Subgroups — #27355928

Order 13568.

592 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285288	C₂	1	1	yes	maximal, characteristic
2	#20283718	C₂	1	2	yes	maximal, complemented
3	#20273630	C₂	1	1	yes	maximal, characteristic, complemented
4	#20273624	C₂	1	1	yes	maximal, characteristic, complemented
5	#20273593	C₂	1	2	yes	maximal, complemented
6	#20273575	C₂	1	2	yes	maximal, complemented
7	#20273538	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273460	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273122	C₂	1	2	yes	maximal, complemented
10	#20273001	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1579	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811160	C₂²	1	2	yes	—
13	#16811024	C₄	1	1	yes	characteristic
14	#16811008	C₄	1	1	yes	characteristic
15	#16810979	C₄	1	2	yes	—
16	#16810861	C₄	1	1	yes	characteristic
17	#16810860	C₄	1	1	yes	characteristic
18	#16810855	C₄	1	2	yes	—
19	#16810754	C₂²	1	2	yes	—
20	C₄×#11719924	C₂²	1	2	yes	—
21	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
22	#16810702	—	2	2	no	—
23	#16810702	C₂²	1	2	yes	—
24	#16810693	—	2	2	no	—
25	#16810688	C₂²	1	2	yes	—
26	#16810683	—	2	2	no	—
27	#16810675	C₂²	1	2	yes	—
28	#16810674	C₂²	1	1	yes	characteristic
29	#16810577	—	2	2	no	—
30	C₄×#11719912	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
32	#16810375	C₂²	1	2	yes	—
33	#16810374	C₂²	1	1	yes	characteristic
34	Q₈⋊#88322	C₂²	1	2	yes	—
35	#16810352	C₂²	1	2	yes	—
36	#16810350	C₂²	1	2	yes	—
37	#16810345	C₂²	1	1	yes	characteristic
38	#16810344	C₂²	1	1	yes	characteristic
39	#16810344	C₂²	1	1	yes	characteristic
40	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
42	C₅₃×#334	C₂²	1	1	yes	characteristic
43	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₅₃×#322	C₂²	1	1	yes	characteristic
45	C₂×#11719924	C₂×C₄	1	2	yes	—
46	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic
47	C₂×#11719912	—	2	2	no	—
48	C₂×#11719912	D₄	1	2	yes	—
49	C₂×C₈×D₅₃	—	2	2	no	—
50	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
52	#13133212	—	2	2	no	—
53	#13133208	C₂×C₄	1	2	yes	—
54	#13133206	C₂×C₄	1	2	yes	—
55	#13133202	—	2	2	no	—
56	#13133193	C₂×C₄	1	2	yes	—
57	#13133192	C₂×C₄	1	2	yes	—
58	C₄×D₂₁₂	—	2	2	no	—
59	C₄×D₂₁₂	C₂³	1	2	yes	—
60	#13133174	—	2	2	no	—
61	C₄²×D₅₃	C₂³	1	1	yes	characteristic
62	#13133151	C₂×C₄	1	1	yes	characteristic
63	#13133148	C₂×C₄	1	2	yes	—
64	C₄⋊#88322	—	2	2	no	—
65	C₄⋊#88322	C₂³	1	2	yes	—
66	#13133146	—	2	1	no	—
67	#13133141	C₂×C₄	1	2	yes	—
68	#13133139	C₂×C₄	1	2	yes	—
69	#13133137	C₂×C₄	1	2	yes	—
70	#13133129	—	2	2	no	—
71	#13133129	—	2	2	no	—
72	#13133129	—	2	2	no	—
73	#13133129	—	2	1	no	—
74	#13133128	—	2	2	no	—
75	#13133127	D₄	1	2	yes	—
76	C₄×#88322	C₂³	1	1	yes	characteristic
77	C₄×#88322	C₂³	1	1	yes	characteristic
78	C₄×#88322	C₂³	1	1	yes	characteristic
79	C₈×Dic₅₃	—	2	2	no	—
80	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
