Subgroups — C₅₃×#10773 #27218285

Order 13568.

1186 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#131×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#421	C₂	1	2	yes	maximal
3	C₂×C₅₃×#344	C₂	1	2	yes	maximal, complemented
4	C₂×C₅₃×#341	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1735	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×#131×C₅₃	—	2	4	no	—
8	C₂×#131×C₅₃	—	2	4	no	—
9	C₂×#131×C₅₃	—	2	2	no	—
10	C₂×#131×C₅₃	C₄	1	4	yes	complemented
11	C₂²×M₁₆×C₅₃	—	2	1	no	—
12	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
13	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₅₃	—	2	1	no	—
15	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#117×C₅₃	C₂²	1	2	yes	—
17	C₅₃×#421	—	2	4	no	—
18	C₅₃×#421	C₄	1	8	yes	complemented
19	C₂×#104×C₅₃	—	2	4	no	—
20	C₅₃×#344	—	2	4	no	—
21	C₅₃×#341	—	2	1	no	—
22	C₅₃×#341	—	2	1	no	—
23	C₅₃×#340	—	2	2	no	—
24	C₂×#41×C₅₃	—	2	1	no	—
25	C₂×#41×C₅₃	—	2	2	no	—
26	C₂×#41×C₅₃	—	2	2	no	—
27	C₂×#41×C₅₃	—	2	1	no	—
28	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×#41×C₅₃	C₂×C₄	1	4	yes	—
30	C₂×#41×C₅₃	D₄	1	4	yes	—
31	C₂×#41×C₅₃	D₄	1	2	yes	—
32	C₂²×Q₈×C₅₃	Q₈	1	1	yes	characteristic
33	C₂²×D₄×C₅₃	—	2	1	no	—
34	C₂²×D₄×C₅₃	Q₈	1	1	yes	characteristic
35	C₂³×C₄×C₅₃	—	2	1	no	—
36	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
37	#131×C₅₃	—	2	16	no	—
38	#131×C₅₃	—	2	8	no	—
39	#131×C₅₃	—	2	8	no	—
40	C₂×M₁₆×C₅₃	—	2	8	no	—
41	C₂×M₁₆×C₅₃	—	2	4	no	—
42	C₂×M₁₆×C₅₃	—	2	2	no	—
43	C₂×M₁₆×C₅₃	—	2	1	no	—
44	C₂×M₁₆×C₅₃	—	2	2	no	—
45	C₂×M₁₆×C₅₃	—	2	2	no	—
46	C₂×M₁₆×C₅₃	—	2	2	no	—
47	C₂×M₁₆×C₅₃	—	2	2	no	—
48	C₂×M₁₆×C₅₃	—	2	2	no	—
49	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
50	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
51	C₂×M₁₆×C₅₃	D₄	1	4	yes	—
52	C₂²×C₈×C₅₃	—	2	8	no	—
53	C₂²×C₈×C₅₃	—	2	4	no	—
54	C₂²×C₈×C₅₃	—	2	1	no	—
55	C₂²×C₈×C₅₃	—	2	2	no	—
56	C₂²×C₈×C₅₃	—	2	2	no	—
57	C₂²×C₈×C₅₃	—	2	2	no	—
58	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
59	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₅₃	D₄	1	2	yes	—
62	#117×C₅₃	—	2	2	no	—
63	#117×C₅₃	—	2	2	no	—
64	#117×C₅₃	—	2	2	no	—
65	#117×C₅₃	C₂×C₄	1	4	yes	—
66	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
68	C₂×C₄²×C₅₃	—	2	2	no	—
69	#104×C₅₃	—	2	16	no	—
70	C₂⁴×C₅₃	—	2	1	no	—
71	C₂⁴×C₅₃	#41	1	1	yes	characteristic
72	#41×C₅₃	—	2	2	no	—
73	#41×C₅₃	—	2	2	no	—
74	#41×C₅₃	—	2	4	no	—
75	#41×C₅₃	—	2	4	no	—
76	#41×C₅₃	—	2	4	no	—
77	#41×C₅₃	—	2	4	no	—
78	#41×C₅₃	—	2	4	no	—
79	#41×C₅₃	C₄²	1	4	yes	—
80	#41×C₅₃	C₂²⋊C₄	1	8	yes	—
81	#41×C₅₃	C₂²⋊C₄	1	4	yes	—
82	C₂×Q₈×C₅₃	—	2	1	no	—
83	C₂×Q₈×C₅₃	—	2	2	no	—
84	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
85	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
86	C₂×D₄×C₅₃	—	2	2	no	—
87	C₂×D₄×C₅₃	—	2	4	no	—
