Subgroups — D₅₃×#1696 #27366722

Order 13568.

618 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#477×D₅₃	C₂	1	2	yes	maximal
3	#446×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20274244	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274243	C₂	1	1	yes	maximal, characteristic, complemented
6	#20274242	C₂	1	2	yes	maximal, complemented
7	#20274241	C₂	1	2	yes	maximal, complemented
8	#20274240	C₂	1	1	yes	maximal, characteristic, complemented
9	#334×D₅₃	C₂	1	1	yes	maximal, characteristic
10	#333×D₅₃	C₂	1	2	yes	maximal
11	C₅₃×#1696	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄⋊Q₈×D₅₃	C₂²	1	2	yes	—
13	C₄×Q₈×D₅₃	C₂²	1	1	yes	characteristic
14	#16811028	C₂²	1	1	yes	characteristic
15	#16811027	—	2	1	no	—
16	#16811027	—	2	1	no	—
17	#16811026	C₂²	1	2	yes	—
18	#16810919	C₂²	1	2	yes	—
19	#16810918	C₂²	1	2	yes	—
20	#16810917	—	2	2	no	—
21	#16810916	—	2	2	no	—
22	#16810915	C₂²	1	2	yes	—
23	#16810754	C₂²	1	1	yes	characteristic
24	#16810753	C₂²	1	1	yes	characteristic
25	#16810752	—	2	1	no	—
26	#16810751	—	2	1	no	—
27	#16810750	C₂²	1	1	yes	characteristic
28	#107×D₅₃	—	2	1	no	—
29	#107×D₅₃	—	2	1	no	—
30	#107×D₅₃	—	2	2	no	—
31	C₄⋊C₈×D₅₃	—	2	1	no	—
32	C₄⋊C₈×D₅₃	C₂²	1	2	yes	—
33	C₄⋊C₈×D₅₃	C₂²	1	1	yes	characteristic
34	#103×D₅₃	—	2	2	no	—
35	#97×D₅₃	—	2	1	no	—
36	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
37	#16810374	C₂²	1	1	yes	characteristic
38	#16810373	C₂²	1	2	yes	—
39	#16810372	C₂²	1	2	yes	—
40	#16810371	C₂²	1	2	yes	—
41	#16810371	C₂²	1	2	yes	—
42	C₅₃×#500	C₂²	1	1	yes	characteristic
43	C₅₃×#477	C₂²	1	2	yes	—
44	C₅₃×#446	C₂²	1	1	yes	characteristic
45	C₅₃×#334	C₂²	1	1	yes	characteristic
46	C₅₃×#333	C₂²	1	2	yes	—
47	C₂×Q₈×D₅₃	—	2	2	no	—
48	C₂×Q₈×D₅₃	D₄	1	1	yes	characteristic
49	C₂×C₈×D₅₃	—	2	1	no	—
50	C₂×C₈×D₅₃	—	2	2	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	#13133216	—	2	2	no	—
55	#13133216	—	2	2	no	—
56	#13133215	—	2	2	no	—
57	Q₈×Dic₅₃	D₄	1	1	yes	characteristic
58	C₄×Dic₁₀₆	—	2	1	no	—
59	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
60	#13133182	—	2	1	no	—
61	#13133182	D₄	1	1	yes	characteristic
62	C₄⋊C₄×D₅₃	—	2	1	no	—
63	C₄⋊C₄×D₅₃	—	2	2	no	—
64	C₄⋊C₄×D₅₃	—	2	2	no	—
65	C₄⋊C₄×D₅₃	D₄	1	1	yes	characteristic
66	C₄²×D₅₃	—	2	1	no	—
67	C₄²×D₅₃	C₂³	1	1	yes	characteristic
68	#13133151	—	2	1	no	—
69	#13133151	—	2	1	no	—
70	#13133151	—	2	2	no	—
71	#13133150	—	2	2	no	—
72	#13133150	—	2	2	no	—
