Subgroups — C₅₃×#10288 #27217800

Order 13568.

554 conjugacy classes of subgroups.

The subgroup lattice has 350 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	C₅₃×#2862	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1962	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1548	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1546	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1276	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×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#500	—	2	1	no	—
12	C₅₃×#492	—	2	1	no	—
13	C₅₃×#487	—	2	1	no	—
14	C₅₃×#486	—	2	1	no	—
15	C₅₃×#480	—	2	2	no	—
16	C₅₃×#474	—	2	2	no	—
17	C₅₃×#461	—	2	2	no	—
18	C₅₃×#448	—	2	2	no	—
19	C₅₃×#431	C₂²	1	1	yes	characteristic
20	C₅₃×#431	C₂²	1	1	yes	characteristic
21	C₅₃×#423	C₂²	1	1	yes	characteristic
22	C₂×#104×C₅₃	—	2	2	no	—
23	C₅₃×#391	—	2	2	no	—
24	C₅₃×#391	—	2	2	no	—
25	C₅₃×#382	—	2	2	no	—
26	C₄³×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#333	—	2	2	no	—
28	C₅₃×#332	—	2	2	no	—
29	C₂×#41×C₅₃	—	2	1	no	—
30	C₂²×Q₈×C₅₃	—	2	1	no	—
31	C₂²×Q₈×C₅₃	D₄	1	2	yes	—
32	C₂×SD₁₆×C₅₃	—	2	4	no	—
33	C₂×M₁₆×C₅₃	—	2	1	no	—
34	C₄⋊Q₈×C₅₃	—	2	1	no	—
35	C₄⋊Q₈×C₅₃	—	2	1	no	—
36	C₄⋊Q₈×C₅₃	—	2	1	no	—
37	C₄⋊Q₈×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	—	2	1	no	—
42	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
43	#127×C₅₃	D₄	1	1	yes	characteristic
44	#124×C₅₃	—	2	1	no	—
45	#121×C₅₃	—	2	1	no	—
46	#121×C₅₃	—	2	1	no	—
47	C₄×D₄×C₅₃	—	2	1	no	—
48	C₄×D₄×C₅₃	—	2	1	no	—
49	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
50	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
51	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
52	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
53	C₂×C₄²×C₅₃	—	2	1	no	—
54	C₂×C₄²×C₅₃	—	2	1	no	—
55	C₂×C₄²×C₅₃	D₄	1	2	yes	—
56	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
57	#107×C₅₃	—	2	4	no	—
58	C₄⋊C₈×C₅₃	—	2	1	no	—
59	C₄⋊C₈×C₅₃	—	2	1	no	—
60	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
61	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
62	#104×C₅₃	—	4	4	no	—
63	#103×C₅₃	—	2	4	no	—
64	#102×C₅₃	—	2	4	no	—
65	C₂²⋊C₈×C₅₃	—	2	1	no	—
66	C₂²⋊C₈×C₅₃	—	2	1	no	—
67	C₄×C₈×C₅₃	—	2	1	no	—
68	C₄×C₈×C₅₃	—	2	1	no	—
69	#95×C₅₃	—	2	8	no	—
70	#41×C₅₃	—	4	2	no	—
71	#41×C₅₃	—	4	2	no	—
72	C₂×Q₈×C₅₃	—	2	2	no	—
73	C₂×Q₈×C₅₃	—	2	1	no	—
74	C₂×Q₈×C₅₃	—	2	2	no	—
75	C₂×Q₈×C₅₃	—	4	4	no	—
76	C₂×Q₈×C₅₃	—	4	1	no	—
77	C₂×Q₈×C₅₃	—	4	1	no	—
78	C₂×Q₈×C₅₃	—	4	2	no	—
79	C₂×Q₈×C₅₃	—	4	2	no	—
80	C₂×D₄×C₅₃	—	2	1	no	—
81	C₂×D₄×C₅₃	—	2	1	no	—
82	C₂×D₄×C₅₃	—	2	1	no	—
83	C₂×D₄×C₅₃	—	4	1	no	—
84	C₂²×C₄×C₅₃	—	2	2	no	—
85	C₂²×C₄×C₅₃	—	2	2	no	—
86	C₂²×C₄×C₅₃	—	2	2	no	—
87	C₂²×C₄×C₅₃	—	4	1	no	—
88	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
91	SD₁₆×C₅₃	—	8	4	no	—
92	M₁₆×C₅₃	—	4	2	no	—
93	C₂×C₈×C₅₃	—	2	2	no	—
94	C₂×C₈×C₅₃	—	2	2	no	—
95	C₂×C₈×C₅₃	—	4	1	no	—
96	C₄⋊C₄×C₅₃	—	2	2	no	—
97	C₄⋊C₄×C₅₃	—	2	1	no	—
98	C₄⋊C₄×C₅₃	—	2	1	no	—
99	C₄⋊C₄×C₅₃	—	2	2	no	—
100	C₄⋊C₄×C₅₃	—	4	1	no	—
101	C₄⋊C₄×C₅₃	—	4	1	no	—
102	C₄⋊C₄×C₅₃	—	4	1	no	—
103	C₄⋊C₄×C₅₃	—	4	1	no	—
104	C₄⋊C₄×C₅₃	—	4	1	no	—
105	C₄⋊C₄×C₅₃	—	4	1	no	—
106	C₂²⋊C₄×C₅₃	—	4	1	no	—
107	C₂²⋊C₄×C₅₃	—	4	1	no	—
108	C₄²×C₅₃	—	2	4	no	—
109	C₄²×C₅₃	—	2	2	no	—
110	C₄²×C₅₃	—	2	2	no	—
111	C₄²×C₅₃	—	4	2	no	—
112	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₅₃	#41	1	1	yes	characteristic
