Subgroups — C₅₃×#19298 #27226810

Order 13568.

586 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3116	C₂	1	2	yes	maximal
3	C₅₃×#2965	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1696	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1695	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1694	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1693	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1692	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1691	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1554	C₂	1	2	yes	maximal, complemented
12	C₅₃×#533	C₂²	1	2	yes	—
13	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#500	C₂²	1	1	yes	characteristic
15	C₅₃×#500	C₂²	1	1	yes	characteristic
16	C₅₃×#499	—	2	1	no	—
17	C₅₃×#499	—	2	1	no	—
18	C₅₃×#498	C₂²	1	2	yes	—
19	C₅₃×#485	—	2	2	no	—
20	C₅₃×#477	C₂²	1	2	yes	—
21	C₅₃×#476	C₂²	1	2	yes	—
22	C₅₃×#462	C₂²	1	2	yes	—
23	C₅₃×#461	C₂²	1	2	yes	—
24	C₅₃×#452	—	2	2	no	—
25	C₅₃×#446	C₂²	1	1	yes	characteristic
26	C₅₃×#446	C₂²	1	1	yes	characteristic
27	C₅₃×#441	—	2	2	no	—
28	C₅₃×#440	—	2	2	no	—
29	C₅₃×#436	C₂²	1	1	yes	characteristic
30	C₅₃×#436	C₂²	1	1	yes	characteristic
31	C₅₃×#435	—	2	1	no	—
32	C₅₃×#435	—	2	1	no	—
33	C₅₃×#432	C₂²	1	2	yes	—
34	C₅₃×#431	—	2	1	no	—
35	C₅₃×#431	C₂²	1	1	yes	characteristic
36	C₅₃×#427	—	2	1	no	—
37	C₅₃×#427	—	2	1	no	—
38	C₄×M₁₆×C₅₃	—	2	1	no	—
39	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#335	C₂²	1	2	yes	—
41	C₅₃×#334	C₂²	1	1	yes	characteristic
42	C₅₃×#334	C₂²	1	1	yes	characteristic
43	C₅₃×#333	C₂²	1	2	yes	—
44	C₅₃×#333	C₂²	1	2	yes	—
45	C₅₃×#332	C₂²	1	2	yes	—
46	C₅₃×#332	C₂²	1	2	yes	—
47	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
48	C₂×Q₁₆×C₅₃	—	2	2	no	—
49	C₂×SD₁₆×C₅₃	—	2	2	no	—
50	C₂×M₁₆×C₅₃	—	2	1	no	—
51	C₂×M₁₆×C₅₃	—	2	1	no	—
52	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
54	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
55	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
56	#125×C₅₃	—	2	2	no	—
57	C₂²⋊Q₈×C₅₃	—	2	2	no	—
58	C₂²⋊Q₈×C₅₃	—	2	2	no	—
59	C₄×Q₈×C₅₃	—	2	2	no	—
60	C₄×Q₈×C₅₃	—	2	2	no	—
61	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
62	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
63	C₄×D₄×C₅₃	—	2	1	no	—
64	C₄×D₄×C₅₃	—	2	1	no	—
65	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
66	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
67	#117×C₅₃	—	2	1	no	—
68	#117×C₅₃	—	2	2	no	—
69	#117×C₅₃	D₄	1	1	yes	characteristic
70	C₂×C₄²×C₅₃	—	2	1	no	—
71	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
72	#107×C₅₃	—	2	1	no	—
73	#107×C₅₃	—	2	1	no	—
74	#107×C₅₃	—	2	1	no	—
75	#107×C₅₃	—	2	1	no	—
76	#107×C₅₃	—	2	2	no	—
77	#106×C₅₃	—	2	2	no	—
78	#106×C₅₃	—	2	2	no	—
79	#106×C₅₃	—	2	2	no	—
