Subgroups — C₅₃×#19430 #27226942

Order 13568.

626 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3228	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3109	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#3101	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2975	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1701	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1697	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1695	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1691	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1572	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1554	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1543	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×D₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#500	C₂²	1	1	yes	characteristic, complemented
16	C₅₃×#498	C₂²	1	1	yes	characteristic, complemented
17	C₅₃×#486	C₂²	1	1	yes	characteristic, complemented
18	C₅₃×#486	C₂²	1	1	yes	characteristic, complemented
19	C₅₃×#483	—	2	1	no	—
20	C₅₃×#482	—	2	2	no	—
21	C₅₃×#481	—	2	1	no	—
22	C₅₃×#478	—	2	2	no	—
23	C₅₃×#476	C₂²	1	2	yes	—
24	C₅₃×#476	C₂²	1	2	yes	—
25	C₅₃×#461	C₂²	1	2	yes	—
26	C₅₃×#459	C₂²	1	2	yes	—
27	C₅₃×#450	—	2	1	no	—
28	C₅₃×#447	—	2	1	no	—
29	C₅₃×#442	—	2	2	no	—
30	C₄×SD₁₆×C₅₃	—	2	2	no	—
31	C₅₃×#436	—	2	2	no	—
32	C₅₃×#436	C₂²	1	2	yes	—
33	C₅₃×#436	C₂²	1	2	yes	—
34	C₅₃×#432	C₂²	1	1	yes	characteristic
35	C₅₃×#431	C₂²	1	1	yes	characteristic
36	C₅₃×#431	C₂²	1	1	yes	characteristic
37	C₅₃×#431	C₂²	1	1	yes	characteristic
38	C₅₃×#407	—	2	2	no	—
39	C₅₃×#335	C₂²	1	2	yes	complemented
40	C₅₃×#334	C₂²	1	2	yes	complemented
41	C₅₃×#333	C₂²	1	2	yes	complemented
42	C₅₃×#333	C₂²	1	2	yes	complemented
43	C₅₃×#332	C₂²	1	2	yes	complemented
44	C₅₃×#331	C₂²	1	2	yes	complemented
45	C₂×#41×C₅₃	—	2	1	no	—
46	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₅₃	—	2	1	no	—
48	C₂×SD₁₆×C₅₃	—	2	2	no	—
49	C₂×D₈×C₅₃	—	2	2	no	—
50	C₂×M₁₆×C₅₃	—	2	2	no	—
51	C₂×M₁₆×C₅₃	—	2	2	no	—
52	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
55	#127×C₅₃	C₂³	1	1	yes	characteristic
56	#125×C₅₃	—	2	1	no	—
57	#124×C₅₃	—	2	1	no	—
58	C₂²⋊Q₈×C₅₃	—	2	1	no	—
59	C₂²⋊Q₈×C₅₃	—	2	1	no	—
60	#121×C₅₃	—	2	1	no	—
61	#121×C₅₃	—	2	1	no	—
62	C₄×Q₈×C₅₃	—	2	1	no	—
63	C₄×Q₈×C₅₃	—	2	1	no	—
64	C₄×D₄×C₅₃	—	2	1	no	—
65	C₄×D₄×C₅₃	—	2	1	no	—
66	C₄×D₄×C₅₃	—	2	1	no	—
67	C₄×D₄×C₅₃	—	2	1	no	—
68	C₄×D₄×C₅₃	D₄	1	2	yes	—
69	C₄×D₄×C₅₃	C₂³	1	2	yes	—
70	#117×C₅₃	—	2	1	no	—
71	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
72	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
73	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
74	#107×C₅₃	—	2	2	no	—
75	#106×C₅₃	—	2	2	no	—
76	#106×C₅₃	—	2	2	no	—
77	#106×C₅₃	—	2	2	no	—
78	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
79	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
80	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
