Subgroups — C₅₃×#18828 #27226340

Order 13568.

906 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3503	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3226	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3216	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3115	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1649	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1648	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1627	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1626	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1556	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1497	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1488	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈²×C₅₃	—	2	1	no	—
13	C₅₃×#557	C₂²	1	2	yes	—
14	C₅₃×#554	—	2	1	no	—
15	C₅₃×#554	—	2	2	no	—
16	D₄×Q₈×C₅₃	—	2	2	no	—
17	D₄×Q₈×C₅₃	C₂²	1	2	yes	—
18	C₅₃×#544	—	2	2	no	—
19	C₅₃×#544	—	2	1	no	—
20	C₅₃×#533	—	2	1	no	—
21	C₅₃×#533	C₂²	1	1	yes	characteristic
22	C₅₃×#532	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
24	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#520	—	2	1	no	—
26	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#500	C₂²	1	2	yes	—
28	C₅₃×#497	—	2	1	no	—
29	C₅₃×#489	—	2	1	no	—
30	C₅₃×#486	C₂²	1	2	yes	—
31	C₅₃×#482	—	2	1	no	—
32	C₅₃×#480	—	2	2	no	—
33	C₅₃×#475	C₂²	1	1	yes	characteristic
34	C₅₃×#474	—	2	2	no	—
35	C₅₃×#474	C₂²	1	1	yes	characteristic
36	C₅₃×#462	C₂²	1	2	yes	—
37	C₅₃×#460	C₂²	1	2	yes	—
38	C₅₃×#460	C₂²	1	1	yes	characteristic
39	C₅₃×#459	C₂²	1	1	yes	characteristic
40	C₅₃×#452	—	2	2	no	—
41	C₅₃×#449	—	2	1	no	—
42	C₅₃×#442	—	2	1	no	—
43	C₅₃×#440	—	2	1	no	—
44	C₅₃×#432	C₂²	1	1	yes	characteristic
45	C₅₃×#432	C₂²	1	1	yes	characteristic
46	C₅₃×#431	C₂²	1	1	yes	characteristic
47	C₅₃×#332	C₂²	1	2	yes	—
48	C₅₃×#332	C₂²	1	2	yes	—
49	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	—
50	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	—
51	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	—
52	C₅₃×D₄⋊C₈	C₂²	1	2	yes	—
53	C₂×#41×C₅₃	—	2	1	no	—
54	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
55	C₂²×Q₈×C₅₃	—	2	2	no	—
56	C₂²×Q₈×C₅₃	—	2	1	no	—
57	C₂×Q₁₆×C₅₃	—	2	2	no	—
58	C₂×SD₁₆×C₅₃	—	2	1	no	—
59	C₂×SD₁₆×C₅₃	—	2	2	no	—
60	C₂×D₈×C₅₃	—	2	1	no	—
61	C₄⋊Q₈×C₅₃	—	2	2	no	—
62	C₄⋊Q₈×C₅₃	—	2	2	no	—
63	C₄⋊Q₈×C₅₃	—	2	4	no	—
64	C₄⋊Q₈×C₅₃	—	2	2	no	—
65	C₄⋊Q₈×C₅₃	—	2	1	no	—
66	C₄⋊Q₈×C₅₃	—	2	1	no	—
67	C₄⋊Q₈×C₅₃	—	2	2	no	—
68	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
69	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
