Subgroups — C₅₃×#17871 #27225383

Order 13568.

738 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×AGL(1,ℤ/8)×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1588	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1586	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1583	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1581	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1565	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1478	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1477	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₅₃	C₄	1	1	yes	characteristic
13	C₂²×M₁₆×C₅₃	—	2	1	no	—
14	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
17	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
18	C₅₃×#442	C₄	1	2	yes	—
19	C₅₃×#440	C₄	1	2	yes	—
20	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	—
21	C₄×D₈×C₅₃	C₂²	1	2	yes	—
22	C₅₃×#436	—	2	1	no	—
23	C₅₃×#436	—	2	1	no	—
24	C₅₃×#436	C₂²	1	2	yes	—
25	C₅₃×#436	C₂²	1	2	yes	—
26	C₈×D₄×C₅₃	—	2	1	no	—
27	C₈×D₄×C₅₃	—	2	1	no	—
28	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
29	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
30	C₅₃×#431	—	2	1	no	—
31	C₅₃×#431	C₂²	1	1	yes	characteristic
32	C₅₃×#431	C₂²	1	1	yes	characteristic
33	C₅₃×#428	C₄	1	1	yes	characteristic
34	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₅₃×#418	C₄	1	1	yes	characteristic
36	C₅₃×#417	C₄	1	1	yes	characteristic
37	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
38	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₅₃	—	2	1	no	—
40	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
41	C₅₃×#335	C₂²	1	2	yes	complemented
42	C₅₃×#334	C₂²	1	2	yes	complemented
43	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
44	C₅₃×D₄⋊C₈	C₂²	1	2	yes	—
45	C₅₃×D₄⋊C₈	C₂²	1	2	yes	—
46	C₅₃×D₄⋊C₈	C₂²	1	2	yes	complemented
47	C₅₃×#322	C₂²	1	2	yes	complemented
48	C₅₃×#322	C₂²	1	2	yes	complemented
49	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂³×C₄×C₅₃	—	2	1	no	—
52	AGL(1,ℤ/8)×C₅₃	—	2	4	no	—
53	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
54	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
55	C₂×M₁₆×C₅₃	—	2	2	no	—
56	C₂×M₁₆×C₅₃	—	2	2	no	—
57	C₂×M₁₆×C₅₃	—	4	4	no	—
58	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₈×C₅₃	—	2	1	no	—
61	C₂²×C₈×C₅₃	—	2	1	no	—
62	C₂²×C₈×C₅₃	—	2	1	no	—
63	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
65	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₄×D₄×C₅₃	—	2	1	no	—
68	C₄×D₄×C₅₃	—	2	1	no	—
69	C₄×D₄×C₅₃	—	2	1	no	—
70	C₄×D₄×C₅₃	—	2	1	no	—
71	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
72	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₄×D₄×C₅₃	C₂³	1	2	yes	—
74	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
75	#117×C₅₃	—	2	1	no	—
76	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
78	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
79	C₂×C₄²×C₅₃	—	2	1	no	—
80	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
81	#107×C₅₃	C₂×C₄	1	2	yes	—
82	#106×C₅₃	C₂×C₄	1	2	yes	—
83	C₄⋊C₈×C₅₃	—	2	1	no	—
