₃	1	2	yes	—
72	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
73	C₈	—	2	2	no	—
74	C₈	C₄²×C₅₃	1	1	yes	characteristic
75	C₈	C₄²×C₅₃	1	1	yes	characteristic
76	C₈	C₂²×C₄×C₅₃	1	2	yes	—
77	C₂²	—	4	1	no	—
78	C₂²	#97×C₅₃	1	2	yes	—
79	C₂²	C₂×M₁₆×C₅₃	1	1	yes	characteristic
80	C₄	—	4	2	no	—
81	C₄	#97×C₅₃	1	2	yes	—
82	C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
83	C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	2	2	no	—
86	C₂	C₂×#97×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₅₃	C₄²
commutator	C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂×C₈	C₂³×C₅₃
socle	C₂×C₅₃	C₂×#97
prime radical	C₂³×C₅₃	M₁₆
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#347	C₂	semidirect
7. M₃₂×C₅₃	C₄	semidirect
9. C₂×C₁₆×C₅₃	C₄	semidirect
40. #2110	C₅₃	direct (N × H)
46. #347	C₂×C₅₃	semidirect
47. C₅₃	#2110	direct (N × H)
51. M₃₂	C₄×C₅₃	semidirect
53. C₂×C₁₆	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×M₃₂×C₅₃	2
C₅₃×#405	2
C₅₃×#347	2
#2110	53

Subgroups — C53×#2110 #20271047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2110 #20271047