Subgroups — #27375694

Order 13568.

638 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287690	C₂	1	1	yes	maximal, characteristic
2	#20287583	C₂	1	1	yes	maximal, characteristic, complemented
3	#20287055	C₂	1	2	yes	maximal, complemented
4	#20274307	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274249	C₂	1	2	yes	maximal, complemented
6	#20274119	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274075	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273844	C₂	1	2	yes	maximal, complemented
9	#20273790	C₂	1	2	yes	maximal, complemented
10	#20273601	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811333	C₂²	1	2	yes	—
13	#16811295	C₂²	1	2	yes	—
14	#16811028	C₂²	1	1	yes	characteristic
15	#16811025	—	2	1	no	—
16	#16811000	—	2	1	no	—
17	#16811000	C₂²	1	1	yes	characteristic
18	#16810996	C₂²	1	2	yes	—
19	#16810970	C₂²	1	2	yes	—
20	#16810938	—	2	2	no	—
21	#16810924	—	2	2	no	—
22	#16810918	C₂²	1	2	yes	—
23	#16810914	C₂²	1	2	yes	—
24	#16810872	—	2	2	no	—
25	#16810823	C₂²	1	2	yes	—
26	#16810819	—	2	2	no	—
27	#16810819	C₂²	1	2	yes	—
28	#16810784	—	2	2	no	—
29	C₄×D₄₂₄	—	2	1	no	—
30	C₄×D₄₂₄	—	2	1	no	—
31	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
32	#16810702	C₂²	1	2	yes	—
33	#16810623	—	2	2	no	—
34	#16810376	C₂²	1	1	yes	characteristic
35	#16810371	C₂²	1	2	yes	—
36	#16810366	C₂²	1	2	yes	—
37	#16810352	C₂²	1	2	yes	—
38	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
41	C₅₃×#335	C₂²	1	1	yes	characteristic
42	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
43	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
44	C₈²×C₅₃	C₂²	1	1	yes	characteristic
45	C₂×#11719949	—	2	2	no	—
46	C₂×#11719913	—	4	2	no	—
47	C₂×#11719924	—	2	2	no	—
48	C₂×D₄₂₄	—	2	1	no	—
49	C₂×D₄₂₄	—	2	1	no	—
50	C₂×D₄₂₄	—	2	2	no	—
51	C₂×D₄₂₄	D₄	1	1	yes	characteristic
52	C₂×#11719912	—	2	2	no	—
53	#13133217	C₂³	1	2	yes	—
54	#13133214	—	2	2	no	—
55	#13133214	C₂³	1	2	yes	—
56	#13133209	—	2	2	no	—
57	#13133195	—	4	2	no	—
58	#13133193	—	2	2	no	—
59	#13133192	—	2	2	no	—
60	#13133192	—	4	2	no	—
61	C₄×D₂₁₂	—	2	2	no	—
62	C₄×D₂₁₂	—	2	2	no	—
63	C₄×D₂₁₂	C₂³	1	2	yes	—
64	C₄⋊C₄×D₅₃	—	2	2	no	—
65	#13133151	—	2	1	no	—
66	#13133151	—	2	1	no	—
67	#13133151	D₄	1	1	yes	characteristic
68	#13133150	—	2	2	no	—
69	#13133149	—	2	2	no	—
70	C₄⋊#88322	C₂³	1	2	yes	—
71	#13133141	—	2	2	no	—
72	#13133139	—	2	2	no	—
73	#13133139	—	2	2	no	—
74	#13133139	—	2	2	no	—
75	#13133139	—	2	2	no	—
76	#13133139	D₄	1	2	yes	—
77	#13133137	—	2	2	no	—
78	#13133129	—	2	2	no	—
79	C₂×Q₁₆×C₅₃	D₄	1	1	yes	characteristic
80	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
81	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
82	#107×C₅₃	D₄	1	1	yes	characteristic
83	C₄⋊C₈×C₅₃	—	2	1	no	—
84	C₄⋊C₈×C₅₃	—	2	1	no	—
85	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
87	#103×C₅₃	D₄	1	1	yes	characteristic
88	#103×C₅₃	D₄	1	1	yes	characteristic
89	C₄×C₈×C₅₃	—	2	1	no	—
90	C₄×C₈×C₅₃	—	2	1	no	—
