Subgroups — C₅₃×#8839 #27216351

Order 13568.

466 conjugacy classes of subgroups.

The subgroup lattice has 378 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×#1842	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	—	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×#104×C₅₃	C₄	1	1	yes	characteristic
20	C₂×#104×C₅₃	C₄	1	1	yes	characteristic
21	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
22	C₄³×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#339	C₄	1	1	yes	characteristic
24	C₅₃×#339	C₄	1	1	yes	characteristic
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	—	4	4	no	—
56	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
57	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
58	#104×C₅₃	—	2	2	no	—
59	#104×C₅₃	C₈	1	4	yes	complemented
60	C₂²⋊C₈×C₅₃	—	2	2	no	—
61	C₂²⋊C₈×C₅₃	—	2	1	no	—
62	C₂²⋊C₈×C₅₃	—	2	1	no	—
63	#97×C₅₃	—	2	1	no	—
64	C₄×C₈×C₅₃	—	2	2	no	—
65	C₄×C₈×C₅₃	—	2	1	no	—
66	C₄×C₈×C₅₃	—	2	1	no	—
67	C₄×C₈×C₅₃	—	2	1	no	—
68	#41×C₅₃	—	2	1	no	—
69	#41×C₅₃	—	2	1	no	—
70	#41×C₅₃	C₂×C₈	1	2	yes	—
71	#41×C₅₃	M₁₆	1	2	yes	—
72	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
73	C₂×D₄×C₅₃	—	2	1	no	—
74	C₂×D₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
80	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
83	M₁₆×C₅₃	—	2	1	no	—
84	M₁₆×C₅₃	—	2	1	no	—
85	M₁₆×C₅₃	—	2	1	no	—
86	M₁₆×C₅₃	C₂×C₈	1	2	yes	complemented
87	C₂×C₈×C₅₃	—	2	1	no	—
88	C₂×C₈×C₅₃	—	2	1	no	—
89	C₂×C₈×C₅₃	—	2	1	no	—
90	C₂×C₈×C₅₃	—	2	2	no	—
91	C₂×C₈×C₅₃	—	2	1	no	—
92	C₂×C₈×C₅₃	—	4	2	no	—
93	C₂×C₈×C₅₃	—	4	2	no	—
94	C₂×C₈×C₅₃	—	4	1	no	—
95	C₂×C₈×C₅₃	—	4	1	no	—
96	C₂×C₈×C₅₃	—	8	2	no	—
97	C₄⋊C₄×C₅₃	—	2	1	no	—
98	C₄⋊C₄×C₅₃	—	2	1	no	—
99	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
100	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
101	C₂²⋊C₄×C₅₃	—	2	1	no	—
102	C₂²⋊C₄×C₅₃	—	2	1	no	—
103	C₂²⋊C₄×C₅₃	—	2	1	no	—
104	C₄²×C₅₃	—	2	2	no	—
105	C₄²×C₅₃	—	2	2	no	—
106	C₄²×C₅₃	—	2	1	no	—
107	C₄²×C₅₃	—	2	1	no	—
108	C₄²×C₅₃	—	2	1	no	—
109	C₄²×C₅₃	—	2	1	no	—
110	C₄²×C₅₃	—	2	1	no	—
111	C₄²×C₅₃	—	2	1	no	—
112	C₄²×C₅₃	—	4	2	no	—
113	C₄²×C₅₃	C₂×C₈	1	2	yes	complemented
114	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
117	C₄²×C₅₃	#41	1	1	yes	characteristic
118	C₂³×C₅₃	—	2	1	no	—
119	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
120	Q₈×C₅₃	—	2	1	no	—
121	Q₈×C₅₃	C₄⋊C₈	1	2	yes	complemented
122	D₄×C₅₃	—	2	2	no	—
123	D₄×C₅₃	—	2	2	no	—
124	D₄×C₅₃	—	2	1	no	—
125	D₄×C₅₃	C₄⋊C₈	1	2	yes	complemented
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	2	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	2	1	no	—
133	C₂×C₄×C₅₃	—	2	1	no	—
134	C₂×C₄×C₅₃	—	2	2	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	1	no	—
140	C₂×C₄×C₅₃	—	4	1	no	—
141	C₂×C₄×C₅₃	—	4	1	no	—
142	C₂×C₄×C₅₃	—	8	1	no	—
143	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
148	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
151	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
152	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
153	C₈×C₅₃	—	2	2	no	—
154	C₈×C₅₃	—	4	1	no	—
155	C₈×C₅₃	—	4	2	no	—
156	C₈×C₅₃	—	4	1	no	—
157	C₈×C₅₃	—	4	1	no	—
158	C₈×C₅₃	—	8	1	no	—
159	C₈×C₅₃	—	8	1	no	—
160	#8839	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂²×C₅₃	—	2	1	no	—
162	C₂²×C₅₃	—	2	1	no	—
163	C₂²×C₅₃	—	2	2	no	—
164	C₂²×C₅₃	—	4	1	no	—
165	C₂²×C₅₃	—	4	1	no	—
166	C₂²×C₅₃	#385	1	1	yes	characteristic
167	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
