Subgroups — C₅₃×#18762 #27226274

Order 13568.

964 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3227	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3100	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#3037	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1652	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1628	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1549	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1486	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1479	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#137×C₅₃	—	2	1	no	—
11	C₅₃×#556	—	2	2	no	—
12	C₅₃×#556	—	2	2	no	—
13	C₅₃×#556	—	2	2	no	—
14	C₅₃×#556	C₂²	1	4	yes	—
15	C₅₃×#532	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#126×C₅₃	—	2	2	no	—
18	C₂×#124×C₅₃	—	2	1	no	—
19	C₂×C₂²⋊Q₈×C₅₃	—	2	1	no	—
20	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
21	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#117×C₅₃	—	2	1	no	—
24	C₅₃×#498	C₂²	1	2	yes	complemented
25	C₅₃×#486	C₂²	1	2	yes	complemented
26	C₅₃×#481	—	2	2	no	—
27	C₅₃×#480	—	2	1	no	—
28	C₅₃×#476	—	2	2	no	—
29	C₅₃×#474	C₂²	1	2	yes	—
30	C₅₃×#464	—	2	2	no	—
31	C₅₃×#462	C₂²	1	2	yes	complemented
32	C₅₃×#460	C₂²	1	2	yes	complemented
33	C₅₃×#459	C₂²	1	2	yes	—
34	C₅₃×#447	—	2	1	no	—
35	C₄×D₈×C₅₃	—	2	2	no	—
36	C₅₃×#431	C₂²	1	1	yes	characteristic
37	C₅₃×#431	C₂²	1	1	yes	characteristic
38	C₅₃×#423	C₂²	1	1	yes	characteristic
39	C₅₃×#417	—	2	1	no	—
40	C₅₃×#332	C₂²	1	4	yes	complemented
41	C₅₃×Q₈⋊C₈	C₂²	1	4	yes	complemented
42	C₅₃×D₄⋊C₈	C₂²	1	4	yes	—
43	C₂×#41×C₅₃	—	2	1	no	—
44	C₂²×Q₈×C₅₃	—	2	1	no	—
45	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
46	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₅₃	—	2	1	no	—
48	#137×C₅₃	—	4	4	no	—
49	C₂×Q₁₆×C₅₃	—	2	2	no	—
50	C₂×SD₁₆×C₅₃	—	2	2	no	—
51	C₂×D₈×C₅₃	—	2	2	no	—
52	C₂×M₁₆×C₅₃	—	2	1	no	—
53	C₄⋊Q₈×C₅₃	—	2	1	no	—
54	C₄⋊Q₈×C₅₃	—	2	1	no	—
55	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
56	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
57	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
58	#127×C₅₃	C₂³	1	1	yes	characteristic
59	#126×C₅₃	—	2	8	no	—
60	#126×C₅₃	—	2	8	no	—
61	#124×C₅₃	—	2	4	no	—
62	#124×C₅₃	—	2	1	no	—
63	#124×C₅₃	—	2	4	no	—
64	C₂²⋊Q₈×C₅₃	—	2	4	no	—
65	C₂²⋊Q₈×C₅₃	—	2	4	no	—
66	#121×C₅₃	—	2	1	no	—
67	#121×C₅₃	—	2	1	no	—
68	C₄×Q₈×C₅₃	—	2	4	no	—
69	C₄×Q₈×C₅₃	—	2	4	no	—
70	C₄×Q₈×C₅₃	—	2	1	no	—
71	C₄×Q₈×C₅₃	—	2	1	no	—
72	C₄×Q₈×C₅₃	D₄	1	2	yes	—
73	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
74	C₄×D₄×C₅₃	—	2	1	no	—
75	C₄×D₄×C₅₃	—	2	1	no	—
76	C₄×D₄×C₅₃	—	2	1	no	—
77	C₄×D₄×C₅₃	—	2	1	no	—
78	C₄×D₄×C₅₃	D₄	1	2	yes	—
