Subgroups — C₅₃×#8973 #27216485

Order 13568.

664 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#476	C₂	1	1	yes	maximal, characteristic
2	D₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1987	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1956	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅₃×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1296	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1279	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₄×C₅₃	—	2	1	no	—
10	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
12	C₅₃×#476	—	2	2	no	—
13	C₅₃×#476	C₄	1	4	yes	complemented
14	C₂×#106×C₅₃	—	2	1	no	—
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#103×C₅₃	—	2	2	no	—
18	C₂×#102×C₅₃	—	2	1	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#389	—	2	1	no	—
21	C₅₃×#385	—	2	2	no	—
22	C₅₃×#384	C₂²	1	1	yes	characteristic
23	C₅₃×#380	—	2	1	no	—
24	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#340	—	2	1	no	—
26	C₅₃×#340	—	2	1	no	—
27	C₅₃×D₄⋊C₈	—	2	2	no	—
28	C₂²×Q₈×C₅₃	—	2	1	no	—
29	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₅₃	—	2	1	no	—
31	C₂³×C₄×C₅₃	—	2	2	no	—
32	C₂²×C₈×C₅₃	—	2	1	no	—
33	C₂²×C₈×C₅₃	—	2	1	no	—
34	C₂²×C₈×C₅₃	—	2	1	no	—
35	C₄⋊Q₈×C₅₃	—	2	1	no	—
36	C₄⋊Q₈×C₅₃	—	2	1	no	—
37	C₄⋊Q₈×C₅₃	—	2	1	no	—
38	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	complemented
39	C₄×D₄×C₅₃	—	2	1	no	—
40	C₄×D₄×C₅₃	—	2	1	no	—
41	C₄×D₄×C₅₃	—	4	2	no	—
42	C₄×D₄×C₅₃	—	4	2	no	—
43	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
44	C₄×D₄×C₅₃	D₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
47	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
48	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
49	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
50	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
51	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
52	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
56	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
57	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
58	C₂×C₄²×C₅₃	—	2	1	no	—
59	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
60	#106×C₅₃	—	2	4	no	—
61	C₄⋊C₈×C₅₃	—	2	1	no	—
62	C₄⋊C₈×C₅₃	—	2	1	no	—
63	C₄⋊C₈×C₅₃	—	2	1	no	—
64	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	complemented
65	#103×C₅₃	—	4	4	no	—
66	#102×C₅₃	—	2	4	no	—
67	C₄×C₈×C₅₃	—	2	1	no	—
68	C₄×C₈×C₅₃	—	2	1	no	—
69	#95×C₅₃	—	2	2	no	—
70	#95×C₅₃	—	2	1	no	—
71	C₂⁴×C₅₃	—	2	1	no	—
72	C₂×Q₈×C₅₃	—	2	2	no	—
73	C₂×Q₈×C₅₃	—	4	2	no	—
74	C₂×Q₈×C₅₃	—	4	2	no	—
75	C₂×Q₈×C₅₃	—	4	1	no	—
76	C₂×D₄×C₅₃	—	2	1	no	—
77	C₂×D₄×C₅₃	—	2	1	no	—
78	C₂×D₄×C₅₃	—	2	1	no	—
79	C₂×D₄×C₅₃	C₂²⋊C₄	1	2	yes	—
80	C₂×D₄×C₅₃	SD₁₆	1	2	yes	—
81	C₂×D₄×C₅₃	Q₁₆	1	2	yes	—
82	C₂²×C₄×C₅₃	—	2	1	no	—
83	C₂²×C₄×C₅₃	—	2	1	no	—
84	C₂²×C₄×C₅₃	—	2	2	no	—
85	C₂²×C₄×C₅₃	—	2	2	no	—
86	C₂²×C₄×C₅₃	—	2	1	no	—
87	C₂²×C₄×C₅₃	—	2	2	no	—
88	C₂²×C₄×C₅₃	—	4	4	no	—