116	C₂³×C₅₃	—	4	1	no	—
117	C₂³×C₅₃	#120	1	1	yes	characteristic
118	Q₈×C₅₃	—	4	4	no	—
119	Q₈×C₅₃	—	4	2	no	—
120	Q₈×C₅₃	—	8	2	no	—
121	Q₈×C₅₃	—	8	2	no	—
122	Q₈×C₅₃	—	8	1	no	—
123	Q₈×C₅₃	—	8	1	no	—
124	D₄×C₅₃	—	4	2	no	—
125	D₄×C₅₃	—	4	2	no	—
126	D₄×C₅₃	—	4	2	no	—
127	D₄×C₅₃	—	8	2	no	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	4	no	—
131	C₂×C₄×C₅₃	—	4	2	no	—
132	C₂×C₄×C₅₃	—	4	2	no	—
133	C₂×C₄×C₅₃	—	4	4	no	—
134	C₂×C₄×C₅₃	—	4	2	no	—
135	C₂×C₄×C₅₃	—	4	2	no	—
136	C₂×C₄×C₅₃	—	4	1	no	—
137	C₂×C₄×C₅₃	—	4	4	no	—
138	C₂×C₄×C₅₃	—	8	1	no	—
139	C₂×C₄×C₅₃	—	8	1	no	—
140	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
141	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
142	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
143	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
144	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
145	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
146	C₂×C₄×C₅₃	#135	1	2	yes	—
147	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
148	C₈×C₅₃	—	4	2	no	—
149	C₈×C₅₃	—	4	2	no	—
150	C₈×C₅₃	—	8	1	no	—
151	#10288	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂²×C₅₃	—	2	1	no	—
153	C₂²×C₅₃	—	2	1	no	—
154	C₂²×C₅₃	—	8	1	no	—
155	C₂²×C₅₃	—	8	1	no	—
156	C₂²×C₅₃	—	8	1	no	—
157	C₂²×C₅₃	#392	1	1	yes	characteristic
158	C₂²×C₅₃	#449	1	2	yes	—
159	C₄×C₅₃	—	2	1	no	—
160	C₄×C₅₃	—	2	1	no	—
161	C₄×C₅₃	—	4	4	no	—
162	C₄×C₅₃	—	8	2	no	—
163	C₄×C₅₃	—	8	2	no	—
164	C₄×C₅₃	—	8	1	no	—
165	C₄×C₅₃	#455	1	2	yes	—
166	C₄×C₅₃	#456	1	2	yes	—
167	C₄×C₅₃	#463	1	2	yes	—
168	C₄×C₅₃	#495	1	1	yes	characteristic
169	C₄×C₅₃	#496	1	1	yes	characteristic
170	#3226	C₂×C₅₃	1	1	yes	characteristic
171	#3206	C₂×C₅₃	1	1	yes	characteristic
172	#2862	C₂×C₅₃	1	1	yes	characteristic, complemented
173	#1962	C₂×C₅₃	1	1	yes	characteristic
174	#1548	C₂×C₅₃	1	1	yes	characteristic, complemented
175	#1546	C₂×C₅₃	1	1	yes	characteristic, complemented
176	#1276	C₂×C₅₃	1	1	yes	characteristic, complemented
177	C₂×C₅₃	—	2	2	no	—
178	C₂×C₅₃	—	16	1	no	—
179	C₂×C₅₃	#2004	1	1	yes	characteristic
180	C₂×C₅₃	#2005	1	1	yes	characteristic
181	C₂×C₅₃	#2012	1	1	yes	characteristic
182	#532	C₂²×C₅₃	1	1	yes	characteristic
183	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
184	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
185	#500	—	2	1	no	—
186	#492	—	2	1	no	—
187	#487	—	2	1	no	—
188	#486	—	2	1	no	—
189	#480	—	2	2	no	—
190	#474	—	2	2	no	—
191	#461	—	2	2	no	—
192	#448	—	2	2	no	—
193	#431	C₂²×C₅₃	1	1	yes	characteristic
194	#431	C₂²×C₅₃	1	1	yes	characteristic
195	#423	C₂²×C₅₃	1	1	yes	characteristic
196	C₂×#104	—	2	2	no	—
197	#391	—	2	2	no	—
198	#391	—	2	2	no	—
199	#382	—	2	2	no	—
200	C₄³	C₂²×C₅₃	1	1	yes	characteristic
201	#333	—	2	2	no	—
202	#332	—	2	2	no	—
203	C₅₃	#10288	1	1	yes	characteristic, direct product factor, Sylow
204	C₂×#41	—	2	1	no	—
205	C₂²×Q₈	—	2	1	no	—
206	C₂²×Q₈	D₄×C₅₃	1	2	yes	—
207	C₂×SD₁₆	—	2	4	no	—
208	C₂×M₁₆	—	2	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
218	#127	D₄×C₅₃	1	1	yes	characteristic
219	#124	—	2	1	no	—
220	#121	—	2	1	no	—
221	#121	—	2	1	no	—
222	C₄×D₄	—	2	1	no	—
223	C₄×D₄	—	2	1	no	—
224	C₂×C₄⋊C₄	—	2	1	no	—
225	C₂×C₄⋊C₄	—	2	1	no	—
226	C₂×C₄⋊C₄	—	2	1	no	—
227	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
228	C₂×C₄²	—	2	1	no	—
229	C₂×C₄²	—	2	1	no	—
230	C₂×C₄²	D₄×C₅₃	1	2	yes	—
231	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
