Subgroups — C₅₃×#19233 #27226745

Order 13568.

640 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3246	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#3227	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#3165	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2935	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1695	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1691	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1675	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1671	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1565	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1542	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅₃×#333	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#131×C₅₃	—	2	1	no	—
13	C₅₃×#533	C₂²	1	1	yes	characteristic
14	C₅₃×#532	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#501	—	2	2	no	—
17	C₅₃×#500	C₂²	1	2	yes	—
18	C₅₃×#498	—	2	1	no	—
19	C₅₃×#498	C₂²	1	2	yes	—
20	C₅₃×#486	—	2	1	no	—
21	C₅₃×#481	—	2	2	no	—
22	C₅₃×#476	C₂²	1	2	yes	—
23	C₅₃×#476	C₂²	1	2	yes	—
24	C₅₃×#476	C₂²	1	2	yes	—
25	C₅₃×#475	—	2	1	no	—
26	C₅₃×#465	—	2	2	no	—
27	C₅₃×#461	C₂²	1	2	yes	—
28	C₅₃×#460	—	2	1	no	—
29	C₅₃×#436	C₂²	1	2	yes	—
30	C₅₃×#435	—	2	2	no	—
31	C₈×D₄×C₅₃	C₂²	1	2	yes	—
32	C₅₃×#431	C₂²	1	1	yes	characteristic
33	C₅₃×#428	—	2	1	no	—
34	C₅₃×#428	—	2	1	no	—
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×#333	C₂²	1	2	yes	—
42	C₅₃×#333	C₂²	1	1	yes	characteristic
43	C₅₃×#333	C₂²	1	2	yes	—
44	C₅₃×#333	C₂²	1	1	yes	characteristic
45	C₅₃×#332	C₂²	1	1	yes	characteristic
46	C₅₃×#332	C₂²	1	1	yes	characteristic
47	C₂×#41×C₅₃	—	2	1	no	—
48	C₂²×Q₈×C₅₃	—	2	1	no	—
49	C₂×SD₁₆×C₅₃	—	2	2	no	—
50	#131×C₅₃	—	4	4	no	—
51	C₂×M₁₆×C₅₃	—	2	2	no	—
52	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
53	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	—	2	2	no	—
55	C₂²×C₈×C₅₃	—	2	1	no	—
56	C₂²×C₈×C₅₃	—	2	1	no	—
57	C₄⋊Q₈×C₅₃	—	2	1	no	—
58	C₄⋊Q₈×C₅₃	—	2	1	no	—
59	C₄⋊Q₈×C₅₃	—	2	1	no	—
60	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
61	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
63	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
64	#127×C₅₃	D₄	1	1	yes	characteristic
65	#125×C₅₃	—	2	1	no	—
66	#124×C₅₃	—	2	1	no	—
67	C₂²⋊Q₈×C₅₃	—	2	1	no	—
68	C₂²⋊Q₈×C₅₃	—	2	1	no	—
69	#121×C₅₃	—	2	1	no	—
70	#121×C₅₃	D₄	1	2	yes	—
71	C₄×Q₈×C₅₃	—	2	1	no	—
72	C₄×Q₈×C₅₃	—	2	1	no	—
73	C₄×D₄×C₅₃	—	2	1	no	—
74	C₄×D₄×C₅₃	C₂³	1	2	yes	—
75	#117×C₅₃	—	2	1	no	—
76	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
77	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
78	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
79	#107×C₅₃	—	2	2	no	—
80	#107×C₅₃	—	2	2	no	—
81	#106×C₅₃	—	2	2	no	—
82	#106×C₅₃	—	2	2	no	—
83	#106×C₅₃	—	2	2	no	—
84	#106×C₅₃	—	2	2	no	—
85	#106×C₅₃	—	2	2	no	—
86	C₄⋊C₈×C₅₃	—	2	1	no	—
