Subgroups — C₅₃×#10160 #27217672

Order 13568.

578 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3205	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#3141	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1965	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1548	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1492	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#532	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#493	—	2	1	no	—
12	C₅₃×#489	—	2	2	no	—
13	C₅₃×#487	—	2	1	no	—
14	C₅₃×#486	—	2	1	no	—
15	C₅₃×#486	—	2	2	no	—
16	C₅₃×#486	—	2	1	no	—
17	C₅₃×#486	—	2	2	no	—
18	C₅₃×#459	—	2	2	no	—
19	C₅₃×#447	—	2	2	no	—
20	C₅₃×#431	C₂²	1	1	yes	characteristic
21	C₅₃×#423	C₂²	1	1	yes	characteristic
22	C₅₃×#417	—	2	2	no	—
23	C₂×#102×C₅₃	—	2	2	no	—
24	C₂×#100×C₅₃	—	2	2	no	—
25	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#333	—	2	2	no	—
28	C₅₃×D₄⋊C₈	—	2	2	no	—
29	C₂×#41×C₅₃	—	2	1	no	—
30	C₂²×Q₈×C₅₃	—	2	1	no	—
31	C₂²×D₄×C₅₃	—	2	1	no	—
32	C₂²×D₄×C₅₃	—	2	1	no	—
33	C₂²×D₄×C₅₃	D₄	1	2	yes	—
34	C₂×D₈×C₅₃	—	2	4	no	—
35	C₂×M₁₆×C₅₃	—	2	1	no	—
36	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
37	C₂²×C₈×C₅₃	—	2	1	no	—
38	C₄⋊Q₈×C₅₃	—	2	1	no	—
39	C₄⋊Q₈×C₅₃	—	2	1	no	—
40	C₄⋊Q₈×C₅₃	—	2	1	no	—
41	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
42	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
43	#127×C₅₃	—	2	1	no	—
44	#127×C₅₃	—	2	1	no	—
45	#127×C₅₃	—	2	1	no	—
46	#127×C₅₃	—	2	1	no	—
47	#127×C₅₃	D₄	1	1	yes	characteristic
48	#124×C₅₃	—	2	1	no	—
49	#121×C₅₃	—	2	1	no	—
50	#121×C₅₃	—	2	1	no	—
51	C₄×D₄×C₅₃	—	2	1	no	—
52	C₄×D₄×C₅₃	—	2	1	no	—
53	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
54	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
55	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₅₃	—	2	1	no	—
57	C₄⋊C₈×C₅₃	—	2	1	no	—
58	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
59	#103×C₅₃	—	2	4	no	—
60	#102×C₅₃	—	2	8	no	—
61	#102×C₅₃	—	2	4	no	—
62	#102×C₅₃	—	2	4	no	—
63	#100×C₅₃	—	4	4	no	—
64	C₂²⋊C₈×C₅₃	—	2	1	no	—
65	#97×C₅₃	—	2	1	no	—
66	C₄×C₈×C₅₃	—	2	1	no	—
67	C₄×C₈×C₅₃	—	2	1	no	—
68	C₄×C₈×C₅₃	—	2	1	no	—
69	C₄×C₈×C₅₃	D₄	1	2	yes	—
70	C₂⁴×C₅₃	—	2	2	no	—
71	#41×C₅₃	—	4	2	no	—
72	#41×C₅₃	—	4	2	no	—
73	C₂×Q₈×C₅₃	—	2	1	no	—
74	C₂×Q₈×C₅₃	—	2	2	no	—
75	C₂×Q₈×C₅₃	—	4	1	no	—
76	C₂×Q₈×C₅₃	—	8	2	no	—
77	C₂×D₄×C₅₃	—	2	1	no	—
78	C₂×D₄×C₅₃	—	2	2	no	—
79	C₂×D₄×C₅₃	—	2	2	no	—
80	C₂×D₄×C₅₃	—	2	1	no	—
81	C₂×D₄×C₅₃	—	2	1	no	—
82	C₂×D₄×C₅₃	—	4	4	no	—
83	C₂×D₄×C₅₃	—	4	1	no	—
84	C₂×D₄×C₅₃	—	4	1	no	—
85	C₂×D₄×C₅₃	—	4	1	no	—
86	C₂×D₄×C₅₃	—	4	1	no	—
87	C₂×D₄×C₅₃	—	4	1	no	—
88	C₂×D₄×C₅₃	—	4	2	no	—
89	C₂×D₄×C₅₃	—	4	2	no	—
90	C₂×D₄×C₅₃	—	8	2	no	—
91	C₂²×C₄×C₅₃	—	4	1	no	—
92	C₂²×C₄×C₅₃	—	4	1	no	—
