Subgroups — C₅₃×#19231 #27226743

Order 13568.

726 conjugacy classes of subgroups.

The subgroup lattice has 472 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×#3226	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3152	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2935	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1689	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1685	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1669	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1665	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1565	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1540	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137×C₅₃	—	2	1	no	—
13	C₂×#137×C₅₃	—	2	1	no	—
14	C₂×#131×C₅₃	—	2	1	no	—
15	C₅₃×#532	C₂²	1	1	yes	characteristic
16	C₅₃×#532	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#500	—	2	1	no	—
19	C₅₃×#497	—	2	2	no	—
20	C₅₃×#494	C₂²	1	2	yes	—
21	C₅₃×#492	C₂²	1	2	yes	—
22	C₅₃×#486	—	2	1	no	—
23	C₅₃×#480	—	2	2	no	—
24	C₅₃×#474	—	2	1	no	—
25	C₅₃×#474	C₂²	1	2	yes	—
26	C₅₃×#474	C₂²	1	2	yes	—
27	C₅₃×#474	C₂²	1	2	yes	—
28	C₅₃×#466	—	2	2	no	—
29	C₅₃×#459	—	2	1	no	—
30	C₅₃×#459	C₂²	1	2	yes	—
31	C₅₃×#436	C₂²	1	2	yes	—
32	C₅₃×#435	—	2	2	no	—
33	C₈×D₄×C₅₃	C₂²	1	2	yes	—
34	C₅₃×#431	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
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×#332	C₂²	1	2	yes	—
42	C₅₃×#332	C₂²	1	1	yes	characteristic
43	C₅₃×#332	C₂²	1	2	yes	—
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	#137×C₅₃	—	4	4	no	—
51	#137×C₅₃	—	4	4	no	—
52	C₂×Q₁₆×C₅₃	—	2	2	no	—
53	C₂×Q₁₆×C₅₃	—	2	2	no	—
54	C₂×SD₁₆×C₅₃	—	2	2	no	—
55	C₂×SD₁₆×C₅₃	—	2	2	no	—
56	C₂×D₈×C₅₃	—	2	2	no	—
57	#131×C₅₃	—	4	4	no	—
58	C₂×M₁₆×C₅₃	—	2	2	no	—
59	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
60	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₅₃	—	2	2	no	—
62	C₂²×C₈×C₅₃	—	2	1	no	—
63	C₂²×C₈×C₅₃	—	2	1	no	—
64	C₄⋊Q₈×C₅₃	—	2	1	no	—
65	C₄⋊Q₈×C₅₃	—	2	1	no	—
66	C₄⋊Q₈×C₅₃	—	2	1	no	—
67	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
68	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
70	#127×C₅₃	D₄	1	1	yes	characteristic
71	#127×C₅₃	C₂³	1	1	yes	characteristic
72	#124×C₅₃	—	2	1	no	—
73	#124×C₅₃	—	2	1	no	—
74	#121×C₅₃	—	2	1	no	—
75	#121×C₅₃	—	2	1	no	—
76	#121×C₅₃	—	2	1	no	—
77	#121×C₅₃	D₄	1	2	yes	—
78	C₄×D₄×C₅₃	—	2	1	no	—
79	C₄×D₄×C₅₃	—	2	1	no	—
80	C₄×D₄×C₅₃	—	2	1	no	—
81	C₄×D₄×C₅₃	C₂³	1	2	yes	—
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	2	no	—
86	#107×C₅₃	—	2	2	no	—
87	#107×C₅₃	—	2	2	no	—
88	C₄⋊C₈×C₅₃	—	2	1	no	—
89	C₄⋊C₈×C₅₃	—	2	1	no	—
90	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
92	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
93	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
95	#102×C₅₃	—	2	2	no	—
96	#102×C₅₃	—	2	2	no	—
97	#102×C₅₃	—	2	2	no	—
98	#102×C₅₃	—	2	2	no	—
99	C₂²⋊C₈×C₅₃	—	2	2	no	—
100	C₂²⋊C₈×C₅₃	—	2	2	no	—
101	C₂²⋊C₈×C₅₃	—	2	1	no	—
102	#97×C₅₃	D₄	1	2	yes	—