73	#13133150	—	2	2	no	—
74	#13133150	—	2	2	no	—
75	C₄⋊#88322	—	2	1	no	—
76	C₄⋊#88322	C₂³	1	1	yes	characteristic
77	C₄⋊#88322	C₂³	1	2	yes	—
78	#13133146	—	2	1	no	—
79	#13133146	—	2	1	no	—
80	#13133146	—	2	2	no	—
81	#13133146	—	2	1	no	—
82	#13133142	—	2	2	no	—
83	#13133141	—	2	2	no	—
84	#13133140	—	2	2	no	—
85	#13133128	—	2	1	no	—
86	#13133127	—	2	1	no	—
87	#13133127	—	2	1	no	—
88	C₄×#88322	C₂³	1	1	yes	characteristic
89	C₈×Dic₅₃	D₄	1	1	yes	characteristic
90	C₈×Dic₅₃	D₄	1	1	yes	characteristic
91	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
92	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
93	#107×C₅₃	—	2	1	no	—
94	#107×C₅₃	—	2	1	no	—
95	#107×C₅₃	—	2	2	no	—
96	C₄⋊C₈×C₅₃	—	2	1	no	—
97	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
99	#103×C₅₃	—	2	2	no	—
100	#97×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	4	no	—
105	Q₈×D₅₃	—	8	4	no	—
106	C₂×C₄×D₅₃	—	2	1	no	—
107	C₂×C₄×D₅₃	—	2	1	no	—
108	C₂×C₄×D₅₃	—	4	2	no	—
109	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
112	C₂×Dic₁₀₆	—	2	1	no	—
113	C₂×Dic₁₀₆	—	2	2	no	—
114	C₂×Dic₁₀₆	—	4	2	no	—
115	C₂×Dic₁₀₆	C₂×D₄	1	1	yes	characteristic
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	C₄⋊Dic₅₃	—	2	1	no	—
127	C₄⋊Dic₅₃	—	2	2	no	—
128	C₄⋊Dic₅₃	—	2	2	no	—
129	C₄⋊Dic₅₃	C₂×D₄	1	1	yes	characteristic
130	#11719919	—	2	1	no	—
131	#11719919	—	2	1	no	—
132	#11719919	—	2	1	no	—
133	#11719919	—	4	2	no	—
134	#11719919	—	4	2	no	—
135	C₄×Dic₅₃	—	2	1	no	—
136	C₄×Dic₅₃	—	2	1	no	—
137	C₄×Dic₅₃	—	2	1	no	—
138	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
139	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
140	C₄×Dic₅₃	#41	1	1	yes	characteristic
141	C₂×#88322	—	2	1	no	—
142	C₂×#88322	—	2	2	no	—
143	C₂×#88322	C₂×D₄	1	1	yes	characteristic
144	C₂×#88322	C₂×D₄	1	1	yes	characteristic
145	C₈×D₅₃	—	2	2	no	—
146	C₈×D₅₃	—	4	2	no	—
147	C₈×D₅₃	—	4	4	no	—
148	C₈×D₅₃	Q₁₆	1	4	yes	—
149	C₂²×D₅₃	#121	1	1	yes	characteristic
150	Q₈×C₅₃	—	2	2	no	—
151	Q₈×C₅₃	—	8	2	no	—
152	C₂×C₄×C₅₃	—	2	1	no	—
153	C₂×C₄×C₅₃	—	2	1	no	—
154	C₂×C₄×C₅₃	—	4	2	no	—
155	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
158	C₂×Dic₅₃	—	2	1	no	—
159	C₂×Dic₅₃	—	2	1	no	—
160	C₂×Dic₅₃	—	4	2	no	—
161	C₂×Dic₅₃	#121	1	1	yes	characteristic
162	C₂×Dic₅₃	#121	1	1	yes	characteristic
163	C₂×Dic₅₃	#121	1	1	yes	characteristic
164	C₄×D₅₃	—	2	2	no	—
165	C₄×D₅₃	—	4	2	no	—
166	C₄×D₅₃	—	4	2	no	—
167	C₄×D₅₃	—	8	4	no	—
