Subgroups — C₅₃×#8132 #27215644

Order 13568.

478 conjugacy classes of subgroups.

The subgroup lattice has 156 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	1	yes	maximal, characteristic
2	C₄×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1273	C₂	1	4	yes	maximal
5	C₂×#97×C₅₃	C₄	1	2	yes	—
6	C₂×#97×C₅₃	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
11	C₄³×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#336	C₄	1	4	yes	—
13	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
14	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
15	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
16	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
21	#97×C₅₃	—	2	4	no	—
22	#97×C₅₃	C₈	1	8	yes	complemented
23	C₄×C₈×C₅₃	—	2	8	no	—
24	C₄×C₈×C₅₃	—	2	4	no	—
25	C₄×C₈×C₅₃	—	2	4	no	—
26	C₄×C₈×C₅₃	—	2	4	no	—
27	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
28	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
29	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	—	2	4	no	—
33	C₂×C₈×C₅₃	—	2	4	no	—
34	C₂×C₈×C₅₃	—	2	4	no	—
35	C₂×C₈×C₅₃	—	2	4	no	—
36	C₂×C₈×C₅₃	—	2	4	no	—
37	C₂×C₈×C₅₃	—	2	8	no	—
38	C₂×C₈×C₅₃	C₂×C₈	1	8	yes	—
39	C₂×C₈×C₅₃	M₁₆	1	8	yes	—
40	C₄²×C₅₃	—	2	4	no	—
41	C₄²×C₅₃	—	2	4	no	—
42	C₄²×C₅₃	C₂×C₈	1	4	yes	—
43	C₄²×C₅₃	M₁₆	1	4	yes	—
44	C₄²×C₅₃	#41	1	4	yes	—
45	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
46	C₂×C₄×C₅₃	—	2	2	no	—
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	—	2	2	no	—
50	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
51	C₂×C₄×C₅₃	#97	1	4	yes	—
52	C₂×C₄×C₅₃	#117	1	2	yes	—
53	C₂×C₄×C₅₃	#117	1	2	yes	—
54	C₂×C₄×C₅₃	#117	1	2	yes	—
55	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
56	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
57	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
58	C₂×C₄×C₅₃	#131	1	4	yes	—
59	C₈×C₅₃	—	2	8	no	—
60	C₈×C₅₃	—	2	8	no	—
61	C₈×C₅₃	—	4	4	no	—
62	#8132	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂²×C₅₃	#376	1	1	yes	characteristic
64	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
65	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
66	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
67	C₂²×C₅₃	#405	1	2	yes	—
68	C₄×C₅₃	—	2	4	no	—
69	C₄×C₅₃	—	2	4	no	—
70	C₄×C₅₃	#431	1	4	yes	—
71	C₄×C₅₃	#432	1	4	yes	—
72	C₄×C₅₃	#433	1	4	yes	—
73	#1744	C₂×C₅₃	1	1	yes	characteristic
74	C₄×#97	C₂×C₅₃	1	1	yes	characteristic
75	C₄²×C₈	C₂×C₅₃	1	1	yes	characteristic
76	#1273	C₂×C₅₃	1	4	yes	—
77	C₂×C₅₃	C₈×M₁₆	1	2	yes	—
78	C₂×C₅₃	#1451	1	2	yes	—
79	C₂×C₅₃	#1744	1	1	yes	characteristic
80	C₂×C₅₃	#1745	1	1	yes	characteristic
81	C₂×C₅₃	#1746	1	1	yes	characteristic
82	C₂×#97	C₄×C₅₃	1	2	yes	—
83	C₂×#97	C₄×C₅₃	1	2	yes	—
84	C₂×C₄×C₈	C₄×C₅₃	1	4	yes	—
85	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
86	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
87	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
88	C₄³	C₂²×C₅₃	1	1	yes	characteristic
89	#336	C₄×C₅₃	1	4	yes	—
90	C₅₃	#8132	1	1	yes	characteristic, direct product factor, Sylow
91	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
92	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
93	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
94	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
95	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
96	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
97	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
98	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
99	#97	—	2	4	no	—
100	#97	C₈×C₅₃	1	8	yes	complemented
101	C₄×C₈	—	2	8	no	—
102	C₄×C₈	—	2	4	no	—
103	C₄×C₈	—	2	4	no	—
104	C₄×C₈	—	2	4	no	—
105	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
106	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
107	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
108	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
109	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	—	2	4	no	—
112	C₂×C₈	—	2	4	no	—
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	2	8	no	—
116	C₂×C₈	C₂×C₈×C₅₃	1	8	yes	—
117	C₂×C₈	M₁₆×C₅₃	1	8	yes	—
118	C₄²	—	2	4	no	—
119	C₄²	—	2	4	no	—
120	C₄²	C₂×C₈×C₅₃	1	4	yes	—
121	C₄²	M₁₆×C₅₃	1	4	yes	—
122	C₄²	#41×C₅₃	1	4	yes	—
123	C₂³	C₂×C₄²×C₅₃	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	C₄×C₈×C₅₃	1	4	yes	—
129	C₂×C₄	#97×C₅₃	1	4	yes	—
130	C₂×C₄	#117×C₅₃	1	2	yes	—
131	C₂×C₄	#117×C₅₃	1	2	yes	—
132	C₂×C₄	#117×C₅₃	1	2	yes	—
133	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
134	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
135	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
136	C₂×C₄	#131×C₅₃	1	4	yes	—
137	C₈	—	2	8	no	—
138	C₈	—	2	8	no	—
139	C₈	—	4	4	no	—
140	C₂²	C₅₃×#376	1	1	yes	characteristic
141	C₂²	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
142	C₂²	C₂×#97×C₅₃	1	1	yes	characteristic
143	C₂²	C₄×M₁₆×C₅₃	1	2	yes	—
144	C₂²	C₅₃×#405	1	2	yes	—
145	C₄	—	2	4	no	—
146	C₄	—	2	4	no	—
147	C₄	C₅₃×#431	1	4	yes	—
148	C₄	C₅₃×#432	1	4	yes	—
149	C₄	C₅₃×#433	1	4	yes	—
150	C₂	C₈×M₁₆×C₅₃	1	2	yes	—
151	C₂	C₅₃×#1451	1	2	yes	—
152	C₂	C₅₃×#1744	1	1	yes	characteristic
153	C₂	C₅₃×#1745	1	1	yes	characteristic
154	C₂	C₅₃×#1746	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²×C₅₃	C₂³
commutator	C₂²	C₂×C₄×C₈×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
22. #97×C₅₃	C₈	semidirect
62. #8132	C₅₃	direct (N × H)
90. C₅₃	#8132	direct (N × H)
100. #97	C₈×C₅₃	semidirect

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
C₄²×C₈×C₅₃	2
C₅₃×#1273	2
#8132	53

Subgroups — C53×#8132 #27215644

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8132 #27215644