Subgroups — #27299017

Order 13568.

570 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#108×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810660	C₂	1	1	yes	maximal, characteristic
4	#20272825	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272824	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272823	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1387	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₅₃	—	2	1	no	—
9	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
10	C₂×#13133175	—	2	1	no	—
11	C₂×#13133157	C₂²	1	1	yes	characteristic
12	C₂×#13133156	—	2	1	no	—
13	M₃₂×D₅₃	—	2	2	no	—
14	M₃₂×D₅₃	C₄	1	4	yes	complemented
15	C₂×C₁₆×D₅₃	—	2	1	no	—
16	C₂×#13133153	C₂²	1	1	yes	characteristic
17	C₂×#13133152	—	2	1	no	—
18	#108×D₅₃	—	2	2	no	—
19	#108×D₅₃	C₄	1	4	yes	complemented
20	#16810660	—	2	1	no	—
21	#16810660	—	2	1	no	—
22	#16810660	C₄	1	2	yes	—
23	#16810660	C₄	1	2	yes	—
24	C₂×#13133151	C₂²	1	1	yes	characteristic
25	C₂×#13133148	—	2	1	no	—
26	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
29	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
30	C₂×#11719917	—	2	1	no	—
31	C₂×#11719912	—	4	1	no	—
32	M₁₆×D₅₃	—	2	4	no	—
33	M₁₆×D₅₃	—	4	2	no	—
34	C₂²×#88322	D₄	1	1	yes	characteristic
35	C₂×C₈×D₅₃	—	4	1	no	—
36	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
41	C₂×C₈×D₅₃	D₄	1	2	yes	—
42	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
43	C₂×C₈×D₅₃	Q₈	1	2	yes	—
44	#13133175	—	2	2	no	—
45	#13133175	—	2	2	no	—
46	#13133175	—	4	1	no	—
47	#13133175	—	4	1	no	—
48	C₂×C₄⋊Dic₅₃	—	2	1	no	—
49	C₂×#11719921	—	4	1	no	—
50	C₂×C₄×Dic₅₃	—	4	1	no	—
51	#13133157	—	2	1	no	—
52	#13133157	C₂×C₄	1	2	yes	—
53	#13133156	—	2	4	no	—
54	C₂×#11719908	—	2	1	no	—
55	C₁₆×D₅₃	—	2	4	no	—
56	#13133153	—	2	1	no	—
57	#13133153	C₂×C₄	1	2	yes	—
58	#13133152	—	2	4	no	—
59	#13133151	—	2	1	no	—
60	#13133151	C₂×C₄	1	2	yes	—
61	#13133148	—	2	2	no	—
62	#13133148	—	2	2	no	—
63	C₂×M₁₆×C₅₃	—	2	1	no	—
64	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
66	M₃₂×C₅₃	—	2	1	no	—
67	M₃₂×C₅₃	C₂×C₄	1	2	yes	—
68	C₂×C₁₆×C₅₃	—	2	1	no	—
69	#108×C₅₃	—	2	1	no	—
70	#108×C₅₃	C₂×C₄	1	2	yes	—
71	#107×C₅₃	—	2	1	no	—
72	#107×C₅₃	C₂×C₄	1	2	yes	—
73	C₂³×D₅₃	SD₁₆	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	—	4	1	no	—
75	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×Dic₅₃	—	4	1	no	—
77	C₂²×Dic₅₃	SD₁₆	1	1	yes	characteristic
78	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
79	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
82	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
83	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
84	C₂×C₄×D₅₃	SD₁₆	1	2	yes	—
85	C₂×C₄×D₅₃	Q₁₆	1	2	yes	—
86	M₁₆×C₅₃	—	2	2	no	—
87	M₁₆×C₅₃	—	4	1	no	—
88	C₂×C₈×C₅₃	—	4	1	no	—
89	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
95	C₄⋊C₄×C₅₃	—	2	2	no	—
96	C₄⋊C₄×C₅₃	—	4	1	no	—