91	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
96	#11719949	—	4	4	no	—
97	#11719949	—	8	2	no	—
98	C₂×D₂₁₂	—	2	2	no	—
99	C₂×D₂₁₂	—	2	2	no	—
100	C₂×D₂₁₂	—	2	2	no	—
101	C₂×D₂₁₂	—	4	2	no	—
102	C₂×D₂₁₂	—	4	2	no	—
103	C₂×D₂₁₂	C₂×D₄	1	2	yes	—
104	C₂×C₄×D₅₃	—	2	2	no	—
105	C₂×C₄×D₅₃	—	4	2	no	—
106	C₂×C₄×D₅₃	—	4	2	no	—
107	C₂×Dic₁₀₆	—	2	2	no	—
108	Q₁₆×C₅₃	—	2	2	no	—
109	C₂×C₈×C₅₃	—	2	1	no	—
110	C₂×C₈×C₅₃	—	2	1	no	—
111	C₂×C₈×C₅₃	—	2	1	no	—
112	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₄⋊C₄×C₅₃	—	2	1	no	—
117	C₄⋊C₄×C₅₃	—	2	1	no	—
118	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
119	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₅₃	—	2	1	no	—
121	C₄²×C₅₃	—	2	1	no	—
122	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
123	#11719924	—	8	2	no	—
124	#11719921	—	2	2	no	—
125	#11719921	—	4	2	no	—
126	#11719921	—	4	2	no	—
127	C₄⋊Dic₅₃	—	2	2	no	—
128	C₄⋊Dic₅₃	—	2	2	no	—
129	C₄⋊Dic₅₃	C₂×D₄	1	2	yes	—
130	#11719919	—	4	2	no	—
131	#11719919	—	4	2	no	—
132	C₂×#88322	—	2	2	no	—
133	D₄₂₄	—	2	2	no	—
134	D₄₂₄	—	4	2	no	—
135	D₄₂₄	—	8	1	no	—
136	D₄₂₄	—	8	2	no	—
137	#11719913	—	8	4	no	—
138	#11719912	—	8	2	no	—
139	C₂²×D₅₃	—	2	2	no	—
140	C₂²×D₅₃	—	4	2	no	—
141	Q₈×C₅₃	—	2	1	no	—
142	Q₈×C₅₃	—	2	1	no	—
143	Q₈×C₅₃	—	2	1	no	—
144	Q₈×C₅₃	#135	1	2	yes	—
145	C₂×C₄×C₅₃	—	2	1	no	—
146	C₂×C₄×C₅₃	—	2	1	no	—
147	C₂×C₄×C₅₃	—	2	1	no	—
148	C₂×C₄×C₅₃	—	2	1	no	—
149	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂×Dic₅₃	—	2	2	no	—
153	C₂×Dic₅₃	—	4	2	no	—
154	D₂₁₂	—	2	2	no	—
155	D₂₁₂	—	2	2	no	—
156	D₂₁₂	—	4	2	no	—
157	D₂₁₂	—	4	2	no	—
158	D₂₁₂	—	4	2	no	—
159	D₂₁₂	—	4	2	no	—
160	D₂₁₂	—	8	4	no	—
161	D₂₁₂	—	8	2	no	—
162	D₂₁₂	—	16	2	no	—
163	C₄×D₅₃	—	4	2	no	—
164	C₄×D₅₃	—	4	2	no	—
165	C₄×D₅₃	—	8	4	no	—
166	C₄×D₅₃	—	16	2	no	—
167	Dic₁₀₆	—	8	2	no	—
168	C₈×C₅₃	—	2	2	no	—
169	C₈×C₅₃	—	2	1	no	—
170	C₈×C₅₃	—	2	1	no	—
171	C₈×C₅₃	—	2	1	no	—
172	C₈×C₅₃	—	4	1	no	—
173	C₈×C₅₃	—	4	1	no	—
174	C₈×C₅₃	#137	1	2	yes	—
175	#88322	—	4	2	no	—
176	#19909	—	53	1	no	maximal, Sylow
177	C₂²×C₅₃	D₄²	1	1	yes	characteristic
178	C₂×D₅₃	—	4	2	no	—
179	C₂×D₅₃	—	4	2	no	—
180	C₂×D₅₃	—	4	2	no	—
181	C₂×D₅₃	—	8	2	no	—
182	C₂×D₅₃	—	8	2	no	—
183	C₂×D₅₃	—	8	2	no	—
184	C₄×C₅₃	—	2	2	no	—
185	C₄×C₅₃	—	2	1	no	—
186	C₄×C₅₃	—	4	1	no	—
187	C₄×C₅₃	—	4	1	no	—
188	C₄×C₅₃	—	4	1	no	—
189	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
190	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
191	C₄×C₅₃	#576	1	1	yes	characteristic
192	C₄×C₅₃	#577	1	1	yes	characteristic
193	Dic₅₃	—	4	2	no	—
194	Dic₅₃	—	8	2	no	—
195	#3391	—	53	1	no	—
196	#3380	—	53	1	no	—
197	#3333	—	53	2	no	—
198	#1706	—	53	1	no	—
199	#1697	—	53	2	no	—
200	#1676	—	53	1	no	—
201	#1668	—	53	1	no	—
202	#1631	—	53	2	no	—
203	#1622	—	53	2	no	—
204	#1589	—	53	1	no	—
205	C₈×Q₁₆	D₅₃	1	1	yes	characteristic, complemented
206	C₂×C₅₃	#3278	1	1	yes	characteristic
207	C₂×C₅₃	#3280	1	1	yes	characteristic
208	C₂×C₅₃	#3292	1	1	yes	characteristic
209	D₅₃	—	8	2	no	—
210	D₅₃	—	16	2	no	—