84	C₄⋊C₈×C₅₃	—	2	1	no	—
85	C₄⋊C₈×C₅₃	—	2	1	no	—
86	C₄⋊C₈×C₅₃	—	2	1	no	—
87	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
88	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
89	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
90	#103×C₅₃	C₂×C₄	1	2	yes	—
91	#102×C₅₃	C₂×C₄	1	2	yes	—
92	#102×C₅₃	C₂×C₄	1	2	yes	—
93	#102×C₅₃	C₂×C₄	1	2	yes	—
94	C₂²⋊C₈×C₅₃	—	2	2	no	—
95	C₂²⋊C₈×C₅₃	—	2	2	no	—
96	C₂²⋊C₈×C₅₃	—	2	1	no	—
97	C₂²⋊C₈×C₅₃	—	2	1	no	—
98	C₂²⋊C₈×C₅₃	—	2	1	no	—
99	C₂²⋊C₈×C₅₃	—	2	1	no	—
100	#97×C₅₃	C₂×C₄	1	2	yes	—
101	C₄×C₈×C₅₃	—	2	1	no	—
102	C₄×C₈×C₅₃	—	2	1	no	—
103	C₄×C₈×C₅₃	—	2	1	no	—
104	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
105	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
106	C₄×C₈×C₅₃	C₂³	1	2	yes	—
107	C₄×C₈×C₅₃	C₂³	1	2	yes	—
108	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
109	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
110	C₂⁴×C₅₃	—	2	1	no	—
111	#41×C₅₃	—	2	2	no	—
112	#41×C₅₃	M₁₆	1	4	yes	complemented
113	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×D₄×C₅₃	—	2	1	no	—
115	C₂×D₄×C₅₃	—	2	1	no	—
116	C₂×D₄×C₅₃	—	2	2	no	—
117	C₂×D₄×C₅₃	—	2	2	no	—
118	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
119	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₄×C₅₃	—	2	1	no	—
121	C₂²×C₄×C₅₃	—	2	1	no	—
122	C₂²×C₄×C₅₃	—	2	1	no	—
123	C₂²×C₄×C₅₃	—	2	2	no	—
124	C₂²×C₄×C₅₃	—	2	1	no	—
125	C₂²×C₄×C₅₃	—	2	2	no	—
126	C₂²×C₄×C₅₃	—	4	2	no	—
127	C₂²×C₄×C₅₃	—	4	2	no	—
128	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
131	SD₁₆×C₅₃	—	2	4	no	—
132	D₈×C₅₃	—	2	4	no	—
133	M₁₆×C₅₃	—	2	2	no	—
134	M₁₆×C₅₃	—	2	2	no	—
135	M₁₆×C₅₃	—	4	4	no	—
136	M₁₆×C₅₃	—	4	4	no	—
137	C₂×C₈×C₅₃	—	2	2	no	—
138	C₂×C₈×C₅₃	—	2	1	no	—
139	C₂×C₈×C₅₃	—	2	1	no	—
140	C₂×C₈×C₅₃	—	2	2	no	—
141	C₂×C₈×C₅₃	—	2	2	no	—
142	C₂×C₈×C₅₃	—	4	1	no	—
143	C₂×C₈×C₅₃	—	4	2	no	—
144	C₂×C₈×C₅₃	—	4	2	no	—
145	C₂×C₈×C₅₃	—	4	2	no	—
146	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
147	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
148	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
149	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
150	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
151	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
152	C₄⋊C₄×C₅₃	—	2	1	no	—
153	C₄⋊C₄×C₅₃	—	2	1	no	—
154	C₄⋊C₄×C₅₃	—	2	1	no	—
155	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
156	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
157	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
158	C₂²⋊C₄×C₅₃	—	2	1	no	—
159	C₂²⋊C₄×C₅₃	—	2	2	no	—
160	C₂²⋊C₄×C₅₃	—	2	1	no	—
161	C₂²⋊C₄×C₅₃	—	2	2	no	—
162	C₂²⋊C₄×C₅₃	—	2	1	no	—
163	C₄²×C₅₃	—	2	1	no	—
164	C₄²×C₅₃	—	2	1	no	—
165	C₄²×C₅₃	—	2	1	no	—
166	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
167	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
168	C₄²×C₅₃	#41	1	1	yes	characteristic
169	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
170	C₂³×C₅₃	—	2	1	no	—
171	C₂³×C₅₃	—	2	2	no	—
172	C₂³×C₅₃	—	4	2	no	—
173	C₂³×C₅₃	—	4	2	no	—
174	C₂³×C₅₃	—	4	2	no	—
