Subgroups — C₅₃×#19197 #27226709

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3226	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#3206	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2931	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1699	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1679	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1549	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1546	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#131×C₅₃	—	2	1	no	—
11	C₅₃×#532	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#500	—	2	1	no	—
15	C₅₃×#492	—	2	1	no	—
16	C₅₃×#487	—	2	1	no	—
17	C₅₃×#487	—	2	1	no	—
18	C₅₃×#487	—	2	1	no	—
19	C₅₃×#487	C₂²	1	2	yes	—
20	C₅₃×#487	C₂²	1	2	yes	—
21	C₅₃×#486	—	2	1	no	—
22	C₅₃×#480	—	2	2	no	—
23	C₅₃×#474	C₂²	1	4	yes	—
24	C₅₃×#461	C₂²	1	4	yes	—
25	C₅₃×#448	—	2	2	no	—
26	C₅₃×#436	—	2	1	no	—
27	C₅₃×#436	C₂²	1	2	yes	—
28	C₈×D₄×C₅₃	—	2	1	no	—
29	C₈×D₄×C₅₃	C₂²	1	2	yes	—
30	C₅₃×#431	C₂²	1	1	yes	characteristic
31	C₅₃×#431	C₂²	1	1	yes	characteristic
32	C₅₃×#423	C₂²	1	1	yes	characteristic
33	C₂×#103×C₅₃	—	2	1	no	—
34	C₂×#103×C₅₃	—	2	1	no	—
35	C₅₃×#408	—	2	1	no	—
36	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#335	C₂²	1	2	yes	—
38	C₅₃×#334	C₂²	1	2	yes	—
39	C₅₃×#332	C₂²	1	4	yes	—
40	C₅₃×#332	C₂²	1	4	yes	—
41	C₂×#41×C₅₃	—	2	1	no	—
42	C₂²×Q₈×C₅₃	—	2	1	no	—
43	C₂²×Q₈×C₅₃	—	2	1	no	—
44	C₂×SD₁₆×C₅₃	—	2	4	no	—
45	#131×C₅₃	—	4	4	no	—
46	C₂×M₁₆×C₅₃	—	2	2	no	—
47	C₂×M₁₆×C₅₃	—	2	1	no	—
48	C₂²×C₈×C₅₃	—	2	2	no	—
49	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
51	C₄⋊Q₈×C₅₃	—	2	1	no	—
52	C₄⋊Q₈×C₅₃	—	2	1	no	—
53	C₄⋊Q₈×C₅₃	—	2	1	no	—
54	C₄⋊Q₈×C₅₃	—	2	1	no	—
55	C₄⋊Q₈×C₅₃	—	2	1	no	—
56	C₄⋊Q₈×C₅₃	—	2	1	no	—
57	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
58	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
59	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
60	#127×C₅₃	D₄	1	1	yes	characteristic
61	#124×C₅₃	—	2	1	no	—
62	#121×C₅₃	—	2	1	no	—
63	#121×C₅₃	D₄	1	2	yes	—
64	C₄×D₄×C₅₃	—	2	1	no	—
65	C₄×D₄×C₅₃	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
67	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
68	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
69	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
70	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
71	#107×C₅₃	—	2	4	no	—
72	C₄⋊C₈×C₅₃	—	2	1	no	—
73	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
74	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
75	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
76	#103×C₅₃	—	2	2	no	—
77	#103×C₅₃	—	2	2	no	—
78	#103×C₅₃	—	2	2	no	—
79	#103×C₅₃	—	2	2	no	—
80	#103×C₅₃	—	2	4	no	—
81	#102×C₅₃	—	2	4	no	—
82	C₂²⋊C₈×C₅₃	—	2	2	no	—
83	C₂²⋊C₈×C₅₃	—	2	2	no	—
84	C₂²⋊C₈×C₅₃	—	2	1	no	—
85	C₂²⋊C₈×C₅₃	—	2	1	no	—
86	C₄×C₈×C₅₃	—	2	1	no	—
87	C₄×C₈×C₅₃	—	2	1	no	—
88	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
89	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
90	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
92	#41×C₅₃	—	4	2	no	—
93	#41×C₅₃	—	4	2	no	—
94	C₂×Q₈×C₅₃	—	2	1	no	—
95	C₂×Q₈×C₅₃	—	2	2	no	—
96	C₂×Q₈×C₅₃	—	2	2	no	—
97	C₂×Q₈×C₅₃	—	4	1	no	—
