Subgroups — C₄×C₅₃×#322 #27215351

Order 13568.

654 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₈×C₅₃	C₂	1	2	yes	maximal
2	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#322	C₂	1	4	yes	maximal
4	C₂×C₄×C₈×C₅₃	C₄	1	8	yes	—
5	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	—
6	C₂×C₄×C₈×C₅₃	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
8	C₄³×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#322	C₄	1	16	yes	direct product factor
10	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
11	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
12	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
13	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
14	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
15	C₄×C₈×C₅₃	—	2	16	no	—
16	C₄×C₈×C₅₃	C₈	1	16	yes	complemented
17	C₄×C₈×C₅₃	C₂×C₄	1	16	yes	—
18	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
19	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
20	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
22	C₂×C₈×C₅₃	—	2	16	no	—
23	C₂×C₈×C₅₃	C₄²	1	16	yes	—
24	C₂×C₈×C₅₃	C₄²	1	8	yes	—
25	C₂×C₈×C₅₃	C₂×C₈	1	16	yes	—
26	C₄²×C₅₃	C₂×C₈	1	8	yes	—
27	C₄²×C₅₃	M₁₆	1	16	yes	—
28	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
29	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₄×C₈	1	8	yes	—
31	C₂×C₄×C₅₃	#97	1	16	yes	—
32	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
33	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
34	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
35	C₂×C₄×C₅₃	C₂×M₁₆	1	8	yes	—
36	C₈×C₅₃	—	2	16	no	—
37	C₈×C₅₃	C₄×C₈	1	16	yes	complemented
38	C₄×#322	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂²×C₅₃	C₄³	1	1	yes	characteristic
40	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
41	C₂²×C₅₃	C₂×#97	1	4	yes	—
42	C₄×C₅₃	#322	1	16	yes	direct product factor
43	C₄×C₅₃	C₂×C₄×C₈	1	4	yes	—
44	C₄×C₅₃	C₄×M₁₆	1	8	yes	—
45	C₄²×C₈	C₂×C₅₃	1	2	yes	—
46	C₄²×C₈	C₂×C₅₃	1	1	yes	characteristic
47	C₂×#322	C₂×C₅₃	1	4	yes	—
48	C₂×C₅₃	C₂×#322	1	4	yes	—
49	C₂×C₅₃	C₄²×C₈	1	1	yes	characteristic
50	C₂×C₅₃	C₄×#97	1	2	yes	—
51	C₂×C₄×C₈	C₄×C₅₃	1	8	yes	—
52	C₂×C₄×C₈	C₄×C₅₃	1	4	yes	—
53	C₂×C₄×C₈	C₂²×C₅₃	1	4	yes	—
54	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
55	C₄³	C₂²×C₅₃	1	1	yes	characteristic
56	#322	C₄×C₅₃	1	16	yes	direct product factor
57	C₅₃	C₄×#322	1	1	yes	characteristic, direct product factor, Sylow
58	C₂²×C₈	C₂×C₄×C₅₃	1	8	yes	—
59	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
60	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
61	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
62	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
63	C₄×C₈	—	2	16	no	—
64	C₄×C₈	C₈×C₅₃	1	16	yes	complemented
65	C₄×C₈	C₂×C₄×C₅₃	1	16	yes	—
66	C₄×C₈	C₂×C₄×C₅₃	1	8	yes	—
67	C₂²×C₄	C₄²×C₅₃	1	4	yes	—
68	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
69	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
70	C₂×C₈	—	2	16	no	—
71	C₂×C₈	C₄²×C₅₃	1	16	yes	—
72	C₂×C₈	C₄²×C₅₃	1	8	yes	—
73	C₂×C₈	C₂×C₈×C₅₃	1	16	yes	—
74	C₄²	C₂×C₈×C₅₃	1	8	yes	—
75	C₄²	M₁₆×C₅₃	1	16	yes	—
76	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
77	C₂³	C₂×C₄²×C₅₃	1	1	yes	characteristic
78	C₂×C₄	C₄×C₈×C₅₃	1	8	yes	—
79	C₂×C₄	#97×C₅₃	1	16	yes	—
80	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
81	C₂×C₄	C₂×C₄²×C₅₃	1	4	yes	—
82	C₂×C₄	C₂²×C₈×C₅₃	1	4	yes	—
83	C₂×C₄	C₂×M₁₆×C₅₃	1	8	yes	—
84	C₈	—	2	16	no	—
85	C₈	C₄×C₈×C₅₃	1	16	yes	complemented
86	C₂²	C₄³×C₅₃	1	1	yes	characteristic
87	C₂²	C₂×C₄×C₈×C₅₃	1	2	yes	—
88	C₂²	C₂×#97×C₅₃	1	4	yes	—
89	C₄	C₅₃×#322	1	16	yes	direct product factor
90	C₄	C₂×C₄×C₈×C₅₃	1	4	yes	—
91	C₄	C₄×M₁₆×C₅₃	1	8	yes	—
92	C₂	C₂×C₅₃×#322	1	4	yes	—
93	C₂	C₄²×C₈×C₅₃	1	1	yes	characteristic
94	C₂	C₄×#97×C₅₃	1	2	yes	—

	subgroup	G/H
center	C₄³×C₅₃	C₂²
commutator	C₂	C₄²×C₈×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle, prime radical	C₂³×C₅₃	C₂×C₄²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₅₃×#322	C₄	direct (N × H)
16. C₄×C₈×C₅₃	C₈	semidirect
37. C₈×C₅₃	C₄×C₈	semidirect
38. C₄×#322	C₅₃	direct (N × H)
42. C₄×C₅₃	#322	direct (N × H)
56. #322	C₄×C₅₃	direct (N × H)
57. C₅₃	C₄×#322	direct (N × H)
64. C₄×C₈	C₈×C₅₃	semidirect
85. C₈	C₄×C₈×C₅₃	semidirect
89. C₄	C₅₃×#322	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄²×C₈×C₅₃	2
C₂×C₅₃×#322	2
C₄×#322	53

Subgroups — C4×C53×#322 #27215351

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅₃×#322 #27215351