Subgroups — C₅₃×#36600 #27244112

Order 13568.

1574 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2859	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2859	C₂	1	2	yes	maximal
3	C₅₃×#2585	C₂	1	8	yes	maximal
4	C₅₃×#2570	C₂	1	4	yes	maximal
5	C₅₃×#2552	C₂	1	2	yes	maximal
6	C₅₃×#2453	C₂	1	4	yes	maximal
7	C₅₃×#2370	C₂	1	4	yes	maximal
8	C₅₃×#2355	C₂	1	2	yes	maximal
9	C₄×#125×C₅₃	C₂	1	1	yes	maximal, characteristic
10	C₄×#125×C₅₃	C₂	1	2	yes	maximal
11	C₄²×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂×#125×C₅₃	C₂²	1	4	yes	—
13	C₂×#125×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#125×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#400	C₂²	1	8	yes	—
18	C₅₃×#398	C₂²	1	8	yes	—
19	C₅₃×#398	C₂²	1	8	yes	—
20	C₅₃×#391	C₂²	1	4	yes	—
21	C₅₃×#389	C₂²	1	4	yes	—
22	C₅₃×#389	C₂²	1	8	yes	—
23	C₅₃×#389	C₂²	1	4	yes	—
24	C₅₃×#389	C₂²	1	4	yes	—
25	C₅₃×#389	C₂²	1	2	yes	—
26	C₅₃×#389	C₂²	1	4	yes	—
27	C₅₃×#389	C₂²	1	8	yes	—
28	C₅₃×#387	C₂²	1	8	yes	—
29	C₅₃×#387	C₂²	1	4	yes	—
30	C₅₃×#387	C₂²	1	8	yes	—
31	C₅₃×#387	C₂²	1	8	yes	—
32	C₅₃×#387	C₂²	1	8	yes	—
33	C₅₃×#384	C₂²	1	4	yes	—
34	C₅₃×#384	C₂²	1	2	yes	—
35	C₅₃×#382	C₂²	1	1	yes	characteristic
36	C₅₃×#382	C₂²	1	2	yes	—
37	C₅₃×#382	C₂²	1	4	yes	—
38	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
39	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
40	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
41	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
42	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
43	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
44	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
45	C₄×C₄⋊C₄×C₅₃	C₂²	1	8	yes	—
46	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
47	C₄³×C₅₃	C₂²	1	1	yes	characteristic
48	C₄³×C₅₃	C₂²	1	2	yes	—
49	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
50	#125×C₅₃	—	2	16	no	—
51	#125×C₅₃	—	2	4	no	—
52	#125×C₅₃	—	2	8	no	—
53	C₄×Q₈×C₅₃	—	2	8	no	—
54	C₄×Q₈×C₅₃	Q₈	1	8	yes	—
55	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
56	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
58	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
59	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
60	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
61	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
62	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
63	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
65	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
66	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
67	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
70	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
71	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
72	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
73	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
75	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
76	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
77	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
78	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
79	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
81	#95×C₅₃	C₂³	1	4	yes	—
82	#95×C₅₃	C₂³	1	4	yes	—
83	#95×C₅₃	C₂³	1	2	yes	—
84	#95×C₅₃	C₂³	1	8	yes	—
85	#95×C₅₃	C₂³	1	4	yes	—
86	#95×C₅₃	C₂³	1	4	yes	—
87	#95×C₅₃	C₂³	1	8	yes	—
88	#95×C₅₃	C₂³	1	8	yes	—
89	#95×C₅₃	C₂³	1	8	yes	—
90	#95×C₅₃	C₂³	1	8	yes	—