88	C₂×D₄×C₅₃	—	2	2	no	—
89	C₂×D₄×C₅₃	—	2	4	no	—
90	C₂×D₄×C₅₃	—	2	2	no	—
91	C₂×D₄×C₅₃	—	2	1	no	—
92	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
93	C₂×D₄×C₅₃	C₄⋊C₄	1	4	yes	—
94	C₂²×C₄×C₅₃	—	2	1	no	—
95	C₂²×C₄×C₅₃	—	2	2	no	—
96	C₂²×C₄×C₅₃	—	2	4	no	—
97	C₂²×C₄×C₅₃	—	2	4	no	—
98	C₂²×C₄×C₅₃	—	2	2	no	—
99	C₂²×C₄×C₅₃	—	4	2	no	—
100	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
102	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	#41	1	2	yes	—
109	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
110	M₁₆×C₅₃	—	2	16	no	—
111	M₁₆×C₅₃	—	2	2	no	—
112	M₁₆×C₅₃	—	2	4	no	—
113	M₁₆×C₅₃	—	2	4	no	—
114	M₁₆×C₅₃	—	2	4	no	—
115	M₁₆×C₅₃	—	4	2	no	—
116	M₁₆×C₅₃	—	4	2	no	—
117	M₁₆×C₅₃	C₂²⋊C₄	1	4	yes	—
118	C₂×C₈×C₅₃	—	2	16	no	—
119	C₂×C₈×C₅₃	—	2	2	no	—
120	C₂×C₈×C₅₃	—	2	2	no	—
121	C₂×C₈×C₅₃	—	2	2	no	—
122	C₂×C₈×C₅₃	—	2	2	no	—
123	C₂×C₈×C₅₃	—	2	4	no	—
124	C₂×C₈×C₅₃	—	2	2	no	—
125	C₂×C₈×C₅₃	—	2	4	no	—
126	C₂×C₈×C₅₃	—	2	4	no	—
127	C₂×C₈×C₅₃	—	4	2	no	—
128	C₂×C₈×C₅₃	—	4	2	no	—
129	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
130	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
131	C₄⋊C₄×C₅₃	—	2	4	no	—
132	C₄⋊C₄×C₅₃	—	4	2	no	—
133	C₂²⋊C₄×C₅₃	—	2	4	no	—
134	C₂²⋊C₄×C₅₃	—	4	2	no	—
135	C₄²×C₅₃	—	2	4	no	—
136	C₄²×C₅₃	—	4	2	no	—
137	C₂³×C₅₃	—	2	2	no	—
138	C₂³×C₅₃	—	2	4	no	—
139	C₂³×C₅₃	—	4	2	no	—
140	C₂³×C₅₃	—	4	4	no	—
141	C₂³×C₅₃	#117	1	1	yes	characteristic
142	C₂³×C₅₃	C₄×D₄	1	2	yes	—
143	C₂³×C₅₃	#121	1	1	yes	characteristic
144	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
145	C₂³×C₅₃	#123	1	2	yes	—
146	Q₈×C₅₃	—	2	2	no	—
147	Q₈×C₅₃	—	2	4	no	—
148	Q₈×C₅₃	#95	1	4	yes	—
149	D₄×C₅₃	—	2	4	no	—
150	D₄×C₅₃	—	2	4	no	—
151	D₄×C₅₃	—	2	8	no	—
152	D₄×C₅₃	—	2	2	no	—
153	D₄×C₅₃	—	2	4	no	—
154	D₄×C₅₃	#95	1	4	yes	—
155	C₂×C₄×C₅₃	—	2	2	no	—
156	C₂×C₄×C₅₃	—	2	4	no	—
157	C₂×C₄×C₅₃	—	2	4	no	—
158	C₂×C₄×C₅₃	—	2	4	no	—
159	C₂×C₄×C₅₃	—	2	8	no	—
160	C₂×C₄×C₅₃	—	4	4	no	—
161	C₂×C₄×C₅₃	—	8	4	no	—
162	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
165	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
166	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
167	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
169	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
170	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
171	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
172	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
173	C₂×C₄×C₅₃	#120	1	2	yes	—
174	C₂×C₄×C₅₃	#120	1	2	yes	—
175	C₂×C₄×C₅₃	#121	1	2	yes	—
176	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
177	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
178	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
179	C₂×C₄×C₅₃	#123	1	2	yes	—
180	C₈×C₅₃	—	2	2	no	—
181	C₈×C₅₃	—	2	4	no	—