168	C₄×D₅₃	C₂×Q₁₆	1	2	yes	—
169	C₄×D₅₃	#137	1	2	yes	—
170	C₄×D₅₃	#137	1	4	yes	—
171	Dic₁₀₆	—	2	2	no	—
172	Dic₁₀₆	—	4	2	no	—
173	Dic₁₀₆	—	8	4	no	—
174	Dic₁₀₆	—	8	2	no	—
175	C₈×C₅₃	—	2	1	no	—
176	C₈×C₅₃	—	4	1	no	—
177	C₈×C₅₃	—	4	2	no	—
178	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
179	#88322	—	2	1	no	—
180	#88322	—	4	1	no	—
181	#88322	—	4	2	no	—
182	#88322	C₂×Q₁₆	1	2	yes	—
183	C₂×#1696	—	53	1	no	maximal, Sylow
184	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
185	C₂×D₅₃	#462	1	2	yes	—
186	C₂×D₅₃	#470	1	2	yes	—
187	C₂×D₅₃	#472	1	2	yes	—
188	C₄×C₅₃	—	2	1	no	—
189	C₄×C₅₃	—	4	1	no	—
190	C₄×C₅₃	—	4	1	no	—
191	C₄×C₅₃	—	8	2	no	—
192	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
193	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
194	C₄×C₅₃	C₂×#137	1	2	yes	—
195	Dic₅₃	—	2	1	no	—
196	Dic₅₃	—	4	1	no	—
197	Dic₅₃	—	4	1	no	—
198	Dic₅₃	—	8	2	no	—
199	Dic₅₃	#462	1	2	yes	—
200	Dic₅₃	#470	1	2	yes	—
201	C₂×#500	—	53	1	no	—
202	C₂×#477	—	53	2	no	—
203	C₂×#446	—	53	1	no	—
204	#1696	—	53	1	no	—
205	#1696	—	53	2	no	—
206	#1696	—	53	1	no	—
207	#1696	—	53	1	no	—
208	#1696	—	53	2	no	—
209	#1696	D₅₃	1	1	yes	characteristic, direct product factor
210	C₂×#334	—	53	1	no	—
211	C₂×#333	—	53	2	no	—
212	C₂×C₅₃	C₂×#462	1	1	yes	characteristic
213	C₂×C₅₃	C₂×#470	1	1	yes	characteristic
214	C₂×C₅₃	C₂×#472	1	1	yes	characteristic
215	D₅₃	#1696	1	4	yes	direct product factor
216	C₂×C₄⋊Q₈	—	53	2	no	—
217	C₂×C₄×Q₈	—	53	1	no	—
218	#500	—	53	2	no	—
219	#500	—	53	1	no	—
220	#500	—	106	1	no	—
221	#500	—	106	1	no	—
222	#500	C₂×D₅₃	1	1	yes	characteristic
223	#477	—	53	2	no	—
224	#477	—	53	2	no	—
225	#477	—	53	2	no	—
226	#477	—	106	2	no	—
227	#477	—	106	2	no	—
228	#477	C₂×D₅₃	1	2	yes	—
229	#446	—	53	1	no	—
230	#446	—	53	1	no	—
231	#446	—	53	1	no	—
232	#446	—	106	1	no	—
233	#446	—	106	1	no	—
234	#446	C₂×D₅₃	1	1	yes	characteristic
235	C₂×#107	—	106	2	no	—
236	C₂×#107	—	106	1	no	—
237	C₂×#107	—	106	1	no	—
238	C₂×C₄⋊C₈	—	53	2	no	—
239	C₂×C₄⋊C₈	—	53	1	no	—
240	C₂×C₄⋊C₈	—	106	1	no	—
241	C₂×#103	—	106	2	no	—
242	C₂×#97	—	106	1	no	—
243	C₂×C₄×C₈	—	53	1	no	—
244	#334	—	53	2	no	—
245	#334	—	53	1	no	—
246	#334	C₂×D₅₃	1	1	yes	characteristic
247	#333	—	53	2	no	—
248	#333	—	53	2	no	—
249	#333	—	53	2	no	—
250	#333	C₂×D₅₃	1	2	yes	—
251	C₅₃	C₂×#1696	1	1	yes	characteristic, complemented, Sylow
252	C₂²×Q₈	—	53	1	no	—
253	C₂²×Q₈	—	106	2	no	—
254	C₂²×C₈	—	53	1	no	—