93	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
96	D₈×C₅₃	—	8	4	no	—
97	M₁₆×C₅₃	—	2	2	no	—
98	M₁₆×C₅₃	—	2	2	no	—
99	M₁₆×C₅₃	—	4	2	no	—
100	C₂×C₈×C₅₃	—	2	2	no	—
101	C₂×C₈×C₅₃	—	2	2	no	—
102	C₂×C₈×C₅₃	—	2	2	no	—
103	C₂×C₈×C₅₃	—	4	2	no	—
104	C₂×C₈×C₅₃	—	4	1	no	—
105	C₄⋊C₄×C₅₃	—	2	2	no	—
106	C₄⋊C₄×C₅₃	—	2	1	no	—
107	C₄⋊C₄×C₅₃	—	2	2	no	—
108	C₄⋊C₄×C₅₃	—	2	1	no	—
109	C₄⋊C₄×C₅₃	—	4	1	no	—
110	C₄⋊C₄×C₅₃	—	4	1	no	—
111	C₂²⋊C₄×C₅₃	—	4	1	no	—
112	C₂²⋊C₄×C₅₃	—	4	1	no	—
113	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₂³×C₅₃	—	4	1	no	—
118	C₂³×C₅₃	—	4	2	no	—
119	C₂³×C₅₃	—	4	4	no	—
120	C₂³×C₅₃	—	8	2	no	—
121	C₂³×C₅₃	—	8	2	no	—
122	C₂³×C₅₃	#121	1	1	yes	characteristic
123	Q₈×C₅₃	—	4	2	no	—
124	Q₈×C₅₃	—	8	2	no	—
125	Q₈×C₅₃	—	8	2	no	—
126	D₄×C₅₃	—	4	4	no	—
127	D₄×C₅₃	—	4	2	no	—
128	D₄×C₅₃	—	4	4	no	—
129	D₄×C₅₃	—	4	2	no	—
130	D₄×C₅₃	—	4	2	no	—
131	D₄×C₅₃	—	8	2	no	—
132	D₄×C₅₃	—	8	2	no	—
133	D₄×C₅₃	—	8	2	no	—
134	D₄×C₅₃	—	8	2	no	—
135	D₄×C₅₃	—	8	1	no	—
136	D₄×C₅₃	—	8	1	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	4	1	no	—
140	C₂×C₄×C₅₃	—	4	2	no	—
141	C₂×C₄×C₅₃	—	8	2	no	—
142	C₂×C₄×C₅₃	—	8	1	no	—
143	C₂×C₄×C₅₃	—	8	1	no	—
144	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
145	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
146	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
147	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
148	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
149	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
150	C₂×C₄×C₅₃	#135	1	2	yes	—
151	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
152	C₈×C₅₃	—	4	2	no	—
153	C₈×C₅₃	—	4	2	no	—
154	C₈×C₅₃	—	4	2	no	—
155	C₈×C₅₃	—	8	1	no	—
156	#10160	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂²×C₅₃	—	2	1	no	—
158	C₂²×C₅₃	—	2	1	no	—
159	C₂²×C₅₃	—	8	2	no	—
160	C₂²×C₅₃	—	8	1	no	—
161	C₂²×C₅₃	—	8	1	no	—
162	C₂²×C₅₃	—	8	2	no	—
163	C₂²×C₅₃	—	8	1	no	—
164	C₂²×C₅₃	—	8	2	no	—
165	C₂²×C₅₃	—	16	4	no	—
166	C₂²×C₅₃	#392	1	1	yes	characteristic
167	C₂²×C₅₃	#463	1	2	yes	—
168	C₄×C₅₃	—	2	1	no	—
169	C₄×C₅₃	—	2	1	no	—
170	C₄×C₅₃	—	8	1	no	—
171	C₄×C₅₃	—	8	2	no	—
172	C₄×C₅₃	#449	1	2	yes	—
173	C₄×C₅₃	#455	1	2	yes	—
174	C₄×C₅₃	#472	1	2	yes	—
175	C₄×C₅₃	#495	1	1	yes	characteristic
176	C₄×C₅₃	#496	1	1	yes	characteristic
177	#3205	C₂×C₅₃	1	1	yes	characteristic
178	#3141	C₂×C₅₃	1	1	yes	characteristic, complemented
179	C₂×#486	C₂×C₅₃	1	1	yes	characteristic, complemented
180	#1965	C₂×C₅₃	1	1	yes	characteristic, complemented
181	#1548	C₂×C₅₃	1	1	yes	characteristic, complemented
182	#1492	C₂×C₅₃	1	1	yes	characteristic, complemented
183	#1278	C₂×C₅₃	1	1	yes	characteristic, complemented
184	C₂×C₅₃	—	2	2	no	—
185	C₂×C₅₃	—	16	1	no	—
186	C₂×C₅₃	—	16	2	no	—
187	C₂×C₅₃	#2004	1	1	yes	characteristic
