Subgroups — C₅₃×#9247 #27216759

Order 13568.

544 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	C₂×C₅₃×#461	C₂	1	1	yes	maximal, characteristic
2	D₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1997	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1548	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1486	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1292	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1283	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₅₃	C₄	1	2	yes	—
9	C₂²×M₁₆×C₅₃	—	2	1	no	—
10	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#461	—	2	1	no	—
13	C₅₃×#461	—	2	1	no	—
14	C₅₃×#461	—	2	1	no	—
15	C₅₃×#461	—	2	1	no	—
16	C₅₃×#436	—	2	1	no	—
17	C₅₃×#435	—	2	2	no	—
18	C₈×D₄×C₅₃	—	2	1	no	—
19	C₅₃×#431	C₂²	1	1	yes	characteristic
20	C₅₃×#423	C₂²	1	1	yes	characteristic
21	C₅₃×#423	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#103×C₅₃	C₄	1	2	yes	—
24	C₅₃×#407	—	2	1	no	—
25	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#343	—	2	1	no	—
27	C₅₃×#343	—	2	1	no	—
28	C₅₃×#333	—	2	2	no	—
29	C₅₃×D₄⋊C₈	—	2	1	no	—
30	C₅₃×D₄⋊C₈	—	2	1	no	—
31	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₅₃	—	2	1	no	—
34	C₂×SD₁₆×C₅₃	—	2	2	no	—
35	C₂×SD₁₆×C₅₃	—	2	2	no	—
36	C₂×SD₁₆×C₅₃	—	4	4	no	—
37	C₂×M₁₆×C₅₃	—	2	2	no	—
38	C₂×M₁₆×C₅₃	—	2	1	no	—
39	C₂×M₁₆×C₅₃	—	2	1	no	—
40	C₂×M₁₆×C₅₃	—	4	2	no	—
41	C₂×M₁₆×C₅₃	—	4	2	no	—
42	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
43	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	—	2	2	no	—
45	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
46	C₄⋊Q₈×C₅₃	—	2	1	no	—
47	C₄⋊Q₈×C₅₃	—	2	1	no	—
48	C₄⋊Q₈×C₅₃	—	2	1	no	—
49	C₄⋊Q₈×C₅₃	—	2	1	no	—
50	C₄×D₄×C₅₃	—	2	1	no	—
51	C₄×D₄×C₅₃	—	2	1	no	—
52	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
53	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
54	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
55	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
57	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
58	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
60	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₅₃	—	2	1	no	—
62	C₄⋊C₈×C₅₃	—	2	1	no	—
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	—	2	1	no	—
67	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
68	#103×C₅₃	—	2	2	no	—
69	#103×C₅₃	—	2	2	no	—
70	C₂²⋊C₈×C₅₃	—	2	2	no	—
71	C₂²⋊C₈×C₅₃	—	2	2	no	—
72	C₂²⋊C₈×C₅₃	—	2	1	no	—
73	#97×C₅₃	—	2	1	no	—
74	C₄×C₈×C₅₃	—	2	1	no	—
75	C₄×C₈×C₅₃	—	2	1	no	—
76	C₂⁴×C₅₃	—	2	1	no	—
77	C₂×Q₈×C₅₃	—	2	2	no	—
78	C₂×Q₈×C₅₃	—	2	2	no	—
79	C₂×Q₈×C₅₃	—	4	2	no	—
80	C₂×Q₈×C₅₃	—	4	2	no	—
81	C₂×D₄×C₅₃	—	2	2	no	—
82	C₂×D₄×C₅₃	—	2	1	no	—
83	C₂×D₄×C₅₃	—	4	1	no	—
