Subgroups — C₅₃×#8840 #27216352

Order 13568.

446 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2988	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#104×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1565	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1477	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1283	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1276	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
9	C₈×Q₈×C₅₃	—	2	1	no	—
10	C₈×D₄×C₅₃	—	2	2	no	—
11	C₈×D₄×C₅₃	—	2	1	no	—
12	C₅₃×#431	—	2	1	no	—
13	C₅₃×#431	C₂²	1	1	yes	characteristic
14	C₅₃×#431	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×#104×C₅₃	C₄	1	1	yes	characteristic
19	C₂×#104×C₅₃	C₄	1	1	yes	characteristic
20	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
21	C₄³×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#339	C₄	1	1	yes	characteristic
23	C₅₃×#339	C₄	1	1	yes	characteristic
24	C₅₃×#336	—	2	2	no	—
25	C₅₃×#335	—	2	1	no	—
26	C₅₃×#334	—	2	1	no	—
27	C₅₃×Q₈⋊C₈	—	2	1	no	—
28	C₅₃×D₄⋊C₈	—	2	1	no	—
29	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	—	2	2	no	—
33	C₂²×C₈×C₅₃	—	2	1	no	—
34	C₂²×C₈×C₅₃	D₄	1	2	yes	—
35	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
36	C₄×Q₈×C₅₃	Q₈	1	1	yes	characteristic
37	C₄×D₄×C₅₃	—	2	1	no	—
38	C₄×D₄×C₅₃	—	2	1	no	—
39	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
40	C₄×D₄×C₅₃	Q₈	1	1	yes	characteristic
41	#117×C₅₃	—	2	1	no	—
42	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	—	2	1	no	—
44	C₂×C₄²×C₅₃	—	2	1	no	—
45	C₂×C₄²×C₅₃	—	2	1	no	—
46	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₅₃	—	2	2	no	—
50	C₄⋊C₈×C₅₃	—	2	1	no	—
51	C₄⋊C₈×C₅₃	—	2	1	no	—
52	C₄⋊C₈×C₅₃	—	2	1	no	—
53	C₄⋊C₈×C₅₃	—	2	1	no	—
54	C₄⋊C₈×C₅₃	—	2	1	no	—
55	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
56	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
57	#104×C₅₃	—	2	2	no	—
58	#104×C₅₃	C₈	1	4	yes	complemented
59	C₂²⋊C₈×C₅₃	—	2	2	no	—
60	C₂²⋊C₈×C₅₃	—	2	1	no	—
61	C₂²⋊C₈×C₅₃	—	2	1	no	—
62	#97×C₅₃	—	2	1	no	—
63	C₄×C₈×C₅₃	—	2	2	no	—
64	C₄×C₈×C₅₃	—	2	1	no	—
65	C₄×C₈×C₅₃	—	2	1	no	—
66	C₄×C₈×C₅₃	—	2	1	no	—
67	#41×C₅₃	—	2	1	no	—
68	#41×C₅₃	—	2	1	no	—
69	#41×C₅₃	C₂×C₈	1	2	yes	—
70	#41×C₅₃	M₁₆	1	2	yes	—
71	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
72	C₂×D₄×C₅₃	—	2	1	no	—
73	C₂×D₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	—	2	1	no	—
75	C₂²×C₄×C₅₃	—	2	1	no	—
76	C₂²×C₄×C₅₃	—	2	1	no	—
77	C₂²×C₄×C₅₃	—	2	1	no	—
78	C₂²×C₄×C₅₃	—	2	1	no	—
79	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
82	M₁₆×C₅₃	—	2	1	no	—
83	M₁₆×C₅₃	—	2	1	no	—
84	M₁₆×C₅₃	—	2	1	no	—
85	M₁₆×C₅₃	C₂×C₈	1	2	yes	complemented
86	C₂×C₈×C₅₃	—	2	1	no	—
87	C₂×C₈×C₅₃	—	2	1	no	—
88	C₂×C₈×C₅₃	—	2	1	no	—
89	C₂×C₈×C₅₃	—	2	2	no	—
90	C₂×C₈×C₅₃	—	2	1	no	—
91	C₂×C₈×C₅₃	—	4	2	no	—
92	C₂×C₈×C₅₃	—	4	2	no	—
93	C₂×C₈×C₅₃	—	4	1	no	—
94	C₂×C₈×C₅₃	—	4	1	no	—
95	C₂×C₈×C₅₃	—	8	2	no	—
96	C₄⋊C₄×C₅₃	—	2	1	no	—
97	C₄⋊C₄×C₅₃	—	2	1	no	—