82	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
83	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
84	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
85	C₄⋊C₈×C₅₃	—	2	2	no	—
86	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
87	#103×C₅₃	C₂×C₄	1	2	yes	—
88	C₄×C₈×C₅₃	—	2	2	no	—
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	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
93	C₂×D₂₁₂	—	2	2	no	—
94	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
95	C₂×C₄×D₅₃	—	2	2	no	—
96	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
99	Q₁₆×C₅₃	—	2	2	no	—
100	C₂×C₈×C₅₃	—	2	2	no	—
101	C₂×C₈×C₅₃	—	2	1	no	—
102	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
105	C₄⋊C₄×C₅₃	—	2	2	no	—
106	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
107	C₄²×C₅₃	—	2	2	no	—
108	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
109	#11719924	—	4	2	no	—
110	#11719921	—	2	2	no	—
111	#11719921	—	2	2	no	—
112	#11719921	—	2	2	no	—
113	#11719921	—	2	2	no	—
114	#11719921	—	2	2	no	—
115	C₄⋊Dic₅₃	—	2	2	no	—
116	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
117	#11719919	—	2	2	no	—
118	#11719919	—	2	2	no	—
119	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
121	C₄×Dic₅₃	#41	1	1	yes	characteristic
122	C₂×#88322	—	2	2	no	—
123	C₂×#88322	—	2	1	no	—
124	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
125	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
126	C₂×#88322	C₂×D₄	1	2	yes	—
127	D₄₂₄	—	2	2	no	—
128	#11719912	—	4	2	no	—
129	#11719912	—	4	4	no	—
130	C₈×D₅₃	—	2	2	no	—
131	C₈×D₅₃	—	4	1	no	—
132	C₈×D₅₃	—	4	4	no	—
133	C₂²×D₅₃	—	2	2	no	—
134	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
135	Q₈×C₅₃	—	2	2	no	—
136	Q₈×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
142	C₂×Dic₅₃	—	2	2	no	—
143	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
144	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
145	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
146	D₂₁₂	—	2	2	no	—
147	D₂₁₂	—	2	2	no	—
148	D₂₁₂	—	8	2	no	—
149	C₄×D₅₃	—	2	1	no	—
150	C₄×D₅₃	—	2	1	no	—
151	C₄×D₅₃	—	2	2	no	—
152	C₄×D₅₃	—	4	1	no	—
153	C₄×D₅₃	—	4	2	no	—
154	C₄×D₅₃	—	4	2	no	—
155	C₈×C₅₃	—	2	1	no	—
156	C₈×C₅₃	—	2	2	no	—
157	C₈×C₅₃	—	4	1	no	—
158	C₈×C₅₃	—	4	2	no	—
159	C₈×C₅₃	#131	1	2	yes	—
160	#88322	—	2	1	no	—
161	#88322	—	2	1	no	—
162	#88322	—	2	2	no	—
163	#88322	—	4	1	no	—
164	#88322	—	4	2	no	—
165	#18075	—	53	1	no	maximal, Sylow
166	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂×D₅₃	—	2	1	no	—
168	C₂×D₅₃	—	2	1	no	—
169	C₂×D₅₃	—	2	1	no	—
170	C₂×D₅₃	—	4	2	no	—
171	C₂×D₅₃	—	4	2	no	—
172	C₂×D₅₃	—	4	2	no	—
173	C₄×C₅₃	—	2	1	no	—
174	C₄×C₅₃	—	4	2	no	—
175	C₄×C₅₃	—	4	2	no	—
176	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
177	C₄×C₅₃	#568	1	1	yes	characteristic
178	C₄×C₅₃	#575	1	2	yes	—
179	Dic₅₃	—	2	1	no	—
180	Dic₅₃	—	2	1	no	—
181	Dic₅₃	—	2	1	no	—
182	Dic₅₃	—	4	2	no	—
