Subgroups — C₅₃×#22274 #27229786

Order 13568.

922 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#495	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2326	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1959	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1880	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1879	C₂	1	2	yes	maximal
6	C₅₃×#1878	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1877	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1834	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1746	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×SD₁₆×C₅₃	C₂²	1	2	yes	—
12	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#124×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
16	C₅₃×#495	C₄	1	8	yes	complemented
17	C₂×#103×C₅₃	C₂²	1	2	yes	—
18	C₂×#103×C₅₃	C₂²	1	2	yes	—
19	C₂×#102×C₅₃	C₂²	1	2	yes	—
20	C₂×#102×C₅₃	C₂²	1	2	yes	—
21	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#391	C₂²	1	2	yes	—
24	C₅₃×#390	C₂²	1	2	yes	—
25	C₅₃×#388	—	2	2	no	—
26	C₅₃×#388	—	2	2	no	—
27	C₅₃×#385	—	2	2	no	—
28	C₅₃×#385	—	2	2	no	—
29	C₅₃×#384	C₂²	1	2	yes	—
30	C₅₃×#376	C₂²	1	1	yes	characteristic
31	C₅₃×#340	C₂²	1	1	yes	characteristic
32	C₅₃×#340	C₂²	1	2	yes	—
33	C₅₃×#340	C₂²	1	1	yes	characteristic
34	C₅₃×#336	C₂²	1	2	yes	—
35	C₅₃×#336	C₂²	1	2	yes	—
36	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
37	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
38	C₂³×C₄×C₅₃	—	2	2	no	—
39	C₂×Q₁₆×C₅₃	—	4	2	no	—
40	C₂×Q₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
41	C₂×SD₁₆×C₅₃	—	4	2	no	—
42	C₂×SD₁₆×C₅₃	—	4	2	no	—
43	C₂×SD₁₆×C₅₃	C₂×C₄	1	8	yes	complemented
44	C₂×D₈×C₅₃	—	4	2	no	—
45	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	—
46	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	#124×C₅₃	—	2	4	no	—
50	#124×C₅₃	C₂×C₄	1	8	yes	complemented
51	C₄×Q₈×C₅₃	—	4	2	no	—
52	C₄×Q₈×C₅₃	—	4	2	no	—
53	C₄×D₄×C₅₃	—	4	2	no	—
54	C₄×D₄×C₅₃	—	4	2	no	—
55	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
56	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
57	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
58	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
60	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
61	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
62	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
63	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
64	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
65	C₂×C₄²×C₅₃	—	2	2	no	—
66	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
67	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	#103×C₅₃	—	4	2	no	—
69	#103×C₅₃	—	4	2	no	—
70	#102×C₅₃	—	4	2	no	—
