Subgroups — C₅₃×#10050 #27217562

Order 13568.

702 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#459	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2547	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1982	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₃×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1279	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₅₃	—	2	1	no	—
9	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×#124×C₅₃	—	2	1	no	—
12	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#486	—	2	2	no	—
14	C₅₃×#486	—	2	2	no	—
15	C₅₃×#459	—	2	2	no	—
16	C₅₃×#459	—	2	2	no	—
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#102×C₅₃	—	2	1	no	—
20	C₂×#102×C₅₃	—	2	2	no	—
21	C₂×#102×C₅₃	—	2	2	no	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#397	—	2	2	no	—
24	C₅₃×#391	—	2	1	no	—
25	C₅₃×#380	—	2	2	no	—
26	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#332	—	2	2	no	—
28	C₅₃×D₄⋊C₈	—	2	2	no	—
29	C₂²×Q₈×C₅₃	—	2	1	no	—
30	C₂²×D₄×C₅₃	—	2	1	no	—
31	C₂²×D₄×C₅₃	—	2	1	no	—
32	C₂²×D₄×C₅₃	—	2	1	no	—
33	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₅₃	—	2	1	no	—
35	C₂×D₈×C₅₃	—	2	4	no	—
36	C₂×D₈×C₅₃	—	4	2	no	—
37	C₂×D₈×C₅₃	—	4	2	no	—
38	C₂²×C₈×C₅₃	—	2	1	no	—
39	C₂²×C₈×C₅₃	—	2	1	no	—
40	C₂²×C₈×C₅₃	—	2	1	no	—
41	C₄⋊Q₈×C₅₃	—	2	1	no	—
42	C₄⋊Q₈×C₅₃	—	2	1	no	—
43	C₄⋊Q₈×C₅₃	—	2	1	no	—
44	C₄⋊Q₈×C₅₃	—	2	1	no	—
45	#127×C₅₃	—	2	1	no	—
46	#127×C₅₃	—	2	1	no	—
47	#127×C₅₃	—	2	1	no	—
48	#127×C₅₃	D₄	1	2	yes	—
49	#124×C₅₃	—	4	2	no	—
50	#124×C₅₃	—	4	2	no	—
51	C₄×D₄×C₅₃	—	2	1	no	—
52	C₄×D₄×C₅₃	—	2	1	no	—
53	C₄×D₄×C₅₃	—	2	1	no	—
54	C₄×D₄×C₅₃	—	2	1	no	—
55	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
56	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
58	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
59	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
60	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
61	C₂×C₄²×C₅₃	—	2	1	no	—
62	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
63	C₄⋊C₈×C₅₃	—	2	1	no	—
64	C₄⋊C₈×C₅₃	—	2	1	no	—
65	C₄⋊C₈×C₅₃	—	2	1	no	—
66	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
67	#102×C₅₃	—	2	8	no	—
68	#102×C₅₃	—	2	4	no	—
69	#102×C₅₃	—	4	4	no	—
70	C₄×C₈×C₅₃	—	2	1	no	—
71	C₄×C₈×C₅₃	D₄	1	2	yes	—
72	#95×C₅₃	—	2	4	no	—
73	#95×C₅₃	—	2	1	no	—
74	C₂⁴×C₅₃	—	2	1	no	—
75	C₂⁴×C₅₃	—	4	1	no	—
76	C₂×Q₈×C₅₃	—	4	1	no	—
77	C₂×Q₈×C₅₃	—	4	1	no	—
78	C₂×Q₈×C₅₃	—	4	1	no	—
79	C₂×Q₈×C₅₃	—	4	1	no	—
80	C₂×Q₈×C₅₃	—	4	1	no	—
81	C₂×Q₈×C₅₃	—	4	1	no	—
82	C₂×D₄×C₅₃	—	2	1	no	—
83	C₂×D₄×C₅₃	—	2	2	no	—
84	C₂×D₄×C₅₃	—	2	2	no	—
85	C₂×D₄×C₅₃	—	2	1	no	—
86	C₂×D₄×C₅₃	—	2	2	no	—
87	C₂×D₄×C₅₃	—	2	1	no	—
88	C₂×D₄×C₅₃	—	2	1	no	—