81	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
82	#103×C₅₃	—	2	2	no	—
83	#102×C₅₃	—	2	2	no	—
84	#102×C₅₃	—	2	2	no	—
85	#102×C₅₃	—	2	2	no	—
86	#102×C₅₃	—	2	2	no	—
87	#102×C₅₃	—	2	2	no	—
88	C₂²⋊C₈×C₅₃	—	2	2	no	—
89	C₂²⋊C₈×C₅₃	—	2	2	no	—
90	C₂²⋊C₈×C₅₃	—	2	1	no	—
91	C₂²⋊C₈×C₅₃	—	2	1	no	—
92	C₂²⋊C₈×C₅₃	—	2	1	no	—
93	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
94	#97×C₅₃	—	2	1	no	—
95	C₄×C₈×C₅₃	—	2	1	no	—
96	C₄×C₈×C₅₃	—	2	1	no	—
97	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
98	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
99	C₂⁴×C₅₃	—	2	1	no	—
100	#41×C₅₃	—	8	2	no	—
101	C₂×Q₈×C₅₃	—	2	1	no	—
102	C₂×Q₈×C₅₃	—	2	1	no	—
103	C₂×Q₈×C₅₃	—	2	1	no	—
104	C₂×D₄×C₅₃	—	2	1	no	—
105	C₂×D₄×C₅₃	—	2	1	no	—
106	C₂×D₄×C₅₃	—	2	1	no	—
107	C₂×D₄×C₅₃	—	2	1	no	—
108	C₂×D₄×C₅₃	—	4	2	no	—
109	C₂×D₄×C₅₃	—	4	1	no	—
110	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₅₃	—	2	1	no	—
112	C₂²×C₄×C₅₃	—	2	2	no	—
113	C₂²×C₄×C₅₃	—	2	2	no	—
114	C₂²×C₄×C₅₃	—	4	1	no	—
115	C₂²×C₄×C₅₃	—	8	2	no	—
116	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
119	SD₁₆×C₅₃	—	4	4	no	—
120	D₈×C₅₃	—	4	4	no	—
121	M₁₆×C₅₃	—	8	2	no	—
122	M₁₆×C₅₃	—	8	2	no	—
123	C₂×C₈×C₅₃	—	2	2	no	—
124	C₂×C₈×C₅₃	—	2	2	no	—
125	C₂×C₈×C₅₃	—	2	2	no	—
126	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
127	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
128	C₄⋊C₄×C₅₃	—	2	1	no	—
129	C₄⋊C₄×C₅₃	—	2	1	no	—
130	C₄⋊C₄×C₅₃	—	2	1	no	—
131	C₄⋊C₄×C₅₃	—	2	1	no	—
132	C₄⋊C₄×C₅₃	—	2	1	no	—
133	C₄⋊C₄×C₅₃	—	2	1	no	—
134	C₄⋊C₄×C₅₃	—	2	1	no	—
135	C₄⋊C₄×C₅₃	—	4	1	no	—
136	C₄⋊C₄×C₅₃	—	4	1	no	—
137	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
138	C₂²⋊C₄×C₅₃	—	2	2	no	—
139	C₂²⋊C₄×C₅₃	—	2	2	no	—
140	C₂²⋊C₄×C₅₃	—	4	1	no	—
141	C₂²⋊C₄×C₅₃	—	4	1	no	—
142	C₂²⋊C₄×C₅₃	—	4	1	no	—
143	C₄²×C₅₃	—	4	1	no	—
144	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
145	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
146	C₄²×C₅₃	#41	1	1	yes	characteristic
147	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
148	C₂³×C₅₃	—	2	2	no	—
149	C₂³×C₅₃	—	4	2	no	—
150	C₂³×C₅₃	—	4	1	no	—
151	C₂³×C₅₃	—	8	1	no	—
152	C₂³×C₅₃	—	8	1	no	—
153	C₂³×C₅₃	#121	1	1	yes	characteristic
154	Q₈×C₅₃	—	4	2	no	—
155	Q₈×C₅₃	—	8	1	no	—
156	Q₈×C₅₃	—	8	1	no	—
157	D₄×C₅₃	—	2	2	no	—
158	D₄×C₅₃	—	2	2	no	—
159	D₄×C₅₃	—	4	2	no	—
160	D₄×C₅₃	—	4	2	no	—
161	D₄×C₅₃	—	4	2	no	—
162	D₄×C₅₃	—	8	1	no	—
163	D₄×C₅₃	—	8	2	no	—
164	C₂×C₄×C₅₃	—	2	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	—	4	1	no	—
173	C₂×C₄×C₅₃	—	8	2	no	—
174	C₂×C₄×C₅₃	—	8	2	no	—
175	C₂×C₄×C₅₃	—	8	1	no	—
176	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
177	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
178	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
179	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
182	C₈×C₅₃	—	2	2	no	—
183	C₈×C₅₃	—	2	2	no	—
184	C₈×C₅₃	—	4	2	no	—
185	C₈×C₅₃	—	4	2	no	—
186	C₈×C₅₃	—	4	2	no	—