98	C₂×Q₈×C₅₃	—	4	1	no	—
99	C₂×Q₈×C₅₃	—	4	2	no	—
100	C₂×Q₈×C₅₃	—	4	2	no	—
101	C₂×Q₈×C₅₃	—	8	2	no	—
102	C₂×D₄×C₅₃	—	2	2	no	—
103	C₂×D₄×C₅₃	—	2	1	no	—
104	C₂×D₄×C₅₃	—	2	1	no	—
105	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₅₃	—	2	2	no	—
107	C₂²×C₄×C₅₃	—	4	1	no	—
108	C₂²×C₄×C₅₃	—	4	1	no	—
109	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
112	SD₁₆×C₅₃	—	8	4	no	—
113	M₁₆×C₅₃	—	4	4	no	—
114	M₁₆×C₅₃	—	4	2	no	—
115	C₂×C₈×C₅₃	—	2	2	no	—
116	C₂×C₈×C₅₃	—	2	2	no	—
117	C₂×C₈×C₅₃	—	2	2	no	—
118	C₂×C₈×C₅₃	—	2	2	no	—
119	C₂×C₈×C₅₃	—	2	2	no	—
120	C₂×C₈×C₅₃	—	4	4	no	—
121	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
125	C₄⋊C₄×C₅₃	—	2	1	no	—
126	C₄⋊C₄×C₅₃	—	2	2	no	—
127	C₄⋊C₄×C₅₃	—	2	2	no	—
128	C₄⋊C₄×C₅₃	—	2	2	no	—
129	C₄⋊C₄×C₅₃	—	2	2	no	—
130	C₄⋊C₄×C₅₃	—	4	1	no	—
131	C₄⋊C₄×C₅₃	—	4	1	no	—
132	C₄⋊C₄×C₅₃	—	4	1	no	—
133	C₄⋊C₄×C₅₃	—	4	1	no	—
134	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
135	C₂²⋊C₄×C₅₃	—	2	2	no	—
136	C₂²⋊C₄×C₅₃	—	2	2	no	—
137	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
138	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
139	C₄²×C₅₃	#41	1	1	yes	characteristic
140	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
141	C₂³×C₅₃	—	2	2	no	—
142	C₂³×C₅₃	#121	1	1	yes	characteristic
143	Q₈×C₅₃	—	4	2	no	—
144	Q₈×C₅₃	—	4	2	no	—
145	Q₈×C₅₃	—	4	2	no	—
146	Q₈×C₅₃	—	4	4	no	—
147	Q₈×C₅₃	—	8	2	no	—
148	Q₈×C₅₃	—	8	2	no	—
149	D₄×C₅₃	—	2	4	no	—
150	D₄×C₅₃	—	4	4	no	—
151	D₄×C₅₃	—	4	2	no	—
152	D₄×C₅₃	—	4	2	no	—
153	C₂×C₄×C₅₃	—	2	2	no	—
154	C₂×C₄×C₅₃	—	2	2	no	—
155	C₂×C₄×C₅₃	—	2	2	no	—
156	C₂×C₄×C₅₃	—	4	2	no	—
157	C₂×C₄×C₅₃	—	4	2	no	—
158	C₂×C₄×C₅₃	—	4	2	no	—
159	C₂×C₄×C₅₃	—	4	2	no	—
160	C₂×C₄×C₅₃	—	8	2	no	—
161	C₂×C₄×C₅₃	—	8	2	no	—
162	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
163	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
164	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
165	C₂×C₄×C₅₃	#135	1	2	yes	—
166	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
167	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
170	C₈×C₅₃	—	2	1	no	—
171	C₈×C₅₃	—	2	1	no	—
172	C₈×C₅₃	—	2	1	no	—
173	C₈×C₅₃	—	2	1	no	—
174	C₈×C₅₃	—	4	2	no	—
175	C₈×C₅₃	—	4	2	no	—
176	C₈×C₅₃	—	4	2	no	—
177	#19197	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
178	C₂²×C₅₃	—	2	1	no	—
179	C₂²×C₅₃	—	2	1	no	—
180	C₂²×C₅₃	—	4	2	no	—
181	C₂²×C₅₃	—	4	2	no	—
182	C₂²×C₅₃	—	4	2	no	—
183	C₂²×C₅₃	#463	1	2	yes	—
184	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
185	C₄×C₅₃	—	2	1	no	—
186	C₄×C₅₃	—	2	1	no	—
187	C₄×C₅₃	—	4	2	no	—
188	C₄×C₅₃	—	8	2	no	—
189	C₄×C₅₃	—	8	2	no	—
190	C₄×C₅₃	#463	1	2	yes	—
191	C₄×C₅₃	#473	1	2	yes	—
192	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
193	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
194	C₄×C₅₃	#577	1	2	yes	—
195	#3226	C₂×C₅₃	1	1	yes	characteristic
196	#3206	C₂×C₅₃	1	1	yes	characteristic
197	C₂×#487	C₂×C₅₃	1	1	yes	characteristic, complemented
198	#2931	C₂×C₅₃	1	1	yes	characteristic
199	#1699	C₂×C₅₃	1	4	yes	complemented
200	#1679	C₂×C₅₃	1	4	yes	complemented
201	#1564	C₂×C₅₃	1	1	yes	characteristic, complemented
202	#1549	C₂×C₅₃	1	1	yes	characteristic, complemented