168	C₂²×C₅₃	#435	1	1	yes	characteristic
169	C₄×C₅₃	—	2	4	no	—
170	C₄×C₅₃	—	2	2	no	—
171	C₄×C₅₃	—	2	1	no	—
172	C₄×C₅₃	—	2	1	no	—
173	C₄×C₅₃	—	2	1	no	—
174	C₄×C₅₃	—	4	2	no	—
175	C₄×C₅₃	—	4	1	no	—
176	C₄×C₅₃	—	4	1	no	—
177	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
178	C₄×C₅₃	#424	1	1	yes	characteristic
179	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
180	C₄×C₅₃	#435	1	1	yes	characteristic
181	C₄×C₅₃	#453	1	1	yes	characteristic
182	C₄×C₅₃	#456	1	1	yes	characteristic
183	#2988	C₂×C₅₃	1	1	yes	characteristic
184	#1842	C₂×C₅₃	1	1	yes	characteristic, complemented
185	C₄×#104	C₂×C₅₃	1	1	yes	characteristic
186	#1565	C₂×C₅₃	1	1	yes	characteristic, complemented
187	#1477	C₂×C₅₃	1	1	yes	characteristic
188	#1283	C₂×C₅₃	1	1	yes	characteristic, complemented
189	#1276	C₂×C₅₃	1	1	yes	characteristic, complemented
190	C₂×C₅₃	—	2	1	no	—
191	C₂×C₅₃	—	2	1	no	—
192	C₂×C₅₃	—	4	2	no	—
193	C₂×C₅₃	#1850	1	1	yes	characteristic
194	C₂×C₅₃	#1859	1	1	yes	characteristic
195	C₂×C₅₃	#1861	1	1	yes	characteristic
196	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
197	C₈×Q₈	—	2	1	no	—
198	C₈×D₄	—	2	2	no	—
199	C₈×D₄	—	2	1	no	—
200	#431	—	2	1	no	—
201	#431	C₂²×C₅₃	1	1	yes	characteristic
202	#431	C₂²×C₅₃	1	1	yes	characteristic
203	C₂×C₄⋊C₈	—	2	1	no	—
204	C₂×C₄⋊C₈	—	2	2	no	—
205	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
206	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
207	C₂×#104	C₄×C₅₃	1	1	yes	characteristic
208	C₂×#104	C₄×C₅₃	1	1	yes	characteristic
209	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
210	C₄³	C₂²×C₅₃	1	1	yes	characteristic
211	#339	C₄×C₅₃	1	1	yes	characteristic
212	#339	C₄×C₅₃	1	1	yes	characteristic
213	#335	—	2	1	no	—
214	#334	—	2	1	no	—
215	Q₈⋊C₈	—	2	1	no	—
216	D₄⋊C₈	—	2	1	no	—
217	C₅₃	#8839	1	1	yes	characteristic, direct product factor, Sylow
218	C₂×#41	C₂×C₄×C₅₃	1	1	yes	characteristic
219	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
220	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
221	C₂²×C₈	—	2	2	no	—
222	C₂²×C₈	—	2	1	no	—
223	C₂²×C₈	D₄×C₅₃	1	2	yes	—
224	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
225	C₄×Q₈	Q₈×C₅₃	1	1	yes	characteristic
226	C₄×D₄	—	2	1	no	—
227	C₄×D₄	—	2	1	no	—
228	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
229	C₄×D₄	Q₈×C₅₃	1	1	yes	characteristic
230	#117	—	2	1	no	—
231	#117	C₂×C₄×C₅₃	1	1	yes	characteristic
232	C₂×C₄²	—	2	1	no	—
233	C₂×C₄²	—	2	1	no	—
234	C₂×C₄²	—	2	1	no	—
235	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
236	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
237	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
238	C₄⋊C₈	—	2	2	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	—	2	1	no	—
244	C₄⋊C₈	—	4	4	no	—
245	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
246	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
247	#104	—	2	2	no	—
248	#104	C₈×C₅₃	1	4	yes	complemented
249	C₂²⋊C₈	—	2	2	no	—
250	C₂²⋊C₈	—	2	1	no	—
251	C₂²⋊C₈	—	2	1	no	—
252	#97	—	2	1	no	—
253	C₄×C₈	—	2	2	no	—
254	C₄×C₈	—	2	1	no	—
255	C₄×C₈	—	2	1	no	—
256	C₄×C₈	—	2	1	no	—
257	#41	—	2	1	no	—
258	#41	—	2	1	no	—
259	#41	C₂×C₈×C₅₃	1	2	yes	—
260	#41	M₁₆×C₅₃	1	2	yes	—
261	C₂×Q₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
262	C₂×D₄	—	2	1	no	—
263	C₂×D₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	—	2	1	no	—
266	C₂²×C₄	—	2	1	no	—
267	C₂²×C₄	—	2	1	no	—
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
271	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
272	M₁₆	—	2	1	no	—
