Subgroups — C₅₃×#13287 #27220799

Order 13568.

370 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic
2	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1844	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1386	C₂	1	4	yes	maximal
5	C₅₃×#533	—	2	1	no	—
6	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
7	C₅₃×#500	C₄	1	2	yes	complemented
8	C₅₃×#499	—	2	1	no	—
9	C₅₃×#499	C₄	1	2	yes	complemented
10	C₅₃×#498	—	2	1	no	—
11	C₅₃×#427	—	2	1	no	—
12	C₅₃×#427	C₂²	1	2	yes	—
13	C₂×#106×C₅₃	—	2	1	no	—
14	C₂×#106×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#384	—	2	1	no	—
17	C₅₃×#346	—	2	1	no	—
18	C₄×C₁₆×C₅₃	—	2	1	no	—
19	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
21	C₄⋊Q₈×C₅₃	—	2	2	no	—
22	#125×C₅₃	—	2	2	no	—
23	C₂²⋊Q₈×C₅₃	—	4	2	no	—
24	C₄×Q₈×C₅₃	—	4	2	no	—
25	#117×C₅₃	—	2	2	no	—
26	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
28	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
29	M₃₂×C₅₃	—	2	2	no	—
30	M₃₂×C₅₃	Q₈	1	4	yes	complemented
31	C₂×C₁₆×C₅₃	—	2	2	no	—
32	#107×C₅₃	—	2	2	no	—
33	#107×C₅₃	C₂×C₄	1	4	yes	—
34	#106×C₅₃	—	2	2	no	—
35	#106×C₅₃	—	2	2	no	—
36	#106×C₅₃	—	2	4	no	—
37	#106×C₅₃	—	4	2	no	—
38	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₅₃	—	4	2	no	—
43	C₂²×C₄×C₅₃	—	4	2	no	—
44	C₂²×C₄×C₅₃	D₈	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
49	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
50	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₅₃	#41	1	2	yes	—
54	C₄⋊C₄×C₅₃	—	2	4	no	—
55	C₄⋊C₄×C₅₃	—	4	4	no	—
56	C₄⋊C₄×C₅₃	—	4	2	no	—
57	C₄⋊C₄×C₅₃	—	4	2	no	—
58	C₄⋊C₄×C₅₃	—	4	2	no	—
59	C₄⋊C₄×C₅₃	—	8	4	no	—
60	C₂²⋊C₄×C₅₃	—	4	2	no	—
61	C₄²×C₅₃	—	4	2	no	—
62	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₅₃	SD₁₆	1	2	yes	—
65	C₁₆×C₅₃	—	2	4	no	—
66	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
67	Q₈×C₅₃	—	8	4	no	—
68	C₂×C₄×C₅₃	—	2	4	no	—
69	C₂×C₄×C₅₃	—	4	4	no	—
70	C₂×C₄×C₅₃	—	8	4	no	—
71	C₂×C₄×C₅₃	—	8	2	no	—
72	C₂×C₄×C₅₃	#102	1	2	yes	—
73	C₂×C₄×C₅₃	#102	1	2	yes	—
74	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
80	C₈×C₅₃	#103	1	2	yes	—
81	C₈×C₅₃	#103	1	2	yes	—
82	C₈×C₅₃	C₄×Q₈	1	2	yes	—
83	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
84	#13287	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂²×C₅₃	—	2	2	no	—
86	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
87	C₂²×C₅₃	#486	1	1	yes	characteristic
88	C₂²×C₅₃	#500	1	1	yes	characteristic
89	C₄×C₅₃	—	2	2	no	—
90	C₄×C₅₃	—	8	4	no	—
91	C₄×C₅₃	—	16	2	no	—
92	C₄×C₅₃	#360	1	4	yes	—
93	C₄×C₅₃	#391	1	1	yes	characteristic
94	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
95	C₄×C₅₃	#487	1	1	yes	characteristic
96	C₄×C₅₃	#498	1	1	yes	characteristic
97	#3160	C₂×C₅₃	1	1	yes	characteristic
98	C₄×M₃₂	C₂×C₅₃	1	1	yes	characteristic
99	#1844	C₂×C₅₃	1	1	yes	characteristic
100	#1386	C₂×C₅₃	1	4	yes	—
101	C₂×C₅₃	—	2	1	no	—
102	C₂×C₅₃	—	2	1	no	—
103	C₂×C₅₃	#1990	1	1	yes	characteristic
104	C₂×C₅₃	C₂×#360	1	2	yes	—
105	#533	—	2	1	no	—
106	C₂×M₃₂	C₂²×C₅₃	1	2	yes	—
107	#500	C₄×C₅₃	1	2	yes	complemented
108	#499	—	2	1	no	—
109	#499	C₄×C₅₃	1	2	yes	complemented
110	#498	—	2	1	no	—
111	#427	—	2	1	no	—
112	#427	C₂²×C₅₃	1	2	yes	—
113	C₂×#106	—	2	1	no	—
114	C₂×#106	C₂²×C₅₃	1	2	yes	—
115	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
116	#384	—	2	1	no	—
117	#346	—	2	1	no	—
118	C₄×C₁₆	—	2	1	no	—
119	C₅₃	#13287	1	1	yes	characteristic, direct product factor, Sylow
