Subgroups — #28317006

Order 13568.

630 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291532	C₂	1	2	yes	maximal, complemented
2	#20290354	C₂	1	1	yes	maximal, characteristic, complemented
3	#20290350	C₂	1	2	yes	maximal, complemented
4	#20290347	C₂	1	1	yes	maximal, characteristic, complemented
5	#20290303	C₂	1	1	yes	maximal, characteristic, complemented
6	#20290300	C₂	1	2	yes	maximal, complemented
7	#20290296	C₂	1	1	yes	maximal, characteristic, complemented
8	#20290276	C₂	1	1	yes	maximal, characteristic, complemented
9	#20290069	C₂	1	2	yes	maximal, complemented
10	C₈×D₅₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	#20285287	C₂	1	1	yes	maximal, characteristic
12	#16811673	C₂²	1	2	yes	—
13	#16811672	C₂²	1	2	yes	—
14	#16811657	C₂²	1	2	yes	—
15	#16811653	—	2	2	no	—
16	#16811651	—	2	2	no	—
17	#16811651	C₂²	1	2	yes	—
18	#16811648	—	2	2	no	—
19	#16811645	C₂²	1	2	yes	—
20	#16811588	—	2	2	no	—
21	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
22	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
23	#16811486	C₂²	1	1	yes	characteristic
24	#16811486	C₂²	1	1	yes	characteristic
25	#16811484	C₂²	1	2	yes	—
26	#16811473	C₂²	1	2	yes	—
27	#16811473	C₂²	1	2	yes	—
28	#16811471	C₂²	1	2	yes	—
29	#16811471	C₂²	1	2	yes	—
30	C₈×#88324	C₂²	1	1	yes	characteristic
31	C₈×#88324	C₂²	1	1	yes	characteristic
32	C₈×#88324	C₂²	1	1	yes	characteristic
33	C₈×#88324	C₂²	1	1	yes	characteristic
34	#16811158	C₂²	1	2	yes	—
35	#16811007	C₄	1	1	yes	characteristic
36	#16810980	C₄	1	2	yes	—
37	#16810865	C₄	1	1	yes	characteristic
38	#16810860	C₄	1	1	yes	characteristic
39	#16810853	C₄	1	2	yes	—
40	#16810717	C₂²	1	1	yes	characteristic
41	#16810710	C₂²	1	2	yes	—
42	#16810705	C₂²	1	1	yes	characteristic
43	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
44	C₅₃×#499	C₄	1	1	yes	characteristic
45	C₂×#11719936	—	2	2	no	—
46	C₂×D₅₃⋊C₈	—	2	1	no	—
47	C₂×D₅₃⋊C₈	—	2	2	no	—
48	C₂×D₅₃⋊C₈	D₄	1	1	yes	characteristic
49	C₂×D₅₃⋊C₈	D₄	1	1	yes	characteristic
50	#13133284	—	2	2	no	—
51	C₄⋊#88324	—	2	2	no	—
52	C₄⋊#88324	C₂³	1	2	yes	—
53	C₄⋊#88324	C₂³	1	2	yes	—
54	#13133274	—	2	2	no	—
55	#13133274	—	2	2	no	—
56	#13133274	—	2	2	no	—
57	#13133272	—	2	2	no	—
58	#13133272	—	2	2	no	—
59	C₄×#88324	—	2	1	no	—
60	C₄×#88324	—	2	2	no	—
61	C₄×#88324	D₄	1	1	yes	characteristic
62	C₄×#88324	D₄	1	1	yes	characteristic
63	C₄×#88324	C₂³	1	1	yes	characteristic
64	C₄×#88324	C₂³	1	1	yes	characteristic
65	C₄×#88324	C₂³	1	1	yes	characteristic