87	C₄⋊C₈×C₅₃	—	2	1	no	—
88	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
89	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
90	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
92	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
93	#103×C₅₃	—	2	2	no	—
94	#102×C₅₃	—	2	2	no	—
95	#102×C₅₃	—	2	2	no	—
96	#102×C₅₃	—	2	2	no	—
97	C₂²⋊C₈×C₅₃	—	2	2	no	—
98	C₂²⋊C₈×C₅₃	—	2	2	no	—
99	C₂²⋊C₈×C₅₃	—	2	1	no	—
100	#97×C₅₃	D₄	1	2	yes	—
101	C₄×C₈×C₅₃	—	2	1	no	—
102	C₄×C₈×C₅₃	C₂³	1	2	yes	—
103	#41×C₅₃	—	4	2	no	—
104	#41×C₅₃	—	4	2	no	—
105	C₂×Q₈×C₅₃	—	2	1	no	—
106	C₂×Q₈×C₅₃	—	2	1	no	—
107	C₂×Q₈×C₅₃	—	2	1	no	—
108	C₂×Q₈×C₅₃	—	2	2	no	—
109	C₂×Q₈×C₅₃	—	4	1	no	—
110	C₂×Q₈×C₅₃	—	8	2	no	—
111	C₂×D₄×C₅₃	—	2	2	no	—
112	C₂×D₄×C₅₃	—	2	1	no	—
113	C₂×D₄×C₅₃	—	2	1	no	—
114	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₅₃	—	2	2	no	—
116	C₂²×C₄×C₅₃	—	4	1	no	—
117	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
120	SD₁₆×C₅₃	—	8	2	no	—
121	M₁₆×C₅₃	—	2	2	no	—
122	M₁₆×C₅₃	—	4	4	no	—
123	M₁₆×C₅₃	SD₁₆	1	4	yes	complemented
124	C₂×C₈×C₅₃	—	2	1	no	—
125	C₂×C₈×C₅₃	—	2	1	no	—
126	C₂×C₈×C₅₃	—	2	1	no	—
127	C₂×C₈×C₅₃	—	2	1	no	—
128	C₂×C₈×C₅₃	—	2	2	no	—
129	C₂×C₈×C₅₃	—	4	2	no	—
130	C₂×C₈×C₅₃	—	4	4	no	—
131	C₂×C₈×C₅₃	—	8	1	no	—
132	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
133	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
134	C₄⋊C₄×C₅₃	—	2	1	no	—
135	C₄⋊C₄×C₅₃	—	2	2	no	—
136	C₄⋊C₄×C₅₃	—	2	1	no	—
137	C₄⋊C₄×C₅₃	—	2	1	no	—
138	C₄⋊C₄×C₅₃	—	2	2	no	—
139	C₄⋊C₄×C₅₃	—	2	1	no	—
140	C₄⋊C₄×C₅₃	—	2	1	no	—
141	C₄⋊C₄×C₅₃	—	4	1	no	—
142	C₄⋊C₄×C₅₃	—	4	1	no	—
143	C₄⋊C₄×C₅₃	—	4	1	no	—
144	C₄⋊C₄×C₅₃	—	4	1	no	—
145	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
146	C₂²⋊C₄×C₅₃	—	2	2	no	—
147	C₂²⋊C₄×C₅₃	—	2	2	no	—
148	C₂²⋊C₄×C₅₃	—	4	1	no	—
149	C₄²×C₅₃	—	4	1	no	—
150	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
151	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
152	C₄²×C₅₃	#41	1	1	yes	characteristic
153	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
154	C₂³×C₅₃	—	2	2	no	—
155	C₂³×C₅₃	#121	1	1	yes	characteristic
156	Q₈×C₅₃	—	4	2	no	—
157	Q₈×C₅₃	—	4	1	no	—
158	Q₈×C₅₃	—	4	1	no	—
159	Q₈×C₅₃	—	8	2	no	—
160	Q₈×C₅₃	—	8	1	no	—
161	Q₈×C₅₃	—	8	2	no	—
162	D₄×C₅₃	—	2	2	no	—
163	D₄×C₅₃	—	2	2	no	—
164	D₄×C₅₃	—	4	4	no	—
165	D₄×C₅₃	—	4	2	no	—
166	D₄×C₅₃	—	8	1	no	—
167	C₂×C₄×C₅₃	—	2	1	no	—
168	C₂×C₄×C₅₃	—	2	1	no	—
169	C₂×C₄×C₅₃	—	2	2	no	—
170	C₂×C₄×C₅₃	—	4	1	no	—
171	C₂×C₄×C₅₃	—	4	2	no	—
172	C₂×C₄×C₅₃	—	4	2	no	—
173	C₂×C₄×C₅₃	—	4	2	no	—
174	C₂×C₄×C₅₃	—	4	1	no	—
175	C₂×C₄×C₅₃	—	4	1	no	—
176	C₂×C₄×C₅₃	—	8	2	no	—
177	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
178	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
179	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
180	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
181	C₂×C₄×C₅₃	#137	1	2	yes	—