183	#3168	—	53	1	no	—
184	#2969	—	53	2	no	—
185	#1594	—	53	1	no	—
186	#1593	—	53	1	no	—
187	#1588	—	53	2	no	—
188	#1585	—	53	2	no	—
189	#1580	—	53	1	no	—
190	#1579	D₅₃	1	1	yes	characteristic, complemented
191	#1564	—	53	1	no	—
192	#1482	—	53	2	no	—
193	#1451	—	53	1	no	—
194	C₂×C₅₃	#2935	1	1	yes	characteristic
195	C₂×C₅₃	#2943	1	1	yes	characteristic
196	C₂×C₅₃	#2957	1	1	yes	characteristic
197	D₅₃	—	4	1	no	—
198	D₅₃	—	8	2	no	—
199	#529	—	53	2	no	—
200	#499	—	53	1	no	—
201	#495	—	53	1	no	—
202	#488	—	53	2	no	—
203	#467	—	53	1	no	—
204	#466	—	53	1	no	—
205	#465	—	53	2	no	—
206	#446	—	53	2	no	—
207	C₈×Q₈	C₂×D₅₃	1	2	yes	—
208	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
209	C₄×SD₁₆	—	53	2	no	—
210	C₄×D₈	—	53	1	no	—
211	#436	—	53	2	no	—
212	#436	—	106	2	no	—
213	#435	—	106	2	no	—
214	C₈×D₄	—	53	2	no	—
215	#433	—	106	2	no	—
216	#431	—	53	1	no	—
217	#431	—	53	2	no	—
218	#405	—	106	2	no	—
219	C₄×M₁₆	—	53	1	no	—
220	C₂×C₄×C₈	—	53	1	no	—
221	#335	—	53	2	no	—
222	#334	—	53	1	no	—
223	#334	C₂×D₅₃	1	1	yes	characteristic
224	Q₈⋊C₈	—	53	2	no	—
225	Q₈⋊C₈	C₂×D₅₃	1	2	yes	—
226	D₄⋊C₈	—	53	2	no	—
227	D₄⋊C₈	—	53	2	no	—
228	#322	—	53	1	no	—
229	#322	—	53	1	no	—
230	#322	—	53	1	no	—
231	#322	C₂×D₅₃	1	1	yes	characteristic
232	C₅₃	#18075	1	1	yes	characteristic, complemented, Sylow
233	C₂×Q₁₆	C₄×D₅₃	1	1	yes	characteristic
234	C₂×SD₁₆	—	53	2	no	—
235	C₂×D₈	—	53	1	no	—
236	C₂×M₁₆	—	53	2	no	—
237	C₂×M₁₆	—	106	2	no	—
238	C₂²×C₈	—	53	1	no	—
239	C₂²×C₈	—	53	1	no	—
240	C₂²×C₈	—	106	2	no	—
241	#126	—	106	2	no	—
242	#125	—	53	2	no	—
243	#124	—	53	2	no	—
244	#123	—	106	2	no	—
245	C₂²⋊Q₈	—	53	2	no	—
246	#121	—	53	2	no	—
247	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
248	C₄×D₄	—	53	2	no	—
249	C₄×D₄	—	106	2	no	—
250	#117	—	106	2	no	—
251	C₂×C₄²	—	53	1	no	—
252	#107	—	53	1	no	—
253	#107	C₄×D₅₃	1	1	yes	characteristic
254	#106	—	53	2	no	—
255	C₄⋊C₈	—	2	2	no	—
256	C₄⋊C₈	—	53	2	no	—
257	C₄⋊C₈	—	106	2	no	—
258	C₄⋊C₈	—	106	1	no	—
259	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
260	#103	—	53	2	no	—
261	#103	C₄×D₅₃	1	2	yes	—
262	#102	—	53	2	no	—
263	#102	—	53	2	no	—
264	C₂²⋊C₈	—	106	2	no	—
265	C₂²⋊C₈	—	106	1	no	—
266	C₂²⋊C₈	—	106	2	no	—
267	C₂²⋊C₈	—	106	2	no	—
268	#97	—	53	2	no	—
269	#97	—	106	2	no	—
270	C₄×C₈	—	2	2	no	—
271	C₄×C₈	—	53	1	no	—
272	C₄×C₈	—	53	1	no	—
273	C₄×C₈	—	53	1	no	—
274	C₄×C₈	—	53	1	no	—
275	C₄×C₈	—	53	1	no	—
276	C₄×C₈	—	106	2	no	—
277	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
278	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
280	C₂×Q₈	C₂×C₄×D₅₃	1	2	yes	—
281	C₂×D₄	—	53	2	no	—
282	C₂×D₄	—	106	2	no	—
283	C₂²×C₄	—	53	1	no	—
284	C₂²×C₄	—	53	1	no	—
285	C₂²×C₄	—	53	1	no	—
286	C₂²×C₄	—	106	2	no	—
287	Q₁₆	—	2	2	no	—
288	SD₁₆	—	212	2	no	—
289	D₈	—	106	2	no	—
290	M₁₆	—	212	2	no	—
291	M₁₆	—	212	4	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	53	1	no	—
295	C₂×C₈	—	53	1	no	—
296	C₂×C₈	—	53	2	no	—