89	C₂×D₄×C₅₃	—	2	1	no	—
90	C₂×D₄×C₅₃	—	4	4	no	—
91	C₂×D₄×C₅₃	—	4	1	no	—
92	C₂×D₄×C₅₃	—	4	1	no	—
93	C₂×D₄×C₅₃	—	4	1	no	—
94	C₂×D₄×C₅₃	—	4	1	no	—
95	C₂×D₄×C₅₃	—	4	1	no	—
96	C₂×D₄×C₅₃	—	4	2	no	—
97	C₂×D₄×C₅₃	—	4	2	no	—
98	C₂×D₄×C₅₃	—	4	1	no	—
99	C₂×D₄×C₅₃	—	4	1	no	—
100	C₂×D₄×C₅₃	D₈	1	2	yes	—
101	C₂²×C₄×C₅₃	—	2	1	no	—
102	C₂²×C₄×C₅₃	—	2	2	no	—
103	C₂²×C₄×C₅₃	—	2	2	no	—
104	C₂²×C₄×C₅₃	—	4	1	no	—
105	C₂²×C₄×C₅₃	—	4	1	no	—
106	C₂²×C₄×C₅₃	—	4	2	no	—
107	C₂²×C₄×C₅₃	—	4	2	no	—
108	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
111	D₈×C₅₃	—	4	8	no	—
112	C₂×C₈×C₅₃	—	2	2	no	—
113	C₂×C₈×C₅₃	—	2	2	no	—
114	C₂×C₈×C₅₃	—	4	2	no	—
115	C₂×C₈×C₅₃	—	4	1	no	—
116	C₂×C₈×C₅₃	—	4	1	no	—
117	C₂×C₈×C₅₃	—	4	1	no	—
118	C₂×C₈×C₅₃	—	4	2	no	—
119	C₄⋊C₄×C₅₃	—	2	1	no	—
120	C₄⋊C₄×C₅₃	—	2	2	no	—
121	C₄⋊C₄×C₅₃	—	2	1	no	—
122	C₄⋊C₄×C₅₃	—	2	2	no	—
123	C₄⋊C₄×C₅₃	—	2	2	no	—
124	C₄⋊C₄×C₅₃	—	2	2	no	—
125	C₄⋊C₄×C₅₃	—	4	1	no	—
126	C₄⋊C₄×C₅₃	—	4	1	no	—
127	C₂²⋊C₄×C₅₃	—	4	8	no	—
128	C₂²⋊C₄×C₅₃	—	4	1	no	—
129	C₂²⋊C₄×C₅₃	—	4	1	no	—
130	C₄²×C₅₃	—	4	1	no	—
131	C₄²×C₅₃	—	4	1	no	—
132	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
133	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
135	C₄²×C₅₃	#41	1	1	yes	characteristic
136	C₂³×C₅₃	—	2	2	no	—
137	C₂³×C₅₃	—	4	1	no	—
138	C₂³×C₅₃	—	4	2	no	—
139	C₂³×C₅₃	—	4	1	no	—
140	C₂³×C₅₃	—	4	2	no	—
141	C₂³×C₅₃	—	4	2	no	—
142	C₂³×C₅₃	—	8	1	no	—
143	C₂³×C₅₃	—	8	1	no	—
144	C₂³×C₅₃	—	8	2	no	—
145	C₂³×C₅₃	—	8	2	no	—
146	C₂³×C₅₃	#120	1	1	yes	characteristic
147	Q₈×C₅₃	—	8	1	no	—
148	Q₈×C₅₃	—	8	1	no	—
149	Q₈×C₅₃	—	8	1	no	—
150	Q₈×C₅₃	—	8	1	no	—
151	D₄×C₅₃	—	2	2	no	—
152	D₄×C₅₃	—	2	2	no	—
153	D₄×C₅₃	—	4	4	no	—
154	D₄×C₅₃	—	4	2	no	—
155	D₄×C₅₃	—	4	1	no	—
156	D₄×C₅₃	—	4	2	no	—
157	D₄×C₅₃	—	4	4	no	—
158	D₄×C₅₃	—	4	1	no	—
159	D₄×C₅₃	—	8	2	no	—
160	D₄×C₅₃	—	8	1	no	—
161	D₄×C₅₃	—	8	1	no	—
162	D₄×C₅₃	—	8	1	no	—
163	D₄×C₅₃	—	8	1	no	—
164	C₂×C₄×C₅₃	—	2	2	no	—
165	C₂×C₄×C₅₃	—	4	1	no	—
166	C₂×C₄×C₅₃	—	4	2	no	—
167	C₂×C₄×C₅₃	—	4	2	no	—
168	C₂×C₄×C₅₃	—	4	2	no	—
169	C₂×C₄×C₅₃	—	4	2	no	—
170	C₂×C₄×C₅₃	—	4	2	no	—
171	C₂×C₄×C₅₃	—	4	2	no	—
172	C₂×C₄×C₅₃	—	4	2	no	—
173	C₂×C₄×C₅₃	—	8	2	no	—
174	C₂×C₄×C₅₃	—	8	1	no	—
175	C₂×C₄×C₅₃	—	8	1	no	—
176	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
177	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
178	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
179	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
180	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
181	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
182	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
183	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
184	C₂×C₄×C₅₃	#135	1	2	yes	—
