Subgroups — C₅₃×#19657 #27227169

Order 13568.

700 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3230	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3229	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3226	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#3206	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#3131	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1719	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1718	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1710	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1708	C₂	1	2	yes	maximal, complemented
11	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#533	C₂²	1	1	yes	characteristic
13	C₅₃×#532	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#500	C₂²	1	2	yes	—
17	C₅₃×#500	C₂²	1	2	yes	—
18	C₅₃×#498	C₂²	1	2	yes	—
19	C₅₃×#492	C₂²	1	2	yes	—
20	C₅₃×#488	—	2	2	no	—
21	C₅₃×#487	—	2	2	no	—
22	C₅₃×#487	C₂²	1	2	yes	—
23	C₅₃×#487	C₂²	1	2	yes	—
24	C₅₃×#487	C₂²	1	2	yes	—
25	C₅₃×#487	C₂²	1	2	yes	—
26	C₅₃×#487	C₂²	1	1	yes	characteristic
27	C₅₃×#487	C₂²	1	1	yes	characteristic
28	C₅₃×#487	C₂²	1	2	yes	—
29	C₅₃×#486	C₂²	1	2	yes	—
30	C₅₃×#486	C₂²	1	1	yes	characteristic
31	C₅₃×#486	C₂²	1	1	yes	characteristic
32	C₅₃×#484	—	2	2	no	—
33	C₅₃×#483	—	2	2	no	—
34	C₅₃×#480	—	2	2	no	—
35	C₅₃×#477	—	2	2	no	—
36	C₅₃×#475	—	2	2	no	—
37	C₅₃×#474	—	2	2	no	—
38	C₅₃×#461	—	2	2	no	—
39	C₅₃×#448	—	2	2	no	—
40	C₅₃×#431	C₂²	1	1	yes	characteristic
41	C₅₃×#431	C₂²	1	1	yes	characteristic
42	C₅₃×#416	—	2	2	no	—
43	C₂×#103×C₅₃	—	2	2	no	—
44	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
45	C₅₃×#322	C₂²	1	2	yes	—
46	C₈²×C₅₃	C₂²	1	2	yes	—
47	C₂×#41×C₅₃	—	2	1	no	—
48	C₂²×Q₈×C₅₃	—	2	1	no	—
49	C₂²×Q₈×C₅₃	—	2	1	no	—
50	C₂×SD₁₆×C₅₃	—	2	4	no	—
51	C₂²×C₈×C₅₃	D₄	1	2	yes	—
52	C₄⋊Q₈×C₅₃	—	2	1	no	—
53	C₄⋊Q₈×C₅₃	—	2	1	no	—
54	C₄⋊Q₈×C₅₃	—	2	1	no	—
55	C₄⋊Q₈×C₅₃	—	2	1	no	—
56	C₄⋊Q₈×C₅₃	—	2	1	no	—
57	C₄⋊Q₈×C₅₃	—	2	1	no	—
58	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
59	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
60	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
62	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
63	#127×C₅₃	C₂³	1	1	yes	characteristic
64	#125×C₅₃	—	2	1	no	—
65	#124×C₅₃	—	2	1	no	—
66	C₂²⋊Q₈×C₅₃	—	2	1	no	—
67	C₂²⋊Q₈×C₅₃	—	2	1	no	—
68	#121×C₅₃	—	2	1	no	—
69	#121×C₅₃	—	2	1	no	—
70	C₄×Q₈×C₅₃	—	2	1	no	—
71	C₄×Q₈×C₅₃	—	2	1	no	—
72	C₄×D₄×C₅₃	—	2	1	no	—
73	C₄×D₄×C₅₃	—	2	1	no	—
74	#117×C₅₃	—	2	1	no	—
75	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
76	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
77	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
78	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
79	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
