Subgroups — C₅₃×#19236 #27226748

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3245	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#3227	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3141	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2935	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1703	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1697	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1683	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1677	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1565	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅₃×#333	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1540	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#131×C₅₃	—	2	1	no	—
13	C₅₃×#532	C₂²	1	1	yes	characteristic
14	C₅₃×#532	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#498	—	2	1	no	—
17	C₅₃×#489	—	2	2	no	—
18	C₅₃×#487	C₂²	1	2	yes	—
19	C₅₃×#486	—	2	1	no	—
20	C₅₃×#486	C₂²	1	2	yes	—
21	C₅₃×#481	—	2	2	no	—
22	C₅₃×#476	C₂²	1	2	yes	—
23	C₅₃×#476	C₂²	1	2	yes	—
24	C₅₃×#474	—	2	1	no	—
25	C₅₃×#474	C₂²	1	2	yes	—
26	C₅₃×#466	—	2	2	no	—
27	C₅₃×#459	—	2	1	no	—
28	C₅₃×#459	C₂²	1	2	yes	—
29	C₅₃×#436	C₂²	1	2	yes	—
30	C₅₃×#435	—	2	2	no	—
31	C₈×D₄×C₅₃	C₂²	1	2	yes	—
32	C₅₃×#431	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₅₃×#417	—	2	1	no	—
36	C₅₃×#417	—	2	1	no	—
37	C₅₃×#408	—	2	1	no	—
38	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
39	C₅₃×#335	C₂²	1	2	yes	—
40	C₅₃×#334	C₂²	1	2	yes	—
41	C₅₃×#333	C₂²	1	2	yes	—
42	C₅₃×#333	C₂²	1	2	yes	—
43	C₅₃×#332	C₂²	1	1	yes	characteristic
44	C₅₃×#332	C₂²	1	1	yes	characteristic
45	C₅₃×#331	C₂²	1	1	yes	characteristic
46	C₅₃×#331	C₂²	1	1	yes	characteristic
47	C₂×#41×C₅₃	—	2	1	no	—
48	C₂×#41×C₅₃	—	2	1	no	—
49	C₂²×Q₈×C₅₃	—	2	1	no	—
50	C₂×D₈×C₅₃	—	2	2	no	—
51	#131×C₅₃	—	4	4	no	—
52	C₂×M₁₆×C₅₃	—	2	2	no	—
53	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
54	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₅₃	—	2	2	no	—
56	C₂²×C₈×C₅₃	—	2	1	no	—
57	C₂²×C₈×C₅₃	—	2	1	no	—
58	C₄⋊Q₈×C₅₃	—	2	1	no	—
59	C₄⋊Q₈×C₅₃	—	2	1	no	—
60	C₄⋊Q₈×C₅₃	—	2	1	no	—
61	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
62	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
63	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
64	#127×C₅₃	D₄	1	1	yes	characteristic
65	#127×C₅₃	C₂³	1	1	yes	characteristic
66	#124×C₅₃	—	2	1	no	—
67	#124×C₅₃	—	2	1	no	—
68	#121×C₅₃	—	2	1	no	—
69	#121×C₅₃	—	2	1	no	—
70	#121×C₅₃	—	2	1	no	—
71	#121×C₅₃	D₄	1	2	yes	—
72	C₄×D₄×C₅₃	—	2	1	no	—
73	C₄×D₄×C₅₃	—	2	1	no	—
74	C₄×D₄×C₅₃	—	2	1	no	—
75	C₄×D₄×C₅₃	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
77	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
78	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
79	#107×C₅₃	—	2	2	no	—
80	#106×C₅₃	—	2	2	no	—
81	#106×C₅₃	—	2	2	no	—
82	C₄⋊C₈×C₅₃	—	2	1	no	—
83	C₄⋊C₈×C₅₃	—	2	1	no	—
84	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
