Subgroups — C₅₃×#19624 #27227136

Order 13568.

772 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3229	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3227	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#3167	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#3165	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#3152	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1716	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1714	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1708	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1707	C₂	1	2	yes	maximal, complemented
11	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137×C₅₃	—	2	2	no	—
13	C₅₃×#533	C₂²	1	1	yes	characteristic
14	C₅₃×#532	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#501	—	2	2	no	—
18	C₅₃×#500	C₂²	1	2	yes	—
19	C₅₃×#498	C₂²	1	2	yes	—
20	C₅₃×#498	C₂²	1	2	yes	—
21	C₅₃×#498	C₂²	1	2	yes	—
22	C₅₃×#498	C₂²	1	1	yes	characteristic
23	C₅₃×#497	—	2	2	no	—
24	C₅₃×#494	C₂²	1	2	yes	—
25	C₅₃×#492	C₂²	1	2	yes	—
26	C₅₃×#492	C₂²	1	1	yes	characteristic
27	C₅₃×#487	—	2	2	no	—
28	C₅₃×#487	C₂²	1	2	yes	—
29	C₅₃×#487	C₂²	1	1	yes	characteristic
30	C₅₃×#487	C₂²	1	2	yes	—
31	C₅₃×#487	C₂²	1	2	yes	—
32	C₅₃×#486	C₂²	1	2	yes	—
33	C₅₃×#486	C₂²	1	1	yes	characteristic
34	C₅₃×#483	—	2	2	no	—
35	C₅₃×#481	—	2	2	no	—
36	C₅₃×#476	—	2	2	no	—
37	C₅₃×#475	—	2	2	no	—
38	C₅₃×#465	—	2	2	no	—
39	C₄×SD₁₆×C₅₃	—	2	2	no	—
40	C₅₃×#431	C₂²	1	1	yes	characteristic
41	C₅₃×#428	—	2	2	no	—
42	C₂×#103×C₅₃	—	2	2	no	—
43	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
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	#137×C₅₃	—	4	8	no	—
51	C₂×Q₁₆×C₅₃	—	2	4	no	—
52	C₂×SD₁₆×C₅₃	—	2	4	no	—
53	C₂×SD₁₆×C₅₃	—	2	2	no	—
54	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
55	C₂²×C₈×C₅₃	D₄	1	2	yes	—
56	C₄⋊Q₈×C₅₃	—	2	1	no	—
57	C₄⋊Q₈×C₅₃	—	2	1	no	—
58	C₄⋊Q₈×C₅₃	—	2	1	no	—
59	C₄⋊Q₈×C₅₃	—	2	1	no	—
60	C₄⋊Q₈×C₅₃	—	2	1	no	—
61	C₄⋊Q₈×C₅₃	—	2	1	no	—
62	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
63	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
64	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
65	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
66	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
67	#127×C₅₃	C₂³	1	1	yes	characteristic
68	#125×C₅₃	—	2	1	no	—
69	#124×C₅₃	—	2	1	no	—
70	C₂²⋊Q₈×C₅₃	—	2	1	no	—
71	C₂²⋊Q₈×C₅₃	—	2	1	no	—
72	#121×C₅₃	—	2	1	no	—
73	#121×C₅₃	—	2	1	no	—
74	C₄×Q₈×C₅₃	—	2	1	no	—
75	C₄×Q₈×C₅₃	—	2	1	no	—
76	C₄×D₄×C₅₃	—	2	1	no	—
77	C₄×D₄×C₅₃	—	2	1	no	—
78	#117×C₅₃	—	2	1	no	—
79	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
80	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
81	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
82	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