89	C₂²×C₄×C₅₃	—	4	1	no	—
90	C₂²×C₄×C₅₃	—	4	1	no	—
91	C₂²×C₄×C₅₃	—	4	1	no	—
92	C₂²×C₄×C₅₃	—	4	1	no	—
93	C₂²×C₄×C₅₃	—	4	4	no	—
94	C₂²×C₄×C₅₃	—	4	1	no	—
95	C₂²×C₄×C₅₃	—	8	2	no	—
96	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₅₃	—	2	2	no	—
100	C₂×C₈×C₅₃	—	2	2	no	—
101	C₂×C₈×C₅₃	—	4	1	no	—
102	C₂×C₈×C₅₃	—	4	1	no	—
103	C₂×C₈×C₅₃	—	4	1	no	—
104	C₂×C₈×C₅₃	—	4	1	no	—
105	C₂×C₈×C₅₃	—	4	1	no	—
106	C₂×C₈×C₅₃	—	4	1	no	—
107	C₂×C₈×C₅₃	—	4	1	no	—
108	C₄⋊C₄×C₅₃	—	2	1	no	—
109	C₄⋊C₄×C₅₃	—	2	1	no	—
110	C₄⋊C₄×C₅₃	—	2	2	no	—
111	C₄⋊C₄×C₅₃	—	2	2	no	—
112	C₄⋊C₄×C₅₃	—	2	1	no	—
113	C₄⋊C₄×C₅₃	—	2	1	no	—
114	C₄⋊C₄×C₅₃	—	4	4	no	—
115	C₄⋊C₄×C₅₃	—	4	2	no	—
116	C₄⋊C₄×C₅₃	—	4	1	no	—
117	C₄⋊C₄×C₅₃	—	4	1	no	—
118	C₄⋊C₄×C₅₃	—	4	1	no	—
119	C₄⋊C₄×C₅₃	—	4	1	no	—
120	C₄⋊C₄×C₅₃	—	4	1	no	—
121	C₄⋊C₄×C₅₃	—	4	1	no	—
122	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
123	C₄⋊C₄×C₅₃	SD₁₆	1	2	yes	—
124	C₂²⋊C₄×C₅₃	—	2	2	no	—
125	C₂²⋊C₄×C₅₃	—	2	2	no	—
126	C₂²⋊C₄×C₅₃	—	4	4	no	—
127	C₄²×C₅₃	—	4	1	no	—
128	C₄²×C₅₃	—	4	1	no	—
129	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₅₃	#41	1	1	yes	characteristic
132	C₄²×C₅₃	#41	1	1	yes	characteristic
133	C₂³×C₅₃	—	2	2	no	—
134	C₂³×C₅₃	—	2	2	no	—
135	C₂³×C₅₃	—	2	2	no	—
136	C₂³×C₅₃	—	4	1	no	—
137	C₂³×C₅₃	—	4	1	no	—
138	C₂³×C₅₃	—	4	1	no	—
139	C₂³×C₅₃	—	4	1	no	—
140	C₂³×C₅₃	#120	1	1	yes	characteristic
141	Q₈×C₅₃	—	4	4	no	—
142	Q₈×C₅₃	—	8	2	no	—
143	D₄×C₅₃	—	2	2	no	—
144	D₄×C₅₃	—	2	2	no	—
145	D₄×C₅₃	—	2	2	no	—
146	D₄×C₅₃	#103	1	4	yes	complemented
147	C₂×C₄×C₅₃	—	2	1	no	—
148	C₂×C₄×C₅₃	—	2	1	no	—
149	C₂×C₄×C₅₃	—	2	2	no	—
150	C₂×C₄×C₅₃	—	2	2	no	—
151	C₂×C₄×C₅₃	—	4	1	no	—
152	C₂×C₄×C₅₃	—	4	1	no	—
153	C₂×C₄×C₅₃	—	4	1	no	—
154	C₂×C₄×C₅₃	—	4	1	no	—
155	C₂×C₄×C₅₃	—	4	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	1	no	—
163	C₂×C₄×C₅₃	—	4	1	no	—
164	C₂×C₄×C₅₃	—	4	2	no	—
165	C₂×C₄×C₅₃	—	4	1	no	—
166	C₂×C₄×C₅₃	—	8	8	no	—
167	C₂×C₄×C₅₃	—	8	1	no	—
168	C₂×C₄×C₅₃	—	8	1	no	—
169	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
170	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
171	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
172	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
173	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
174	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
175	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
176	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
177	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
178	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
179	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
180	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
181	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
182	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
183	C₈×C₅₃	—	4	2	no	—
184	C₈×C₅₃	—	4	2	no	—
185	C₈×C₅₃	—	8	1	no	—
186	#8973	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
187	C₂²×C₅₃	—	2	4	no	—
