Subgroups — C₅₃×#8567 #27216079

Order 13568.

568 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2293	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1842	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×D₄⋊C₈	C₂	1	2	yes	maximal
5	C₅₃×#1279	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×#117×C₅₃	—	2	1	no	—
8	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
9	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
10	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
11	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#390	C₄	1	1	yes	characteristic
14	C₅₃×#390	C₄	1	1	yes	characteristic
15	C₅₃×#389	C₄	1	1	yes	characteristic
16	C₅₃×#389	C₄	1	1	yes	characteristic
17	C₅₃×#388	—	2	1	no	—
18	C₅₃×#388	—	2	1	no	—
19	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₄×C₂²⋊C₄×C₅₃	—	2	1	no	—
21	C₄³×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#336	—	2	1	no	—
23	C₅₃×D₄⋊C₈	—	2	4	no	—
24	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₃	—	2	1	no	—
26	C₂²×C₈×C₅₃	—	2	2	no	—
27	C₂²×C₈×C₅₃	—	2	2	no	—
28	C₄×D₄×C₅₃	—	2	1	no	—
29	C₄×D₄×C₅₃	—	2	1	no	—
30	C₄×D₄×C₅₃	D₄	1	2	yes	—
31	C₄×D₄×C₅₃	Q₈	1	2	yes	—
32	#117×C₅₃	—	4	2	no	—
33	#117×C₅₃	—	4	2	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
36	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
40	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
41	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
42	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
43	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
44	C₂×C₄²×C₅₃	—	2	1	no	—
45	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
52	C₄⋊C₈×C₅₃	—	2	4	no	—
53	C₄⋊C₈×C₅₃	—	2	2	no	—
54	C₄⋊C₈×C₅₃	—	2	2	no	—
55	C₄⋊C₈×C₅₃	—	4	2	no	—
56	C₄⋊C₈×C₅₃	—	4	2	no	—
57	C₄×C₈×C₅₃	—	2	2	no	—
58	C₄×C₈×C₅₃	—	2	2	no	—
59	#95×C₅₃	—	2	1	no	—
60	#95×C₅₃	—	2	1	no	—
61	#95×C₅₃	—	2	1	no	—
62	C₂⁴×C₅₃	—	2	1	no	—
63	C₂×D₄×C₅₃	—	2	1	no	—
64	C₂×D₄×C₅₃	—	2	2	no	—
65	C₂×D₄×C₅₃	—	2	2	no	—
66	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	—	2	1	no	—
68	C₂²×C₄×C₅₃	—	2	2	no	—
69	C₂²×C₄×C₅₃	—	2	2	no	—
70	C₂²×C₄×C₅₃	—	2	1	no	—
71	C₂²×C₄×C₅₃	—	2	1	no	—
72	C₂²×C₄×C₅₃	—	4	1	no	—
73	C₂²×C₄×C₅₃	—	4	1	no	—
74	C₂²×C₄×C₅₃	—	4	1	no	—
75	C₂²×C₄×C₅₃	—	4	1	no	—
76	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	—	2	4	no	—
84	C₂×C₈×C₅₃	—	4	2	no	—
85	C₂×C₈×C₅₃	—	4	2	no	—
86	C₂×C₈×C₅₃	—	4	2	no	—
87	C₂×C₈×C₅₃	—	4	2	no	—
88	C₂×C₈×C₅₃	—	4	2	no	—
89	C₄⋊C₄×C₅₃	—	2	1	no	—
90	C₄⋊C₄×C₅₃	—	2	2	no	—
91	C₄⋊C₄×C₅₃	—	2	2	no	—
92	C₄⋊C₄×C₅₃	—	4	2	no	—
93	C₄⋊C₄×C₅₃	—	4	2	no	—
94	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
95	C₂²⋊C₄×C₅₃	—	2	2	no	—
96	C₂²⋊C₄×C₅₃	—	2	2	no	—
97	C₂²⋊C₄×C₅₃	—	4	2	no	—
98	C₂²⋊C₄×C₅₃	—	4	2	no	—
99	C₂²⋊C₄×C₅₃	—	4	2	no	—
100	C₂²⋊C₄×C₅₃	—	4	2	no	—
101	C₄²×C₅₃	—	2	2	no	—
102	C₄²×C₅₃	—	2	2	no	—
103	C₄²×C₅₃	—	2	2	no	—
104	C₄²×C₅₃	—	2	1	no	—
105	C₄²×C₅₃	—	2	1	no	—
106	C₄²×C₅₃	—	4	1	no	—
107	C₄²×C₅₃	—	4	1	no	—
108	C₄²×C₅₃	—	4	1	no	—
109	C₄²×C₅₃	D₈	1	2	yes	—
110	C₄²×C₅₃	SD₁₆	1	2	yes	—
111	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