80	C₄⋊C₈×C₅₃	—	2	1	no	—
81	C₄⋊C₈×C₅₃	—	2	1	no	—
82	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
84	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
85	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
86	#103×C₅₃	—	2	2	no	—
87	#103×C₅₃	—	2	2	no	—
88	#103×C₅₃	—	2	2	no	—
89	#102×C₅₃	—	2	2	no	—
90	C₂²⋊C₈×C₅₃	—	2	1	no	—
91	C₂²⋊C₈×C₅₃	—	2	1	no	—
92	C₂²⋊C₈×C₅₃	—	2	2	no	—
93	C₂²⋊C₈×C₅₃	—	2	1	no	—
94	#97×C₅₃	—	2	1	no	—
95	#97×C₅₃	—	2	1	no	—
96	#97×C₅₃	—	2	2	no	—
97	C₄×C₈×C₅₃	—	2	1	no	—
98	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
99	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
100	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
102	#41×C₅₃	—	4	2	no	—
103	C₂×Q₈×C₅₃	—	2	2	no	—
104	C₂×Q₈×C₅₃	—	2	2	no	—
105	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂×D₄×C₅₃	—	2	1	no	—
107	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	—	2	1	no	—
109	C₂²×C₄×C₅₃	—	2	1	no	—
110	C₂²×C₄×C₅₃	—	2	1	no	—
111	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
114	Q₁₆×C₅₃	—	4	4	no	—
115	SD₁₆×C₅₃	—	4	4	no	—
116	M₁₆×C₅₃	—	4	2	no	—
117	M₁₆×C₅₃	—	4	2	no	—
118	C₂×C₈×C₅₃	—	2	2	no	—
119	C₂×C₈×C₅₃	—	2	2	no	—
120	C₂×C₈×C₅₃	—	2	1	no	—
121	C₂×C₈×C₅₃	—	2	1	no	—
122	C₂×C₈×C₅₃	—	2	2	no	—
123	C₂×C₈×C₅₃	—	2	2	no	—
124	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
128	C₄⋊C₄×C₅₃	—	2	1	no	—
129	C₄⋊C₄×C₅₃	—	2	1	no	—
130	C₄⋊C₄×C₅₃	—	2	2	no	—
131	C₄⋊C₄×C₅₃	—	2	2	no	—
132	C₄⋊C₄×C₅₃	—	2	2	no	—
133	C₄⋊C₄×C₅₃	—	2	2	no	—
134	C₄⋊C₄×C₅₃	—	4	2	no	—
135	C₄⋊C₄×C₅₃	—	4	2	no	—
136	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
137	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
138	C₂²⋊C₄×C₅₃	—	2	1	no	—
139	C₂²⋊C₄×C₅₃	—	2	1	no	—
140	C₂²⋊C₄×C₅₃	—	2	1	no	—
141	C₂²⋊C₄×C₅₃	—	4	2	no	—
142	C₄²×C₅₃	—	2	1	no	—
143	C₄²×C₅₃	—	2	1	no	—
144	C₄²×C₅₃	—	2	1	no	—
145	C₄²×C₅₃	—	4	2	no	—
146	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
147	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
148	C₄²×C₅₃	#41	1	1	yes	characteristic
149	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
150	C₂³×C₅₃	—	2	1	no	—
151	C₂³×C₅₃	#121	1	1	yes	characteristic
152	Q₈×C₅₃	—	2	2	no	—
153	Q₈×C₅₃	—	4	2	no	—
154	Q₈×C₅₃	—	4	2	no	—
155	Q₈×C₅₃	—	8	2	no	—
156	D₄×C₅₃	—	2	2	no	—
157	D₄×C₅₃	—	4	2	no	—
158	C₂×C₄×C₅₃	—	2	1	no	—
159	C₂×C₄×C₅₃	—	2	1	no	—
160	C₂×C₄×C₅₃	—	2	2	no	—
161	C₂×C₄×C₅₃	—	2	1	no	—
162	C₂×C₄×C₅₃	—	2	1	no	—
163	C₂×C₄×C₅₃	—	2	1	no	—
164	C₂×C₄×C₅₃	—	4	1	no	—
165	C₂×C₄×C₅₃	—	4	2	no	—
166	C₂×C₄×C₅₃	—	4	2	no	—
167	C₂×C₄×C₅₃	—	4	1	no	—
168	C₂×C₄×C₅₃	—	4	1	no	—
169	C₂×C₄×C₅₃	—	4	2	no	—
170	C₂×C₄×C₅₃	—	4	2	no	—
171	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
172	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
173	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