188	C₂²×C₅₃	—	4	2	no	—
189	C₂²×C₅₃	—	4	2	no	—
190	C₂²×C₅₃	—	4	2	no	—
191	C₂²×C₅₃	—	4	2	no	—
192	C₂²×C₅₃	—	4	2	no	—
193	C₂²×C₅₃	—	4	2	no	—
194	C₂²×C₅₃	#386	1	1	yes	characteristic
195	C₂²×C₅₃	#450	1	1	yes	characteristic
196	C₂²×C₅₃	#450	1	1	yes	characteristic
197	C₂²×C₅₃	#451	1	1	yes	characteristic
198	C₂²×C₅₃	#452	1	1	yes	characteristic
199	C₂²×C₅₃	#453	1	1	yes	characteristic
200	C₂²×C₅₃	#454	1	1	yes	characteristic
201	C₄×C₅₃	—	2	1	no	—
202	C₄×C₅₃	—	2	1	no	—
203	C₄×C₅₃	—	4	2	no	—
204	C₄×C₅₃	—	4	2	no	—
205	C₄×C₅₃	—	8	2	no	—
206	C₄×C₅₃	—	8	2	no	—
207	C₄×C₅₃	—	8	1	no	—
208	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
209	C₄×C₅₃	#417	1	1	yes	characteristic
210	C₄×C₅₃	C₄×SD₁₆	1	1	yes	characteristic
211	C₄×C₅₃	#440	1	1	yes	characteristic
212	C₄×C₅₃	#461	1	1	yes	characteristic
213	C₄×C₅₃	#463	1	1	yes	characteristic
214	C₄×C₅₃	#483	1	1	yes	characteristic
215	C₄×C₅₃	#484	1	1	yes	characteristic
216	C₂×#476	C₂×C₅₃	1	1	yes	characteristic
217	D₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic
218	#1987	C₂×C₅₃	1	1	yes	characteristic, complemented
219	#1956	C₂×C₅₃	1	1	yes	characteristic, complemented
220	C₂×D₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic
221	#1296	C₂×C₅₃	1	1	yes	characteristic, complemented
222	#1279	C₂×C₅₃	1	1	yes	characteristic, complemented
223	C₂×C₅₃	—	4	4	no	—
224	C₂×C₅₃	#1629	1	1	yes	characteristic
225	C₂×C₅₃	#1632	1	1	yes	characteristic
226	C₂×C₅₃	#1635	1	1	yes	characteristic
227	C₂×C₅₃	#1637	1	1	yes	characteristic
228	C₂×C₅₃	#1872	1	1	yes	characteristic
229	C₂×C₅₃	#1877	1	1	yes	characteristic
230	C₂×C₅₃	#1883	1	1	yes	characteristic
231	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
232	C₂×C₄×D₄	—	2	1	no	—
233	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
234	C₂²×C₄⋊C₄	—	2	1	no	—
235	#476	—	2	2	no	—
236	#476	C₄×C₅₃	1	4	yes	complemented
237	C₂×#106	—	2	1	no	—
238	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
239	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
240	C₂×#103	—	2	2	no	—
241	C₂×#102	—	2	1	no	—
242	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
243	#389	—	2	1	no	—
244	#385	—	2	2	no	—
245	#384	C₂²×C₅₃	1	1	yes	characteristic
246	#380	—	2	1	no	—
247	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
248	#340	—	2	1	no	—
249	#340	—	2	1	no	—
250	D₄⋊C₈	—	2	2	no	—
251	C₅₃	#8973	1	1	yes	characteristic, direct product factor, Sylow
252	C₂²×Q₈	—	2	1	no	—
253	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
254	C₂³×C₄	—	2	1	no	—
255	C₂³×C₄	—	2	2	no	—
256	C₂²×C₈	—	2	1	no	—
257	C₂²×C₈	—	2	1	no	—
258	C₂²×C₈	—	2	1	no	—
259	C₄⋊Q₈	—	2	1	no	—
260	C₄⋊Q₈	—	2	1	no	—
261	C₄⋊Q₈	—	2	1	no	—
262	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	complemented
263	C₄×D₄	—	2	1	no	—
264	C₄×D₄	—	2	1	no	—
265	C₄×D₄	—	4	2	no	—
266	C₄×D₄	—	4	2	no	—
267	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
268	C₄×D₄	D₄×C₅₃	1	2	yes	—
269	C₂×C₄⋊C₄	—	2	1	no	—
