Subgroups — #27355842

Order 13568.

652 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283839	C₂	1	2	yes	maximal, complemented
2	C₄×#13133240	C₂	1	1	yes	maximal, characteristic
3	#20273624	C₂	1	1	yes	maximal, characteristic, complemented
4	#20273554	C₂	1	2	yes	maximal, complemented
5	#20273549	C₂	1	2	yes	maximal, complemented
6	#20273540	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273538	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273469	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273153	C₂	1	2	yes	maximal, complemented
10	#20273001	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1593	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133240	C₄	1	1	yes	characteristic
13	C₄×#11719949	C₂²	1	2	yes	—
14	#16810754	C₂²	1	2	yes	—
15	Q₁₆×Dic₅₃	C₄	1	1	yes	characteristic
16	C₄×#11719924	C₂²	1	2	yes	—
17	#16810708	C₄	1	2	yes	—
18	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
19	#16810705	C₄	1	1	yes	characteristic
20	#16810696	—	2	2	no	—
21	#16810693	—	2	2	no	—
22	C₈×D₂₁₂	C₂²	1	2	yes	—
23	#16810688	C₂²	1	2	yes	—
24	#16810678	—	2	2	no	—
25	#16810674	C₂²	1	1	yes	characteristic
26	#16810660	C₄	1	1	yes	characteristic
27	#16810645	C₂²	1	2	yes	—
28	C₄⋊C₈×D₅₃	—	2	2	no	—
29	#16810618	C₄	1	2	yes	—
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	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	C₅₃×#446	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×#11719949	C₂×C₄	1	2	yes	—
46	#13133240	—	2	4	no	—
47	C₂×#11719924	C₂×C₄	1	2	yes	—
48	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic
49	C₂×#11719912	—	2	2	no	—
50	C₂×#11719912	—	2	1	no	—
51	C₂×C₈×D₅₃	—	2	2	no	—
52	C₂×C₈×D₅₃	—	2	2	no	—
53	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
55	Q₈×Dic₅₃	C₂×C₄	1	2	yes	—
56	C₄×D₂₁₂	—	2	2	no	—
57	C₄×D₂₁₂	C₂³	1	2	yes	—
58	#13133179	—	2	2	no	—
59	#13133179	C₂×C₄	1	2	yes	—
60	#13133175	—	2	2	no	—
61	#13133175	C₂×C₄	1	2	yes	—
62	C₄²×D₅₃	—	2	2	no	—
63	C₄²×D₅₃	C₂³	1	1	yes	characteristic
64	#13133151	C₂×C₄	1	1	yes	characteristic
65	#13133148	C₂×C₄	1	2	yes	—
66	C₄⋊#88322	—	2	2	no	—
67	C₄⋊#88322	C₂³	1	2	yes	—
68	#13133146	—	2	2	no	—
69	#13133146	D₄	1	2	yes	—
70	#13133141	C₂×C₄	1	2	yes	—
71	#13133139	C₂×C₄	1	2	yes	—
72	#13133137	C₂×C₄	1	2	yes	—
73	#13133129	—	2	2	no	—
74	#13133129	—	2	2	no	—
75	#13133129	D₄	1	2	yes	—
76	#13133128	—	2	2	no	—
77	#13133127	—	2	1	no	—
78	C₄×#88322	C₂³	1	1	yes	characteristic
79	C₄×#88322	C₂³	1	1	yes	characteristic
80	C₄×#88322	C₂³	1	1	yes	characteristic
81	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
82	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
83	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
84	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
85	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
86	C₄⋊C₈×C₅₃	—	2	2	no	—
87	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
88	#103×C₅₃	C₂×C₄	1	2	yes	—
89	#97×C₅₃	—	2	2	no	—
90	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