70	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
71	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
72	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
73	#127×C₅₃	C₂³	1	1	yes	characteristic
74	#125×C₅₃	—	2	2	no	—
75	#125×C₅₃	—	2	4	no	—
76	#125×C₅₃	—	2	2	no	—
77	#125×C₅₃	—	2	1	no	—
78	#124×C₅₃	—	2	1	no	—
79	C₂²⋊Q₈×C₅₃	—	2	2	no	—
80	C₂²⋊Q₈×C₅₃	—	2	4	no	—
81	C₂²⋊Q₈×C₅₃	—	2	4	no	—
82	C₂²⋊Q₈×C₅₃	—	2	2	no	—
83	C₂²⋊Q₈×C₅₃	—	2	4	no	—
84	C₂²⋊Q₈×C₅₃	—	2	1	no	—
85	C₂²⋊Q₈×C₅₃	D₄	1	2	yes	—
86	#121×C₅₃	—	2	1	no	—
87	#121×C₅₃	—	2	1	no	—
88	C₄×Q₈×C₅₃	—	2	2	no	—
89	C₄×Q₈×C₅₃	—	2	2	no	—
90	C₄×Q₈×C₅₃	—	2	1	no	—
91	C₄×Q₈×C₅₃	—	2	2	no	—
92	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
93	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
95	C₄×D₄×C₅₃	—	2	4	no	—
96	C₄×D₄×C₅₃	—	2	2	no	—
97	C₄×D₄×C₅₃	—	2	1	no	—
98	C₄×D₄×C₅₃	—	2	1	no	—
99	C₄×D₄×C₅₃	—	2	1	no	—
100	C₄×D₄×C₅₃	—	2	1	no	—
101	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
102	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
103	#117×C₅₃	—	2	2	no	—
104	#117×C₅₃	—	2	1	no	—
105	#117×C₅₃	—	2	1	no	—
106	#117×C₅₃	D₄	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
108	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
109	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
110	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
111	C₂×C₄²×C₅₃	—	2	1	no	—
112	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
113	#107×C₅₃	—	2	4	no	—
114	#107×C₅₃	—	2	1	no	—
115	#106×C₅₃	—	2	1	no	—
116	C₄⋊C₈×C₅₃	—	2	1	no	—
117	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
118	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
119	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
120	#103×C₅₃	—	2	2	no	—
121	#103×C₅₃	—	2	1	no	—
122	#102×C₅₃	—	2	4	no	—
123	#102×C₅₃	—	2	2	no	—
124	#102×C₅₃	—	2	1	no	—
125	#102×C₅₃	—	2	1	no	—
126	#102×C₅₃	—	2	1	no	—
127	C₂²⋊C₈×C₅₃	—	2	1	no	—
128	C₂²⋊C₈×C₅₃	—	2	1	no	—
129	C₂²⋊C₈×C₅₃	—	2	1	no	—
130	#97×C₅₃	—	2	1	no	—
131	#97×C₅₃	—	2	1	no	—
132	C₄×C₈×C₅₃	C₂³	1	2	yes	—
133	#41×C₅₃	—	2	2	no	—
134	#41×C₅₃	—	2	2	no	—
135	#41×C₅₃	—	8	2	no	—
136	C₂×Q₈×C₅₃	—	2	4	no	—
137	C₂×Q₈×C₅₃	—	2	1	no	—
138	C₂×Q₈×C₅₃	—	2	2	no	—
139	C₂×Q₈×C₅₃	—	2	2	no	—
140	C₂×Q₈×C₅₃	—	2	1	no	—
141	C₂×Q₈×C₅₃	—	8	2	no	—
142	C₂×Q₈×C₅₃	—	8	2	no	—
143	C₂×Q₈×C₅₃	—	8	2	no	—
144	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
145	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
146	C₂×D₄×C₅₃	—	2	1	no	—
147	C₂×D₄×C₅₃	—	2	1	no	—
148	C₂×D₄×C₅₃	—	2	1	no	—
149	C₂×D₄×C₅₃	—	2	1	no	—
150	C₂×D₄×C₅₃	—	4	1	no	—
151	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₅₃	—	2	1	no	—
