Subgroups — #27367965

Order 13568.

606 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285299	C₂	1	1	yes	maximal, characteristic
2	#20284847	C₂	1	2	yes	maximal
3	C₈×Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20274144	C₂	1	2	yes	maximal, complemented
5	#20274143	C₂	1	2	yes	maximal, complemented
6	#20274110	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274109	C₂	1	1	yes	maximal, characteristic, complemented
8	#20274082	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273477	C₂	1	1	yes	maximal, characteristic
10	#20273414	C₂	1	2	yes	maximal
11	C₅₃×#1670	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811178	C₂²	1	2	yes	—
13	C₄×Q₈×D₅₃	C₂²	1	1	yes	characteristic
14	#16811028	C₂²	1	1	yes	characteristic
15	#16810973	C₂²	1	2	yes	—
16	#16810955	—	2	2	no	—
17	#16810905	C₂²	1	2	yes	—
18	#16810861	—	2	1	no	—
19	#16810861	—	2	1	no	—
20	#16810834	C₂²	1	2	yes	—
21	#16810833	C₂²	1	2	yes	—
22	#16810780	—	2	2	no	—
23	Q₈×#88322	C₂²	1	1	yes	characteristic
24	C₈×Dic₁₀₆	—	2	1	no	—
25	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
26	#16810747	—	2	1	no	—
27	#16810747	C₂²	1	1	yes	characteristic
28	#16810729	—	2	2	no	—
29	#16810727	—	2	2	no	—
30	Q₁₆×Dic₅₃	—	2	1	no	—
31	Q₁₆×Dic₅₃	—	2	1	no	—
32	#16810679	C₂²	1	2	yes	—
33	C₄⋊C₈×D₅₃	—	2	1	no	—
34	C₄⋊C₈×D₅₃	C₂²	1	1	yes	characteristic
35	C₄×C₈×D₅₃	—	2	1	no	—
36	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
37	#16810376	C₂²	1	1	yes	characteristic
38	#16810373	C₂²	1	2	yes	—
39	#16810370	C₂²	1	2	yes	—
40	#16810369	C₂²	1	2	yes	—
41	#16810368	C₂²	1	2	yes	—
42	C₅₃×#494	C₂²	1	1	yes	characteristic
43	C₅₃×#475	C₂²	1	2	yes	—
44	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
45	C₅₃×#335	C₂²	1	1	yes	characteristic
46	C₅₃×#332	C₂²	1	2	yes	—
47	C₂×Q₈×D₅₃	D₄	1	1	yes	characteristic
48	C₂×Dic₂₁₂	—	2	2	no	—
49	C₂×#11719925	—	2	2	no	—
50	C₂×C₈×D₅₃	—	2	1	no	—
51	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
52	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
53	#13133217	C₂³	1	2	yes	—
54	#13133208	—	2	2	no	—
55	#13133197	—	2	2	no	—
56	#13133193	—	2	2	no	—
57	Q₈×Dic₅₃	—	2	2	no	—
58	Q₈×Dic₅₃	D₄	1	1	yes	characteristic
59	C₄×Dic₁₀₆	—	2	1	no	—
60	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
61	#13133182	—	2	1	no	—
62	#13133182	—	2	2	no	—
63	#13133182	D₄	1	1	yes	characteristic
64	#13133175	—	2	2	no	—
65	C₄⋊C₄×D₅₃	—	2	1	no	—
66	C₄⋊C₄×D₅₃	D₄	1	1	yes	characteristic
67	C₄²×D₅₃	—	2	1	no	—
68	C₄²×D₅₃	C₂³	1	1	yes	characteristic
69	#13133151	—	2	1	no	—
70	#13133151	—	2	1	no	—
71	#13133148	—	2	2	no	—
