Subgroups — #27355903

Order 13568.

578 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285298	C₂	1	1	yes	maximal, characteristic
2	#20283657	C₂	1	2	yes	maximal
3	#20273625	C₂	1	1	yes	maximal, characteristic, complemented
4	#20273624	C₂	1	1	yes	maximal, characteristic, complemented
5	#20273589	C₂	1	2	yes	maximal, complemented
6	#20273586	C₂	1	2	yes	maximal, complemented
7	#20273533	C₂	1	1	yes	maximal, characteristic, complemented
8	#334×D₅₃	C₂	1	1	yes	maximal, characteristic
9	#20273116	C₂	1	2	yes	maximal
10	#20273001	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#1579	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811177	C₂²	1	2	yes	—
13	#16811027	C₄	1	1	yes	characteristic
14	#16811001	C₄	1	1	yes	characteristic
15	#16810972	C₄	1	2	yes	—
16	#16810865	C₄	1	1	yes	characteristic
17	#16810863	C₄	1	2	yes	—
18	#16810861	C₄	1	1	yes	characteristic
19	#16810754	C₂²	1	2	yes	—
20	#16810753	—	2	2	no	—
21	#16810753	C₂²	1	2	yes	—
22	#16810752	—	2	2	no	—
23	#16810747	C₂²	1	2	yes	—
24	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
25	C₄×#11719925	C₂²	1	2	yes	—
26	#16810674	C₂²	1	1	yes	characteristic
27	#16810648	—	2	2	no	—
28	C₄⋊C₈×D₅₃	C₂²	1	2	yes	—
29	#16810577	—	2	2	no	—
30	C₄×#11719912	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
32	#16810374	C₂²	1	1	yes	characteristic
33	#16810373	C₂²	1	2	yes	—
34	#16810355	C₂²	1	2	yes	—
35	Q₈⋊#88322	C₂²	1	2	yes	—
36	#16810353	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×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
46	C₂×#11719925	C₂×C₄	1	2	yes	—
47	C₂×#11719912	D₄	1	2	yes	—
48	C₂×C₈×D₅₃	—	2	2	no	—
49	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
51	#13133216	C₂×C₄	1	2	yes	—
52	#13133215	C₂×C₄	1	2	yes	—
53	#13133207	—	2	2	no	—
54	#13133197	C₂×C₄	1	2	yes	—
55	#13133195	—	2	2	no	—
56	#13133193	C₂×C₄	1	2	yes	—
57	C₄×Dic₁₀₆	—	2	2	no	—
58	C₄×Dic₁₀₆	C₂³	1	2	yes	—
59	#13133174	—	2	2	no	—
60	C₄²×D₅₃	C₂³	1	1	yes	characteristic
61	#13133151	C₂×C₄	1	1	yes	characteristic
62	#13133150	C₂×C₄	1	2	yes	—
63	C₄⋊#88322	—	2	2	no	—
64	C₄⋊#88322	C₂³	1	2	yes	—
65	#13133146	—	2	2	no	—
66	#13133146	—	2	2	no	—
67	#13133146	—	2	1	no	—
68	#13133146	—	2	2	no	—
69	#13133142	C₂×C₄	1	2	yes	—
70	#13133141	C₂×C₄	1	2	yes	—
71	#13133140	C₂×C₄	1	2	yes	—
72	#13133129	—	2	1	no	—
73	#13133128	—	2	2	no	—
74	#13133127	—	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×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
96	C₂×Dic₁₀₆	—	2	2	no	—
97	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