182	C₈×C₅₃	—	2	2	no	—
183	C₈×C₅₃	—	4	4	no	—
184	#10773	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
185	C₂²×C₅₃	—	2	2	no	—
186	C₂²×C₅₃	—	2	2	no	—
187	C₂²×C₅₃	—	2	2	no	—
188	C₂²×C₅₃	—	2	4	no	—
189	C₂²×C₅₃	—	4	2	no	—
190	C₂²×C₅₃	—	4	2	no	—
191	C₂²×C₅₃	—	4	8	no	—
192	C₂²×C₅₃	—	4	4	no	—
193	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
194	C₂²×C₅₃	#380	1	1	yes	characteristic
195	C₂²×C₅₃	#381	1	1	yes	characteristic
196	C₂²×C₅₃	#385	1	2	yes	—
197	C₂²×C₅₃	#386	1	2	yes	—
198	C₄×C₅₃	—	2	4	no	—
199	C₄×C₅₃	—	8	4	no	—
200	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
201	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
202	C₄×C₅₃	#379	1	1	yes	characteristic
203	C₄×C₅₃	#380	1	1	yes	characteristic
204	C₄×C₅₃	#385	1	2	yes	—
205	C₄×C₅₃	#386	1	2	yes	—
206	C₂²×#131	C₂×C₅₃	1	1	yes	characteristic
207	C₂×#421	C₂×C₅₃	1	2	yes	—
208	C₂×#344	C₂×C₅₃	1	2	yes	complemented
209	C₂×#341	C₂×C₅₃	1	1	yes	characteristic, complemented
210	#1735	C₂×C₅₃	1	1	yes	characteristic, complemented
211	C₂×C₅₃	—	2	2	no	—
212	C₂×C₅₃	—	2	2	no	—
213	C₂×C₅₃	—	2	1	no	—
214	C₂×C₅₃	—	2	1	no	—
215	C₂×C₅₃	—	4	4	no	—
216	C₂×C₅₃	C₂²⋊#95	1	1	yes	characteristic
217	C₂×C₅₃	#1814	1	2	yes	—
218	C₂²×#41	C₂²×C₅₃	1	1	yes	characteristic
219	C₂×#131	—	2	4	no	—
220	C₂×#131	—	2	4	no	—
221	C₂×#131	—	2	2	no	—
222	C₂×#131	C₄×C₅₃	1	4	yes	complemented
223	C₂²×M₁₆	—	2	1	no	—
224	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
225	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
226	C₂³×C₈	—	2	1	no	—
227	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
228	C₂×#117	C₂²×C₅₃	1	2	yes	—
229	#421	—	2	4	no	—
230	#421	C₄×C₅₃	1	8	yes	complemented
231	C₂×#104	—	2	4	no	—
232	#344	—	2	4	no	—
233	#341	—	2	1	no	—
234	#341	—	2	1	no	—
235	#340	—	2	2	no	—
236	C₅₃	#10773	1	1	yes	characteristic, direct product factor, Sylow
237	C₂×#41	—	2	1	no	—
238	C₂×#41	—	2	2	no	—
239	C₂×#41	—	2	2	no	—
240	C₂×#41	—	2	1	no	—
241	C₂×#41	C₂×C₄×C₅₃	1	2	yes	—
242	C₂×#41	C₂×C₄×C₅₃	1	4	yes	—
243	C₂×#41	D₄×C₅₃	1	4	yes	—
244	C₂×#41	D₄×C₅₃	1	2	yes	—
245	C₂²×Q₈	Q₈×C₅₃	1	1	yes	characteristic
246	C₂²×D₄	—	2	1	no	—
247	C₂²×D₄	Q₈×C₅₃	1	1	yes	characteristic
248	C₂³×C₄	—	2	1	no	—
249	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
250	#131	—	2	16	no	—
251	#131	—	2	8	no	—
252	#131	—	2	8	no	—
253	C₂×M₁₆	—	2	8	no	—
254	C₂×M₁₆	—	2	1	no	—
255	C₂×M₁₆	—	2	2	no	—
256	C₂×M₁₆	—	2	2	no	—
257	C₂×M₁₆	—	2	4	no	—
258	C₂×M₁₆	—	2	2	no	—
259	C₂×M₁₆	—	2	2	no	—
260	C₂×M₁₆	—	2	2	no	—
261	C₂×M₁₆	—	2	2	no	—
262	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
263	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
264	C₂×M₁₆	D₄×C₅₃	1	4	yes	—
265	C₂²×C₈	—	2	8	no	—
266	C₂²×C₈	—	2	1	no	—
267	C₂²×C₈	—	2	2	no	—
268	C₂²×C₈	—	2	2	no	—
269	C₂²×C₈	—	2	4	no	—