72	C₄⋊#88322	—	2	1	no	—
73	C₄⋊#88322	C₂³	1	2	yes	—
74	C₄⋊#88322	C₂³	1	1	yes	characteristic
75	#13133146	—	2	1	no	—
76	#13133146	—	2	1	no	—
77	#13133146	—	2	1	no	—
78	#13133142	—	2	2	no	—
79	#13133141	—	2	2	no	—
80	#13133141	—	2	2	no	—
81	#13133140	—	2	2	no	—
82	#13133140	—	2	2	no	—
83	#13133129	—	2	2	no	—
84	#13133127	—	2	2	no	—
85	C₄×#88322	—	2	1	no	—
86	C₄×#88322	C₂³	1	1	yes	characteristic
87	C₈×Dic₅₃	—	2	1	no	—
88	C₈×Dic₅₃	—	2	1	no	—
89	C₈×Dic₅₃	D₄	1	1	yes	characteristic
90	C₈×Dic₅₃	D₄	1	1	yes	characteristic
91	C₂×Q₁₆×C₅₃	—	2	1	no	—
92	C₂×Q₁₆×C₅₃	—	2	1	no	—
93	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
94	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
95	#106×C₅₃	—	2	2	no	—
96	C₄⋊C₈×C₅₃	—	2	1	no	—
97	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
98	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
99	#103×C₅₃	—	2	2	no	—
100	C₄×C₈×C₅₃	—	2	1	no	—
101	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
102	C₂×Q₈×C₅₃	—	2	2	no	—
103	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
104	Q₈×D₅₃	—	2	2	no	—
105	Q₈×D₅₃	—	2	2	no	—
106	C₂×C₄×D₅₃	—	2	1	no	—
107	C₂×C₄×D₅₃	—	2	1	no	—
108	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
111	C₂×Dic₁₀₆	—	2	1	no	—
112	C₂×Dic₁₀₆	—	2	2	no	—
113	C₂×Dic₁₀₆	C₂×D₄	1	1	yes	characteristic
114	Q₁₆×C₅₃	—	4	2	no	—
115	Q₁₆×C₅₃	—	4	2	no	—
116	C₂×C₈×C₅₃	—	2	1	no	—
117	C₂×C₈×C₅₃	—	2	2	no	—
118	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
120	C₄⋊C₄×C₅₃	—	2	1	no	—
121	C₄⋊C₄×C₅₃	—	2	2	no	—
122	C₄⋊C₄×C₅₃	—	2	2	no	—
123	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₅₃	—	2	1	no	—
125	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
126	#11719925	—	4	4	no	—
127	#11719921	—	4	2	no	—
128	C₄⋊Dic₅₃	—	2	1	no	—
129	C₄⋊Dic₅₃	—	2	2	no	—
130	C₄⋊Dic₅₃	—	2	2	no	—
131	C₄⋊Dic₅₃	—	4	2	no	—
132	C₄⋊Dic₅₃	C₂×D₄	1	1	yes	characteristic
133	#11719919	—	2	1	no	—
134	#11719919	—	2	1	no	—
135	#11719919	—	2	1	no	—
136	#11719919	—	4	2	no	—
137	C₄×Dic₅₃	—	2	1	no	—
138	C₄×Dic₅₃	—	2	1	no	—
139	C₄×Dic₅₃	—	2	1	no	—
140	C₄×Dic₅₃	—	4	2	no	—
141	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
142	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
143	C₄×Dic₅₃	#41	1	1	yes	characteristic
144	C₂×#88322	—	2	1	no	—
145	C₂×#88322	—	2	2	no	—
146	C₂×#88322	C₂×D₄	1	1	yes	characteristic
147	C₂×#88322	C₂×D₄	1	1	yes	characteristic
148	Dic₂₁₂	—	4	4	no	—
149	C₈×D₅₃	—	2	2	no	—
150	C₈×D₅₃	—	4	1	no	—
151	C₈×D₅₃	—	4	2	no	—
152	C₂²×D₅₃	#121	1	1	yes	characteristic
153	Q₈×C₅₃	—	2	1	no	—
154	Q₈×C₅₃	—	4	2	no	—
155	Q₈×C₅₃	—	4	2	no	—
156	Q₈×C₅₃	#135	1	2	yes	—