85	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
88	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
89	#103×C₅₃	—	2	2	no	—
90	#103×C₅₃	—	2	2	no	—
91	#102×C₅₃	—	2	2	no	—
92	#102×C₅₃	—	2	2	no	—
93	#102×C₅₃	—	2	2	no	—
94	#102×C₅₃	—	2	2	no	—
95	#102×C₅₃	—	2	2	no	—
96	#102×C₅₃	—	2	2	no	—
97	C₂²⋊C₈×C₅₃	—	2	2	no	—
98	C₂²⋊C₈×C₅₃	—	2	2	no	—
99	C₂²⋊C₈×C₅₃	—	2	1	no	—
100	#97×C₅₃	D₄	1	2	yes	—
101	C₄×C₈×C₅₃	—	2	1	no	—
102	C₄×C₈×C₅₃	C₂³	1	2	yes	—
103	#41×C₅₃	—	4	2	no	—
104	#41×C₅₃	—	4	2	no	—
105	#41×C₅₃	—	4	2	no	—
106	#41×C₅₃	—	4	2	no	—
107	C₂×Q₈×C₅₃	—	2	1	no	—
108	C₂×Q₈×C₅₃	—	2	1	no	—
109	C₂×Q₈×C₅₃	—	2	2	no	—
110	C₂×Q₈×C₅₃	—	4	1	no	—
111	C₂×Q₈×C₅₃	—	8	2	no	—
112	C₂×D₄×C₅₃	—	2	2	no	—
113	C₂×D₄×C₅₃	—	2	1	no	—
114	C₂×D₄×C₅₃	—	2	1	no	—
115	C₂×D₄×C₅₃	—	2	1	no	—
116	C₂×D₄×C₅₃	—	2	1	no	—
117	C₂×D₄×C₅₃	—	2	1	no	—
118	C₂×D₄×C₅₃	—	4	1	no	—
119	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₅₃	—	2	2	no	—
121	C₂²×C₄×C₅₃	—	4	1	no	—
122	C₂²×C₄×C₅₃	—	4	1	no	—
123	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
126	D₈×C₅₃	—	8	2	no	—
127	M₁₆×C₅₃	—	2	2	no	—
128	M₁₆×C₅₃	—	2	2	no	—
129	M₁₆×C₅₃	—	4	4	no	—
130	C₂×C₈×C₅₃	—	2	1	no	—
131	C₂×C₈×C₅₃	—	2	1	no	—
132	C₂×C₈×C₅₃	—	2	1	no	—
133	C₂×C₈×C₅₃	—	2	1	no	—
134	C₂×C₈×C₅₃	—	2	2	no	—
135	C₂×C₈×C₅₃	—	4	4	no	—
136	C₂×C₈×C₅₃	—	4	2	no	—
137	C₂×C₈×C₅₃	—	8	1	no	—
138	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
139	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
140	C₄⋊C₄×C₅₃	—	2	1	no	—
141	C₄⋊C₄×C₅₃	—	2	2	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	—	4	1	no	—
146	C₄⋊C₄×C₅₃	—	4	1	no	—
147	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
148	C₂²⋊C₄×C₅₃	—	2	2	no	—
149	C₂²⋊C₄×C₅₃	—	2	2	no	—
150	C₂²⋊C₄×C₅₃	—	4	1	no	—
151	C₂²⋊C₄×C₅₃	—	4	1	no	—
152	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
153	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
154	C₄²×C₅₃	#41	1	1	yes	characteristic
155	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
156	C₂³×C₅₃	—	2	2	no	—
157	C₂³×C₅₃	—	4	1	no	—
158	C₂³×C₅₃	#121	1	1	yes	characteristic
159	Q₈×C₅₃	—	4	2	no	—
160	Q₈×C₅₃	—	4	1	no	—
161	Q₈×C₅₃	—	4	1	no	—
162	Q₈×C₅₃	—	8	2	no	—
163	Q₈×C₅₃	—	8	2	no	—
164	D₄×C₅₃	—	2	2	no	—
165	D₄×C₅₃	—	2	2	no	—
166	D₄×C₅₃	—	4	4	no	—
167	D₄×C₅₃	—	4	1	no	—
168	D₄×C₅₃	—	4	1	no	—
169	D₄×C₅₃	—	4	1	no	—
170	D₄×C₅₃	—	4	1	no	—
171	D₄×C₅₃	—	4	2	no	—
172	D₄×C₅₃	—	8	1	no	—
173	D₄×C₅₃	—	8	1	no	—
174	D₄×C₅₃	—	8	2	no	—
175	C₂×C₄×C₅₃	—	2	1	no	—
176	C₂×C₄×C₅₃	—	2	1	no	—
177	C₂×C₄×C₅₃	—	2	2	no	—
178	C₂×C₄×C₅₃	—	4	1	no	—
179	C₂×C₄×C₅₃	—	4	2	no	—
180	C₂×C₄×C₅₃	—	4	2	no	—
181	C₂×C₄×C₅₃	—	4	1	no	—
182	C₂×C₄×C₅₃	—	4	2	no	—
183	C₂×C₄×C₅₃	—	8	2	no	—
184	C₂×C₄×C₅₃	—	8	2	no	—
185	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
186	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
187	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
188	C₂×C₄×C₅₃	#135	1	2	yes	—
189	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