80	#107×C₅₃	—	2	4	no	—
81	#107×C₅₃	—	2	4	no	—
82	#106×C₅₃	—	2	4	no	—
83	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
84	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
85	#103×C₅₃	—	2	4	no	—
86	#103×C₅₃	—	2	4	no	—
87	#103×C₅₃	—	2	2	no	—
88	#103×C₅₃	—	2	2	no	—
89	#103×C₅₃	—	2	4	no	—
90	#103×C₅₃	—	2	4	no	—
91	#103×C₅₃	—	2	4	no	—
92	#102×C₅₃	—	2	2	no	—
93	#102×C₅₃	—	2	2	no	—
94	#102×C₅₃	—	2	4	no	—
95	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
96	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
97	C₄×C₈×C₅₃	D₄	1	2	yes	—
98	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
99	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₅₃	C₂³	1	2	yes	—
101	C₄×C₈×C₅₃	C₂³	1	2	yes	—
102	#41×C₅₃	—	4	4	no	—
103	C₂×Q₈×C₅₃	—	2	1	no	—
104	C₂×Q₈×C₅₃	—	2	1	no	—
105	C₂×Q₈×C₅₃	—	2	1	no	—
106	C₂×Q₈×C₅₃	—	2	2	no	—
107	C₂×Q₈×C₅₃	—	2	2	no	—
108	C₂×Q₈×C₅₃	—	4	1	no	—
109	C₂×Q₈×C₅₃	—	4	1	no	—
110	C₂×Q₈×C₅₃	—	4	4	no	—
111	C₂×Q₈×C₅₃	—	8	2	no	—
112	C₂×D₄×C₅₃	—	2	1	no	—
113	C₂×D₄×C₅₃	—	2	1	no	—
114	C₂×D₄×C₅₃	—	2	1	no	—
115	C₂×D₄×C₅₃	—	4	1	no	—
116	C₂²×C₄×C₅₃	—	4	1	no	—
117	C₂²×C₄×C₅₃	—	4	1	no	—
118	C₂²×C₄×C₅₃	—	4	1	no	—
119	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
122	SD₁₆×C₅₃	—	8	4	no	—
123	C₂×C₈×C₅₃	—	2	4	no	—
124	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
125	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
126	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
127	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
128	C₄⋊C₄×C₅₃	—	2	1	no	—
129	C₄⋊C₄×C₅₃	—	2	1	no	—
130	C₄⋊C₄×C₅₃	—	2	2	no	—
131	C₄⋊C₄×C₅₃	—	2	2	no	—
132	C₄⋊C₄×C₅₃	—	2	1	no	—
133	C₄⋊C₄×C₅₃	—	2	2	no	—
134	C₄⋊C₄×C₅₃	—	2	2	no	—
135	C₄⋊C₄×C₅₃	—	2	1	no	—
136	C₄⋊C₄×C₅₃	—	2	1	no	—
137	C₄⋊C₄×C₅₃	—	2	1	no	—
138	C₄⋊C₄×C₅₃	—	4	1	no	—
139	C₄⋊C₄×C₅₃	—	4	1	no	—
140	C₄⋊C₄×C₅₃	—	4	1	no	—
141	C₄⋊C₄×C₅₃	—	4	1	no	—
142	C₄⋊C₄×C₅₃	—	4	1	no	—
143	C₄⋊C₄×C₅₃	—	4	1	no	—
144	C₂²⋊C₄×C₅₃	—	4	1	no	—
145	C₂²⋊C₄×C₅₃	—	4	1	no	—
146	C₂²⋊C₄×C₅₃	—	4	1	no	—
147	C₄²×C₅₃	—	4	1	no	—
148	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
149	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
150	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
151	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
152	C₂³×C₅₃	—	4	1	no	—
153	C₂³×C₅₃	#127	1	1	yes	characteristic
154	Q₈×C₅₃	—	4	4	no	—
155	Q₈×C₅₃	—	4	2	no	—
156	Q₈×C₅₃	—	4	2	no	—
157	Q₈×C₅₃	—	4	4	no	—
158	Q₈×C₅₃	—	4	2	no	—
159	Q₈×C₅₃	—	8	2	no	—
160	Q₈×C₅₃	—	8	2	no	—
161	D₄×C₅₃	—	4	2	no	—
162	D₄×C₅₃	—	4	2	no	—
163	D₄×C₅₃	—	4	2	no	—
164	D₄×C₅₃	—	8	2	no	—
165	C₂×C₄×C₅₃	—	2	2	no	—
166	C₂×C₄×C₅₃	—	2	2	no	—
167	C₂×C₄×C₅₃	—	4	1	no	—
168	C₂×C₄×C₅₃	—	4	2	no	—
169	C₂×C₄×C₅₃	—	4	2	no	—
170	C₂×C₄×C₅₃	—	4	1	no	—
171	C₂×C₄×C₅₃	—	4	1	no	—
172	C₂×C₄×C₅₃	—	8	2	no	—
