Subgroups — C₅₃×#19678 #27227190

Order 13568.

828 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3226	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3208	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3167	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3151	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#3139	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1719	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1717	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1717	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1714	C₂	1	2	yes	maximal, complemented
11	C₅₃×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
13	C₅₃×#533	C₂²	1	1	yes	characteristic
14	C₅₃×#532	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#500	C₂²	1	2	yes	complemented
18	C₅₃×#498	C₂²	1	1	yes	characteristic, complemented
19	C₅₃×#496	—	2	2	no	—
20	C₅₃×#493	C₂²	1	2	yes	—
21	C₅₃×#492	C₂²	1	2	yes	—
22	C₅₃×#492	C₂²	1	2	yes	—
23	C₅₃×#492	C₂²	1	1	yes	characteristic, complemented
24	C₅₃×#491	—	2	2	no	—
25	C₅₃×#487	C₂²	1	2	yes	complemented
26	C₅₃×#487	C₂²	1	2	yes	complemented
27	C₅₃×#487	C₂²	1	1	yes	characteristic, complemented
28	C₅₃×#486	—	2	2	no	—
29	C₅₃×#486	C₂²	1	2	yes	—
30	C₅₃×#486	C₂²	1	2	yes	—
31	C₅₃×#486	C₂²	1	2	yes	—
32	C₅₃×#486	C₂²	1	2	yes	complemented
33	C₅₃×#486	C₂²	1	1	yes	characteristic, complemented
34	C₅₃×#480	—	2	2	no	—
35	C₅₃×#474	—	2	2	no	—
36	C₅₃×#465	—	2	2	no	—
37	C₅₃×#460	—	2	2	no	—
38	C₅₃×#450	—	2	2	no	—
39	C₄×SD₁₆×C₅₃	—	2	2	no	—
40	C₅₃×#431	C₂²	1	1	yes	characteristic
41	C₅₃×#419	—	2	2	no	—
42	C₂×#102×C₅₃	—	2	2	no	—
43	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
45	C₅₃×#322	C₂²	1	2	yes	complemented
46	C₅₃×#322	C₂²	1	2	yes	complemented
47	C₂×#41×C₅₃	—	2	1	no	—
48	C₂²×Q₈×C₅₃	—	2	1	no	—
49	C₂²×D₄×C₅₃	—	2	1	no	—
50	C₂²×D₄×C₅₃	—	2	1	no	—
51	C₂²×D₄×C₅₃	—	2	1	no	—
52	AGL(1,ℤ/8)×C₅₃	—	4	8	no	—
53	C₂×SD₁₆×C₅₃	—	2	4	no	—
54	C₂×SD₁₆×C₅₃	—	2	2	no	—
55	C₂×SD₁₆×C₅₃	—	2	4	no	—
56	C₂×D₈×C₅₃	—	2	4	no	—
57	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
58	C₂²×C₈×C₅₃	D₄	1	2	yes	—
59	C₄⋊Q₈×C₅₃	—	2	1	no	—
60	C₄⋊Q₈×C₅₃	—	2	1	no	—
61	C₄⋊Q₈×C₅₃	—	2	1	no	—
62	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
63	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
65	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
66	#127×C₅₃	—	2	1	no	—
67	#127×C₅₃	—	2	1	no	—
68	#127×C₅₃	—	2	1	no	—
69	#127×C₅₃	C₂³	1	2	yes	—
70	#127×C₅₃	C₂³	1	1	yes	characteristic
71	#125×C₅₃	—	2	1	no	—
72	#124×C₅₃	—	2	1	no	—
73	C₂²⋊Q₈×C₅₃	—	2	1	no	—
74	C₂²⋊Q₈×C₅₃	—	2	1	no	—