83	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
84	#107×C₅₃	—	2	4	no	—
85	#106×C₅₃	—	2	4	no	—
86	#106×C₅₃	—	2	2	no	—
87	#106×C₅₃	—	2	4	no	—
88	#106×C₅₃	—	2	4	no	—
89	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
90	#103×C₅₃	—	2	4	no	—
91	#103×C₅₃	—	2	4	no	—
92	#103×C₅₃	—	2	2	no	—
93	#103×C₅₃	—	2	4	no	—
94	#102×C₅₃	—	2	2	no	—
95	#102×C₅₃	—	2	4	no	—
96	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
97	#97×C₅₃	D₄	1	2	yes	—
98	C₄×C₈×C₅₃	D₄	1	2	yes	—
99	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₅₃	C₂³	1	2	yes	—
101	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
102	C₄×C₈×C₅₃	C₂³	1	2	yes	—
103	#41×C₅₃	—	4	4	no	—
104	C₂×Q₈×C₅₃	—	2	1	no	—
105	C₂×Q₈×C₅₃	—	2	2	no	—
106	C₂×Q₈×C₅₃	—	2	1	no	—
107	C₂×Q₈×C₅₃	—	2	1	no	—
108	C₂×Q₈×C₅₃	—	2	2	no	—
109	C₂×Q₈×C₅₃	—	4	4	no	—
110	C₂×Q₈×C₅₃	—	4	1	no	—
111	C₂×Q₈×C₅₃	—	4	1	no	—
112	C₂×Q₈×C₅₃	—	4	4	no	—
113	C₂×D₄×C₅₃	—	2	1	no	—
114	C₂×D₄×C₅₃	—	2	1	no	—
115	C₂×D₄×C₅₃	—	2	1	no	—
116	C₂×D₄×C₅₃	—	4	1	no	—
117	C₂²×C₄×C₅₃	—	4	1	no	—
118	C₂²×C₄×C₅₃	—	4	1	no	—
119	C₂²×C₄×C₅₃	—	4	1	no	—
120	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
123	Q₁₆×C₅₃	—	4	8	no	—
124	SD₁₆×C₅₃	—	4	4	no	—
125	SD₁₆×C₅₃	—	4	8	no	—
126	M₁₆×C₅₃	SD₁₆	1	8	yes	complemented
127	C₂×C₈×C₅₃	—	4	2	no	—
128	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
129	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
130	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
131	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
132	C₄⋊C₄×C₅₃	—	2	2	no	—
133	C₄⋊C₄×C₅₃	—	2	1	no	—
134	C₄⋊C₄×C₅₃	—	2	2	no	—
135	C₄⋊C₄×C₅₃	—	2	2	no	—
136	C₄⋊C₄×C₅₃	—	2	1	no	—
137	C₄⋊C₄×C₅₃	—	2	1	no	—
138	C₄⋊C₄×C₅₃	—	2	1	no	—
139	C₄⋊C₄×C₅₃	—	2	1	no	—
140	C₄⋊C₄×C₅₃	—	2	2	no	—
141	C₄⋊C₄×C₅₃	—	2	1	no	—
142	C₄⋊C₄×C₅₃	—	4	1	no	—
143	C₄⋊C₄×C₅₃	—	4	1	no	—
144	C₄⋊C₄×C₅₃	—	4	1	no	—
145	C₄⋊C₄×C₅₃	—	4	1	no	—
146	C₄⋊C₄×C₅₃	—	4	1	no	—
147	C₄⋊C₄×C₅₃	—	4	1	no	—
148	C₂²⋊C₄×C₅₃	—	4	1	no	—
149	C₂²⋊C₄×C₅₃	—	4	1	no	—
150	C₂²⋊C₄×C₅₃	—	4	1	no	—
151	C₄²×C₅₃	—	4	1	no	—
152	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
153	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
154	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
155	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
156	C₂³×C₅₃	—	4	1	no	—
157	C₂³×C₅₃	#127	1	1	yes	characteristic
158	Q₈×C₅₃	—	4	4	no	—
159	Q₈×C₅₃	—	4	2	no	—
160	Q₈×C₅₃	—	4	4	no	—
161	Q₈×C₅₃	—	4	2	no	—
162	Q₈×C₅₃	—	4	2	no	—
163	Q₈×C₅₃	—	8	2	no	—
164	Q₈×C₅₃	—	8	2	no	—
165	D₄×C₅₃	—	4	2	no	—
166	D₄×C₅₃	—	4	2	no	—
167	D₄×C₅₃	—	4	2	no	—
168	D₄×C₅₃	—	8	2	no	—
169	C₂×C₄×C₅₃	—	2	2	no	—
170	C₂×C₄×C₅₃	—	2	2	no	—
171	C₂×C₄×C₅₃	—	4	1	no	—
172	C₂×C₄×C₅₃	—	4	2	no	—
173	C₂×C₄×C₅₃	—	4	2	no	—
174	C₂×C₄×C₅₃	—	4	1	no	—
175	C₂×C₄×C₅₃	—	4	1	no	—
176	C₂×C₄×C₅₃	—	8	2	no	—