190	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
192	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
193	C₈×C₅₃	—	2	2	no	—
194	C₈×C₅₃	—	2	2	no	—
195	C₈×C₅₃	—	4	1	no	—
196	C₈×C₅₃	—	4	1	no	—
197	C₈×C₅₃	—	4	1	no	—
198	C₈×C₅₃	—	4	2	no	—
199	C₈×C₅₃	—	4	1	no	—
200	#19236	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
201	C₂²×C₅₃	—	2	1	no	—
202	C₂²×C₅₃	—	2	1	no	—
203	C₂²×C₅₃	—	4	2	no	—
204	C₂²×C₅₃	—	4	2	no	—
205	C₂²×C₅₃	—	4	2	no	—
206	C₂²×C₅₃	—	8	1	no	—
207	C₂²×C₅₃	—	8	1	no	—
208	C₂²×C₅₃	—	8	1	no	—
209	C₂²×C₅₃	#463	1	2	yes	—
210	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
211	C₄×C₅₃	—	2	1	no	—
212	C₄×C₅₃	—	2	1	no	—
213	C₄×C₅₃	—	2	1	no	—
214	C₄×C₅₃	—	4	2	no	—
215	C₄×C₅₃	—	8	1	no	—
216	C₄×C₅₃	—	8	2	no	—
217	C₄×C₅₃	#463	1	2	yes	—
218	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
219	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
220	C₄×C₅₃	#577	1	1	yes	characteristic
221	C₄×C₅₃	#578	1	1	yes	characteristic
222	#3245	C₂×C₅₃	1	1	yes	characteristic
223	#3227	C₂×C₅₃	1	1	yes	characteristic, complemented
224	#3141	C₂×C₅₃	1	1	yes	characteristic, complemented
225	#2935	C₂×C₅₃	1	1	yes	characteristic, complemented
226	#1703	C₂×C₅₃	1	2	yes	complemented
227	#1697	C₂×C₅₃	1	2	yes	complemented
228	#1683	C₂×C₅₃	1	2	yes	complemented
229	#1677	C₂×C₅₃	1	2	yes	complemented
230	#1565	C₂×C₅₃	1	1	yes	characteristic, complemented
231	C₂×#333	C₂×C₅₃	1	1	yes	characteristic, complemented
232	#1540	C₂×C₅₃	1	1	yes	characteristic, complemented
233	C₂×C₅₃	—	2	2	no	—
234	C₂×C₅₃	—	8	2	no	—
235	C₂×C₅₃	—	16	1	no	—
236	C₂×C₅₃	C₂×#463	1	1	yes	characteristic
237	C₂×C₅₃	#3063	1	1	yes	characteristic
238	C₂×C₅₃	#3068	1	1	yes	characteristic
239	C₂×#131	—	2	1	no	—
240	#532	C₂²×C₅₃	1	1	yes	characteristic
241	#532	C₂²×C₅₃	1	1	yes	characteristic
242	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
243	#498	—	2	1	no	—
244	#489	—	2	2	no	—
245	#487	C₂²×C₅₃	1	2	yes	—
246	#486	—	2	1	no	—
247	#486	C₂²×C₅₃	1	2	yes	—
248	#481	—	2	2	no	—
249	#476	C₂²×C₅₃	1	2	yes	—
250	#476	C₂²×C₅₃	1	2	yes	—
251	#474	—	2	1	no	—
252	#474	C₂²×C₅₃	1	2	yes	—
253	#466	—	2	2	no	—
254	#459	—	2	1	no	—
255	#459	C₂²×C₅₃	1	2	yes	—
256	#436	C₂²×C₅₃	1	2	yes	—
257	#435	—	2	2	no	—
258	C₈×D₄	C₂²×C₅₃	1	2	yes	—
259	#431	C₂²×C₅₃	1	1	yes	characteristic
260	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
261	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
262	#417	—	2	1	no	—
263	#417	—	2	1	no	—
264	#408	—	2	1	no	—
265	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
266	#335	C₂²×C₅₃	1	2	yes	—
267	#334	C₂²×C₅₃	1	2	yes	—
268	#333	C₂²×C₅₃	1	2	yes	—
269	#333	C₂²×C₅₃	1	2	yes	—
270	#332	C₂²×C₅₃	1	1	yes	characteristic
271	#332	C₂²×C₅₃	1	1	yes	characteristic
272	#331	C₂²×C₅₃	1	1	yes	characteristic
273	#331	C₂²×C₅₃	1	1	yes	characteristic
274	C₅₃	#19236	1	1	yes	characteristic, direct product factor, Sylow
275	C₂×#41	—	2	1	no	—
276	C₂×#41	—	2	1	no	—
277	C₂²×Q₈	—	2	1	no	—
278	C₂×D₈	—	2	2	no	—
279	#131	—	4	4	no	—