75	#121×C₅₃	—	2	1	no	—
76	#121×C₅₃	—	2	1	no	—
77	C₄×Q₈×C₅₃	—	2	1	no	—
78	C₄×Q₈×C₅₃	—	2	1	no	—
79	C₄×D₄×C₅₃	—	2	1	no	—
80	C₄×D₄×C₅₃	—	2	1	no	—
81	#117×C₅₃	—	2	1	no	—
82	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
83	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
84	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
85	#107×C₅₃	—	2	4	no	—
86	#106×C₅₃	—	2	2	no	—
87	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
88	#103×C₅₃	—	2	4	no	—
89	#103×C₅₃	—	2	4	no	—
90	#103×C₅₃	—	2	2	no	—
91	#102×C₅₃	—	2	4	no	—
92	#102×C₅₃	—	2	4	no	—
93	#102×C₅₃	—	2	4	no	—
94	#102×C₅₃	—	2	2	no	—
95	#102×C₅₃	—	2	4	no	—
96	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
97	#97×C₅₃	D₄	1	2	yes	—
98	C₄×C₈×C₅₃	D₄	1	2	yes	—
99	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₅₃	C₂³	1	2	yes	—
102	C₄×C₈×C₅₃	C₂³	1	2	yes	—
103	C₂⁴×C₅₃	—	4	1	no	—
104	#41×C₅₃	—	4	4	no	—
105	C₂×Q₈×C₅₃	—	2	2	no	—
106	C₂×Q₈×C₅₃	—	2	1	no	—
107	C₂×Q₈×C₅₃	—	2	1	no	—
108	C₂×Q₈×C₅₃	—	2	1	no	—
109	C₂×Q₈×C₅₃	—	4	1	no	—
110	C₂×Q₈×C₅₃	—	4	4	no	—
111	C₂×D₄×C₅₃	—	2	1	no	—
112	C₂×D₄×C₅₃	—	2	2	no	—
113	C₂×D₄×C₅₃	—	2	1	no	—
114	C₂×D₄×C₅₃	—	2	2	no	—
115	C₂×D₄×C₅₃	—	2	1	no	—
116	C₂×D₄×C₅₃	—	2	2	no	—
117	C₂×D₄×C₅₃	—	4	4	no	—
118	C₂×D₄×C₅₃	—	4	1	no	—
119	C₂×D₄×C₅₃	—	4	1	no	—
120	C₂×D₄×C₅₃	—	4	1	no	—
121	C₂×D₄×C₅₃	—	4	1	no	—
122	C₂×D₄×C₅₃	—	4	4	no	—
123	C₂×D₄×C₅₃	—	8	2	no	—
124	C₂²×C₄×C₅₃	—	4	1	no	—
125	C₂²×C₄×C₅₃	—	4	1	no	—
126	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
129	SD₁₆×C₅₃	—	4	4	no	—
130	SD₁₆×C₅₃	—	4	8	no	—
131	SD₁₆×C₅₃	—	8	4	no	—
132	D₈×C₅₃	—	4	8	no	—
133	M₁₆×C₅₃	—	2	4	no	—
134	C₂×C₈×C₅₃	—	4	2	no	—
135	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
136	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
137	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
138	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
139	C₄⋊C₄×C₅₃	—	2	2	no	—
140	C₄⋊C₄×C₅₃	—	2	1	no	—
141	C₄⋊C₄×C₅₃	—	2	1	no	—
142	C₄⋊C₄×C₅₃	—	2	2	no	—
143	C₄⋊C₄×C₅₃	—	2	1	no	—
144	C₄⋊C₄×C₅₃	—	2	1	no	—
145	C₄⋊C₄×C₅₃	—	2	1	no	—
146	C₄⋊C₄×C₅₃	—	2	1	no	—
147	C₄⋊C₄×C₅₃	—	4	1	no	—
148	C₄⋊C₄×C₅₃	—	4	1	no	—
149	C₄⋊C₄×C₅₃	—	4	1	no	—
150	C₄⋊C₄×C₅₃	—	4	1	no	—
151	C₂²⋊C₄×C₅₃	—	4	1	no	—
152	C₂²⋊C₄×C₅₃	—	4	1	no	—
153	C₂²⋊C₄×C₅₃	—	4	1	no	—
154	C₄²×C₅₃	—	4	1	no	—
155	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
156	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
157	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
158	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
159	C₂³×C₅₃	—	4	1	no	—
160	C₂³×C₅₃	—	4	2	no	—