175	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
176	Q₈×C₅₃	—	2	1	no	—
177	Q₈×C₅₃	C₂×M₁₆	1	2	yes	complemented
178	D₄×C₅₃	—	2	4	no	—
179	D₄×C₅₃	—	2	2	no	—
180	D₄×C₅₃	—	2	1	no	—
181	D₄×C₅₃	—	2	2	no	—
182	D₄×C₅₃	—	4	2	no	—
183	D₄×C₅₃	C₂×M₁₆	1	2	yes	complemented
184	C₂×C₄×C₅₃	—	2	1	no	—
185	C₂×C₄×C₅₃	—	2	1	no	—
186	C₂×C₄×C₅₃	—	2	1	no	—
187	C₂×C₄×C₅₃	—	2	4	no	—
188	C₂×C₄×C₅₃	—	2	2	no	—
189	C₂×C₄×C₅₃	—	2	1	no	—
190	C₂×C₄×C₅₃	—	2	1	no	—
191	C₂×C₄×C₅₃	—	4	1	no	—
192	C₂×C₄×C₅₃	—	4	2	no	—
193	C₂×C₄×C₅₃	—	4	4	no	—
194	C₂×C₄×C₅₃	—	4	2	no	—
195	C₂×C₄×C₅₃	—	4	2	no	—
196	C₂×C₄×C₅₃	—	4	1	no	—
197	C₂×C₄×C₅₃	—	4	2	no	—
198	C₂×C₄×C₅₃	—	4	1	no	—
199	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
200	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
201	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
202	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
203	C₂×C₄×C₅₃	#131	1	2	yes	—
204	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
205	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
206	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
207	C₈×C₅₃	—	2	2	no	—
208	C₈×C₅₃	—	2	2	no	—
209	C₈×C₅₃	—	2	1	no	—
210	C₈×C₅₃	—	2	1	no	—
211	C₈×C₅₃	—	2	1	no	—
212	C₈×C₅₃	—	4	2	no	—
213	C₈×C₅₃	—	4	1	no	—
214	C₈×C₅₃	—	4	2	no	—
215	C₈×C₅₃	—	4	1	no	—
216	C₈×C₅₃	AGL(1,ℤ/8)	1	2	yes	complemented
217	#17871	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
218	C₂²×C₅₃	—	2	1	no	—
219	C₂²×C₅₃	—	2	1	no	—
220	C₂²×C₅₃	—	2	2	no	—
221	C₂²×C₅₃	—	4	1	no	—
222	C₂²×C₅₃	—	4	1	no	—
223	C₂²×C₅₃	—	4	2	no	—
224	C₂²×C₅₃	—	4	2	no	—
225	C₂²×C₅₃	—	4	2	no	—
226	C₂²×C₅₃	—	8	4	no	—
227	C₂²×C₅₃	#435	1	2	yes	—
228	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
229	C₄×C₅₃	—	2	2	no	—
230	C₄×C₅₃	—	2	1	no	—
231	C₄×C₅₃	—	2	1	no	—
232	C₄×C₅₃	—	2	1	no	—
233	C₄×C₅₃	—	4	1	no	—
234	C₄×C₅₃	—	4	2	no	—
235	C₄×C₅₃	—	4	1	no	—
236	C₄×C₅₃	#435	1	2	yes	—
237	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
238	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
239	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
240	C₄×C₅₃	#575	1	1	yes	characteristic
241	C₈×#41	C₂×C₅₃	1	1	yes	characteristic, complemented
242	C₄×AGL(1,ℤ/8)	C₂×C₅₃	1	1	yes	characteristic
243	C₂×#436	C₂×C₅₃	1	1	yes	characteristic, complemented
244	#1588	C₂×C₅₃	1	2	yes	complemented
245	#1586	C₂×C₅₃	1	2	yes	complemented
246	#1583	C₂×C₅₃	1	2	yes	complemented
247	#1581	C₂×C₅₃	1	2	yes	complemented
248	#1565	C₂×C₅₃	1	1	yes	characteristic, complemented
249	#1478	C₂×C₅₃	1	1	yes	characteristic, complemented
250	#1477	C₂×C₅₃	1	1	yes	characteristic, complemented
251	#1451	C₂×C₅₃	1	1	yes	characteristic, complemented
252	C₂×C₅₃	—	2	2	no	—
253	C₂×C₅₃	—	4	2	no	—
254	C₂×C₅₃	—	8	2	no	—
255	C₂×C₅₃	C₂×#435	1	1	yes	characteristic
256	C₂×C₅₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
257	C₂×C₅₃	#2955	1	1	yes	characteristic
258	C₂×AGL(1,ℤ/8)	C₄×C₅₃	1	1	yes	characteristic
259	C₂²×M₁₆	—	2	1	no	—
260	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
261	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
