Subgroups — C₅₃×#8269 #27215781

Order 13568.

366 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1564	C₂	1	2	yes	maximal
3	C₅₃×#1452	C₂	1	2	yes	maximal
4	C₅₃×#1276	C₂	1	2	yes	maximal
5	C₅₃×#431	C₂²	1	4	yes	—
6	C₂×#97×C₅₃	C₄	1	2	yes	—
7	C₂×#97×C₅₃	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₄³×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#335	C₄	1	4	yes	complemented
12	C₅₃×#334	C₄	1	4	yes	complemented
13	C₅₃×#322	—	2	4	no	—
14	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	—	2	2	no	—
18	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
20	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
21	C₂²⋊C₈×C₅₃	—	2	4	no	—
22	#97×C₅₃	—	2	4	no	—
23	#97×C₅₃	—	2	4	no	—
24	C₄×C₈×C₅₃	—	2	4	no	—
25	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₅₃	D₄	1	2	yes	—
28	C₄×C₈×C₅₃	Q₈	1	2	yes	—
29	C₂²×C₄×C₅₃	—	2	2	no	—
30	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	#41	1	2	yes	—
32	C₂×C₈×C₅₃	—	2	2	no	—
33	C₂×C₈×C₅₃	—	2	2	no	—
34	C₂×C₈×C₅₃	—	2	2	no	—
35	C₂×C₈×C₅₃	—	2	8	no	—
36	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₄²	1	2	yes	—
39	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
40	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
41	C₄²×C₅₃	—	2	2	no	—
42	C₄²×C₅₃	—	4	4	no	—
43	C₄²×C₅₃	M₁₆	1	4	yes	—
44	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
45	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
47	C₂³×C₅₃	#117	1	1	yes	characteristic
48	C₂×C₄×C₅₃	—	2	4	no	—
49	C₂×C₄×C₅₃	—	2	4	no	—
50	C₂×C₄×C₅₃	—	4	4	no	—
51	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
54	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
55	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
56	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
57	C₈×C₅₃	—	2	1	no	—
58	C₈×C₅₃	—	2	1	no	—
59	C₈×C₅₃	—	2	4	no	—
60	C₈×C₅₃	—	4	8	no	—
61	C₈×C₅₃	#97	1	4	yes	—
62	#8269	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂²×C₅₃	—	2	2	no	—
64	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₅₃	#432	1	2	yes	—
66	C₄×C₅₃	—	2	2	no	—
67	C₄×C₅₃	—	4	4	no	—
68	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
69	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
70	C₄×C₅₃	#423	1	2	yes	—
71	C₄×C₅₃	#444	1	4	yes	—
72	C₄×#97	C₂×C₅₃	1	1	yes	characteristic
73	#1564	C₂×C₅₃	1	2	yes	—
74	#1452	C₂×C₅₃	1	2	yes	—
75	#1276	C₂×C₅₃	1	2	yes	—
76	C₂×C₅₃	—	2	2	no	—
77	C₂×C₅₃	#1767	1	1	yes	characteristic
78	C₂×C₅₃	#1773	1	2	yes	—
79	#431	C₂²×C₅₃	1	4	yes	—
80	C₂×#97	C₄×C₅₃	1	2	yes	—
81	C₂×#97	C₄×C₅₃	1	2	yes	—
82	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
84	C₄³	C₂²×C₅₃	1	1	yes	characteristic
85	#335	C₄×C₅₃	1	4	yes	complemented
86	#334	C₄×C₅₃	1	4	yes	complemented
87	#322	—	2	4	no	—
88	C₅₃	#8269	1	1	yes	characteristic, direct product factor, Sylow
89	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
90	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
91	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
92	C₂×C₄²	—	2	2	no	—
93	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
94	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
95	C₄⋊C₈	C₂×C₄×C₅₃	1	8	yes	—
96	C₂²⋊C₈	—	2	4	no	—
97	#97	—	2	4	no	—
98	#97	—	2	4	no	—
99	C₄×C₈	—	2	4	no	—
100	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
101	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
102	C₄×C₈	D₄×C₅₃	1	2	yes	—
103	C₄×C₈	Q₈×C₅₃	1	2	yes	—
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
106	C₂²×C₄	#41×C₅₃	1	2	yes	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	8	no	—
111	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
112	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
113	C₂×C₈	C₄²×C₅₃	1	2	yes	—
114	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
115	C₂×C₈	M₁₆×C₅₃	1	4	yes	—
116	C₄²	—	2	2	no	—
117	C₄²	—	4	4	no	—
118	C₄²	M₁₆×C₅₃	1	4	yes	—
119	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
120	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
121	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
122	C₂³	#117×C₅₃	1	1	yes	characteristic
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	4	4	no	—
126	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
128	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
129	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
130	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
131	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	—	2	4	no	—
135	C₈	—	4	8	no	—
136	C₈	#97×C₅₃	1	4	yes	—
137	C₂²	—	2	2	no	—
138	C₂²	C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
139	C₂²	C₅₃×#432	1	2	yes	—
140	C₄	—	2	2	no	—
141	C₄	—	4	4	no	—
142	C₄	C₂×#97×C₅₃	1	1	yes	characteristic
143	C₄	C₄×M₁₆×C₅₃	1	1	yes	characteristic
144	C₄	C₅₃×#423	1	2	yes	—
145	C₄	C₅₃×#444	1	4	yes	—
146	C₂	—	2	2	no	—
147	C₂	C₅₃×#1767	1	1	yes	characteristic
148	C₂	C₅₃×#1773	1	2	yes	—

	subgroup	G/H
center	C₄²×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	C₄×C₄⋊C₄
prime radical	C₂³×C₅₃	#117
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₅₃×#335	C₄	semidirect
12. C₅₃×#334	C₄	semidirect
62. #8269	C₅₃	direct (N × H)
85. #335	C₄×C₅₃	semidirect
86. #334	C₄×C₅₃	semidirect
88. C₅₃	#8269	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#97×C₅₃	2
C₅₃×#1564	2
C₅₃×#1452	2
C₅₃×#1276	2
#8269	53

Subgroups — C53×#8269 #27215781

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8269 #27215781