255	C₂²×C₈	—	53	1	no	—
256	C₂²×C₈	—	106	1	no	—
257	C₂²×C₈	—	106	2	no	—
258	C₄⋊Q₈	—	53	2	no	—
259	C₄⋊Q₈	—	106	2	no	—
260	C₄⋊Q₈	—	106	2	no	—
261	C₄⋊Q₈	—	106	2	no	—
262	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
263	C₄×Q₈	—	53	1	no	—
264	C₄×Q₈	—	53	1	no	—
265	C₄×Q₈	—	53	1	no	—
266	C₄×Q₈	—	106	1	no	—
267	C₄×Q₈	—	106	1	no	—
268	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
269	C₂×C₄⋊C₄	—	53	1	no	—
270	C₂×C₄⋊C₄	—	106	1	no	—
271	C₂×C₄⋊C₄	—	106	2	no	—
272	C₂×C₄⋊C₄	—	106	2	no	—
273	C₂×C₄²	—	53	1	no	—
274	C₂×C₄²	—	106	1	no	—
275	#107	—	2	1	no	—
276	#107	—	2	1	no	—
277	#107	—	2	2	no	—
278	#107	—	106	2	no	—
279	#107	—	106	2	no	—
280	#107	—	106	2	no	—
281	#107	—	106	2	no	—
282	#107	—	106	1	no	—
283	#107	—	106	1	no	—
284	#107	—	106	2	no	—
285	C₄⋊C₈	—	2	1	no	—
286	C₄⋊C₈	—	53	1	no	—
287	C₄⋊C₈	—	53	2	no	—
288	C₄⋊C₈	—	106	1	no	—
289	C₄⋊C₈	—	106	1	no	—
290	C₄⋊C₈	—	106	2	no	—
291	C₄⋊C₈	—	106	1	no	—
292	C₄⋊C₈	—	106	1	no	—
293	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
294	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
295	#103	—	2	2	no	—
296	#103	—	106	2	no	—
297	#103	—	106	2	no	—
298	#103	—	106	2	no	—
299	#97	—	2	1	no	—
300	#97	—	106	1	no	—
301	#97	—	106	1	no	—
302	#97	—	106	1	no	—
303	C₄×C₈	—	53	1	no	—
304	C₄×C₈	—	53	1	no	—
305	C₄×C₈	—	53	1	no	—
306	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
307	C₂×Q₈	—	2	2	no	—
308	C₂×Q₈	—	53	1	no	—
309	C₂×Q₈	—	106	1	no	—
310	C₂×Q₈	—	106	4	no	—
311	C₂×Q₈	—	106	2	no	—
312	C₂×Q₈	—	424	4	no	—
313	C₂×Q₈	—	212	2	no	—
314	C₂×Q₈	D₄×D₅₃	1	1	yes	characteristic
315	C₂²×C₄	—	53	1	no	—
316	C₂²×C₄	—	53	1	no	—
317	C₂²×C₄	—	53	1	no	—
318	C₂²×C₄	—	106	1	no	—
319	C₂²×C₄	—	106	1	no	—
320	C₂²×C₄	—	212	2	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	53	1	no	—
324	C₂×C₈	—	53	4	no	—
325	C₂×C₈	—	53	1	no	—
326	C₂×C₈	—	106	2	no	—
327	C₂×C₈	—	106	1	no	—
328	C₂×C₈	—	106	2	no	—
329	C₂×C₈	—	212	2	no	—
330	C₂×C₈	—	212	4	no	—
331	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
332	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
333	C₄⋊C₄	—	2	1	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	53	1	no	—
337	C₄⋊C₄	—	106	1	no	—
338	C₄⋊C₄	—	106	1	no	—
339	C₄⋊C₄	—	106	1	no	—
340	C₄⋊C₄	—	106	1	no	—
341	C₄⋊C₄	—	106	2	no	—
342	C₄⋊C₄	—	106	2	no	—
343	C₄⋊C₄	—	212	2	no	—
344	C₄⋊C₄	—	212	2	no	—
345	C₄⋊C₄	D₄×D₅₃	1	1	yes	characteristic
346	C₄²	—	2	1	no	—
347	C₄²	—	53	1	no	—