280	C₂×M₁₆	—	2	2	no	—
281	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
282	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
283	C₂²×C₈	—	2	1	no	—
284	C₂²×C₈	—	2	1	no	—
285	C₂²×C₈	—	2	2	no	—
286	C₄⋊Q₈	—	2	1	no	—
287	C₄⋊Q₈	—	2	1	no	—
288	C₄⋊Q₈	—	2	1	no	—
289	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
290	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
291	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
292	#127	D₄×C₅₃	1	1	yes	characteristic
293	#127	C₂³×C₅₃	1	1	yes	characteristic
294	#124	—	2	1	no	—
295	#124	—	2	1	no	—
296	#121	—	2	1	no	—
297	#121	—	2	1	no	—
298	#121	—	2	1	no	—
299	#121	D₄×C₅₃	1	2	yes	—
300	C₄×D₄	—	2	1	no	—
301	C₄×D₄	—	2	1	no	—
302	C₄×D₄	—	2	1	no	—
303	C₄×D₄	C₂³×C₅₃	1	2	yes	—
304	C₂×C₄⋊C₄	—	2	1	no	—
305	C₂×C₄⋊C₄	—	2	1	no	—
306	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
307	#107	—	2	2	no	—
308	#106	—	2	2	no	—
309	#106	—	2	2	no	—
310	C₄⋊C₈	—	2	1	no	—
311	C₄⋊C₈	—	2	1	no	—
312	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
313	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
314	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
315	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
316	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
317	#103	—	2	2	no	—
318	#103	—	2	2	no	—
319	#102	—	2	2	no	—
320	#102	—	2	2	no	—
321	#102	—	2	2	no	—
322	#102	—	2	2	no	—
323	#102	—	2	2	no	—
324	#102	—	2	2	no	—
325	C₂²⋊C₈	—	2	1	no	—
326	C₂²⋊C₈	—	2	2	no	—
327	C₂²⋊C₈	—	2	2	no	—
328	#97	D₄×C₅₃	1	2	yes	—
329	C₄×C₈	—	2	1	no	—
330	C₄×C₈	C₂³×C₅₃	1	2	yes	—
331	#41	—	4	2	no	—
332	#41	—	4	2	no	—
333	#41	—	4	2	no	—
334	#41	—	4	2	no	—
335	C₂×Q₈	—	2	1	no	—
336	C₂×Q₈	—	2	1	no	—
337	C₂×Q₈	—	2	2	no	—
338	C₂×Q₈	—	4	1	no	—
339	C₂×Q₈	—	8	2	no	—
340	C₂×D₄	—	2	2	no	—
341	C₂×D₄	—	2	1	no	—
342	C₂×D₄	—	2	1	no	—
343	C₂×D₄	—	2	1	no	—
344	C₂×D₄	—	2	1	no	—
345	C₂×D₄	—	2	1	no	—
346	C₂×D₄	—	4	1	no	—
347	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	4	1	no	—
350	C₂²×C₄	—	4	1	no	—
351	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
352	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
353	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
354	D₈	—	8	2	no	—
355	M₁₆	—	2	2	no	—
356	M₁₆	—	2	2	no	—
357	M₁₆	—	4	4	no	—
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	2	1	no	—
361	C₂×C₈	—	2	1	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	4	4	no	—
364	C₂×C₈	—	4	2	no	—
365	C₂×C₈	—	8	1	no	—
366	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
367	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
368	C₄⋊C₄	—	2	1	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	2	2	no	—
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	1	no	—
373	C₄⋊C₄	—	4	1	no	—
374	C₄⋊C₄	—	4	1	no	—
375	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
376	C₂²⋊C₄	—	2	2	no	—
377	C₂²⋊C₄	—	2	2	no	—
378	C₂²⋊C₄	—	4	1	no	—
379	C₂²⋊C₄	—	4	1	no	—
380	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