185	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
186	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
187	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
188	C₈×C₅₃	—	4	2	no	—
189	C₈×C₅₃	—	4	2	no	—
190	C₈×C₅₃	—	8	1	no	—
191	#10050	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
192	C₂²×C₅₃	—	4	4	no	—
193	C₂²×C₅₃	—	4	2	no	—
194	C₂²×C₅₃	—	8	2	no	—
195	C₂²×C₅₃	—	8	1	no	—
196	C₂²×C₅₃	—	8	1	no	—
197	C₂²×C₅₃	—	8	2	no	—
198	C₂²×C₅₃	—	8	2	no	—
199	C₂²×C₅₃	—	8	2	no	—
200	C₂²×C₅₃	—	16	2	no	—
201	C₂²×C₅₃	—	16	2	no	—
202	C₂²×C₅₃	#392	1	1	yes	characteristic
203	C₂²×C₅₃	#447	1	1	yes	characteristic
204	C₂²×C₅₃	#449	1	2	yes	—
205	C₂²×C₅₃	#449	1	1	yes	characteristic
206	C₂²×C₅₃	#454	1	2	yes	—
207	C₄×C₅₃	—	2	1	no	—
208	C₄×C₅₃	—	2	1	no	—
209	C₄×C₅₃	—	8	2	no	—
210	C₄×C₅₃	—	8	2	no	—
211	C₄×C₅₃	—	8	1	no	—
212	C₄×C₅₃	#447	1	1	yes	characteristic
213	C₄×C₅₃	#449	1	1	yes	characteristic
214	C₄×C₅₃	#463	1	2	yes	—
215	C₄×C₅₃	#466	1	1	yes	characteristic
216	C₄×C₅₃	#469	1	1	yes	characteristic
217	C₄×C₅₃	#495	1	1	yes	characteristic
218	C₄×C₅₃	#496	1	1	yes	characteristic
219	C₂×#486	C₂×C₅₃	1	1	yes	characteristic, complemented
220	C₂×#459	C₂×C₅₃	1	1	yes	characteristic
221	#2547	C₂×C₅₃	1	1	yes	characteristic, complemented
222	#1982	C₂×C₅₃	1	1	yes	characteristic, complemented
223	C₂×#332	C₂×C₅₃	1	1	yes	characteristic, complemented
224	C₂×D₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
225	#1279	C₂×C₅₃	1	1	yes	characteristic, complemented
226	C₂×C₅₃	—	8	2	no	—
227	C₂×C₅₃	—	16	2	no	—
228	C₂×C₅₃	#1625	1	2	yes	—
229	C₂×C₅₃	#1628	1	2	yes	—
230	C₂×C₅₃	#1999	1	1	yes	characteristic
231	C₂×C₅₃	#2003	1	1	yes	characteristic
232	C₂×C₅₃	#2012	1	1	yes	characteristic
233	C₂²×D₈	—	2	1	no	—
234	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
235	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
236	C₂×#124	—	2	1	no	—
237	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
238	#486	—	2	2	no	—
239	#486	—	2	2	no	—
240	#459	—	2	2	no	—
241	#459	—	2	2	no	—
242	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
243	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
244	C₂×#102	—	2	2	no	—
245	C₂×#102	—	2	1	no	—
246	C₂×#102	—	2	2	no	—
247	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
248	#397	—	2	2	no	—
249	#391	—	2	1	no	—
250	#380	—	2	2	no	—
251	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
252	#332	—	2	2	no	—
253	D₄⋊C₈	—	2	2	no	—
254	C₅₃	#10050	1	1	yes	characteristic, direct product factor, Sylow
255	C₂²×Q₈	—	2	1	no	—
256	C₂²×D₄	—	2	1	no	—
257	C₂²×D₄	—	2	1	no	—
258	C₂²×D₄	—	2	1	no	—
259	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
260	C₂³×C₄	—	2	1	no	—