182	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
185	C₈×C₅₃	—	2	2	no	—
186	C₈×C₅₃	—	4	1	no	—
187	C₈×C₅₃	—	4	1	no	—
188	C₈×C₅₃	—	4	1	no	—
189	C₈×C₅₃	—	4	2	no	—
190	C₈×C₅₃	—	4	1	no	—
191	C₈×C₅₃	C₂×SD₁₆	1	4	yes	—
192	#19233	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
193	C₂²×C₅₃	—	2	1	no	—
194	C₂²×C₅₃	—	2	1	no	—
195	C₂²×C₅₃	—	4	2	no	—
196	C₂²×C₅₃	—	4	2	no	—
197	C₂²×C₅₃	—	4	2	no	—
198	C₂²×C₅₃	#461	1	2	yes	—
199	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
200	C₄×C₅₃	—	2	1	no	—
201	C₄×C₅₃	—	2	1	no	—
202	C₄×C₅₃	—	2	1	no	—
203	C₄×C₅₃	—	4	2	no	—
204	C₄×C₅₃	—	8	1	no	—
205	C₄×C₅₃	—	8	2	no	—
206	C₄×C₅₃	—	8	1	no	—
207	C₄×C₅₃	#461	1	2	yes	—
208	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
209	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
210	C₄×C₅₃	#577	1	1	yes	characteristic
211	C₄×C₅₃	#578	1	1	yes	characteristic
212	#3246	C₂×C₅₃	1	1	yes	characteristic
213	#3227	C₂×C₅₃	1	1	yes	characteristic
214	#3165	C₂×C₅₃	1	1	yes	characteristic, complemented
215	#2935	C₂×C₅₃	1	1	yes	characteristic
216	#1695	C₂×C₅₃	1	2	yes	complemented
217	#1691	C₂×C₅₃	1	2	yes	complemented
218	#1675	C₂×C₅₃	1	2	yes	complemented
219	#1671	C₂×C₅₃	1	2	yes	complemented
220	#1565	C₂×C₅₃	1	1	yes	characteristic, complemented
221	#1542	C₂×C₅₃	1	1	yes	characteristic, complemented
222	C₂×#333	C₂×C₅₃	1	1	yes	characteristic, complemented
223	C₂×C₅₃	—	2	2	no	—
224	C₂×C₅₃	—	8	2	no	—
225	C₂×C₅₃	C₂×#461	1	1	yes	characteristic
226	C₂×C₅₃	#3063	1	1	yes	characteristic
227	C₂×C₅₃	#3068	1	1	yes	characteristic
228	C₂×#131	—	2	1	no	—
229	#533	C₂²×C₅₃	1	1	yes	characteristic
230	#532	C₂²×C₅₃	1	1	yes	characteristic
231	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
232	#501	—	2	2	no	—
233	#500	C₂²×C₅₃	1	2	yes	—
234	#498	—	2	1	no	—
235	#498	C₂²×C₅₃	1	2	yes	—
236	#486	—	2	1	no	—
237	#481	—	2	2	no	—
238	#476	C₂²×C₅₃	1	2	yes	—
239	#476	C₂²×C₅₃	1	2	yes	—
240	#476	C₂²×C₅₃	1	2	yes	—
241	#475	—	2	1	no	—
242	#465	—	2	2	no	—
243	#461	C₂²×C₅₃	1	2	yes	—
244	#460	—	2	1	no	—
245	#436	C₂²×C₅₃	1	2	yes	—
246	#435	—	2	2	no	—
247	C₈×D₄	C₂²×C₅₃	1	2	yes	—
248	#431	C₂²×C₅₃	1	1	yes	characteristic
249	#428	—	2	1	no	—
250	#428	—	2	1	no	—
251	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
252	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
253	#408	—	2	1	no	—
254	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
255	#335	C₂²×C₅₃	1	2	yes	—
256	#334	C₂²×C₅₃	1	2	yes	—
257	#333	C₂²×C₅₃	1	2	yes	—
258	#333	C₂²×C₅₃	1	2	yes	—
259	#333	C₂²×C₅₃	1	1	yes	characteristic
260	#333	C₂²×C₅₃	1	1	yes	characteristic
261	#332	C₂²×C₅₃	1	1	yes	characteristic
262	#332	C₂²×C₅₃	1	1	yes	characteristic
263	C₅₃	#19233	1	1	yes	characteristic, direct product factor, Sylow
264	C₂×#41	—	2	1	no	—
265	C₂²×Q₈	—	2	1	no	—
266	C₂×SD₁₆	—	2	2	no	—
267	#131	—	4	4	no	—
268	C₂×M₁₆	—	2	2	no	—