113	C₄²×C₅₃	#41	1	1	yes	characteristic
114	C₄²×C₅₃	#41	1	1	yes	characteristic
115	C₂³×C₅₃	—	2	2	no	—
116	C₂³×C₅₃	—	4	1	no	—
117	C₂³×C₅₃	—	4	1	no	—
118	C₂³×C₅₃	—	4	1	no	—
119	C₂³×C₅₃	—	4	1	no	—
120	C₂³×C₅₃	—	4	1	no	—
121	C₂³×C₅₃	—	4	1	no	—
122	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	D₄×C₅₃	—	2	4	no	—
124	D₄×C₅₃	—	4	2	no	—
125	C₂×C₄×C₅₃	—	2	1	no	—
126	C₂×C₄×C₅₃	—	2	1	no	—
127	C₂×C₄×C₅₃	—	2	1	no	—
128	C₂×C₄×C₅₃	—	2	1	no	—
129	C₂×C₄×C₅₃	—	2	1	no	—
130	C₂×C₄×C₅₃	—	2	1	no	—
131	C₂×C₄×C₅₃	—	2	1	no	—
132	C₂×C₄×C₅₃	—	2	1	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	1	no	—
135	C₂×C₄×C₅₃	—	2	1	no	—
136	C₂×C₄×C₅₃	—	2	1	no	—
137	C₂×C₄×C₅₃	—	4	1	no	—
138	C₂×C₄×C₅₃	—	4	1	no	—
139	C₂×C₄×C₅₃	—	4	2	no	—
140	C₂×C₄×C₅₃	—	4	2	no	—
141	C₂×C₄×C₅₃	—	4	2	no	—
142	C₂×C₄×C₅₃	—	4	2	no	—
143	C₂×C₄×C₅₃	—	4	1	no	—
144	C₂×C₄×C₅₃	—	4	1	no	—
145	C₂×C₄×C₅₃	—	4	1	no	—
146	C₂×C₄×C₅₃	—	4	1	no	—
147	C₂×C₄×C₅₃	—	4	1	no	—
148	C₂×C₄×C₅₃	—	4	1	no	—
149	C₂×C₄×C₅₃	#102	1	2	yes	—
150	C₂×C₄×C₅₃	#102	1	2	yes	—
151	C₂×C₄×C₅₃	#104	1	2	yes	—
152	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
154	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
155	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
158	C₂×C₄×C₅₃	#131	1	2	yes	—
159	C₂×C₄×C₅₃	#131	1	2	yes	—
160	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
162	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
163	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
164	C₈×C₅₃	—	4	4	no	—
165	C₈×C₅₃	—	8	2	no	—
166	#8567	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂²×C₅₃	—	4	2	no	—
168	C₂²×C₅₃	—	4	2	no	—
169	C₂²×C₅₃	—	4	2	no	—
170	C₂²×C₅₃	—	8	1	no	—
171	C₂²×C₅₃	—	8	1	no	—
172	C₂²×C₅₃	—	8	1	no	—
173	C₂²×C₅₃	—	8	1	no	—
174	C₂²×C₅₃	#380	1	1	yes	characteristic
175	C₂²×C₅₃	#408	1	1	yes	characteristic
176	C₂²×C₅₃	#408	1	1	yes	characteristic
177	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
178	C₂²×C₅₃	#417	1	1	yes	characteristic
179	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
180	C₂²×C₅₃	#421	1	1	yes	characteristic
181	C₄×C₅₃	—	2	1	no	—
182	C₄×C₅₃	—	2	1	no	—
183	C₄×C₅₃	—	2	2	no	—
184	C₄×C₅₃	—	2	2	no	—
185	C₄×C₅₃	—	2	2	no	—
186	C₄×C₅₃	—	4	2	no	—
187	C₄×C₅₃	—	4	1	no	—
188	C₄×C₅₃	—	8	1	no	—
189	C₄×C₅₃	—	8	1	no	—
190	C₄×C₅₃	#424	1	2	yes	—
191	C₄×C₅₃	#433	1	2	yes	—
192	C₄×C₅₃	#459	1	1	yes	characteristic
193	C₄×C₅₃	#461	1	1	yes	characteristic
194	C₄×C₅₃	#474	1	1	yes	characteristic
195	C₄×C₅₃	#476	1	1	yes	characteristic
196	#2293	C₂×C₅₃	1	1	yes	characteristic
197	#1842	C₂×C₅₃	1	1	yes	characteristic, complemented
198	C₄×C₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
199	C₂×D₄⋊C₈	C₂×C₅₃	1	2	yes	—
200	#1279	C₂×C₅₃	1	2	yes	complemented
201	C₂×C₅₃	—	8	2	no	—
202	C₂×C₅₃	#1480	1	1	yes	characteristic
203	C₂×C₅₃	#1483	1	1	yes	characteristic
204	C₂×C₅₃	#1492	1	1	yes	characteristic
205	C₂×C₅₃	#1495	1	1	yes	characteristic
206	C₂×C₅₃	#1798	1	1	yes	characteristic
207	C₂×C₅₃	#1802	1	1	yes	characteristic
208	C₂×C₅₃	#1806	1	1	yes	characteristic
209	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
210	C₂×#117	—	2	1	no	—