261	C₂×D₈	—	2	4	no	—
262	C₂×D₈	—	4	2	no	—
263	C₂×D₈	—	4	2	no	—
264	C₂²×C₈	—	2	1	no	—
265	C₂²×C₈	—	2	1	no	—
266	C₂²×C₈	—	2	1	no	—
267	C₄⋊Q₈	—	2	1	no	—
268	C₄⋊Q₈	—	2	1	no	—
269	C₄⋊Q₈	—	2	1	no	—
270	C₄⋊Q₈	—	2	1	no	—
271	#127	—	2	1	no	—
272	#127	—	2	1	no	—
273	#127	—	2	1	no	—
274	#127	D₄×C₅₃	1	2	yes	—
275	#124	—	4	2	no	—
276	#124	—	4	2	no	—
277	C₄×D₄	—	2	1	no	—
278	C₄×D₄	—	2	1	no	—
279	C₄×D₄	—	2	1	no	—
280	C₄×D₄	—	2	1	no	—
281	C₂×C₄⋊C₄	—	2	1	no	—
282	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
283	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
284	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
285	C₂×C₂²⋊C₄	—	2	4	no	—
286	C₂×C₂²⋊C₄	—	2	1	no	—
287	C₂×C₄²	—	2	1	no	—
288	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
289	C₄⋊C₈	—	2	1	no	—
290	C₄⋊C₈	—	2	1	no	—
291	C₄⋊C₈	—	2	1	no	—
292	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
293	#102	—	2	4	no	—
294	#102	—	2	8	no	—
295	#102	—	4	4	no	—
296	C₄×C₈	—	2	1	no	—
297	C₄×C₈	D₄×C₅₃	1	2	yes	—
298	#95	—	2	4	no	—
299	#95	—	2	1	no	—
300	C₂⁴	—	2	1	no	—
301	C₂⁴	—	4	1	no	—
302	C₂×Q₈	—	4	1	no	—
303	C₂×Q₈	—	4	1	no	—
304	C₂×Q₈	—	4	1	no	—
305	C₂×Q₈	—	4	1	no	—
306	C₂×Q₈	—	4	1	no	—
307	C₂×Q₈	—	4	1	no	—
308	C₂×D₄	—	2	1	no	—
309	C₂×D₄	—	2	2	no	—
310	C₂×D₄	—	2	2	no	—
311	C₂×D₄	—	2	1	no	—
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	—	2	1	no	—
314	C₂×D₄	—	2	1	no	—
315	C₂×D₄	—	2	1	no	—
316	C₂×D₄	—	4	4	no	—
317	C₂×D₄	—	4	1	no	—
318	C₂×D₄	—	4	1	no	—
319	C₂×D₄	—	4	1	no	—
320	C₂×D₄	—	4	1	no	—
321	C₂×D₄	—	4	2	no	—
322	C₂×D₄	—	4	1	no	—
323	C₂×D₄	—	4	2	no	—
324	C₂×D₄	—	4	1	no	—
325	C₂×D₄	—	4	1	no	—
326	C₂×D₄	D₈×C₅₃	1	2	yes	—
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	4	1	no	—
331	C₂²×C₄	—	4	1	no	—
332	C₂²×C₄	—	4	2	no	—
333	C₂²×C₄	—	4	2	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	D₈	—	4	8	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	4	2	no	—
341	C₂×C₈	—	4	1	no	—
342	C₂×C₈	—	4	1	no	—
343	C₂×C₈	—	4	1	no	—
344	C₂×C₈	—	4	2	no	—
345	C₄⋊C₄	—	2	1	no	—
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	2	no	—
351	C₄⋊C₄	—	4	1	no	—
352	C₄⋊C₄	—	4	1	no	—
353	C₂²⋊C₄	—	4	8	no	—
354	C₂²⋊C₄	—	4	1	no	—
355	C₂²⋊C₄	—	4	1	no	—
356	C₄²	—	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	#41×C₅₃	1	1	yes	characteristic
362	C₂³	—	2	2	no	—
363	C₂³	—	4	1	no	—
364	C₂³	—	4	2	no	—
365	C₂³	—	4	1	no	—
366	C₂³	—	4	2	no	—
367	C₂³	—	4	2	no	—
368	C₂³	—	8	1	no	—
369	C₂³	—	8	1	no	—
370	C₂³	—	8	2	no	—
371	C₂³	—	8	2	no	—
372	C₂³	#120×C₅₃	1	1	yes	characteristic
373	Q₈	—	8	1	no	—
374	Q₈	—	8	1	no	—
375	Q₈	—	8	1	no	—
376	Q₈	—	8	1	no	—
377	D₄	—	2	2	no	—
378	D₄	—	2	2	no	—
379	D₄	—	4	4	no	—
380	D₄	—	4	2	no	—
381	D₄	—	4	1	no	—