188	C₂×C₅₃	#2012	1	1	yes	characteristic
189	C₂×C₅₃	#2018	1	1	yes	characteristic
190	#532	C₂²×C₅₃	1	1	yes	characteristic
191	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
192	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
193	#493	—	2	1	no	—
194	#489	—	2	2	no	—
195	#487	—	2	1	no	—
196	#486	—	2	2	no	—
197	#486	—	2	1	no	—
198	#486	—	2	1	no	—
199	#486	—	2	2	no	—
200	#459	—	2	2	no	—
201	#447	—	2	2	no	—
202	#431	C₂²×C₅₃	1	1	yes	characteristic
203	#423	C₂²×C₅₃	1	1	yes	characteristic
204	#417	—	2	2	no	—
205	C₂×#102	—	2	2	no	—
206	C₂×#100	—	2	2	no	—
207	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
208	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
209	#333	—	2	2	no	—
210	D₄⋊C₈	—	2	2	no	—
211	C₅₃	#10160	1	1	yes	characteristic, direct product factor, Sylow
212	C₂×#41	—	2	1	no	—
213	C₂²×Q₈	—	2	1	no	—
214	C₂²×D₄	—	2	1	no	—
215	C₂²×D₄	—	2	1	no	—
216	C₂²×D₄	D₄×C₅₃	1	2	yes	—
217	C₂×D₈	—	2	4	no	—
218	C₂×M₁₆	—	2	1	no	—
219	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
220	C₂²×C₈	—	2	1	no	—
221	C₄⋊Q₈	—	2	1	no	—
222	C₄⋊Q₈	—	2	1	no	—
223	C₄⋊Q₈	—	2	1	no	—
224	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
225	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
226	#127	—	2	1	no	—
227	#127	—	2	1	no	—
228	#127	—	2	1	no	—
229	#127	—	2	1	no	—
230	#127	D₄×C₅₃	1	1	yes	characteristic
231	#124	—	2	1	no	—
232	#121	—	2	1	no	—
233	#121	—	2	1	no	—
234	C₄×D₄	—	2	1	no	—
235	C₄×D₄	—	2	1	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	2	1	no	—
238	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	—	2	1	no	—
241	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
242	#103	—	2	4	no	—
243	#102	—	2	8	no	—
244	#102	—	2	4	no	—
245	#102	—	2	4	no	—
246	#100	—	4	4	no	—
247	C₂²⋊C₈	—	2	1	no	—
248	#97	—	2	1	no	—
249	C₄×C₈	—	2	1	no	—
250	C₄×C₈	—	2	1	no	—
251	C₄×C₈	—	2	1	no	—
252	C₄×C₈	D₄×C₅₃	1	2	yes	—
253	C₂⁴	—	2	2	no	—
254	#41	—	4	2	no	—
255	#41	—	4	2	no	—
256	C₂×Q₈	—	2	1	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×Q₈	—	8	2	no	—
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	2	1	no	—
264	C₂×D₄	—	2	1	no	—
265	C₂×D₄	—	4	4	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂×D₄	—	4	1	no	—
268	C₂×D₄	—	4	1	no	—
269	C₂×D₄	—	4	1	no	—
270	C₂×D₄	—	4	1	no	—
271	C₂×D₄	—	4	2	no	—
272	C₂×D₄	—	4	2	no	—
273	C₂×D₄	—	8	2	no	—
274	C₂²×C₄	—	4	1	no	—
275	C₂²×C₄	—	4	1	no	—
276	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
277	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
278	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
279	D₈	—	8	4	no	—
280	M₁₆	—	2	2	no	—
281	M₁₆	—	2	2	no	—
282	M₁₆	—	4	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	4	2	no	—