173	C₂×C₄×C₅₃	—	8	2	no	—
174	C₂×C₄×C₅₃	—	8	2	no	—
175	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
176	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
177	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
178	C₂×C₄×C₅₃	#135	1	4	yes	—
179	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
182	C₈×C₅₃	—	2	2	no	—
183	C₈×C₅₃	—	2	2	no	—
184	C₈×C₅₃	—	2	4	no	—
185	C₈×C₅₃	—	2	4	no	—
186	#19657	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
187	C₂²×C₅₃	—	2	1	no	—
188	C₂²×C₅₃	—	2	1	no	—
189	C₂²×C₅₃	—	8	1	no	—
190	C₂²×C₅₃	—	8	1	no	—
191	C₂²×C₅₃	—	8	1	no	—
192	C₂²×C₅₃	#495	1	2	yes	—
193	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
194	C₄×C₅₃	—	2	1	no	—
195	C₄×C₅₃	—	2	1	no	—
196	C₄×C₅₃	—	8	1	no	—
197	C₄×C₅₃	—	8	2	no	—
198	C₄×C₅₃	—	8	2	no	—
199	C₄×C₅₃	—	8	1	no	—
200	C₄×C₅₃	—	8	1	no	—
201	C₄×C₅₃	#495	1	2	yes	—
202	C₄×C₅₃	C₂×#135	1	2	yes	—
203	C₄×C₅₃	#577	1	2	yes	—
204	C₄×C₅₃	#578	1	2	yes	—
205	#3230	C₂×C₅₃	1	1	yes	characteristic, complemented
206	#3229	C₂×C₅₃	1	1	yes	characteristic, complemented
207	#3226	C₂×C₅₃	1	1	yes	characteristic
208	#3206	C₂×C₅₃	1	1	yes	characteristic
209	#3131	C₂×C₅₃	1	1	yes	characteristic
210	C₂×#487	C₂×C₅₃	1	1	yes	characteristic, complemented
211	#1719	C₂×C₅₃	1	2	yes	complemented
212	#1718	C₂×C₅₃	1	2	yes	complemented
213	#1710	C₂×C₅₃	1	2	yes	complemented
214	#1708	C₂×C₅₃	1	2	yes	complemented
215	#1450	C₂×C₅₃	1	1	yes	characteristic, complemented
216	C₂×C₅₃	—	2	2	no	—
217	C₂×C₅₃	—	16	1	no	—
218	C₂×C₅₃	C₂×#495	1	1	yes	characteristic
219	C₂×C₅₃	#3154	1	1	yes	characteristic
220	C₂×C₅₃	#3156	1	1	yes	characteristic
221	#533	C₂²×C₅₃	1	1	yes	characteristic
222	#532	C₂²×C₅₃	1	1	yes	characteristic
223	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
224	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
225	#500	C₂²×C₅₃	1	2	yes	—
226	#500	C₂²×C₅₃	1	2	yes	—
227	#498	C₂²×C₅₃	1	2	yes	—
228	#492	C₂²×C₅₃	1	2	yes	—
229	#488	—	2	2	no	—
230	#487	—	2	2	no	—
231	#487	C₂²×C₅₃	1	2	yes	—
232	#487	C₂²×C₅₃	1	2	yes	—
233	#487	C₂²×C₅₃	1	1	yes	characteristic
234	#487	C₂²×C₅₃	1	1	yes	characteristic
235	#487	C₂²×C₅₃	1	2	yes	—
236	#487	C₂²×C₅₃	1	2	yes	—
237	#487	C₂²×C₅₃	1	2	yes	—
238	#486	C₂²×C₅₃	1	1	yes	characteristic
239	#486	C₂²×C₅₃	1	1	yes	characteristic
240	#486	C₂²×C₅₃	1	2	yes	—
241	#484	—	2	2	no	—
242	#483	—	2	2	no	—
243	#480	—	2	2	no	—
244	#477	—	2	2	no	—
245	#475	—	2	2	no	—
246	#474	—	2	2	no	—
247	#461	—	2	2	no	—
248	#448	—	2	2	no	—
249	#431	C₂²×C₅₃	1	1	yes	characteristic
250	#431	C₂²×C₅₃	1	1	yes	characteristic
251	#416	—	2	2	no	—
252	C₂×#103	—	2	2	no	—
253	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
254	#322	C₂²×C₅₃	1	2	yes	—
255	C₈²	C₂²×C₅₃	1	2	yes	—
256	C₅₃	#19657	1	1	yes	characteristic, direct product factor, Sylow