153	C₂²×C₄×C₅₃	—	2	4	no	—
154	C₂²×C₄×C₅₃	—	2	2	no	—
155	C₂²×C₄×C₅₃	—	2	1	no	—
156	C₂²×C₄×C₅₃	—	2	1	no	—
157	C₂²×C₄×C₅₃	—	4	1	no	—
158	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
161	Q₁₆×C₅₃	—	8	2	no	—
162	SD₁₆×C₅₃	—	4	2	no	—
163	SD₁₆×C₅₃	—	8	2	no	—
164	D₈×C₅₃	—	4	2	no	—
165	C₂×C₈×C₅₃	—	2	2	no	—
166	C₂×C₈×C₅₃	—	2	1	no	—
167	C₂×C₈×C₅₃	—	2	1	no	—
168	C₂×C₈×C₅₃	—	2	2	no	—
169	C₄⋊C₄×C₅₃	—	2	4	no	—
170	C₄⋊C₄×C₅₃	—	2	4	no	—
171	C₄⋊C₄×C₅₃	—	2	4	no	—
172	C₄⋊C₄×C₅₃	—	2	4	no	—
173	C₄⋊C₄×C₅₃	—	2	2	no	—
174	C₄⋊C₄×C₅₃	—	2	1	no	—
175	C₄⋊C₄×C₅₃	—	2	1	no	—
176	C₄⋊C₄×C₅₃	—	2	2	no	—
177	C₄⋊C₄×C₅₃	—	2	1	no	—
178	C₄⋊C₄×C₅₃	—	2	1	no	—
179	C₄⋊C₄×C₅₃	—	2	2	no	—
180	C₄⋊C₄×C₅₃	—	2	2	no	—
181	C₄⋊C₄×C₅₃	—	2	1	no	—
182	C₄⋊C₄×C₅₃	—	2	2	no	—
183	C₄⋊C₄×C₅₃	—	2	1	no	—
184	C₄⋊C₄×C₅₃	—	2	1	no	—
185	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
186	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
187	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
188	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
189	C₂²⋊C₄×C₅₃	—	2	4	no	—
190	C₂²⋊C₄×C₅₃	—	2	2	no	—
191	C₂²⋊C₄×C₅₃	—	2	1	no	—
192	C₂²⋊C₄×C₅₃	—	2	1	no	—
193	C₂²⋊C₄×C₅₃	—	2	1	no	—
194	C₂²⋊C₄×C₅₃	—	4	1	no	—
195	C₂²⋊C₄×C₅₃	—	4	1	no	—
196	C₄²×C₅₃	—	2	4	no	—
197	C₄²×C₅₃	—	2	1	no	—
198	C₄²×C₅₃	—	2	1	no	—
199	C₄²×C₅₃	—	2	1	no	—
200	C₄²×C₅₃	—	2	2	no	—
201	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
202	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
203	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
204	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
205	C₂³×C₅₃	—	2	1	no	—
206	C₂³×C₅₃	—	4	1	no	—
207	C₂³×C₅₃	#120	1	1	yes	characteristic
208	Q₈×C₅₃	—	2	1	no	—
209	Q₈×C₅₃	—	2	2	no	—
210	Q₈×C₅₃	—	2	2	no	—
211	Q₈×C₅₃	—	4	2	no	—
212	Q₈×C₅₃	—	4	2	no	—
213	Q₈×C₅₃	—	4	2	no	—
214	Q₈×C₅₃	—	4	2	no	—
215	Q₈×C₅₃	—	8	2	no	—
216	Q₈×C₅₃	—	8	4	no	—
217	Q₈×C₅₃	#137	1	2	yes	—
218	D₄×C₅₃	—	2	4	no	—
219	D₄×C₅₃	—	2	1	no	—
220	D₄×C₅₃	—	4	2	no	—
221	D₄×C₅₃	—	4	1	no	—
222	D₄×C₅₃	—	4	1	no	—
223	D₄×C₅₃	—	4	2	no	—
224	D₄×C₅₃	—	8	2	no	—
225	D₄×C₅₃	#137	1	2	yes	—
226	C₂×C₄×C₅₃	—	2	4	no	—
227	C₂×C₄×C₅₃	—	2	1	no	—
228	C₂×C₄×C₅₃	—	2	1	no	—
229	C₂×C₄×C₅₃	—	2	1	no	—
230	C₂×C₄×C₅₃	—	2	2	no	—
231	C₂×C₄×C₅₃	—	2	4	no	—
232	C₂×C₄×C₅₃	—	2	2	no	—
233	C₂×C₄×C₅₃	—	2	1	no	—
234	C₂×C₄×C₅₃	—	2	1	no	—
235	C₂×C₄×C₅₃	—	2	2	no	—
236	C₂×C₄×C₅₃	—	4	1	no	—
237	C₂×C₄×C₅₃	—	4	2	no	—
238	C₂×C₄×C₅₃	—	4	2	no	—
239	C₂×C₄×C₅₃	—	4	1	no	—
240	C₂×C₄×C₅₃	—	8	2	no	—
241	C₂×C₄×C₅₃	—	8	2	no	—
242	C₂×C₄×C₅₃	—	8	4	no	—
243	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