79	C₄×D₄×C₅₃	C₂³	1	2	yes	—
80	#117×C₅₃	—	2	4	no	—
81	#117×C₅₃	—	2	4	no	—
82	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
83	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
84	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
85	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
88	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
89	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
90	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
91	C₂×C₄²×C₅₃	—	2	2	no	—
92	C₂×C₄²×C₅₃	—	2	1	no	—
93	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
94	#107×C₅₃	—	2	2	no	—
95	#106×C₅₃	—	2	4	no	—
96	C₄⋊C₈×C₅₃	—	2	1	no	—
97	C₄⋊C₈×C₅₃	—	2	1	no	—
98	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
99	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
100	#103×C₅₃	—	2	2	no	—
101	#102×C₅₃	—	2	4	no	—
102	#102×C₅₃	—	2	2	no	—
103	#102×C₅₃	—	2	2	no	—
104	#102×C₅₃	—	2	2	no	—
105	C₂²⋊C₈×C₅₃	—	2	1	no	—
106	C₂²⋊C₈×C₅₃	—	2	1	no	—
107	C₄×C₈×C₅₃	—	2	1	no	—
108	C₄×C₈×C₅₃	C₂³	1	2	yes	—
109	C₂⁴×C₅₃	—	2	1	no	—
110	#41×C₅₃	—	4	2	no	—
111	#41×C₅₃	—	4	2	no	—
112	C₂×Q₈×C₅₃	—	2	1	no	—
113	C₂×Q₈×C₅₃	—	2	2	no	—
114	C₂×Q₈×C₅₃	—	2	2	no	—
115	C₂×Q₈×C₅₃	—	2	2	no	—
116	C₂×Q₈×C₅₃	—	2	2	no	—
117	C₂×Q₈×C₅₃	—	2	1	no	—
118	C₂×Q₈×C₅₃	—	4	4	no	—
119	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
120	C₂×D₄×C₅₃	—	2	1	no	—
121	C₂×D₄×C₅₃	—	2	1	no	—
122	C₂×D₄×C₅₃	—	2	1	no	—
123	C₂×D₄×C₅₃	—	2	1	no	—
124	C₂×D₄×C₅₃	—	4	2	no	—
125	C₂×D₄×C₅₃	—	4	1	no	—
126	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₅₃	—	2	1	no	—
128	C₂²×C₄×C₅₃	—	2	1	no	—
129	C₂²×C₄×C₅₃	—	2	1	no	—
130	C₂²×C₄×C₅₃	—	2	2	no	—
131	C₂²×C₄×C₅₃	—	2	2	no	—
132	C₂²×C₄×C₅₃	—	2	2	no	—
133	C₂²×C₄×C₅₃	—	4	2	no	—
134	C₂²×C₄×C₅₃	—	4	2	no	—
135	C₂²×C₄×C₅₃	—	4	1	no	—
136	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
139	Q₁₆×C₅₃	—	4	4	no	—
140	SD₁₆×C₅₃	—	4	4	no	—
141	D₈×C₅₃	—	4	4	no	—
142	M₁₆×C₅₃	—	4	2	no	—
143	C₂×C₈×C₅₃	—	2	2	no	—
144	C₂×C₈×C₅₃	—	2	2	no	—
145	C₂×C₈×C₅₃	—	2	2	no	—
146	C₄⋊C₄×C₅₃	—	2	8	no	—
147	C₄⋊C₄×C₅₃	—	2	2	no	—
148	C₄⋊C₄×C₅₃	—	2	2	no	—
149	C₄⋊C₄×C₅₃	—	2	1	no	—
150	C₄⋊C₄×C₅₃	—	2	1	no	—
151	C₄⋊C₄×C₅₃	—	2	2	no	—
152	C₄⋊C₄×C₅₃	—	2	2	no	—
153	C₄⋊C₄×C₅₃	—	2	2	no	—
154	C₄⋊C₄×C₅₃	—	2	1	no	—
155	C₄⋊C₄×C₅₃	—	2	1	no	—
156	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
157	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
158	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
159	C₂²⋊C₄×C₅₃	—	2	8	no	—
160	C₂²⋊C₄×C₅₃	—	2	8	no	—
161	C₂²⋊C₄×C₅₃	—	2	2	no	—
162	C₂²⋊C₄×C₅₃	—	2	2	no	—
163	C₂²⋊C₄×C₅₃	—	4	1	no	—
164	C₂²⋊C₄×C₅₃	—	4	1	no	—
165	C₄²×C₅₃	—	2	8	no	—
166	C₄²×C₅₃	—	2	2	no	—
167	C₄²×C₅₃	—	2	2	no	—
