Subgroups — C₅₃×#18816 #27226328

Order 13568.

918 conjugacy classes of subgroups.

The subgroup lattice has 566 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×#3216	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3206	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3115	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1645	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1644	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1623	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1622	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1558	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	C₂²	1	2	yes	—
13	C₅₃×#557	—	2	1	no	—
14	C₅₃×#554	—	2	2	no	—
15	C₅₃×#554	C₂²	1	2	yes	—
16	D₄×Q₈×C₅₃	—	2	2	no	—
17	D₄×Q₈×C₅₃	—	2	1	no	—
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×#497	—	2	1	no	—
28	C₅₃×#492	C₂²	1	2	yes	—
29	C₅₃×#489	—	2	1	no	—
30	C₅₃×#487	C₂²	1	2	yes	—
31	C₅₃×#482	—	2	1	no	—
32	C₅₃×#475	C₂²	1	1	yes	characteristic
33	C₅₃×#474	C₂²	1	1	yes	characteristic
34	C₅₃×#462	C₂²	1	2	yes	—
35	C₅₃×#461	—	2	2	no	—
36	C₅₃×#460	C₂²	1	2	yes	—
37	C₅₃×#460	C₂²	1	1	yes	characteristic
38	C₅₃×#459	C₂²	1	1	yes	characteristic
39	C₅₃×#452	—	2	2	no	—
40	C₅₃×#449	—	2	1	no	—
41	C₅₃×#448	—	2	2	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×#333	C₂²	1	2	yes	—
48	C₅₃×#333	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	D₄	1	2	yes	—
57	C₂×Q₁₆×C₅₃	—	2	2	no	—
58	C₂×SD₁₆×C₅₃	—	2	1	no	—
59	C₂×SD₁₆×C₅₃	—	2	4	no	—
60	C₂×SD₁₆×C₅₃	—	2	2	no	—
61	C₂×D₈×C₅₃	—	2	1	no	—
62	C₄⋊Q₈×C₅₃	—	2	2	no	—
63	C₄⋊Q₈×C₅₃	—	2	2	no	—
64	C₄⋊Q₈×C₅₃	—	2	2	no	—
65	C₄⋊Q₈×C₅₃	—	2	2	no	—
66	C₄⋊Q₈×C₅₃	—	2	1	no	—
67	C₄⋊Q₈×C₅₃	—	2	1	no	—
68	C₄⋊Q₈×C₅₃	—	2	4	no	—
69	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
70	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
71	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
72	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
73	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
74	#127×C₅₃	C₂³	1	1	yes	characteristic
75	#125×C₅₃	—	2	2	no	—
76	#125×C₅₃	—	2	4	no	—
77	#125×C₅₃	—	2	2	no	—
78	#125×C₅₃	—	2	1	no	—
79	#124×C₅₃	—	2	1	no	—
80	C₂²⋊Q₈×C₅₃	—	2	2	no	—
81	C₂²⋊Q₈×C₅₃	—	2	4	no	—
82	C₂²⋊Q₈×C₅₃	—	2	4	no	—
83	C₂²⋊Q₈×C₅₃	—	2	2	no	—
84	C₂²⋊Q₈×C₅₃	—	2	4	no	—
85	C₂²⋊Q₈×C₅₃	—	2	1	no	—
86	C₂²⋊Q₈×C₅₃	D₄	1	2	yes	—
87	#121×C₅₃	—	2	1	no	—
88	#121×C₅₃	—	2	1	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	—	2	2	no	—
93	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
94	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
95	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
96	C₄×D₄×C₅₃	—	2	4	no	—
97	C₄×D₄×C₅₃	—	2	2	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	—	2	1	no	—
102	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
