Subgroups — C₅₃×#13281 #27220793

Order 13568.

366 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#499	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1846	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1380	C₂	1	4	yes	maximal
5	C₂×#125×C₅₃	—	2	1	no	—
6	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
7	C₅₃×#499	—	2	2	no	—
8	C₅₃×#499	C₄	1	4	yes	complemented
9	C₂×#107×C₅₃	C₂²	1	2	yes	—
10	C₂×#107×C₅₃	C₂²	1	2	yes	—
11	C₂×#106×C₅₃	—	2	1	no	—
12	C₂×#106×C₅₃	—	2	1	no	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#382	—	2	1	no	—
15	C₄×C₁₆×C₅₃	—	2	2	no	—
16	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
18	#125×C₅₃	—	2	4	no	—
19	#125×C₅₃	—	4	2	no	—
20	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
21	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
22	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
23	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₅₃	—	2	2	no	—
25	C₂×C₁₆×C₅₃	—	2	2	no	—
26	C₂×C₁₆×C₅₃	Q₈	1	4	yes	—
27	#107×C₅₃	—	2	2	no	—
28	#107×C₅₃	—	2	4	no	—
29	#107×C₅₃	C₂×C₄	1	4	yes	—
30	#106×C₅₃	—	2	4	no	—
31	#106×C₅₃	—	4	2	no	—
32	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₅₃	D₄	1	2	yes	—
34	#95×C₅₃	—	4	2	no	—
35	C₂²×C₄×C₅₃	—	4	2	no	—
36	C₂²×C₄×C₅₃	—	4	2	no	—
37	C₂²×C₄×C₅₃	D₈	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
41	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
42	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
45	C₂×C₈×C₅₃	#41	1	2	yes	—
46	C₄⋊C₄×C₅₃	—	2	4	no	—
47	C₄⋊C₄×C₅₃	—	4	8	no	—
48	C₄⋊C₄×C₅₃	—	4	2	no	—
49	C₄⋊C₄×C₅₃	—	4	2	no	—
50	C₄⋊C₄×C₅₃	—	4	2	no	—
51	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
52	C₄²×C₅₃	SD₁₆	1	2	yes	—
53	C₁₆×C₅₃	—	2	4	no	—
54	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	2	4	no	—
56	C₂×C₄×C₅₃	—	4	4	no	—
57	C₂×C₄×C₅₃	—	8	4	no	—
58	C₂×C₄×C₅₃	—	8	2	no	—
59	C₂×C₄×C₅₃	—	8	4	no	—
60	C₂×C₄×C₅₃	#102	1	4	yes	—
61	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
67	C₈×C₅₃	#103	1	4	yes	—
68	C₈×C₅₃	C₄×Q₈	1	2	yes	—
69	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
70	#13281	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
72	C₂²×C₅₃	#486	1	1	yes	characteristic
73	C₂²×C₅₃	#500	1	1	yes	characteristic
74	C₂²×C₅₃	#508	1	4	yes	—
75	C₄×C₅₃	—	2	4	no	—
76	C₄×C₅₃	—	8	4	no	—
77	C₄×C₅₃	—	16	2	no	—
78	C₄×C₅₃	#391	1	1	yes	characteristic
79	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
80	C₄×C₅₃	#487	1	1	yes	characteristic
81	C₄×C₅₃	#498	1	1	yes	characteristic
82	C₂×#499	C₂×C₅₃	1	1	yes	characteristic
83	C₂×C₄×C₁₆	C₂×C₅₃	1	1	yes	characteristic
84	#1846	C₂×C₅₃	1	1	yes	characteristic
85	#1380	C₂×C₅₃	1	4	yes	—
86	C₂×C₅₃	#1990	1	1	yes	characteristic
87	C₂×C₅₃	#2138	1	2	yes	—
88	C₂×C₅₃	#2242	1	2	yes	—
89	C₂×C₅₃	#2253	1	2	yes	—
90	C₂×#125	—	2	1	no	—
91	C₂²×C₁₆	C₂²×C₅₃	1	2	yes	—
92	#499	—	2	2	no	—
93	#499	C₄×C₅₃	1	4	yes	complemented
94	C₂×#107	C₂²×C₅₃	1	2	yes	—
95	C₂×#107	C₂²×C₅₃	1	2	yes	—
96	C₂×#106	—	2	1	no	—
97	C₂×#106	—	2	1	no	—
98	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
99	#382	—	2	1	no	—
100	C₄×C₁₆	—	2	2	no	—
101	C₅₃	#13281	1	1	yes	characteristic, direct product factor, Sylow
102	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
103	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
104	#125	—	2	4	no	—
105	#125	—	4	2	no	—
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄⋊C₄	—	2	2	no	—
108	C₂×C₄⋊C₄	—	4	2	no	—
109	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
110	C₂×C₁₆	—	2	2	no	—
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	Q₈×C₅₃	1	4	yes	—
113	#107	—	2	2	no	—
114	#107	—	2	4	no	—
115	#107	C₂×C₄×C₅₃	1	4	yes	—
116	#106	—	2	4	no	—
117	#106	—	4	2	no	—
118	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
119	C₄×C₈	D₄×C₅₃	1	2	yes	—
120	#95	—	4	2	no	—
121	C₂²×C₄	—	4	2	no	—
122	C₂²×C₄	—	4	2	no	—
123	C₂²×C₄	D₈×C₅₃	1	2	yes	—
124	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
125	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
126	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
127	C₂×C₈	Q₁₆×C₅₃	1	2	yes	—
128	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
129	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
130	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
131	C₂×C₈	#41×C₅₃	1	2	yes	—
132	C₄⋊C₄	—	2	4	no	—
133	C₄⋊C₄	—	4	8	no	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	—	4	2	no	—
137	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
138	C₄²	SD₁₆×C₅₃	1	2	yes	—
139	C₁₆	—	2	4	no	—
140	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	4	4	no	—
143	C₂×C₄	—	8	4	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	—	8	4	no	—
146	C₂×C₄	#102×C₅₃	1	4	yes	—
147	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
148	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
149	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
150	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
151	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
152	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
153	C₈	#103×C₅₃	1	4	yes	—
154	C₈	C₄×Q₈×C₅₃	1	2	yes	—
155	C₈	C₂²⋊Q₈×C₅₃	1	2	yes	—
156	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
157	C₂²	C₅₃×#486	1	1	yes	characteristic
158	C₂²	C₅₃×#500	1	1	yes	characteristic
159	C₂²	C₅₃×#508	1	4	yes	—
160	C₄	—	2	4	no	—
161	C₄	—	8	4	no	—
162	C₄	—	16	2	no	—
163	C₄	C₅₃×#391	1	1	yes	characteristic
164	C₄	C₂×#103×C₅₃	1	1	yes	characteristic
165	C₄	C₅₃×#487	1	1	yes	characteristic
166	C₄	C₅₃×#498	1	1	yes	characteristic
167	C₂	C₅₃×#1990	1	1	yes	characteristic
168	C₂	C₅₃×#2138	1	2	yes	—
169	C₂	C₅₃×#2242	1	2	yes	—
170	C₂	C₅₃×#2253	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₅₃×#499	C₄	semidirect
70. #13281	C₅₃	direct (N × H)
93. #499	C₄×C₅₃	semidirect
101. C₅₃	#13281	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#499	2
C₂×C₄×C₁₆×C₅₃	2
C₅₃×#1846	2
C₅₃×#1380	2
#13281	53

Subgroups — C53×#13281 #27220793

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13281 #27220793