Subgroups — C₅₃×#7842 #27215354

Order 13568.

558 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	C₂×C₅₃×#322	C₂	1	2	yes	maximal
5	C₂×C₅₃×#322	C₂	1	2	yes	maximal
6	C₂×#97×C₅₃	C₄	1	4	yes	—
7	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
12	C₄³×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#336	C₄	1	4	yes	—
14	C₅₃×#322	C₄	1	8	yes	complemented
15	C₅₃×#322	C₄	1	8	yes	complemented
16	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
17	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
19	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
24	#97×C₅₃	—	2	8	no	—
25	C₄×C₈×C₅₃	—	2	8	no	—
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	4	yes	—
31	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
32	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
33	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₅₃	—	2	4	no	—
37	C₂×C₈×C₅₃	—	2	8	no	—
38	C₂×C₈×C₅₃	—	2	8	no	—
39	C₂×C₈×C₅₃	C₄²	1	4	yes	—
40	C₂×C₈×C₅₃	C₄²	1	8	yes	—
41	C₂×C₈×C₅₃	C₄²	1	8	yes	—
42	C₂×C₈×C₅₃	C₄²	1	4	yes	—
43	C₂×C₈×C₅₃	M₁₆	1	8	yes	—
44	C₄²×C₅₃	—	2	4	no	—
45	C₄²×C₅₃	—	2	4	no	—
46	C₄²×C₅₃	M₁₆	1	8	yes	—
47	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
48	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
49	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₄×C₅₃	—	2	4	no	—
51	C₂×C₄×C₅₃	—	2	4	no	—
52	C₂×C₄×C₅₃	#97	1	8	yes	—
53	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
56	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
57	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
58	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
59	C₂×C₄×C₅₃	#131	1	4	yes	—
60	C₈×C₅₃	—	2	4	no	—
61	C₈×C₅₃	—	2	8	no	—
62	C₈×C₅₃	—	2	8	no	—
63	C₈×C₅₃	#97	1	8	yes	complemented
64	#7842	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂²×C₅₃	C₄³	1	1	yes	characteristic
66	C₂²×C₅₃	C₂×#97	1	2	yes	—
67	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
68	C₂²×C₅₃	#405	1	2	yes	—
69	C₄×C₅₃	—	2	8	no	—
70	C₄×C₅₃	C₂×#97	1	2	yes	—
71	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
72	C₄×C₅₃	C₄×M₁₆	1	4	yes	—
73	C₄×C₅₃	#405	1	4	yes	—
74	#1744	C₂×C₅₃	1	1	yes	characteristic
75	C₄×#97	C₂×C₅₃	1	1	yes	characteristic
76	C₄²×C₈	C₂×C₅₃	1	1	yes	characteristic
77	C₂×#322	C₂×C₅₃	1	2	yes	—
78	C₂×#322	C₂×C₅₃	1	2	yes	—
79	C₂×C₅₃	#1451	1	4	yes	—
80	C₂×C₅₃	C₄×#97	1	1	yes	characteristic
81	C₂×C₅₃	C₄×#97	1	1	yes	characteristic
82	C₂×C₅₃	#1726	1	1	yes	characteristic
83	C₂×#97	C₄×C₅₃	1	4	yes	—
84	C₂×C₄×C₈	C₄×C₅₃	1	4	yes	—
85	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
86	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
87	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
88	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
89	C₄³	C₂²×C₅₃	1	1	yes	characteristic
90	#336	C₄×C₅₃	1	4	yes	—
91	#322	C₄×C₅₃	1	8	yes	complemented
92	#322	C₄×C₅₃	1	8	yes	complemented
93	C₅₃	#7842	1	1	yes	characteristic, direct product factor, Sylow
94	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
95	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
96	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
97	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
98	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
99	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
100	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
101	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
102	#97	—	2	8	no	—
103	C₄×C₈	—	2	8	no	—
104	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
105	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
106	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
107	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
108	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
109	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
110	C₂²×C₄	C₄²×C₅₃	1	4	yes	—
111	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
112	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
113	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	2	8	no	—
116	C₂×C₈	—	2	8	no	—
117	C₂×C₈	C₄²×C₅₃	1	4	yes	—
118	C₂×C₈	C₄²×C₅₃	1	8	yes	—
119	C₂×C₈	C₄²×C₅₃	1	8	yes	—
120	C₂×C₈	C₄²×C₅₃	1	4	yes	—
121	C₂×C₈	M₁₆×C₅₃	1	8	yes	—
122	C₄²	—	2	4	no	—
123	C₄²	—	2	4	no	—
124	C₄²	M₁₆×C₅₃	1	8	yes	—
125	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
126	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
127	C₂³	C₂×C₄²×C₅₃	1	1	yes	characteristic
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	#97×C₅₃	1	8	yes	—
131	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
132	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
133	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
134	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
135	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
136	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
137	C₂×C₄	#131×C₅₃	1	4	yes	—
138	C₈	—	2	4	no	—
139	C₈	—	2	8	no	—
140	C₈	—	2	8	no	—
141	C₈	#97×C₅₃	1	8	yes	complemented
142	C₂²	C₄³×C₅₃	1	1	yes	characteristic
143	C₂²	C₂×#97×C₅₃	1	2	yes	—
144	C₂²	C₄×M₁₆×C₅₃	1	2	yes	—
145	C₂²	C₅₃×#405	1	2	yes	—
146	C₄	—	2	8	no	—
147	C₄	C₂×#97×C₅₃	1	2	yes	—
148	C₄	C₄×M₁₆×C₅₃	1	2	yes	—
149	C₄	C₄×M₁₆×C₅₃	1	4	yes	—
150	C₄	C₅₃×#405	1	4	yes	—
151	C₂	C₅₃×#1451	1	4	yes	—
152	C₂	C₄×#97×C₅₃	1	1	yes	characteristic
153	C₂	C₄×#97×C₅₃	1	1	yes	characteristic
154	C₂	C₅₃×#1726	1	1	yes	characteristic

	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
14. C₅₃×#322	C₄	semidirect
15. C₅₃×#322	C₄	semidirect
63. C₈×C₅₃	#97	semidirect
64. #7842	C₅₃	direct (N × H)
91. #322	C₄×C₅₃	semidirect
92. #322	C₄×C₅₃	semidirect
93. C₅₃	#7842	direct (N × H)
141. C₈	#97×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
C₂×C₅₃×#322	2
#7842	53

Subgroups — C53×#7842 #27215354

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7842 #27215354