98	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
99	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
100	C₂²⋊C₄×C₅₃	—	2	1	no	—
101	C₂²⋊C₄×C₅₃	—	2	1	no	—
102	C₂²⋊C₄×C₅₃	—	2	1	no	—
103	C₄²×C₅₃	—	2	1	no	—
104	C₄²×C₅₃	—	2	1	no	—
105	C₄²×C₅₃	—	2	1	no	—
106	C₄²×C₅₃	—	2	1	no	—
107	C₄²×C₅₃	—	2	1	no	—
108	C₄²×C₅₃	—	2	1	no	—
109	C₄²×C₅₃	—	4	1	no	—
110	C₄²×C₅₃	—	4	1	no	—
111	C₄²×C₅₃	—	4	2	no	—
112	C₄²×C₅₃	C₂×C₈	1	2	yes	complemented
113	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
116	C₄²×C₅₃	#41	1	1	yes	characteristic
117	C₂³×C₅₃	—	2	1	no	—
118	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
119	Q₈×C₅₃	—	2	1	no	—
120	Q₈×C₅₃	C₄⋊C₈	1	2	yes	—
121	D₄×C₅₃	—	2	2	no	—
122	D₄×C₅₃	—	2	2	no	—
123	D₄×C₅₃	—	2	1	no	—
124	D₄×C₅₃	C₄⋊C₈	1	2	yes	—
125	C₂×C₄×C₅₃	—	2	1	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	2	1	no	—
129	C₂×C₄×C₅₃	—	2	1	no	—
130	C₂×C₄×C₅₃	—	2	2	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	—	4	1	no	—
135	C₂×C₄×C₅₃	—	4	1	no	—
136	C₂×C₄×C₅₃	—	4	1	no	—
137	C₂×C₄×C₅₃	—	4	1	no	—
138	C₂×C₄×C₅₃	—	4	1	no	—
139	C₂×C₄×C₅₃	—	4	1	no	—
140	C₂×C₄×C₅₃	—	4	1	no	—
141	C₂×C₄×C₅₃	—	8	1	no	—
142	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
147	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
150	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
151	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
152	C₈×C₅₃	—	2	2	no	—
153	C₈×C₅₃	—	4	1	no	—
154	C₈×C₅₃	—	4	1	no	—
155	C₈×C₅₃	—	4	2	no	—
156	C₈×C₅₃	—	4	1	no	—
157	C₈×C₅₃	—	8	1	no	—
158	C₈×C₅₃	—	8	1	no	—
159	#8840	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂²×C₅₃	—	2	1	no	—
161	C₂²×C₅₃	—	2	1	no	—
162	C₂²×C₅₃	—	2	2	no	—
163	C₂²×C₅₃	—	4	1	no	—
164	C₂²×C₅₃	—	4	1	no	—
165	C₂²×C₅₃	#385	1	1	yes	characteristic
166	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
167	C₂²×C₅₃	#435	1	1	yes	characteristic
168	C₄×C₅₃	—	2	2	no	—
169	C₄×C₅₃	—	2	1	no	—
170	C₄×C₅₃	—	2	1	no	—
171	C₄×C₅₃	—	2	1	no	—
172	C₄×C₅₃	—	4	2	no	—
173	C₄×C₅₃	—	4	2	no	—
174	C₄×C₅₃	—	4	1	no	—
175	C₄×C₅₃	—	4	1	no	—
176	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
177	C₄×C₅₃	#424	1	1	yes	characteristic
178	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
179	C₄×C₅₃	#435	1	1	yes	characteristic
180	C₄×C₅₃	#454	1	1	yes	characteristic
181	C₄×C₅₃	#455	1	1	yes	characteristic
182	#2988	C₂×C₅₃	1	1	yes	characteristic
183	#1831	C₂×C₅₃	1	1	yes	characteristic, complemented
184	C₄×#104	C₂×C₅₃	1	1	yes	characteristic
185	#1565	C₂×C₅₃	1	1	yes	characteristic, complemented
186	#1477	C₂×C₅₃	1	1	yes	characteristic
187	#1283	C₂×C₅₃	1	1	yes	characteristic, complemented
188	#1276	C₂×C₅₃	1	1	yes	characteristic, complemented
189	C₂×C₅₃	—	2	1	no	—
190	C₂×C₅₃	—	2	1	no	—
191	C₂×C₅₃	—	4	2	no	—
192	C₂×C₅₃	#1850	1	1	yes	characteristic
193	C₂×C₅₃	#1860	1	1	yes	characteristic
194	C₂×C₅₃	#1861	1	1	yes	characteristic
195	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
196	C₈×Q₈	—	2	1	no	—
197	C₈×D₄	—	2	2	no	—
198	C₈×D₄	—	2	1	no	—
199	#431	—	2	1	no	—
