Subgroups — #27349805

Order 13568.

806 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283346	C₂	1	2	yes	maximal
2	C₄×M₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#20273018	C₂	1	1	yes	maximal, characteristic, complemented
4	#20273018	C₂	1	2	yes	maximal, complemented
5	#20273013	C₂	1	1	yes	maximal, characteristic, complemented
6	#20273011	C₂	1	2	yes	maximal, complemented
7	#20273009	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273006	C₂	1	2	yes	maximal, complemented
9	#20273000	C₂	1	1	yes	maximal, characteristic, complemented
10	#20272997	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1455	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×M₁₆×D₅₃	C₄	1	2	yes	—
13	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
14	#16810678	C₄	1	4	yes	—
15	#16810678	C₄	1	4	yes	—
16	#16810674	C₂²	1	2	yes	—
17	#16810674	C₂²	1	2	yes	—
18	#16810674	C₂²	1	2	yes	—
19	#16810674	C₂²	1	2	yes	—
20	#16810646	C₄	1	4	yes	—
21	#16810586	C₄	1	4	yes	—
22	C₄×#11719917	C₂²	1	2	yes	—
23	C₄×#11719917	C₂²	1	1	yes	characteristic
24	#16810575	C₄	1	2	yes	—
25	M₁₆×Dic₅₃	C₄	1	2	yes	—
26	C₄×#11719912	C₂²	1	1	yes	characteristic
27	C₄×#11719912	C₂²	1	1	yes	characteristic
28	#97×D₅₃	C₄	1	2	yes	—
29	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
31	#16810345	C₂²	1	2	yes	complemented
32	#16810345	C₂²	1	2	yes	complemented
33	#16810345	C₂²	1	2	yes	complemented
34	#16810344	C₂²	1	1	yes	characteristic, complemented
35	#16810344	C₂²	1	2	yes	complemented
36	C₈×#88322	C₂²	1	1	yes	characteristic, complemented
37	C₈×#88322	C₂²	1	2	yes	complemented
38	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
39	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#322	C₂²	1	1	yes	characteristic, complemented
41	C₅₃×#322	C₂²	1	2	yes	complemented
42	C₈²×C₅₃	C₂²	1	1	yes	characteristic, complemented
43	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×D₅₃	C₂×C₄	1	2	yes	—
45	C₂×#11719917	C₂×C₄	1	2	yes	—
46	C₂×#11719917	C₂×C₄	1	4	yes	—
47	C₂×#11719912	C₂×C₄	1	2	yes	—
48	C₂×#11719912	C₂×C₄	1	2	yes	—
49	M₁₆×D₅₃	—	2	8	no	—
50	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
51	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
54	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
55	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
56	C₄²×D₅₃	C₂×C₄	1	2	yes	—
57	C₄²×D₅₃	C₂×C₄	1	2	yes	—
58	C₄²×D₅₃	C₂³	1	1	yes	characteristic
59	C₄²×D₅₃	C₂³	1	1	yes	characteristic
60	#13133146	C₂×C₄	1	4	yes	—
61	#13133146	C₂×C₄	1	4	yes	—
62	#13133146	C₂×C₄	1	4	yes	—
63	#13133146	C₂×C₄	1	4	yes	—
64	#13133129	C₂×C₄	1	4	yes	—
65	#13133129	C₂×C₄	1	4	yes	—
66	#13133129	C₂×C₄	1	4	yes	—
67	#13133129	C₂×C₄	1	4	yes	—
68	#13133128	C₂×C₄	1	2	yes	—
69	#13133128	C₂×C₄	1	4	yes	—
70	#13133127	C₂×C₄	1	2	yes	—
71	#13133127	C₂×C₄	1	2	yes	—
72	C₄×#88322	C₂³	1	1	yes	characteristic
73	C₄×#88322	C₂³	1	1	yes	characteristic
74	C₄×#88322	C₂³	1	2	yes	—
75	C₄×#88322	C₂³	1	2	yes	—
76	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
77	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
78	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
79	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
80	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
81	#97×C₅₃	C₂×C₄	1	2	yes	—
82	#97×C₅₃	C₂×C₄	1	4	yes	—
83	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₅₃	C₂³	1	2	yes	—
86	C₄×C₈×C₅₃	C₂³	1	2	yes	—
87	C₂³×D₅₃	C₄²	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
90	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
91	C₂²×Dic₅₃	C₄²	1	2	yes	—
92	C₂×C₄×D₅₃	—	2	4	no	—
93	C₂×C₄×D₅₃	—	4	4	no	—
94	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
95	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
96	C₂×C₄×D₅₃	C₄²	1	2	yes	—
97	C₂×C₄×D₅₃	C₄²	1	2	yes	—
98	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
99	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
