Subgroups — C₅₃×#11410 #27218922

Order 13568.

306 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2162	C₂	1	2	yes	maximal
2	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1312	C₂	1	2	yes	maximal, complemented
4	C₈×C₁₆×C₅₃	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×C₄⋊C₁₆	C₄	1	4	yes	—
9	C₅₃×C₂²⋊C₁₆	C₄	1	4	yes	—
10	C₄×C₁₆×C₅₃	C₂²	1	4	yes	—
11	C₈²×C₅₃	C₄	1	1	yes	characteristic
12	C₈²×C₅₃	C₄	1	1	yes	characteristic
13	C₈²×C₅₃	C₂²	1	1	yes	characteristic
14	C₈²×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₅₃	C₈	1	2	yes	—
16	C₂²×C₈×C₅₃	C₈	1	2	yes	—
17	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆×C₅₃	C₂×C₄	1	8	yes	—
20	C₄×C₈×C₅₃	C₈	1	2	yes	—
21	C₄×C₈×C₅₃	C₈	1	2	yes	—
22	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₅₃	—	2	4	no	—
33	C₂×C₈×C₅₃	—	2	4	no	—
34	C₂×C₈×C₅₃	C₁₆	1	8	yes	complemented
35	C₂×C₈×C₅₃	C₄²	1	2	yes	—
36	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
37	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
38	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
39	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
41	C₄²×C₅₃	C₄²	1	1	yes	characteristic
42	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
43	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
44	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₁₆×C₅₃	—	2	8	no	—
46	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	—
53	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₅₃	—	2	2	no	—
57	C₈×C₅₃	—	2	2	no	—
58	C₈×C₅₃	C₂×C₁₆	1	4	yes	—
59	C₈×C₅₃	C₂×C₁₆	1	4	yes	—
60	C₈×C₅₃	M₃₂	1	4	yes	—
61	C₈×C₅₃	#131	1	4	yes	—
62	#11410	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂²×C₅₃	—	2	1	no	—
64	C₂²×C₅₃	—	2	1	no	—
65	C₂²×C₅₃	C₄×C₁₆	1	2	yes	—
66	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
67	C₄×C₅₃	—	2	1	no	—
68	C₄×C₅₃	—	2	1	no	—
69	C₄×C₅₃	C₄×C₁₆	1	2	yes	—
70	C₄×C₅₃	#405	1	2	yes	—
71	C₄×C₅₃	C₂²×C₁₆	1	2	yes	—
72	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
73	#2162	C₂×C₅₃	1	2	yes	—
74	C₂×C₈²	C₂×C₅₃	1	1	yes	characteristic
75	#1312	C₂×C₅₃	1	2	yes	complemented
76	C₈×C₁₆	C₂×C₅₃	1	2	yes	complemented
77	C₂×C₅₃	—	2	2	no	—
78	C₂×C₅₃	#1450	1	1	yes	characteristic
79	C₂×C₅₃	C₂×C₄×C₁₆	1	1	yes	characteristic
80	C₂×C₅₃	C₄×M₃₂	1	1	yes	characteristic
81	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
82	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
84	C₄⋊C₁₆	C₄×C₅₃	1	4	yes	—
85	C₂²⋊C₁₆	C₄×C₅₃	1	4	yes	—
86	C₄×C₁₆	C₂²×C₅₃	1	4	yes	—
87	C₈²	C₄×C₅₃	1	1	yes	characteristic
88	C₈²	C₄×C₅₃	1	1	yes	characteristic
89	C₈²	C₂²×C₅₃	1	1	yes	characteristic
90	C₈²	C₂²×C₅₃	1	1	yes	characteristic
91	C₅₃	#11410	1	1	yes	characteristic, direct product factor, Sylow
92	C₂²×C₈	C₈×C₅₃	1	2	yes	—
93	C₂²×C₈	C₈×C₅₃	1	2	yes	—
94	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
95	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
96	C₂×C₁₆	C₂×C₄×C₅₃	1	8	yes	—
97	C₄×C₈	C₈×C₅₃	1	2	yes	—
98	C₄×C₈	C₈×C₅₃	1	2	yes	—
99	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
100	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
101	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
102	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
103	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
104	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
105	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
106	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
107	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
108	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
109	C₂×C₈	—	2	4	no	—
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	C₁₆×C₅₃	1	8	yes	complemented
112	C₂×C₈	C₄²×C₅₃	1	2	yes	—
113	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
114	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
115	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
116	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
117	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
118	C₄²	C₄²×C₅₃	1	1	yes	characteristic
119	C₄²	C₂×C₈×C₅₃	1	1	yes	characteristic
120	C₄²	C₂×C₈×C₅₃	1	1	yes	characteristic
121	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
122	C₁₆	—	2	8	no	—
123	C₂³	C₄×C₈×C₅₃	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	C₄×C₈×C₅₃	1	1	yes	characteristic
127	C₂×C₄	C₄×C₈×C₅₃	1	1	yes	characteristic
128	C₂×C₄	C₄×C₈×C₅₃	1	1	yes	characteristic
129	C₂×C₄	C₂×C₁₆×C₅₃	1	4	yes	—
130	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
131	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
132	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	2	2	no	—
135	C₈	C₂×C₁₆×C₅₃	1	4	yes	—
136	C₈	C₂×C₁₆×C₅₃	1	4	yes	—
137	C₈	M₃₂×C₅₃	1	4	yes	—
138	C₈	#131×C₅₃	1	4	yes	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	C₄×C₁₆×C₅₃	1	2	yes	—
142	C₂²	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	C₄×C₁₆×C₅₃	1	2	yes	—
146	C₄	C₅₃×#405	1	2	yes	—
147	C₄	C₂²×C₁₆×C₅₃	1	2	yes	—
148	C₄	C₂×M₃₂×C₅₃	1	2	yes	—
149	C₂	—	2	2	no	—
150	C₂	C₅₃×#1450	1	1	yes	characteristic
151	C₂	C₂×C₄×C₁₆×C₅₃	1	1	yes	characteristic
152	C₂	C₄×M₃₂×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈²×C₅₃	C₂²
commutator	C₂	C₂×C₄×C₁₆×C₅₃
Frattini, cosocle	C₄×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×C₄×C₈
prime radical	C₂³×C₅₃	C₄×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1312	C₂	semidirect
4. C₈×C₁₆×C₅₃	C₂	semidirect
34. C₂×C₈×C₅₃	C₁₆	semidirect
62. #11410	C₅₃	direct (N × H)
75. #1312	C₂×C₅₃	semidirect
76. C₈×C₁₆	C₂×C₅₃	semidirect
91. C₅₃	#11410	direct (N × H)
111. C₂×C₈	C₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2162	2
C₂×C₈²×C₅₃	2
C₅₃×#1312	2
C₈×C₁₆×C₅₃	2
#11410	53

Subgroups — C53×#11410 #27218922

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11410 #27218922