71	#102×C₅₃	—	4	2	no	—
72	#97×C₅₃	—	2	2	no	—
73	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
74	#95×C₅₃	—	2	2	no	—
75	#95×C₅₃	—	2	2	no	—
76	#95×C₅₃	D₄	1	1	yes	characteristic
77	#95×C₅₃	D₄	1	2	yes	—
78	#95×C₅₃	D₄	1	1	yes	characteristic
79	C₂⁴×C₅₃	—	2	2	no	—
80	C₂×Q₈×C₅₃	—	2	2	no	—
81	C₂×Q₈×C₅₃	—	2	2	no	—
82	C₂×Q₈×C₅₃	—	2	2	no	—
83	C₂×Q₈×C₅₃	—	2	2	no	—
84	C₂×Q₈×C₅₃	—	2	2	no	—
85	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
86	C₂×D₄×C₅₃	—	2	2	no	—
87	C₂×D₄×C₅₃	—	2	2	no	—
88	C₂×D₄×C₅₃	—	2	2	no	—
89	C₂×D₄×C₅₃	—	2	2	no	—
90	C₂×D₄×C₅₃	—	2	2	no	—
91	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
92	C₂²×C₄×C₅₃	—	2	2	no	—
93	C₂²×C₄×C₅₃	—	2	2	no	—
94	C₂²×C₄×C₅₃	—	2	2	no	—
95	C₂²×C₄×C₅₃	—	4	4	no	—
96	C₂²×C₄×C₅₃	—	8	2	no	—
97	C₂²×C₄×C₅₃	—	8	2	no	—
98	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
100	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
103	Q₁₆×C₅₃	—	4	4	no	—
104	SD₁₆×C₅₃	—	4	8	no	—
105	D₈×C₅₃	—	4	4	no	—
106	C₂×C₈×C₅₃	—	2	2	no	—
107	C₂×C₈×C₅₃	—	2	2	no	—
108	C₂×C₈×C₅₃	—	2	2	no	—
109	C₂×C₈×C₅₃	—	2	2	no	—
110	C₂×C₈×C₅₃	—	2	2	no	—
111	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
113	C₄⋊C₄×C₅₃	—	2	2	no	—
114	C₄⋊C₄×C₅₃	—	2	2	no	—
115	C₄⋊C₄×C₅₃	—	2	2	no	—
116	C₄⋊C₄×C₅₃	—	2	2	no	—
117	C₄⋊C₄×C₅₃	—	4	4	no	—
118	C₄⋊C₄×C₅₃	—	8	2	no	—
119	C₄⋊C₄×C₅₃	—	8	2	no	—
120	C₂²⋊C₄×C₅₃	—	2	8	no	—
121	C₂²⋊C₄×C₅₃	—	2	8	no	—
122	C₂²⋊C₄×C₅₃	—	4	4	no	—
123	C₂²⋊C₄×C₅₃	—	8	2	no	—
124	C₂²⋊C₄×C₅₃	—	8	2	no	—
125	C₄²×C₅₃	—	2	2	no	—
126	C₄²×C₅₃	—	2	2	no	—
127	C₄²×C₅₃	—	4	4	no	—
128	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
129	C₄²×C₅₃	#41	1	2	yes	—
130	C₂³×C₅₃	—	2	4	no	—
131	C₂³×C₅₃	—	4	2	no	—
132	C₂³×C₅₃	—	4	2	no	—
133	C₂³×C₅₃	—	4	2	no	—
134	C₂³×C₅₃	—	4	2	no	—
135	C₂³×C₅₃	—	4	2	no	—
136	C₂³×C₅₃	—	4	2	no	—
137	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
138	Q₈×C₅₃	—	2	4	no	—
139	Q₈×C₅₃	—	2	4	no	—
140	Q₈×C₅₃	—	4	2	no	—
141	Q₈×C₅₃	—	4	2	no	—
142	D₄×C₅₃	—	2	4	no	—
143	D₄×C₅₃	—	2	4	no	—
144	D₄×C₅₃	—	4	2	no	—
145	D₄×C₅₃	—	4	2	no	—
146	C₂×C₄×C₅₃	—	2	1	no	—
147	C₂×C₄×C₅₃	—	2	2	no	—
148	C₂×C₄×C₅₃	—	2	1	no	—
149	C₂×C₄×C₅₃	—	2	4	no	—
150	C₂×C₄×C₅₃	—	2	2	no	—
151	C₂×C₄×C₅₃	—	2	2	no	—
152	C₂×C₄×C₅₃	—	2	2	no	—
153	C₂×C₄×C₅₃	—	2	2	no	—
154	C₂×C₄×C₅₃	—	2	2	no	—
155	C₂×C₄×C₅₃	—	2	2	no	—
156	C₂×C₄×C₅₃	—	4	2	no	—
157	C₂×C₄×C₅₃	—	4	2	no	—
158	C₂×C₄×C₅₃	—	4	2	no	—
159	C₂×C₄×C₅₃	—	4	2	no	—
160	C₂×C₄×C₅₃	—	4	2	no	—
161	C₂×C₄×C₅₃	—	4	2	no	—
162	C₂×C₄×C₅₃	—	4	2	no	—
163	C₂×C₄×C₅₃	—	4	2	no	—
164	C₂×C₄×C₅₃	—	8	8	no	—
165	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
166	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
167	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