211	#554	—	53	2	no	—
212	#550	—	53	2	no	—
213	#500	—	53	1	no	—
214	#499	—	106	1	no	—
215	#493	—	53	1	no	—
216	#493	—	106	1	no	—
217	#492	—	53	2	no	—
218	#486	—	53	2	no	—
219	#480	—	106	2	no	—
220	#478	—	106	2	no	—
221	#477	—	53	2	no	—
222	#476	—	53	2	no	—
223	#468	—	106	2	no	—
224	#460	—	53	2	no	—
225	#459	—	53	2	no	—
226	#459	—	106	2	no	—
227	#452	—	106	2	no	—
228	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
229	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
230	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
231	C₄×D₈	—	53	1	no	—
232	C₄×D₈	—	106	1	no	—
233	C₄×D₈	—	106	1	no	—
234	#436	—	53	2	no	—
235	#417	—	106	2	no	—
236	#335	—	53	1	no	—
237	#335	C₂×D₅₃	1	1	yes	characteristic
238	#333	—	53	2	no	—
239	#331	—	53	2	no	—
240	Q₈⋊C₈	C₂×D₅₃	1	1	yes	characteristic
241	Q₈⋊C₈	C₂×D₅₃	1	1	yes	characteristic
242	D₄⋊C₈	—	53	2	no	—
243	C₈²	C₂×D₅₃	1	1	yes	characteristic
244	C₅₃	#19909	1	1	yes	characteristic, complemented, Sylow
245	C₂×#41	—	106	2	no	—
246	C₂×Q₁₆	D₂₁₂	1	1	yes	characteristic
247	C₂×SD₁₆	—	106	2	no	—
248	C₂×SD₁₆	—	212	2	no	—
249	C₂×D₈	—	53	1	no	—
250	C₂×D₈	—	106	2	no	—
251	C₂×D₈	—	106	1	no	—
252	C₂×D₈	—	106	1	no	—
253	C₂×M₁₆	—	106	2	no	—
254	C₄⋊Q₈	—	53	2	no	—
255	#127	—	53	2	no	—
256	#127	—	106	2	no	—
257	#125	—	106	2	no	—
258	C₂²⋊Q₈	—	106	2	no	—
259	C₂²⋊Q₈	—	212	2	no	—
260	#121	—	106	2	no	—
261	#121	—	212	2	no	—
262	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
263	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
264	C₄×D₄	—	53	2	no	—
265	C₄×D₄	—	106	2	no	—
266	C₄×D₄	—	106	2	no	—
267	C₂×C₄⋊C₄	—	106	2	no	—
268	#107	—	53	1	no	—
269	#107	—	106	1	no	—
270	#107	—	106	1	no	—
271	#107	—	106	2	no	—
272	#107	D₂₁₂	1	1	yes	characteristic
273	#106	—	106	2	no	—
274	C₄⋊C₈	—	2	1	no	—
275	C₄⋊C₈	—	2	1	no	—
276	C₄⋊C₈	—	53	2	no	—
277	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
278	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
279	#103	—	106	2	no	—
280	#103	D₂₁₂	1	1	yes	characteristic
281	#103	D₂₁₂	1	1	yes	characteristic
282	#102	—	53	2	no	—
283	#102	—	106	2	no	—
284	#102	—	106	2	no	—
285	#102	—	106	2	no	—
286	#102	—	106	2	no	—
287	#102	—	106	2	no	—
288	C₂²⋊C₈	—	106	2	no	—
289	C₄×C₈	—	2	1	no	—
290	C₄×C₈	—	2	1	no	—
291	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
292	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
293	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
294	#41	—	424	2	no	—
295	#41	—	212	4	no	—
296	C₂×Q₈	—	106	2	no	—
297	C₂×Q₈	C₂×D₂₁₂	1	1	yes	characteristic
298	C₂×Q₈	C₂×D₂₁₂	1	1	yes	characteristic
299	C₂×D₄	—	53	2	no	—
300	C₂×D₄	—	106	2	no	—
301	C₂×D₄	—	106	2	no	—
302	C₂×D₄	—	106	2	no	—
303	C₂×D₄	—	212	2	no	—
304	C₂×D₄	—	212	2	no	—
305	C₂²×C₄	—	106	2	no	—
306	C₂²×C₄	—	212	2	no	—
307	C₂²×C₄	—	212	2	no	—
308	Q₁₆	—	2	2	no	—
309	SD₁₆	—	424	4	no	—
310	SD₁₆	—	424	2	no	—
311	D₈	—	106	2	no	—
312	D₈	—	424	1	no	—
313	D₈	—	424	2	no	—
314	D₈	—	212	2	no	—
315	M₁₆	—	424	2	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	2	1	no	—