91	C₂×Q₈×C₅₃	—	2	4	no	—
92	C₂×Q₈×C₅₃	C₂×Q₈	1	4	yes	—
93	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
94	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
95	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
96	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
97	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
98	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
99	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
101	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
102	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
103	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄×C₅₃	—	2	8	no	—
105	C₄⋊C₄×C₅₃	—	2	16	no	—
106	C₄⋊C₄×C₅₃	—	2	16	no	—
107	C₄⋊C₄×C₅₃	—	2	8	no	—
108	C₄⋊C₄×C₅₃	—	2	16	no	—
109	C₄⋊C₄×C₅₃	—	2	8	no	—
110	C₄⋊C₄×C₅₃	—	2	4	no	—
111	C₄⋊C₄×C₅₃	—	2	8	no	—
112	C₄⋊C₄×C₅₃	—	2	4	no	—
113	C₄⋊C₄×C₅₃	—	4	4	no	—
114	C₄⋊C₄×C₅₃	—	4	8	no	—
115	C₄⋊C₄×C₅₃	—	4	4	no	—
116	C₄⋊C₄×C₅₃	C₂×Q₈	1	8	yes	—
117	C₄⋊C₄×C₅₃	C₂×Q₈	1	4	yes	—
118	C₄⋊C₄×C₅₃	#41	1	8	yes	—
119	C₄²×C₅₃	—	2	8	no	—
120	C₄²×C₅₃	—	2	4	no	—
121	C₄²×C₅₃	—	2	8	no	—
122	C₄²×C₅₃	—	2	4	no	—
123	C₄²×C₅₃	—	2	2	no	—
124	C₄²×C₅₃	—	4	4	no	—
125	C₄²×C₅₃	—	4	2	no	—
126	C₄²×C₅₃	—	4	8	no	—
127	C₄²×C₅₃	C₂×Q₈	1	8	yes	—
128	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
129	C₄²×C₅₃	#41	1	4	yes	—
130	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
131	Q₈×C₅₃	—	2	8	no	—
132	C₂×C₄×C₅₃	—	2	4	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	4	no	—
135	C₂×C₄×C₅₃	—	2	2	no	—
136	C₂×C₄×C₅₃	—	2	4	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	2	4	no	—
139	C₂×C₄×C₅₃	—	2	4	no	—
140	C₂×C₄×C₅₃	—	2	2	no	—
141	C₂×C₄×C₅₃	—	2	4	no	—
142	C₂×C₄×C₅₃	—	2	8	no	—
143	C₂×C₄×C₅₃	—	2	4	no	—
144	C₂×C₄×C₅₃	—	2	8	no	—
145	C₂×C₄×C₅₃	—	2	2	no	—
146	C₂×C₄×C₅₃	—	2	4	no	—
147	C₂×C₄×C₅₃	—	2	4	no	—
148	C₂×C₄×C₅₃	—	2	8	no	—
149	C₂×C₄×C₅₃	—	2	2	no	—
150	C₂×C₄×C₅₃	—	2	4	no	—
151	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
152	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
153	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
154	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
155	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
156	C₂×C₄×C₅₃	C₂×#41	1	8	yes	—
157	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
158	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
159	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
160	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
161	#36600	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
163	C₂²×C₅₃	C₂²×#41	1	2	yes	—
164	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
165	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
166	C₂²×C₅₃	#585	1	2	yes	—
167	C₄×C₅₃	—	2	4	no	—
168	C₄×C₅₃	—	2	4	no	—
169	C₄×C₅₃	—	2	4	no	—
170	C₄×C₅₃	—	2	2	no	—
171	C₄×C₅₃	—	2	8	no	—
172	C₄×C₅₃	—	4	4	no	—
173	C₄×C₅₃	—	4	8	no	—
174	C₄×C₅₃	—	4	2	no	—
175	C₄×C₅₃	—	4	4	no	—
176	C₄×C₅₃	#533	1	4	yes	—
177	C₄×C₅₃	#557	1	4	yes	—
178	#2859	C₂×C₅₃	1	1	yes	characteristic
179	#2859	C₂×C₅₃	1	2	yes	—
180	#2585	C₂×C₅₃	1	8	yes	—
181	#2570	C₂×C₅₃	1	4	yes	—
182	#2552	C₂×C₅₃	1	2	yes	—
183	#2453	C₂×C₅₃	1	4	yes	—
184	#2370	C₂×C₅₃	1	4	yes	—
185	#2355	C₂×C₅₃	1	2	yes	—
186	C₄×#125	C₂×C₅₃	1	2	yes	—
187	C₄×#125	C₂×C₅₃	1	1	yes	characteristic
188	C₄²×Q₈	C₂×C₅₃	1	1	yes	characteristic
189	C₂×C₅₃	C₂×#533	1	1	yes	characteristic
