Subgroups — C₅₃×#22228 #27229740

Order 13568.

774 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#488	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2330	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2326	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1992	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1959	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1928	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1925	C₂	1	2	yes	maximal
8	C₅₃×#1863	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1862	C₂	1	2	yes	maximal
10	C₅₃×#1845	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1746	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#125×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#124×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#488	C₄	1	8	yes	complemented
17	C₂×#107×C₅₃	C₂²	1	2	yes	—
18	C₂×#107×C₅₃	C₂²	1	2	yes	—
19	C₂×#103×C₅₃	C₂²	1	2	yes	—
20	C₂×#103×C₅₃	C₂²	1	2	yes	—
21	C₂×#102×C₅₃	C₂²	1	2	yes	—
22	C₂×#102×C₅₃	C₂²	1	2	yes	—
23	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#391	C₂²	1	1	yes	characteristic
26	C₅₃×#390	C₂²	1	1	yes	characteristic
27	C₅₃×#389	—	2	2	no	—
28	C₅₃×#389	C₂²	1	2	yes	—
29	C₅₃×#388	—	2	2	no	—
30	C₅₃×#387	—	2	2	no	—
31	C₅₃×#385	—	2	2	no	—
32	C₅₃×#384	C₂²	1	1	yes	characteristic
33	C₅₃×#384	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
35	C₅₃×#376	C₂²	1	1	yes	characteristic
36	C₅₃×#340	C₂²	1	2	yes	—
37	C₅₃×#340	C₂²	1	2	yes	—
38	C₅₃×#336	C₂²	1	2	yes	—
39	C₅₃×#336	C₂²	1	2	yes	—
40	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
41	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₅₃	—	2	1	no	—
43	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
44	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
45	#125×C₅₃	—	2	2	no	—
46	#125×C₅₃	C₂×C₄	1	4	yes	complemented
47	#124×C₅₃	—	2	2	no	—
48	#124×C₅₃	C₂×C₄	1	4	yes	—
49	C₄×Q₈×C₅₃	—	4	2	no	—
50	C₄×D₄×C₅₃	—	4	2	no	—
51	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
52	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
53	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
54	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
55	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
56	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
60	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
61	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
62	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
63	C₂×C₄²×C₅₃	—	2	1	no	—
64	C₂×C₄²×C₅₃	—	2	2	no	—
65	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	#107×C₅₃	—	4	2	no	—
69	#107×C₅₃	—	4	2	no	—
70	#103×C₅₃	—	4	2	no	—
71	#103×C₅₃	C₂×C₄	1	8	yes	complemented
72	#102×C₅₃	—	4	2	no	—
73	#102×C₅₃	C₂×C₄	1	8	yes	—
74	#97×C₅₃	—	2	2	no	—
75	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
76	#95×C₅₃	—	2	2	no	—
77	#95×C₅₃	—	2	2	no	—
78	#95×C₅₃	—	2	2	no	—
79	#95×C₅₃	D₄	1	2	yes	—
80	#95×C₅₃	D₄	1	2	yes	—
81	C₂⁴×C₅₃	—	2	1	no	—
82	C₂×Q₈×C₅₃	—	2	1	no	—
83	C₂×Q₈×C₅₃	—	2	2	no	—
84	C₂×Q₈×C₅₃	—	2	2	no	—
85	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
86	C₂×D₄×C₅₃	—	2	1	no	—
87	C₂×D₄×C₅₃	—	2	2	no	—
88	C₂×D₄×C₅₃	—	2	2	no	—
89	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₅₃	—	2	1	no	—