270	C₂×C₄⋊C₄	—	2	1	no	—
271	C₂×C₄⋊C₄	—	2	2	no	—
272	C₂×C₄⋊C₄	—	2	1	no	—
273	C₂×C₄⋊C₄	—	2	1	no	—
274	C₂×C₄⋊C₄	—	4	2	no	—
275	C₂×C₄⋊C₄	—	4	2	no	—
276	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
277	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
278	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
279	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
280	C₂×C₂²⋊C₄	—	2	1	no	—
281	C₂×C₂²⋊C₄	—	2	2	no	—
282	C₂×C₄²	—	2	1	no	—
283	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
284	#106	—	2	4	no	—
285	C₄⋊C₈	—	2	1	no	—
286	C₄⋊C₈	—	2	1	no	—
287	C₄⋊C₈	—	2	1	no	—
288	C₄⋊C₈	C₂×C₄×C₅₃	1	2	yes	complemented
289	#103	—	4	4	no	—
290	#102	—	2	4	no	—
291	C₄×C₈	—	2	1	no	—
292	C₄×C₈	—	2	1	no	—
293	#95	—	2	2	no	—
294	#95	—	2	1	no	—
295	C₂⁴	—	2	1	no	—
296	C₂×Q₈	—	2	2	no	—
297	C₂×Q₈	—	4	2	no	—
298	C₂×Q₈	—	4	2	no	—
299	C₂×Q₈	—	4	1	no	—
300	C₂×D₄	—	2	1	no	—
301	C₂×D₄	—	2	1	no	—
302	C₂×D₄	—	2	1	no	—
303	C₂×D₄	C₂²⋊C₄×C₅₃	1	2	yes	—
304	C₂×D₄	SD₁₆×C₅₃	1	2	yes	—
305	C₂×D₄	Q₁₆×C₅₃	1	2	yes	—
306	C₂²×C₄	—	2	1	no	—
307	C₂²×C₄	—	2	1	no	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	4	4	no	—
313	C₂²×C₄	—	4	1	no	—
314	C₂²×C₄	—	4	1	no	—
315	C₂²×C₄	—	4	1	no	—
316	C₂²×C₄	—	4	1	no	—
317	C₂²×C₄	—	4	4	no	—
318	C₂²×C₄	—	4	1	no	—
319	C₂²×C₄	—	8	2	no	—
320	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
321	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
322	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	4	1	no	—
326	C₂×C₈	—	4	1	no	—
327	C₂×C₈	—	4	1	no	—
328	C₂×C₈	—	4	1	no	—
329	C₂×C₈	—	4	1	no	—
330	C₂×C₈	—	4	1	no	—
331	C₂×C₈	—	4	1	no	—
332	C₄⋊C₄	—	2	1	no	—
333	C₄⋊C₄	—	2	1	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	1	no	—
337	C₄⋊C₄	—	2	1	no	—
338	C₄⋊C₄	—	4	2	no	—
339	C₄⋊C₄	—	4	4	no	—
340	C₄⋊C₄	—	4	1	no	—
341	C₄⋊C₄	—	4	1	no	—
342	C₄⋊C₄	—	4	1	no	—
343	C₄⋊C₄	—	4	1	no	—
344	C₄⋊C₄	—	4	1	no	—
345	C₄⋊C₄	—	4	1	no	—
346	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
347	C₄⋊C₄	SD₁₆×C₅₃	1	2	yes	—
348	C₂²⋊C₄	—	2	2	no	—
349	C₂²⋊C₄	—	2	2	no	—
350	C₂²⋊C₄	—	4	4	no	—
351	C₄²	—	4	1	no	—
352	C₄²	—	4	1	no	—
353	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
354	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
355	C₄²	#41×C₅₃	1	1	yes	characteristic
356	C₄²	#41×C₅₃	1	1	yes	characteristic
357	C₂³	—	2	2	no	—
358	C₂³	—	2	2	no	—
359	C₂³	—	2	2	no	—
360	C₂³	—	4	1	no	—
361	C₂³	—	4	1	no	—
362	C₂³	—	4	1	no	—
363	C₂³	—	4	1	no	—
364	C₂³	#120×C₅₃	1	1	yes	characteristic
365	Q₈	—	4	4	no	—
366	Q₈	—	8	2	no	—
367	D₄	—	2	2	no	—
368	D₄	—	2	2	no	—
369	D₄	—	2	2	no	—
370	D₄	#103×C₅₃	1	4	yes	complemented
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	4	1	no	—
376	C₂×C₄	—	4	1	no	—
377	C₂×C₄	—	4	1	no	—
378	C₂×C₄	—	4	1	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	1	no	—
390	C₂×C₄	—	8	8	no	—
391	C₂×C₄	—	8	1	no	—
392	C₂×C₄	—	8	1	no	—
393	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