270	C₂²×C₈	—	2	2	no	—
271	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
272	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
273	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
274	C₂²×C₈	D₄×C₅₃	1	2	yes	—
275	#117	—	2	2	no	—
276	#117	—	2	2	no	—
277	#117	—	2	2	no	—
278	#117	C₂×C₄×C₅₃	1	4	yes	—
279	C₂×C₄⋊C₄	—	2	2	no	—
280	C₂×C₂²⋊C₄	—	2	2	no	—
281	C₂×C₄²	—	2	2	no	—
282	#104	—	2	16	no	—
283	C₂⁴	—	2	1	no	—
284	C₂⁴	#41×C₅₃	1	1	yes	characteristic
285	#41	—	2	2	no	—
286	#41	—	2	2	no	—
287	#41	—	2	4	no	—
288	#41	—	2	4	no	—
289	#41	—	2	4	no	—
290	#41	—	2	4	no	—
291	#41	—	2	4	no	—
292	#41	C₄²×C₅₃	1	4	yes	—
293	#41	C₂²⋊C₄×C₅₃	1	8	yes	—
294	#41	C₂²⋊C₄×C₅₃	1	4	yes	—
295	C₂×Q₈	—	2	1	no	—
296	C₂×Q₈	—	2	2	no	—
297	C₂×Q₈	C₄⋊C₄×C₅₃	1	2	yes	—
298	C₂×Q₈	C₄⋊C₄×C₅₃	1	4	yes	—
299	C₂×D₄	—	2	2	no	—
300	C₂×D₄	—	2	4	no	—
301	C₂×D₄	—	2	2	no	—
302	C₂×D₄	—	2	4	no	—
303	C₂×D₄	—	2	2	no	—
304	C₂×D₄	—	2	1	no	—
305	C₂×D₄	C₄⋊C₄×C₅₃	1	2	yes	—
306	C₂×D₄	C₄⋊C₄×C₅₃	1	4	yes	—
307	C₂²×C₄	—	2	1	no	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	2	4	no	—
310	C₂²×C₄	—	2	4	no	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	4	2	no	—
313	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
314	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
315	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
316	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
317	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
318	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
319	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
320	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
321	C₂²×C₄	#41×C₅₃	1	2	yes	—
322	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
323	M₁₆	—	2	16	no	—
324	M₁₆	—	2	2	no	—
325	M₁₆	—	2	4	no	—
326	M₁₆	—	2	4	no	—
327	M₁₆	—	2	4	no	—
328	M₁₆	—	4	2	no	—
329	M₁₆	—	4	2	no	—
330	M₁₆	C₂²⋊C₄×C₅₃	1	4	yes	—
331	C₂×C₈	—	2	16	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	2	4	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	4	no	—
339	C₂×C₈	—	2	4	no	—
340	C₂×C₈	—	4	2	no	—
341	C₂×C₈	—	4	2	no	—
342	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
343	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
344	C₄⋊C₄	—	2	4	no	—
345	C₄⋊C₄	—	4	2	no	—
346	C₂²⋊C₄	—	2	4	no	—
347	C₂²⋊C₄	—	4	2	no	—
348	C₄²	—	2	4	no	—
349	C₄²	—	4	2	no	—
350	C₂³	—	2	2	no	—
351	C₂³	—	2	4	no	—
352	C₂³	—	4	2	no	—
353	C₂³	—	4	4	no	—
354	C₂³	#117×C₅₃	1	1	yes	characteristic
355	C₂³	C₄×D₄×C₅₃	1	2	yes	—
356	C₂³	#121×C₅₃	1	1	yes	characteristic
357	C₂³	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
358	C₂³	#123×C₅₃	1	2	yes	—
359	Q₈	—	2	2	no	—
360	Q₈	—	2	4	no	—
361	Q₈	#95×C₅₃	1	4	yes	—
362	D₄	—	2	4	no	—
363	D₄	—	2	4	no	—
364	D₄	—	2	8	no	—
365	D₄	—	2	2	no	—
366	D₄	—	2	4	no	—