200	#431	C₂²×C₅₃	1	1	yes	characteristic
201	#431	C₂²×C₅₃	1	1	yes	characteristic
202	C₂×C₄⋊C₈	—	2	1	no	—
203	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
204	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
205	C₂×#104	C₄×C₅₃	1	1	yes	characteristic
206	C₂×#104	C₄×C₅₃	1	1	yes	characteristic
207	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
208	C₄³	C₂²×C₅₃	1	1	yes	characteristic
209	#339	C₄×C₅₃	1	1	yes	characteristic
210	#339	C₄×C₅₃	1	1	yes	characteristic
211	#336	—	2	2	no	—
212	#335	—	2	1	no	—
213	#334	—	2	1	no	—
214	Q₈⋊C₈	—	2	1	no	—
215	D₄⋊C₈	—	2	1	no	—
216	C₅₃	#8840	1	1	yes	characteristic, direct product factor, Sylow
217	C₂×#41	C₂×C₄×C₅₃	1	1	yes	characteristic
218	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
219	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
220	C₂²×C₈	—	2	2	no	—
221	C₂²×C₈	—	2	1	no	—
222	C₂²×C₈	D₄×C₅₃	1	2	yes	—
223	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
224	C₄×Q₈	Q₈×C₅₃	1	1	yes	characteristic
225	C₄×D₄	—	2	1	no	—
226	C₄×D₄	—	2	1	no	—
227	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
228	C₄×D₄	Q₈×C₅₃	1	1	yes	characteristic
229	#117	—	2	1	no	—
230	#117	C₂×C₄×C₅₃	1	1	yes	characteristic
231	C₂×C₄²	—	2	1	no	—
232	C₂×C₄²	—	2	1	no	—
233	C₂×C₄²	—	2	1	no	—
234	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
235	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
236	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
237	C₄⋊C₈	—	2	2	no	—
238	C₄⋊C₈	—	2	1	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	—	2	1	no	—
241	C₄⋊C₈	—	2	1	no	—
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
244	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
245	#104	—	2	2	no	—
246	#104	C₈×C₅₃	1	4	yes	complemented
247	C₂²⋊C₈	—	2	2	no	—
248	C₂²⋊C₈	—	2	1	no	—
249	C₂²⋊C₈	—	2	1	no	—
250	#97	—	2	1	no	—
251	C₄×C₈	—	2	2	no	—
252	C₄×C₈	—	2	1	no	—
253	C₄×C₈	—	2	1	no	—
254	C₄×C₈	—	2	1	no	—
255	#41	—	2	1	no	—
256	#41	—	2	1	no	—
257	#41	C₂×C₈×C₅₃	1	2	yes	—
258	#41	M₁₆×C₅₃	1	2	yes	—
259	C₂×Q₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
262	C₂²×C₄	—	2	1	no	—
263	C₂²×C₄	—	2	1	no	—
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	—	2	1	no	—
266	C₂²×C₄	—	2	1	no	—
267	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
269	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
270	M₁₆	—	2	1	no	—
271	M₁₆	—	2	1	no	—
272	M₁₆	—	2	1	no	—
273	M₁₆	C₂×C₈×C₅₃	1	2	yes	complemented
274	C₂×C₈	—	2	1	no	—
275	C₂×C₈	—	2	1	no	—
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	4	2	no	—
280	C₂×C₈	—	4	2	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	4	1	no	—
283	C₂×C₈	—	8	2	no	—
284	C₄⋊C₄	—	2	1	no	—
285	C₄⋊C₄	—	2	1	no	—
286	C₄⋊C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
287	C₄⋊C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
288	C₂²⋊C₄	—	2	1	no	—
289	C₂²⋊C₄	—	2	1	no	—
290	C₂²⋊C₄	—	2	1	no	—
291	C₄²	—	2	1	no	—
292	C₄²	—	2	1	no	—
293	C₄²	—	2	1	no	—
294	C₄²	—	2	1	no	—
