Subgroups — C₅₃×#10820 #27218332

Order 13568.

894 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#340	C₂	1	2	yes	maximal
3	C₂×C₅₃×#340	C₂	1	2	yes	maximal
4	C₅₃×C₄⋊#95	C₂	1	2	yes	maximal
5	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
8	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₅₃	—	2	4	no	—
10	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	complemented
11	C₂×#95×C₅₃	—	2	2	no	—
12	C₂×#95×C₅₃	—	2	2	no	—
13	C₅₃×#340	—	2	4	no	—
14	C₅₃×#340	C₄	1	8	yes	complemented
15	C₂³×C₄×C₅₃	—	2	4	no	—
16	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
17	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
18	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	—	2	2	no	—
20	C₂²×C₈×C₅₃	—	2	2	no	—
21	C₂²×C₈×C₅₃	—	2	2	no	—
22	C₂²×C₈×C₅₃	—	2	2	no	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₅₃	D₄	1	4	yes	—
27	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
32	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
33	C₂×C₄²×C₅₃	—	2	2	no	—
34	C₂×C₄²×C₅₃	—	2	2	no	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
37	C₄×C₈×C₅₃	—	2	8	no	—
38	#95×C₅₃	—	4	4	no	—
39	#95×C₅₃	—	4	4	no	—
40	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	—	2	2	no	—
42	C₂²×C₄×C₅₃	—	2	2	no	—
43	C₂²×C₄×C₅₃	—	2	2	no	—
44	C₂²×C₄×C₅₃	—	2	2	no	—
45	C₂²×C₄×C₅₃	—	2	8	no	—
46	C₂²×C₄×C₅₃	—	4	4	no	—
47	C₂²×C₄×C₅₃	—	4	4	no	—
48	C₂²×C₄×C₅₃	—	4	4	no	—
49	C₂²×C₄×C₅₃	—	4	4	no	—
50	C₂²×C₄×C₅₃	—	4	4	no	—
51	C₂²×C₄×C₅₃	—	4	4	no	—
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	D₈	1	2	yes	—
55	C₂²×C₄×C₅₃	SD₁₆	1	4	yes	—
56	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	#41	1	2	yes	—
62	C₂²×C₄×C₅₃	#41	1	2	yes	—
63	C₂²×C₄×C₅₃	#41	1	2	yes	—
64	C₂²×C₄×C₅₃	#41	1	2	yes	—
65	C₂×C₈×C₅₃	—	2	2	no	—
66	C₂×C₈×C₅₃	—	2	2	no	—
67	C₂×C₈×C₅₃	—	2	2	no	—
68	C₂×C₈×C₅₃	—	2	2	no	—
69	C₂×C₈×C₅₃	—	2	8	no	—
70	C₂×C₈×C₅₃	—	2	8	no	—
71	C₂×C₈×C₅₃	C₄²	1	4	yes	—
72	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
73	C₄⋊C₄×C₅₃	—	2	8	no	—
74	C₄⋊C₄×C₅₃	—	2	8	no	—
75	C₄⋊C₄×C₅₃	—	4	4	no	—
76	C₄⋊C₄×C₅₃	—	4	4	no	—
77	C₄²×C₅₃	—	2	2	no	—
78	C₄²×C₅₃	—	2	2	no	—
79	C₄²×C₅₃	—	2	4	no	—
80	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂³×C₅₃	C₄×D₄	1	2	yes	—
82	C₂³×C₅₃	#121	1	2	yes	—
83	C₂³×C₅₃	#124	1	2	yes	—
84	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
85	C₂³×C₅₃	C₂×SD₁₆	1	2	yes	—
86	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
87	C₂³×C₅₃	#135	1	2	yes	—
88	C₂³×C₅₃	#135	1	2	yes	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	8	no	—
94	C₂×C₄×C₅₃	—	2	4	no	—
95	C₂×C₄×C₅₃	—	2	4	no	—
96	C₂×C₄×C₅₃	—	4	8	no	—
97	C₂×C₄×C₅₃	—	4	8	no	—
98	C₂×C₄×C₅₃	—	4	8	no	—
99	C₂×C₄×C₅₃	—	8	4	no	—
100	C₂×C₄×C₅₃	—	8	4	no	—
101	C₂×C₄×C₅₃	—	8	4	no	—
102	C₂×C₄×C₅₃	—	8	4	no	—
103	C₂×C₄×C₅₃	#106	1	4	yes	—
104	C₂×C₄×C₅₃	#107	1	4	yes	—
105	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
108	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#117	1	4	yes	—
110	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
111	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
113	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
114	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₅₃	#123	1	4	yes	—
116	C₂×C₄×C₅₃	#125	1	2	yes	—
117	C₂×C₄×C₅₃	#126	1	4	yes	—
118	C₈×C₅₃	—	2	4	no	—
119	C₈×C₅₃	—	2	4	no	—
