Subgroups — C₅₃×#7844 #27215356

Order 13568.

518 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#1744	C₂	1	2	yes	maximal
2	C₄×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×#322	C₂	1	2	yes	maximal
5	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#97×C₅₃	C₄	1	4	yes	—
7	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
11	C₄³×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#336	C₄	1	8	yes	—
13	C₅₃×#322	C₄	1	8	yes	complemented
14	C₅₃×#322	C₄	1	4	yes	complemented
15	C₈²×C₅₃	C₄	1	4	yes	complemented
16	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
17	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
18	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
21	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
23	#97×C₅₃	—	2	8	no	—
24	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
25	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
27	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
32	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
34	C₂×C₈×C₅₃	—	2	4	no	—
35	C₂×C₈×C₅₃	—	2	8	no	—
36	C₂×C₈×C₅₃	—	2	8	no	—
37	C₂×C₈×C₅₃	C₄²	1	4	yes	—
38	C₂×C₈×C₅₃	C₄²	1	8	yes	—
39	C₂×C₈×C₅₃	C₄²	1	8	yes	—
40	C₂×C₈×C₅₃	C₄²	1	4	yes	—
41	C₂×C₈×C₅₃	M₁₆	1	8	yes	—
42	C₄²×C₅₃	—	2	4	no	—
43	C₄²×C₅₃	—	2	8	no	—
44	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
47	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₄×C₅₃	—	2	4	no	—
49	C₂×C₄×C₅₃	—	2	8	no	—
50	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
53	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
54	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
55	C₂×C₄×C₅₃	#131	1	8	yes	—
56	C₈×C₅₃	—	2	4	no	—
57	C₈×C₅₃	—	2	8	no	—
58	C₈×C₅₃	—	2	8	no	—
59	C₈×C₅₃	#97	1	8	yes	—
60	#7844	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂²×C₅₃	C₄³	1	1	yes	characteristic
62	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
63	C₂²×C₅₃	#405	1	4	yes	—
64	C₄×C₅₃	—	4	4	no	—
65	C₄×C₅₃	C₂×#97	1	2	yes	—
66	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
67	C₄×C₅₃	#405	1	4	yes	—
68	C₄×C₅₃	#405	1	4	yes	—
69	#1744	C₂×C₅₃	1	2	yes	—
70	C₄×#97	C₂×C₅₃	1	1	yes	characteristic
71	C₂×#322	C₂×C₅₃	1	1	yes	characteristic
72	C₂×#322	C₂×C₅₃	1	2	yes	—
73	C₂×C₈²	C₂×C₅₃	1	1	yes	characteristic
74	C₂×C₅₃	#1453	1	4	yes	—
75	C₂×C₅₃	C₄×#97	1	1	yes	characteristic
76	C₂×C₅₃	#1726	1	2	yes	—
77	C₂×#97	C₄×C₅₃	1	4	yes	—
78	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
79	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
80	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
81	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
82	C₄³	C₂²×C₅₃	1	1	yes	characteristic
83	#336	C₄×C₅₃	1	8	yes	—
84	#322	C₄×C₅₃	1	8	yes	complemented
85	#322	C₄×C₅₃	1	4	yes	complemented
86	C₈²	C₄×C₅₃	1	4	yes	complemented
87	C₅₃	#7844	1	1	yes	characteristic, direct product factor, Sylow
88	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
89	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
90	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
91	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
92	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
93	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
94	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
95	#97	—	2	8	no	—
96	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
97	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
98	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
99	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
100	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
101	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
102	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
103	C₂²×C₄	C₄²×C₅₃	1	4	yes	—
104	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
105	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
106	C₂×C₈	—	2	4	no	—
107	C₂×C₈	—	2	8	no	—
108	C₂×C₈	—	2	8	no	—
109	C₂×C₈	C₄²×C₅₃	1	4	yes	—
110	C₂×C₈	C₄²×C₅₃	1	8	yes	—
111	C₂×C₈	C₄²×C₅₃	1	8	yes	—
112	C₂×C₈	C₄²×C₅₃	1	4	yes	—
113	C₂×C₈	M₁₆×C₅₃	1	8	yes	—
114	C₄²	—	2	4	no	—
115	C₄²	—	2	8	no	—
116	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
117	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
118	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
119	C₂³	C₂×C₄²×C₅₃	1	1	yes	characteristic
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	2	8	no	—
122	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
123	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
124	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
125	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
126	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
127	C₂×C₄	#131×C₅₃	1	8	yes	—
128	C₈	—	2	4	no	—
129	C₈	—	2	8	no	—
130	C₈	—	2	8	no	—
131	C₈	#97×C₅₃	1	8	yes	—
132	C₂²	C₄³×C₅₃	1	1	yes	characteristic
133	C₂²	C₄×M₁₆×C₅₃	1	2	yes	—
134	C₂²	C₅₃×#405	1	4	yes	—
135	C₄	—	4	4	no	—
136	C₄	C₂×#97×C₅₃	1	2	yes	—
137	C₄	C₄×M₁₆×C₅₃	1	2	yes	—
138	C₄	C₅₃×#405	1	4	yes	—
139	C₄	C₅₃×#405	1	4	yes	—
140	C₂	C₅₃×#1453	1	4	yes	—
141	C₂	C₄×#97×C₅₃	1	1	yes	characteristic
142	C₂	C₅₃×#1726	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
13. C₅₃×#322	C₄	semidirect
14. C₅₃×#322	C₄	semidirect
15. C₈²×C₅₃	C₄	semidirect
60. #7844	C₅₃	direct (N × H)
84. #322	C₄×C₅₃	semidirect
85. #322	C₄×C₅₃	semidirect
86. C₈²	C₄×C₅₃	semidirect
87. C₅₃	#7844	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#1744	2
C₄×#97×C₅₃	2
C₂×C₅₃×#322	2
C₂×C₈²×C₅₃	2
#7844	53

Subgroups — C53×#7844 #27215356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7844 #27215356