295	C₄²	—	2	1	no	—
296	C₄²	—	2	1	no	—
297	C₄²	—	4	1	no	—
298	C₄²	—	4	1	no	—
299	C₄²	—	4	2	no	—
300	C₄²	C₂×C₈×C₅₃	1	2	yes	complemented
301	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
302	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
303	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
304	C₄²	#41×C₅₃	1	1	yes	characteristic
305	C₂³	—	2	1	no	—
306	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
307	Q₈	—	2	1	no	—
308	Q₈	C₄⋊C₈×C₅₃	1	2	yes	—
309	D₄	—	2	2	no	—
310	D₄	—	2	2	no	—
311	D₄	—	2	1	no	—
312	D₄	C₄⋊C₈×C₅₃	1	2	yes	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	8	1	no	—
330	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
331	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
332	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
333	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
334	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
335	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
336	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
337	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
338	C₂×C₄	#131×C₅₃	1	1	yes	characteristic
339	C₂×C₄	#131×C₅₃	1	1	yes	characteristic
340	C₈	—	2	2	no	—
341	C₈	—	4	1	no	—
342	C₈	—	4	1	no	—
343	C₈	—	4	2	no	—
344	C₈	—	4	1	no	—
345	C₈	—	8	1	no	—
346	C₈	—	8	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	2	no	—
350	C₂²	—	4	1	no	—
351	C₂²	—	4	1	no	—
352	C₂²	C₅₃×#385	1	1	yes	characteristic
353	C₂²	C₈×D₄×C₅₃	1	1	yes	characteristic
354	C₂²	C₅₃×#435	1	1	yes	characteristic
355	C₄	—	2	2	no	—
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	4	2	no	—
360	C₄	—	4	2	no	—
361	C₄	—	4	1	no	—
362	C₄	—	4	1	no	—
363	C₄	C₂×C₄⋊C₈×C₅₃	1	1	yes	characteristic
364	C₄	C₅₃×#424	1	1	yes	characteristic
365	C₄	C₈×D₄×C₅₃	1	1	yes	characteristic
366	C₄	C₅₃×#435	1	1	yes	characteristic
367	C₄	C₅₃×#454	1	1	yes	characteristic
368	C₄	C₅₃×#455	1	1	yes	characteristic
369	C₂	—	2	1	no	—
370	C₂	—	2	1	no	—
371	C₂	—	4	2	no	—
372	C₂	C₅₃×#1850	1	1	yes	characteristic
373	C₂	C₅₃×#1860	1	1	yes	characteristic
374	C₂	C₅₃×#1861	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#120
commutator	C₂×C₄	C₂²×C₈×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#385
prime radical	C₂×#41×C₅₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#1831	C₂	semidirect
4. C₅₃×#1565	C₂	semidirect
6. C₅₃×#1283	C₂	semidirect
7. C₅₃×#1276	C₂	semidirect
58. #104×C₅₃	C₈	semidirect
85. M₁₆×C₅₃	C₂×C₈	semidirect
112. C₄²×C₅₃	C₂×C₈	semidirect
159. #8840	C₅₃	direct (N × H)
183. #1831	C₂×C₅₃	semidirect
185. #1565	C₂×C₅₃	semidirect
187. #1283	C₂×C₅₃	semidirect
188. #1276	C₂×C₅₃	semidirect
216. C₅₃	#8840	direct (N × H)
246. #104	C₈×C₅₃	semidirect
273. M₁₆	C₂×C₈×C₅₃	semidirect
300. C₄²	C₂×C₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2988	2
C₅₃×#1831	2
C₄×#104×C₅₃	2
C₅₃×#1565	2
C₅₃×#1477	2
C₅₃×#1283	2
C₅₃×#1276	2
#8840	53

Subgroups — C53×#8840 #27216352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8840 #27216352