Subgroups — #27353563

Order 13568.

678 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283794	C₂	1	2	yes	maximal
2	C₄×Q₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#20273535	C₂	1	1	yes	maximal, characteristic, complemented
4	#20273534	C₂	1	1	yes	maximal, characteristic, complemented
5	#20273533	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#11719925	C₂	1	2	yes	maximal, complemented
7	#20273519	C₂	1	2	yes	maximal, complemented
8	#20273460	C₂	1	1	yes	maximal, characteristic
9	#20273108	C₂	1	2	yes	maximal
10	#20273000	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#1579	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₅₃	C₄	1	1	yes	characteristic
13	C₄×Q₈×D₅₃	C₂²	1	2	yes	—
14	Q₈×#88322	C₂²	1	2	yes	—
15	C₈×Dic₁₀₆	—	2	2	no	—
16	C₈×Dic₁₀₆	C₂²	1	2	yes	—
17	#16810747	—	2	2	no	—
18	#16810747	C₂²	1	2	yes	—
19	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
20	Q₁₆×Dic₅₃	C₄	1	1	yes	characteristic
21	#16810717	C₄	1	1	yes	characteristic
22	C₄×#11719925	C₂²	1	2	yes	—
23	#16810715	C₄	1	2	yes	—
24	#16810675	—	2	2	no	—
25	#16810675	C₂²	1	2	yes	—
26	#16810674	—	2	2	no	—
27	#16810674	C₂²	1	1	yes	characteristic
28	#16810674	C₂²	1	1	yes	characteristic
29	#107×D₅₃	C₄	1	1	yes	characteristic
30	#103×D₅₃	C₄	1	2	yes	—
31	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
32	#16810375	C₂²	1	2	yes	—
33	#16810374	C₂²	1	1	yes	characteristic
34	#16810355	C₂²	1	2	yes	—
35	Q₈⋊#88322	C₂²	1	2	yes	—
36	#16810353	C₂²	1	2	yes	—
37	#16810344	C₂²	1	1	yes	characteristic
38	C₈×#88322	C₂²	1	1	yes	characteristic
39	C₈×#88322	C₂²	1	1	yes	characteristic
40	C₈×Q₈×C₅₃	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×Q₈×D₅₃	C₂×C₄	1	2	yes	—
46	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
47	C₂×#11719925	C₂×C₄	1	2	yes	—
48	Q₁₆×D₅₃	C₈	1	8	yes	complemented
49	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
51	Q₈×Dic₅₃	C₂×C₄	1	2	yes	—
52	C₄×Dic₁₀₆	—	2	2	no	—
53	C₄×Dic₁₀₆	C₂³	1	2	yes	—
54	#13133182	—	2	2	no	—
55	#13133182	C₂×C₄	1	2	yes	—
56	C₄⋊C₄×D₅₃	—	2	2	no	—
57	C₄⋊C₄×D₅₃	C₂×C₄	1	2	yes	—
58	C₄²×D₅₃	—	2	2	no	—
59	C₄²×D₅₃	C₂³	1	1	yes	characteristic
60	#13133151	C₂×C₄	1	1	yes	characteristic
61	#13133150	C₂×C₄	1	2	yes	—
62	C₄⋊#88322	—	2	2	no	—
63	C₄⋊#88322	C₂³	1	2	yes	—
64	#13133146	—	2	2	no	—
65	#13133146	—	2	1	no	—
66	#13133146	—	2	2	no	—
67	#13133146	D₄	1	2	yes	—
68	#13133142	C₂×C₄	1	2	yes	—
69	#13133141	C₂×C₄	1	2	yes	—
70	#13133140	C₂×C₄	1	2	yes	—
71	#13133129	—	2	2	no	—
72	#13133129	—	2	1	no	—
73	#13133129	D₄	1	2	yes	—
74	C₄×#88322	—	2	2	no	—
75	C₄×#88322	C₂³	1	1	yes	characteristic
76	C₄×#88322	C₂³	1	1	yes	characteristic
77	C₄×#88322	C₂³	1	1	yes	characteristic
78	C₈×Dic₅₃	—	2	2	no	—
79	C₈×Dic₅₃	—	2	2	no	—
80	C₈×Dic₅₃	—	2	2	no	—