244	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
245	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
246	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
247	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
248	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
249	C₈×C₅₃	—	4	2	no	—
250	C₈×C₅₃	—	4	1	no	—
251	C₈×C₅₃	—	4	1	no	—
252	C₈×C₅₃	—	4	2	no	—
253	#18828	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
254	C₂²×C₅₃	—	2	1	no	—
255	C₂²×C₅₃	—	2	2	no	—
256	C₂²×C₅₃	—	2	1	no	—
257	C₂²×C₅₃	—	2	1	no	—
258	C₂²×C₅₃	—	4	1	no	—
259	C₂²×C₅₃	—	4	1	no	—
260	C₂²×C₅₃	—	8	1	no	—
261	C₂²×C₅₃	—	8	1	no	—
262	C₂²×C₅₃	—	8	1	no	—
263	C₂²×C₅₃	C₂×#120	1	1	yes	characteristic
264	C₄×C₅₃	—	2	1	no	—
265	C₄×C₅₃	—	2	2	no	—
266	C₄×C₅₃	—	2	1	no	—
267	C₄×C₅₃	—	2	1	no	—
268	C₄×C₅₃	—	2	1	no	—
269	C₄×C₅₃	—	4	1	no	—
270	C₄×C₅₃	—	4	2	no	—
271	C₄×C₅₃	—	4	4	no	—
272	C₄×C₅₃	—	4	1	no	—
273	C₄×C₅₃	—	4	2	no	—
274	C₄×C₅₃	—	8	1	no	—
275	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
276	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
277	C₄×C₅₃	#576	1	2	yes	—
278	#3503	C₂×C₅₃	1	1	yes	characteristic, complemented
279	#3226	C₂×C₅₃	1	1	yes	characteristic, complemented
280	#3216	C₂×C₅₃	1	1	yes	characteristic, complemented
281	#3115	C₂×C₅₃	1	1	yes	characteristic
282	#1649	C₂×C₅₃	1	2	yes	complemented
283	#1648	C₂×C₅₃	1	2	yes	complemented
284	#1627	C₂×C₅₃	1	2	yes	complemented
285	#1626	C₂×C₅₃	1	2	yes	complemented
286	#1556	C₂×C₅₃	1	1	yes	characteristic, complemented
287	#1497	C₂×C₅₃	1	1	yes	characteristic, complemented
288	#1488	C₂×C₅₃	1	1	yes	characteristic, complemented
289	C₂×C₅₃	—	4	1	no	—
290	C₂×C₅₃	—	4	2	no	—
291	C₂×C₅₃	—	16	1	no	—
292	C₂×C₅₃	#3004	1	1	yes	characteristic
293	C₂×C₅₃	#3008	1	1	yes	characteristic
294	C₂×C₅₃	#3020	1	1	yes	characteristic
295	Q₈²	—	2	1	no	—
296	#557	C₂²×C₅₃	1	2	yes	—
297	#554	—	2	1	no	—
298	#554	—	2	2	no	—
299	D₄×Q₈	—	2	2	no	—
300	D₄×Q₈	C₂²×C₅₃	1	2	yes	—
301	#544	—	2	2	no	—
302	#544	—	2	1	no	—
303	#533	—	2	1	no	—
304	#533	C₂²×C₅₃	1	1	yes	characteristic
305	#532	C₂²×C₅₃	1	1	yes	characteristic
306	C₂×C₄⋊Q₈	—	2	1	no	—
307	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
308	#520	—	2	1	no	—
309	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
310	#500	C₂²×C₅₃	1	2	yes	—
311	#497	—	2	1	no	—
312	#489	—	2	1	no	—
313	#486	C₂²×C₅₃	1	2	yes	—
314	#482	—	2	1	no	—
315	#480	—	2	2	no	—
316	#475	C₂²×C₅₃	1	1	yes	characteristic
317	#474	—	2	2	no	—
318	#474	C₂²×C₅₃	1	1	yes	characteristic
319	#462	C₂²×C₅₃	1	2	yes	—
320	#460	C₂²×C₅₃	1	1	yes	characteristic
321	#460	C₂²×C₅₃	1	2	yes	—