203	#1546	C₂×C₅₃	1	1	yes	characteristic, complemented
204	C₂×C₅₃	—	2	2	no	—
205	C₂×C₅₃	—	8	2	no	—
206	C₂×C₅₃	C₂×#463	1	1	yes	characteristic
207	C₂×C₅₃	#3057	1	1	yes	characteristic
208	C₂×C₅₃	C₂×#473	1	1	yes	characteristic
209	C₂×#131	—	2	1	no	—
210	#532	C₂²×C₅₃	1	1	yes	characteristic
211	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
212	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
213	#500	—	2	1	no	—
214	#492	—	2	1	no	—
215	#487	—	2	1	no	—
216	#487	—	2	1	no	—
217	#487	—	2	1	no	—
218	#487	C₂²×C₅₃	1	2	yes	—
219	#487	C₂²×C₅₃	1	2	yes	—
220	#486	—	2	1	no	—
221	#480	—	2	2	no	—
222	#474	C₂²×C₅₃	1	4	yes	—
223	#461	C₂²×C₅₃	1	4	yes	—
224	#448	—	2	2	no	—
225	#436	—	2	1	no	—
226	#436	C₂²×C₅₃	1	2	yes	—
227	C₈×D₄	—	2	1	no	—
228	C₈×D₄	C₂²×C₅₃	1	2	yes	—
229	#431	C₂²×C₅₃	1	1	yes	characteristic
230	#431	C₂²×C₅₃	1	1	yes	characteristic
231	#423	C₂²×C₅₃	1	1	yes	characteristic
232	C₂×#103	—	2	1	no	—
233	C₂×#103	—	2	1	no	—
234	#408	—	2	1	no	—
235	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
236	#335	C₂²×C₅₃	1	2	yes	—
237	#334	C₂²×C₅₃	1	2	yes	—
238	#332	C₂²×C₅₃	1	4	yes	—
239	#332	C₂²×C₅₃	1	4	yes	—
240	C₅₃	#19197	1	1	yes	characteristic, direct product factor, Sylow
241	C₂×#41	—	2	1	no	—
242	C₂²×Q₈	—	2	1	no	—
243	C₂²×Q₈	—	2	1	no	—
244	C₂×SD₁₆	—	2	4	no	—
245	#131	—	4	4	no	—
246	C₂×M₁₆	—	2	1	no	—
247	C₂×M₁₆	—	2	2	no	—
248	C₂²×C₈	—	2	2	no	—
249	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
250	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
251	C₄⋊Q₈	—	2	1	no	—
252	C₄⋊Q₈	—	2	1	no	—
253	C₄⋊Q₈	—	2	1	no	—
254	C₄⋊Q₈	—	2	1	no	—
255	C₄⋊Q₈	—	2	1	no	—
256	C₄⋊Q₈	—	2	1	no	—
257	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
258	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
259	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
260	#127	D₄×C₅₃	1	1	yes	characteristic
261	#124	—	2	1	no	—
262	#121	—	2	1	no	—
263	#121	D₄×C₅₃	1	2	yes	—
264	C₄×D₄	—	2	1	no	—
265	C₄×D₄	C₂³×C₅₃	1	2	yes	—
266	C₂×C₄⋊C₄	—	2	1	no	—
267	C₂×C₄⋊C₄	—	2	1	no	—
268	C₂×C₄⋊C₄	—	2	1	no	—
269	C₂×C₄⋊C₄	—	2	1	no	—
270	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
271	#107	—	2	4	no	—
272	C₄⋊C₈	—	2	1	no	—
273	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
274	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
275	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
276	#103	—	2	2	no	—
277	#103	—	2	2	no	—
278	#103	—	2	2	no	—
279	#103	—	2	2	no	—
280	#103	—	2	4	no	—
281	#102	—	2	4	no	—
282	C₂²⋊C₈	—	2	1	no	—
283	C₂²⋊C₈	—	2	1	no	—
284	C₂²⋊C₈	—	2	2	no	—
285	C₂²⋊C₈	—	2	2	no	—
286	C₄×C₈	—	2	1	no	—
287	C₄×C₈	—	2	1	no	—
288	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
289	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
290	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
291	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
292	#41	—	4	2	no	—
293	#41	—	4	2	no	—
294	C₂×Q₈	—	2	1	no	—
295	C₂×Q₈	—	2	2	no	—
296	C₂×Q₈	—	2	2	no	—
297	C₂×Q₈	—	4	1	no	—
298	C₂×Q₈	—	4	1	no	—
299	C₂×Q₈	—	4	2	no	—
300	C₂×Q₈	—	4	2	no	—
301	C₂×Q₈	—	8	2	no	—
302	C₂×D₄	—	2	2	no	—
303	C₂×D₄	—	2	1	no	—
304	C₂×D₄	—	2	1	no	—
305	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
306	C₂²×C₄	—	2	2	no	—