367	D₄	#95×C₅₃	1	4	yes	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	4	no	—
370	C₂×C₄	—	2	4	no	—
371	C₂×C₄	—	2	4	no	—
372	C₂×C₄	—	2	8	no	—
373	C₂×C₄	—	4	4	no	—
374	C₂×C₄	—	8	4	no	—
375	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
376	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
377	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
378	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
379	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
380	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
381	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
382	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
383	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
384	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
385	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
386	C₂×C₄	#120×C₅₃	1	2	yes	—
387	C₂×C₄	#120×C₅₃	1	2	yes	—
388	C₂×C₄	#121×C₅₃	1	2	yes	—
389	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
390	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
391	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
392	C₂×C₄	#123×C₅₃	1	2	yes	—
393	C₈	—	2	2	no	—
394	C₈	—	2	4	no	—
395	C₈	—	2	2	no	—
396	C₈	—	4	4	no	—
397	C₂²	—	2	2	no	—
398	C₂²	—	2	2	no	—
399	C₂²	—	2	2	no	—
400	C₂²	—	2	4	no	—
401	C₂²	—	4	2	no	—
402	C₂²	—	4	2	no	—
403	C₂²	—	4	8	no	—
404	C₂²	—	4	4	no	—
405	C₂²	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
406	C₂²	C₅₃×#380	1	1	yes	characteristic
407	C₂²	C₅₃×#381	1	1	yes	characteristic
408	C₂²	C₅₃×#385	1	2	yes	—
409	C₂²	C₅₃×#386	1	2	yes	—
410	C₄	—	2	4	no	—
411	C₄	—	8	4	no	—
412	C₄	C₂×#95×C₅₃	1	1	yes	characteristic
413	C₄	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
414	C₄	C₅₃×#379	1	1	yes	characteristic
415	C₄	C₅₃×#380	1	1	yes	characteristic
416	C₄	C₅₃×#385	1	2	yes	—
417	C₄	C₅₃×#386	1	2	yes	—
418	C₂	—	2	2	no	—
419	C₂	—	2	2	no	—
420	C₂	—	2	1	no	—
421	C₂	—	2	1	no	—
422	C₂	—	4	4	no	—
423	C₂	C₅₃×C₂²⋊#95	1	1	yes	characteristic
424	C₂	C₅₃×#1814	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	#121
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
socle	C₂²×C₅₃	#380
prime radical	C₂²×#41×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₅₃×#344	C₂	semidirect
4. C₂×C₅₃×#341	C₂	semidirect
5. C₅₃×#1735	C₂	semidirect
10. C₂×#131×C₅₃	C₄	semidirect
18. C₅₃×#421	C₄	semidirect
184. #10773	C₅₃	direct (N × H)
208. C₂×#344	C₂×C₅₃	semidirect
209. C₂×#341	C₂×C₅₃	semidirect
210. #1735	C₂×C₅₃	semidirect
222. C₂×#131	C₄×C₅₃	semidirect
230. #421	C₄×C₅₃	semidirect
236. C₅₃	#10773	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#131×C₅₃	2
C₂×C₅₃×#421	2
C₂×C₅₃×#344	2
C₂×C₅₃×#341	2
C₅₃×#1735	2
#10773	53

Subgroups — C53×#10773 #27218285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10773 #27218285