91	C₂²×C₄×C₅₃	—	2	2	no	—
92	C₂²×C₄×C₅₃	—	2	2	no	—
93	C₂²×C₄×C₅₃	—	2	1	no	—
94	C₂²×C₄×C₅₃	—	2	1	no	—
95	C₂²×C₄×C₅₃	—	4	2	no	—
96	C₂²×C₄×C₅₃	—	8	2	no	—
97	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
102	C₂×C₈×C₅₃	—	2	2	no	—
103	C₂×C₈×C₅₃	—	2	2	no	—
104	C₂×C₈×C₅₃	—	2	2	no	—
105	C₂×C₈×C₅₃	—	2	2	no	—
106	C₂×C₈×C₅₃	—	2	2	no	—
107	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
108	C₄⋊C₄×C₅₃	—	2	8	no	—
109	C₄⋊C₄×C₅₃	—	2	2	no	—
110	C₄⋊C₄×C₅₃	—	2	2	no	—
111	C₄⋊C₄×C₅₃	—	2	2	no	—
112	C₄⋊C₄×C₅₃	—	2	2	no	—
113	C₄⋊C₄×C₅₃	—	2	2	no	—
114	C₄⋊C₄×C₅₃	—	4	2	no	—
115	C₄⋊C₄×C₅₃	—	8	2	no	—
116	C₄⋊C₄×C₅₃	—	8	2	no	—
117	C₄⋊C₄×C₅₃	—	8	2	no	—
118	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
119	C₂²⋊C₄×C₅₃	—	2	8	no	—
120	C₂²⋊C₄×C₅₃	—	4	2	no	—
121	C₂²⋊C₄×C₅₃	—	8	2	no	—
122	C₄²×C₅₃	—	2	2	no	—
123	C₄²×C₅₃	—	2	2	no	—
124	C₄²×C₅₃	—	4	2	no	—
125	C₄²×C₅₃	—	8	2	no	—
126	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
127	C₄²×C₅₃	#41	1	2	yes	—
128	C₂³×C₅₃	—	2	2	no	—
129	C₂³×C₅₃	—	4	1	no	—
130	C₂³×C₅₃	—	4	1	no	—
131	C₂³×C₅₃	—	4	2	no	—
132	C₂³×C₅₃	—	4	2	no	—
133	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
134	Q₈×C₅₃	—	2	4	no	—
135	Q₈×C₅₃	—	4	2	no	—
136	D₄×C₅₃	—	2	4	no	—
137	D₄×C₅₃	—	4	2	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	—	2	2	no	—
141	C₂×C₄×C₅₃	—	2	4	no	—
142	C₂×C₄×C₅₃	—	2	1	no	—
143	C₂×C₄×C₅₃	—	2	1	no	—
144	C₂×C₄×C₅₃	—	2	2	no	—
145	C₂×C₄×C₅₃	—	2	2	no	—
146	C₂×C₄×C₅₃	—	2	2	no	—
147	C₂×C₄×C₅₃	—	2	2	no	—
148	C₂×C₄×C₅₃	—	2	2	no	—
149	C₂×C₄×C₅₃	—	4	1	no	—
150	C₂×C₄×C₅₃	—	4	1	no	—
151	C₂×C₄×C₅₃	—	4	2	no	—
152	C₂×C₄×C₅₃	—	4	2	no	—
153	C₂×C₄×C₅₃	—	4	1	no	—
154	C₂×C₄×C₅₃	—	4	1	no	—
155	C₂×C₄×C₅₃	—	4	2	no	—
156	C₂×C₄×C₅₃	—	4	2	no	—
157	C₂×C₄×C₅₃	—	4	2	no	—
158	C₂×C₄×C₅₃	—	4	2	no	—
159	C₂×C₄×C₅₃	—	4	2	no	—
160	C₂×C₄×C₅₃	—	8	4	no	—
161	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
162	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
164	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
165	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
166	C₈×C₅₃	—	2	4	no	—
167	C₈×C₅₃	—	4	2	no	—
168	#22228	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
169	C₂²×C₅₃	—	4	2	no	—
170	C₂²×C₅₃	—	4	4	no	—
171	C₂²×C₅₃	—	8	1	no	—
172	C₂²×C₅₃	—	8	1	no	—
173	C₂²×C₅₃	—	8	2	no	—
174	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
175	C₂²×C₅₃	#535	1	1	yes	characteristic
176	C₂²×C₅₃	#536	1	1	yes	characteristic
177	C₂²×C₅₃	#576	1	2	yes	—
178	C₂²×C₅₃	#578	1	2	yes	—
179	C₄×C₅₃	—	2	4	no	—
180	C₄×C₅₃	—	4	2	no	—
181	C₄×C₅₃	—	4	4	no	—
182	C₄×C₅₃	—	4	1	no	—
183	C₄×C₅₃	—	4	1	no	—
184	C₄×C₅₃	—	4	2	no	—
185	C₄×C₅₃	—	8	1	no	—
186	C₄×C₅₃	—	8	2	no	—
187	C₄×C₅₃	—	8	1	no	—
188	C₄×C₅₃	#520	1	2	yes	—
189	C₄×C₅₃	#542	1	2	yes	—
190	C₂×#488	C₂×C₅₃	1	1	yes	characteristic
191	#2330	C₂×C₅₃	1	1	yes	characteristic, complemented
192	#2326	C₂×C₅₃	1	1	yes	characteristic
193	#1992	C₂×C₅₃	1	1	yes	characteristic, complemented
194	#1959	C₂×C₅₃	1	1	yes	characteristic
195	#1928	C₂×C₅₃	1	2	yes	complemented
196	#1925	C₂×C₅₃	1	2	yes	—
197	#1863	C₂×C₅₃	1	2	yes	complemented
198	#1862	C₂×C₅₃	1	2	yes	—
199	#1845	C₂×C₅₃	1	1	yes	characteristic, complemented
200	#1746	C₂×C₅₃	1	1	yes	characteristic, complemented
