Subgroups — #27356149

Order 13568.

598 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283875	C₂	1	2	yes	maximal, complemented
2	#20273629	C₂	1	1	yes	maximal, characteristic, complemented
3	#20273624	C₂	1	1	yes	maximal, characteristic, complemented
4	#20273559	C₂	1	2	yes	maximal, complemented
5	#20273552	C₂	1	2	yes	maximal, complemented
6	#20273539	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273534	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#16810374	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273128	C₂	1	2	yes	maximal, complemented
10	#20273004	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#3168	C₂	1	1	yes	maximal, characteristic
12	#16810754	C₂²	1	2	yes	—
13	Q₈×#88322	C₂²	1	2	yes	—
14	#16810701	C₂²	1	2	yes	—
15	#16810695	—	2	2	no	—
16	D₄×#88322	—	2	2	no	—
17	D₄×#88322	C₂²	1	2	yes	—
18	#16810684	—	2	2	no	—
19	#16810676	C₂²	1	1	yes	characteristic
20	#16810650	—	2	2	no	—
21	C₂×C₄⋊#88322	C₂²	1	2	yes	—
22	C₄×#11719917	C₂²	1	1	yes	characteristic
23	#16810374	C₂²	1	1	yes	characteristic
24	#16810374	C₂²	1	2	yes	—
25	#16810374	C₂²	1	1	yes	characteristic
26	Q₈⋊#88322	C₂²	1	2	yes	—
27	Q₈⋊#88322	C₂²	1	2	yes	—
28	D₄⋊#88322	C₂²	1	2	yes	—
29	D₄⋊#88322	C₂²	1	2	yes	—
30	#16810345	C₂²	1	1	yes	characteristic
31	#16810345	C₂²	1	1	yes	characteristic
32	#16810344	C₂²	1	1	yes	characteristic
33	#16810344	C₂²	1	1	yes	characteristic
34	C₅₃×#529	C₂²	1	2	yes	—
35	C₅₃×#499	C₄	1	1	yes	characteristic
36	C₅₃×#495	C₄	1	1	yes	characteristic
37	C₅₃×#488	C₄	1	2	yes	—
38	C₅₃×#467	C₄	1	1	yes	characteristic
39	C₅₃×#466	C₄	1	1	yes	characteristic
40	C₅₃×#465	C₄	1	2	yes	—
41	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
42	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	—
43	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
45	C₂×#11719917	—	2	2	no	—
46	C₂×#11719917	—	2	1	no	—
47	C₂²×#88322	—	2	2	no	—
48	C₂²×#88322	—	2	2	no	—
49	C₄⋊#88322	—	2	2	no	—
50	C₄⋊#88322	—	2	2	no	—
51	C₄⋊#88322	—	2	2	no	—
52	C₄⋊#88322	D₄	1	2	yes	—
53	C₄⋊#88322	C₂³	1	2	yes	—
54	C₄⋊#88322	C₂³	1	2	yes	—
55	C₂²⋊#88322	—	2	2	no	—
56	C₂²⋊#88322	—	2	2	no	—
57	C₂²⋊#88322	D₄	1	2	yes	—
58	#13133128	—	2	1	no	—
59	#13133128	—	2	2	no	—
60	C₄×#88322	—	2	2	no	—
61	C₄×#88322	C₂³	1	1	yes	characteristic
62	C₄×#88322	C₂³	1	1	yes	characteristic
63	C₄×#88322	C₂³	1	1	yes	characteristic
64	C₄×#88322	C₂³	1	1	yes	characteristic
65	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
67	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
70	#126×C₅₃	—	2	2	no	—
71	#125×C₅₃	C₂×C₄	1	2	yes	—
72	#124×C₅₃	C₂×C₄	1	2	yes	—
73	#123×C₅₃	—	2	2	no	—
74	C₂²⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
75	#121×C₅₃	C₂×C₄	1	2	yes	—
76	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
77	C₄×D₄×C₅₃	—	2	2	no	—
78	C₄×D₄×C₅₃	C₂³	1	2	yes	—
79	#117×C₅₃	—	2	2	no	—
80	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
81	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
82	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
83	#106×C₅₃	C₂×C₄	1	2	yes	—
84	#103×C₅₃	C₂×C₄	1	2	yes	—
85	#103×C₅₃	C₂×C₄	1	2	yes	—
86	#102×C₅₃	C₂×C₄	1	2	yes	—
87	#102×C₅₃	C₂×C₄	1	2	yes	—