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	C₄×C₈×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	Q₈×D₅₃	—	2	4	no	—
95	Q₈×D₅₃	C₂×C₈	1	8	yes	—
96	C₂×C₄×D₅₃	—	2	2	no	—
97	C₂×C₄×D₅₃	—	2	2	no	—
98	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
101	C₂×Dic₁₀₆	—	2	2	no	—
102	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
103	Q₁₆×C₅₃	C₂×C₈	1	4	yes	—
104	C₂×C₈×C₅₃	—	2	2	no	—
105	C₂×C₈×C₅₃	—	2	1	no	—
106	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
109	C₄⋊C₄×C₅₃	—	2	2	no	—
110	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
111	C₄²×C₅₃	—	2	2	no	—
112	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
113	#11719925	C₂×C₈	1	8	yes	—
114	C₄⋊Dic₅₃	—	2	2	no	—
115	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
116	#11719919	—	2	2	no	—
117	#11719919	—	2	2	no	—
118	#11719919	—	2	2	no	—
119	C₄×Dic₅₃	—	2	2	no	—
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	Dic₂₁₂	C₂×C₈	1	4	yes	—
131	C₈×D₅₃	—	2	2	no	—
132	C₈×D₅₃	C₂×C₈	1	4	yes	—
133	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
134	Q₈×C₅₃	—	2	2	no	—
135	Q₈×C₅₃	C₂²×C₈	1	4	yes	—
136	C₂×C₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
141	C₂×Dic₅₃	—	2	2	no	—
142	C₂×Dic₅₃	—	2	2	no	—
143	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
144	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
145	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
146	C₄×D₅₃	—	2	8	no	—
147	C₄×D₅₃	—	2	2	no	—
148	C₄×D₅₃	—	4	1	no	—
149	C₄×D₅₃	—	8	2	no	—
150	C₄×D₅₃	C₂²×C₈	1	2	yes	—
151	C₄×D₅₃	#131	1	2	yes	—
152	Dic₁₀₆	—	2	2	no	—
153	Dic₁₀₆	—	2	8	no	—
154	Dic₁₀₆	C₂²×C₈	1	4	yes	—
155	C₈×C₅₃	—	2	1	no	—
156	C₈×C₅₃	—	2	2	no	—
157	C₈×C₅₃	—	4	1	no	—
158	C₈×C₅₃	—	4	2	no	—
159	C₈×C₅₃	C₂²×C₈	1	2	yes	—
160	#88322	—	2	1	no	—
161	#88322	—	2	2	no	—
162	#88322	—	4	1	no	—
163	#88322	—	4	2	no	—
164	#88322	C₂²×C₈	1	2	yes	—
165	#17856	—	53	1	no	maximal, Sylow
166	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂×D₅₃	—	2	1	no	—
168	C₂×D₅₃	—	2	1	no	—
169	C₂×D₅₃	C₈×D₄	1	2	yes	—
170	C₄×C₅₃	—	2	4	no	—
171	C₄×C₅₃	—	2	1	no	—
172	C₄×C₅₃	—	4	2	no	—
173	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
174	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
175	C₄×C₅₃	#578	1	2	yes	—
176	Dic₅₃	—	2	4	no	—
177	Dic₅₃	—	2	1	no	—
178	Dic₅₃	—	2	1	no	—
179	Dic₅₃	—	4	2	no	—
180	Dic₅₃	C₈×D₄	1	2	yes	—
181	#2982	—	53	2	no	—
182	C₂×C₄×Q₁₆	—	53	1	no	—
183	#1579	—	53	1	no	—
184	#1579	—	53	1	no	—
185	#1579	—	53	1	no	—
186	#1579	D₅₃	1	1	yes	characteristic, complemented
187	C₈×Q₁₆	—	53	2	no	—
188	C₈×Q₁₆	—	53	2	no	—
189	#1564	—	53	1	no	—
190	#1479	—	53	2	no	—
191	#1450	—	53	1	no	—
192	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
193	C₂×C₅₃	#2950	1	1	yes	characteristic