93	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
94	#11719949	—	2	4	no	—
95	#11719949	—	4	2	no	—
96	C₂×D₂₁₂	—	2	2	no	—
97	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
98	C₂×C₄×D₅₃	—	2	2	no	—
99	C₂×C₄×D₅₃	—	2	2	no	—
100	C₂×C₄×D₅₃	—	2	2	no	—
101	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
104	Q₁₆×C₅₃	—	2	2	no	—
105	C₂×C₈×C₅₃	—	2	2	no	—
106	C₂×C₈×C₅₃	—	2	1	no	—
107	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
110	C₄⋊C₄×C₅₃	—	2	2	no	—
111	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
112	C₄²×C₅₃	—	2	2	no	—
113	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
114	#11719924	—	2	4	no	—
115	#11719921	—	2	2	no	—
116	#11719921	—	2	2	no	—
117	#11719921	—	2	2	no	—
118	C₄⋊Dic₅₃	—	2	2	no	—
119	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
120	C₄×Dic₅₃	—	2	2	no	—
121	C₄×Dic₅₃	—	2	2	no	—
122	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
123	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
124	C₄×Dic₅₃	#41	1	1	yes	characteristic
125	C₂×#88322	—	2	2	no	—
126	C₂×#88322	—	2	1	no	—
127	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
128	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
129	C₂×#88322	C₂×D₄	1	2	yes	—
130	D₄₂₄	M₁₆	1	4	yes	complemented
131	#11719912	—	4	2	no	—
132	#11719912	—	8	2	no	—
133	C₈×D₅₃	—	2	2	no	—
134	C₈×D₅₃	—	4	1	no	—
135	C₈×D₅₃	—	8	2	no	—
136	C₈×D₅₃	—	8	2	no	—
137	C₂²×D₅₃	—	2	2	no	—
138	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
139	Q₈×C₅₃	—	2	2	no	—
140	Q₈×C₅₃	—	2	2	no	—
141	C₂×C₄×C₅₃	—	2	2	no	—
142	C₂×C₄×C₅₃	—	2	2	no	—
143	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
146	C₂×Dic₅₃	—	2	2	no	—
147	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
148	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
149	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
150	D₂₁₂	—	2	2	no	—
151	D₂₁₂	—	4	4	no	—
152	D₂₁₂	C₂×M₁₆	1	4	yes	—
153	C₄×D₅₃	—	2	1	no	—
154	C₄×D₅₃	—	2	1	no	—
155	C₄×D₅₃	—	2	2	no	—
156	C₄×D₅₃	—	2	2	no	—
157	C₄×D₅₃	—	4	4	no	—
158	C₄×D₅₃	—	4	2	no	—
159	C₄×D₅₃	—	4	2	no	—
160	C₄×D₅₃	—	4	4	no	—
161	C₈×C₅₃	—	2	1	no	—
162	C₈×C₅₃	—	2	2	no	—
163	C₈×C₅₃	—	4	1	no	—
164	C₈×C₅₃	—	4	2	no	—
165	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
166	#88322	—	2	1	no	—
167	#88322	—	2	1	no	—
168	#88322	—	2	2	no	—
169	#88322	—	4	1	no	—
170	#88322	—	4	2	no	—
171	#18065	—	53	1	no	maximal, Sylow
172	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
173	C₂×D₅₃	—	2	1	no	—
174	C₂×D₅₃	—	2	4	no	—
175	C₂×D₅₃	—	2	1	no	—
176	C₂×D₅₃	—	2	1	no	—
177	C₂×D₅₃	—	4	2	no	—
178	C₂×D₅₃	—	4	2	no	—
179	C₄×C₅₃	—	2	1	no	—
180	C₄×C₅₃	—	4	2	no	—
181	C₄×C₅₃	—	4	2	no	—
182	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
183	C₄×C₅₃	#568	1	1	yes	characteristic
184	C₄×C₅₃	#577	1	2	yes	—
185	Dic₅₃	—	2	1	no	—
186	Dic₅₃	—	2	1	no	—
187	Dic₅₃	—	4	2	no	—
188	Dic₅₃	#444	1	2	yes	—
189	#2990	—	53	2	no	—
190	C₄×#135	—	53	1	no	—
191	#1593	—	53	1	no	—
192	#1593	D₅₃	1	1	yes	characteristic, complemented