84	C₂×D₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×D₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	—	2	1	no	—
87	C₂²×C₄×C₅₃	—	2	1	no	—
88	C₂²×C₄×C₅₃	—	2	1	no	—
89	C₂²×C₄×C₅₃	—	2	1	no	—
90	C₂²×C₄×C₅₃	—	2	1	no	—
91	C₂²×C₄×C₅₃	—	2	2	no	—
92	C₂²×C₄×C₅₃	—	4	1	no	—
93	C₂²×C₄×C₅₃	—	4	1	no	—
94	C₂²×C₄×C₅₃	—	4	1	no	—
95	C₂²×C₄×C₅₃	—	4	1	no	—
96	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
99	SD₁₆×C₅₃	—	4	4	no	—
100	SD₁₆×C₅₃	—	4	4	no	—
101	M₁₆×C₅₃	—	2	1	no	—
102	M₁₆×C₅₃	—	2	1	no	—
103	M₁₆×C₅₃	—	2	1	no	—
104	M₁₆×C₅₃	—	4	4	no	—
105	M₁₆×C₅₃	—	4	1	no	—
106	M₁₆×C₅₃	—	4	1	no	—
107	M₁₆×C₅₃	—	4	1	no	—
108	M₁₆×C₅₃	—	4	1	no	—
109	M₁₆×C₅₃	SD₁₆	1	2	yes	complemented
110	C₂×C₈×C₅₃	—	2	1	no	—
111	C₂×C₈×C₅₃	—	2	2	no	—
112	C₂×C₈×C₅₃	—	2	1	no	—
113	C₂×C₈×C₅₃	—	2	2	no	—
114	C₂×C₈×C₅₃	—	4	4	no	—
115	C₂×C₈×C₅₃	—	4	2	no	—
116	C₂×C₈×C₅₃	—	4	1	no	—
117	C₂×C₈×C₅₃	—	4	1	no	—
118	C₄⋊C₄×C₅₃	—	2	1	no	—
119	C₄⋊C₄×C₅₃	—	4	1	no	—
120	C₄⋊C₄×C₅₃	—	4	1	no	—
121	C₄⋊C₄×C₅₃	—	4	1	no	—
122	C₄⋊C₄×C₅₃	—	4	1	no	—
123	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
124	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
125	C₂²⋊C₄×C₅₃	—	2	1	no	—
126	C₂²⋊C₄×C₅₃	—	2	1	no	—
127	C₂²⋊C₄×C₅₃	—	2	1	no	—
128	C₂²⋊C₄×C₅₃	—	2	1	no	—
129	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
132	C₄²×C₅₃	#41	1	1	yes	characteristic
133	C₂³×C₅₃	—	2	1	no	—
134	C₂³×C₅₃	—	2	1	no	—
135	C₂³×C₅₃	—	4	2	no	—
136	C₂³×C₅₃	—	4	1	no	—
137	C₂³×C₅₃	—	4	1	no	—
138	C₂³×C₅₃	—	8	1	no	—
139	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
140	Q₈×C₅₃	—	4	2	no	—
141	Q₈×C₅₃	—	4	2	no	—
142	Q₈×C₅₃	—	8	2	no	—
143	D₄×C₅₃	—	2	1	no	—
144	D₄×C₅₃	—	2	1	no	—
145	D₄×C₅₃	—	2	1	no	—
146	D₄×C₅₃	—	4	2	no	—
147	D₄×C₅₃	#100	1	2	yes	—
148	C₂×C₄×C₅₃	—	2	1	no	—
149	C₂×C₄×C₅₃	—	2	1	no	—
150	C₂×C₄×C₅₃	—	2	1	no	—
151	C₂×C₄×C₅₃	—	2	1	no	—
152	C₂×C₄×C₅₃	—	4	1	no	—
153	C₂×C₄×C₅₃	—	4	1	no	—
154	C₂×C₄×C₅₃	—	4	1	no	—
155	C₂×C₄×C₅₃	—	4	1	no	—
156	C₂×C₄×C₅₃	—	4	1	no	—
157	C₂×C₄×C₅₃	—	4	1	no	—
158	C₂×C₄×C₅₃	—	4	1	no	—
159	C₂×C₄×C₅₃	—	4	1	no	—
160	C₂×C₄×C₅₃	—	4	1	no	—
161	C₂×C₄×C₅₃	—	4	1	no	—
162	C₂×C₄×C₅₃	—	4	1	no	—
163	C₂×C₄×C₅₃	—	4	2	no	—
164	C₂×C₄×C₅₃	—	8	2	no	—
165	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
166	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
167	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
168	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
169	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
170	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
171	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
172	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
173	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
174	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
175	C₈×C₅₃	—	2	2	no	—
176	C₈×C₅₃	—	4	2	no	—
177	C₈×C₅₃	—	4	2	no	—