394	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
395	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
396	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
397	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
398	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
399	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
400	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
401	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
402	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
403	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
404	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
405	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
406	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
407	C₈	—	4	2	no	—
408	C₈	—	4	2	no	—
409	C₈	—	8	1	no	—
410	C₂²	—	2	4	no	—
411	C₂²	—	4	2	no	—
412	C₂²	—	4	2	no	—
413	C₂²	—	4	2	no	—
414	C₂²	—	4	2	no	—
415	C₂²	—	4	2	no	—
416	C₂²	—	4	2	no	—
417	C₂²	C₅₃×#386	1	1	yes	characteristic
418	C₂²	C₅₃×#450	1	1	yes	characteristic
419	C₂²	C₅₃×#450	1	1	yes	characteristic
420	C₂²	C₅₃×#451	1	1	yes	characteristic
421	C₂²	C₅₃×#452	1	1	yes	characteristic
422	C₂²	C₅₃×#453	1	1	yes	characteristic
423	C₂²	C₅₃×#454	1	1	yes	characteristic
424	C₄	—	2	1	no	—
425	C₄	—	2	1	no	—
426	C₄	—	4	2	no	—
427	C₄	—	4	2	no	—
428	C₄	—	8	2	no	—
429	C₄	—	8	2	no	—
430	C₄	—	8	1	no	—
431	C₄	C₂×#103×C₅₃	1	1	yes	characteristic
432	C₄	C₅₃×#417	1	1	yes	characteristic
433	C₄	C₄×SD₁₆×C₅₃	1	1	yes	characteristic
434	C₄	C₅₃×#440	1	1	yes	characteristic
435	C₄	C₅₃×#461	1	1	yes	characteristic
436	C₄	C₅₃×#463	1	1	yes	characteristic
437	C₄	C₅₃×#483	1	1	yes	characteristic
438	C₄	C₅₃×#484	1	1	yes	characteristic
439	C₂	—	4	4	no	—
440	C₂	C₅₃×#1629	1	1	yes	characteristic
441	C₂	C₅₃×#1632	1	1	yes	characteristic
442	C₂	C₅₃×#1635	1	1	yes	characteristic
443	C₂	C₅₃×#1637	1	1	yes	characteristic
444	C₂	C₅₃×#1872	1	1	yes	characteristic
445	C₂	C₅₃×#1877	1	1	yes	characteristic
446	C₂	C₅₃×#1883	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	#120
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂²×D₄×C₅₃	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1987	C₂	semidirect
4. C₅₃×#1956	C₂	semidirect
6. C₅₃×#1296	C₂	semidirect
7. C₅₃×#1279	C₂	semidirect
13. C₅₃×#476	C₄	semidirect
38. C₄⋊Q₈×C₅₃	C₂×C₄	semidirect
64. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
146. D₄×C₅₃	#103	semidirect
186. #8973	C₅₃	direct (N × H)
218. #1987	C₂×C₅₃	semidirect
219. #1956	C₂×C₅₃	semidirect
221. #1296	C₂×C₅₃	semidirect
222. #1279	C₂×C₅₃	semidirect
236. #476	C₄×C₅₃	semidirect
251. C₅₃	#8973	direct (N × H)
262. C₄⋊Q₈	C₂×C₄×C₅₃	semidirect
288. C₄⋊C₈	C₂×C₄×C₅₃	semidirect
370. D₄	#103×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×#476	2
D₄×C₄⋊C₄×C₅₃	2
C₅₃×#1987	2
C₅₃×#1956	2
C₂×C₅₃×D₄⋊C₈	2
C₅₃×#1296	2
C₅₃×#1279	2
#8973	53

Subgroups — C53×#8973 #27216485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8973 #27216485