120	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
121	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
122	C₄⋊Q₈	—	2	2	no	—
123	#125	—	2	2	no	—
124	C₂²⋊Q₈	—	4	2	no	—
125	C₄×Q₈	—	4	2	no	—
126	#117	—	2	2	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	4	2	no	—
129	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
130	M₃₂	—	2	2	no	—
131	M₃₂	Q₈×C₅₃	1	4	yes	complemented
132	C₂×C₁₆	—	2	2	no	—
133	#107	—	2	2	no	—
134	#107	C₂×C₄×C₅₃	1	4	yes	—
135	#106	—	2	2	no	—
136	#106	—	2	2	no	—
137	#106	—	2	4	no	—
138	#106	—	4	2	no	—
139	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
140	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
141	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
142	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
143	C₂×Q₈	—	4	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	D₈×C₅₃	1	2	yes	—
146	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
147	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
148	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
149	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
150	C₂×C₈	Q₁₆×C₅₃	1	2	yes	—
151	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
152	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
153	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
154	C₂×C₈	#41×C₅₃	1	2	yes	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	4	4	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	8	4	no	—
161	C₂²⋊C₄	—	4	2	no	—
162	C₄²	—	4	2	no	—
163	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
164	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
165	C₄²	SD₁₆×C₅₃	1	2	yes	—
166	C₁₆	—	2	4	no	—
167	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
168	Q₈	—	8	4	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	8	4	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	#102×C₅₃	1	2	yes	—
174	C₂×C₄	#102×C₅₃	1	2	yes	—
175	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
176	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
177	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
178	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
179	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
180	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
181	C₈	#103×C₅₃	1	2	yes	—
182	C₈	#103×C₅₃	1	2	yes	—
183	C₈	C₄×Q₈×C₅₃	1	2	yes	—
184	C₈	C₂²⋊Q₈×C₅₃	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
187	C₂²	C₅₃×#486	1	1	yes	characteristic
188	C₂²	C₅₃×#500	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	8	4	no	—
191	C₄	—	16	2	no	—
192	C₄	C₅₃×#360	1	4	yes	—
193	C₄	C₅₃×#391	1	1	yes	characteristic
194	C₄	C₂×#103×C₅₃	1	1	yes	characteristic
195	C₄	C₅₃×#487	1	1	yes	characteristic
196	C₄	C₅₃×#498	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	C₅₃×#1990	1	1	yes	characteristic
200	C₂	C₂×C₅₃×#360	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₂³×C₅₃	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
7. C₅₃×#500	C₄	semidirect
9. C₅₃×#499	C₄	semidirect
30. M₃₂×C₅₃	Q₈	semidirect
84. #13287	C₅₃	direct (N × H)
107. #500	C₄×C₅₃	semidirect
109. #499	C₄×C₅₃	semidirect
119. C₅₃	#13287	direct (N × H)
131. M₃₂	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3160	2
C₄×M₃₂×C₅₃	2
C₅₃×#1844	2
C₅₃×#1386	2
#13287	53

Subgroups — C53×#13287 #27220799

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13287 #27220799