257	C₂×#41	—	2	1	no	—
258	C₂²×Q₈	—	2	1	no	—
259	C₂²×Q₈	—	2	1	no	—
260	C₂×SD₁₆	—	2	4	no	—
261	C₂²×C₈	D₄×C₅₃	1	2	yes	—
262	C₄⋊Q₈	—	2	1	no	—
263	C₄⋊Q₈	—	2	1	no	—
264	C₄⋊Q₈	—	2	1	no	—
265	C₄⋊Q₈	—	2	1	no	—
266	C₄⋊Q₈	—	2	1	no	—
267	C₄⋊Q₈	—	2	1	no	—
268	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
269	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
270	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
271	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
272	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
273	#127	C₂³×C₅₃	1	1	yes	characteristic
274	#125	—	2	1	no	—
275	#124	—	2	1	no	—
276	C₂²⋊Q₈	—	2	1	no	—
277	C₂²⋊Q₈	—	2	1	no	—
278	#121	—	2	1	no	—
279	#121	—	2	1	no	—
280	C₄×Q₈	—	2	1	no	—
281	C₄×Q₈	—	2	1	no	—
282	C₄×D₄	—	2	1	no	—
283	C₄×D₄	—	2	1	no	—
284	#117	—	2	1	no	—
285	C₂×C₄⋊C₄	—	2	1	no	—
286	C₂×C₄⋊C₄	—	2	1	no	—
287	C₂×C₄⋊C₄	—	2	1	no	—
288	C₂×C₄⋊C₄	—	2	1	no	—
289	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
290	#107	—	2	4	no	—
291	#107	—	2	4	no	—
292	#106	—	2	4	no	—
293	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
294	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
295	#103	—	2	4	no	—
296	#103	—	2	4	no	—
297	#103	—	2	2	no	—
298	#103	—	2	2	no	—
299	#103	—	2	4	no	—
300	#103	—	2	4	no	—
301	#103	—	2	4	no	—
302	#102	—	2	2	no	—
303	#102	—	2	2	no	—
304	#102	—	2	4	no	—
305	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
306	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
307	C₄×C₈	D₄×C₅₃	1	2	yes	—
308	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
309	C₄×C₈	C₂³×C₅₃	1	2	yes	—
310	C₄×C₈	C₂³×C₅₃	1	2	yes	—
311	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
312	#41	—	4	4	no	—
313	C₂×Q₈	—	2	1	no	—
314	C₂×Q₈	—	2	1	no	—
315	C₂×Q₈	—	2	1	no	—
316	C₂×Q₈	—	2	2	no	—
317	C₂×Q₈	—	2	2	no	—
318	C₂×Q₈	—	4	1	no	—
319	C₂×Q₈	—	4	1	no	—
320	C₂×Q₈	—	4	4	no	—
321	C₂×Q₈	—	8	2	no	—
322	C₂×D₄	—	2	1	no	—
323	C₂×D₄	—	2	1	no	—
324	C₂×D₄	—	2	1	no	—
325	C₂×D₄	—	4	1	no	—
326	C₂²×C₄	—	4	1	no	—
327	C₂²×C₄	—	4	1	no	—
328	C₂²×C₄	—	4	1	no	—
329	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
330	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
331	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
332	SD₁₆	—	8	4	no	—
333	C₂×C₈	—	2	4	no	—
334	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
335	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
336	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
337	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
338	C₄⋊C₄	—	2	1	no	—
339	C₄⋊C₄	—	2	1	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	2	1	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	1	no	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	—	4	1	no	—
349	C₄⋊C₄	—	4	1	no	—
350	C₄⋊C₄	—	4	1	no	—