262	C₈×Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
263	C₈×Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
264	#442	C₄×C₅₃	1	2	yes	—
265	#440	C₄×C₅₃	1	2	yes	—
266	C₄×SD₁₆	C₂²×C₅₃	1	2	yes	—
267	C₄×D₈	C₂²×C₅₃	1	2	yes	—
268	#436	—	2	1	no	—
269	#436	—	2	1	no	—
270	#436	C₂²×C₅₃	1	2	yes	—
271	#436	C₂²×C₅₃	1	2	yes	—
272	C₈×D₄	—	2	1	no	—
273	C₈×D₄	—	2	1	no	—
274	C₈×D₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
275	C₈×D₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
276	#431	—	2	1	no	—
277	#431	C₂²×C₅₃	1	1	yes	characteristic
278	#431	C₂²×C₅₃	1	1	yes	characteristic
279	#428	C₄×C₅₃	1	1	yes	characteristic
280	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
281	#418	C₄×C₅₃	1	1	yes	characteristic
282	#417	C₄×C₅₃	1	1	yes	characteristic
283	C₂×C₂²⋊C₈	—	2	1	no	—
284	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
285	C₂×C₄×C₈	—	2	1	no	—
286	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
287	#335	C₂²×C₅₃	1	2	yes	complemented
288	#334	C₂²×C₅₃	1	2	yes	complemented
289	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	complemented
290	D₄⋊C₈	C₂²×C₅₃	1	2	yes	—
291	D₄⋊C₈	C₂²×C₅₃	1	2	yes	—
292	D₄⋊C₈	C₂²×C₅₃	1	2	yes	complemented
293	#322	C₂²×C₅₃	1	2	yes	complemented
294	#322	C₂²×C₅₃	1	2	yes	complemented
295	C₅₃	#17871	1	1	yes	characteristic, direct product factor, Sylow
296	C₂×#41	C₂×C₄×C₅₃	1	1	yes	characteristic
297	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
298	C₂³×C₄	—	2	1	no	—
299	AGL(1,ℤ/8)	—	2	4	no	—
300	C₂×SD₁₆	C₂×C₄×C₅₃	1	2	yes	—
301	C₂×D₈	C₂×C₄×C₅₃	1	2	yes	—
302	C₂×M₁₆	—	2	2	no	—
303	C₂×M₁₆	—	2	2	no	—
304	C₂×M₁₆	—	4	4	no	—
305	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
306	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
307	C₂²×C₈	—	2	1	no	—
308	C₂²×C₈	—	2	1	no	—
309	C₂²×C₈	—	2	1	no	—
310	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
311	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
312	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
313	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
314	C₄×D₄	—	2	1	no	—
315	C₄×D₄	—	2	1	no	—
316	C₄×D₄	—	2	1	no	—
317	C₄×D₄	—	2	1	no	—
318	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
319	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
320	C₄×D₄	C₂³×C₅₃	1	2	yes	—
321	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
322	#117	—	2	1	no	—
323	#117	C₂×C₄×C₅₃	1	1	yes	characteristic
324	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
325	C₂×C₂²⋊C₄	—	2	1	no	—
326	C₂×C₄²	—	2	1	no	—
327	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
328	#107	C₂×C₄×C₅₃	1	2	yes	—
329	#106	C₂×C₄×C₅₃	1	2	yes	—
330	C₄⋊C₈	—	2	1	no	—
331	C₄⋊C₈	—	2	1	no	—
332	C₄⋊C₈	—	2	1	no	—
333	C₄⋊C₈	—	2	1	no	—
334	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
335	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
336	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
337	#103	C₂×C₄×C₅₃	1	2	yes	—
338	#102	C₂×C₄×C₅₃	1	2	yes	—
339	#102	C₂×C₄×C₅₃	1	2	yes	—
340	#102	C₂×C₄×C₅₃	1	2	yes	—
341	C₂²⋊C₈	—	2	1	no	—
342	C₂²⋊C₈	—	2	1	no	—
343	C₂²⋊C₈	—	2	2	no	—