98	Q₁₆×C₅₃	—	2	2	no	—
99	C₂×C₈×C₅₃	—	2	2	no	—
100	C₂×C₈×C₅₃	—	2	1	no	—
101	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
104	C₄⋊C₄×C₅₃	—	2	2	no	—
105	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
106	C₄²×C₅₃	—	2	2	no	—
107	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
108	#11719925	—	4	2	no	—
109	#11719921	—	2	2	no	—
110	#11719921	—	2	2	no	—
111	C₄⋊Dic₅₃	—	2	2	no	—
112	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
113	#11719919	—	2	2	no	—
114	#11719919	—	2	2	no	—
115	#11719919	—	2	2	no	—
116	#11719919	—	2	2	no	—
117	#11719919	—	2	2	no	—
118	C₄×Dic₅₃	—	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	Dic₂₁₂	—	2	2	no	—
128	#11719912	—	2	4	no	—
129	C₈×D₅₃	—	2	2	no	—
130	C₈×D₅₃	—	4	1	no	—
131	C₈×D₅₃	—	4	4	no	—
132	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
133	Q₈×C₅₃	—	2	2	no	—
134	Q₈×C₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	—	2	2	no	—
136	C₂×C₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
140	C₂×Dic₅₃	—	2	2	no	—
141	C₂×Dic₅₃	—	2	2	no	—
142	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
143	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
144	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
145	C₄×D₅₃	—	2	1	no	—
146	C₄×D₅₃	—	2	1	no	—
147	C₄×D₅₃	—	4	1	no	—
148	C₄×D₅₃	#137	1	4	yes	—
149	Dic₁₀₆	—	2	2	no	—
150	Dic₁₀₆	—	2	2	no	—
151	Dic₁₀₆	—	8	2	no	—
152	C₈×C₅₃	—	2	1	no	—
153	C₈×C₅₃	—	2	2	no	—
154	C₈×C₅₃	—	4	1	no	—
155	C₈×C₅₃	—	4	2	no	—
156	C₈×C₅₃	#131	1	2	yes	—
157	#88322	—	2	1	no	—
158	#88322	—	2	1	no	—
159	#88322	—	2	2	no	—
160	#88322	—	4	1	no	—
161	#88322	—	4	2	no	—
162	#18072	—	53	1	no	maximal, Sylow
163	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
164	C₂×D₅₃	—	2	1	no	—
165	C₂×D₅₃	—	2	1	no	—
166	C₂×D₅₃	#441	1	2	yes	—
167	C₄×C₅₃	—	2	1	no	—
168	C₄×C₅₃	—	4	2	no	—
169	C₄×C₅₃	—	4	2	no	—
170	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
171	C₄×C₅₃	#568	1	1	yes	characteristic
172	C₄×C₅₃	C₂×#137	1	2	yes	—
173	Dic₅₃	—	2	1	no	—
174	Dic₅₃	—	2	1	no	—
175	Dic₅₃	—	4	2	no	—
176	Dic₅₃	—	4	2	no	—
177	Dic₅₃	#441	1	2	yes	—
178	#3170	—	53	1	no	—
179	#2962	—	53	2	no	—
180	#1593	—	53	1	no	—
181	#1593	—	53	1	no	—
182	#1587	—	53	2	no	—
183	#1587	—	53	2	no	—
184	#1579	—	53	1	no	—
185	#1579	D₅₃	1	1	yes	characteristic, complemented
186	C₂×#334	—	53	1	no	—
187	#1481	—	53	2	no	—
188	#1451	—	53	1	no	—
189	C₂×C₅₃	#2935	1	1	yes	characteristic
190	C₂×C₅₃	C₂×#441	1	1	yes	characteristic
191	C₂×C₅₃	#2957	1	1	yes	characteristic
192	D₅₃	—	2	2	no	—