100	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
102	M₁₆×C₅₃	—	2	4	no	—
103	M₁₆×C₅₃	—	4	4	no	—
104	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
105	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
106	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
107	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
108	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
110	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
111	C₄×Dic₅₃	C₄²	1	2	yes	—
112	C₄×Dic₅₃	C₄²	1	2	yes	—
113	C₄×Dic₅₃	C₄²	1	2	yes	—
114	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
117	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
118	#11719917	—	2	4	no	—
119	#11719917	—	4	4	no	—
120	C₂×#88322	C₂²×C₄	1	2	yes	—
121	C₂×#88322	C₂²×C₄	1	4	yes	—
122	C₂×#88322	C₂²×C₄	1	4	yes	—
123	C₂×#88322	C₂²×C₄	1	2	yes	—
124	#11719912	—	2	4	no	—
125	#11719912	—	2	4	no	—
126	C₈×D₅₃	—	2	4	no	—
127	C₈×D₅₃	M₁₆	1	8	yes	complemented
128	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
129	C₂²×D₅₃	—	2	2	no	—
130	C₂²×D₅₃	—	4	2	no	—
131	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
132	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	4	no	—
135	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
137	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
139	C₂×Dic₅₃	—	2	2	no	—
140	C₂×Dic₅₃	—	4	2	no	—
141	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
142	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
143	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
144	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
145	C₄×D₅₃	—	2	2	no	—
146	C₄×D₅₃	—	2	2	no	—
147	C₄×D₅₃	—	2	4	no	—
148	C₄×D₅₃	—	2	4	no	—
149	C₄×D₅₃	—	2	2	no	—
150	C₄×D₅₃	—	4	2	no	—
151	C₄×D₅₃	—	4	2	no	—
152	C₄×D₅₃	C₂×M₁₆	1	4	yes	—
153	C₈×C₅₃	—	2	2	no	—
154	C₈×C₅₃	—	2	4	no	—
155	C₈×C₅₃	—	2	4	no	—
156	C₈×C₅₃	C₂×M₁₆	1	4	yes	—
157	#88322	—	2	2	no	—
158	#88322	—	2	4	no	—
159	#88322	—	2	4	no	—
160	#88322	C₂×M₁₆	1	4	yes	—
161	#17334	—	53	1	no	maximal, Sylow
162	C₂²×C₅₃	—	2	1	no	—
163	C₂²×C₅₃	—	2	2	no	—
164	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
165	C₂×D₅₃	—	2	2	no	—
166	C₂×D₅₃	—	2	1	no	—
167	C₂×D₅₃	—	2	2	no	—
168	C₂×D₅₃	—	4	4	no	—
169	C₂×D₅₃	C₄×M₁₆	1	2	yes	—
170	C₄×C₅₃	—	2	1	no	—
171	C₄×C₅₃	—	2	2	no	—
172	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
173	C₄×C₅₃	#568	1	4	yes	—
174	Dic₅₃	—	2	1	no	—
175	Dic₅₃	—	2	2	no	—
176	Dic₅₃	—	2	2	no	—
177	Dic₅₃	C₄×M₁₆	1	2	yes	—
178	#2920	—	53	2	no	—
179	C₂×C₄×M₁₆	—	53	1	no	—
180	#1455	—	53	1	no	—
181	#1455	—	53	2	no	—
182	#1455	D₅₃	1	1	yes	characteristic, complemented
183	#1453	—	53	2	no	—
184	#1453	—	53	1	no	—
185	#1453	—	53	1	no	—
186	#1452	—	53	2	no	—
187	#1450	—	53	1	no	—
188	#1450	—	53	1	no	—
189	C₂×C₅₃	—	4	1	no	—
190	C₂×C₅₃	C₂×C₄×M₁₆	1	1	yes	characteristic
191	C₂×C₅₃	#2873	1	2	yes	—
192	D₅₃	—	2	2	no	—
193	D₅₃	—	4	1	no	—
194	C₂²×M₁₆	—	53	2	no	—
195	C₂²×C₄²	—	53	1	no	—
196	#432	—	53	4	no	—
197	#432	—	53	4	no	—
198	#431	—	53	2	no	—
199	#431	—	53	2	no	—
200	#431	—	53	2	no	—
201	#431	—	53	2	no	—
202	#423	—	53	4	no	—
203	#407	—	53	4	no	—
204	C₄×M₁₆	—	53	2	no	—
205	C₄×M₁₆	—	53	1	no	—
206	C₄×M₁₆	—	53	2	no	—
207	C₄×M₁₆	—	53	2	no	—
208	C₄×M₁₆	—	53	1	no	—
209	C₄×M₁₆	—	53	1	no	—
210	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
211	C₄×M₁₆	C₂×D₅₃	1	2	yes	—
212	C₂×#97	—	53	2	no	—
213	C₂×C₄×C₈	—	53	1	no	—
214	C₂×C₄×C₈	—	53	1	no	—
215	#322	—	53	1	no	—
216	#322	—	53	2	no	—
217	#322	—	53	2	no	—
218	#322	—	53	2	no	—
219	#322	—	53	2	no	—
220	#322	C₂×D₅₃	1	1	yes	characteristic, complemented
221	#322	C₂×D₅₃	1	2	yes	complemented
222	C₈²	—	53	2	no	—
223	C₈²	—	53	1	no	—
224	C₈²	C₂×D₅₃	1	1	yes	characteristic, complemented
225	C₅₃	#17334	1	1	yes	characteristic, complemented, Sylow
226	C₂³×C₄	—	53	1	no	—
227	C₂³×C₄	—	53	2	no	—
228	C₂×M₁₆	—	53	2	no	—
229	C₂×M₁₆	—	53	4	no	—