382	D₄	—	4	2	no	—
383	D₄	—	4	4	no	—
384	D₄	—	4	1	no	—
385	D₄	—	8	2	no	—
386	D₄	—	8	1	no	—
387	D₄	—	8	1	no	—
388	D₄	—	8	1	no	—
389	D₄	—	8	1	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	8	1	no	—
400	C₂×C₄	—	8	1	no	—
401	C₂×C₄	—	8	2	no	—
402	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
403	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
404	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
405	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
406	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
407	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
408	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
409	C₂×C₄	#135×C₅₃	1	2	yes	—
410	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
411	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
412	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
413	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
414	C₈	—	4	2	no	—
415	C₈	—	4	2	no	—
416	C₈	—	8	1	no	—
417	C₂²	—	4	2	no	—
418	C₂²	—	4	4	no	—
419	C₂²	—	8	1	no	—
420	C₂²	—	8	1	no	—
421	C₂²	—	8	2	no	—
422	C₂²	—	8	2	no	—
423	C₂²	—	8	2	no	—
424	C₂²	—	8	2	no	—
425	C₂²	—	16	2	no	—
426	C₂²	—	16	2	no	—
427	C₂²	C₅₃×#392	1	1	yes	characteristic
428	C₂²	C₅₃×#447	1	1	yes	characteristic
429	C₂²	C₅₃×#449	1	1	yes	characteristic
430	C₂²	C₅₃×#449	1	2	yes	—
431	C₂²	C₅₃×#454	1	2	yes	—
432	C₄	—	2	1	no	—
433	C₄	—	2	1	no	—
434	C₄	—	8	2	no	—
435	C₄	—	8	1	no	—
436	C₄	—	8	2	no	—
437	C₄	C₅₃×#447	1	1	yes	characteristic
438	C₄	C₅₃×#449	1	1	yes	characteristic
439	C₄	C₅₃×#463	1	2	yes	—
440	C₄	C₅₃×#466	1	1	yes	characteristic
441	C₄	C₅₃×#469	1	1	yes	characteristic
442	C₄	C₅₃×#495	1	1	yes	characteristic
443	C₄	C₅₃×#496	1	1	yes	characteristic
444	C₂	—	8	2	no	—
445	C₂	—	16	2	no	—
446	C₂	C₅₃×#1625	1	2	yes	—
447	C₂	C₅₃×#1628	1	2	yes	—
448	C₂	C₅₃×#1999	1	1	yes	characteristic
449	C₂	C₅₃×#2003	1	1	yes	characteristic
450	C₂	C₅₃×#2012	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂×C₅₃×#459	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#486	C₂	semidirect
3. C₅₃×#2547	C₂	semidirect
4. C₅₃×#1982	C₂	semidirect
5. C₂×C₅₃×#332	C₂	semidirect
6. C₂×C₅₃×D₄⋊C₈	C₂	semidirect
7. C₅₃×#1279	C₂	semidirect
191. #10050	C₅₃	direct (N × H)
219. C₂×#486	C₂×C₅₃	semidirect
221. #2547	C₂×C₅₃	semidirect
222. #1982	C₂×C₅₃	semidirect
223. C₂×#332	C₂×C₅₃	semidirect
224. C₂×D₄⋊C₈	C₂×C₅₃	semidirect
225. #1279	C₂×C₅₃	semidirect
254. C₅₃	#10050	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#486	2
C₂×C₅₃×#459	2
C₅₃×#2547	2
C₅₃×#1982	2
C₂×C₅₃×#332	2
C₂×C₅₃×D₄⋊C₈	2
C₅₃×#1279	2
#10050	53

Subgroups — C53×#10050 #27217562

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10050 #27217562