193	#533	—	53	2	no	—
194	#500	—	53	1	no	—
195	#494	—	53	1	no	—
196	#487	—	53	2	no	—
197	#467	—	53	1	no	—
198	#467	—	53	1	no	—
199	#467	—	53	2	no	—
200	#446	—	53	2	no	—
201	#446	—	53	2	no	—
202	#446	—	106	2	no	—
203	#446	—	106	2	no	—
204	C₈×Q₈	—	53	2	no	—
205	C₈×Q₈	C₂×D₅₃	1	2	yes	—
206	C₄×Q₁₆	—	53	2	no	—
207	C₄×Q₁₆	—	53	1	no	—
208	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
209	#431	—	53	1	no	—
210	#424	—	106	2	no	—
211	C₂×C₄⋊C₈	—	53	2	no	—
212	#405	—	106	2	no	—
213	C₄×M₁₆	—	53	1	no	—
214	C₂×C₄×C₈	—	53	1	no	—
215	#334	—	53	2	no	—
216	#334	—	53	1	no	—
217	#334	C₂×D₅₃	1	1	yes	characteristic
218	Q₈⋊C₈	—	53	2	no	—
219	Q₈⋊C₈	—	53	2	no	—
220	Q₈⋊C₈	—	53	2	no	—
221	Q₈⋊C₈	C₂×D₅₃	1	2	yes	—
222	#322	—	53	1	no	—
223	#322	—	53	1	no	—
224	#322	—	53	1	no	—
225	#322	C₂×D₅₃	1	1	yes	characteristic
226	C₅₃	#18072	1	1	yes	characteristic, complemented, Sylow
227	C₂×Q₁₆	—	53	2	no	—
228	C₂×Q₁₆	—	53	1	no	—
229	C₂×Q₁₆	C₄×D₅₃	1	1	yes	characteristic
230	C₂×M₁₆	—	53	2	no	—
231	C₂²×C₈	—	53	1	no	—
232	C₂²×C₈	—	53	1	no	—
233	C₂²×C₈	—	106	2	no	—
234	C₄⋊Q₈	—	53	2	no	—
235	C₄⋊Q₈	—	53	2	no	—
236	#125	—	106	2	no	—
237	C₂²⋊Q₈	—	53	2	no	—
238	C₂²⋊Q₈	—	53	2	no	—
239	C₂²⋊Q₈	—	106	2	no	—
240	C₄×Q₈	—	53	2	no	—
241	C₄×Q₈	—	106	2	no	—
242	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
243	#117	—	106	2	no	—
244	C₂×C₄²	—	53	1	no	—
245	#107	—	53	1	no	—
246	#107	—	53	2	no	—
247	#107	C₄×D₅₃	1	1	yes	characteristic
248	C₄⋊C₈	—	2	2	no	—
249	C₄⋊C₈	—	53	2	no	—
250	C₄⋊C₈	—	106	2	no	—
251	C₄⋊C₈	—	106	2	no	—
252	C₄⋊C₈	—	106	1	no	—
253	C₄⋊C₈	—	106	2	no	—
254	C₄⋊C₈	—	106	2	no	—
255	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
256	#103	—	53	2	no	—
257	#103	—	53	2	no	—
258	#103	—	53	2	no	—
259	#103	C₄×D₅₃	1	2	yes	—
260	C₂²⋊C₈	—	106	1	no	—
261	#97	—	53	2	no	—
262	#97	—	106	2	no	—
263	#97	—	106	2	no	—
264	C₄×C₈	—	2	2	no	—
265	C₄×C₈	—	53	1	no	—
266	C₄×C₈	—	53	1	no	—
267	C₄×C₈	—	53	1	no	—
268	C₄×C₈	—	53	1	no	—
269	C₄×C₈	—	53	1	no	—
270	C₄×C₈	—	106	2	no	—
271	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
272	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
273	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
274	C₂×Q₈	—	53	2	no	—
275	C₂×Q₈	—	106	2	no	—
276	C₂×Q₈	C₂×C₄×D₅₃	1	2	yes	—
277	C₂²×C₄	—	53	1	no	—
278	C₂²×C₄	—	53	1	no	—
279	C₂²×C₄	—	53	1	no	—
280	Q₁₆	—	2	2	no	—
281	Q₁₆	—	106	2	no	—
282	Q₁₆	—	212	2	no	—
283	M₁₆	—	106	4	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	53	1	no	—
287	C₂×C₈	—	53	1	no	—
288	C₂×C₈	—	53	2	no	—
289	C₂×C₈	—	106	2	no	—