103	C₄×C₈×C₅₃	—	2	1	no	—
104	C₄×C₈×C₅₃	C₂³	1	2	yes	—
105	#41×C₅₃	—	4	2	no	—
106	#41×C₅₃	—	4	2	no	—
107	#41×C₅₃	—	4	2	no	—
108	#41×C₅₃	—	4	2	no	—
109	C₂×Q₈×C₅₃	—	2	1	no	—
110	C₂×Q₈×C₅₃	—	2	1	no	—
111	C₂×Q₈×C₅₃	—	2	2	no	—
112	C₂×Q₈×C₅₃	—	4	1	no	—
113	C₂×Q₈×C₅₃	—	4	2	no	—
114	C₂×Q₈×C₅₃	—	4	2	no	—
115	C₂×D₄×C₅₃	—	2	2	no	—
116	C₂×D₄×C₅₃	—	2	1	no	—
117	C₂×D₄×C₅₃	—	2	1	no	—
118	C₂×D₄×C₅₃	—	2	1	no	—
119	C₂×D₄×C₅₃	—	2	1	no	—
120	C₂×D₄×C₅₃	—	2	1	no	—
121	C₂×D₄×C₅₃	—	4	1	no	—
122	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
123	C₂²×C₄×C₅₃	—	2	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	#41	1	1	yes	characteristic
129	Q₁₆×C₅₃	—	4	4	no	—
130	Q₁₆×C₅₃	—	4	4	no	—
131	SD₁₆×C₅₃	—	4	4	no	—
132	SD₁₆×C₅₃	—	4	4	no	—
133	D₈×C₅₃	—	8	2	no	—
134	M₁₆×C₅₃	—	2	2	no	—
135	M₁₆×C₅₃	—	4	4	no	—
136	M₁₆×C₅₃	D₈	1	4	yes	complemented
137	C₂×C₈×C₅₃	—	2	1	no	—
138	C₂×C₈×C₅₃	—	2	1	no	—
139	C₂×C₈×C₅₃	—	2	1	no	—
140	C₂×C₈×C₅₃	—	2	1	no	—
141	C₂×C₈×C₅₃	—	2	2	no	—
142	C₂×C₈×C₅₃	—	4	2	no	—
143	C₂×C₈×C₅₃	—	4	4	no	—
144	C₂×C₈×C₅₃	—	8	1	no	—
145	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
146	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
147	C₄⋊C₄×C₅₃	—	2	1	no	—
148	C₄⋊C₄×C₅₃	—	2	2	no	—
149	C₄⋊C₄×C₅₃	—	2	1	no	—
150	C₄⋊C₄×C₅₃	—	2	2	no	—
151	C₄⋊C₄×C₅₃	—	2	1	no	—
152	C₄⋊C₄×C₅₃	—	4	1	no	—
153	C₄⋊C₄×C₅₃	—	4	1	no	—
154	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
155	C₂²⋊C₄×C₅₃	—	2	2	no	—
156	C₂²⋊C₄×C₅₃	—	2	2	no	—
157	C₂²⋊C₄×C₅₃	—	4	1	no	—
158	C₂²⋊C₄×C₅₃	—	4	1	no	—
159	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
160	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
161	C₄²×C₅₃	#41	1	1	yes	characteristic
162	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
163	C₂³×C₅₃	—	2	2	no	—
164	C₂³×C₅₃	—	4	1	no	—
165	C₂³×C₅₃	#121	1	1	yes	characteristic
166	Q₈×C₅₃	—	4	2	no	—
167	Q₈×C₅₃	—	4	2	no	—
168	Q₈×C₅₃	—	4	1	no	—
169	Q₈×C₅₃	—	4	2	no	—
170	Q₈×C₅₃	—	4	1	no	—
171	Q₈×C₅₃	—	8	2	no	—
172	D₄×C₅₃	—	2	2	no	—
173	D₄×C₅₃	—	2	2	no	—
174	D₄×C₅₃	—	4	4	no	—
175	D₄×C₅₃	—	4	1	no	—
176	D₄×C₅₃	—	4	1	no	—
177	D₄×C₅₃	—	4	1	no	—
178	D₄×C₅₃	—	4	1	no	—
179	D₄×C₅₃	—	4	2	no	—
180	D₄×C₅₃	—	8	1	no	—
181	D₄×C₅₃	—	8	1	no	—
182	D₄×C₅₃	—	8	2	no	—
183	C₂×C₄×C₅₃	—	2	1	no	—
184	C₂×C₄×C₅₃	—	2	1	no	—
185	C₂×C₄×C₅₃	—	2	2	no	—
186	C₂×C₄×C₅₃	—	4	1	no	—
187	C₂×C₄×C₅₃	—	4	2	no	—
188	C₂×C₄×C₅₃	—	4	2	no	—
189	C₂×C₄×C₅₃	—	4	1	no	—
190	C₂×C₄×C₅₃	—	4	2	no	—
191	C₂×C₄×C₅₃	—	8	2	no	—
192	C₂×C₄×C₅₃	—	8	2	no	—
193	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
194	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
195	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
196	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
197	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
198	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
200	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
201	C₈×C₅₃	—	2	2	no	—
202	C₈×C₅₃	—	4	1	no	—
203	C₈×C₅₃	—	4	1	no	—
204	C₈×C₅₃	—	4	1	no	—