174	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
177	C₈×C₅₃	—	2	1	no	—
178	C₈×C₅₃	—	2	1	no	—
179	C₈×C₅₃	—	4	1	no	—
180	C₈×C₅₃	—	4	1	no	—
181	C₈×C₅₃	—	4	2	no	—
182	C₈×C₅₃	—	4	2	no	—
183	C₈×C₅₃	#137	1	2	yes	—
184	C₈×C₅₃	#137	1	2	yes	—
185	#19298	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
186	C₂²×C₅₃	—	2	1	no	—
187	C₂²×C₅₃	—	2	1	no	—
188	C₂²×C₅₃	—	2	1	no	—
189	C₂²×C₅₃	—	4	1	no	—
190	C₂²×C₅₃	—	4	1	no	—
191	C₂²×C₅₃	—	4	1	no	—
192	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
193	C₄×C₅₃	—	2	1	no	—
194	C₄×C₅₃	—	2	1	no	—
195	C₄×C₅₃	—	2	1	no	—
196	C₄×C₅₃	—	2	1	no	—
197	C₄×C₅₃	—	4	1	no	—
198	C₄×C₅₃	—	4	1	no	—
199	C₄×C₅₃	—	4	1	no	—
200	C₄×C₅₃	—	8	2	no	—
201	C₄×C₅₃	—	8	2	no	—
202	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
203	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
204	C₄×C₅₃	#575	1	2	yes	—
205	#3160	C₂×C₅₃	1	1	yes	characteristic, complemented
206	#3116	C₂×C₅₃	1	2	yes	—
207	#2965	C₂×C₅₃	1	1	yes	characteristic
208	#1696	C₂×C₅₃	1	1	yes	characteristic, complemented
209	#1695	C₂×C₅₃	1	1	yes	characteristic, complemented
210	#1694	C₂×C₅₃	1	1	yes	characteristic, complemented
211	#1693	C₂×C₅₃	1	1	yes	characteristic, complemented
212	#1692	C₂×C₅₃	1	2	yes	complemented
213	#1691	C₂×C₅₃	1	2	yes	complemented
214	#1568	C₂×C₅₃	1	1	yes	characteristic, complemented
215	#1554	C₂×C₅₃	1	2	yes	complemented
216	C₂×C₅₃	—	4	1	no	—
217	C₂×C₅₃	—	8	1	no	—
218	C₂×C₅₃	#3039	1	1	yes	characteristic
219	C₂×C₅₃	#3054	1	1	yes	characteristic
220	C₂×C₅₃	#3067	1	1	yes	characteristic
221	#533	C₂²×C₅₃	1	2	yes	—
222	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
223	#500	C₂²×C₅₃	1	1	yes	characteristic
224	#500	C₂²×C₅₃	1	1	yes	characteristic
225	#499	—	2	1	no	—
226	#499	—	2	1	no	—
227	#498	C₂²×C₅₃	1	2	yes	—
228	#485	—	2	2	no	—
229	#477	C₂²×C₅₃	1	2	yes	—
230	#476	C₂²×C₅₃	1	2	yes	—
231	#462	C₂²×C₅₃	1	2	yes	—
232	#461	C₂²×C₅₃	1	2	yes	—
233	#452	—	2	2	no	—
234	#446	C₂²×C₅₃	1	1	yes	characteristic
235	#446	C₂²×C₅₃	1	1	yes	characteristic
236	#441	—	2	2	no	—
237	#440	—	2	2	no	—
238	#436	C₂²×C₅₃	1	1	yes	characteristic
239	#436	C₂²×C₅₃	1	1	yes	characteristic
240	#435	—	2	1	no	—
241	#435	—	2	1	no	—
242	#432	C₂²×C₅₃	1	2	yes	—
243	#431	—	2	1	no	—
244	#431	C₂²×C₅₃	1	1	yes	characteristic
245	#427	—	2	1	no	—
246	#427	—	2	1	no	—
247	C₄×M₁₆	—	2	1	no	—
248	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
249	#335	C₂²×C₅₃	1	2	yes	—
250	#334	C₂²×C₅₃	1	1	yes	characteristic
251	#334	C₂²×C₅₃	1	1	yes	characteristic
252	#333	C₂²×C₅₃	1	2	yes	—
253	#333	C₂²×C₅₃	1	2	yes	—
254	#332	C₂²×C₅₃	1	2	yes	—
255	#332	C₂²×C₅₃	1	2	yes	—
256	C₅₃	#19298	1	1	yes	characteristic, direct product factor, Sylow
257	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