230	C₂×M₁₆	—	53	2	no	—
231	C₂×M₁₆	—	53	2	no	—
232	C₂×M₁₆	—	106	8	no	—
233	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
234	C₂×M₁₆	C₄×D₅₃	1	4	yes	—
235	C₂²×C₈	—	53	2	no	—
236	C₂²×C₈	—	53	2	no	—
237	C₂×C₄²	—	53	1	no	—
238	C₂×C₄²	—	53	2	no	—
239	C₂×C₄²	—	53	1	no	—
240	C₂×C₄²	—	53	1	no	—
241	C₂×C₄²	—	53	2	no	—
242	C₂×C₄²	—	53	2	no	—
243	C₂×C₄²	—	53	1	no	—
244	C₂×C₄²	—	53	1	no	—
245	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
246	C₄⋊C₈	—	53	4	no	—
247	C₄⋊C₈	—	53	4	no	—
248	C₄⋊C₈	—	53	4	no	—
249	C₄⋊C₈	—	53	4	no	—
250	C₂²⋊C₈	—	53	4	no	—
251	C₂²⋊C₈	—	53	4	no	—
252	C₂²⋊C₈	—	53	4	no	—
253	C₂²⋊C₈	—	53	4	no	—
254	#97	—	53	2	no	—
255	#97	—	53	2	no	—
256	#97	—	53	2	no	—
257	#97	—	53	4	no	—
258	#97	C₄×D₅₃	1	2	yes	—
259	#97	C₄×D₅₃	1	4	yes	—
260	C₄×C₈	—	53	2	no	—
261	C₄×C₈	—	53	2	no	—
262	C₄×C₈	—	53	1	no	—
263	C₄×C₈	—	53	2	no	—
264	C₄×C₈	—	53	2	no	—
265	C₄×C₈	—	53	1	no	—
266	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₅₃	1	2	yes	—
268	C₄×C₈	C₂²×D₅₃	1	2	yes	—
269	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
270	C₂⁴	—	53	1	no	—
271	C₂²×C₄	—	53	1	no	—
272	C₂²×C₄	—	53	1	no	—
273	C₂²×C₄	—	53	2	no	—
274	C₂²×C₄	—	53	1	no	—
275	C₂²×C₄	—	53	1	no	—
276	C₂²×C₄	—	53	2	no	—
277	C₂²×C₄	—	53	2	no	—
278	C₂²×C₄	—	53	1	no	—
279	C₂²×C₄	—	53	2	no	—
280	C₂²×C₄	—	53	2	no	—
281	C₂²×C₄	—	106	4	no	—
282	C₂²×C₄	—	212	4	no	—
283	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
284	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
285	M₁₆	—	2	4	no	—
286	M₁₆	—	4	4	no	—
287	M₁₆	—	106	4	no	—
288	M₁₆	—	106	4	no	—
289	M₁₆	—	106	4	no	—
290	M₁₆	—	212	4	no	—
291	C₂×C₈	—	53	2	no	—
292	C₂×C₈	—	53	8	no	—
293	C₂×C₈	—	53	2	no	—
294	C₂×C₈	—	53	4	no	—
295	C₂×C₈	—	53	4	no	—
296	C₂×C₈	—	106	4	no	—
297	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
298	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
299	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
300	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
301	C₄²	—	53	2	no	—
302	C₄²	—	53	2	no	—
303	C₄²	—	53	2	no	—
304	C₄²	—	53	1	no	—
305	C₄²	—	53	1	no	—
306	C₄²	—	53	2	no	—
307	C₄²	—	53	2	no	—
308	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
309	C₄²	C₂×C₄×D₅₃	1	2	yes	—
310	C₄²	C₂³×D₅₃	1	1	yes	characteristic
311	C₂³	—	53	1	no	—
312	C₂³	—	53	1	no	—
313	C₂³	—	106	2	no	—
314	C₂³	—	212	2	no	—
315	C₂³	C₄²×D₅₃	1	1	yes	characteristic
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	53	1	no	—
319	C₂×C₄	—	53	4	no	—
320	C₂×C₄	—	53	1	no	—
321	C₂×C₄	—	53	2	no	—
322	C₂×C₄	—	53	2	no	—
323	C₂×C₄	—	106	2	no	—
324	C₂×C₄	—	106	2	no	—
325	C₂×C₄	—	106	2	no	—
326	C₂×C₄	—	106	2	no	—
327	C₂×C₄	—	106	4	no	—
328	C₂×C₄	—	106	4	no	—
329	C₂×C₄	—	212	2	no	—
330	C₂×C₄	—	212	2	no	—
331	C₂×C₄	—	212	2	no	—
332	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
333	C₂×C₄	C₄²×D₅₃	1	2	yes	—
334	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
335	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
336	C₈	—	2	2	no	—
337	C₈	—	2	4	no	—
338	C₈	—	2	4	no	—
339	C₈	—	53	4	no	—
340	C₈	—	106	2	no	—
341	C₈	—	106	4	no	—
342	C₈	—	106	4	no	—
343	C₈	M₁₆×D₅₃	1	4	yes	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	2	no	—
346	C₂²	—	53	2	no	—
347	C₂²	—	106	2	no	—
348	C₂²	—	106	1	no	—
349	C₂²	—	106	2	no	—
350	C₂²	—	212	4	no	—
351	C₂²	C₂×C₄²×D₅₃	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	2	2	no	—
354	C₄	—	53	2	no	—
355	C₄	—	106	1	no	—
356	C₄	—	106	2	no	—
357	C₄	—	106	2	no	—
358	C₄	C₂×M₁₆×D₅₃	1	2	yes	—
359	C₄	#16811389	1	4	yes	—
360	C₂	—	4	1	no	—
361	C₂	—	106	2	no	—
362	C₂	—	212	1	no	—
363	C₂	C₄×M₁₆×D₅₃	1	1	yes	characteristic
364	C₂	#20283132	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₄²
Fitting	C₅₃×#1455	C₂
prime radical	C₂³×D₅₃	C₄²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1455	D₅₃
hypocenter	C₅₃	#17334
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ #1455