177	C₂×C₄×C₅₃	—	8	2	no	—
178	C₂×C₄×C₅₃	—	8	2	no	—
179	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
180	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
181	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
182	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
183	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
186	C₈×C₅₃	—	2	2	no	—
187	C₈×C₅₃	—	2	2	no	—
188	C₈×C₅₃	—	2	4	no	—
189	C₈×C₅₃	C₂×SD₁₆	1	8	yes	—
190	#19624	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
191	C₂²×C₅₃	—	2	1	no	—
192	C₂²×C₅₃	—	2	1	no	—
193	C₂²×C₅₃	—	8	1	no	—
194	C₂²×C₅₃	—	8	1	no	—
195	C₂²×C₅₃	—	8	1	no	—
196	C₂²×C₅₃	#492	1	2	yes	—
197	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
198	C₄×C₅₃	—	2	1	no	—
199	C₄×C₅₃	—	2	1	no	—
200	C₄×C₅₃	—	8	1	no	—
201	C₄×C₅₃	—	8	2	no	—
202	C₄×C₅₃	—	8	2	no	—
203	C₄×C₅₃	—	8	1	no	—
204	C₄×C₅₃	—	8	1	no	—
205	C₄×C₅₃	#492	1	2	yes	—
206	C₄×C₅₃	C₂²×SD₁₆	1	2	yes	—
207	C₄×C₅₃	#575	1	2	yes	—
208	C₄×C₅₃	#578	1	2	yes	—
209	#3229	C₂×C₅₃	1	1	yes	characteristic, complemented
210	#3227	C₂×C₅₃	1	1	yes	characteristic
211	#3167	C₂×C₅₃	1	1	yes	characteristic
212	#3165	C₂×C₅₃	1	1	yes	characteristic, complemented
213	#3152	C₂×C₅₃	1	1	yes	characteristic
214	C₂×#487	C₂×C₅₃	1	1	yes	characteristic, complemented
215	#1716	C₂×C₅₃	1	2	yes	complemented
216	#1714	C₂×C₅₃	1	2	yes	complemented
217	#1708	C₂×C₅₃	1	2	yes	complemented
218	#1707	C₂×C₅₃	1	2	yes	complemented
219	C₈×M₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
220	C₂×C₅₃	—	2	2	no	—
221	C₂×C₅₃	—	16	1	no	—
222	C₂×C₅₃	C₂×#492	1	1	yes	characteristic
223	C₂×C₅₃	#3153	1	1	yes	characteristic
224	C₂×C₅₃	#3156	1	1	yes	characteristic
225	C₂×#137	—	2	2	no	—
226	#533	C₂²×C₅₃	1	1	yes	characteristic
227	#532	C₂²×C₅₃	1	1	yes	characteristic
228	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
229	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
230	#501	—	2	2	no	—
231	#500	C₂²×C₅₃	1	2	yes	—
232	#498	C₂²×C₅₃	1	2	yes	—
233	#498	C₂²×C₅₃	1	1	yes	characteristic
234	#498	C₂²×C₅₃	1	2	yes	—
235	#498	C₂²×C₅₃	1	2	yes	—
236	#497	—	2	2	no	—
237	#494	C₂²×C₅₃	1	2	yes	—
238	#492	C₂²×C₅₃	1	2	yes	—
239	#492	C₂²×C₅₃	1	1	yes	characteristic
240	#487	—	2	2	no	—
241	#487	C₂²×C₅₃	1	2	yes	—
242	#487	C₂²×C₅₃	1	2	yes	—
243	#487	C₂²×C₅₃	1	1	yes	characteristic
244	#487	C₂²×C₅₃	1	2	yes	—
245	#486	C₂²×C₅₃	1	1	yes	characteristic
246	#486	C₂²×C₅₃	1	2	yes	—
247	#483	—	2	2	no	—
248	#481	—	2	2	no	—
249	#476	—	2	2	no	—
250	#475	—	2	2	no	—
251	#465	—	2	2	no	—
252	C₄×SD₁₆	—	2	2	no	—
253	#431	C₂²×C₅₃	1	1	yes	characteristic
254	#428	—	2	2	no	—
255	C₂×#103	—	2	2	no	—
256	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
257	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
258	#322	C₂²×C₅₃	1	2	yes	—
259	C₈²	C₂²×C₅₃	1	2	yes	—