258	C₂×Q₁₆	—	2	2	no	—
259	C₂×SD₁₆	—	2	2	no	—
260	C₂×M₁₆	—	2	1	no	—
261	C₂×M₁₆	—	2	1	no	—
262	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
263	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
264	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
265	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
266	#125	—	2	2	no	—
267	C₂²⋊Q₈	—	2	2	no	—
268	C₂²⋊Q₈	—	2	2	no	—
269	C₄×Q₈	—	2	2	no	—
270	C₄×Q₈	—	2	2	no	—
271	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
272	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
273	C₄×D₄	—	2	1	no	—
274	C₄×D₄	—	2	1	no	—
275	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
276	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
277	#117	—	2	1	no	—
278	#117	—	2	2	no	—
279	#117	D₄×C₅₃	1	1	yes	characteristic
280	C₂×C₄²	—	2	1	no	—
281	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
282	#107	—	2	1	no	—
283	#107	—	2	1	no	—
284	#107	—	2	1	no	—
285	#107	—	2	1	no	—
286	#107	—	2	2	no	—
287	#106	—	2	2	no	—
288	#106	—	2	2	no	—
289	#106	—	2	2	no	—
290	C₄⋊C₈	—	2	1	no	—
291	C₄⋊C₈	—	2	1	no	—
292	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
293	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
294	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
295	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
296	#103	—	2	2	no	—
297	#103	—	2	2	no	—
298	#103	—	2	2	no	—
299	#102	—	2	2	no	—
300	C₂²⋊C₈	—	2	1	no	—
301	C₂²⋊C₈	—	2	2	no	—
302	C₂²⋊C₈	—	2	1	no	—
303	C₂²⋊C₈	—	2	1	no	—
304	#97	—	2	1	no	—
305	#97	—	2	1	no	—
306	#97	—	2	2	no	—
307	C₄×C₈	—	2	1	no	—
308	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
309	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
310	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
311	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
312	#41	—	4	2	no	—
313	C₂×Q₈	—	2	2	no	—
314	C₂×Q₈	—	2	2	no	—
315	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
316	C₂×D₄	—	2	1	no	—
317	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
318	C₂²×C₄	—	2	1	no	—
319	C₂²×C₄	—	2	1	no	—
320	C₂²×C₄	—	2	1	no	—
321	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
322	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
323	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
324	Q₁₆	—	4	4	no	—
325	SD₁₆	—	4	4	no	—
326	M₁₆	—	4	2	no	—
327	M₁₆	—	4	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
335	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
336	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
337	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
338	C₄⋊C₄	—	2	1	no	—
339	C₄⋊C₄	—	2	1	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	4	2	no	—
345	C₄⋊C₄	—	4	2	no	—
346	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