201	C₂×C₅₃	—	8	2	no	—
202	C₂×C₅₃	#2326	1	1	yes	characteristic
203	C₂×C₅₃	#2951	1	2	yes	—
204	C₂×C₅₃	#3220	1	1	yes	characteristic
205	C₂×C₅₃	#3223	1	1	yes	characteristic
206	C₂×C₅₃	#3243	1	2	yes	—
207	C₂×#125	C₂²×C₅₃	1	1	yes	characteristic
208	C₂×#124	C₂²×C₅₃	1	1	yes	characteristic
209	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
210	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
211	#488	C₄×C₅₃	1	8	yes	complemented
212	C₂×#107	C₂²×C₅₃	1	2	yes	—
213	C₂×#107	C₂²×C₅₃	1	2	yes	—
214	C₂×#103	C₂²×C₅₃	1	2	yes	—
215	C₂×#103	C₂²×C₅₃	1	2	yes	—
216	C₂×#102	C₂²×C₅₃	1	2	yes	—
217	C₂×#102	C₂²×C₅₃	1	2	yes	—
218	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
219	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
220	#391	C₂²×C₅₃	1	1	yes	characteristic
221	#390	C₂²×C₅₃	1	1	yes	characteristic
222	#389	—	2	2	no	—
223	#389	C₂²×C₅₃	1	2	yes	—
224	#388	—	2	2	no	—
225	#387	—	2	2	no	—
226	#385	—	2	2	no	—
227	#384	C₂²×C₅₃	1	1	yes	characteristic
228	#384	C₂²×C₅₃	1	1	yes	characteristic
229	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
230	#376	C₂²×C₅₃	1	1	yes	characteristic
231	#340	C₂²×C₅₃	1	2	yes	—
232	#340	C₂²×C₅₃	1	2	yes	—
233	#336	C₂²×C₅₃	1	2	yes	—
234	#336	C₂²×C₅₃	1	2	yes	—
235	C₅₃	#22228	1	1	yes	characteristic, direct product factor, Sylow
236	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
237	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
238	C₂³×C₄	—	2	1	no	—
239	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
240	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
241	#125	—	2	2	no	—
242	#125	C₂×C₄×C₅₃	1	4	yes	complemented
243	#124	—	2	2	no	—
244	#124	C₂×C₄×C₅₃	1	4	yes	—
245	C₄×Q₈	—	4	2	no	—
246	C₄×D₄	—	4	2	no	—
247	C₂×C₄⋊C₄	—	2	2	no	—
248	C₂×C₄⋊C₄	—	2	1	no	—
249	C₂×C₄⋊C₄	—	2	2	no	—
250	C₂×C₄⋊C₄	—	2	2	no	—
251	C₂×C₄⋊C₄	—	2	2	no	—
252	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
253	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
254	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
255	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
256	C₂×C₂²⋊C₄	—	2	2	no	—
257	C₂×C₂²⋊C₄	—	2	1	no	—
258	C₂×C₂²⋊C₄	—	2	2	no	—
259	C₂×C₄²	—	2	1	no	—
260	C₂×C₄²	—	2	2	no	—
261	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
262	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
263	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
264	#107	—	4	2	no	—
265	#107	—	4	2	no	—
266	#103	—	4	2	no	—
267	#103	C₂×C₄×C₅₃	1	8	yes	complemented
268	#102	—	4	2	no	—
269	#102	C₂×C₄×C₅₃	1	8	yes	—
270	#97	—	2	2	no	—
271	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	complemented
272	#95	—	2	2	no	—
273	#95	—	2	2	no	—
274	#95	—	2	2	no	—
275	#95	D₄×C₅₃	1	2	yes	—
276	#95	D₄×C₅₃	1	2	yes	—
277	C₂⁴	—	2	1	no	—
278	C₂×Q₈	—	2	1	no	—
279	C₂×Q₈	—	2	2	no	—
280	C₂×Q₈	—	2	2	no	—
281	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
282	C₂×D₄	—	2	1	no	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
286	C₂²×C₄	—	2	1	no	—
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	1	no	—
290	C₂²×C₄	—	2	1	no	—
291	C₂²×C₄	—	4	2	no	—
292	C₂²×C₄	—	8	2	no	—
293	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
294	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