211	C₂×C₄⋊C₈	—	2	1	no	—
212	C₂×C₄⋊C₈	—	2	1	no	—
213	C₂×C₄⋊C₈	—	2	1	no	—
214	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
215	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
216	#390	C₄×C₅₃	1	1	yes	characteristic
217	#390	C₄×C₅₃	1	1	yes	characteristic
218	#389	C₄×C₅₃	1	1	yes	characteristic
219	#389	C₄×C₅₃	1	1	yes	characteristic
220	#388	—	2	1	no	—
221	#388	—	2	1	no	—
222	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
223	C₄×C₂²⋊C₄	—	2	1	no	—
224	C₄³	C₂²×C₅₃	1	1	yes	characteristic
225	#336	—	2	1	no	—
226	D₄⋊C₈	—	2	4	no	—
227	C₅₃	#8567	1	1	yes	characteristic, direct product factor, Sylow
228	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
229	C₂³×C₄	—	2	1	no	—
230	C₂²×C₈	—	2	2	no	—
231	C₂²×C₈	—	2	2	no	—
232	C₄×D₄	—	2	1	no	—
233	C₄×D₄	—	2	1	no	—
234	C₄×D₄	D₄×C₅₃	1	2	yes	—
235	C₄×D₄	Q₈×C₅₃	1	2	yes	—
236	#117	—	4	2	no	—
237	#117	—	4	2	no	—
238	C₂×C₄⋊C₄	—	2	1	no	—
239	C₂×C₄⋊C₄	—	2	1	no	—
240	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
241	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
242	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
243	C₂×C₂²⋊C₄	—	2	1	no	—
244	C₂×C₂²⋊C₄	—	2	1	no	—
245	C₂×C₂²⋊C₄	—	2	1	no	—
246	C₂×C₂²⋊C₄	—	2	1	no	—
247	C₂×C₂²⋊C₄	—	2	1	no	—
248	C₂×C₄²	—	2	1	no	—
249	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
250	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
251	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
252	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
253	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
254	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
255	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
256	C₄⋊C₈	—	2	4	no	—
257	C₄⋊C₈	—	2	2	no	—
258	C₄⋊C₈	—	2	2	no	—
259	C₄⋊C₈	—	4	2	no	—
260	C₄⋊C₈	—	4	2	no	—
261	C₄×C₈	—	2	2	no	—
262	C₄×C₈	—	2	2	no	—
263	#95	—	2	1	no	—
264	#95	—	2	1	no	—
265	#95	—	2	1	no	—
266	C₂⁴	—	2	1	no	—
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	C₄⋊C₄×C₅₃	1	2	yes	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	2	1	no	—
275	C₂²×C₄	—	2	1	no	—
276	C₂²×C₄	—	4	1	no	—
277	C₂²×C₄	—	4	1	no	—
278	C₂²×C₄	—	4	1	no	—
279	C₂²×C₄	—	4	1	no	—
280	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
281	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
282	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
283	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
284	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
285	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
287	C₂×C₈	—	2	4	no	—
288	C₂×C₈	—	4	2	no	—
289	C₂×C₈	—	4	2	no	—
290	C₂×C₈	—	4	2	no	—
291	C₂×C₈	—	4	2	no	—
292	C₂×C₈	—	4	2	no	—
293	C₄⋊C₄	—	2	1	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	4	2	no	—
297	C₄⋊C₄	—	4	2	no	—
298	C₄⋊C₄	C₄⋊C₄×C₅₃	1	2	yes	—
299	C₂²⋊C₄	—	2	2	no	—
300	C₂²⋊C₄	—	2	2	no	—
301	C₂²⋊C₄	—	4	2	no	—
302	C₂²⋊C₄	—	4	2	no	—
303	C₂²⋊C₄	—	4	2	no	—
304	C₂²⋊C₄	—	4	2	no	—
305	C₄²	—	2	2	no	—
306	C₄²	—	2	2	no	—
307	C₄²	—	2	2	no	—
308	C₄²	—	2	1	no	—
309	C₄²	—	2	1	no	—
310	C₄²	—	4	1	no	—
311	C₄²	—	4	1	no	—
312	C₄²	—	4	1	no	—