307	C₂²×C₄	—	4	1	no	—
308	C₂²×C₄	—	4	1	no	—
309	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
310	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
311	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
312	SD₁₆	—	8	4	no	—
313	M₁₆	—	4	4	no	—
314	M₁₆	—	4	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	4	4	no	—
321	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
322	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
323	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
324	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	2	2	no	—
330	C₄⋊C₄	—	4	1	no	—
331	C₄⋊C₄	—	4	1	no	—
332	C₄⋊C₄	—	4	1	no	—
333	C₄⋊C₄	—	4	1	no	—
334	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
335	C₂²⋊C₄	—	2	2	no	—
336	C₂²⋊C₄	—	2	2	no	—
337	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
338	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
339	C₄²	#41×C₅₃	1	1	yes	characteristic
340	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
341	C₂³	—	2	2	no	—
342	C₂³	#121×C₅₃	1	1	yes	characteristic
343	Q₈	—	4	2	no	—
344	Q₈	—	4	2	no	—
345	Q₈	—	4	2	no	—
346	Q₈	—	4	4	no	—
347	Q₈	—	8	2	no	—
348	Q₈	—	8	2	no	—
349	D₄	—	2	4	no	—
350	D₄	—	4	4	no	—
351	D₄	—	4	2	no	—
352	D₄	—	4	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	8	2	no	—
361	C₂×C₄	—	8	2	no	—
362	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
363	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
364	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
365	C₂×C₄	#135×C₅₃	1	2	yes	—
366	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
367	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
368	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
369	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	1	no	—
373	C₈	—	2	1	no	—
374	C₈	—	4	2	no	—
375	C₈	—	4	2	no	—
376	C₈	—	4	2	no	—
377	C₂²	—	2	1	no	—
378	C₂²	—	2	1	no	—
379	C₂²	—	4	2	no	—
380	C₂²	—	4	2	no	—
381	C₂²	—	4	2	no	—
382	C₂²	C₅₃×#463	1	2	yes	—
383	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
384	C₄	—	2	1	no	—
385	C₄	—	2	1	no	—
386	C₄	—	4	2	no	—
387	C₄	—	8	2	no	—
388	C₄	—	8	2	no	—
389	C₄	C₅₃×#463	1	2	yes	—
390	C₄	C₅₃×#473	1	2	yes	—
391	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
392	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
393	C₄	C₅₃×#577	1	2	yes	—
394	C₂	—	2	2	no	—
395	C₂	—	8	2	no	—
396	C₂	C₂×C₅₃×#463	1	1	yes	characteristic
397	C₂	C₅₃×#3057	1	1	yes	characteristic
398	C₂	C₂×C₅₃×#473	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. C₂×C₅₃×#487	C₂	semidirect
5. C₅₃×#1699	C₂	semidirect
6. C₅₃×#1679	C₂	semidirect
7. C₅₃×#1564	C₂	semidirect
8. C₅₃×#1549	C₂	semidirect
9. C₅₃×#1546	C₂	semidirect
177. #19197	C₅₃	direct (N × H)
197. C₂×#487	C₂×C₅₃	semidirect
199. #1699	C₂×C₅₃	semidirect
200. #1679	C₂×C₅₃	semidirect
201. #1564	C₂×C₅₃	semidirect
202. #1549	C₂×C₅₃	semidirect
203. #1546	C₂×C₅₃	semidirect
240. C₅₃	#19197	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×#3226	2
C₅₃×#3206	2
C₂×C₅₃×#487	2
C₅₃×#2931	2
C₅₃×#1699	2
C₅₃×#1679	2
C₅₃×#1564	2
C₅₃×#1549	2
C₅₃×#1546	2
#19197	53

Subgroups — C53×#19197 #27226709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19197 #27226709