161	C₂³×C₅₃	—	8	2	no	—
162	C₂³×C₅₃	—	8	2	no	—
163	C₂³×C₅₃	—	8	2	no	—
164	C₂³×C₅₃	#127	1	1	yes	characteristic
165	Q₈×C₅₃	—	4	4	no	—
166	Q₈×C₅₃	—	4	2	no	—
167	Q₈×C₅₃	—	4	2	no	—
168	Q₈×C₅₃	—	4	2	no	—
169	Q₈×C₅₃	—	8	2	no	—
170	D₄×C₅₃	—	4	4	no	—
171	D₄×C₅₃	—	4	2	no	—
172	D₄×C₅₃	—	4	4	no	—
173	D₄×C₅₃	—	4	2	no	—
174	D₄×C₅₃	—	4	2	no	—
175	D₄×C₅₃	—	4	4	no	—
176	D₄×C₅₃	—	8	2	no	—
177	D₄×C₅₃	—	8	2	no	—
178	D₄×C₅₃	—	8	2	no	—
179	D₄×C₅₃	—	8	2	no	—
180	C₂×C₄×C₅₃	—	2	2	no	—
181	C₂×C₄×C₅₃	—	2	2	no	—
182	C₂×C₄×C₅₃	—	4	1	no	—
183	C₂×C₄×C₅₃	—	4	2	no	—
184	C₂×C₄×C₅₃	—	4	1	no	—
185	C₂×C₄×C₅₃	—	4	1	no	—
186	C₂×C₄×C₅₃	—	8	2	no	—
187	C₂×C₄×C₅₃	—	8	2	no	—
188	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
189	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
190	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
191	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
192	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
195	C₈×C₅₃	—	2	4	no	—
196	C₈×C₅₃	—	2	2	no	—
197	C₈×C₅₃	—	2	2	no	—
198	C₈×C₅₃	—	2	4	no	—
199	#19678	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
200	C₂²×C₅₃	—	2	1	no	—
201	C₂²×C₅₃	—	2	1	no	—
202	C₂²×C₅₃	—	8	1	no	—
203	C₂²×C₅₃	—	8	2	no	—
204	C₂²×C₅₃	—	8	2	no	—
205	C₂²×C₅₃	—	8	1	no	—
206	C₂²×C₅₃	—	8	2	no	—
207	C₂²×C₅₃	—	8	1	no	—
208	C₂²×C₅₃	—	16	4	no	—
209	C₂²×C₅₃	#496	1	2	yes	—
210	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
211	C₄×C₅₃	—	2	1	no	—
212	C₄×C₅₃	—	2	1	no	—
213	C₄×C₅₃	—	8	1	no	—
214	C₄×C₅₃	—	8	2	no	—
215	C₄×C₅₃	—	8	1	no	—
216	C₄×C₅₃	—	8	1	no	—
217	C₄×C₅₃	#496	1	2	yes	—
218	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
219	C₄×C₅₃	#575	1	2	yes	—
220	C₄×C₅₃	#577	1	2	yes	—
221	#3226	C₂×C₅₃	1	1	yes	characteristic, complemented
222	#3208	C₂×C₅₃	1	1	yes	characteristic, complemented
223	#3167	C₂×C₅₃	1	1	yes	characteristic, complemented
224	#3151	C₂×C₅₃	1	1	yes	characteristic
225	#3139	C₂×C₅₃	1	1	yes	characteristic, complemented
226	C₂×#486	C₂×C₅₃	1	1	yes	characteristic, complemented
227	#1719	C₂×C₅₃	1	2	yes	complemented
228	#1717	C₂×C₅₃	1	2	yes	complemented
229	#1717	C₂×C₅₃	1	2	yes	complemented
230	#1714	C₂×C₅₃	1	2	yes	complemented
231	#1451	C₂×C₅₃	1	1	yes	characteristic, complemented
232	C₂×C₅₃	—	2	2	no	—
233	C₂×C₅₃	—	16	1	no	—
234	C₂×C₅₃	—	16	2	no	—
235	C₂×C₅₃	C₂×#496	1	1	yes	characteristic
236	C₂×C₅₃	#3153	1	1	yes	characteristic
237	C₂×C₅₃	#3154	1	1	yes	characteristic
238	C₂×AGL(1,ℤ/8)	—	2	2	no	—
239	#533	C₂²×C₅₃	1	1	yes	characteristic
240	#532	C₂²×C₅₃	1	1	yes	characteristic
241	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