190	C₂×C₅₃	C₂×#557	1	1	yes	characteristic
191	C₂×C₅₃	#3480	1	2	yes	—
192	C₂×C₅₃	#3504	1	1	yes	characteristic
193	C₂×C₅₃	#3515	1	2	yes	—
194	C₂×#125	C₂²×C₅₃	1	2	yes	—
195	C₂×#125	C₂²×C₅₃	1	1	yes	characteristic
196	C₂×#125	C₂²×C₅₃	1	4	yes	—
197	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
198	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
199	#400	C₂²×C₅₃	1	8	yes	—
200	#398	C₂²×C₅₃	1	8	yes	—
201	#398	C₂²×C₅₃	1	8	yes	—
202	#391	C₂²×C₅₃	1	4	yes	—
203	#389	C₂²×C₅₃	1	2	yes	—
204	#389	C₂²×C₅₃	1	4	yes	—
205	#389	C₂²×C₅₃	1	4	yes	—
206	#389	C₂²×C₅₃	1	8	yes	—
207	#389	C₂²×C₅₃	1	8	yes	—
208	#389	C₂²×C₅₃	1	4	yes	—
209	#389	C₂²×C₅₃	1	4	yes	—
210	#387	C₂²×C₅₃	1	4	yes	—
211	#387	C₂²×C₅₃	1	8	yes	—
212	#387	C₂²×C₅₃	1	8	yes	—
213	#387	C₂²×C₅₃	1	8	yes	—
214	#387	C₂²×C₅₃	1	8	yes	—
215	#384	C₂²×C₅₃	1	4	yes	—
216	#384	C₂²×C₅₃	1	2	yes	—
217	#382	C₂²×C₅₃	1	1	yes	characteristic
218	#382	C₂²×C₅₃	1	2	yes	—
219	#382	C₂²×C₅₃	1	4	yes	—
220	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
221	C₄×C₄⋊C₄	C₂²×C₅₃	1	8	yes	—
222	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
223	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
224	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
225	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
226	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
227	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
228	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
229	C₄³	C₂²×C₅₃	1	1	yes	characteristic
230	C₄³	C₂²×C₅₃	1	2	yes	—
231	C₅₃	#36600	1	1	yes	characteristic, direct product factor, Sylow
232	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
233	#125	—	2	16	no	—
234	#125	—	2	8	no	—
235	#125	—	2	4	no	—
236	C₄×Q₈	—	2	8	no	—
237	C₄×Q₈	Q₈×C₅₃	1	8	yes	—
238	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
239	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
240	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
241	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
242	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
243	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
244	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
245	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
246	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
247	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
248	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
249	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
250	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
251	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
252	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
253	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
254	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
255	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
256	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
257	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
258	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
259	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
260	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
261	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
262	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
263	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
264	#95	C₂³×C₅₃	1	4	yes	—
265	#95	C₂³×C₅₃	1	4	yes	—
266	#95	C₂³×C₅₃	1	2	yes	—
267	#95	C₂³×C₅₃	1	8	yes	—