347	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
348	C₂²⋊C₄	—	2	1	no	—
349	C₂²⋊C₄	—	2	1	no	—
350	C₂²⋊C₄	—	2	1	no	—
351	C₂²⋊C₄	—	4	2	no	—
352	C₄²	—	2	1	no	—
353	C₄²	—	2	1	no	—
354	C₄²	—	2	1	no	—
355	C₄²	—	4	2	no	—
356	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
357	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
358	C₄²	#41×C₅₃	1	1	yes	characteristic
359	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
360	C₂³	—	2	1	no	—
361	C₂³	#121×C₅₃	1	1	yes	characteristic
362	Q₈	—	2	2	no	—
363	Q₈	—	4	2	no	—
364	Q₈	—	4	2	no	—
365	Q₈	—	8	2	no	—
366	D₄	—	2	2	no	—
367	D₄	—	4	2	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	4	1	no	—
378	C₂×C₄	—	4	1	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
382	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
383	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
384	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
385	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
386	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
387	C₈	—	2	1	no	—
388	C₈	—	2	1	no	—
389	C₈	—	4	1	no	—
390	C₈	—	4	1	no	—
391	C₈	—	4	2	no	—
392	C₈	—	4	2	no	—
393	C₈	#137×C₅₃	1	2	yes	—
394	C₈	#137×C₅₃	1	2	yes	—
395	C₂²	—	2	1	no	—
396	C₂²	—	2	1	no	—
397	C₂²	—	2	1	no	—
398	C₂²	—	4	1	no	—
399	C₂²	—	4	1	no	—
400	C₂²	—	4	1	no	—
401	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
402	C₄	—	2	1	no	—
403	C₄	—	2	1	no	—
404	C₄	—	2	1	no	—
405	C₄	—	2	1	no	—
406	C₄	—	4	1	no	—
407	C₄	—	4	1	no	—
408	C₄	—	4	1	no	—
409	C₄	—	8	2	no	—
410	C₄	—	8	2	no	—
411	C₄	C₂×#137×C₅₃	1	1	yes	characteristic
412	C₄	C₂×#137×C₅₃	1	1	yes	characteristic
413	C₄	C₅₃×#575	1	2	yes	—
414	C₂	—	4	1	no	—
415	C₂	—	8	1	no	—
416	C₂	C₅₃×#3039	1	1	yes	characteristic
417	C₂	C₅₃×#3054	1	1	yes	characteristic
418	C₂	C₅₃×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₂×#41×C₅₃	D₄
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×#3160	C₂	semidirect
4. C₅₃×#1696	C₂	semidirect
5. C₅₃×#1695	C₂	semidirect
6. C₅₃×#1694	C₂	semidirect
7. C₅₃×#1693	C₂	semidirect
8. C₅₃×#1692	C₂	semidirect
9. C₅₃×#1691	C₂	semidirect
10. C₅₃×#1568	C₂	semidirect
11. C₅₃×#1554	C₂	semidirect
185. #19298	C₅₃	direct (N × H)
205. #3160	C₂×C₅₃	semidirect
208. #1696	C₂×C₅₃	semidirect
209. #1695	C₂×C₅₃	semidirect
210. #1694	C₂×C₅₃	semidirect
211. #1693	C₂×C₅₃	semidirect
212. #1692	C₂×C₅₃	semidirect
213. #1691	C₂×C₅₃	semidirect
214. #1568	C₂×C₅₃	semidirect
215. #1554	C₂×C₅₃	semidirect
256. C₅₃	#19298	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×#3160	2
C₅₃×#3116	2
C₅₃×#2965	2
C₅₃×#1696	2
C₅₃×#1695	2
C₅₃×#1694	2
C₅₃×#1693	2
C₅₃×#1692	2
C₅₃×#1691	2
C₅₃×#1568	2
C₅₃×#1554	2
#19298	53

Subgroups — C53×#19298 #27226810

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19298 #27226810