120	#10820	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
122	C₂²×C₅₃	#380	1	1	yes	characteristic
123	C₂²×C₅₃	#382	1	1	yes	characteristic
124	C₂²×C₅₃	#388	1	4	yes	—
125	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
126	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
127	C₂²×C₅₃	#427	1	1	yes	characteristic
128	C₂²×C₅₃	#427	1	1	yes	characteristic
129	C₂²×C₅₃	C₄×D₈	1	2	yes	—
130	C₂²×C₅₃	C₄×SD₁₆	1	4	yes	—
131	C₂²×C₅₃	C₄×Q₁₆	1	2	yes	—
132	C₂²×C₅₃	#465	1	4	yes	—
133	C₂²×C₅₃	#466	1	2	yes	—
134	C₂²×C₅₃	#467	1	2	yes	—
135	C₂²×C₅₃	#486	1	2	yes	—
136	C₂²×C₅₃	#487	1	2	yes	—
137	C₂²×C₅₃	#488	1	4	yes	—
138	C₄×C₅₃	—	2	4	no	—
139	C₄×C₅₃	—	2	4	no	—
140	C₄×C₅₃	—	8	8	no	—
141	C₄×C₅₃	#376	1	1	yes	characteristic
142	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
143	C₄×C₅₃	#381	1	1	yes	characteristic
144	C₄×C₅₃	#383	1	1	yes	characteristic
145	C₄×C₅₃	#387	1	4	yes	—
146	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic
147	C₂×#340	C₂×C₅₃	1	2	yes	—
148	C₂×#340	C₂×C₅₃	1	2	yes	—
149	C₄⋊#95	C₂×C₅₃	1	2	yes	—
150	C₂×C₅₃	#1434	1	1	yes	characteristic
151	C₂×C₅₃	C₄×#106	1	1	yes	characteristic
152	C₂×C₅₃	C₄×#107	1	1	yes	characteristic
153	C₂×C₅₃	#1807	1	1	yes	characteristic
154	C₂×C₅₃	#1808	1	1	yes	characteristic
155	C₂×C₅₃	#1834	1	1	yes	characteristic
156	C₂×C₅₃	#1835	1	1	yes	characteristic
157	C₂×C₅₃	#1933	1	4	yes	—
158	C₂×C₅₃	#1934	1	4	yes	—
159	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
160	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
161	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
162	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
163	C₂×C₄×C₈	—	2	4	no	—
164	C₂×C₄×C₈	C₄×C₅₃	1	4	yes	complemented
165	C₂×#95	—	2	2	no	—
166	C₂×#95	—	2	2	no	—
167	#340	—	2	4	no	—
168	#340	C₄×C₅₃	1	8	yes	complemented
169	C₅₃	#10820	1	1	yes	characteristic, direct product factor, Sylow
170	C₂³×C₄	—	2	4	no	—
171	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
172	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
173	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
179	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
180	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
181	C₂²×C₈	D₄×C₅₃	1	4	yes	—
182	C₂×C₄⋊C₄	—	2	4	no	—
183	C₂×C₄⋊C₄	—	2	4	no	—
184	C₂×C₄⋊C₄	—	2	4	no	—
185	C₂×C₄⋊C₄	—	2	4	no	—
186	C₂×C₄⋊C₄	—	2	4	no	—
187	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	8	yes	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
191	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
192	C₄×C₈	—	2	8	no	—
193	#95	—	4	4	no	—
194	#95	—	4	4	no	—
195	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	8	no	—
201	C₂²×C₄	—	4	4	no	—
202	C₂²×C₄	—	4	4	no	—
203	C₂²×C₄	—	4	4	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	—	4	4	no	—
206	C₂²×C₄	—	4	4	no	—
207	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
208	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
209	C₂²×C₄	D₈×C₅₃	1	2	yes	—
210	C₂²×C₄	SD₁₆×C₅₃	1	4	yes	—
211	C₂²×C₄	Q₁₆×C₅₃	1	2	yes	—
212	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
213	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
214	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
215	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
216	C₂²×C₄	#41×C₅₃	1	2	yes	—