103	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
104	#117×C₅₃	—	2	2	no	—
105	#117×C₅₃	—	2	1	no	—
106	#117×C₅₃	—	2	1	no	—
107	#117×C₅₃	D₄	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
109	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
110	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
111	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
112	C₂×C₄²×C₅₃	—	2	1	no	—
113	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
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	4	no	—
122	#103×C₅₃	—	2	1	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	2	no	—
138	C₂×Q₈×C₅₃	—	2	1	no	—
139	C₂×Q₈×C₅₃	—	2	1	no	—
140	C₂×Q₈×C₅₃	—	8	4	no	—
141	C₂×Q₈×C₅₃	—	8	2	no	—
142	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
143	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
144	C₂×Q₈×C₅₃	C₂×D₄	1	4	yes	—
145	C₂×D₄×C₅₃	—	2	1	no	—
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	—	4	1	no	—
150	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₅₃	—	2	1	no	—
152	C₂²×C₄×C₅₃	—	2	4	no	—
153	C₂²×C₄×C₅₃	—	2	2	no	—
154	C₂²×C₄×C₅₃	—	2	1	no	—
155	C₂²×C₄×C₅₃	—	2	1	no	—
156	C₂²×C₄×C₅₃	—	4	1	no	—
157	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
160	Q₁₆×C₅₃	—	8	2	no	—
161	SD₁₆×C₅₃	—	4	2	no	—
162	SD₁₆×C₅₃	—	8	2	no	—
163	SD₁₆×C₅₃	—	8	4	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	1	no	—
174	C₄⋊C₄×C₅₃	—	2	2	no	—
175	C₄⋊C₄×C₅₃	—	2	2	no	—
176	C₄⋊C₄×C₅₃	—	2	1	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	2	no	—
182	C₄⋊C₄×C₅₃	—	2	1	no	—
183	C₄⋊C₄×C₅₃	—	2	2	no	—
184	C₄⋊C₄×C₅₃	—	2	1	no	—
185	C₄⋊C₄×C₅₃	—	2	1	no	—
186	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
187	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
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	4	no	—
212	Q₈×C₅₃	—	4	2	no	—
213	Q₈×C₅₃	—	4	4	no	—
214	Q₈×C₅₃	—	4	2	no	—
215	Q₈×C₅₃	—	8	4	no	—
216	Q₈×C₅₃	#137	1	2	yes	—
217	D₄×C₅₃	—	2	4	no	—
218	D₄×C₅₃	—	2	1	no	—
219	D₄×C₅₃	—	4	2	no	—
220	D₄×C₅₃	—	4	1	no	—
221	D₄×C₅₃	—	4	1	no	—
222	D₄×C₅₃	—	4	2	no	—
223	D₄×C₅₃	—	8	2	no	—
224	D₄×C₅₃	#137	1	2	yes	—
225	C₂×C₄×C₅₃	—	2	4	no	—
226	C₂×C₄×C₅₃	—	2	1	no	—
227	C₂×C₄×C₅₃	—	2	1	no	—
228	C₂×C₄×C₅₃	—	2	1	no	—
229	C₂×C₄×C₅₃	—	2	2	no	—
230	C₂×C₄×C₅₃	—	2	2	no	—
231	C₂×C₄×C₅₃	—	2	4	no	—
232	C₂×C₄×C₅₃	—	2	1	no	—
233	C₂×C₄×C₅₃	—	2	1	no	—
234	C₂×C₄×C₅₃	—	2	2	no	—
235	C₂×C₄×C₅₃	—	4	1	no	—
236	C₂×C₄×C₅₃	—	4	2	no	—
237	C₂×C₄×C₅₃	—	4	2	no	—
238	C₂×C₄×C₅₃	—	4	1	no	—
239	C₂×C₄×C₅₃	—	8	2	no	—
240	C₂×C₄×C₅₃	—	8	2	no	—
241	C₂×C₄×C₅₃	—	8	4	no	—
242	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
243	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
244	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
245	C₂×C₄×C₅₃	C₂²×D₄	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	C₈×C₅₃	—	4	2	no	—