193	#1582	—	53	2	no	—
194	#1582	—	53	2	no	—
195	#1580	—	53	1	no	—
196	#1580	—	53	1	no	—
197	#1566	—	53	1	no	—
198	#1489	—	53	2	no	—
199	#1451	—	53	1	no	—
200	C₂×C₅₃	D₄×M₁₆	1	1	yes	characteristic
201	C₂×C₅₃	#2952	1	1	yes	characteristic
202	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
203	D₅₃	—	4	1	no	—
204	D₅₃	—	4	4	no	—
205	C₂×#135	—	53	1	no	—
206	C₄×#41	—	53	2	no	—
207	#446	—	53	2	no	—
208	#446	C₂×D₅₃	1	2	yes	—
209	C₄×Q₁₆	—	53	1	no	—
210	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
211	C₄×SD₁₆	—	53	2	no	—
212	C₄×SD₁₆	—	53	2	no	—
213	C₄×D₈	—	53	1	no	—
214	C₄×D₈	—	53	1	no	—
215	#435	—	106	2	no	—
216	#435	—	106	2	no	—
217	C₈×D₄	—	53	2	no	—
218	C₈×D₄	—	53	2	no	—
219	#432	—	106	2	no	—
220	#431	—	53	1	no	—
221	#427	—	53	1	no	—
222	#423	—	53	2	no	—
223	C₂×C₄⋊C₈	—	106	2	no	—
224	#416	—	53	2	no	—
225	C₄×M₁₆	—	53	1	no	—
226	C₂×C₄×C₈	—	53	1	no	—
227	#334	—	53	2	no	—
228	#334	—	53	1	no	—
229	#334	C₂×D₅₃	1	1	yes	characteristic
230	Q₈⋊C₈	—	53	2	no	—
231	Q₈⋊C₈	C₂×D₅₃	1	2	yes	—
232	D₄⋊C₈	—	53	2	no	—
233	D₄⋊C₈	—	53	2	no	—
234	#322	—	53	1	no	—
235	#322	—	53	1	no	—
236	#322	—	53	1	no	—
237	#322	C₂×D₅₃	1	1	yes	characteristic
238	C₅₃	#18065	1	1	yes	characteristic, complemented, Sylow
239	C₂×#41	—	53	2	no	—
240	#135	—	106	4	no	—
241	C₂×Q₁₆	C₄×D₅₃	1	1	yes	characteristic
242	C₂×SD₁₆	—	53	2	no	—
243	C₂×D₈	—	53	1	no	—
244	C₂×M₁₆	—	106	2	no	—
245	C₂×M₁₆	—	106	1	no	—
246	C₂²×C₈	—	53	1	no	—
247	C₂²×C₈	—	53	1	no	—
248	C₂²×C₈	—	106	2	no	—
249	C₂²×C₈	—	106	2	no	—
250	C₄×Q₈	—	53	2	no	—
251	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
252	C₄×D₄	—	53	2	no	—
253	C₄×D₄	—	53	2	no	—
254	C₄×D₄	—	106	2	no	—
255	C₄×D₄	—	106	2	no	—
256	#117	—	53	2	no	—
257	#117	—	106	2	no	—
258	C₂×C₄²	—	53	1	no	—
259	C₂×C₄²	—	106	2	no	—
260	#107	—	53	1	no	—
261	#107	C₄×D₅₃	1	1	yes	characteristic
262	#106	—	53	2	no	—
263	C₄⋊C₈	—	2	2	no	—
264	C₄⋊C₈	—	53	2	no	—
265	C₄⋊C₈	—	53	2	no	—
266	C₄⋊C₈	—	106	2	no	—
267	C₄⋊C₈	—	106	2	no	—
268	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
269	#103	—	53	2	no	—
270	#103	C₄×D₅₃	1	2	yes	—
271	#102	—	53	2	no	—
272	#102	—	53	2	no	—
273	C₂²⋊C₈	—	53	2	no	—
274	C₂²⋊C₈	—	106	2	no	—
275	C₂²⋊C₈	—	106	2	no	—
276	#97	—	2	2	no	—
277	#97	—	106	2	no	—
278	#97	—	106	1	no	—
279	C₄×C₈	—	53	1	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	C₂²×D₅₃	1	1	yes	characteristic
285	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
286	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
287	#41	—	106	4	no	—
288	#41	—	212	2	no	—
289	C₂×Q₈	C₂×C₄×D₅₃	1	2	yes	—
290	C₂×D₄	—	53	2	no	—
291	C₂×D₄	—	106	2	no	—
292	C₂²×C₄	—	53	1	no	—
293	C₂²×C₄	—	53	1	no	—
294	C₂²×C₄	—	53	1	no	—
295	C₂²×C₄	—	106	2	no	—
296	C₂²×C₄	—	106	2	no	—
297	C₂²×C₄	—	106	2	no	—
298	Q₁₆	—	2	2	no	—
299	SD₁₆	—	106	4	no	—
300	D₈	—	53	4	no	—
301	M₁₆	—	424	2	no	—
302	M₁₆	—	212	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	53	1	no	—
306	C₂×C₈	—	53	1	no	—
307	C₂×C₈	—	53	2	no	—
308	C₂×C₈	—	106	2	no	—