194	C₂×C₅₃	#2957	1	1	yes	characteristic
195	D₅₃	—	2	2	no	—
196	C₂²×Q₁₆	—	53	1	no	—
197	C₂×C₄×Q₈	—	53	2	no	—
198	C₈×Q₈	—	53	2	no	—
199	C₈×Q₈	—	53	2	no	—
200	C₈×Q₈	—	53	2	no	—
201	C₈×Q₈	—	106	2	no	—
202	C₈×Q₈	—	106	2	no	—
203	C₈×Q₈	C₂×D₅₃	1	2	yes	—
204	C₄×Q₁₆	—	53	1	no	—
205	C₄×Q₁₆	—	53	2	no	—
206	C₄×Q₁₆	—	53	1	no	—
207	C₄×Q₁₆	—	53	2	no	—
208	C₄×Q₁₆	—	53	1	no	—
209	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
210	#431	—	53	1	no	—
211	#431	—	53	1	no	—
212	#431	—	53	2	no	—
213	#431	—	106	2	no	—
214	#431	—	106	2	no	—
215	C₂×#107	—	53	1	no	—
216	C₂×#103	—	53	2	no	—
217	C₂×C₄×C₈	—	53	1	no	—
218	#335	—	53	2	no	—
219	#334	—	53	1	no	—
220	#334	C₂×D₅₃	1	1	yes	characteristic
221	Q₈⋊C₈	—	53	2	no	—
222	Q₈⋊C₈	—	53	2	no	—
223	Q₈⋊C₈	—	53	2	no	—
224	Q₈⋊C₈	C₂×D₅₃	1	2	yes	—
225	#322	—	53	1	no	—
226	#322	C₂×D₅₃	1	1	yes	characteristic
227	C₈²	—	53	1	no	—
228	C₈²	—	53	1	no	—
229	C₅₃	#17856	1	1	yes	characteristic, complemented, Sylow
230	C₂²×Q₈	—	53	2	no	—
231	C₂×Q₁₆	—	53	2	no	—
232	C₂×Q₁₆	—	53	8	no	—
233	C₂×Q₁₆	—	53	1	no	—
234	C₂×Q₁₆	C₄×D₅₃	1	1	yes	characteristic
235	C₂²×C₈	—	53	1	no	—
236	C₂²×C₈	—	53	1	no	—
237	C₄×Q₈	—	53	2	no	—
238	C₄×Q₈	—	53	2	no	—
239	C₄×Q₈	—	53	2	no	—
240	C₄×Q₈	—	106	2	no	—
241	C₄×Q₈	—	106	2	no	—
242	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
243	C₂×C₄⋊C₄	—	53	2	no	—
244	C₂×C₄⋊C₄	—	106	2	no	—
245	C₂×C₄²	—	53	1	no	—
246	C₂×C₄²	—	106	2	no	—
247	#107	—	53	1	no	—
248	#107	—	53	2	no	—
249	#107	C₄×D₅₃	1	1	yes	characteristic
250	C₄⋊C₈	—	2	2	no	—
251	C₄⋊C₈	—	53	2	no	—
252	C₄⋊C₈	—	53	2	no	—
253	C₄⋊C₈	—	106	2	no	—
254	C₄⋊C₈	—	106	1	no	—
255	C₄⋊C₈	—	106	2	no	—
256	C₄⋊C₈	—	106	2	no	—
257	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
258	#103	—	53	2	no	—
259	#103	—	53	2	no	—
260	#103	—	53	2	no	—
261	#103	C₄×D₅₃	1	2	yes	—
262	C₂²⋊C₈	—	53	2	no	—
263	C₂²⋊C₈	—	106	1	no	—
264	C₂²⋊C₈	—	106	2	no	—
265	C₄×C₈	—	2	2	no	—
266	C₄×C₈	—	53	1	no	—
267	C₄×C₈	—	53	1	no	—
268	C₄×C₈	—	53	1	no	—
269	C₄×C₈	—	53	1	no	—
270	C₄×C₈	—	53	1	no	—
271	C₄×C₈	—	106	2	no	—
272	C₄×C₈	—	106	2	no	—
273	C₄×C₈	—	106	2	no	—
274	C₄×C₈	—	106	2	no	—
275	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
276	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
277	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
278	C₂×Q₈	—	53	8	no	—
279	C₂×Q₈	—	53	2	no	—
280	C₂×Q₈	—	106	2	no	—
281	C₂×Q₈	—	106	4	no	—
282	C₂×Q₈	C₂×C₄×D₅₃	1	2	yes	—
283	C₂²×C₄	—	53	1	no	—
284	C₂²×C₄	—	53	1	no	—
285	C₂²×C₄	—	53	1	no	—
286	C₂²×C₄	—	106	2	no	—
287	C₂²×C₄	—	106	2	no	—
288	Q₁₆	—	53	8	no	—
289	Q₁₆	—	53	4	no	—
290	Q₁₆	C₈×D₅₃	1	4	yes	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	53	1	no	—
294	C₂×C₈	—	53	4	no	—
295	C₂×C₈	—	53	1	no	—
296	C₂×C₈	—	53	2	no	—