178	C₈×C₅₃	—	4	1	no	—
179	C₈×C₅₃	—	8	1	no	—
180	C₈×C₅₃	—	8	1	no	—
181	#9247	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
182	C₂²×C₅₃	—	2	1	no	—
183	C₂²×C₅₃	—	2	1	no	—
184	C₂²×C₅₃	—	2	2	no	—
185	C₂²×C₅₃	—	4	1	no	—
186	C₂²×C₅₃	—	4	1	no	—
187	C₂²×C₅₃	—	4	1	no	—
188	C₂²×C₅₃	—	4	1	no	—
189	C₂²×C₅₃	—	4	1	no	—
190	C₂²×C₅₃	—	8	2	no	—
191	C₂²×C₅₃	—	8	2	no	—
192	C₂²×C₅₃	#386	1	1	yes	characteristic
193	C₂²×C₅₃	C₄×SD₁₆	1	1	yes	characteristic
194	C₂²×C₅₃	#440	1	1	yes	characteristic
195	C₄×C₅₃	—	2	1	no	—
196	C₄×C₅₃	—	2	1	no	—
197	C₄×C₅₃	—	2	1	no	—
198	C₄×C₅₃	—	4	1	no	—
199	C₄×C₅₃	—	4	1	no	—
200	C₄×C₅₃	—	8	2	no	—
201	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
202	C₄×C₅₃	#413	1	1	yes	characteristic
203	C₄×C₅₃	#450	1	1	yes	characteristic
204	C₄×C₅₃	#452	1	1	yes	characteristic
205	C₄×C₅₃	#461	1	1	yes	characteristic
206	C₄×C₅₃	#463	1	1	yes	characteristic
207	C₂×#461	C₂×C₅₃	1	1	yes	characteristic
208	D₄×M₁₆	C₂×C₅₃	1	1	yes	characteristic
209	#1997	C₂×C₅₃	1	1	yes	characteristic, complemented
210	#1548	C₂×C₅₃	1	1	yes	characteristic, complemented
211	#1486	C₂×C₅₃	1	1	yes	characteristic
212	#1292	C₂×C₅₃	1	1	yes	characteristic, complemented
213	#1283	C₂×C₅₃	1	1	yes	characteristic, complemented
214	C₂×C₅₃	—	2	1	no	—
215	C₂×C₅₃	—	2	1	no	—
216	C₂×C₅₃	—	4	2	no	—
217	C₂×C₅₃	—	8	1	no	—
218	C₂×C₅₃	#1877	1	1	yes	characteristic
219	C₂×C₅₃	#1900	1	1	yes	characteristic
220	C₂×C₅₃	#1902	1	1	yes	characteristic
221	C₂²×SD₁₆	C₄×C₅₃	1	2	yes	—
222	C₂²×M₁₆	—	2	1	no	—
223	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
224	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
225	#461	—	2	1	no	—
226	#461	—	2	1	no	—
227	#461	—	2	1	no	—
228	#461	—	2	1	no	—
229	#436	—	2	1	no	—
230	#435	—	2	2	no	—
231	C₈×D₄	—	2	1	no	—
232	#431	C₂²×C₅₃	1	1	yes	characteristic
233	#423	C₂²×C₅₃	1	1	yes	characteristic
234	#423	C₂²×C₅₃	1	1	yes	characteristic
235	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
236	C₂×#103	C₄×C₅₃	1	2	yes	—
237	#407	—	2	1	no	—
238	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
239	#343	—	2	1	no	—
240	#343	—	2	1	no	—
241	#333	—	2	2	no	—
242	D₄⋊C₈	—	2	1	no	—
243	D₄⋊C₈	—	2	1	no	—
244	C₅₃	#9247	1	1	yes	characteristic, direct product factor, Sylow
245	C₂²×Q₈	C₂×C₄×C₅₃	1	2	yes	—
246	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
247	C₂³×C₄	—	2	1	no	—
248	C₂×SD₁₆	—	2	2	no	—
249	C₂×SD₁₆	—	2	2	no	—
250	C₂×SD₁₆	—	4	4	no	—
251	C₂×M₁₆	—	2	2	no	—
252	C₂×M₁₆	—	2	1	no	—
253	C₂×M₁₆	—	2	1	no	—
254	C₂×M₁₆	—	4	2	no	—
255	C₂×M₁₆	—	4	2	no	—
256	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
257	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
258	C₂²×C₈	—	2	2	no	—
259	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
260	C₄⋊Q₈	—	2	1	no	—
261	C₄⋊Q₈	—	2	1	no	—