242	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
243	#500	C₂²×C₅₃	1	2	yes	complemented
244	#498	C₂²×C₅₃	1	1	yes	characteristic, complemented
245	#496	—	2	2	no	—
246	#493	C₂²×C₅₃	1	2	yes	—
247	#492	C₂²×C₅₃	1	2	yes	—
248	#492	C₂²×C₅₃	1	2	yes	—
249	#492	C₂²×C₅₃	1	1	yes	characteristic, complemented
250	#491	—	2	2	no	—
251	#487	C₂²×C₅₃	1	2	yes	complemented
252	#487	C₂²×C₅₃	1	2	yes	complemented
253	#487	C₂²×C₅₃	1	1	yes	characteristic, complemented
254	#486	—	2	2	no	—
255	#486	C₂²×C₅₃	1	2	yes	—
256	#486	C₂²×C₅₃	1	2	yes	—
257	#486	C₂²×C₅₃	1	2	yes	—
258	#486	C₂²×C₅₃	1	1	yes	characteristic, complemented
259	#486	C₂²×C₅₃	1	2	yes	complemented
260	#480	—	2	2	no	—
261	#474	—	2	2	no	—
262	#465	—	2	2	no	—
263	#460	—	2	2	no	—
264	#450	—	2	2	no	—
265	C₄×SD₁₆	—	2	2	no	—
266	#431	C₂²×C₅₃	1	1	yes	characteristic
267	#419	—	2	2	no	—
268	C₂×#102	—	2	2	no	—
269	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
270	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
271	#322	C₂²×C₅₃	1	2	yes	complemented
272	#322	C₂²×C₅₃	1	2	yes	complemented
273	C₅₃	#19678	1	1	yes	characteristic, direct product factor, Sylow
274	C₂×#41	—	2	1	no	—
275	C₂²×Q₈	—	2	1	no	—
276	C₂²×D₄	—	2	1	no	—
277	C₂²×D₄	—	2	1	no	—
278	C₂²×D₄	—	2	1	no	—
279	AGL(1,ℤ/8)	—	4	8	no	—
280	C₂×SD₁₆	—	2	4	no	—
281	C₂×SD₁₆	—	2	2	no	—
282	C₂×SD₁₆	—	2	4	no	—
283	C₂×D₈	—	2	4	no	—
284	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
285	C₂²×C₈	D₄×C₅₃	1	2	yes	—
286	C₄⋊Q₈	—	2	1	no	—
287	C₄⋊Q₈	—	2	1	no	—
288	C₄⋊Q₈	—	2	1	no	—
289	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
290	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
291	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
292	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
293	#127	—	2	1	no	—
294	#127	—	2	1	no	—
295	#127	—	2	1	no	—
296	#127	C₂³×C₅₃	1	2	yes	—
297	#127	C₂³×C₅₃	1	1	yes	characteristic
298	#125	—	2	1	no	—
299	#124	—	2	1	no	—
300	C₂²⋊Q₈	—	2	1	no	—
301	C₂²⋊Q₈	—	2	1	no	—
302	#121	—	2	1	no	—
303	#121	—	2	1	no	—
304	C₄×Q₈	—	2	1	no	—
305	C₄×Q₈	—	2	1	no	—
306	C₄×D₄	—	2	1	no	—
307	C₄×D₄	—	2	1	no	—
308	#117	—	2	1	no	—
309	C₂×C₄⋊C₄	—	2	1	no	—
310	C₂×C₄⋊C₄	—	2	1	no	—
311	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
312	#107	—	2	4	no	—
313	#106	—	2	2	no	—
314	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
315	#103	—	2	4	no	—
316	#103	—	2	4	no	—
317	#103	—	2	2	no	—
318	#102	—	2	4	no	—
319	#102	—	2	4	no	—
320	#102	—	2	2	no	—
321	#102	—	2	4	no	—
322	#102	—	2	4	no	—
323	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
324	#97	D₄×C₅₃	1	2	yes	—
325	C₄×C₈	D₄×C₅₃	1	2	yes	—