249	C₈×C₅₃	—	4	1	no	—
250	C₈×C₅₃	—	4	1	no	—
251	C₈×C₅₃	—	4	2	no	—
252	#18816	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
253	C₂²×C₅₃	—	2	1	no	—
254	C₂²×C₅₃	—	2	2	no	—
255	C₂²×C₅₃	—	2	1	no	—
256	C₂²×C₅₃	—	2	1	no	—
257	C₂²×C₅₃	—	4	1	no	—
258	C₂²×C₅₃	—	4	1	no	—
259	C₂²×C₅₃	—	8	1	no	—
260	C₂²×C₅₃	—	8	1	no	—
261	C₂²×C₅₃	—	8	1	no	—
262	C₂²×C₅₃	C₂×#120	1	1	yes	characteristic
263	C₄×C₅₃	—	2	1	no	—
264	C₄×C₅₃	—	2	2	no	—
265	C₄×C₅₃	—	2	1	no	—
266	C₄×C₅₃	—	2	1	no	—
267	C₄×C₅₃	—	2	1	no	—
268	C₄×C₅₃	—	4	1	no	—
269	C₄×C₅₃	—	4	2	no	—
270	C₄×C₅₃	—	4	1	no	—
271	C₄×C₅₃	—	4	4	no	—
272	C₄×C₅₃	—	4	2	no	—
273	C₄×C₅₃	—	8	1	no	—
274	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
275	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
276	C₄×C₅₃	#577	1	2	yes	—
277	#3503	C₂×C₅₃	1	1	yes	characteristic, complemented
278	#3216	C₂×C₅₃	1	1	yes	characteristic, complemented
279	#3206	C₂×C₅₃	1	1	yes	characteristic, complemented
280	#3115	C₂×C₅₃	1	1	yes	characteristic
281	#1645	C₂×C₅₃	1	2	yes	complemented
282	#1644	C₂×C₅₃	1	2	yes	complemented
283	#1623	C₂×C₅₃	1	2	yes	complemented
284	#1622	C₂×C₅₃	1	2	yes	complemented
285	#1558	C₂×C₅₃	1	1	yes	characteristic, complemented
286	#1497	C₂×C₅₃	1	1	yes	characteristic, complemented
287	#1488	C₂×C₅₃	1	1	yes	characteristic, complemented
288	C₂×C₅₃	—	4	1	no	—
289	C₂×C₅₃	—	4	2	no	—
290	C₂×C₅₃	—	16	1	no	—
291	C₂×C₅₃	#3004	1	1	yes	characteristic
292	C₂×C₅₃	#3010	1	1	yes	characteristic
293	C₂×C₅₃	#3019	1	1	yes	characteristic
294	Q₈²	C₂²×C₅₃	1	2	yes	—
295	#557	—	2	1	no	—
296	#554	—	2	2	no	—
297	#554	C₂²×C₅₃	1	2	yes	—
298	D₄×Q₈	—	2	2	no	—
299	D₄×Q₈	—	2	1	no	—
300	#544	—	2	2	no	—
301	#544	—	2	1	no	—
302	#533	—	2	1	no	—
303	#533	C₂²×C₅₃	1	1	yes	characteristic
304	#532	C₂²×C₅₃	1	1	yes	characteristic
305	C₂×C₄⋊Q₈	—	2	1	no	—
306	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
307	#520	—	2	1	no	—
308	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
309	#497	—	2	1	no	—
310	#492	C₂²×C₅₃	1	2	yes	—
311	#489	—	2	1	no	—
312	#487	C₂²×C₅₃	1	2	yes	—
313	#482	—	2	1	no	—
314	#475	C₂²×C₅₃	1	1	yes	characteristic
315	#474	C₂²×C₅₃	1	1	yes	characteristic
316	#462	C₂²×C₅₃	1	2	yes	—
317	#461	—	2	2	no	—
318	#460	C₂²×C₅₃	1	1	yes	characteristic
319	#460	C₂²×C₅₃	1	2	yes	—
320	#459	C₂²×C₅₃	1	1	yes	characteristic
321	#452	—	2	2	no	—
322	#449	—	2	1	no	—
323	#448	—	2	2	no	—
324	#442	—	2	1	no	—
325	#440	—	2	1	no	—
326	#432	C₂²×C₅₃	1	1	yes	characteristic
327	#432	C₂²×C₅₃	1	1	yes	characteristic
328	#431	C₂²×C₅₃	1	1	yes	characteristic
329	#333	C₂²×C₅₃	1	2	yes	—
330	#333	C₂²×C₅₃	1	2	yes	—
331	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	—
332	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	—
333	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	—
334	D₄⋊C₈	C₂²×C₅₃	1	2	yes	—