287	C₂×C₈	—	4	1	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₂²⋊C₄	—	4	1	no	—
295	C₂²⋊C₄	—	4	1	no	—
296	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
297	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
298	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
299	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
300	C₂³	—	4	1	no	—
301	C₂³	—	4	2	no	—
302	C₂³	—	4	4	no	—
303	C₂³	—	8	2	no	—
304	C₂³	—	8	2	no	—
305	C₂³	#121×C₅₃	1	1	yes	characteristic
306	Q₈	—	4	2	no	—
307	Q₈	—	8	2	no	—
308	Q₈	—	8	2	no	—
309	D₄	—	4	4	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	2	no	—
312	D₄	—	4	4	no	—
313	D₄	—	4	2	no	—
314	D₄	—	8	2	no	—
315	D₄	—	8	2	no	—
316	D₄	—	8	1	no	—
317	D₄	—	8	1	no	—
318	D₄	—	8	2	no	—
319	D₄	—	8	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	8	2	no	—
325	C₂×C₄	—	8	1	no	—
326	C₂×C₄	—	8	1	no	—
327	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
328	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
329	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
330	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
331	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
332	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
333	C₂×C₄	#135×C₅₃	1	2	yes	—
334	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
335	C₈	—	4	2	no	—
336	C₈	—	4	2	no	—
337	C₈	—	4	2	no	—
338	C₈	—	8	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	1	no	—
343	C₂²	—	8	1	no	—
344	C₂²	—	8	2	no	—
345	C₂²	—	8	1	no	—
346	C₂²	—	8	2	no	—
347	C₂²	—	16	4	no	—
348	C₂²	C₅₃×#392	1	1	yes	characteristic
349	C₂²	C₅₃×#463	1	2	yes	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	8	1	no	—
353	C₄	—	8	2	no	—
354	C₄	C₅₃×#449	1	2	yes	—
355	C₄	C₅₃×#455	1	2	yes	—
356	C₄	C₅₃×#472	1	2	yes	—
357	C₄	C₅₃×#495	1	1	yes	characteristic
358	C₄	C₅₃×#496	1	1	yes	characteristic
359	C₂	—	2	2	no	—
360	C₂	—	16	2	no	—
361	C₂	—	16	1	no	—
362	C₂	C₅₃×#2004	1	1	yes	characteristic
363	C₂	C₅₃×#2012	1	1	yes	characteristic
364	C₂	C₅₃×#2018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₅₃×#3205	C₂
LCS	G ▷ C₂×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
2. C₅₃×#3141	C₂	semidirect
3. C₂×C₅₃×#486	C₂	semidirect
4. C₅₃×#1965	C₂	semidirect
5. C₅₃×#1548	C₂	semidirect
6. C₅₃×#1492	C₂	semidirect
7. C₅₃×#1278	C₂	semidirect
156. #10160	C₅₃	direct (N × H)
178. #3141	C₂×C₅₃	semidirect
179. C₂×#486	C₂×C₅₃	semidirect
180. #1965	C₂×C₅₃	semidirect
181. #1548	C₂×C₅₃	semidirect
182. #1492	C₂×C₅₃	semidirect
183. #1278	C₂×C₅₃	semidirect
211. C₅₃	#10160	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×#3205	2
C₅₃×#3141	2
C₂×C₅₃×#486	2
C₅₃×#1965	2
C₅₃×#1548	2
C₅₃×#1492	2
C₅₃×#1278	2
#10160	53

Subgroups — C53×#10160 #27217672

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10160 #27217672