348	C₄²	—	53	1	no	—
349	C₄²	—	53	1	no	—
350	C₄²	—	106	1	no	—
351	C₄²	—	106	1	no	—
352	C₄²	—	106	1	no	—
353	C₄²	C₂³×D₅₃	1	1	yes	characteristic
354	C₂³	—	53	1	no	—
355	Q₈	—	2	2	no	—
356	Q₈	—	8	2	no	—
357	Q₈	—	106	2	no	—
358	Q₈	—	424	2	no	—
359	Q₈	—	424	4	no	—
360	Q₈	—	212	2	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	53	1	no	—
365	C₂×C₄	—	53	1	no	—
366	C₂×C₄	—	53	2	no	—
367	C₂×C₄	—	53	2	no	—
368	C₂×C₄	—	53	4	no	—
369	C₂×C₄	—	53	1	no	—
370	C₂×C₄	—	106	2	no	—
371	C₂×C₄	—	106	1	no	—
372	C₂×C₄	—	106	1	no	—
373	C₂×C₄	—	424	4	no	—
374	C₂×C₄	—	212	2	no	—
375	C₂×C₄	—	212	2	no	—
376	C₂×C₄	—	212	2	no	—
377	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
378	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
379	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
380	C₈	—	2	1	no	—
381	C₈	—	4	2	no	—
382	C₈	—	4	1	no	—
383	C₈	—	53	2	no	—
384	C₈	—	106	1	no	—
385	C₈	—	212	2	no	—
386	C₈	—	212	1	no	—
387	C₈	Q₁₆×D₅₃	1	2	yes	—
388	C₂²	—	53	2	no	—
389	C₂²	—	53	2	no	—
390	C₂²	—	53	2	no	—
391	C₂²	#121×D₅₃	1	1	yes	characteristic
392	C₄	—	2	1	no	—
393	C₄	—	4	1	no	—
394	C₄	—	4	1	no	—
395	C₄	—	8	2	no	—
396	C₄	—	53	2	no	—
397	C₄	—	53	2	no	—
398	C₄	—	106	1	no	—
399	C₄	—	424	2	no	—
400	C₄	—	212	1	no	—
401	C₄	—	212	1	no	—
402	C₄	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
403	C₄	#137×D₅₃	1	2	yes	—
404	C₄	#137×D₅₃	1	1	yes	characteristic
405	C₂	—	53	4	no	—
406	C₂	#462×D₅₃	1	1	yes	characteristic
407	C₂	#470×D₅₃	1	1	yes	characteristic
408	C₂	#472×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1696	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂²×D₅₃	#121
hypercenter	#1696	D₅₃
hypocenter	C₅₃	C₂×#1696
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1696

Complements

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

N	H ≅ G/N	type
4. #20274244	C₂	semidirect
5. #20274243	C₂	semidirect
6. #20274242	C₂	semidirect
7. #20274241	C₂	semidirect
8. #20274240	C₂	semidirect
11. C₅₃×#1696	C₂	semidirect
209. #1696	D₅₃	direct (N × H)
215. D₅₃	#1696	direct (N × H)
251. C₅₃	C₂×#1696	semidirect

Maximal subgroups

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

subgroup	index
#500×D₅₃	2
#477×D₅₃	2
#446×D₅₃	2
#20274244	2
#20274243	2
#20274242	2
#20274241	2
#20274240	2
#334×D₅₃	2
#333×D₅₃	2
C₅₃×#1696	2
C₂×#1696	53

Subgroups — D53×#1696 #27366722

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#1696 #27366722