187	#19430	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
188	C₂²×C₅₃	—	2	1	no	—
189	C₂²×C₅₃	—	2	1	no	—
190	C₂²×C₅₃	—	2	1	no	—
191	C₂²×C₅₃	—	4	2	no	—
192	C₂²×C₅₃	—	4	2	no	—
193	C₂²×C₅₃	—	4	2	no	—
194	C₂²×C₅₃	—	8	4	no	—
195	C₂²×C₅₃	—	8	1	no	—
196	C₂²×C₅₃	—	8	1	no	—
197	C₂²×C₅₃	—	8	1	no	—
198	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
199	C₄×C₅₃	—	2	1	no	—
200	C₄×C₅₃	—	2	1	no	—
201	C₄×C₅₃	—	2	1	no	—
202	C₄×C₅₃	—	2	1	no	—
203	C₄×C₅₃	—	4	2	no	—
204	C₄×C₅₃	—	8	1	no	—
205	C₄×C₅₃	—	8	1	no	—
206	C₄×C₅₃	—	8	1	no	—
207	C₄×C₅₃	#575	1	2	yes	—
208	C₄×C₅₃	#577	1	2	yes	—
209	#3228	C₂×C₅₃	1	1	yes	characteristic, complemented
210	#3109	C₂×C₅₃	1	1	yes	characteristic
211	#3101	C₂×C₅₃	1	1	yes	characteristic, complemented
212	#2975	C₂×C₅₃	1	1	yes	characteristic, complemented
213	#1701	C₂×C₅₃	1	2	yes	complemented
214	#1697	C₂×C₅₃	1	2	yes	complemented
215	#1695	C₂×C₅₃	1	2	yes	complemented
216	#1691	C₂×C₅₃	1	2	yes	complemented
217	#1572	C₂×C₅₃	1	1	yes	characteristic, complemented
218	#1554	C₂×C₅₃	1	1	yes	characteristic, complemented
219	#1543	C₂×C₅₃	1	1	yes	characteristic, complemented
220	C₂×C₅₃	—	4	1	no	—
221	C₂×C₅₃	—	8	2	no	—
222	C₂×C₅₃	—	16	1	no	—
223	C₂×C₅₃	#3042	1	1	yes	characteristic
224	C₂×C₅₃	#3054	1	1	yes	characteristic
225	C₂×C₅₃	#3068	1	1	yes	characteristic
226	#533	C₂²×C₅₃	1	1	yes	characteristic
227	#532	C₂²×C₅₃	1	1	yes	characteristic
228	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
229	#500	C₂²×C₅₃	1	1	yes	characteristic, complemented
230	#498	C₂²×C₅₃	1	1	yes	characteristic, complemented
231	#486	C₂²×C₅₃	1	1	yes	characteristic, complemented
232	#486	C₂²×C₅₃	1	1	yes	characteristic, complemented
233	#483	—	2	1	no	—
234	#482	—	2	2	no	—
235	#481	—	2	1	no	—
236	#478	—	2	2	no	—
237	#476	C₂²×C₅₃	1	2	yes	—
238	#476	C₂²×C₅₃	1	2	yes	—
239	#461	C₂²×C₅₃	1	2	yes	—
240	#459	C₂²×C₅₃	1	2	yes	—
241	#450	—	2	1	no	—
242	#447	—	2	1	no	—
243	#442	—	2	2	no	—
244	C₄×SD₁₆	—	2	2	no	—
245	#436	—	2	2	no	—
246	#436	C₂²×C₅₃	1	2	yes	—
247	#436	C₂²×C₅₃	1	2	yes	—
248	#432	C₂²×C₅₃	1	1	yes	characteristic
249	#431	C₂²×C₅₃	1	1	yes	characteristic
250	#431	C₂²×C₅₃	1	1	yes	characteristic
251	#431	C₂²×C₅₃	1	1	yes	characteristic
252	#407	—	2	2	no	—
253	#335	C₂²×C₅₃	1	2	yes	complemented
254	#334	C₂²×C₅₃	1	2	yes	complemented
255	#333	C₂²×C₅₃	1	2	yes	complemented
256	#333	C₂²×C₅₃	1	2	yes	complemented
257	#332	C₂²×C₅₃	1	2	yes	complemented
258	#331	C₂²×C₅₃	1	2	yes	complemented
259	C₅₃	#19430	1	1	yes	characteristic, direct product factor, Sylow
260	C₂×#41	—	2	1	no	—
261	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
262	C₂³×C₄	—	2	1	no	—
263	C₂×SD₁₆	—	2	2	no	—
264	C₂×D₈	—	2	2	no	—
265	C₂×M₁₆	—	2	2	no	—
266	C₂×M₁₆	—	2	2	no	—
267	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
268	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
269	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
270	#127	C₂³×C₅₃	1	1	yes	characteristic