381	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
382	C₄²	#41×C₅₃	1	1	yes	characteristic
383	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
384	C₂³	—	2	2	no	—
385	C₂³	—	4	1	no	—
386	C₂³	#121×C₅₃	1	1	yes	characteristic
387	Q₈	—	4	2	no	—
388	Q₈	—	4	1	no	—
389	Q₈	—	4	1	no	—
390	Q₈	—	8	2	no	—
391	Q₈	—	8	2	no	—
392	D₄	—	2	2	no	—
393	D₄	—	2	2	no	—
394	D₄	—	4	4	no	—
395	D₄	—	4	1	no	—
396	D₄	—	4	1	no	—
397	D₄	—	4	1	no	—
398	D₄	—	4	1	no	—
399	D₄	—	4	2	no	—
400	D₄	—	8	1	no	—
401	D₄	—	8	1	no	—
402	D₄	—	8	2	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	8	2	no	—
412	C₂×C₄	—	8	2	no	—
413	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
414	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
415	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
416	C₂×C₄	#135×C₅₃	1	2	yes	—
417	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
418	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
419	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
420	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
421	C₈	—	2	2	no	—
422	C₈	—	2	2	no	—
423	C₈	—	4	1	no	—
424	C₈	—	4	1	no	—
425	C₈	—	4	1	no	—
426	C₈	—	4	2	no	—
427	C₈	—	4	1	no	—
428	C₂²	—	2	1	no	—
429	C₂²	—	2	1	no	—
430	C₂²	—	4	2	no	—
431	C₂²	—	4	2	no	—
432	C₂²	—	4	2	no	—
433	C₂²	—	8	1	no	—
434	C₂²	—	8	1	no	—
435	C₂²	—	8	1	no	—
436	C₂²	C₅₃×#463	1	2	yes	—
437	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
438	C₄	—	2	1	no	—
439	C₄	—	2	1	no	—
440	C₄	—	2	1	no	—
441	C₄	—	4	2	no	—
442	C₄	—	8	1	no	—
443	C₄	—	8	2	no	—
444	C₄	C₅₃×#463	1	2	yes	—
445	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
446	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
447	C₄	C₅₃×#577	1	1	yes	characteristic
448	C₄	C₅₃×#578	1	1	yes	characteristic
449	C₂	—	2	2	no	—
450	C₂	—	8	2	no	—
451	C₂	—	16	1	no	—
452	C₂	C₂×C₅₃×#463	1	1	yes	characteristic
453	C₂	C₅₃×#3063	1	1	yes	characteristic
454	C₂	C₅₃×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₅₃×#3245	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
2. C₅₃×#3227	C₂	semidirect
3. C₅₃×#3141	C₂	semidirect
4. C₅₃×#2935	C₂	semidirect
5. C₅₃×#1703	C₂	semidirect
6. C₅₃×#1697	C₂	semidirect
7. C₅₃×#1683	C₂	semidirect
8. C₅₃×#1677	C₂	semidirect
9. C₅₃×#1565	C₂	semidirect
10. C₂×C₅₃×#333	C₂	semidirect
11. C₅₃×#1540	C₂	semidirect
200. #19236	C₅₃	direct (N × H)
223. #3227	C₂×C₅₃	semidirect
224. #3141	C₂×C₅₃	semidirect
225. #2935	C₂×C₅₃	semidirect
226. #1703	C₂×C₅₃	semidirect
227. #1697	C₂×C₅₃	semidirect
228. #1683	C₂×C₅₃	semidirect
229. #1677	C₂×C₅₃	semidirect
230. #1565	C₂×C₅₃	semidirect
231. C₂×#333	C₂×C₅₃	semidirect
232. #1540	C₂×C₅₃	semidirect
274. C₅₃	#19236	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×#3245	2
C₅₃×#3227	2
C₅₃×#3141	2
C₅₃×#2935	2
C₅₃×#1703	2
C₅₃×#1697	2
C₅₃×#1683	2
C₅₃×#1677	2
C₅₃×#1565	2
C₂×C₅₃×#333	2
C₅₃×#1540	2
#19236	53

Subgroups — C53×#19236 #27226748

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19236 #27226748