Complements

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

N	H ≅ G/N	type
3. #20273018	C₂	semidirect
4. #20273018	C₂	semidirect
5. #20273013	C₂	semidirect
6. #20273011	C₂	semidirect
7. #20273009	C₂	semidirect
8. #20273006	C₂	semidirect
9. #20273000	C₂	semidirect
10. #20272997	C₂	semidirect
11. C₅₃×#1455	C₂	semidirect
31. #16810345	C₂²	semidirect
32. #16810345	C₂²	semidirect
33. #16810345	C₂²	semidirect
34. #16810344	C₂²	semidirect
35. #16810344	C₂²	semidirect
36. C₈×#88322	C₂²	semidirect
37. C₈×#88322	C₂²	semidirect
40. C₅₃×#322	C₂²	semidirect
41. C₅₃×#322	C₂²	semidirect
42. C₈²×C₅₃	C₂²	semidirect
127. C₈×D₅₃	M₁₆	semidirect
182. #1455	D₅₃	semidirect
220. #322	C₂×D₅₃	semidirect
221. #322	C₂×D₅₃	semidirect
224. C₈²	C₂×D₅₃	semidirect
225. C₅₃	#17334	semidirect

Maximal subgroups

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

subgroup	index
#20283346	2
C₄×M₁₆×D₅₃	2
#20273018	2
#20273013	2
#20273011	2
#20273009	2
#20273006	2
#20273000	2
#20272997	2
C₅₃×#1455	2
#17334	53