66	C₄×#88324	C₂³	1	1	yes	characteristic
67	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
68	C₂×#11719913	C₂×C₄	1	2	yes	—
69	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic
70	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
71	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
72	#13133212	—	2	2	no	—
73	#13133205	C₂×C₄	1	2	yes	—
74	#13133202	—	2	2	no	—
75	#13133197	C₂×C₄	1	2	yes	—
76	#13133192	C₂×C₄	1	2	yes	—
77	#13133182	C₂³	1	2	yes	—
78	#13133179	—	2	2	no	—
79	#13133179	C₂³	1	2	yes	—
80	#13133175	—	2	2	no	—
81	C₄²×D₅₃	C₂³	1	1	yes	characteristic
82	#13133140	C₂×C₄	1	2	yes	—
83	#13133140	C₂×C₄	1	2	yes	—
84	#13133137	C₂×C₄	1	2	yes	—
85	#13133137	C₂×C₄	1	2	yes	—
86	C₄×#88322	C₂×C₄	1	1	yes	characteristic
87	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
88	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
89	#125×C₅₃	C₂×C₄	1	2	yes	—
90	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
91	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
92	#106×C₅₃	C₂×C₄	1	2	yes	—
93	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
94	C₂×D₂₁₂	—	2	2	no	—
95	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
96	C₂×C₄×D₅₃	—	2	2	no	—
97	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
100	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
101	C₂×#88324	—	2	1	no	—
102	C₂×#88324	—	2	1	no	—
103	C₂×#88324	—	2	2	no	—
104	C₂×#88324	—	2	2	no	—
105	C₂×#88324	C₂×D₄	1	1	yes	characteristic
106	C₂×#88324	C₂×D₄	1	1	yes	characteristic
107	C₂×#88324	C₂×D₄	1	1	yes	characteristic
108	C₂×#88324	C₂×D₄	1	1	yes	characteristic
109	#11719936	—	4	4	no	—
110	D₅₃⋊C₈	—	2	4	no	—
111	D₅₃⋊C₈	—	2	2	no	—
112	D₅₃⋊C₈	—	2	2	no	—
113	D₅₃⋊C₈	—	4	4	no	—
114	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₄⋊C₄×C₅₃	—	2	2	no	—
117	C₄⋊C₄×C₅₃	—	2	2	no	—
118	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
119	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
120	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
121	#11719921	—	2	2	no	—
122	#11719921	—	2	2	no	—
123	#11719921	—	2	2	no	—
124	#11719921	—	2	2	no	—
125	#11719921	—	2	2	no	—
126	C₄⋊Dic₅₃	—	2	2	no	—
127	#11719919	—	2	2	no	—
128	C₄×Dic₅₃	—	2	2	no	—
129	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
130	C₄×Dic₅₃	#41	1	1	yes	characteristic
131	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
132	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
133	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
134	Dic₂₁₂	—	2	2	no	—
135	D₄₂₄	—	2	2	no	—