260	C₅₃	#19624	1	1	yes	characteristic, direct product factor, Sylow
261	C₂×#41	—	2	1	no	—
262	C₂²×Q₈	—	2	1	no	—
263	C₂²×Q₈	—	2	1	no	—
264	#137	—	4	8	no	—
265	C₂×Q₁₆	—	2	4	no	—
266	C₂×SD₁₆	—	2	4	no	—
267	C₂×SD₁₆	—	2	2	no	—
268	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
269	C₂²×C₈	D₄×C₅₃	1	2	yes	—
270	C₄⋊Q₈	—	2	1	no	—
271	C₄⋊Q₈	—	2	1	no	—
272	C₄⋊Q₈	—	2	1	no	—
273	C₄⋊Q₈	—	2	1	no	—
274	C₄⋊Q₈	—	2	1	no	—
275	C₄⋊Q₈	—	2	1	no	—
276	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
277	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
278	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
279	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
280	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
281	#127	C₂³×C₅₃	1	1	yes	characteristic
282	#125	—	2	1	no	—
283	#124	—	2	1	no	—
284	C₂²⋊Q₈	—	2	1	no	—
285	C₂²⋊Q₈	—	2	1	no	—
286	#121	—	2	1	no	—
287	#121	—	2	1	no	—
288	C₄×Q₈	—	2	1	no	—
289	C₄×Q₈	—	2	1	no	—
290	C₄×D₄	—	2	1	no	—
291	C₄×D₄	—	2	1	no	—
292	#117	—	2	1	no	—
293	C₂×C₄⋊C₄	—	2	1	no	—
294	C₂×C₄⋊C₄	—	2	1	no	—
295	C₂×C₄⋊C₄	—	2	1	no	—
296	C₂×C₄⋊C₄	—	2	1	no	—
297	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
298	#107	—	2	4	no	—
299	#106	—	2	4	no	—
300	#106	—	2	2	no	—
301	#106	—	2	4	no	—
302	#106	—	2	4	no	—
303	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
304	#103	—	2	4	no	—
305	#103	—	2	4	no	—
306	#103	—	2	2	no	—
307	#103	—	2	4	no	—
308	#102	—	2	2	no	—
309	#102	—	2	4	no	—
310	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
311	#97	D₄×C₅₃	1	2	yes	—
312	C₄×C₈	D₄×C₅₃	1	2	yes	—
313	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
314	C₄×C₈	C₂³×C₅₃	1	2	yes	—
315	C₄×C₈	C₂³×C₅₃	1	2	yes	—
316	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
317	#41	—	4	4	no	—
318	C₂×Q₈	—	2	1	no	—
319	C₂×Q₈	—	2	2	no	—
320	C₂×Q₈	—	2	1	no	—
321	C₂×Q₈	—	2	1	no	—
322	C₂×Q₈	—	2	2	no	—
323	C₂×Q₈	—	4	4	no	—
324	C₂×Q₈	—	4	1	no	—
325	C₂×Q₈	—	4	1	no	—
326	C₂×Q₈	—	4	4	no	—
327	C₂×D₄	—	2	1	no	—
328	C₂×D₄	—	2	1	no	—
329	C₂×D₄	—	2	1	no	—
330	C₂×D₄	—	4	1	no	—
331	C₂²×C₄	—	4	1	no	—
332	C₂²×C₄	—	4	1	no	—
333	C₂²×C₄	—	4	1	no	—
334	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
335	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
336	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
337	Q₁₆	—	4	8	no	—
338	SD₁₆	—	4	4	no	—
339	SD₁₆	—	4	8	no	—
340	M₁₆	SD₁₆×C₅₃	1	8	yes	complemented
341	C₂×C₈	—	4	2	no	—
342	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
343	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
344	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
345	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	2	1	no	—
352	C₄⋊C₄	—	2	1	no	—
353	C₄⋊C₄	—	2	1	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	2	1	no	—
356	C₄⋊C₄	—	4	1	no	—
357	C₄⋊C₄	—	4	1	no	—