271	#125	—	2	1	no	—
272	#124	—	2	1	no	—
273	C₂²⋊Q₈	—	2	1	no	—
274	C₂²⋊Q₈	—	2	1	no	—
275	#121	—	2	1	no	—
276	#121	—	2	1	no	—
277	C₄×Q₈	—	2	1	no	—
278	C₄×Q₈	—	2	1	no	—
279	C₄×D₄	—	2	1	no	—
280	C₄×D₄	—	2	1	no	—
281	C₄×D₄	—	2	1	no	—
282	C₄×D₄	—	2	1	no	—
283	C₄×D₄	D₄×C₅₃	1	2	yes	—
284	C₄×D₄	C₂³×C₅₃	1	2	yes	—
285	#117	—	2	1	no	—
286	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
287	C₂×C₂²⋊C₄	—	2	1	no	—
288	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
289	#107	—	2	2	no	—
290	#106	—	2	2	no	—
291	#106	—	2	2	no	—
292	#106	—	2	2	no	—
293	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
294	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
295	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
296	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
297	#103	—	2	2	no	—
298	#102	—	2	2	no	—
299	#102	—	2	2	no	—
300	#102	—	2	2	no	—
301	#102	—	2	2	no	—
302	#102	—	2	2	no	—
303	C₂²⋊C₈	—	2	1	no	—
304	C₂²⋊C₈	—	2	1	no	—
305	C₂²⋊C₈	—	2	1	no	—
306	C₂²⋊C₈	—	2	2	no	—
307	C₂²⋊C₈	—	2	2	no	—
308	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
309	#97	—	2	1	no	—
310	C₄×C₈	—	2	1	no	—
311	C₄×C₈	—	2	1	no	—
312	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
313	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
314	C₂⁴	—	2	1	no	—
315	#41	—	8	2	no	—
316	C₂×Q₈	—	2	1	no	—
317	C₂×Q₈	—	2	1	no	—
318	C₂×Q₈	—	2	1	no	—
319	C₂×D₄	—	2	1	no	—
320	C₂×D₄	—	2	1	no	—
321	C₂×D₄	—	2	1	no	—
322	C₂×D₄	—	2	1	no	—
323	C₂×D₄	—	4	2	no	—
324	C₂×D₄	—	4	1	no	—
325	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	—	2	2	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	4	1	no	—
330	C₂²×C₄	—	8	2	no	—
331	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
332	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
333	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
334	SD₁₆	—	4	4	no	—
335	D₈	—	4	4	no	—
336	M₁₆	—	8	2	no	—
337	M₁₆	—	8	2	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
342	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
343	C₄⋊C₄	—	2	1	no	—
344	C₄⋊C₄	—	2	1	no	—
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	1	no	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	1	no	—
350	C₄⋊C₄	—	4	1	no	—
351	C₄⋊C₄	—	4	1	no	—
352	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
353	C₂²⋊C₄	—	2	2	no	—
354	C₂²⋊C₄	—	2	2	no	—
355	C₂²⋊C₄	—	4	1	no	—
356	C₂²⋊C₄	—	4	1	no	—
357	C₂²⋊C₄	—	4	1	no	—
358	C₄²	—	4	1	no	—
359	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
360	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
361	C₄²	#41×C₅₃	1	1	yes	characteristic
362	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
363	C₂³	—	2	2	no	—
364	C₂³	—	4	2	no	—
365	C₂³	—	4	1	no	—
366	C₂³	—	8	1	no	—
367	C₂³	—	8	1	no	—
368	C₂³	#121×C₅₃	1	1	yes	characteristic
369	Q₈	—	4	2	no	—
370	Q₈	—	8	1	no	—
371	Q₈	—	8	1	no	—
372	D₄	—	2	2	no	—
373	D₄	—	2	2	no	—
374	D₄	—	4	2	no	—
375	D₄	—	4	2	no	—
376	D₄	—	4	2	no	—
377	D₄	—	8	1	no	—