322	#459	C₂²×C₅₃	1	1	yes	characteristic
323	#452	—	2	2	no	—
324	#449	—	2	1	no	—
325	#442	—	2	1	no	—
326	#440	—	2	1	no	—
327	#432	C₂²×C₅₃	1	1	yes	characteristic
328	#432	C₂²×C₅₃	1	1	yes	characteristic
329	#431	C₂²×C₅₃	1	1	yes	characteristic
330	#332	C₂²×C₅₃	1	2	yes	—
331	#332	C₂²×C₅₃	1	2	yes	—
332	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	—
333	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	—
334	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	—
335	D₄⋊C₈	C₂²×C₅₃	1	2	yes	—
336	C₅₃	#18828	1	1	yes	characteristic, direct product factor, Sylow
337	C₂×#41	—	2	1	no	—
338	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
339	C₂²×Q₈	—	2	2	no	—
340	C₂²×Q₈	—	2	1	no	—
341	C₂×Q₁₆	—	2	2	no	—
342	C₂×SD₁₆	—	2	1	no	—
343	C₂×SD₁₆	—	2	2	no	—
344	C₂×D₈	—	2	1	no	—
345	C₄⋊Q₈	—	2	2	no	—
346	C₄⋊Q₈	—	2	2	no	—
347	C₄⋊Q₈	—	2	4	no	—
348	C₄⋊Q₈	—	2	2	no	—
349	C₄⋊Q₈	—	2	2	no	—
350	C₄⋊Q₈	—	2	1	no	—
351	C₄⋊Q₈	—	2	1	no	—
352	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
353	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
354	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
355	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
356	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
357	#127	C₂³×C₅₃	1	1	yes	characteristic
358	#125	—	2	2	no	—
359	#125	—	2	4	no	—
360	#125	—	2	2	no	—
361	#125	—	2	1	no	—
362	#124	—	2	1	no	—
363	C₂²⋊Q₈	—	2	2	no	—
364	C₂²⋊Q₈	—	2	4	no	—
365	C₂²⋊Q₈	—	2	4	no	—
366	C₂²⋊Q₈	—	2	1	no	—
367	C₂²⋊Q₈	—	2	2	no	—
368	C₂²⋊Q₈	—	2	4	no	—
369	C₂²⋊Q₈	D₄×C₅₃	1	2	yes	—
370	#121	—	2	1	no	—
371	#121	—	2	1	no	—
372	C₄×Q₈	—	2	2	no	—
373	C₄×Q₈	—	2	2	no	—
374	C₄×Q₈	—	2	2	no	—
375	C₄×Q₈	—	2	1	no	—
376	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
377	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
378	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
379	C₄×D₄	—	2	4	no	—
380	C₄×D₄	—	2	2	no	—
381	C₄×D₄	—	2	1	no	—
382	C₄×D₄	—	2	1	no	—
383	C₄×D₄	—	2	1	no	—
384	C₄×D₄	—	2	1	no	—
385	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
386	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
387	#117	—	2	2	no	—
388	#117	—	2	1	no	—
389	#117	—	2	1	no	—
390	#117	D₄×C₅₃	1	1	yes	characteristic
391	C₂×C₄⋊C₄	—	2	2	no	—
392	C₂×C₄⋊C₄	—	2	2	no	—
393	C₂×C₄⋊C₄	—	2	1	no	—
394	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
395	C₂×C₄²	—	2	1	no	—
396	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
397	#107	—	2	1	no	—
398	#107	—	2	4	no	—
399	#106	—	2	1	no	—
400	C₄⋊C₈	—	2	1	no	—
401	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
402	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