351	C₄⋊C₄	—	4	1	no	—
352	C₄⋊C₄	—	4	1	no	—
353	C₄⋊C₄	—	4	1	no	—
354	C₂²⋊C₄	—	4	1	no	—
355	C₂²⋊C₄	—	4	1	no	—
356	C₂²⋊C₄	—	4	1	no	—
357	C₄²	—	4	1	no	—
358	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
359	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
360	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
361	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
362	C₂³	—	4	1	no	—
363	C₂³	#127×C₅₃	1	1	yes	characteristic
364	Q₈	—	4	4	no	—
365	Q₈	—	4	2	no	—
366	Q₈	—	4	4	no	—
367	Q₈	—	4	2	no	—
368	Q₈	—	4	2	no	—
369	Q₈	—	8	2	no	—
370	Q₈	—	8	2	no	—
371	D₄	—	4	2	no	—
372	D₄	—	4	2	no	—
373	D₄	—	4	2	no	—
374	D₄	—	8	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	4	1	no	—
378	C₂×C₄	—	4	1	no	—
379	C₂×C₄	—	4	1	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	8	2	no	—
383	C₂×C₄	—	8	2	no	—
384	C₂×C₄	—	8	2	no	—
385	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
386	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
387	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
388	C₂×C₄	#135×C₅₃	1	4	yes	—
389	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
390	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
391	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
392	C₈	—	2	2	no	—
393	C₈	—	2	2	no	—
394	C₈	—	2	4	no	—
395	C₈	—	2	4	no	—
396	C₂²	—	2	1	no	—
397	C₂²	—	2	1	no	—
398	C₂²	—	8	1	no	—
399	C₂²	—	8	1	no	—
400	C₂²	—	8	1	no	—
401	C₂²	C₅₃×#495	1	2	yes	—
402	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
403	C₄	—	2	1	no	—
404	C₄	—	2	1	no	—
405	C₄	—	8	1	no	—
406	C₄	—	8	2	no	—
407	C₄	—	8	2	no	—
408	C₄	—	8	1	no	—
409	C₄	—	8	1	no	—
410	C₄	C₅₃×#495	1	2	yes	—
411	C₄	C₂×#135×C₅₃	1	2	yes	—
412	C₄	C₅₃×#577	1	2	yes	—
413	C₄	C₅₃×#578	1	2	yes	—
414	C₂	—	2	2	no	—
415	C₂	—	16	1	no	—
416	C₂	C₂×C₅₃×#495	1	1	yes	characteristic
417	C₂	C₅₃×#3154	1	1	yes	characteristic
418	C₂	C₅₃×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₅₃×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3230	C₂	semidirect
2. C₅₃×#3229	C₂	semidirect
6. C₂×C₅₃×#487	C₂	semidirect
7. C₅₃×#1719	C₂	semidirect
8. C₅₃×#1718	C₂	semidirect
9. C₅₃×#1710	C₂	semidirect
10. C₅₃×#1708	C₂	semidirect
11. C₅₃×#1450	C₂	semidirect
186. #19657	C₅₃	direct (N × H)
205. #3230	C₂×C₅₃	semidirect
206. #3229	C₂×C₅₃	semidirect
210. C₂×#487	C₂×C₅₃	semidirect
211. #1719	C₂×C₅₃	semidirect
212. #1718	C₂×C₅₃	semidirect
213. #1710	C₂×C₅₃	semidirect
214. #1708	C₂×C₅₃	semidirect
215. #1450	C₂×C₅₃	semidirect
256. C₅₃	#19657	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×#3230	2
C₅₃×#3229	2
C₅₃×#3226	2
C₅₃×#3206	2
C₅₃×#3131	2
C₂×C₅₃×#487	2
C₅₃×#1719	2
C₅₃×#1718	2
C₅₃×#1710	2
C₅₃×#1708	2
C₅₃×#1450	2
#19657	53

Subgroups — C53×#19657 #27227169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19657 #27227169