378	D₄	—	8	2	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	4	1	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	1	no	—
386	C₂×C₄	—	4	1	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	8	2	no	—
389	C₂×C₄	—	8	2	no	—
390	C₂×C₄	—	8	1	no	—
391	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
392	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
393	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
394	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
395	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
396	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
397	C₈	—	2	2	no	—
398	C₈	—	2	2	no	—
399	C₈	—	4	2	no	—
400	C₈	—	4	2	no	—
401	C₈	—	4	2	no	—
402	C₂²	—	2	1	no	—
403	C₂²	—	2	1	no	—
404	C₂²	—	2	1	no	—
405	C₂²	—	4	2	no	—
406	C₂²	—	4	2	no	—
407	C₂²	—	4	2	no	—
408	C₂²	—	8	4	no	—
409	C₂²	—	8	1	no	—
410	C₂²	—	8	1	no	—
411	C₂²	—	8	1	no	—
412	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
413	C₄	—	2	1	no	—
414	C₄	—	2	1	no	—
415	C₄	—	2	1	no	—
416	C₄	—	2	1	no	—
417	C₄	—	4	2	no	—
418	C₄	—	8	1	no	—
419	C₄	—	8	1	no	—
420	C₄	—	8	1	no	—
421	C₄	C₅₃×#575	1	2	yes	—
422	C₄	C₅₃×#577	1	2	yes	—
423	C₂	—	4	1	no	—
424	C₂	—	8	2	no	—
425	C₂	—	16	1	no	—
426	C₂	C₅₃×#3042	1	1	yes	characteristic
427	C₂	C₅₃×#3054	1	1	yes	characteristic
428	C₂	C₅₃×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₅₃×#3109	C₂
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×#3228	C₂	semidirect
3. C₅₃×#3101	C₂	semidirect
4. C₅₃×#2975	C₂	semidirect
5. C₅₃×#1701	C₂	semidirect
6. C₅₃×#1697	C₂	semidirect
7. C₅₃×#1695	C₂	semidirect
8. C₅₃×#1691	C₂	semidirect
9. C₅₃×#1572	C₂	semidirect
10. C₅₃×#1554	C₂	semidirect
11. C₅₃×#1543	C₂	semidirect
15. C₅₃×#500	C₂²	semidirect
16. C₅₃×#498	C₂²	semidirect
17. C₅₃×#486	C₂²	semidirect
18. C₅₃×#486	C₂²	semidirect
39. C₅₃×#335	C₂²	semidirect
40. C₅₃×#334	C₂²	semidirect
41. C₅₃×#333	C₂²	semidirect
42. C₅₃×#333	C₂²	semidirect
43. C₅₃×#332	C₂²	semidirect
44. C₅₃×#331	C₂²	semidirect
187. #19430	C₅₃	direct (N × H)
209. #3228	C₂×C₅₃	semidirect
211. #3101	C₂×C₅₃	semidirect
212. #2975	C₂×C₅₃	semidirect
213. #1701	C₂×C₅₃	semidirect
214. #1697	C₂×C₅₃	semidirect
215. #1695	C₂×C₅₃	semidirect
216. #1691	C₂×C₅₃	semidirect
217. #1572	C₂×C₅₃	semidirect
218. #1554	C₂×C₅₃	semidirect
219. #1543	C₂×C₅₃	semidirect
229. #500	C₂²×C₅₃	semidirect
230. #498	C₂²×C₅₃	semidirect
231. #486	C₂²×C₅₃	semidirect
232. #486	C₂²×C₅₃	semidirect
253. #335	C₂²×C₅₃	semidirect
254. #334	C₂²×C₅₃	semidirect
255. #333	C₂²×C₅₃	semidirect
256. #333	C₂²×C₅₃	semidirect
257. #332	C₂²×C₅₃	semidirect
258. #331	C₂²×C₅₃	semidirect
259. C₅₃	#19430	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×#3228	2
C₅₃×#3109	2
C₅₃×#3101	2
C₅₃×#2975	2
C₅₃×#1701	2
C₅₃×#1697	2
C₅₃×#1695	2
C₅₃×#1691	2
C₅₃×#1572	2
C₅₃×#1554	2
C₅₃×#1543	2
#19430	53

Subgroups — C53×#19430 #27226942

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19430 #27226942