297	C₂×C₈	—	106	2	no	—
298	C₂×C₈	—	106	2	no	—
299	C₂×C₈	—	106	1	no	—
300	C₂×C₈	—	212	1	no	—
301	C₂×C₈	—	212	4	no	—
302	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
303	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
304	C₂×C₈	D₄×D₅₃	1	2	yes	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	53	2	no	—
307	C₄⋊C₄	—	106	2	no	—
308	C₄⋊C₄	—	106	2	no	—
309	C₄⋊C₄	—	106	2	no	—
310	C₄⋊C₄	C₂×C₄×D₅₃	1	2	yes	—
311	C₂²⋊C₄	—	106	2	no	—
312	C₂²⋊C₄	—	106	2	no	—
313	C₂²⋊C₄	—	106	2	no	—
314	C₂²⋊C₄	—	106	2	no	—
315	C₂²⋊C₄	—	106	2	no	—
316	C₄²	—	2	2	no	—
317	C₄²	—	53	1	no	—
318	C₄²	—	53	1	no	—
319	C₄²	—	53	1	no	—
320	C₄²	C₂³×D₅₃	1	1	yes	characteristic
321	C₂³	—	53	1	no	—
322	C₂³	—	106	2	no	—
323	Q₈	—	2	2	no	—
324	Q₈	—	2	2	no	—
325	D₄	—	106	2	no	—
326	D₄	—	106	2	no	—
327	D₄	—	424	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	53	1	no	—
331	C₂×C₄	—	53	1	no	—
332	C₂×C₄	—	53	1	no	—
333	C₂×C₄	—	106	1	no	—
334	C₂×C₄	—	106	1	no	—
335	C₂×C₄	—	106	2	no	—
336	C₂×C₄	—	106	2	no	—
337	C₂×C₄	—	212	1	no	—
338	C₂×C₄	—	212	2	no	—
339	C₂×C₄	—	212	2	no	—
340	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
341	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
342	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
343	C₈	—	2	1	no	—
344	C₈	—	2	2	no	—
345	C₈	—	4	1	no	—
346	C₈	—	4	2	no	—
347	C₈	—	106	1	no	—
348	C₈	—	106	1	no	—
349	C₈	—	106	2	no	—
350	C₈	—	212	1	no	—
351	C₈	—	212	2	no	—
352	C₈	#13133223	1	2	yes	—
353	C₂²	—	106	1	no	—
354	C₂²	—	106	1	no	—
355	C₂²	—	106	1	no	—
356	C₂²	—	212	2	no	—
357	C₂²	—	212	2	no	—
358	C₂²	—	212	2	no	—
359	C₂²	C₄×D₄×D₅₃	1	1	yes	characteristic
360	C₄	—	2	1	no	—
361	C₄	—	4	2	no	—
362	C₄	—	4	2	no	—
363	C₄	—	106	1	no	—
364	C₄	—	106	1	no	—
365	C₄	—	106	1	no	—
366	C₄	—	212	2	no	—
367	C₄	C₂×#13133223	1	1	yes	characteristic
368	C₄	#16811388	1	1	yes	characteristic
369	C₄	#16811422	1	2	yes	—
370	C₂	—	424	2	no	—
371	C₂	—	212	1	no	—
372	C₂	#20283454	1	1	yes	characteristic
373	C₂	#20283514	1	1	yes	characteristic
374	C₂	#20283629	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1579	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#16810860	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1579	D₅₃
hypocenter	C₅₃	#18075
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1579

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #20283718	C₂	semidirect
3. #20273630	C₂	semidirect
4. #20273624	C₂	semidirect
5. #20273593	C₂	semidirect
6. #20273575	C₂	semidirect
7. #20273538	C₂	semidirect
8. #20273460	C₂	semidirect
9. #20273122	C₂	semidirect
10. #20273001	C₂	semidirect
11. C₅₃×#1579	C₂	semidirect
190. #1579	D₅₃	semidirect
232. C₅₃	#18075	semidirect

Maximal subgroups

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

subgroup	index
#20285288	2
#20283718	2
#20273630	2
#20273624	2
#20273593	2
#20273575	2
#20273538	2
#20273460	2
#20273122	2
#20273001	2
C₅₃×#1579	2
#18075	53