157	C₂×C₄×C₅₃	—	2	1	no	—
158	C₂×C₄×C₅₃	—	2	1	no	—
159	C₂×C₄×C₅₃	—	4	2	no	—
160	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
163	C₂×Dic₅₃	—	2	1	no	—
164	C₂×Dic₅₃	—	2	1	no	—
165	C₂×Dic₅₃	—	4	2	no	—
166	C₂×Dic₅₃	#121	1	1	yes	characteristic
167	C₂×Dic₅₃	#121	1	1	yes	characteristic
168	C₂×Dic₅₃	#121	1	1	yes	characteristic
169	C₄×D₅₃	—	2	4	no	—
170	C₄×D₅₃	—	2	2	no	—
171	C₄×D₅₃	—	2	1	no	—
172	C₄×D₅₃	—	2	1	no	—
173	C₄×D₅₃	—	2	1	no	—
174	C₄×D₅₃	—	2	1	no	—
175	C₄×D₅₃	—	4	2	no	—
176	Dic₁₀₆	—	2	4	no	—
177	Dic₁₀₆	—	2	1	no	—
178	Dic₁₀₆	—	4	2	no	—
179	Dic₁₀₆	—	4	2	no	—
180	Dic₁₀₆	#135	1	2	yes	—
181	C₈×C₅₃	—	2	1	no	—
182	C₈×C₅₃	—	4	2	no	—
183	C₈×C₅₃	—	4	1	no	—
184	C₈×C₅₃	#137	1	2	yes	—
185	#88322	—	2	1	no	—
186	#88322	—	2	1	no	—
187	#88322	—	4	2	no	—
188	#88322	—	4	1	no	—
189	#19265	—	53	1	no	maximal, Sylow
190	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
191	C₂×D₅₃	—	2	1	no	—
192	C₂×D₅₃	—	2	1	no	—
193	C₂×D₅₃	#473	1	2	yes	—
194	C₄×C₅₃	—	2	2	no	—
195	C₄×C₅₃	—	2	1	no	—
196	C₄×C₅₃	—	4	1	no	—
197	C₄×C₅₃	—	8	2	no	—
198	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
199	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
200	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
201	C₄×C₅₃	#575	1	1	yes	characteristic
202	Dic₅₃	—	2	1	no	—
203	Dic₅₃	—	2	1	no	—
204	Dic₅₃	—	2	2	no	—
205	Dic₅₃	—	2	1	no	—
206	Dic₅₃	—	4	1	no	—
207	Dic₅₃	—	8	2	no	—
208	#3170	—	53	1	no	—
209	#3113	—	53	2	no	—
210	C₂×C₈×Q₈	—	53	1	no	—
211	#1680	—	53	2	no	—
212	#1680	—	53	2	no	—
213	#1674	—	53	1	no	—
214	#1674	—	53	1	no	—
215	#1670	—	53	1	no	—
216	#1670	D₅₃	1	1	yes	characteristic, complemented
217	#1568	—	53	1	no	—
218	#1552	—	53	2	no	—
219	C₂×C₅₃	#3037	1	1	yes	characteristic
220	C₂×C₅₃	#3050	1	1	yes	characteristic
221	C₂×C₅₃	C₂×#473	1	1	yes	characteristic
222	D₅₃	—	2	2	no	—
223	#533	—	53	2	no	—
224	C₂×C₄×Q₈	—	53	1	no	—
225	#500	—	53	1	no	—
226	#494	C₂×D₅₃	1	1	yes	characteristic
227	#487	—	53	2	no	—
228	#483	—	106	2	no	—
229	#475	—	53	2	no	—
230	#475	C₂×D₅₃	1	2	yes	—
231	#467	—	106	1	no	—
232	#467	—	106	1	no	—
233	#462	—	53	2	no	—
234	#462	—	53	2	no	—
235	#451	—	106	2	no	—
236	C₈×Q₈	—	53	1	no	—
237	C₈×Q₈	—	53	1	no	—
238	C₈×Q₈	—	53	1	no	—
239	C₈×Q₈	—	106	1	no	—
240	C₈×Q₈	—	106	1	no	—
241	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
242	#441	—	106	2	no	—
243	#441	—	106	2	no	—
244	C₄×Q₁₆	—	106	1	no	—
245	C₄×Q₁₆	—	106	1	no	—
246	#432	—	53	2	no	—
247	C₂×C₄⋊C₈	—	53	1	no	—
248	C₂×C₄⋊C₈	—	106	1	no	—
249	C₂×C₄×C₈	—	53	1	no	—
250	C₂×C₄×C₈	—	106	1	no	—
251	#335	—	53	1	no	—