262	C₄⋊Q₈	—	2	1	no	—
263	C₄⋊Q₈	—	2	1	no	—
264	C₄×D₄	—	2	1	no	—
265	C₄×D₄	—	2	1	no	—
266	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
267	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
268	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
269	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
270	C₂×C₄⋊C₄	—	2	1	no	—
271	C₂×C₄⋊C₄	—	2	1	no	—
272	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
273	C₂×C₂²⋊C₄	—	2	1	no	—
274	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
275	C₄⋊C₈	—	2	1	no	—
276	C₄⋊C₈	—	2	1	no	—
277	C₄⋊C₈	—	2	1	no	—
278	C₄⋊C₈	—	2	1	no	—
279	C₄⋊C₈	—	2	1	no	—
280	C₄⋊C₈	—	2	1	no	—
281	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
282	#103	—	2	2	no	—
283	#103	—	2	2	no	—
284	C₂²⋊C₈	—	2	2	no	—
285	C₂²⋊C₈	—	2	2	no	—
286	C₂²⋊C₈	—	2	1	no	—
287	#97	—	2	1	no	—
288	C₄×C₈	—	2	1	no	—
289	C₄×C₈	—	2	1	no	—
290	C₂⁴	—	2	1	no	—
291	C₂×Q₈	—	2	2	no	—
292	C₂×Q₈	—	2	2	no	—
293	C₂×Q₈	—	4	2	no	—
294	C₂×Q₈	—	4	2	no	—
295	C₂×D₄	—	2	2	no	—
296	C₂×D₄	—	2	1	no	—
297	C₂×D₄	—	4	1	no	—
298	C₂×D₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
299	C₂×D₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
300	C₂²×C₄	—	2	1	no	—
301	C₂²×C₄	—	2	1	no	—
302	C₂²×C₄	—	2	1	no	—
303	C₂²×C₄	—	2	1	no	—
304	C₂²×C₄	—	2	1	no	—
305	C₂²×C₄	—	2	2	no	—
306	C₂²×C₄	—	4	1	no	—
307	C₂²×C₄	—	4	1	no	—
308	C₂²×C₄	—	4	1	no	—
309	C₂²×C₄	—	4	1	no	—
310	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
311	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
312	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
313	SD₁₆	—	4	4	no	—
314	SD₁₆	—	4	4	no	—
315	M₁₆	—	2	1	no	—
316	M₁₆	—	2	1	no	—
317	M₁₆	—	2	1	no	—
318	M₁₆	—	4	4	no	—
319	M₁₆	—	4	1	no	—
320	M₁₆	—	4	1	no	—
321	M₁₆	—	4	1	no	—
322	M₁₆	—	4	1	no	—
323	M₁₆	SD₁₆×C₅₃	1	2	yes	complemented
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	4	4	no	—
329	C₂×C₈	—	4	2	no	—
330	C₂×C₈	—	4	1	no	—
331	C₂×C₈	—	4	1	no	—
332	C₄⋊C₄	—	2	1	no	—
333	C₄⋊C₄	—	4	1	no	—
334	C₄⋊C₄	—	4	1	no	—
335	C₄⋊C₄	—	4	1	no	—
336	C₄⋊C₄	—	4	1	no	—
337	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
338	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
339	C₂²⋊C₄	—	2	1	no	—
340	C₂²⋊C₄	—	2	1	no	—
341	C₂²⋊C₄	—	2	1	no	—
342	C₂²⋊C₄	—	2	1	no	—
343	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
344	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
345	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
346	C₄²	#41×C₅₃	1	1	yes	characteristic
347	C₂³	—	2	1	no	—
348	C₂³	—	2	1	no	—
349	C₂³	—	4	2	no	—
350	C₂³	—	4	1	no	—
351	C₂³	—	4	1	no	—
352	C₂³	—	8	1	no	—
353	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
354	Q₈	—	4	2	no	—
355	Q₈	—	4	2	no	—
356	Q₈	—	8	2	no	—
357	D₄	—	2	1	no	—
358	D₄	—	2	1	no	—
359	D₄	—	2	1	no	—