269	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
270	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
271	C₂²×C₈	—	2	1	no	—
272	C₂²×C₈	—	2	1	no	—
273	C₂²×C₈	—	2	2	no	—
274	C₄⋊Q₈	—	2	1	no	—
275	C₄⋊Q₈	—	2	1	no	—
276	C₄⋊Q₈	—	2	1	no	—
277	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
278	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
279	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
280	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
281	#127	D₄×C₅₃	1	1	yes	characteristic
282	#125	—	2	1	no	—
283	#124	—	2	1	no	—
284	C₂²⋊Q₈	—	2	1	no	—
285	C₂²⋊Q₈	—	2	1	no	—
286	#121	—	2	1	no	—
287	#121	D₄×C₅₃	1	2	yes	—
288	C₄×Q₈	—	2	1	no	—
289	C₄×Q₈	—	2	1	no	—
290	C₄×D₄	—	2	1	no	—
291	C₄×D₄	C₂³×C₅₃	1	2	yes	—
292	#117	—	2	1	no	—
293	C₂×C₄⋊C₄	—	2	1	no	—
294	C₂×C₄⋊C₄	—	2	1	no	—
295	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
296	#107	—	2	2	no	—
297	#107	—	2	2	no	—
298	#106	—	2	2	no	—
299	#106	—	2	2	no	—
300	#106	—	2	2	no	—
301	#106	—	2	2	no	—
302	#106	—	2	2	no	—
303	C₄⋊C₈	—	2	1	no	—
304	C₄⋊C₈	—	2	1	no	—
305	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
306	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
307	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
308	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
309	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
310	#103	—	2	2	no	—
311	#102	—	2	2	no	—
312	#102	—	2	2	no	—
313	#102	—	2	2	no	—
314	C₂²⋊C₈	—	2	1	no	—
315	C₂²⋊C₈	—	2	2	no	—
316	C₂²⋊C₈	—	2	2	no	—
317	#97	D₄×C₅₃	1	2	yes	—
318	C₄×C₈	—	2	1	no	—
319	C₄×C₈	C₂³×C₅₃	1	2	yes	—
320	#41	—	4	2	no	—
321	#41	—	4	2	no	—
322	C₂×Q₈	—	2	1	no	—
323	C₂×Q₈	—	2	1	no	—
324	C₂×Q₈	—	2	1	no	—
325	C₂×Q₈	—	2	2	no	—
326	C₂×Q₈	—	4	1	no	—
327	C₂×Q₈	—	8	2	no	—
328	C₂×D₄	—	2	2	no	—
329	C₂×D₄	—	2	1	no	—
330	C₂×D₄	—	2	1	no	—
331	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	—	4	1	no	—
334	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
335	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
336	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
337	SD₁₆	—	8	2	no	—
338	M₁₆	—	2	2	no	—
339	M₁₆	—	4	4	no	—
340	M₁₆	SD₁₆×C₅₃	1	4	yes	complemented
341	C₂×C₈	—	2	1	no	—
342	C₂×C₈	—	2	1	no	—
343	C₂×C₈	—	2	1	no	—
344	C₂×C₈	—	2	1	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	—	4	2	no	—
347	C₂×C₈	—	4	4	no	—
348	C₂×C₈	—	8	1	no	—
349	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
350	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
351	C₄⋊C₄	—	2	1	no	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	2	1	no	—
354	C₄⋊C₄	—	2	1	no	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	—	2	1	no	—
357	C₄⋊C₄	—	2	1	no	—
358	C₄⋊C₄	—	4	1	no	—
359	C₄⋊C₄	—	4	1	no	—
360	C₄⋊C₄	—	4	1	no	—
361	C₄⋊C₄	—	4	1	no	—
362	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
363	C₂²⋊C₄	—	2	2	no	—
364	C₂²⋊C₄	—	2	2	no	—
365	C₂²⋊C₄	—	4	1	no	—
