Subgroups — C₂×C₈×C₁₆×C₅₃ #27218908

Order 13568.

398 conjugacy classes of subgroups.

The subgroup lattice has 106 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₆×C₅₃	C₂	1	2	yes	maximal
2	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₈×C₁₆×C₅₃	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×C₅₃	C₄	1	4	yes	—
5	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
6	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₄×C₁₆×C₅₃	C₄	1	4	yes	—
8	C₄×C₁₆×C₅₃	C₂²	1	4	yes	—
9	C₈²×C₅₃	C₄	1	2	yes	—
10	C₈²×C₅₃	C₂²	1	2	yes	—
11	C₂²×C₈×C₅₃	C₈	1	4	yes	—
12	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
13	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₁₆×C₅₃	C₈	1	16	yes	direct product factor
15	C₂×C₁₆×C₅₃	C₂×C₄	1	8	yes	—
16	C₄×C₈×C₅₃	C₈	1	4	yes	—
17	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
20	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
23	C₂×C₈×C₅₃	C₁₆	1	16	yes	direct product factor
24	C₂×C₈×C₅₃	C₄²	1	2	yes	—
25	C₂×C₈×C₅₃	C₂×C₈	1	8	yes	—
26	C₂×C₈×C₅₃	C₂×C₈	1	8	yes	—
27	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
28	C₄²×C₅₃	C₄²	1	1	yes	characteristic
29	C₄²×C₅₃	C₂×C₈	1	2	yes	—
30	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₁₆×C₅₃	C₂×C₈	1	16	yes	direct product factor
32	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
35	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
36	C₂×C₄×C₅₃	C₂×C₁₆	1	8	yes	—
37	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
39	C₈×C₅₃	C₄×C₈	1	4	yes	—
40	C₈×C₅₃	C₂×C₁₆	1	16	yes	direct product factor
41	C₈×C₅₃	C₂²×C₈	1	4	yes	—
42	C₂×C₈×C₁₆	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂²×C₅₃	C₈²	1	2	yes	—
44	C₂²×C₅₃	C₄×C₁₆	1	4	yes	—
45	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
46	C₄×C₅₃	C₈²	1	2	yes	—
47	C₄×C₅₃	C₄×C₁₆	1	4	yes	—
48	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
49	C₄×C₅₃	C₂²×C₁₆	1	4	yes	—
50	C₂×C₄×C₁₆	C₂×C₅₃	1	2	yes	—
51	C₂×C₈²	C₂×C₅₃	1	1	yes	characteristic
52	C₈×C₁₆	C₂×C₅₃	1	4	yes	direct product factor
53	C₂×C₅₃	C₈×C₁₆	1	4	yes	direct product factor
54	C₂×C₅₃	C₂×C₈²	1	1	yes	characteristic
55	C₂×C₅₃	C₂×C₄×C₁₆	1	2	yes	—
56	C₂²×C₁₆	C₄×C₅₃	1	4	yes	—
57	C₂×C₄×C₈	C₄×C₅₃	1	2	yes	—
58	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
59	C₄×C₁₆	C₄×C₅₃	1	4	yes	—
60	C₄×C₁₆	C₂²×C₅₃	1	4	yes	—
61	C₈²	C₄×C₅₃	1	2	yes	—
62	C₈²	C₂²×C₅₃	1	2	yes	—
63	C₅₃	C₂×C₈×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
64	C₂²×C₈	C₈×C₅₃	1	4	yes	—
65	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
66	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
67	C₂×C₁₆	C₈×C₅₃	1	16	yes	direct product factor
68	C₂×C₁₆	C₂×C₄×C₅₃	1	8	yes	—
69	C₄×C₈	C₈×C₅₃	1	4	yes	—
70	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
71	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
72	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
73	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
74	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
75	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
76	C₂×C₈	C₁₆×C₅₃	1	16	yes	direct product factor
77	C₂×C₈	C₄²×C₅₃	1	2	yes	—
78	C₂×C₈	C₂×C₈×C₅₃	1	8	yes	—
79	C₂×C₈	C₂×C₈×C₅₃	1	8	yes	—
80	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
81	C₄²	C₄²×C₅₃	1	1	yes	characteristic
82	C₄²	C₂×C₈×C₅₃	1	2	yes	—
83	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
84	C₁₆	C₂×C₈×C₅₃	1	16	yes	direct product factor
85	C₂³	C₄×C₈×C₅₃	1	1	yes	characteristic
86	C₂×C₄	C₄×C₈×C₅₃	1	1	yes	characteristic
87	C₂×C₄	C₄×C₈×C₅₃	1	2	yes	—
88	C₂×C₄	C₄×C₈×C₅₃	1	4	yes	—
89	C₂×C₄	C₂×C₁₆×C₅₃	1	8	yes	—
90	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
91	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
92	C₈	C₄×C₈×C₅₃	1	4	yes	—
93	C₈	C₂×C₁₆×C₅₃	1	16	yes	direct product factor
94	C₈	C₂²×C₈×C₅₃	1	4	yes	—
95	C₂²	C₈²×C₅₃	1	2	yes	—
96	C₂²	C₄×C₁₆×C₅₃	1	4	yes	—
97	C₂²	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
98	C₄	C₈²×C₅₃	1	2	yes	—
99	C₄	C₄×C₁₆×C₅₃	1	4	yes	—
100	C₄	C₂×C₄×C₈×C₅₃	1	2	yes	—
101	C₄	C₂²×C₁₆×C₅₃	1	4	yes	—
102	C₂	C₈×C₁₆×C₅₃	1	4	yes	direct product factor
103	C₂	C₂×C₈²×C₅₃	1	1	yes	characteristic
104	C₂	C₂×C₄×C₁₆×C₅₃	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₈×C₁₆×C₅₃	C₂	direct (N × H)
14. C₂×C₁₆×C₅₃	C₈	direct (N × H)
23. C₂×C₈×C₅₃	C₁₆	direct (N × H)
31. C₁₆×C₅₃	C₂×C₈	direct (N × H)
40. C₈×C₅₃	C₂×C₁₆	direct (N × H)
42. C₂×C₈×C₁₆	C₅₃	direct (N × H)
52. C₈×C₁₆	C₂×C₅₃	direct (N × H)
53. C₂×C₅₃	C₈×C₁₆	direct (N × H)
63. C₅₃	C₂×C₈×C₁₆	direct (N × H)
67. C₂×C₁₆	C₈×C₅₃	direct (N × H)
76. C₂×C₈	C₁₆×C₅₃	direct (N × H)
84. C₁₆	C₂×C₈×C₅₃	direct (N × H)
93. C₈	C₂×C₁₆×C₅₃	direct (N × H)
102. C₂	C₈×C₁₆×C₅₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₁₆×C₅₃	2
C₂×C₈²×C₅₃	2
C₈×C₁₆×C₅₃	2
C₂×C₈×C₁₆	53

Subgroups — C2×C8×C16×C53 #27218908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×C₁₆×C₅₃ #27218908