97	#11719921	—	4	2	no	—
98	#11719921	—	4	2	no	—
99	C₄⋊Dic₅₃	—	2	2	no	—
100	C₄⋊Dic₅₃	—	4	1	no	—
101	C₄×Dic₅₃	—	4	2	no	—
102	C₄×Dic₅₃	—	4	2	no	—
103	#11719917	—	2	2	no	—
104	#11719917	—	4	1	no	—
105	C₂×#88322	—	4	1	no	—
106	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×#88322	C₂²⋊C₄	1	2	yes	—
109	C₂×#88322	C₄⋊C₄	1	2	yes	—
110	#11719912	—	4	4	no	—
111	C₈×D₅₃	—	4	4	no	—
112	C₈×D₅₃	C₄²	1	2	yes	—
113	C₈×D₅₃	C₄²	1	2	yes	—
114	C₈×D₅₃	C₂²⋊C₄	1	2	yes	—
115	C₈×D₅₃	C₂²⋊C₄	1	2	yes	—
116	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
117	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
118	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
119	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
120	C₁₆×C₅₃	—	2	2	no	—
121	#11719908	—	2	2	no	—
122	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
123	C₂²×D₅₃	—	4	1	no	—
124	C₂²×D₅₃	—	4	1	no	—
125	C₂²×D₅₃	#102	1	1	yes	characteristic
126	C₂²×D₅₃	#102	1	1	yes	characteristic
127	C₂²×D₅₃	#103	1	2	yes	—
128	C₂²×D₅₃	#106	1	2	yes	—
129	C₂×C₄×C₅₃	—	4	2	no	—
130	C₂×C₄×C₅₃	—	8	2	no	—
131	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
137	C₂×Dic₅₃	—	4	2	no	—
138	C₂×Dic₅₃	—	8	2	no	—
139	C₂×Dic₅₃	#102	1	1	yes	characteristic
140	C₂×Dic₅₃	#102	1	1	yes	characteristic
141	C₂×Dic₅₃	#103	1	2	yes	—
142	C₂×Dic₅₃	#106	1	2	yes	—
143	C₄×D₅₃	#95	1	2	yes	—
144	C₄×D₅₃	#95	1	2	yes	—
145	C₄×D₅₃	#102	1	2	yes	—
146	C₄×D₅₃	#102	1	2	yes	—
147	C₄×D₅₃	#103	1	2	yes	—
148	C₄×D₅₃	#103	1	2	yes	—
149	C₄×D₅₃	#107	1	2	yes	—
150	C₄×D₅₃	#107	1	2	yes	—
151	C₈×C₅₃	—	4	2	no	—
152	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
153	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
156	#88322	—	4	2	no	—
157	#88322	#95	1	2	yes	—
158	#88322	#95	1	2	yes	—
159	#12569	—	53	1	no	maximal, Sylow
160	C₂²×C₅₃	—	2	1	no	—
161	C₂²×C₅₃	—	2	1	no	—
162	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
163	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
164	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
165	C₂×D₅₃	—	2	1	no	—
166	C₂×D₅₃	—	2	1	no	—
167	C₂×D₅₃	—	2	1	no	—
168	C₂×D₅₃	—	2	1	no	—
169	C₂×D₅₃	—	4	2	no	—
170	C₂×D₅₃	—	4	2	no	—
171	C₂×D₅₃	#340	1	2	yes	—
172	C₂×D₅₃	#340	1	2	yes	—
173	C₄×C₅₃	—	8	2	no	—
174	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
175	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
176	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
177	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
178	Dic₅₃	—	8	2	no	—
179	Dic₅₃	#340	1	2	yes	—
180	Dic₅₃	#340	1	2	yes	—
181	C₂²×M₃₂	—	53	1	no	—
182	C₂²×#108	—	53	1	no	—
183	C₂×#427	—	53	1	no	—
184	#1387	—	53	1	no	—
185	#1387	—	53	1	no	—
186	#1387	—	53	1	no	—
187	#1387	D₅₃	1	1	yes	characteristic, complemented
188	C₂×C₅₃	—	2	1	no	—
189	C₂×C₅₃	—	2	1	no	—
190	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
191	C₂×C₅₃	#2148	1	1	yes	characteristic
192	C₂×C₅₃	#2148	1	1	yes	characteristic