335	C₅₃	#18816	1	1	yes	characteristic, direct product factor, Sylow
336	C₂×#41	—	2	1	no	—
337	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
338	C₂²×Q₈	—	2	2	no	—
339	C₂²×Q₈	D₄×C₅₃	1	2	yes	—
340	C₂×Q₁₆	—	2	2	no	—
341	C₂×SD₁₆	—	2	1	no	—
342	C₂×SD₁₆	—	2	4	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	2	yes	—
353	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
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	2	no	—
367	C₂²⋊Q₈	—	2	4	no	—
368	C₂²⋊Q₈	—	2	1	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	1	no	—
393	C₂×C₄⋊C₄	—	2	1	no	—
394	C₂×C₄⋊C₄	—	2	2	no	—
395	C₂×C₄²	—	2	1	no	—
396	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
397	#107	—	2	1	no	—
398	#106	—	2	1	no	—
399	C₄⋊C₈	—	2	1	no	—
400	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
401	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
402	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
403	#103	—	2	1	no	—
404	#103	—	2	4	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	C₂²⋊C₈	—	2	1	no	—
411	C₂²⋊C₈	—	2	1	no	—
412	C₂²⋊C₈	—	2	1	no	—
413	#97	—	2	1	no	—
414	#97	—	2	1	no	—
415	C₄×C₈	C₂³×C₅₃	1	2	yes	—
416	#41	—	2	2	no	—
417	#41	—	2	2	no	—
418	#41	—	8	2	no	—
419	C₂×Q₈	—	2	4	no	—
420	C₂×Q₈	—	2	2	no	—
421	C₂×Q₈	—	2	1	no	—
422	C₂×Q₈	—	2	1	no	—
423	C₂×Q₈	—	8	4	no	—
424	C₂×Q₈	—	8	2	no	—
425	C₂×Q₈	C₂×D₄×C₅₃	1	2	yes	—
426	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
427	C₂×Q₈	C₂×D₄×C₅₃	1	4	yes	—
428	C₂×D₄	—	2	1	no	—
429	C₂×D₄	—	2	1	no	—
430	C₂×D₄	—	2	1	no	—
431	C₂×D₄	—	2	1	no	—
432	C₂×D₄	—	4	1	no	—
433	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
434	C₂²×C₄	—	2	1	no	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	2	2	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	4	1	no	—
440	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
441	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
442	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
443	Q₁₆	—	8	2	no	—
444	SD₁₆	—	4	2	no	—
445	SD₁₆	—	8	2	no	—
446	SD₁₆	—	8	4	no	—
447	D₈	—	4	2	no	—
448	C₂×C₈	—	2	2	no	—
449	C₂×C₈	—	2	1	no	—
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	2	no	—
452	C₄⋊C₄	—	2	4	no	—
453	C₄⋊C₄	—	2	4	no	—
454	C₄⋊C₄	—	2	4	no	—
455	C₄⋊C₄	—	2	4	no	—
456	C₄⋊C₄	—	2	2	no	—
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	—	2	2	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	—	2	2	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	1	yes	characteristic
472	C₂²⋊C₄	—	2	4	no	—
473	C₂²⋊C₄	—	2	2	no	—
474	C₂²⋊C₄	—	2	1	no	—
475	C₂²⋊C₄	—	2	1	no	—
476	C₂²⋊C₄	—	2	1	no	—
477	C₂²⋊C₄	—	4	1	no	—
478	C₂²⋊C₄	—	4	1	no	—
479	C₄²	—	2	4	no	—
480	C₄²	—	2	1	no	—
481	C₄²	—	2	1	no	—
482	C₄²	—	2	1	no	—
483	C₄²	—	2	2	no	—
484	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
485	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