168	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
169	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
170	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
171	C₈×C₅₃	—	2	2	no	—
172	C₈×C₅₃	—	2	2	no	—
173	C₈×C₅₃	—	4	2	no	—
174	#22274	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
175	C₂²×C₅₃	—	4	4	no	—
176	C₂²×C₅₃	—	4	4	no	—
177	C₂²×C₅₃	—	4	4	no	—
178	C₂²×C₅₃	—	8	2	no	—
179	C₂²×C₅₃	—	8	2	no	—
180	C₂²×C₅₃	—	8	2	no	—
181	C₂²×C₅₃	—	8	2	no	—
182	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
183	C₂²×C₅₃	#535	1	2	yes	—
184	C₂²×C₅₃	#576	1	1	yes	characteristic
185	C₂²×C₅₃	#577	1	2	yes	—
186	C₂²×C₅₃	#578	1	1	yes	characteristic
187	C₄×C₅₃	—	2	2	no	—
188	C₄×C₅₃	—	2	2	no	—
189	C₄×C₅₃	—	4	4	no	—
190	C₄×C₅₃	—	4	2	no	—
191	C₄×C₅₃	—	4	2	no	—
192	C₄×C₅₃	—	8	2	no	—
193	C₄×C₅₃	—	8	2	no	—
194	C₄×C₅₃	#518	1	1	yes	characteristic
195	C₄×C₅₃	#518	1	1	yes	characteristic
196	C₄×C₅₃	#540	1	2	yes	—
197	C₂×#495	C₂×C₅₃	1	1	yes	characteristic
198	#2326	C₂×C₅₃	1	2	yes	complemented
199	#1959	C₂×C₅₃	1	2	yes	complemented
200	#1880	C₂×C₅₃	1	2	yes	complemented
201	#1879	C₂×C₅₃	1	2	yes	—
202	#1878	C₂×C₅₃	1	2	yes	complemented
203	#1877	C₂×C₅₃	1	2	yes	complemented
204	#1834	C₂×C₅₃	1	1	yes	characteristic, complemented
205	#1746	C₂×C₅₃	1	1	yes	characteristic, complemented
206	C₂×C₅₃	—	8	4	no	—
207	C₂×C₅₃	#2328	1	1	yes	characteristic
208	C₂×C₅₃	#2951	1	1	yes	characteristic
209	C₂×C₅₃	#2952	1	1	yes	characteristic
210	C₂×C₅₃	#3224	1	2	yes	—
211	C₂×C₅₃	#3225	1	2	yes	—
212	C₂²×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
213	C₂²×SD₁₆	C₂²×C₅₃	1	2	yes	—
214	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
215	C₂×#124	C₂²×C₅₃	1	2	yes	—
216	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
217	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
218	#495	C₄×C₅₃	1	8	yes	complemented
219	C₂×#103	C₂²×C₅₃	1	2	yes	—
220	C₂×#103	C₂²×C₅₃	1	2	yes	—
221	C₂×#102	C₂²×C₅₃	1	2	yes	—
222	C₂×#102	C₂²×C₅₃	1	2	yes	—
223	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
224	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
225	#391	C₂²×C₅₃	1	2	yes	—
226	#390	C₂²×C₅₃	1	2	yes	—
227	#388	—	2	2	no	—
228	#388	—	2	2	no	—
229	#385	—	2	2	no	—
230	#385	—	2	2	no	—
231	#384	C₂²×C₅₃	1	2	yes	—
232	#376	C₂²×C₅₃	1	1	yes	characteristic
233	#340	C₂²×C₅₃	1	1	yes	characteristic
234	#340	C₂²×C₅₃	1	2	yes	—
235	#340	C₂²×C₅₃	1	1	yes	characteristic
236	#336	C₂²×C₅₃	1	2	yes	—
237	#336	C₂²×C₅₃	1	2	yes	—
238	C₅₃	#22274	1	1	yes	characteristic, direct product factor, Sylow
239	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
240	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
241	C₂³×C₄	—	2	2	no	—