273	M₁₆	—	2	1	no	—
274	M₁₆	—	2	1	no	—
275	M₁₆	C₂×C₈×C₅₃	1	2	yes	complemented
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	4	2	no	—
282	C₂×C₈	—	4	2	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	8	2	no	—
286	C₄⋊C₄	—	2	1	no	—
287	C₄⋊C₄	—	2	1	no	—
288	C₄⋊C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
289	C₄⋊C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
290	C₂²⋊C₄	—	2	1	no	—
291	C₂²⋊C₄	—	2	1	no	—
292	C₂²⋊C₄	—	2	1	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	2	2	no	—
295	C₄²	—	2	1	no	—
296	C₄²	—	2	1	no	—
297	C₄²	—	2	1	no	—
298	C₄²	—	2	1	no	—
299	C₄²	—	2	1	no	—
300	C₄²	—	2	1	no	—
301	C₄²	—	4	2	no	—
302	C₄²	C₂×C₈×C₅₃	1	2	yes	complemented
303	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
304	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
305	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
306	C₄²	#41×C₅₃	1	1	yes	characteristic
307	C₂³	—	2	1	no	—
308	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
309	Q₈	—	2	1	no	—
310	Q₈	C₄⋊C₈×C₅₃	1	2	yes	complemented
311	D₄	—	2	2	no	—
312	D₄	—	2	2	no	—
313	D₄	—	2	1	no	—
314	D₄	C₄⋊C₈×C₅₃	1	2	yes	complemented
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	8	1	no	—
332	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
333	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
334	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
335	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
336	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
337	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
338	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
339	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
340	C₂×C₄	#131×C₅₃	1	1	yes	characteristic
341	C₂×C₄	#131×C₅₃	1	1	yes	characteristic
342	C₈	—	2	2	no	—
343	C₈	—	4	1	no	—
344	C₈	—	4	2	no	—
345	C₈	—	4	1	no	—
346	C₈	—	4	1	no	—
347	C₈	—	8	1	no	—
348	C₈	—	8	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	2	no	—
352	C₂²	—	4	1	no	—
353	C₂²	—	4	1	no	—
354	C₂²	C₅₃×#385	1	1	yes	characteristic
355	C₂²	C₈×D₄×C₅₃	1	1	yes	characteristic
356	C₂²	C₅₃×#435	1	1	yes	characteristic
357	C₄	—	2	4	no	—
358	C₄	—	2	2	no	—
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	4	2	no	—
363	C₄	—	4	1	no	—
364	C₄	—	4	1	no	—
365	C₄	C₂×C₄⋊C₈×C₅₃	1	1	yes	characteristic
366	C₄	C₅₃×#424	1	1	yes	characteristic
367	C₄	C₈×D₄×C₅₃	1	1	yes	characteristic
368	C₄	C₅₃×#435	1	1	yes	characteristic
369	C₄	C₅₃×#453	1	1	yes	characteristic
370	C₄	C₅₃×#456	1	1	yes	characteristic
371	C₂	—	2	1	no	—
372	C₂	—	2	1	no	—
373	C₂	—	4	2	no	—
374	C₂	C₅₃×#1850	1	1	yes	characteristic
375	C₂	C₅₃×#1859	1	1	yes	characteristic
376	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×#1842	C₂	semidirect
4. C₅₃×#1565	C₂	semidirect
6. C₅₃×#1283	C₂	semidirect
7. C₅₃×#1276	C₂	semidirect
59. #104×C₅₃	C₈	semidirect
86. M₁₆×C₅₃	C₂×C₈	semidirect
113. C₄²×C₅₃	C₂×C₈	semidirect
121. Q₈×C₅₃	C₄⋊C₈	semidirect
125. D₄×C₅₃	C₄⋊C₈	semidirect
160. #8839	C₅₃	direct (N × H)
184. #1842	C₂×C₅₃	semidirect
186. #1565	C₂×C₅₃	semidirect
188. #1283	C₂×C₅₃	semidirect
189. #1276	C₂×C₅₃	semidirect
217. C₅₃	#8839	direct (N × H)
248. #104	C₈×C₅₃	semidirect
275. M₁₆	C₂×C₈×C₅₃	semidirect
302. C₄²	C₂×C₈×C₅₃	semidirect
310. Q₈	C₄⋊C₈×C₅₃	semidirect
314. D₄	C₄⋊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×#1842	2
C₄×#104×C₅₃	2
C₅₃×#1565	2
C₅₃×#1477	2
C₅₃×#1283	2
C₅₃×#1276	2
#8839	53

Subgroups — C53×#8839 #27216351

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8839 #27216351