486	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
487	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
488	C₂³	—	2	1	no	—
489	C₂³	—	4	1	no	—
490	C₂³	#120×C₅₃	1	1	yes	characteristic
491	Q₈	—	2	1	no	—
492	Q₈	—	2	2	no	—
493	Q₈	—	2	2	no	—
494	Q₈	—	4	4	no	—
495	Q₈	—	4	4	no	—
496	Q₈	—	4	2	no	—
497	Q₈	—	4	2	no	—
498	Q₈	—	8	4	no	—
499	Q₈	#137×C₅₃	1	2	yes	—
500	D₄	—	2	4	no	—
501	D₄	—	2	1	no	—
502	D₄	—	4	2	no	—
503	D₄	—	4	1	no	—
504	D₄	—	4	1	no	—
505	D₄	—	4	2	no	—
506	D₄	—	8	2	no	—
507	D₄	#137×C₅₃	1	2	yes	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	4	1	no	—
519	C₂×C₄	—	4	2	no	—
520	C₂×C₄	—	4	2	no	—
521	C₂×C₄	—	4	1	no	—
522	C₂×C₄	—	8	2	no	—
523	C₂×C₄	—	8	2	no	—
524	C₂×C₄	—	8	4	no	—
525	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
526	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
527	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
528	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
529	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
530	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
531	C₈	—	4	2	no	—
532	C₈	—	4	1	no	—
533	C₈	—	4	1	no	—
534	C₈	—	4	2	no	—
535	C₂²	—	2	1	no	—
536	C₂²	—	2	2	no	—
537	C₂²	—	2	1	no	—
538	C₂²	—	2	1	no	—
539	C₂²	—	4	1	no	—
540	C₂²	—	4	1	no	—
541	C₂²	—	8	1	no	—
542	C₂²	—	8	1	no	—
543	C₂²	—	8	1	no	—
544	C₂²	C₂×#120×C₅₃	1	1	yes	characteristic
545	C₄	—	2	1	no	—
546	C₄	—	2	2	no	—
547	C₄	—	2	1	no	—
548	C₄	—	2	1	no	—
549	C₄	—	2	1	no	—
550	C₄	—	4	1	no	—
551	C₄	—	4	2	no	—
552	C₄	—	4	1	no	—
553	C₄	—	4	4	no	—
554	C₄	—	4	2	no	—
555	C₄	—	8	1	no	—
556	C₄	C₂×#137×C₅₃	1	1	yes	characteristic
557	C₄	C₂×#137×C₅₃	1	1	yes	characteristic
558	C₄	C₅₃×#577	1	2	yes	—
559	C₂	—	4	1	no	—
560	C₂	—	4	2	no	—
561	C₂	—	16	1	no	—
562	C₂	C₅₃×#3004	1	1	yes	characteristic
563	C₂	C₅₃×#3010	1	1	yes	characteristic
564	C₂	C₅₃×#3019	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×#3216	C₂	semidirect
3. C₅₃×#3206	C₂	semidirect
5. C₅₃×#1645	C₂	semidirect
6. C₅₃×#1644	C₂	semidirect
7. C₅₃×#1623	C₂	semidirect
8. C₅₃×#1622	C₂	semidirect
9. C₅₃×#1558	C₂	semidirect
10. C₅₃×#1497	C₂	semidirect
11. C₅₃×#1488	C₂	semidirect
252. #18816	C₅₃	direct (N × H)
277. #3503	C₂×C₅₃	semidirect
278. #3216	C₂×C₅₃	semidirect
279. #3206	C₂×C₅₃	semidirect
281. #1645	C₂×C₅₃	semidirect
282. #1644	C₂×C₅₃	semidirect
283. #1623	C₂×C₅₃	semidirect
284. #1622	C₂×C₅₃	semidirect
285. #1558	C₂×C₅₃	semidirect
286. #1497	C₂×C₅₃	semidirect
287. #1488	C₂×C₅₃	semidirect
335. C₅₃	#18816	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×#3216	2
C₅₃×#3206	2
C₅₃×#3115	2
C₅₃×#1645	2
C₅₃×#1644	2
C₅₃×#1623	2
C₅₃×#1622	2
C₅₃×#1558	2
C₅₃×#1497	2
C₅₃×#1488	2
#18816	53

Subgroups — C53×#18816 #27226328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#18816 #27226328