168	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
169	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
170	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
171	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
172	C₂³×C₅₃	—	2	2	no	—
173	C₂³×C₅₃	—	4	2	no	—
174	C₂³×C₅₃	—	4	2	no	—
175	C₂³×C₅₃	—	4	2	no	—
176	C₂³×C₅₃	—	4	1	no	—
177	C₂³×C₅₃	#120	1	1	yes	characteristic
178	Q₈×C₅₃	—	2	2	no	—
179	Q₈×C₅₃	—	2	2	no	—
180	Q₈×C₅₃	—	4	4	no	—
181	Q₈×C₅₃	—	4	4	no	—
182	Q₈×C₅₃	—	4	2	no	—
183	Q₈×C₅₃	—	4	1	no	—
184	Q₈×C₅₃	—	4	1	no	—
185	D₄×C₅₃	—	2	2	no	—
186	D₄×C₅₃	—	2	2	no	—
187	D₄×C₅₃	—	4	2	no	—
188	D₄×C₅₃	—	4	2	no	—
189	D₄×C₅₃	—	4	1	no	—
190	D₄×C₅₃	—	4	1	no	—
191	D₄×C₅₃	—	4	2	no	—
192	D₄×C₅₃	—	8	2	no	—
193	C₂×C₄×C₅₃	—	2	2	no	—
194	C₂×C₄×C₅₃	—	2	2	no	—
195	C₂×C₄×C₅₃	—	2	4	no	—
196	C₂×C₄×C₅₃	—	2	2	no	—
197	C₂×C₄×C₅₃	—	2	2	no	—
198	C₂×C₄×C₅₃	—	4	1	no	—
199	C₂×C₄×C₅₃	—	4	2	no	—
200	C₂×C₄×C₅₃	—	4	4	no	—
201	C₂×C₄×C₅₃	—	4	2	no	—
202	C₂×C₄×C₅₃	—	4	2	no	—
203	C₂×C₄×C₅₃	—	4	2	no	—
204	C₂×C₄×C₅₃	—	4	2	no	—
205	C₂×C₄×C₅₃	—	4	4	no	—
206	C₂×C₄×C₅₃	—	4	1	no	—
207	C₂×C₄×C₅₃	—	8	2	no	—
208	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
209	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
210	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
211	C₂×C₄×C₅₃	#135	1	2	yes	—
212	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
213	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
215	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
216	C₈×C₅₃	—	4	2	no	—
217	C₈×C₅₃	—	4	2	no	—
218	C₈×C₅₃	—	4	2	no	—
219	#18762	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
220	C₂²×C₅₃	—	2	1	no	—
221	C₂²×C₅₃	—	2	1	no	—
222	C₂²×C₅₃	—	4	2	no	—
223	C₂²×C₅₃	—	4	2	no	—
224	C₂²×C₅₃	—	4	2	no	—
225	C₂²×C₅₃	—	8	4	no	—
226	C₂²×C₅₃	—	8	1	no	—
227	C₂²×C₅₃	—	8	1	no	—
228	C₂²×C₅₃	—	8	1	no	—
229	C₂²×C₅₃	#449	1	2	yes	—
230	C₂²×C₅₃	C₂×#120	1	1	yes	characteristic
231	C₄×C₅₃	—	2	1	no	—
232	C₄×C₅₃	—	2	1	no	—
233	C₄×C₅₃	—	2	1	no	—
234	C₄×C₅₃	—	4	2	no	—
235	C₄×C₅₃	—	4	2	no	—
236	C₄×C₅₃	—	4	4	no	—
237	C₄×C₅₃	—	4	2	no	—
238	C₄×C₅₃	—	8	1	no	—
239	C₄×C₅₃	#449	1	2	yes	—
240	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
241	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
242	C₄×C₅₃	#575	1	2	yes	—
243	C₂×#556	C₂×C₅₃	1	1	yes	characteristic, complemented
244	#3227	C₂×C₅₃	1	1	yes	characteristic, complemented
245	#3100	C₂×C₅₃	1	1	yes	characteristic
246	#3037	C₂×C₅₃	1	1	yes	characteristic, complemented
247	#1652	C₂×C₅₃	1	4	yes	complemented
248	#1628	C₂×C₅₃	1	4	yes	complemented
249	#1549	C₂×C₅₃	1	1	yes	characteristic, complemented
250	#1486	C₂×C₅₃	1	1	yes	characteristic, complemented