242	C₂×Q₁₆	—	4	2	no	—
243	C₂×Q₁₆	C₂×C₄×C₅₃	1	4	yes	complemented
244	C₂×SD₁₆	—	4	2	no	—
245	C₂×SD₁₆	—	4	2	no	—
246	C₂×SD₁₆	C₂×C₄×C₅₃	1	8	yes	complemented
247	C₂×D₈	—	4	2	no	—
248	C₂×D₈	C₂×C₄×C₅₃	1	4	yes	—
249	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
250	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
251	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
252	#124	—	2	4	no	—
253	#124	C₂×C₄×C₅₃	1	8	yes	complemented
254	C₄×Q₈	—	4	2	no	—
255	C₄×Q₈	—	4	2	no	—
256	C₄×D₄	—	4	2	no	—
257	C₄×D₄	—	4	2	no	—
258	C₂×C₄⋊C₄	—	2	2	no	—
259	C₂×C₄⋊C₄	—	2	2	no	—
260	C₂×C₄⋊C₄	—	2	2	no	—
261	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
262	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
263	C₂×C₂²⋊C₄	—	2	2	no	—
264	C₂×C₂²⋊C₄	—	2	2	no	—
265	C₂×C₂²⋊C₄	—	2	2	no	—
266	C₂×C₂²⋊C₄	—	2	2	no	—
267	C₂×C₂²⋊C₄	—	2	2	no	—
268	C₂×C₄²	—	2	2	no	—
269	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
270	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
271	#103	—	4	2	no	—
272	#103	—	4	2	no	—
273	#102	—	4	2	no	—
274	#102	—	4	2	no	—
275	#97	—	2	2	no	—
276	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	complemented
277	#95	—	2	2	no	—
278	#95	—	2	2	no	—
279	#95	D₄×C₅₃	1	1	yes	characteristic
280	#95	D₄×C₅₃	1	2	yes	—
281	#95	D₄×C₅₃	1	1	yes	characteristic
282	C₂⁴	—	2	2	no	—
283	C₂×Q₈	—	2	2	no	—
284	C₂×Q₈	—	2	2	no	—
285	C₂×Q₈	—	2	2	no	—
286	C₂×Q₈	—	2	2	no	—
287	C₂×Q₈	—	2	2	no	—
288	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
289	C₂×D₄	—	2	2	no	—
290	C₂×D₄	—	2	2	no	—
291	C₂×D₄	—	2	2	no	—
292	C₂×D₄	—	2	2	no	—
293	C₂×D₄	—	2	2	no	—
294	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
295	C₂²×C₄	—	2	2	no	—
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	4	4	no	—
299	C₂²×C₄	—	8	2	no	—
300	C₂²×C₄	—	8	2	no	—
301	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
302	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
303	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
304	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
305	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
306	Q₁₆	—	4	4	no	—
307	SD₁₆	—	4	8	no	—
308	D₈	—	4	4	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
315	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	4	4	no	—
321	C₄⋊C₄	—	8	2	no	—
322	C₄⋊C₄	—	8	2	no	—
323	C₂²⋊C₄	—	2	8	no	—
324	C₂²⋊C₄	—	2	8	no	—
325	C₂²⋊C₄	—	4	4	no	—
326	C₂²⋊C₄	—	8	2	no	—
327	C₂²⋊C₄	—	8	2	no	—
328	C₄²	—	2	2	no	—
329	C₄²	—	2	2	no	—
330	C₄²	—	4	4	no	—
331	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
332	C₄²	#41×C₅₃	1	2	yes	—
333	C₂³	—	2	4	no	—
334	C₂³	—	4	2	no	—
335	C₂³	—	4	2	no	—
336	C₂³	—	4	2	no	—
337	C₂³	—	4	2	no	—
338	C₂³	—	4	2	no	—
339	C₂³	—	4	2	no	—