205	C₈×C₅₃	—	4	2	no	—
206	C₈×C₅₃	—	4	1	no	—
207	C₈×C₅₃	C₂×D₈	1	4	yes	—
208	#19231	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
209	C₂²×C₅₃	—	2	1	no	—
210	C₂²×C₅₃	—	2	1	no	—
211	C₂²×C₅₃	—	4	2	no	—
212	C₂²×C₅₃	—	4	2	no	—
213	C₂²×C₅₃	—	4	2	no	—
214	C₂²×C₅₃	—	8	1	no	—
215	C₂²×C₅₃	—	8	1	no	—
216	C₂²×C₅₃	—	8	1	no	—
217	C₂²×C₅₃	#459	1	2	yes	—
218	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
219	C₄×C₅₃	—	2	1	no	—
220	C₄×C₅₃	—	2	1	no	—
221	C₄×C₅₃	—	2	1	no	—
222	C₄×C₅₃	—	4	2	no	—
223	C₄×C₅₃	—	8	1	no	—
224	C₄×C₅₃	—	8	2	no	—
225	C₄×C₅₃	#459	1	2	yes	—
226	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
227	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
228	C₄×C₅₃	#577	1	1	yes	characteristic
229	C₄×C₅₃	#578	1	1	yes	characteristic
230	#3245	C₂×C₅₃	1	1	yes	characteristic
231	#3226	C₂×C₅₃	1	1	yes	characteristic, complemented
232	#3152	C₂×C₅₃	1	1	yes	characteristic, complemented
233	#2935	C₂×C₅₃	1	1	yes	characteristic, complemented
234	#1689	C₂×C₅₃	1	2	yes	complemented
235	#1685	C₂×C₅₃	1	2	yes	complemented
236	#1669	C₂×C₅₃	1	2	yes	complemented
237	#1665	C₂×C₅₃	1	2	yes	complemented
238	#1565	C₂×C₅₃	1	1	yes	characteristic, complemented
239	#1540	C₂×C₅₃	1	1	yes	characteristic, complemented
240	C₂×#332	C₂×C₅₃	1	1	yes	characteristic, complemented
241	C₂×C₅₃	—	2	2	no	—
242	C₂×C₅₃	—	8	2	no	—
243	C₂×C₅₃	—	16	1	no	—
244	C₂×C₅₃	C₂×#459	1	1	yes	characteristic
245	C₂×C₅₃	#3063	1	1	yes	characteristic
246	C₂×C₅₃	#3069	1	1	yes	characteristic
247	C₂×#137	—	2	1	no	—
248	C₂×#137	—	2	1	no	—
249	C₂×#131	—	2	1	no	—
250	#532	C₂²×C₅₃	1	1	yes	characteristic
251	#532	C₂²×C₅₃	1	1	yes	characteristic
252	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
253	#500	—	2	1	no	—
254	#497	—	2	2	no	—
255	#494	C₂²×C₅₃	1	2	yes	—
256	#492	C₂²×C₅₃	1	2	yes	—
257	#486	—	2	1	no	—
258	#480	—	2	2	no	—
259	#474	—	2	1	no	—
260	#474	C₂²×C₅₃	1	2	yes	—
261	#474	C₂²×C₅₃	1	2	yes	—
262	#474	C₂²×C₅₃	1	2	yes	—
263	#466	—	2	2	no	—
264	#459	—	2	1	no	—
265	#459	C₂²×C₅₃	1	2	yes	—
266	#436	C₂²×C₅₃	1	2	yes	—
267	#435	—	2	2	no	—
268	C₈×D₄	C₂²×C₅₃	1	2	yes	—
269	#431	C₂²×C₅₃	1	1	yes	characteristic
270	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
271	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
272	#408	—	2	1	no	—
273	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
274	#335	C₂²×C₅₃	1	2	yes	—
275	#334	C₂²×C₅₃	1	2	yes	—
276	#332	C₂²×C₅₃	1	2	yes	—
277	#332	C₂²×C₅₃	1	2	yes	—
278	#332	C₂²×C₅₃	1	1	yes	characteristic
279	#332	C₂²×C₅₃	1	1	yes	characteristic
280	#331	C₂²×C₅₃	1	1	yes	characteristic
281	#331	C₂²×C₅₃	1	1	yes	characteristic
282	C₅₃	#19231	1	1	yes	characteristic, direct product factor, Sylow
283	C₂×#41	—	2	1	no	—
284	C₂×#41	—	2	1	no	—
285	C₂²×Q₈	—	2	1	no	—
286	#137	—	4	4	no	—
287	#137	—	4	4	no	—
288	C₂×Q₁₆	—	2	2	no	—
289	C₂×Q₁₆	—	2	2	no	—
290	C₂×SD₁₆	—	2	2	no	—
291	C₂×SD₁₆	—	2	2	no	—
292	C₂×D₈	—	2	2	no	—
293	#131	—	4	4	no	—
294	C₂×M₁₆	—	2	2	no	—
295	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