251	#1479	C₂×C₅₃	1	1	yes	characteristic, complemented
252	C₂×C₅₃	—	2	2	no	—
253	C₂×C₅₃	—	8	2	no	—
254	C₂×C₅₃	—	16	1	no	—
255	C₂×C₅₃	C₂×#449	1	1	yes	characteristic
256	C₂×C₅₃	#3005	1	1	yes	characteristic
257	C₂×C₅₃	#3020	1	1	yes	characteristic
258	C₂×#137	—	2	1	no	—
259	#556	—	2	2	no	—
260	#556	—	2	2	no	—
261	#556	—	2	2	no	—
262	#556	C₂²×C₅₃	1	4	yes	—
263	#532	C₂²×C₅₃	1	1	yes	characteristic
264	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
265	C₂×#126	—	2	2	no	—
266	C₂×#124	—	2	1	no	—
267	C₂×C₂²⋊Q₈	—	2	1	no	—
268	C₂×C₄×Q₈	—	2	1	no	—
269	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
270	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
271	C₂×#117	—	2	1	no	—
272	#498	C₂²×C₅₃	1	2	yes	complemented
273	#486	C₂²×C₅₃	1	2	yes	complemented
274	#481	—	2	2	no	—
275	#480	—	2	1	no	—
276	#476	—	2	2	no	—
277	#474	C₂²×C₅₃	1	2	yes	—
278	#464	—	2	2	no	—
279	#462	C₂²×C₅₃	1	2	yes	complemented
280	#460	C₂²×C₅₃	1	2	yes	complemented
281	#459	C₂²×C₅₃	1	2	yes	—
282	#447	—	2	1	no	—
283	C₄×D₈	—	2	2	no	—
284	#431	C₂²×C₅₃	1	1	yes	characteristic
285	#431	C₂²×C₅₃	1	1	yes	characteristic
286	#423	C₂²×C₅₃	1	1	yes	characteristic
287	#417	—	2	1	no	—
288	#332	C₂²×C₅₃	1	4	yes	complemented
289	Q₈⋊C₈	C₂²×C₅₃	1	4	yes	complemented
290	D₄⋊C₈	C₂²×C₅₃	1	4	yes	—
291	C₅₃	#18762	1	1	yes	characteristic, direct product factor, Sylow
292	C₂×#41	—	2	1	no	—
293	C₂²×Q₈	—	2	1	no	—
294	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
295	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
296	C₂³×C₄	—	2	1	no	—
297	#137	—	4	4	no	—
298	C₂×Q₁₆	—	2	2	no	—
299	C₂×SD₁₆	—	2	2	no	—
300	C₂×D₈	—	2	2	no	—
301	C₂×M₁₆	—	2	1	no	—
302	C₄⋊Q₈	—	2	1	no	—
303	C₄⋊Q₈	—	2	1	no	—
304	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
305	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
306	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
307	#127	C₂³×C₅₃	1	1	yes	characteristic
308	#126	—	2	8	no	—
309	#126	—	2	8	no	—
310	#124	—	2	4	no	—
311	#124	—	2	4	no	—
312	#124	—	2	1	no	—
313	C₂²⋊Q₈	—	2	4	no	—
314	C₂²⋊Q₈	—	2	4	no	—
315	#121	—	2	1	no	—
316	#121	—	2	1	no	—
317	C₄×Q₈	—	2	4	no	—
318	C₄×Q₈	—	2	4	no	—
319	C₄×Q₈	—	2	1	no	—
320	C₄×Q₈	—	2	1	no	—
321	C₄×Q₈	D₄×C₅₃	1	2	yes	—
322	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
323	C₄×D₄	—	2	1	no	—
324	C₄×D₄	—	2	1	no	—
325	C₄×D₄	—	2	1	no	—
326	C₄×D₄	—	2	1	no	—
327	C₄×D₄	D₄×C₅₃	1	2	yes	—
328	C₄×D₄	C₂³×C₅₃	1	2	yes	—
329	#117	—	2	4	no	—
330	#117	—	2	4	no	—
331	C₂×C₄⋊C₄	—	2	2	no	—
332	C₂×C₄⋊C₄	—	2	1	no	—
333	C₂×C₄⋊C₄	—	2	1	no	—
334	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
335	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