295	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
296	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
297	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
304	C₄⋊C₄	—	2	8	no	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	4	2	no	—
311	C₄⋊C₄	—	8	2	no	—
312	C₄⋊C₄	—	8	2	no	—
313	C₄⋊C₄	—	8	2	no	—
314	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
315	C₂²⋊C₄	—	2	8	no	—
316	C₂²⋊C₄	—	4	2	no	—
317	C₂²⋊C₄	—	8	2	no	—
318	C₄²	—	2	2	no	—
319	C₄²	—	2	2	no	—
320	C₄²	—	4	2	no	—
321	C₄²	—	8	2	no	—
322	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
323	C₄²	#41×C₅₃	1	2	yes	—
324	C₂³	—	2	2	no	—
325	C₂³	—	4	1	no	—
326	C₂³	—	4	1	no	—
327	C₂³	—	4	2	no	—
328	C₂³	—	4	2	no	—
329	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
330	Q₈	—	2	4	no	—
331	Q₈	—	4	2	no	—
332	D₄	—	2	4	no	—
333	D₄	—	4	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	4	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	—	4	1	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	4	2	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	8	4	no	—
357	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
358	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
359	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
360	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
361	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
362	C₈	—	2	4	no	—
363	C₈	—	4	2	no	—
364	C₂²	—	4	4	no	—
365	C₂²	—	4	2	no	—
366	C₂²	—	8	1	no	—
367	C₂²	—	8	1	no	—
368	C₂²	—	8	2	no	—
369	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
370	C₂²	C₅₃×#535	1	1	yes	characteristic
371	C₂²	C₅₃×#536	1	1	yes	characteristic
372	C₂²	C₅₃×#576	1	2	yes	—
373	C₂²	C₅₃×#578	1	2	yes	—
374	C₄	—	2	4	no	—
375	C₄	—	4	4	no	—
376	C₄	—	4	1	no	—
377	C₄	—	4	1	no	—
378	C₄	—	4	2	no	—
379	C₄	—	4	2	no	—
380	C₄	—	8	1	no	—
381	C₄	—	8	1	no	—
382	C₄	—	8	2	no	—
383	C₄	C₅₃×#520	1	2	yes	—
384	C₄	C₅₃×#542	1	2	yes	—
385	C₂	—	8	2	no	—
386	C₂	C₅₃×#2326	1	1	yes	characteristic
387	C₂	C₅₃×#2951	1	2	yes	—
388	C₂	C₅₃×#3220	1	1	yes	characteristic
389	C₂	C₅₃×#3223	1	1	yes	characteristic
390	C₂	C₅₃×#3243	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂²×D₄×C₅₃	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₅₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2330	C₂	semidirect
4. C₅₃×#1992	C₂	semidirect
6. C₅₃×#1928	C₂	semidirect
8. C₅₃×#1863	C₂	semidirect
10. C₅₃×#1845	C₂	semidirect
11. C₅₃×#1746	C₂	semidirect
16. C₅₃×#488	C₄	semidirect
46. #125×C₅₃	C₂×C₄	semidirect
71. #103×C₅₃	C₂×C₄	semidirect
75. C₄×C₈×C₅₃	C₂×C₄	semidirect
168. #22228	C₅₃	direct (N × H)
191. #2330	C₂×C₅₃	semidirect
193. #1992	C₂×C₅₃	semidirect
195. #1928	C₂×C₅₃	semidirect
197. #1863	C₂×C₅₃	semidirect
199. #1845	C₂×C₅₃	semidirect
200. #1746	C₂×C₅₃	semidirect
211. #488	C₄×C₅₃	semidirect
235. C₅₃	#22228	direct (N × H)
242. #125	C₂×C₄×C₅₃	semidirect
267. #103	C₂×C₄×C₅₃	semidirect
271. C₄×C₈	C₂×C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#488	2
C₅₃×#2330	2
C₅₃×#2326	2
C₅₃×#1992	2
C₅₃×#1959	2
C₅₃×#1928	2
C₅₃×#1925	2
C₅₃×#1863	2
C₅₃×#1862	2
C₅₃×#1845	2
C₅₃×#1746	2
#22228	53

Subgroups — C53×#22228 #27229740

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22228 #27229740