252	#335	C₂×D₅₃	1	1	yes	characteristic
253	#334	—	53	2	no	—
254	#332	—	53	2	no	—
255	#332	—	53	2	no	—
256	#332	—	53	2	no	—
257	#332	C₂×D₅₃	1	2	yes	—
258	C₅₃	#19265	1	1	yes	characteristic, complemented, Sylow
259	C₂²×Q₈	—	53	1	no	—
260	C₂×Q₁₆	—	2	1	no	—
261	C₂×Q₁₆	—	2	1	no	—
262	C₂×Q₁₆	—	106	2	no	—
263	C₂×Q₁₆	—	106	2	no	—
264	C₂²×C₈	—	53	1	no	—
265	C₂²×C₈	—	53	1	no	—
266	C₂²×C₈	—	106	1	no	—
267	C₄⋊Q₈	—	53	2	no	—
268	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
269	#125	—	106	2	no	—
270	C₂²⋊Q₈	—	106	2	no	—
271	C₂²⋊Q₈	—	106	2	no	—
272	C₄×Q₈	—	53	1	no	—
273	C₄×Q₈	—	53	1	no	—
274	C₄×Q₈	—	53	1	no	—
275	C₄×Q₈	—	106	1	no	—
276	C₄×Q₈	—	106	1	no	—
277	C₄×Q₈	—	106	2	no	—
278	C₄×Q₈	—	106	2	no	—
279	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
280	#117	—	106	2	no	—
281	C₂×C₄⋊C₄	—	53	1	no	—
282	C₂×C₄⋊C₄	—	106	1	no	—
283	C₂×C₄²	—	53	1	no	—
284	C₂×C₄²	—	106	1	no	—
285	#107	—	106	1	no	—
286	#107	—	106	1	no	—
287	#106	—	2	2	no	—
288	#106	—	106	2	no	—
289	C₄⋊C₈	—	2	1	no	—
290	C₄⋊C₈	—	53	1	no	—
291	C₄⋊C₈	—	53	2	no	—
292	C₄⋊C₈	—	106	1	no	—
293	C₄⋊C₈	—	106	1	no	—
294	C₄⋊C₈	—	106	1	no	—
295	C₄⋊C₈	—	106	1	no	—
296	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
297	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
298	#103	—	2	2	no	—
299	#103	—	106	2	no	—
300	#103	—	106	2	no	—
301	#103	—	106	2	no	—
302	#103	—	106	2	no	—
303	#103	—	106	2	no	—
304	C₂²⋊C₈	—	106	2	no	—
305	#97	—	106	2	no	—
306	C₄×C₈	—	2	1	no	—
307	C₄×C₈	—	53	1	no	—
308	C₄×C₈	—	53	1	no	—
309	C₄×C₈	—	53	1	no	—
310	C₄×C₈	—	106	1	no	—
311	C₄×C₈	—	106	1	no	—
312	C₄×C₈	—	106	1	no	—
313	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
314	C₂×Q₈	—	2	2	no	—
315	C₂×Q₈	—	53	1	no	—
316	C₂×Q₈	—	106	1	no	—
317	C₂×Q₈	—	106	2	no	—
318	C₂×Q₈	—	106	2	no	—
319	C₂×Q₈	—	106	2	no	—
320	C₂×Q₈	D₄×D₅₃	1	1	yes	characteristic
321	C₂²×C₄	—	53	1	no	—
322	C₂²×C₄	—	53	1	no	—
323	C₂²×C₄	—	53	1	no	—
324	C₂²×C₄	—	106	1	no	—
325	C₂²×C₄	—	106	1	no	—
326	Q₁₆	—	4	2	no	—
327	Q₁₆	—	4	2	no	—
328	Q₁₆	—	212	4	no	—
329	Q₁₆	—	212	4	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	53	1	no	—
333	C₂×C₈	—	53	1	no	—
334	C₂×C₈	—	106	2	no	—
335	C₂×C₈	—	106	1	no	—
336	C₂×C₈	—	106	2	no	—
337	C₂×C₈	—	212	1	no	—
338	C₂×C₈	—	212	2	no	—
339	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
340	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
341	C₄⋊C₄	—	2	1	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	53	1	no	—
345	C₄⋊C₄	—	106	1	no	—
346	C₄⋊C₄	—	106	1	no	—
347	C₄⋊C₄	—	106	1	no	—
348	C₄⋊C₄	—	106	1	no	—
349	C₄⋊C₄	—	106	2	no	—
350	C₄⋊C₄	—	106	2	no	—