358	C₄⋊C₄	—	4	1	no	—
359	C₄⋊C₄	—	4	1	no	—
360	C₄⋊C₄	—	4	1	no	—
361	C₄⋊C₄	—	4	1	no	—
362	C₂²⋊C₄	—	4	1	no	—
363	C₂²⋊C₄	—	4	1	no	—
364	C₂²⋊C₄	—	4	1	no	—
365	C₄²	—	4	1	no	—
366	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
367	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
368	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
369	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
370	C₂³	—	4	1	no	—
371	C₂³	#127×C₅₃	1	1	yes	characteristic
372	Q₈	—	4	4	no	—
373	Q₈	—	4	2	no	—
374	Q₈	—	4	2	no	—
375	Q₈	—	4	4	no	—
376	Q₈	—	4	2	no	—
377	Q₈	—	8	2	no	—
378	Q₈	—	8	2	no	—
379	D₄	—	4	2	no	—
380	D₄	—	4	2	no	—
381	D₄	—	4	2	no	—
382	D₄	—	8	2	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	4	1	no	—
386	C₂×C₄	—	4	1	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	8	2	no	—
391	C₂×C₄	—	8	2	no	—
392	C₂×C₄	—	8	2	no	—
393	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
394	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
395	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
396	C₂×C₄	C₂×SD₁₆×C₅₃	1	4	yes	—
397	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
398	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
399	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
400	C₈	—	2	2	no	—
401	C₈	—	2	2	no	—
402	C₈	—	2	4	no	—
403	C₈	C₂×SD₁₆×C₅₃	1	8	yes	—
404	C₂²	—	2	1	no	—
405	C₂²	—	2	1	no	—
406	C₂²	—	8	1	no	—
407	C₂²	—	8	1	no	—
408	C₂²	—	8	1	no	—
409	C₂²	C₅₃×#492	1	2	yes	—
410	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
411	C₄	—	2	1	no	—
412	C₄	—	2	1	no	—
413	C₄	—	8	1	no	—
414	C₄	—	8	2	no	—
415	C₄	—	8	2	no	—
416	C₄	—	8	1	no	—
417	C₄	—	8	1	no	—
418	C₄	C₅₃×#492	1	2	yes	—
419	C₄	C₂²×SD₁₆×C₅₃	1	2	yes	—
420	C₄	C₅₃×#575	1	2	yes	—
421	C₄	C₅₃×#578	1	2	yes	—
422	C₂	—	2	2	no	—
423	C₂	—	16	1	no	—
424	C₂	C₂×C₅₃×#492	1	1	yes	characteristic
425	C₂	C₅₃×#3153	1	1	yes	characteristic
426	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×#3229	C₂	semidirect
4. C₅₃×#3165	C₂	semidirect
6. C₂×C₅₃×#487	C₂	semidirect
7. C₅₃×#1716	C₂	semidirect
8. C₅₃×#1714	C₂	semidirect
9. C₅₃×#1708	C₂	semidirect
10. C₅₃×#1707	C₂	semidirect
11. C₈×M₁₆×C₅₃	C₂	semidirect
126. M₁₆×C₅₃	SD₁₆	semidirect
190. #19624	C₅₃	direct (N × H)
209. #3229	C₂×C₅₃	semidirect
212. #3165	C₂×C₅₃	semidirect
214. C₂×#487	C₂×C₅₃	semidirect
215. #1716	C₂×C₅₃	semidirect
216. #1714	C₂×C₅₃	semidirect
217. #1708	C₂×C₅₃	semidirect
218. #1707	C₂×C₅₃	semidirect
219. C₈×M₁₆	C₂×C₅₃	semidirect
260. C₅₃	#19624	direct (N × H)
340. M₁₆	SD₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3229	2
C₅₃×#3227	2
C₅₃×#3167	2
C₅₃×#3165	2
C₅₃×#3152	2
C₂×C₅₃×#487	2
C₅₃×#1716	2
C₅₃×#1714	2
C₅₃×#1708	2
C₅₃×#1707	2
C₈×M₁₆×C₅₃	2
#19624	53

Subgroups — C53×#19624 #27227136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19624 #27227136