336	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
337	C₂×C₂²⋊C₄	—	2	2	no	—
338	C₂×C₂²⋊C₄	—	2	2	no	—
339	C₂×C₂²⋊C₄	—	2	1	no	—
340	C₂×C₄²	—	2	2	no	—
341	C₂×C₄²	—	2	1	no	—
342	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
343	#107	—	2	2	no	—
344	#106	—	2	4	no	—
345	C₄⋊C₈	—	2	1	no	—
346	C₄⋊C₈	—	2	1	no	—
347	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
348	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
349	#103	—	2	2	no	—
350	#102	—	2	2	no	—
351	#102	—	2	2	no	—
352	#102	—	2	2	no	—
353	#102	—	2	4	no	—
354	C₂²⋊C₈	—	2	1	no	—
355	C₂²⋊C₈	—	2	1	no	—
356	C₄×C₈	—	2	1	no	—
357	C₄×C₈	C₂³×C₅₃	1	2	yes	—
358	C₂⁴	—	2	1	no	—
359	#41	—	4	2	no	—
360	#41	—	4	2	no	—
361	C₂×Q₈	—	2	1	no	—
362	C₂×Q₈	—	2	2	no	—
363	C₂×Q₈	—	2	2	no	—
364	C₂×Q₈	—	2	2	no	—
365	C₂×Q₈	—	2	2	no	—
366	C₂×Q₈	—	2	1	no	—
367	C₂×Q₈	—	4	4	no	—
368	C₂×Q₈	C₂×D₄×C₅₃	1	2	yes	—
369	C₂×D₄	—	2	1	no	—
370	C₂×D₄	—	2	1	no	—
371	C₂×D₄	—	2	1	no	—
372	C₂×D₄	—	2	1	no	—
373	C₂×D₄	—	4	2	no	—
374	C₂×D₄	—	4	1	no	—
375	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	4	2	no	—
383	C₂²×C₄	—	4	2	no	—
384	C₂²×C₄	—	4	1	no	—
385	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
386	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
387	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
388	Q₁₆	—	4	4	no	—
389	SD₁₆	—	4	4	no	—
390	D₈	—	4	4	no	—
391	M₁₆	—	4	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	2	no	—
395	C₄⋊C₄	—	2	8	no	—
396	C₄⋊C₄	—	2	2	no	—
397	C₄⋊C₄	—	2	2	no	—
398	C₄⋊C₄	—	2	1	no	—
399	C₄⋊C₄	—	2	1	no	—
400	C₄⋊C₄	—	2	2	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	—	2	1	no	—
404	C₄⋊C₄	—	2	1	no	—
405	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
406	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
407	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
408	C₂²⋊C₄	—	2	8	no	—
409	C₂²⋊C₄	—	2	8	no	—
410	C₂²⋊C₄	—	2	2	no	—
411	C₂²⋊C₄	—	2	2	no	—
412	C₂²⋊C₄	—	4	1	no	—
413	C₂²⋊C₄	—	4	1	no	—
414	C₄²	—	2	8	no	—
415	C₄²	—	2	2	no	—
416	C₄²	—	2	2	no	—
417	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
418	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
419	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
420	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
421	C₂³	—	2	2	no	—
422	C₂³	—	4	2	no	—
423	C₂³	—	4	2	no	—
424	C₂³	—	4	2	no	—
425	C₂³	—	4	1	no	—
426	C₂³	#120×C₅₃	1	1	yes	characteristic
427	Q₈	—	2	2	no	—
428	Q₈	—	2	2	no	—
429	Q₈	—	4	4	no	—
430	Q₈	—	4	4	no	—
431	Q₈	—	4	2	no	—
432	Q₈	—	4	1	no	—
433	Q₈	—	4	1	no	—
434	D₄	—	2	2	no	—
435	D₄	—	2	2	no	—
436	D₄	—	4	2	no	—
437	D₄	—	4	2	no	—
438	D₄	—	4	1	no	—
439	D₄	—	4	1	no	—
440	D₄	—	4	2	no	—