351	C₄⋊C₄	—	212	2	no	—
352	C₄⋊C₄	—	212	2	no	—
353	C₄⋊C₄	D₄×D₅₃	1	1	yes	characteristic
354	C₂²⋊C₄	—	212	2	no	—
355	C₄²	—	2	1	no	—
356	C₄²	—	53	1	no	—
357	C₄²	—	53	1	no	—
358	C₄²	—	53	1	no	—
359	C₄²	—	106	1	no	—
360	C₄²	—	106	1	no	—
361	C₄²	—	106	1	no	—
362	C₄²	—	212	2	no	—
363	C₄²	C₂³×D₅₃	1	1	yes	characteristic
364	C₂³	—	53	1	no	—
365	Q₈	—	2	1	no	—
366	Q₈	—	4	2	no	—
367	Q₈	—	4	2	no	—
368	Q₈	—	53	2	no	—
369	Q₈	—	106	4	no	—
370	Q₈	—	106	1	no	—
371	Q₈	—	212	2	no	—
372	Q₈	—	212	2	no	—
373	Q₈	#13133240	1	2	yes	—
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	53	1	no	—
378	C₂×C₄	—	53	1	no	—
379	C₂×C₄	—	53	1	no	—
380	C₂×C₄	—	106	2	no	—
381	C₂×C₄	—	106	4	no	—
382	C₂×C₄	—	106	1	no	—
383	C₂×C₄	—	106	1	no	—
384	C₂×C₄	—	106	1	no	—
385	C₂×C₄	—	106	1	no	—
386	C₂×C₄	—	106	1	no	—
387	C₂×C₄	—	106	1	no	—
388	C₂×C₄	—	212	2	no	—
389	C₂×C₄	—	212	2	no	—
390	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
391	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
392	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
393	C₈	—	2	1	no	—
394	C₈	—	4	2	no	—
395	C₈	—	4	1	no	—
396	C₈	—	106	1	no	—
397	C₈	—	106	1	no	—
398	C₈	—	212	2	no	—
399	C₈	—	212	1	no	—
400	C₈	#13133246	1	2	yes	—
401	C₂²	—	53	2	no	—
402	C₂²	—	106	1	no	—
403	C₂²	—	106	1	no	—
404	C₂²	#121×D₅₃	1	1	yes	characteristic
405	C₄	—	2	1	no	—
406	C₄	—	2	2	no	—
407	C₄	—	4	1	no	—
408	C₄	—	8	2	no	—
409	C₄	—	106	1	no	—
410	C₄	—	106	1	no	—
411	C₄	—	106	1	no	—
412	C₄	—	106	2	no	—
413	C₄	—	424	2	no	—
414	C₄	—	212	1	no	—
415	C₄	#135×D₅₃	1	1	yes	characteristic
416	C₄	C₂×#13133240	1	1	yes	characteristic
417	C₄	C₂×#13133246	1	1	yes	characteristic
418	C₄	#16811419	1	1	yes	characteristic
419	C₂	—	106	2	no	—
420	C₂	#20284182	1	1	yes	characteristic
421	C₂	#20284298	1	1	yes	characteristic
422	C₂	#473×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1670	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂²×D₅₃	#121
hypercenter	#1670	D₅₃
hypocenter	C₅₃	#19265
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1670

Complements

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

N	H ≅ G/N	type
4. #20274144	C₂	semidirect
5. #20274143	C₂	semidirect
6. #20274110	C₂	semidirect
7. #20274109	C₂	semidirect
8. #20274082	C₂	semidirect
11. C₅₃×#1670	C₂	semidirect
216. #1670	D₅₃	semidirect
258. C₅₃	#19265	semidirect

Maximal subgroups

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

subgroup	index
#20285299	2
#20284847	2
C₈×Q₈×D₅₃	2
#20274144	2
#20274143	2
#20274110	2
#20274109	2
#20274082	2
#20273477	2
#20273414	2
C₅₃×#1670	2
#19265	53