344	C₂²⋊C₈	—	2	2	no	—
345	C₂²⋊C₈	—	2	1	no	—
346	C₂²⋊C₈	—	2	1	no	—
347	#97	C₂×C₄×C₅₃	1	2	yes	—
348	C₄×C₈	—	2	1	no	—
349	C₄×C₈	—	2	1	no	—
350	C₄×C₈	—	2	1	no	—
351	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
352	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
353	C₄×C₈	C₂³×C₅₃	1	2	yes	—
354	C₄×C₈	C₂³×C₅₃	1	2	yes	—
355	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
356	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
357	C₂⁴	—	2	1	no	—
358	#41	—	2	2	no	—
359	#41	M₁₆×C₅₃	1	4	yes	complemented
360	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
361	C₂×D₄	—	2	1	no	—
362	C₂×D₄	—	2	1	no	—
363	C₂×D₄	—	2	2	no	—
364	C₂×D₄	—	2	2	no	—
365	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
366	C₂×D₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	2	no	—
373	C₂²×C₄	—	4	2	no	—
374	C₂²×C₄	—	4	2	no	—
375	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
376	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
377	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
378	SD₁₆	—	2	4	no	—
379	D₈	—	2	4	no	—
380	M₁₆	—	2	2	no	—
381	M₁₆	—	2	2	no	—
382	M₁₆	—	4	4	no	—
383	M₁₆	—	4	4	no	—
384	C₂×C₈	—	2	2	no	—
385	C₂×C₈	—	2	1	no	—
386	C₂×C₈	—	2	1	no	—
387	C₂×C₈	—	2	2	no	—
388	C₂×C₈	—	2	2	no	—
389	C₂×C₈	—	4	1	no	—
390	C₂×C₈	—	4	2	no	—
391	C₂×C₈	—	4	2	no	—
392	C₂×C₈	—	4	2	no	—
393	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
394	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
395	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
396	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
397	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
398	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
399	C₄⋊C₄	—	2	1	no	—
400	C₄⋊C₄	—	2	1	no	—
401	C₄⋊C₄	—	2	1	no	—
402	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
403	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
404	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
405	C₂²⋊C₄	—	2	1	no	—
406	C₂²⋊C₄	—	2	2	no	—
407	C₂²⋊C₄	—	2	1	no	—
408	C₂²⋊C₄	—	2	2	no	—
409	C₂²⋊C₄	—	2	1	no	—
410	C₄²	—	2	1	no	—
411	C₄²	—	2	1	no	—
412	C₄²	—	2	1	no	—
413	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
414	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
415	C₄²	#41×C₅₃	1	1	yes	characteristic
416	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
417	C₂³	—	2	1	no	—
418	C₂³	—	2	2	no	—
419	C₂³	—	4	2	no	—
420	C₂³	—	4	2	no	—
421	C₂³	—	4	2	no	—
422	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
423	Q₈	—	2	1	no	—
424	Q₈	C₂×M₁₆×C₅₃	1	2	yes	complemented
425	D₄	—	2	4	no	—
426	D₄	—	2	2	no	—
427	D₄	—	2	1	no	—
428	D₄	—	2	2	no	—
429	D₄	—	4	2	no	—
430	D₄	C₂×M₁₆×C₅₃	1	2	yes	complemented
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	4	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	4	4	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
447	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
448	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
449	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
450	C₂×C₄	#131×C₅₃	1	2	yes	—