309	C₂×C₈	—	106	2	no	—
310	C₂×C₈	—	106	1	no	—
311	C₂×C₈	—	424	2	no	—
312	C₂×C₈	—	424	2	no	—
313	C₂×C₈	—	212	1	no	—
314	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
315	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
316	C₂×C₈	D₄×D₅₃	1	2	yes	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	53	2	no	—
319	C₄⋊C₄	—	106	2	no	—
320	C₄⋊C₄	C₂×C₄×D₅₃	1	2	yes	—
321	C₂²⋊C₄	—	106	2	no	—
322	C₂²⋊C₄	—	106	2	no	—
323	C₂²⋊C₄	—	106	2	no	—
324	C₄²	—	2	2	no	—
325	C₄²	—	53	1	no	—
326	C₄²	—	53	1	no	—
327	C₄²	—	53	1	no	—
328	C₄²	—	106	2	no	—
329	C₄²	—	106	2	no	—
330	C₄²	C₂³×D₅₃	1	1	yes	characteristic
331	C₂³	—	53	1	no	—
332	C₂³	—	106	2	no	—
333	Q₈	—	2	2	no	—
334	Q₈	—	2	2	no	—
335	D₄	—	53	4	no	—
336	D₄	—	106	2	no	—
337	D₄	—	212	4	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	53	1	no	—
341	C₂×C₄	—	53	1	no	—
342	C₂×C₄	—	53	1	no	—
343	C₂×C₄	—	106	2	no	—
344	C₂×C₄	—	106	1	no	—
345	C₂×C₄	—	106	1	no	—
346	C₂×C₄	—	106	2	no	—
347	C₂×C₄	—	106	2	no	—
348	C₂×C₄	—	212	4	no	—
349	C₂×C₄	—	212	4	no	—
350	C₂×C₄	—	212	2	no	—
351	C₂×C₄	—	212	2	no	—
352	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
353	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
354	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
355	C₈	—	2	1	no	—
356	C₈	—	2	2	no	—
357	C₈	—	4	1	no	—
358	C₈	—	4	2	no	—
359	C₈	—	106	1	no	—
360	C₈	—	106	1	no	—
361	C₈	—	106	2	no	—
362	C₈	—	212	1	no	—
363	C₈	—	212	2	no	—
364	C₈	M₁₆×D₅₃	1	2	yes	—
365	C₂²	—	106	1	no	—
366	C₂²	—	106	4	no	—
367	C₂²	—	106	1	no	—
368	C₂²	—	106	1	no	—
369	C₂²	—	212	2	no	—
370	C₂²	—	212	2	no	—
371	C₂²	C₄×D₄×D₅₃	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	4	2	no	—
374	C₄	—	4	2	no	—
375	C₄	—	53	2	no	—
376	C₄	—	106	1	no	—
377	C₄	—	106	1	no	—
378	C₄	—	212	2	no	—
379	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
380	C₄	#16811388	1	1	yes	characteristic
381	C₄	#16811432	1	2	yes	—
382	C₂	—	212	1	no	—
383	C₂	—	212	4	no	—
384	C₂	#20283451	1	1	yes	characteristic
385	C₂	#20283595	1	1	yes	characteristic
386	C₂	#444×D₅₃	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×#1593	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#13133240	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1593	D₅₃
hypocenter	C₅₃	#18065
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1593

Complements

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

N	H ≅ G/N	type
1. #20283839	C₂	semidirect
3. #20273624	C₂	semidirect
4. #20273554	C₂	semidirect
5. #20273549	C₂	semidirect
6. #20273540	C₂	semidirect
7. #20273538	C₂	semidirect
8. #20273469	C₂	semidirect
9. #20273153	C₂	semidirect
10. #20273001	C₂	semidirect
11. C₅₃×#1593	C₂	semidirect
130. D₄₂₄	M₁₆	semidirect
192. #1593	D₅₃	semidirect
238. C₅₃	#18065	semidirect

Maximal subgroups

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

subgroup	index
#20283839	2
C₄×#13133240	2
#20273624	2
#20273554	2
#20273549	2
#20273540	2
#20273538	2
#20273469	2
#20273153	2
#20273001	2
C₅₃×#1593	2
#18065	53