441	D₄	—	8	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	4	1	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	4	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	4	4	no	—
455	C₂×C₄	—	4	1	no	—
456	C₂×C₄	—	8	2	no	—
457	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
458	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
459	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
460	C₂×C₄	#135×C₅₃	1	2	yes	—
461	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
462	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
463	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
464	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
465	C₈	—	4	2	no	—
466	C₈	—	4	2	no	—
467	C₈	—	4	2	no	—
468	C₂²	—	2	1	no	—
469	C₂²	—	2	1	no	—
470	C₂²	—	4	2	no	—
471	C₂²	—	4	2	no	—
472	C₂²	—	4	2	no	—
473	C₂²	—	8	4	no	—
474	C₂²	—	8	1	no	—
475	C₂²	—	8	1	no	—
476	C₂²	—	8	1	no	—
477	C₂²	C₅₃×#449	1	2	yes	—
478	C₂²	C₂×#120×C₅₃	1	1	yes	characteristic
479	C₄	—	2	1	no	—
480	C₄	—	2	1	no	—
481	C₄	—	2	1	no	—
482	C₄	—	4	2	no	—
483	C₄	—	4	2	no	—
484	C₄	—	4	4	no	—
485	C₄	—	4	2	no	—
486	C₄	—	8	1	no	—
487	C₄	C₅₃×#449	1	2	yes	—
488	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
489	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
490	C₄	C₅₃×#575	1	2	yes	—
491	C₂	—	2	2	no	—
492	C₂	—	8	2	no	—
493	C₂	—	16	1	no	—
494	C₂	C₂×C₅₃×#449	1	1	yes	characteristic
495	C₂	C₅₃×#3005	1	1	yes	characteristic
496	C₂	C₅₃×#3020	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₅₃×#3100	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. C₂×C₅₃×#556	C₂	semidirect
2. C₅₃×#3227	C₂	semidirect
4. C₅₃×#3037	C₂	semidirect
5. C₅₃×#1652	C₂	semidirect
6. C₅₃×#1628	C₂	semidirect
7. C₅₃×#1549	C₂	semidirect
8. C₅₃×#1486	C₂	semidirect
9. C₅₃×#1479	C₂	semidirect
24. C₅₃×#498	C₂²	semidirect
25. C₅₃×#486	C₂²	semidirect
31. C₅₃×#462	C₂²	semidirect
32. C₅₃×#460	C₂²	semidirect
40. C₅₃×#332	C₂²	semidirect
41. C₅₃×Q₈⋊C₈	C₂²	semidirect
219. #18762	C₅₃	direct (N × H)
243. C₂×#556	C₂×C₅₃	semidirect
244. #3227	C₂×C₅₃	semidirect
246. #3037	C₂×C₅₃	semidirect
247. #1652	C₂×C₅₃	semidirect
248. #1628	C₂×C₅₃	semidirect
249. #1549	C₂×C₅₃	semidirect
250. #1486	C₂×C₅₃	semidirect
251. #1479	C₂×C₅₃	semidirect
272. #498	C₂²×C₅₃	semidirect
273. #486	C₂²×C₅₃	semidirect
279. #462	C₂²×C₅₃	semidirect
280. #460	C₂²×C₅₃	semidirect
288. #332	C₂²×C₅₃	semidirect
289. Q₈⋊C₈	C₂²×C₅₃	semidirect
291. C₅₃	#18762	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#556	2
C₅₃×#3227	2
C₅₃×#3100	2
C₅₃×#3037	2
C₅₃×#1652	2
C₅₃×#1628	2
C₅₃×#1549	2
C₅₃×#1486	2
C₅₃×#1479	2
#18762	53

Subgroups — C53×#18762 #27226274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#18762 #27226274