296	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
297	C₂²×C₈	—	2	1	no	—
298	C₂²×C₈	—	2	1	no	—
299	C₂²×C₈	—	2	2	no	—
300	C₄⋊Q₈	—	2	1	no	—
301	C₄⋊Q₈	—	2	1	no	—
302	C₄⋊Q₈	—	2	1	no	—
303	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
304	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
305	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
306	#127	D₄×C₅₃	1	1	yes	characteristic
307	#127	C₂³×C₅₃	1	1	yes	characteristic
308	#124	—	2	1	no	—
309	#124	—	2	1	no	—
310	#121	—	2	1	no	—
311	#121	—	2	1	no	—
312	#121	—	2	1	no	—
313	#121	D₄×C₅₃	1	2	yes	—
314	C₄×D₄	—	2	1	no	—
315	C₄×D₄	—	2	1	no	—
316	C₄×D₄	—	2	1	no	—
317	C₄×D₄	C₂³×C₅₃	1	2	yes	—
318	C₂×C₄⋊C₄	—	2	1	no	—
319	C₂×C₄⋊C₄	—	2	1	no	—
320	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
321	#107	—	2	2	no	—
322	#107	—	2	2	no	—
323	#107	—	2	2	no	—
324	C₄⋊C₈	—	2	1	no	—
325	C₄⋊C₈	—	2	1	no	—
326	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
327	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
328	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
329	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
330	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
331	#102	—	2	2	no	—
332	#102	—	2	2	no	—
333	#102	—	2	2	no	—
334	#102	—	2	2	no	—
335	C₂²⋊C₈	—	2	1	no	—
336	C₂²⋊C₈	—	2	2	no	—
337	C₂²⋊C₈	—	2	2	no	—
338	#97	D₄×C₅₃	1	2	yes	—
339	C₄×C₈	—	2	1	no	—
340	C₄×C₈	C₂³×C₅₃	1	2	yes	—
341	#41	—	4	2	no	—
342	#41	—	4	2	no	—
343	#41	—	4	2	no	—
344	#41	—	4	2	no	—
345	C₂×Q₈	—	2	1	no	—
346	C₂×Q₈	—	2	1	no	—
347	C₂×Q₈	—	2	2	no	—
348	C₂×Q₈	—	4	1	no	—
349	C₂×Q₈	—	4	2	no	—
350	C₂×Q₈	—	4	2	no	—
351	C₂×D₄	—	2	2	no	—
352	C₂×D₄	—	2	1	no	—
353	C₂×D₄	—	2	1	no	—
354	C₂×D₄	—	2	1	no	—
355	C₂×D₄	—	2	1	no	—
356	C₂×D₄	—	2	1	no	—
357	C₂×D₄	—	4	1	no	—
358	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	—	4	1	no	—
361	C₂²×C₄	—	4	1	no	—
362	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
363	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
364	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
365	Q₁₆	—	4	4	no	—
366	Q₁₆	—	4	4	no	—
367	SD₁₆	—	4	4	no	—
368	SD₁₆	—	4	4	no	—
369	D₈	—	8	2	no	—
370	M₁₆	—	2	2	no	—
371	M₁₆	—	4	4	no	—
372	M₁₆	D₈×C₅₃	1	4	yes	complemented
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	1	no	—
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	4	2	no	—
379	C₂×C₈	—	4	4	no	—
380	C₂×C₈	—	8	1	no	—
381	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
382	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
383	C₄⋊C₄	—	2	1	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	2	1	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	1	no	—
388	C₄⋊C₄	—	4	1	no	—
389	C₄⋊C₄	—	4	1	no	—
390	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
391	C₂²⋊C₄	—	2	2	no	—
392	C₂²⋊C₄	—	2	2	no	—
393	C₂²⋊C₄	—	4	1	no	—
394	C₂²⋊C₄	—	4	1	no	—
395	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
396	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
397	C₄²	#41×C₅₃	1	1	yes	characteristic