313	C₄²	D₈×C₅₃	1	2	yes	—
314	C₄²	SD₁₆×C₅₃	1	2	yes	—
315	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
316	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
317	C₄²	#41×C₅₃	1	1	yes	characteristic
318	C₄²	#41×C₅₃	1	1	yes	characteristic
319	C₂³	—	2	2	no	—
320	C₂³	—	4	1	no	—
321	C₂³	—	4	1	no	—
322	C₂³	—	4	1	no	—
323	C₂³	—	4	1	no	—
324	C₂³	—	4	1	no	—
325	C₂³	—	4	1	no	—
326	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
327	D₄	—	2	4	no	—
328	D₄	—	4	2	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	4	1	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	—	4	1	no	—
351	C₂×C₄	—	4	1	no	—
352	C₂×C₄	—	4	1	no	—
353	C₂×C₄	#102×C₅₃	1	2	yes	—
354	C₂×C₄	#102×C₅₃	1	2	yes	—
355	C₂×C₄	#104×C₅₃	1	2	yes	—
356	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
357	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
358	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
359	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
360	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
361	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
362	C₂×C₄	#131×C₅₃	1	2	yes	—
363	C₂×C₄	#131×C₅₃	1	2	yes	—
364	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
365	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
366	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
367	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
368	C₈	—	4	4	no	—
369	C₈	—	8	2	no	—
370	C₂²	—	4	2	no	—
371	C₂²	—	4	2	no	—
372	C₂²	—	4	2	no	—
373	C₂²	—	8	1	no	—
374	C₂²	—	8	1	no	—
375	C₂²	—	8	1	no	—
376	C₂²	—	8	1	no	—
377	C₂²	C₅₃×#380	1	1	yes	characteristic
378	C₂²	C₅₃×#408	1	1	yes	characteristic
379	C₂²	C₅₃×#408	1	1	yes	characteristic
380	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
381	C₂²	C₅₃×#417	1	1	yes	characteristic
382	C₂²	C₂×#104×C₅₃	1	1	yes	characteristic
383	C₂²	C₅₃×#421	1	1	yes	characteristic
384	C₄	—	2	2	no	—
385	C₄	—	2	2	no	—
386	C₄	—	2	2	no	—
387	C₄	—	2	1	no	—
388	C₄	—	2	1	no	—
389	C₄	—	4	1	no	—
390	C₄	—	4	2	no	—
391	C₄	—	8	1	no	—
392	C₄	—	8	1	no	—
393	C₄	C₅₃×#424	1	2	yes	—
394	C₄	C₅₃×#433	1	2	yes	—
395	C₄	C₅₃×#459	1	1	yes	characteristic
396	C₄	C₅₃×#461	1	1	yes	characteristic
397	C₄	C₅₃×#474	1	1	yes	characteristic
398	C₄	C₅₃×#476	1	1	yes	characteristic
399	C₂	—	8	2	no	—
400	C₂	C₅₃×#1480	1	1	yes	characteristic
401	C₂	C₅₃×#1483	1	1	yes	characteristic
402	C₂	C₅₃×#1492	1	1	yes	characteristic
403	C₂	C₅₃×#1495	1	1	yes	characteristic
404	C₂	C₅₃×#1798	1	1	yes	characteristic
405	C₂	C₅₃×#1802	1	1	yes	characteristic
406	C₂	C₅₃×#1806	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₅₃	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#1842	C₂	semidirect
3. C₄×C₄⋊C₈×C₅₃	C₂	semidirect
5. C₅₃×#1279	C₂	semidirect
166. #8567	C₅₃	direct (N × H)
197. #1842	C₂×C₅₃	semidirect
198. C₄×C₄⋊C₈	C₂×C₅₃	semidirect
200. #1279	C₂×C₅₃	semidirect
227. C₅₃	#8567	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×#2293	2
C₅₃×#1842	2
C₄×C₄⋊C₈×C₅₃	2
C₂×C₅₃×D₄⋊C₈	2
C₅₃×#1279	2
#8567	53

Subgroups — C53×#8567 #27216079

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8567 #27216079