136	#11719913	—	2	4	no	—
137	C₈×D₅₃	—	2	2	no	—
138	C₈×D₅₃	—	2	2	no	—
139	C₂²×D₅₃	—	2	2	no	—
140	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₅₃	—	2	2	no	—
142	C₂×C₄×C₅₃	—	2	2	no	—
143	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
146	C₂×Dic₅₃	—	2	2	no	—
147	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
148	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×Dic₅₃	C₂×#41	1	1	yes	characteristic
150	D₂₁₂	—	2	2	no	—
151	D₂₁₂	—	2	2	no	—
152	D₂₁₂	—	4	4	no	—
153	C₄×D₅₃	—	2	2	no	—
154	C₄×D₅₃	—	2	1	no	—
155	C₄×D₅₃	—	2	1	no	—
156	C₄×D₅₃	—	2	1	no	—
157	C₄×D₅₃	—	2	1	no	—
158	C₄×D₅₃	—	4	2	no	—
159	C₄×D₅₃	—	4	2	no	—
160	Dic₁₀₆	—	2	2	no	—
161	Dic₁₀₆	—	2	2	no	—
162	#88324	—	2	2	no	—
163	#88324	—	2	2	no	—
164	#88324	—	2	1	no	—
165	#88324	—	2	1	no	—
166	#88324	—	4	2	no	—
167	#88324	—	4	2	no	—
168	#88324	#135	1	2	yes	—
169	#88324	#135	1	2	yes	—
170	C₈×C₅₃	#131	1	2	yes	—
171	C₈×C₅₃	#131	1	2	yes	—
172	#88322	—	2	1	no	—
173	#88322	—	2	1	no	—
174	#17959	—	53	1	no	maximal, Sylow
175	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
176	C₂×D₅₃	—	2	1	no	—
177	C₂×D₅₃	—	2	1	no	—
178	C₂×D₅₃	—	4	2	no	—
179	C₂×D₅₃	—	4	2	no	—
180	C₂×D₅₃	—	4	2	no	—
181	C₂×D₅₃	#443	1	2	yes	—
182	C₄×C₅₃	—	2	1	no	—
183	C₄×C₅₃	—	2	1	no	—
184	C₄×C₅₃	—	4	2	no	—
185	C₄×C₅₃	—	4	2	no	—
186	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
187	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
188	Dic₅₃	—	2	1	no	—
189	Dic₅₃	—	2	1	no	—
190	Dic₅₃	—	4	2	no	—
191	Dic₅₃	C₂×#135	1	1	yes	characteristic
192	Dic₅₃	C₂×#135	1	1	yes	characteristic
193	#3168	—	53	1	no	—
194	#2966	—	53	2	no	—
195	#1591	—	53	1	no	—
196	#1590	—	53	2	no	—
197	#1589	—	53	1	no	—
198	C₈×Q₁₆	—	53	1	no	—
199	C₈×SD₁₆	—	53	2	no	—
200	C₈×D₈	—	53	1	no	—
201	#1568	—	53	1	no	—
202	#1494	—	53	2	no	—
203	C₂×C₈²	—	53	1	no	—
204	C₂×C₅₃	#2931	1	1	yes	characteristic
205	C₂×C₅₃	C₄×#135	1	1	yes	characteristic
206	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
207	D₅₃	—	2	2	no	—
208	D₅₃	—	8	2	no	—
209	#529	—	53	2	no	—
210	#499	#6157	1	1	yes	characteristic
211	#495	—	53	1	no	—
212	#488	—	53	2	no	—
213	#467	—	53	1	no	—
214	#466	—	53	1	no	—
215	#465	—	53	2	no	—
216	#446	—	53	2	no	—
217	C₈×Q₈	—	53	2	no	—
218	C₄×Q₁₆	—	53	1	no	—
219	C₄×SD₁₆	—	53	2	no	—
220	C₄×D₈	—	53	1	no	—
221	#436	—	53	2	no	—
222	#435	—	106	2	no	—
223	C₈×D₄	—	53	2	no	—
224	C₈×D₄	—	106	2	no	—
225	#433	—	106	2	no	—
226	#432	—	53	2	no	—
227	#405	—	106	2	no	—
228	C₂×C₄×C₈	—	53	1	no	—
229	C₂×C₄×C₈	—	53	1	no	—