398	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
399	C₂³	—	2	2	no	—
400	C₂³	—	4	1	no	—
401	C₂³	#121×C₅₃	1	1	yes	characteristic
402	Q₈	—	4	2	no	—
403	Q₈	—	4	2	no	—
404	Q₈	—	4	1	no	—
405	Q₈	—	4	2	no	—
406	Q₈	—	4	1	no	—
407	Q₈	—	8	2	no	—
408	D₄	—	2	2	no	—
409	D₄	—	2	2	no	—
410	D₄	—	4	4	no	—
411	D₄	—	4	1	no	—
412	D₄	—	4	1	no	—
413	D₄	—	4	1	no	—
414	D₄	—	4	1	no	—
415	D₄	—	4	2	no	—
416	D₄	—	8	1	no	—
417	D₄	—	8	1	no	—
418	D₄	—	8	2	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	4	1	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	8	2	no	—
428	C₂×C₄	—	8	2	no	—
429	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
430	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
431	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
432	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
433	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
434	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
435	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
436	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
437	C₈	—	2	2	no	—
438	C₈	—	4	1	no	—
439	C₈	—	4	1	no	—
440	C₈	—	4	1	no	—
441	C₈	—	4	2	no	—
442	C₈	—	4	1	no	—
443	C₈	C₂×D₈×C₅₃	1	4	yes	—
444	C₂²	—	2	1	no	—
445	C₂²	—	2	1	no	—
446	C₂²	—	4	2	no	—
447	C₂²	—	4	2	no	—
448	C₂²	—	4	2	no	—
449	C₂²	—	8	1	no	—
450	C₂²	—	8	1	no	—
451	C₂²	—	8	1	no	—
452	C₂²	C₅₃×#459	1	2	yes	—
453	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
454	C₄	—	2	1	no	—
455	C₄	—	2	1	no	—
456	C₄	—	2	1	no	—
457	C₄	—	4	2	no	—
458	C₄	—	8	1	no	—
459	C₄	—	8	2	no	—
460	C₄	C₅₃×#459	1	2	yes	—
461	C₄	C₂²×D₈×C₅₃	1	1	yes	characteristic
462	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
463	C₄	C₅₃×#577	1	1	yes	characteristic
464	C₄	C₅₃×#578	1	1	yes	characteristic
465	C₂	—	2	2	no	—
466	C₂	—	8	2	no	—
467	C₂	—	16	1	no	—
468	C₂	C₂×C₅₃×#459	1	1	yes	characteristic
469	C₂	C₅₃×#3063	1	1	yes	characteristic
470	C₂	C₅₃×#3069	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×#3226	C₂	semidirect
3. C₅₃×#3152	C₂	semidirect
4. C₅₃×#2935	C₂	semidirect
5. C₅₃×#1689	C₂	semidirect
6. C₅₃×#1685	C₂	semidirect
7. C₅₃×#1669	C₂	semidirect
8. C₅₃×#1665	C₂	semidirect
9. C₅₃×#1565	C₂	semidirect
10. C₅₃×#1540	C₂	semidirect
11. C₂×C₅₃×#332	C₂	semidirect
136. M₁₆×C₅₃	D₈	semidirect
208. #19231	C₅₃	direct (N × H)
231. #3226	C₂×C₅₃	semidirect
232. #3152	C₂×C₅₃	semidirect
233. #2935	C₂×C₅₃	semidirect
234. #1689	C₂×C₅₃	semidirect
235. #1685	C₂×C₅₃	semidirect
236. #1669	C₂×C₅₃	semidirect
237. #1665	C₂×C₅₃	semidirect
238. #1565	C₂×C₅₃	semidirect
239. #1540	C₂×C₅₃	semidirect
240. C₂×#332	C₂×C₅₃	semidirect
282. C₅₃	#19231	direct (N × H)
372. M₁₆	D₈×C₅₃	semidirect

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×#3226	2
C₅₃×#3152	2
C₅₃×#2935	2
C₅₃×#1689	2
C₅₃×#1685	2
C₅₃×#1669	2
C₅₃×#1665	2
C₅₃×#1565	2
C₅₃×#1540	2
C₂×C₅₃×#332	2
#19231	53

Subgroups — C53×#19231 #27226743

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19231 #27226743