217	C₂²×C₄	#41×C₅₃	1	2	yes	—
218	C₂²×C₄	#41×C₅₃	1	2	yes	—
219	C₂²×C₄	#41×C₅₃	1	2	yes	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	8	no	—
225	C₂×C₈	—	2	8	no	—
226	C₂×C₈	C₄²×C₅₃	1	4	yes	—
227	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
228	C₄⋊C₄	—	2	8	no	—
229	C₄⋊C₄	—	2	8	no	—
230	C₄⋊C₄	—	4	4	no	—
231	C₄⋊C₄	—	4	4	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	4	no	—
235	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
236	C₂³	C₄×D₄×C₅₃	1	2	yes	—
237	C₂³	#121×C₅₃	1	2	yes	—
238	C₂³	#124×C₅₃	1	2	yes	—
239	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
240	C₂³	C₂×SD₁₆×C₅₃	1	2	yes	—
241	C₂³	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
242	C₂³	#135×C₅₃	1	2	yes	—
243	C₂³	#135×C₅₃	1	2	yes	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	4	8	no	—
252	C₂×C₄	—	4	8	no	—
253	C₂×C₄	—	4	8	no	—
254	C₂×C₄	—	8	4	no	—
255	C₂×C₄	—	8	4	no	—
256	C₂×C₄	—	8	4	no	—
257	C₂×C₄	—	8	4	no	—
258	C₂×C₄	#106×C₅₃	1	4	yes	—
259	C₂×C₄	#107×C₅₃	1	4	yes	—
260	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
261	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
262	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
263	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
264	C₂×C₄	#117×C₅₃	1	4	yes	—
265	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
266	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
267	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
268	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
269	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
270	C₂×C₄	#123×C₅₃	1	4	yes	—
271	C₂×C₄	#125×C₅₃	1	2	yes	—
272	C₂×C₄	#126×C₅₃	1	4	yes	—
273	C₈	—	2	4	no	—
274	C₈	—	2	4	no	—
275	C₂²	C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
276	C₂²	C₅₃×#380	1	1	yes	characteristic
277	C₂²	C₅₃×#382	1	1	yes	characteristic
278	C₂²	C₅₃×#388	1	4	yes	—
279	C₂²	C₂×#106×C₅₃	1	1	yes	characteristic
280	C₂²	C₂×#107×C₅₃	1	1	yes	characteristic
281	C₂²	C₅₃×#427	1	1	yes	characteristic
282	C₂²	C₅₃×#427	1	1	yes	characteristic
283	C₂²	C₄×D₈×C₅₃	1	2	yes	—
284	C₂²	C₄×SD₁₆×C₅₃	1	4	yes	—
285	C₂²	C₄×Q₁₆×C₅₃	1	2	yes	—
286	C₂²	C₅₃×#465	1	4	yes	—
287	C₂²	C₅₃×#466	1	2	yes	—
288	C₂²	C₅₃×#467	1	2	yes	—
289	C₂²	C₅₃×#486	1	2	yes	—
290	C₂²	C₅₃×#487	1	2	yes	—
291	C₂²	C₅₃×#488	1	4	yes	—
292	C₄	—	2	4	no	—
293	C₄	—	2	4	no	—
294	C₄	—	8	8	no	—
295	C₄	C₅₃×#376	1	1	yes	characteristic
296	C₄	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
297	C₄	C₅₃×#381	1	1	yes	characteristic
298	C₄	C₅₃×#383	1	1	yes	characteristic
299	C₄	C₅₃×#387	1	4	yes	—
300	C₂	C₅₃×#1434	1	1	yes	characteristic
301	C₂	C₄×#106×C₅₃	1	1	yes	characteristic
302	C₂	C₄×#107×C₅₃	1	1	yes	characteristic
303	C₂	C₅₃×#1807	1	1	yes	characteristic
304	C₂	C₅₃×#1808	1	1	yes	characteristic
305	C₂	C₅₃×#1834	1	1	yes	characteristic
306	C₂	C₅₃×#1835	1	1	yes	characteristic
307	C₂	C₅₃×#1933	1	4	yes	—
308	C₂	C₅₃×#1934	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₂×C₄×C₈×C₅₃	C₄	semidirect
14. C₅₃×#340	C₄	semidirect
120. #10820	C₅₃	direct (N × H)
164. C₂×C₄×C₈	C₄×C₅₃	semidirect
168. #340	C₄×C₅₃	semidirect
169. C₅₃	#10820	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₈×C₅₃	2
C₂×C₅₃×#340	2
C₅₃×C₄⋊#95	2
#10820	53

Subgroups — C53×#10820 #27218332

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10820 #27218332