403	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
404	#103	—	2	1	no	—
405	#103	—	2	2	no	—
406	#102	—	2	1	no	—
407	#102	—	2	1	no	—
408	#102	—	2	1	no	—
409	#102	—	2	2	no	—
410	#102	—	2	4	no	—
411	C₂²⋊C₈	—	2	1	no	—
412	C₂²⋊C₈	—	2	1	no	—
413	C₂²⋊C₈	—	2	1	no	—
414	#97	—	2	1	no	—
415	#97	—	2	1	no	—
416	C₄×C₈	C₂³×C₅₃	1	2	yes	—
417	#41	—	2	2	no	—
418	#41	—	2	2	no	—
419	#41	—	8	2	no	—
420	C₂×Q₈	—	2	4	no	—
421	C₂×Q₈	—	2	1	no	—
422	C₂×Q₈	—	2	2	no	—
423	C₂×Q₈	—	2	2	no	—
424	C₂×Q₈	—	2	1	no	—
425	C₂×Q₈	—	8	2	no	—
426	C₂×Q₈	—	8	2	no	—
427	C₂×Q₈	—	8	2	no	—
428	C₂×Q₈	C₂×D₄×C₅₃	1	2	yes	—
429	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
430	C₂×D₄	—	2	1	no	—
431	C₂×D₄	—	2	1	no	—
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	—	2	1	no	—
434	C₂×D₄	—	4	1	no	—
435	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
436	C₂²×C₄	—	2	1	no	—
437	C₂²×C₄	—	2	4	no	—
438	C₂²×C₄	—	2	2	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	—	4	1	no	—
442	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
443	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
444	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
445	Q₁₆	—	8	2	no	—
446	SD₁₆	—	4	2	no	—
447	SD₁₆	—	8	2	no	—
448	D₈	—	4	2	no	—
449	C₂×C₈	—	2	2	no	—
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	1	no	—
452	C₂×C₈	—	2	2	no	—
453	C₄⋊C₄	—	2	4	no	—
454	C₄⋊C₄	—	2	4	no	—
455	C₄⋊C₄	—	2	4	no	—
456	C₄⋊C₄	—	2	4	no	—
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	—	2	2	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	2	no	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	—	2	1	no	—
463	C₄⋊C₄	—	2	2	no	—
464	C₄⋊C₄	—	2	2	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	2	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
470	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
471	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
472	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
473	C₂²⋊C₄	—	2	4	no	—
474	C₂²⋊C₄	—	2	2	no	—
475	C₂²⋊C₄	—	2	1	no	—
476	C₂²⋊C₄	—	2	1	no	—
477	C₂²⋊C₄	—	2	1	no	—
478	C₂²⋊C₄	—	4	1	no	—
479	C₂²⋊C₄	—	4	1	no	—
480	C₄²	—	2	4	no	—
481	C₄²	—	2	1	no	—
482	C₄²	—	2	1	no	—
483	C₄²	—	2	1	no	—
484	C₄²	—	2	2	no	—
485	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
486	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
487	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
488	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
489	C₂³	—	2	1	no	—
490	C₂³	—	4	1	no	—
491	C₂³	#120×C₅₃	1	1	yes	characteristic
492	Q₈	—	2	1	no	—
493	Q₈	—	2	2	no	—
494	Q₈	—	2	2	no	—
495	Q₈	—	4	2	no	—