366	C₄²	—	4	1	no	—
367	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
368	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
369	C₄²	#41×C₅₃	1	1	yes	characteristic
370	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
371	C₂³	—	2	2	no	—
372	C₂³	#121×C₅₃	1	1	yes	characteristic
373	Q₈	—	4	2	no	—
374	Q₈	—	4	1	no	—
375	Q₈	—	4	1	no	—
376	Q₈	—	8	1	no	—
377	Q₈	—	8	2	no	—
378	Q₈	—	8	2	no	—
379	D₄	—	2	2	no	—
380	D₄	—	2	2	no	—
381	D₄	—	4	4	no	—
382	D₄	—	4	2	no	—
383	D₄	—	8	1	no	—
384	C₂×C₄	—	2	1	no	—
385	C₂×C₄	—	2	1	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	1	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	8	2	no	—
394	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
395	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
396	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
397	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
398	C₂×C₄	#137×C₅₃	1	2	yes	—
399	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
400	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
401	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
402	C₈	—	2	2	no	—
403	C₈	—	4	1	no	—
404	C₈	—	4	1	no	—
405	C₈	—	4	1	no	—
406	C₈	—	4	2	no	—
407	C₈	—	4	1	no	—
408	C₈	C₂×SD₁₆×C₅₃	1	4	yes	—
409	C₂²	—	2	1	no	—
410	C₂²	—	2	1	no	—
411	C₂²	—	4	2	no	—
412	C₂²	—	4	2	no	—
413	C₂²	—	4	2	no	—
414	C₂²	C₅₃×#461	1	2	yes	—
415	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
416	C₄	—	2	1	no	—
417	C₄	—	2	1	no	—
418	C₄	—	2	1	no	—
419	C₄	—	4	2	no	—
420	C₄	—	8	1	no	—
421	C₄	—	8	2	no	—
422	C₄	—	8	1	no	—
423	C₄	C₅₃×#461	1	2	yes	—
424	C₄	C₂²×SD₁₆×C₅₃	1	1	yes	characteristic
425	C₄	C₂×#137×C₅₃	1	1	yes	characteristic
426	C₄	C₅₃×#577	1	1	yes	characteristic
427	C₄	C₅₃×#578	1	1	yes	characteristic
428	C₂	—	2	2	no	—
429	C₂	—	8	2	no	—
430	C₂	C₂×C₅₃×#461	1	1	yes	characteristic
431	C₂	C₅₃×#3063	1	1	yes	characteristic
432	C₂	C₅₃×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₅₃×#532	C₂²
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
3. C₅₃×#3165	C₂	semidirect
5. C₅₃×#1695	C₂	semidirect
6. C₅₃×#1691	C₂	semidirect
7. C₅₃×#1675	C₂	semidirect
8. C₅₃×#1671	C₂	semidirect
9. C₅₃×#1565	C₂	semidirect
10. C₅₃×#1542	C₂	semidirect
11. C₂×C₅₃×#333	C₂	semidirect
123. M₁₆×C₅₃	SD₁₆	semidirect
192. #19233	C₅₃	direct (N × H)
214. #3165	C₂×C₅₃	semidirect
216. #1695	C₂×C₅₃	semidirect
217. #1691	C₂×C₅₃	semidirect
218. #1675	C₂×C₅₃	semidirect
219. #1671	C₂×C₅₃	semidirect
220. #1565	C₂×C₅₃	semidirect
221. #1542	C₂×C₅₃	semidirect
222. C₂×#333	C₂×C₅₃	semidirect
263. C₅₃	#19233	direct (N × H)
340. M₁₆	SD₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3246	2
C₅₃×#3227	2
C₅₃×#3165	2
C₅₃×#2935	2
C₅₃×#1695	2
C₅₃×#1691	2
C₅₃×#1675	2
C₅₃×#1671	2
C₅₃×#1565	2
C₅₃×#1542	2
C₂×C₅₃×#333	2
#19233	53

Subgroups — C53×#19233 #27226745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19233 #27226745