193	D₅₃	—	2	2	no	—
194	D₅₃	—	2	2	no	—
195	C₂²×M₁₆	—	106	1	no	—
196	C₂³×C₈	—	53	1	no	—
197	C₂×#117	—	106	1	no	—
198	C₂×M₃₂	—	53	1	no	—
199	C₂×M₃₂	—	53	4	no	—
200	C₂×M₃₂	—	106	1	no	—
201	C₂×M₃₂	—	106	2	no	—
202	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
203	C₂²×C₁₆	—	106	1	no	—
204	C₂×#108	—	53	1	no	—
205	C₂×#108	—	53	4	no	—
206	C₂×#108	—	106	2	no	—
207	C₂×#108	—	106	1	no	—
208	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
209	#427	—	53	2	no	—
210	#427	—	53	2	no	—
211	#427	—	106	1	no	—
212	#427	—	106	1	no	—
213	C₂×#107	—	53	1	no	—
214	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
215	C₂×#106	—	106	1	no	—
216	C₅₃	#12569	1	1	yes	characteristic, complemented, Sylow
217	C₂³×C₄	—	53	1	no	—
218	C₂×M₁₆	—	2	1	no	—
219	C₂×M₁₆	—	106	4	no	—
220	C₂×M₁₆	—	106	1	no	—
221	C₂×M₁₆	—	212	1	no	—
222	C₂×M₁₆	—	212	2	no	—
223	C₂²×C₈	—	53	1	no	—
224	C₂²×C₈	—	53	1	no	—
225	C₂²×C₈	—	53	1	no	—
226	C₂²×C₈	—	53	2	no	—
227	C₂²×C₈	—	53	2	no	—
228	C₂²×C₈	—	53	1	no	—
229	C₂²×C₈	—	53	1	no	—
230	C₂²×C₈	—	53	2	no	—
231	C₂²×C₈	—	53	2	no	—
232	C₂²×C₈	—	212	1	no	—
233	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
234	#117	—	106	2	no	—
235	#117	—	106	2	no	—
236	#117	—	212	1	no	—
237	#117	—	212	1	no	—
238	C₂×C₄⋊C₄	—	2	1	no	—
239	C₂×C₄⋊C₄	—	106	1	no	—
240	C₂×C₂²⋊C₄	—	212	1	no	—
241	C₂×C₄²	—	212	1	no	—
242	M₃₂	—	2	1	no	—
243	M₃₂	—	53	2	no	—
244	M₃₂	—	106	4	no	—
245	M₃₂	—	106	1	no	—
246	M₃₂	C₄×D₅₃	1	2	yes	—
247	C₂×C₁₆	—	2	1	no	—
248	C₂×C₁₆	—	106	4	no	—
249	C₂×C₁₆	—	106	1	no	—
250	#108	—	2	1	no	—
251	#108	—	53	2	no	—
252	#108	—	106	4	no	—
253	#108	—	106	1	no	—
254	#108	C₄×D₅₃	1	2	yes	—
255	#107	—	2	1	no	—
256	#107	—	53	2	no	—
257	#107	—	106	1	no	—
258	#107	C₄×D₅₃	1	2	yes	—
259	#106	—	106	2	no	—
260	#106	—	106	2	no	—
261	C₂⁴	—	53	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	53	1	no	—
264	C₂²×C₄	—	53	1	no	—
265	C₂²×C₄	—	53	1	no	—
266	C₂²×C₄	—	53	2	no	—
267	C₂²×C₄	—	53	2	no	—
268	C₂²×C₄	—	53	1	no	—
269	C₂²×C₄	—	53	1	no	—
270	C₂²×C₄	—	53	2	no	—
271	C₂²×C₄	—	53	2	no	—
272	C₂²×C₄	—	212	1	no	—
273	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
274	M₁₆	—	2	2	no	—
275	M₁₆	—	4	1	no	—
276	M₁₆	—	106	2	no	—
277	M₁₆	—	212	4	no	—
278	M₁₆	—	212	1	no	—
279	C₂×C₈	—	4	1	no	—
280	C₂×C₈	—	53	1	no	—
281	C₂×C₈	—	53	1	no	—
282	C₂×C₈	—	53	2	no	—
283	C₂×C₈	—	53	2	no	—
284	C₂×C₈	—	53	2	no	—
285	C₂×C₈	—	53	2	no	—
286	C₂×C₈	—	53	2	no	—
287	C₂×C₈	—	53	2	no	—
288	C₂×C₈	—	53	2	no	—
289	C₂×C₈	—	53	2	no	—
290	C₂×C₈	—	53	2	no	—
291	C₂×C₈	—	53	2	no	—
292	C₂×C₈	—	212	4	no	—
293	C₂×C₈	—	212	1	no	—
294	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
295	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
296	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
297	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
298	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