88	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
89	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
90	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
92	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
93	C₂×D₄×C₅₃	—	2	2	no	—
94	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₅₃	—	2	2	no	—
96	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
99	Q₁₆×C₅₃	—	2	2	no	—
100	SD₁₆×C₅₃	—	4	2	no	—
101	D₈×C₅₃	—	2	2	no	—
102	C₂×C₈×C₅₃	—	2	2	no	—
103	C₂×C₈×C₅₃	—	4	1	no	—
104	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₄⋊C₄×C₅₃	—	2	2	no	—
109	C₄⋊C₄×C₅₃	—	2	2	no	—
110	C₄⋊C₄×C₅₃	—	2	2	no	—
111	C₄⋊C₄×C₅₃	—	2	2	no	—
112	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
113	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
114	C₂²⋊C₄×C₅₃	—	2	2	no	—
115	C₂²⋊C₄×C₅₃	—	2	2	no	—
116	C₂²⋊C₄×C₅₃	—	2	2	no	—
117	C₂²⋊C₄×C₅₃	—	2	2	no	—
118	C₂²⋊C₄×C₅₃	—	2	2	no	—
119	C₄²×C₅₃	—	2	2	no	—
120	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₅₃	#41	1	1	yes	characteristic
123	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
124	#11719917	—	4	2	no	—
125	#11719917	—	8	2	no	—
126	C₂×#88322	—	2	2	no	—
127	C₂×#88322	—	2	2	no	—
128	C₂×#88322	—	2	1	no	—
129	C₂×#88322	—	2	1	no	—
130	C₂×#88322	—	8	2	no	—
131	C₂×#88322	—	8	2	no	—
132	C₂×#88322	C₂×D₄	1	2	yes	—
133	C₂×#88322	C₂×D₄	1	2	yes	—
134	C₂³×C₅₃	—	2	2	no	—
135	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
136	Q₈×C₅₃	—	2	2	no	—
137	Q₈×C₅₃	—	2	2	no	—
138	D₄×C₅₃	—	2	2	no	—
139	D₄×C₅₃	—	2	2	no	—
140	D₄×C₅₃	—	4	4	no	—
141	C₂×C₄×C₅₃	—	2	2	no	—
142	C₂×C₄×C₅₃	—	2	1	no	—
143	C₂×C₄×C₅₃	—	2	1	no	—
144	C₂×C₄×C₅₃	—	2	2	no	—
145	C₂×C₄×C₅₃	—	2	2	no	—
146	C₂×C₄×C₅₃	—	2	2	no	—
147	C₂×C₄×C₅₃	—	2	2	no	—
148	C₂×C₄×C₅₃	—	4	2	no	—
149	C₂×C₄×C₅₃	—	4	2	no	—
150	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
156	C₈×C₅₃	—	2	1	no	—
157	C₈×C₅₃	—	2	1	no	—
158	C₈×C₅₃	—	2	1	no	—
159	C₈×C₅₃	#131	1	2	yes	—
160	#88322	—	2	2	no	—
161	#88322	—	2	2	no	—
162	#88322	—	4	1	no	—
163	#88322	—	4	1	no	—
164	#88322	—	4	2	no	—
165	#88322	—	4	2	no	—
166	#18093	—	53	1	no	maximal, Sylow
167	C₂²×C₅₃	—	2	1	no	—
168	C₂²×C₅₃	—	2	1	no	—
169	C₂²×C₅₃	—	4	2	no	—
170	C₂²×C₅₃	—	4	2	no	—
171	C₂²×C₅₃	—	4	2	no	—
172	C₂²×C₅₃	#444	1	2	yes	—
173	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
174	C₄×C₅₃	—	2	1	no	—
175	C₄×C₅₃	—	2	1	no	—
176	C₄×C₅₃	—	2	1	no	—
177	C₄×C₅₃	—	2	1	no	—
178	C₄×C₅₃	—	4	2	no	—
179	C₄×C₅₃	—	4	2	no	—
180	C₄×C₅₃	—	4	2	no	—
181	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
182	C₄×C₅₃	#568	1	1	yes	characteristic
183	C₄×C₅₃	#577	1	2	yes	—
184	#3168	D₅₃	1	1	yes	characteristic
185	#2994	—	53	2	no	—
186	#1594	—	53	1	no	—
187	#1593	—	53	1	no	—
188	#1583	—	53	2	no	—
189	#1582	—	53	2	no	—
190	#1580	—	53	1	no	—
191	#1579	—	53	1	no	—
192	C₂×#334	—	53	1	no	—
193	#1483	—	53	2	no	—
194	#1451	—	53	1	no	—
195	C₂×C₅₃	—	2	2	no	—
196	C₂×C₅₃	—	8	2	no	—
197	C₂×C₅₃	#2935	1	1	yes	characteristic
198	C₂×C₅₃	#2952	1	1	yes	characteristic
199	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
200	#529	C₂×D₅₃	1	2	yes	—
201	#499	Dic₅₃	1	1	yes	characteristic
202	#495	Dic₅₃	1	1	yes	characteristic
203	#488	Dic₅₃	1	2	yes	—
