Subgroups — C₅₃×#13627 #27221139

Order 13568.

218 conjugacy classes of subgroups.

The subgroup lattice has 124 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×M₆₄	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×C₄⋊C₃₂	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
7	C₅₃×M₆₄	—	2	4	no	—
8	C₂×C₃₂×C₅₃	C₂²	1	4	yes	—
9	C₄×C₁₆×C₅₃	C₄	1	2	yes	—
10	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
11	C₂²×C₈×C₅₃	C₈	1	1	yes	characteristic
12	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄²×C₅₃	C₈	1	1	yes	characteristic
14	C₂×C₁₆×C₅₃	—	2	2	no	—
15	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
16	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
18	C₂×C₁₆×C₅₃	Q₈	1	2	yes	—
19	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
20	C₄×C₈×C₅₃	C₈	1	2	yes	—
21	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₃₂×C₅₃	—	2	4	no	—
23	C₂²×C₄×C₅₃	C₁₆	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
25	C₂×C₈×C₅₃	—	2	2	no	—
26	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
29	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
30	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₄²×C₅₃	C₁₆	1	1	yes	characteristic
32	C₄²×C₅₃	C₁₆	1	1	yes	characteristic
33	C₄²×C₅₃	C₂×C₈	1	2	yes	—
34	C₁₆×C₅₃	—	2	2	no	—
35	C₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
36	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₁₆×C₅₃	C₂×Q₈	1	1	yes	characteristic
38	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	2	4	no	—
40	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
41	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
42	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
44	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
45	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
46	C₈×C₅₃	—	2	2	no	—
47	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
48	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
50	#13627	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂²×C₅₃	—	2	2	no	—
52	C₂²×C₅₃	C₄⋊C₁₆	1	2	yes	—
53	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
54	C₄×C₅₃	—	2	2	no	—
55	C₄×C₅₃	C₄⋊C₁₆	1	2	yes	—
56	C₄×C₅₃	M₆₄	1	4	yes	complemented
57	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
58	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
59	C₂×M₆₄	C₂×C₅₃	1	2	yes	—
60	C₂×C₄×C₁₆	C₂×C₅₃	1	1	yes	characteristic
61	C₄⋊C₃₂	C₂×C₅₃	1	4	yes	complemented
62	C₂×C₅₃	—	2	2	no	—
63	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
64	C₂×C₅₃	C₂×M₆₄	1	2	yes	—
65	C₂²×C₁₆	C₄×C₅₃	1	1	yes	characteristic
66	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
67	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
68	M₆₄	—	2	4	no	—
69	C₂×C₃₂	C₂²×C₅₃	1	4	yes	—
70	C₄×C₁₆	C₄×C₅₃	1	2	yes	—
71	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
72	C₅₃	#13627	1	1	yes	characteristic, direct product factor, Sylow
73	C₂²×C₈	C₈×C₅₃	1	1	yes	characteristic
74	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
75	C₂×C₄²	C₈×C₅₃	1	1	yes	characteristic
76	C₂×C₁₆	—	2	2	no	—
77	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
78	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
79	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
80	C₂×C₁₆	Q₈×C₅₃	1	2	yes	—
81	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
82	C₄×C₈	C₈×C₅₃	1	2	yes	—
83	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
84	C₃₂	—	2	4	no	—
85	C₂²×C₄	C₁₆×C₅₃	1	2	yes	—
86	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
89	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
90	C₂×C₈	C₂×C₈×C₅₃	1	1	yes	characteristic
91	C₂×C₈	M₁₆×C₅₃	1	2	yes	—
92	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
93	C₄²	C₁₆×C₅₃	1	1	yes	characteristic
94	C₄²	C₁₆×C₅₃	1	1	yes	characteristic
95	C₄²	C₂×C₈×C₅₃	1	2	yes	—
96	C₁₆	—	2	2	no	—
97	C₁₆	C₄⋊C₄×C₅₃	1	2	yes	—
98	C₁₆	C₂×D₄×C₅₃	1	1	yes	characteristic
99	C₁₆	C₂×Q₈×C₅₃	1	1	yes	characteristic
100	C₂³	C₂×C₁₆×C₅₃	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	C₄⋊C₈×C₅₃	1	2	yes	—
103	C₂×C₄	C₂×C₁₆×C₅₃	1	2	yes	—
104	C₂×C₄	C₂×C₁₆×C₅₃	1	1	yes	characteristic
105	C₂×C₄	C₂×C₁₆×C₅₃	1	2	yes	—
106	C₂×C₄	M₃₂×C₅₃	1	2	yes	—
107	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
108	C₈	—	2	2	no	—
109	C₈	C₄⋊C₈×C₅₃	1	2	yes	—
110	C₈	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
111	C₈	C₂×M₁₆×C₅₃	1	1	yes	characteristic
112	C₂²	—	2	2	no	—
113	C₂²	C₅₃×C₄⋊C₁₆	1	2	yes	—
114	C₂²	C₂²×C₁₆×C₅₃	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	C₅₃×C₄⋊C₁₆	1	2	yes	—
117	C₄	C₅₃×M₆₄	1	4	yes	complemented
118	C₄	C₂×C₄⋊C₈×C₅₃	1	1	yes	characteristic
119	C₄	C₂×M₃₂×C₅₃	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	C₂×C₅₃×C₄⋊C₁₆	1	1	yes	characteristic
122	C₂	C₂×C₅₃×M₆₄	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆×C₅₃	C₂³
commutator	C₂²	C₂²×C₁₆×C₅₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₅₃
socle	C₂²×C₅₃	C₂²×C₁₆
prime radical	C₂³×C₅₃	C₂×C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×C₄⋊C₃₂	C₂	semidirect
50. #13627	C₅₃	direct (N × H)
56. C₄×C₅₃	M₆₄	semidirect
61. C₄⋊C₃₂	C₂×C₅₃	semidirect
72. C₅₃	#13627	direct (N × H)
117. C₄	C₅₃×M₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×M₆₄	2
C₂×C₄×C₁₆×C₅₃	2
C₅₃×C₄⋊C₃₂	2
#13627	53

Subgroups — C53×#13627 #27221139

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13627 #27221139