340	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
341	Q₈	—	2	4	no	—
342	Q₈	—	2	4	no	—
343	Q₈	—	4	2	no	—
344	Q₈	—	4	2	no	—
345	D₄	—	2	4	no	—
346	D₄	—	2	4	no	—
347	D₄	—	4	2	no	—
348	D₄	—	4	2	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	4	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	8	8	no	—
368	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
369	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
370	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
371	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
372	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
373	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
374	C₈	—	2	2	no	—
375	C₈	—	2	2	no	—
376	C₈	—	4	2	no	—
377	C₂²	—	4	4	no	—
378	C₂²	—	4	4	no	—
379	C₂²	—	4	4	no	—
380	C₂²	—	8	2	no	—
381	C₂²	—	8	2	no	—
382	C₂²	—	8	2	no	—
383	C₂²	—	8	2	no	—
384	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
385	C₂²	C₅₃×#535	1	2	yes	—
386	C₂²	C₅₃×#576	1	1	yes	characteristic
387	C₂²	C₅₃×#577	1	2	yes	—
388	C₂²	C₅₃×#578	1	1	yes	characteristic
389	C₄	—	2	2	no	—
390	C₄	—	2	2	no	—
391	C₄	—	4	2	no	—
392	C₄	—	4	2	no	—
393	C₄	—	4	4	no	—
394	C₄	—	8	2	no	—
395	C₄	—	8	2	no	—
396	C₄	C₅₃×#518	1	1	yes	characteristic
397	C₄	C₅₃×#518	1	1	yes	characteristic
398	C₄	C₅₃×#540	1	2	yes	—
399	C₂	—	8	4	no	—
400	C₂	C₅₃×#2328	1	1	yes	characteristic
401	C₂	C₅₃×#2951	1	1	yes	characteristic
402	C₂	C₅₃×#2952	1	1	yes	characteristic
403	C₂	C₅₃×#3224	1	2	yes	—
404	C₂	C₅₃×#3225	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂²×D₈×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₅₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2326	C₂	semidirect
3. C₅₃×#1959	C₂	semidirect
4. C₅₃×#1880	C₂	semidirect
6. C₅₃×#1878	C₂	semidirect
7. C₅₃×#1877	C₂	semidirect
8. C₅₃×#1834	C₂	semidirect
9. C₅₃×#1746	C₂	semidirect
16. C₅₃×#495	C₄	semidirect
40. C₂×Q₁₆×C₅₃	C₂×C₄	semidirect
43. C₂×SD₁₆×C₅₃	C₂×C₄	semidirect
50. #124×C₅₃	C₂×C₄	semidirect
73. C₄×C₈×C₅₃	C₂×C₄	semidirect
174. #22274	C₅₃	direct (N × H)
198. #2326	C₂×C₅₃	semidirect
199. #1959	C₂×C₅₃	semidirect
200. #1880	C₂×C₅₃	semidirect
202. #1878	C₂×C₅₃	semidirect
203. #1877	C₂×C₅₃	semidirect
204. #1834	C₂×C₅₃	semidirect
205. #1746	C₂×C₅₃	semidirect
218. #495	C₄×C₅₃	semidirect
238. C₅₃	#22274	direct (N × H)
243. C₂×Q₁₆	C₂×C₄×C₅₃	semidirect
246. C₂×SD₁₆	C₂×C₄×C₅₃	semidirect
253. #124	C₂×C₄×C₅₃	semidirect
276. C₄×C₈	C₂×C₄×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×#495	2
C₅₃×#2326	2
C₅₃×#1959	2
C₅₃×#1880	2
C₅₃×#1879	2
C₅₃×#1878	2
C₅₃×#1877	2
C₅₃×#1834	2
C₅₃×#1746	2
#22274	53

Subgroups — C53×#22274 #27229786

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22274 #27229786