232	#107	—	2	4	no	—
233	C₄⋊C₈	—	2	1	no	—
234	C₄⋊C₈	—	2	1	no	—
235	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
236	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
237	#104	—	4	4	no	—
238	#103	—	2	4	no	—
239	#102	—	2	4	no	—
240	C₂²⋊C₈	—	2	1	no	—
241	C₂²⋊C₈	—	2	1	no	—
242	C₄×C₈	—	2	1	no	—
243	C₄×C₈	—	2	1	no	—
244	#95	—	2	8	no	—
245	#41	—	4	2	no	—
246	#41	—	4	2	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	1	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	—	4	4	no	—
251	C₂×Q₈	—	4	1	no	—
252	C₂×Q₈	—	4	1	no	—
253	C₂×Q₈	—	4	2	no	—
254	C₂×Q₈	—	4	2	no	—
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	2	1	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
264	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
265	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
266	SD₁₆	—	8	4	no	—
267	M₁₆	—	4	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	4	1	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₂²⋊C₄	—	4	1	no	—
282	C₂²⋊C₄	—	4	1	no	—
283	C₄²	—	2	4	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	4	2	no	—
287	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
288	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
289	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
290	C₄²	#41×C₅₃	1	1	yes	characteristic
291	C₂³	—	4	1	no	—
292	C₂³	#120×C₅₃	1	1	yes	characteristic
293	Q₈	—	4	4	no	—
294	Q₈	—	4	2	no	—
295	Q₈	—	8	2	no	—
296	Q₈	—	8	1	no	—
297	Q₈	—	8	1	no	—
298	Q₈	—	8	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	8	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	4	no	—
313	C₂×C₄	—	8	1	no	—
314	C₂×C₄	—	8	1	no	—
315	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
316	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
317	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
318	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
319	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
320	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
321	C₂×C₄	#135×C₅₃	1	2	yes	—
322	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
323	C₈	—	4	2	no	—
324	C₈	—	4	2	no	—
325	C₈	—	8	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	8	1	no	—
329	C₂²	—	8	1	no	—
330	C₂²	—	8	1	no	—
331	C₂²	C₅₃×#392	1	1	yes	characteristic
332	C₂²	C₅₃×#449	1	2	yes	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	4	4	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	2	no	—
338	C₄	—	8	1	no	—
339	C₄	C₅₃×#455	1	2	yes	—
340	C₄	C₅₃×#456	1	2	yes	—
341	C₄	C₅₃×#463	1	2	yes	—
342	C₄	C₅₃×#495	1	1	yes	characteristic
343	C₄	C₅₃×#496	1	1	yes	characteristic
344	C₂	—	2	2	no	—
345	C₂	—	16	1	no	—
346	C₂	C₅₃×#2004	1	1	yes	characteristic
347	C₂	C₅₃×#2005	1	1	yes	characteristic
348	C₂	C₅₃×#2012	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₅₃×#532	C₂²
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
3. C₅₃×#2862	C₂	semidirect
5. C₅₃×#1548	C₂	semidirect
6. C₅₃×#1546	C₂	semidirect
7. C₅₃×#1276	C₂	semidirect
151. #10288	C₅₃	direct (N × H)
172. #2862	C₂×C₅₃	semidirect
174. #1548	C₂×C₅₃	semidirect
175. #1546	C₂×C₅₃	semidirect
176. #1276	C₂×C₅₃	semidirect
203. C₅₃	#10288	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
C₅₃×#2862	2
C₅₃×#1962	2
C₅₃×#1548	2
C₅₃×#1546	2
C₅₃×#1276	2
#10288	53

Subgroups — C53×#10288 #27217800

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10288 #27217800