299	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	106	2	no	—
303	C₄⋊C₄	—	212	1	no	—
304	C₂²⋊C₄	—	212	2	no	—
305	C₂²⋊C₄	—	212	2	no	—
306	C₄²	—	212	2	no	—
307	C₄²	—	212	2	no	—
308	C₁₆	—	2	2	no	—
309	C₁₆	—	106	2	no	—
310	C₂³	—	53	1	no	—
311	C₂³	—	53	1	no	—
312	C₂³	—	53	2	no	—
313	C₂³	—	53	2	no	—
314	C₂³	—	212	1	no	—
315	C₂³	—	212	1	no	—
316	C₂³	SD₁₆×D₅₃	1	1	yes	characteristic
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	8	2	no	—
319	C₂×C₄	—	53	1	no	—
320	C₂×C₄	—	53	1	no	—
321	C₂×C₄	—	53	2	no	—
322	C₂×C₄	—	53	2	no	—
323	C₂×C₄	—	53	2	no	—
324	C₂×C₄	—	53	2	no	—
325	C₂×C₄	—	53	2	no	—
326	C₂×C₄	—	53	2	no	—
327	C₂×C₄	—	53	2	no	—
328	C₂×C₄	—	53	2	no	—
329	C₂×C₄	—	53	2	no	—
330	C₂×C₄	—	53	2	no	—
331	C₂×C₄	—	424	2	no	—
332	C₂×C₄	—	212	2	no	—
333	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
334	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
335	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
336	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
337	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
338	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
339	C₈	—	4	2	no	—
340	C₈	—	53	2	no	—
341	C₈	—	53	2	no	—
342	C₈	—	212	2	no	—
343	C₈	C₄²×D₅₃	1	1	yes	characteristic
344	C₈	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
345	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
346	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	53	2	no	—
350	C₂²	—	53	2	no	—
351	C₂²	—	106	1	no	—
352	C₂²	—	106	1	no	—
353	C₂²	—	106	1	no	—
354	C₂²	—	106	1	no	—
355	C₂²	—	212	2	no	—
356	C₂²	—	212	2	no	—
357	C₂²	#102×D₅₃	1	1	yes	characteristic
358	C₂²	#103×D₅₃	1	1	yes	characteristic
359	C₂²	#106×D₅₃	1	1	yes	characteristic
360	C₄	—	8	2	no	—
361	C₄	—	53	2	no	—
362	C₄	—	53	2	no	—
363	C₄	—	424	2	no	—
364	C₄	#95×D₅₃	1	1	yes	characteristic
365	C₄	#102×D₅₃	1	1	yes	characteristic
366	C₄	#103×D₅₃	1	1	yes	characteristic
367	C₄	#107×D₅₃	1	1	yes	characteristic
368	C₂	—	2	1	no	—
369	C₂	—	2	1	no	—
370	C₂	—	106	2	no	—
371	C₂	—	106	2	no	—
372	C₂	#340×D₅₃	1	1	yes	characteristic
373	C₂	#20276461	1	1	yes	characteristic
374	C₂	#20276463	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#102×D₅₃
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1387	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×D₅₃	SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1387	D₅₃
hypocenter	C₅₃	#12569
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1387

Complements

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

N	H ≅ G/N	type
4. #20272825	C₂	semidirect
5. #20272824	C₂	semidirect
6. #20272823	C₂	semidirect
7. C₅₃×#1387	C₂	semidirect
14. M₃₂×D₅₃	C₄	semidirect
19. #108×D₅₃	C₄	semidirect
187. #1387	D₅₃	semidirect
216. C₅₃	#12569	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂×D₅₃	2
C₂×#108×D₅₃	2
C₂×#16810660	2
#20272825	2
#20272824	2
#20272823	2
C₅₃×#1387	2
#12569	53