230	C₂×C₄×C₈	—	53	1	no	—
231	#335	—	53	1	no	—
232	#335	—	53	1	no	—
233	#334	—	53	2	no	—
234	Q₈⋊C₈	—	53	2	no	—
235	Q₈⋊C₈	—	53	2	no	—
236	D₄⋊C₈	—	53	2	no	—
237	D₄⋊C₈	—	53	2	no	—
238	C₈²	—	53	1	no	—
239	C₈²	—	53	1	no	—
240	C₈²	—	53	1	no	—
241	C₈²	—	53	1	no	—
242	C₅₃	#17959	1	1	yes	characteristic, complemented, Sylow
243	C₂×Q₁₆	—	53	1	no	—
244	C₂×SD₁₆	—	53	2	no	—
245	C₂×D₈	—	53	1	no	—
246	C₂×M₁₆	—	106	2	no	—
247	C₂²×C₈	—	53	1	no	—
248	C₂²×C₈	—	53	1	no	—
249	C₂²×C₈	—	53	1	no	—
250	C₂²×C₈	—	53	1	no	—
251	C₂²×C₈	—	106	2	no	—
252	C₂²×C₈	—	106	1	no	—
253	#126	—	106	2	no	—
254	#125	C₂×#6157	1	2	yes	—
255	#124	—	53	2	no	—
256	#123	—	106	2	no	—
257	C₂²⋊Q₈	—	53	2	no	—
258	#121	—	53	2	no	—
259	C₄×Q₈	—	53	2	no	—
260	C₄×D₄	—	53	2	no	—
261	C₄×D₄	—	106	2	no	—
262	#117	—	106	2	no	—
263	C₂×C₄²	—	53	1	no	—
264	#107	C₂×#6157	1	1	yes	characteristic
265	#107	C₂×#6157	1	1	yes	characteristic
266	#106	C₂×#6157	1	2	yes	—
267	C₄⋊C₈	—	53	2	no	—
268	C₄⋊C₈	—	53	2	no	—
269	C₄⋊C₈	—	106	2	no	—
270	C₄⋊C₈	—	106	2	no	—
271	#103	—	53	2	no	—
272	#103	—	53	2	no	—
273	#102	—	53	2	no	—
274	#102	—	53	2	no	—
275	C₂²⋊C₈	—	106	2	no	—
276	C₂²⋊C₈	—	106	2	no	—
277	C₂²⋊C₈	—	106	2	no	—
278	#97	—	106	2	no	—
279	#97	—	106	2	no	—
280	C₄×C₈	—	53	1	no	—
281	C₄×C₈	—	53	1	no	—
282	C₄×C₈	—	53	1	no	—
283	C₄×C₈	—	53	1	no	—
284	C₄×C₈	—	53	1	no	—
285	C₄×C₈	—	53	1	no	—
286	C₄×C₈	—	53	1	no	—
287	C₄×C₈	—	53	1	no	—
288	C₄×C₈	—	53	1	no	—
289	C₄×C₈	—	106	1	no	—
290	C₄×C₈	—	106	2	no	—
291	C₄×C₈	C₂×#6157	1	1	yes	characteristic
292	C₂×Q₈	—	53	2	no	—
293	C₂×D₄	—	53	2	no	—
294	C₂×D₄	—	106	2	no	—
295	C₂²×C₄	—	53	1	no	—
296	C₂²×C₄	—	53	1	no	—
297	C₂²×C₄	—	53	1	no	—
298	C₂²×C₄	—	106	2	no	—
299	Q₁₆	—	106	2	no	—
300	SD₁₆	—	106	4	no	—
301	D₈	—	106	2	no	—
302	M₁₆	—	212	4	no	—
303	C₂×C₈	—	53	1	no	—
304	C₂×C₈	—	53	1	no	—
305	C₂×C₈	—	53	1	no	—
306	C₂×C₈	—	53	1	no	—
307	C₂×C₈	—	53	1	no	—
308	C₂×C₈	—	53	1	no	—
309	C₂×C₈	—	106	2	no	—
310	C₂×C₈	—	106	2	no	—
311	C₂×C₈	—	106	1	no	—
312	C₂×C₈	—	106	1	no	—
313	C₂×C₈	—	106	4	no	—
314	C₂×C₈	—	106	2	no	—
315	C₂×C₈	—	106	2	no	—
316	C₂×C₈	—	106	2	no	—
317	C₂×C₈	—	106	2	no	—
318	C₂×C₈	—	212	4	no	—
319	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
320	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	106	2	no	—
324	C₄⋊C₄	—	106	2	no	—
325	C₄⋊C₄	C₂²×#6157	1	2	yes	—
326	C₄⋊C₄	C₂²×#6157	1	2	yes	—
327	C₂²⋊C₄	—	106	2	no	—
328	C₂²⋊C₄	—	106	2	no	—
329	C₂²⋊C₄	—	106	2	no	—
330	C₂²⋊C₄	—	106	2	no	—
331	C₂²⋊C₄	—	106	2	no	—
332	C₄²	—	53	1	no	—