319	C₂×C₈	—	106	2	no	—
320	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
321	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
322	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
323	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
324	C₄⋊C₄	—	2	1	no	—
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	—	53	2	no	—
327	C₄⋊C₄	—	106	2	no	—
328	C₄⋊C₄	—	106	2	no	—
329	C₄⋊C₄	—	212	2	no	—
330	C₄⋊C₄	—	212	2	no	—
331	C₄⋊C₄	C₂×D₂₁₂	1	1	yes	characteristic
332	C₄⋊C₄	C₂×D₂₁₂	1	1	yes	characteristic
333	C₂²⋊C₄	—	106	2	no	—
334	C₂²⋊C₄	—	212	2	no	—
335	C₂²⋊C₄	—	212	2	no	—
336	C₄²	—	2	1	no	—
337	C₄²	—	2	1	no	—
338	C₄²	C₂³×D₅₃	1	1	yes	characteristic
339	C₂³	—	106	2	no	—
340	C₂³	—	212	2	no	—
341	Q₈	—	2	1	no	—
342	Q₈	—	2	1	no	—
343	Q₈	—	2	1	no	—
344	Q₈	—	424	2	no	—
345	Q₈	#13133239	1	2	yes	—
346	D₄	—	848	2	no	—
347	D₄	—	106	2	no	—
348	D₄	—	106	2	no	—
349	D₄	—	424	2	no	—
350	D₄	—	424	4	no	—
351	D₄	—	212	2	no	—
352	D₄	—	212	2	no	—
353	D₄	—	212	2	no	—
354	D₄	—	212	2	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	848	2	no	—
360	C₂×C₄	—	106	2	no	—
361	C₂×C₄	—	424	4	no	—
362	C₂×C₄	—	212	2	no	—
363	C₂×C₄	—	212	2	no	—
364	C₂×C₄	—	212	2	no	—
365	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
366	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
367	C₂×C₄	#13133263	1	1	yes	characteristic
368	C₈	—	2	2	no	—
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	4	1	no	—
373	C₈	—	4	1	no	—
374	C₈	—	212	2	no	—
375	C₈	#13133249	1	2	yes	—
376	C₂²	—	424	2	no	—
377	C₂²	—	424	2	no	—
378	C₂²	—	424	2	no	—
379	C₂²	—	212	2	no	—
380	C₂²	—	212	2	no	—
381	C₂²	—	212	2	no	—
382	C₂²	D₄×D₂₁₂	1	1	yes	characteristic
383	C₄	—	2	1	no	—
384	C₄	—	2	2	no	—
385	C₄	—	4	1	no	—
386	C₄	—	4	1	no	—
387	C₄	—	4	1	no	—
388	C₄	—	424	2	no	—
389	C₄	—	212	2	no	—
390	C₄	C₂×#13133239	1	1	yes	characteristic
391	C₄	C₂×#13133249	1	1	yes	characteristic
392	C₄	#16811428	1	1	yes	characteristic
393	C₄	#16811430	1	1	yes	characteristic
394	C₂	—	848	2	no	—
395	C₂	—	424	2	no	—
396	C₂	#20286382	1	1	yes	characteristic
397	C₂	#20286402	1	1	yes	characteristic
398	C₂	#20286515	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₁₂
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₈×Q₁₆×C₅₃	C₂
socle	C₂²×C₅₃	D₄²
prime radical	#20287690	C₂
hypercenter	C₈×Q₁₆	D₅₃
hypocenter	C₅₃	#19909
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #20287583	C₂	semidirect
3. #20287055	C₂	semidirect
4. #20274307	C₂	semidirect
5. #20274249	C₂	semidirect
6. #20274119	C₂	semidirect
7. #20274075	C₂	semidirect
8. #20273844	C₂	semidirect
9. #20273790	C₂	semidirect
10. #20273601	C₂	semidirect
11. C₈×Q₁₆×C₅₃	C₂	semidirect
205. C₈×Q₁₆	D₅₃	semidirect
244. C₅₃	#19909	semidirect

Maximal subgroups

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

subgroup	index
#20287690	2
#20287583	2
#20287055	2
#20274307	2
#20274249	2
#20274119	2
#20274075	2
#20273844	2
#20273790	2
#20273601	2
C₈×Q₁₆×C₅₃	2
#19909	53