297	C₂×C₈	—	106	2	no	—
298	C₂×C₈	—	106	2	no	—
299	C₂×C₈	—	106	1	no	—
300	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
301	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
302	C₂×C₈	D₄×D₅₃	1	2	yes	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	53	2	no	—
305	C₄⋊C₄	—	106	2	no	—
306	C₄⋊C₄	—	106	2	no	—
307	C₄⋊C₄	—	106	2	no	—
308	C₄⋊C₄	—	106	2	no	—
309	C₄⋊C₄	C₂×C₄×D₅₃	1	2	yes	—
310	C₄²	—	2	2	no	—
311	C₄²	—	53	1	no	—
312	C₄²	—	53	1	no	—
313	C₄²	—	53	1	no	—
314	C₄²	—	106	2	no	—
315	C₄²	—	106	2	no	—
316	C₄²	—	106	2	no	—
317	C₄²	C₂³×D₅₃	1	1	yes	characteristic
318	C₂³	—	53	1	no	—
319	Q₈	—	2	2	no	—
320	Q₈	—	53	4	no	—
321	Q₈	—	106	8	no	—
322	Q₈	—	106	2	no	—
323	Q₈	C₂×C₈×D₅₃	1	4	yes	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	53	1	no	—
327	C₂×C₄	—	53	2	no	—
328	C₂×C₄	—	53	2	no	—
329	C₂×C₄	—	53	1	no	—
330	C₂×C₄	—	53	1	no	—
331	C₂×C₄	—	106	8	no	—
332	C₂×C₄	—	106	2	no	—
333	C₂×C₄	—	106	2	no	—
334	C₂×C₄	—	106	2	no	—
335	C₂×C₄	—	424	2	no	—
336	C₂×C₄	—	212	1	no	—
337	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
338	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
339	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
340	C₈	—	2	1	no	—
341	C₈	—	2	2	no	—
342	C₈	—	4	1	no	—
343	C₈	—	4	2	no	—
344	C₈	—	53	2	no	—
345	C₈	—	106	1	no	—
346	C₈	—	106	2	no	—
347	C₈	—	212	1	no	—
348	C₈	—	212	2	no	—
349	C₈	C₂×C₈×D₅₃	1	2	yes	—
350	C₂²	—	53	2	no	—
351	C₂²	—	106	1	no	—
352	C₂²	—	106	1	no	—
353	C₂²	C₄×D₄×D₅₃	1	1	yes	characteristic
354	C₄	—	2	1	no	—
355	C₄	—	2	4	no	—
356	C₄	—	4	2	no	—
357	C₄	—	53	2	no	—
358	C₄	—	106	1	no	—
359	C₄	—	106	1	no	—
360	C₄	—	106	4	no	—
361	C₄	—	212	2	no	—
362	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
363	C₄	#131×D₅₃	1	1	yes	characteristic
364	C₄	#16811439	1	2	yes	—
365	C₂	—	106	2	no	—
366	C₂	C₈×D₄×D₅₃	1	1	yes	characteristic
367	C₂	#20283580	1	1	yes	characteristic
368	C₂	#20283629	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1579	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂²×D₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1579	D₅₃
hypocenter	C₅₃	#17856
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1579

Complements

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

N	H ≅ G/N	type
3. #20273535	C₂	semidirect
4. #20273534	C₂	semidirect
5. #20273533	C₂	semidirect
6. C₈×#11719925	C₂	semidirect
7. #20273519	C₂	semidirect
11. C₅₃×#1579	C₂	semidirect
48. Q₁₆×D₅₃	C₈	semidirect
186. #1579	D₅₃	semidirect
229. C₅₃	#17856	semidirect

Maximal subgroups

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

subgroup	index
#20283794	2
C₄×Q₁₆×D₅₃	2
#20273535	2
#20273534	2
#20273533	2
C₈×#11719925	2
#20273519	2
#20273460	2
#20273108	2
#20273000	2
C₅₃×#1579	2
#17856	53