360	D₄	—	4	2	no	—
361	D₄	#100×C₅₃	1	2	yes	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	4	1	no	—
368	C₂×C₄	—	4	1	no	—
369	C₂×C₄	—	4	1	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	—	4	1	no	—
372	C₂×C₄	—	4	1	no	—
373	C₂×C₄	—	4	1	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	4	1	no	—
376	C₂×C₄	—	4	1	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	8	2	no	—
379	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
380	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
381	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
382	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
383	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
384	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
385	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
386	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
387	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
388	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
389	C₈	—	2	2	no	—
390	C₈	—	4	2	no	—
391	C₈	—	4	2	no	—
392	C₈	—	4	1	no	—
393	C₈	—	8	1	no	—
394	C₈	—	8	1	no	—
395	C₂²	—	2	1	no	—
396	C₂²	—	2	1	no	—
397	C₂²	—	2	2	no	—
398	C₂²	—	4	1	no	—
399	C₂²	—	4	1	no	—
400	C₂²	—	4	1	no	—
401	C₂²	—	4	1	no	—
402	C₂²	—	4	1	no	—
403	C₂²	—	8	2	no	—
404	C₂²	—	8	2	no	—
405	C₂²	C₅₃×#386	1	1	yes	characteristic
406	C₂²	C₄×SD₁₆×C₅₃	1	1	yes	characteristic
407	C₂²	C₅₃×#440	1	1	yes	characteristic
408	C₄	—	2	1	no	—
409	C₄	—	2	1	no	—
410	C₄	—	2	1	no	—
411	C₄	—	4	1	no	—
412	C₄	—	4	1	no	—
413	C₄	—	8	2	no	—
414	C₄	C₂×#100×C₅₃	1	1	yes	characteristic
415	C₄	C₅₃×#413	1	1	yes	characteristic
416	C₄	C₅₃×#450	1	1	yes	characteristic
417	C₄	C₅₃×#452	1	1	yes	characteristic
418	C₄	C₅₃×#461	1	1	yes	characteristic
419	C₄	C₅₃×#463	1	1	yes	characteristic
420	C₂	—	2	1	no	—
421	C₂	—	2	1	no	—
422	C₂	—	4	2	no	—
423	C₂	—	8	1	no	—
424	C₂	C₅₃×#1877	1	1	yes	characteristic
425	C₂	C₅₃×#1900	1	1	yes	characteristic
426	C₂	C₅₃×#1902	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#386
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂²×D₄×C₅₃	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ 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
3. C₅₃×#1997	C₂	semidirect
4. C₅₃×#1548	C₂	semidirect
6. C₅₃×#1292	C₂	semidirect
7. C₅₃×#1283	C₂	semidirect
109. M₁₆×C₅₃	SD₁₆	semidirect
181. #9247	C₅₃	direct (N × H)
209. #1997	C₂×C₅₃	semidirect
210. #1548	C₂×C₅₃	semidirect
212. #1292	C₂×C₅₃	semidirect
213. #1283	C₂×C₅₃	semidirect
244. C₅₃	#9247	direct (N × H)
323. 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
C₂×C₅₃×#461	2
D₄×M₁₆×C₅₃	2
C₅₃×#1997	2
C₅₃×#1548	2
C₅₃×#1486	2
C₅₃×#1292	2
C₅₃×#1283	2
#9247	53

Subgroups — C53×#9247 #27216759

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9247 #27216759