333	C₄²	—	53	1	no	—
334	C₄²	—	53	1	no	—
335	C₄²	—	106	2	no	—
336	C₄²	C₂²×#6157	1	1	yes	characteristic
337	C₂³	—	53	1	no	—
338	C₂³	—	106	2	no	—
339	Q₈	—	106	2	no	—
340	Q₈	—	106	2	no	—
341	D₄	—	106	2	no	—
342	D₄	—	106	2	no	—
343	D₄	—	212	4	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	53	1	no	—
347	C₂×C₄	—	53	1	no	—
348	C₂×C₄	—	53	1	no	—
349	C₂×C₄	—	106	1	no	—
350	C₂×C₄	—	106	1	no	—
351	C₂×C₄	—	106	2	no	—
352	C₂×C₄	—	106	1	no	—
353	C₂×C₄	—	106	1	no	—
354	C₂×C₄	—	106	2	no	—
355	C₂×C₄	—	212	2	no	—
356	C₂×C₄	—	212	2	no	—
357	C₂×C₄	D₄×#6157	1	1	yes	characteristic
358	C₂×C₄	D₄×#6157	1	1	yes	characteristic
359	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
360	C₈	—	53	2	no	—
361	C₈	—	53	2	no	—
362	C₈	—	106	1	no	—
363	C₈	—	106	1	no	—
364	C₈	—	106	2	no	—
365	C₈	—	106	2	no	—
366	C₈	—	106	1	no	—
367	C₈	—	106	1	no	—
368	C₈	—	212	2	no	—
369	C₈	—	212	2	no	—
370	C₈	#13133305	1	2	yes	—
371	C₈	#13133305	1	2	yes	—
372	C₂²	—	53	2	no	—
373	C₂²	—	106	1	no	—
374	C₂²	—	106	1	no	—
375	C₂²	—	212	2	no	—
376	C₂²	—	212	2	no	—
377	C₂²	—	212	2	no	—
378	C₂²	C₂×D₄×#6157	1	1	yes	characteristic
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	4	2	no	—
382	C₄	—	4	2	no	—
383	C₄	—	53	1	no	—
384	C₄	—	53	1	no	—
385	C₄	—	106	1	no	—
386	C₄	—	106	1	no	—
387	C₄	—	212	2	no	—
388	C₄	C₂×#13133305	1	1	yes	characteristic
389	C₄	C₂×#13133305	1	1	yes	characteristic
390	C₂	—	106	2	no	—
391	C₂	—	424	2	no	—
392	C₂	#20291440	1	1	yes	characteristic
393	C₂	#135×#6157	1	1	yes	characteristic
394	C₂	D₅₃⋊#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#6157
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₅₃×#499	C₄
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#16810860	C₄
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	#499	#6157
hypocenter	C₅₃	#17959
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

Complements

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

N	H ≅ G/N	type
1. #20291532	C₂	semidirect
2. #20290354	C₂	semidirect
3. #20290350	C₂	semidirect
4. #20290347	C₂	semidirect
5. #20290303	C₂	semidirect
6. #20290300	C₂	semidirect
7. #20290296	C₂	semidirect
8. #20290276	C₂	semidirect
9. #20290069	C₂	semidirect
10. C₈×D₅₃⋊C₈	C₂	semidirect
242. C₅₃	#17959	semidirect

Maximal subgroups

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

subgroup	index
#20291532	2
#20290354	2
#20290350	2
#20290347	2
#20290303	2
#20290300	2
#20290296	2
#20290276	2
#20290069	2
C₈×D₅₃⋊C₈	2
#20285287	2
#17959	53