268	#95	C₂³×C₅₃	1	4	yes	—
269	#95	C₂³×C₅₃	1	4	yes	—
270	#95	C₂³×C₅₃	1	8	yes	—
271	#95	C₂³×C₅₃	1	8	yes	—
272	#95	C₂³×C₅₃	1	8	yes	—
273	#95	C₂³×C₅₃	1	8	yes	—
274	C₂×Q₈	—	2	4	no	—
275	C₂×Q₈	C₂×Q₈×C₅₃	1	4	yes	—
276	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
277	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
278	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
279	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
280	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
281	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
282	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
283	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
284	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
285	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
286	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
287	C₄⋊C₄	—	2	8	no	—
288	C₄⋊C₄	—	2	16	no	—
289	C₄⋊C₄	—	2	16	no	—
290	C₄⋊C₄	—	2	16	no	—
291	C₄⋊C₄	—	2	8	no	—
292	C₄⋊C₄	—	2	8	no	—
293	C₄⋊C₄	—	2	8	no	—
294	C₄⋊C₄	—	2	4	no	—
295	C₄⋊C₄	—	2	4	no	—
296	C₄⋊C₄	—	4	4	no	—
297	C₄⋊C₄	—	4	8	no	—
298	C₄⋊C₄	—	4	4	no	—
299	C₄⋊C₄	C₂×Q₈×C₅₃	1	8	yes	—
300	C₄⋊C₄	C₂×Q₈×C₅₃	1	4	yes	—
301	C₄⋊C₄	#41×C₅₃	1	8	yes	—
302	C₄²	—	2	8	no	—
303	C₄²	—	2	4	no	—
304	C₄²	—	2	8	no	—
305	C₄²	—	2	4	no	—
306	C₄²	—	2	2	no	—
307	C₄²	—	4	4	no	—
308	C₄²	—	4	2	no	—
309	C₄²	—	4	8	no	—
310	C₄²	C₂×Q₈×C₅₃	1	8	yes	—
311	C₄²	C₂×Q₈×C₅₃	1	4	yes	—
312	C₄²	#41×C₅₃	1	4	yes	—
313	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
314	Q₈	—	2	8	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	4	no	—
322	C₂×C₄	—	2	4	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	4	no	—
325	C₂×C₄	—	2	4	no	—
326	C₂×C₄	—	2	8	no	—
327	C₂×C₄	—	2	8	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	4	no	—
330	C₂×C₄	—	2	4	no	—
331	C₂×C₄	—	2	8	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	4	no	—
334	C₂×C₄	C₂²×Q₈×C₅₃	1	4	yes	—
335	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
336	C₂×C₄	C₂²×Q₈×C₅₃	1	4	yes	—
337	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
338	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
339	C₂×C₄	C₂×#41×C₅₃	1	8	yes	—
340	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
341	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
342	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
343	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
344	C₂²	C₂³×Q₈×C₅₃	1	1	yes	characteristic
345	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
346	C₂²	C₂²×#41×C₅₃	1	2	yes	—
347	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
348	C₂²	C₅₃×#585	1	2	yes	—
349	C₄	—	2	4	no	—
350	C₄	—	2	4	no	—
351	C₄	—	2	2	no	—
352	C₄	—	2	8	no	—
353	C₄	—	2	4	no	—
354	C₄	—	4	4	no	—
355	C₄	—	4	8	no	—
356	C₄	—	4	2	no	—
357	C₄	—	4	4	no	—
358	C₄	C₅₃×#533	1	4	yes	—
359	C₄	C₅₃×#557	1	4	yes	—
360	C₂	C₂×C₅₃×#533	1	1	yes	characteristic
361	C₂	C₂×C₅₃×#557	1	1	yes	characteristic
362	C₂	C₅₃×#3480	1	2	yes	—
363	C₂	C₅₃×#3504	1	1	yes	characteristic
364	C₂	C₅₃×#3515	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
161. #36600	C₅₃	direct (N × H)
231. C₅₃	#36600	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#2859	2
C₅₃×#2585	2
C₅₃×#2570	2
C₅₃×#2552	2
C₅₃×#2453	2
C₅₃×#2370	2
C₅₃×#2355	2
C₄×#125×C₅₃	2
C₄²×Q₈×C₅₃	2
#36600	53

Subgroups — C53×#36600 #27244112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#36600 #27244112