326	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
327	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
328	C₄×C₈	C₂³×C₅₃	1	2	yes	—
329	C₄×C₈	C₂³×C₅₃	1	2	yes	—
330	C₂⁴	—	4	1	no	—
331	#41	—	4	4	no	—
332	C₂×Q₈	—	2	2	no	—
333	C₂×Q₈	—	2	1	no	—
334	C₂×Q₈	—	2	1	no	—
335	C₂×Q₈	—	2	1	no	—
336	C₂×Q₈	—	4	1	no	—
337	C₂×Q₈	—	4	4	no	—
338	C₂×D₄	—	2	1	no	—
339	C₂×D₄	—	2	2	no	—
340	C₂×D₄	—	2	1	no	—
341	C₂×D₄	—	2	2	no	—
342	C₂×D₄	—	2	1	no	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	—	4	4	no	—
345	C₂×D₄	—	4	1	no	—
346	C₂×D₄	—	4	1	no	—
347	C₂×D₄	—	4	1	no	—
348	C₂×D₄	—	4	1	no	—
349	C₂×D₄	—	4	4	no	—
350	C₂×D₄	—	8	2	no	—
351	C₂²×C₄	—	4	1	no	—
352	C₂²×C₄	—	4	1	no	—
353	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
354	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
355	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
356	SD₁₆	—	4	8	no	—
357	SD₁₆	—	4	4	no	—
358	SD₁₆	—	8	4	no	—
359	D₈	—	4	8	no	—
360	M₁₆	—	2	4	no	—
361	C₂×C₈	—	4	2	no	—
362	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
363	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
364	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
365	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	1	no	—
368	C₄⋊C₄	—	2	1	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	2	1	no	—
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	1	no	—
373	C₄⋊C₄	—	2	1	no	—
374	C₄⋊C₄	—	4	1	no	—
375	C₄⋊C₄	—	4	1	no	—
376	C₄⋊C₄	—	4	1	no	—
377	C₄⋊C₄	—	4	1	no	—
378	C₂²⋊C₄	—	4	1	no	—
379	C₂²⋊C₄	—	4	1	no	—
380	C₂²⋊C₄	—	4	1	no	—
381	C₄²	—	4	1	no	—
382	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
383	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
384	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
385	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
386	C₂³	—	4	1	no	—
387	C₂³	—	4	2	no	—
388	C₂³	—	8	2	no	—
389	C₂³	—	8	2	no	—
390	C₂³	—	8	2	no	—
391	C₂³	#127×C₅₃	1	1	yes	characteristic
392	Q₈	—	4	2	no	—
393	Q₈	—	4	4	no	—
394	Q₈	—	4	2	no	—
395	Q₈	—	4	2	no	—
396	Q₈	—	8	2	no	—
397	D₄	—	4	2	no	—
398	D₄	—	4	4	no	—
399	D₄	—	4	2	no	—
400	D₄	—	4	4	no	—
401	D₄	—	4	4	no	—
402	D₄	—	4	2	no	—
403	D₄	—	8	2	no	—
404	D₄	—	8	2	no	—
405	D₄	—	8	2	no	—
406	D₄	—	8	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	1	no	—
411	C₂×C₄	—	4	1	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	8	2	no	—
414	C₂×C₄	—	8	2	no	—
415	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
416	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
417	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