451	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
452	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
453	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
454	C₈	—	2	2	no	—
455	C₈	—	2	2	no	—
456	C₈	—	2	1	no	—
457	C₈	—	2	1	no	—
458	C₈	—	2	1	no	—
459	C₈	—	4	2	no	—
460	C₈	—	4	1	no	—
461	C₈	—	4	2	no	—
462	C₈	—	4	1	no	—
463	C₈	AGL(1,ℤ/8)×C₅₃	1	2	yes	complemented
464	C₂²	—	2	1	no	—
465	C₂²	—	2	1	no	—
466	C₂²	—	2	2	no	—
467	C₂²	—	4	1	no	—
468	C₂²	—	4	1	no	—
469	C₂²	—	4	2	no	—
470	C₂²	—	4	2	no	—
471	C₂²	—	4	2	no	—
472	C₂²	—	8	4	no	—
473	C₂²	C₅₃×#435	1	2	yes	—
474	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
475	C₄	—	2	1	no	—
476	C₄	—	2	1	no	—
477	C₄	—	2	1	no	—
478	C₄	—	2	2	no	—
479	C₄	—	4	2	no	—
480	C₄	—	4	1	no	—
481	C₄	—	4	1	no	—
482	C₄	C₅₃×#435	1	2	yes	—
483	C₄	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
484	C₄	C₂×#131×C₅₃	1	1	yes	characteristic
485	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
486	C₄	C₅₃×#575	1	1	yes	characteristic
487	C₂	—	2	2	no	—
488	C₂	—	4	2	no	—
489	C₂	—	8	2	no	—
490	C₂	C₂×C₅₃×#435	1	1	yes	characteristic
491	C₂	C₄×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
492	C₂	C₅₃×#2955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×AGL(1,ℤ/8)×C₅₃	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈×#41×C₅₃	C₂	semidirect
3. C₂×C₅₃×#436	C₂	semidirect
4. C₅₃×#1588	C₂	semidirect
5. C₅₃×#1586	C₂	semidirect
6. C₅₃×#1583	C₂	semidirect
7. C₅₃×#1581	C₂	semidirect
8. C₅₃×#1565	C₂	semidirect
9. C₅₃×#1478	C₂	semidirect
10. C₅₃×#1477	C₂	semidirect
11. C₅₃×#1451	C₂	semidirect
16. C₈×Q₈×C₅₃	C₂²	semidirect
17. C₈×Q₈×C₅₃	C₂²	semidirect
28. C₈×D₄×C₅₃	C₂²	semidirect
29. C₈×D₄×C₅₃	C₂²	semidirect
41. C₅₃×#335	C₂²	semidirect
42. C₅₃×#334	C₂²	semidirect
43. C₅₃×Q₈⋊C₈	C₂²	semidirect
46. C₅₃×D₄⋊C₈	C₂²	semidirect
47. C₅₃×#322	C₂²	semidirect
48. C₅₃×#322	C₂²	semidirect
112. #41×C₅₃	M₁₆	semidirect
177. Q₈×C₅₃	C₂×M₁₆	semidirect
183. D₄×C₅₃	C₂×M₁₆	semidirect
216. C₈×C₅₃	AGL(1,ℤ/8)	semidirect
217. #17871	C₅₃	direct (N × H)
241. C₈×#41	C₂×C₅₃	semidirect
243. C₂×#436	C₂×C₅₃	semidirect
244. #1588	C₂×C₅₃	semidirect
245. #1586	C₂×C₅₃	semidirect
246. #1583	C₂×C₅₃	semidirect
247. #1581	C₂×C₅₃	semidirect
248. #1565	C₂×C₅₃	semidirect
249. #1478	C₂×C₅₃	semidirect
250. #1477	C₂×C₅₃	semidirect
251. #1451	C₂×C₅₃	semidirect
262. C₈×Q₈	C₂²×C₅₃	semidirect
263. C₈×Q₈	C₂²×C₅₃	semidirect
274. C₈×D₄	C₂²×C₅₃	semidirect
275. C₈×D₄	C₂²×C₅₃	semidirect
287. #335	C₂²×C₅₃	semidirect
288. #334	C₂²×C₅₃	semidirect
289. Q₈⋊C₈	C₂²×C₅₃	semidirect
292. D₄⋊C₈	C₂²×C₅₃	semidirect
293. #322	C₂²×C₅₃	semidirect
294. #322	C₂²×C₅₃	semidirect
295. C₅₃	#17871	direct (N × H)
359. #41	M₁₆×C₅₃	semidirect
424. Q₈	C₂×M₁₆×C₅₃	semidirect
430. D₄	C₂×M₁₆×C₅₃	semidirect
463. C₈	AGL(1,ℤ/8)×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₈×#41×C₅₃	2
C₄×AGL(1,ℤ/8)×C₅₃	2
C₂×C₅₃×#436	2
C₅₃×#1588	2
C₅₃×#1586	2
C₅₃×#1583	2
C₅₃×#1581	2
C₅₃×#1565	2
C₅₃×#1478	2
C₅₃×#1477	2
C₅₃×#1451	2
#17871	53

Subgroups — C53×#17871 #27225383

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17871 #27225383