496	Q₈	—	4	2	no	—
497	Q₈	—	4	2	no	—
498	Q₈	—	4	2	no	—
499	Q₈	—	8	2	no	—
500	Q₈	—	8	4	no	—
501	Q₈	#137×C₅₃	1	2	yes	—
502	D₄	—	2	4	no	—
503	D₄	—	2	1	no	—
504	D₄	—	4	2	no	—
505	D₄	—	4	1	no	—
506	D₄	—	4	1	no	—
507	D₄	—	4	2	no	—
508	D₄	—	8	2	no	—
509	D₄	#137×C₅₃	1	2	yes	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	4	1	no	—
521	C₂×C₄	—	4	2	no	—
522	C₂×C₄	—	4	2	no	—
523	C₂×C₄	—	4	1	no	—
524	C₂×C₄	—	8	2	no	—
525	C₂×C₄	—	8	2	no	—
526	C₂×C₄	—	8	4	no	—
527	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
528	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
529	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
530	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
531	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
532	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
533	C₈	—	4	2	no	—
534	C₈	—	4	1	no	—
535	C₈	—	4	1	no	—
536	C₈	—	4	2	no	—
537	C₂²	—	2	1	no	—
538	C₂²	—	2	2	no	—
539	C₂²	—	2	1	no	—
540	C₂²	—	2	1	no	—
541	C₂²	—	4	1	no	—
542	C₂²	—	4	1	no	—
543	C₂²	—	8	1	no	—
544	C₂²	—	8	1	no	—
545	C₂²	—	8	1	no	—
546	C₂²	C₂×#120×C₅₃	1	1	yes	characteristic
547	C₄	—	2	1	no	—
548	C₄	—	2	2	no	—
549	C₄	—	2	1	no	—
550	C₄	—	2	1	no	—
551	C₄	—	2	1	no	—
552	C₄	—	4	1	no	—
553	C₄	—	4	2	no	—
554	C₄	—	4	4	no	—
555	C₄	—	4	1	no	—
556	C₄	—	4	2	no	—
557	C₄	—	8	1	no	—
558	C₄	C₂×#137×C₅₃	1	1	yes	characteristic
559	C₄	C₂×#137×C₅₃	1	1	yes	characteristic
560	C₄	C₅₃×#576	1	2	yes	—
561	C₂	—	4	1	no	—
562	C₂	—	4	2	no	—
563	C₂	—	16	1	no	—
564	C₂	C₅₃×#3004	1	1	yes	characteristic
565	C₂	C₅₃×#3008	1	1	yes	characteristic
566	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×#3115	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×#3503	C₂	semidirect
2. C₅₃×#3226	C₂	semidirect
3. C₅₃×#3216	C₂	semidirect
5. C₅₃×#1649	C₂	semidirect
6. C₅₃×#1648	C₂	semidirect
7. C₅₃×#1627	C₂	semidirect
8. C₅₃×#1626	C₂	semidirect
9. C₅₃×#1556	C₂	semidirect
10. C₅₃×#1497	C₂	semidirect
11. C₅₃×#1488	C₂	semidirect
253. #18828	C₅₃	direct (N × H)
278. #3503	C₂×C₅₃	semidirect
279. #3226	C₂×C₅₃	semidirect
280. #3216	C₂×C₅₃	semidirect
282. #1649	C₂×C₅₃	semidirect
283. #1648	C₂×C₅₃	semidirect
284. #1627	C₂×C₅₃	semidirect
285. #1626	C₂×C₅₃	semidirect
286. #1556	C₂×C₅₃	semidirect
287. #1497	C₂×C₅₃	semidirect
288. #1488	C₂×C₅₃	semidirect
336. C₅₃	#18828	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×#3503	2
C₅₃×#3226	2
C₅₃×#3216	2
C₅₃×#3115	2
C₅₃×#1649	2
C₅₃×#1648	2
C₅₃×#1627	2
C₅₃×#1626	2
C₅₃×#1556	2
C₅₃×#1497	2
C₅₃×#1488	2
#18828	53

Subgroups — C53×#18828 #27226340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#18828 #27226340