418	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
419	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
420	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
421	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
422	C₈	—	2	4	no	—
423	C₈	—	2	2	no	—
424	C₈	—	2	2	no	—
425	C₈	—	2	4	no	—
426	C₂²	—	2	1	no	—
427	C₂²	—	2	1	no	—
428	C₂²	—	8	1	no	—
429	C₂²	—	8	2	no	—
430	C₂²	—	8	2	no	—
431	C₂²	—	8	1	no	—
432	C₂²	—	8	2	no	—
433	C₂²	—	8	1	no	—
434	C₂²	—	16	4	no	—
435	C₂²	C₅₃×#496	1	2	yes	—
436	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
437	C₄	—	2	1	no	—
438	C₄	—	2	1	no	—
439	C₄	—	8	1	no	—
440	C₄	—	8	2	no	—
441	C₄	—	8	1	no	—
442	C₄	—	8	1	no	—
443	C₄	C₅₃×#496	1	2	yes	—
444	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
445	C₄	C₅₃×#575	1	2	yes	—
446	C₄	C₅₃×#577	1	2	yes	—
447	C₂	—	2	2	no	—
448	C₂	—	16	2	no	—
449	C₂	—	16	1	no	—
450	C₂	C₂×C₅₃×#496	1	1	yes	characteristic
451	C₂	C₅₃×#3153	1	1	yes	characteristic
452	C₂	C₅₃×#3154	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₅₃×#3151	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3226	C₂	semidirect
2. C₅₃×#3208	C₂	semidirect
3. C₅₃×#3167	C₂	semidirect
5. C₅₃×#3139	C₂	semidirect
6. C₂×C₅₃×#486	C₂	semidirect
7. C₅₃×#1719	C₂	semidirect
8. C₅₃×#1717	C₂	semidirect
9. C₅₃×#1717	C₂	semidirect
10. C₅₃×#1714	C₂	semidirect
11. C₅₃×#1451	C₂	semidirect
17. C₅₃×#500	C₂²	semidirect
18. C₅₃×#498	C₂²	semidirect
23. C₅₃×#492	C₂²	semidirect
25. C₅₃×#487	C₂²	semidirect
26. C₅₃×#487	C₂²	semidirect
27. C₅₃×#487	C₂²	semidirect
32. C₅₃×#486	C₂²	semidirect
33. C₅₃×#486	C₂²	semidirect
45. C₅₃×#322	C₂²	semidirect
46. C₅₃×#322	C₂²	semidirect
199. #19678	C₅₃	direct (N × H)
221. #3226	C₂×C₅₃	semidirect
222. #3208	C₂×C₅₃	semidirect
223. #3167	C₂×C₅₃	semidirect
225. #3139	C₂×C₅₃	semidirect
226. C₂×#486	C₂×C₅₃	semidirect
227. #1719	C₂×C₅₃	semidirect
228. #1717	C₂×C₅₃	semidirect
229. #1717	C₂×C₅₃	semidirect
230. #1714	C₂×C₅₃	semidirect
231. #1451	C₂×C₅₃	semidirect
243. #500	C₂²×C₅₃	semidirect
244. #498	C₂²×C₅₃	semidirect
249. #492	C₂²×C₅₃	semidirect
251. #487	C₂²×C₅₃	semidirect
252. #487	C₂²×C₅₃	semidirect
253. #487	C₂²×C₅₃	semidirect
258. #486	C₂²×C₅₃	semidirect
259. #486	C₂²×C₅₃	semidirect
271. #322	C₂²×C₅₃	semidirect
272. #322	C₂²×C₅₃	semidirect
273. C₅₃	#19678	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×#3226	2
C₅₃×#3208	2
C₅₃×#3167	2
C₅₃×#3151	2
C₅₃×#3139	2
C₂×C₅₃×#486	2
C₅₃×#1719	2
C₅₃×#1717	2
C₅₃×#1714	2
C₅₃×#1451	2
#19678	53

Subgroups — C53×#19678 #27227190

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19678 #27227190