290	C₂×C₈	—	106	2	no	—
291	C₂×C₈	—	106	1	no	—
292	C₂×C₈	—	212	1	no	—
293	C₂×C₈	—	212	4	no	—
294	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
295	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
296	C₂×C₈	D₄×D₅₃	1	2	yes	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	53	2	no	—
299	C₄⋊C₄	—	106	2	no	—
300	C₄⋊C₄	—	106	2	no	—
301	C₄⋊C₄	—	106	2	no	—
302	C₄⋊C₄	—	106	2	no	—
303	C₄⋊C₄	—	106	2	no	—
304	C₄⋊C₄	—	106	2	no	—
305	C₄⋊C₄	C₂×C₄×D₅₃	1	2	yes	—
306	C₂²⋊C₄	—	106	2	no	—
307	C₂²⋊C₄	—	106	2	no	—
308	C₄²	—	2	2	no	—
309	C₄²	—	53	1	no	—
310	C₄²	—	53	1	no	—
311	C₄²	—	53	1	no	—
312	C₄²	—	106	2	no	—
313	C₄²	C₂³×D₅₃	1	1	yes	characteristic
314	C₂³	—	53	1	no	—
315	Q₈	—	2	2	no	—
316	Q₈	—	2	2	no	—
317	Q₈	—	106	2	no	—
318	Q₈	—	106	2	no	—
319	Q₈	—	424	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	53	1	no	—
323	C₂×C₄	—	53	4	no	—
324	C₂×C₄	—	53	1	no	—
325	C₂×C₄	—	53	1	no	—
326	C₂×C₄	—	106	1	no	—
327	C₂×C₄	—	106	1	no	—
328	C₂×C₄	—	106	2	no	—
329	C₂×C₄	—	106	2	no	—
330	C₂×C₄	—	212	1	no	—
331	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
332	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
333	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	2	no	—
336	C₈	—	4	1	no	—
337	C₈	—	4	2	no	—
338	C₈	—	106	1	no	—
339	C₈	—	106	1	no	—
340	C₈	—	106	2	no	—
341	C₈	—	212	1	no	—
342	C₈	—	212	2	no	—
343	C₈	#13133223	1	2	yes	—
344	C₂²	—	53	2	no	—
345	C₂²	—	106	1	no	—
346	C₂²	—	106	1	no	—
347	C₂²	C₄×D₄×D₅₃	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	4	2	no	—
350	C₄	—	4	2	no	—
351	C₄	—	53	2	no	—
352	C₄	—	106	1	no	—
353	C₄	—	106	1	no	—
354	C₄	—	212	2	no	—
355	C₄	—	212	2	no	—
356	C₄	C₂×#13133223	1	1	yes	characteristic
357	C₄	#16811388	1	1	yes	characteristic
358	C₄	#137×D₅₃	1	2	yes	—
359	C₂	—	106	2	no	—
360	C₂	#20283454	1	1	yes	characteristic
361	C₂	#441×D₅₃	1	1	yes	characteristic
362	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	C₂²×D₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1579	D₅₃
hypocenter	C₅₃	#18072
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
3. #20273625	C₂	semidirect
4. #20273624	C₂	semidirect
5. #20273589	C₂	semidirect
6. #20273586	C₂	semidirect
7. #20273533	C₂	semidirect
11. C₅₃×#1579	C₂	semidirect
185. #1579	D₅₃	semidirect
226. C₅₃	#18072	semidirect

Maximal subgroups

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

subgroup	index
#20285298	2
#20283657	2
#20273625	2
#20273624	2
#20273589	2
#20273586	2
#20273533	2
#334×D₅₃	2
#20273116	2
#20273001	2
C₅₃×#1579	2
#18072	53