204	#467	Dic₅₃	1	1	yes	characteristic
205	#466	Dic₅₃	1	1	yes	characteristic
206	#465	Dic₅₃	1	2	yes	—
207	#446	—	53	2	no	—
208	C₈×Q₈	—	53	2	no	—
209	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
210	C₄×SD₁₆	C₂×D₅₃	1	2	yes	—
211	C₄×D₈	C₂×D₅₃	1	1	yes	characteristic
212	#436	—	53	2	no	—
213	#435	—	106	2	no	—
214	C₈×D₄	—	53	2	no	—
215	C₈×D₄	—	106	2	no	—
216	#433	—	106	2	no	—
217	#431	—	53	1	no	—
218	#424	—	106	2	no	—
219	C₂×C₄⋊C₈	—	53	2	no	—
220	C₄×M₁₆	—	53	1	no	—
221	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
222	#334	—	53	1	no	—
223	#334	—	53	2	no	—
224	#334	—	53	1	no	—
225	Q₈⋊C₈	—	53	2	no	—
226	Q₈⋊C₈	—	53	2	no	—
227	D₄⋊C₈	—	53	2	no	—
228	D₄⋊C₈	—	53	2	no	—
229	#322	—	53	1	no	—
230	#322	—	53	1	no	—
231	#322	—	53	1	no	—
232	#322	—	53	1	no	—
233	C₅₃	#18093	1	1	yes	characteristic, complemented, Sylow
234	C₂×Q₁₆	C₂×Dic₅₃	1	1	yes	characteristic
235	C₂×SD₁₆	C₂×Dic₅₃	1	2	yes	—
236	C₂×D₈	C₂×Dic₅₃	1	1	yes	characteristic
237	C₂×M₁₆	—	106	2	no	—
238	C₂×M₁₆	—	106	1	no	—
239	C₂²×C₈	—	106	2	no	—
240	C₂²×C₈	—	106	2	no	—
241	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
242	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
243	#126	—	2	2	no	—
244	#125	C₂×Dic₅₃	1	2	yes	—
245	#124	C₂×Dic₅₃	1	2	yes	—
246	#123	—	2	2	no	—
247	C₂²⋊Q₈	C₂×Dic₅₃	1	2	yes	—
248	#121	C₂×Dic₅₃	1	2	yes	—
249	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
250	C₄×D₄	—	2	2	no	—
251	C₄×D₄	C₂²×D₅₃	1	2	yes	—
252	#117	—	2	2	no	—
253	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
254	#107	C₂×Dic₅₃	1	1	yes	characteristic
255	#107	C₂×Dic₅₃	1	1	yes	characteristic
256	#106	C₂×Dic₅₃	1	2	yes	—
257	C₄⋊C₈	—	53	2	no	—
258	C₄⋊C₈	—	53	2	no	—
259	C₄⋊C₈	—	53	2	no	—
260	C₄⋊C₈	—	106	2	no	—
261	C₄⋊C₈	—	106	2	no	—
262	C₄⋊C₈	—	106	2	no	—
263	#103	C₂×Dic₅₃	1	2	yes	—
264	#103	C₂×Dic₅₃	1	2	yes	—
265	#102	C₂×Dic₅₃	1	2	yes	—
266	#102	C₂×Dic₅₃	1	2	yes	—
267	C₂²⋊C₈	—	53	2	no	—
268	C₂²⋊C₈	—	106	2	no	—
269	C₂²⋊C₈	—	106	2	no	—
270	#97	—	106	2	no	—
271	#97	—	106	1	no	—
272	C₄×C₈	—	53	1	no	—
273	C₄×C₈	—	53	1	no	—
274	C₄×C₈	—	53	1	no	—
275	C₄×C₈	—	53	1	no	—
276	C₄×C₈	—	106	2	no	—
277	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
278	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
281	C₂×Q₈	C₂²×Dic₅₃	1	2	yes	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	C₂²×Dic₅₃	1	2	yes	—
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
286	C₂²×C₄	#11719947	1	1	yes	characteristic
287	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
288	Q₁₆	—	2	2	no	—
289	SD₁₆	—	4	2	no	—
290	D₈	—	2	2	no	—
291	M₁₆	—	424	2	no	—
292	M₁₆	—	212	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	4	1	no	—
295	C₂×C₈	—	53	2	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	—	106	1	no	—
301	C₂×C₈	—	424	2	no	—
302	C₂×C₈	—	424	2	no	—
303	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
304	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
305	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
306	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
312	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
313	C₂²⋊C₄	—	2	2	no	—
314	C₂²⋊C₄	—	2	2	no	—
315	C₂²⋊C₄	—	2	2	no	—
316	C₂²⋊C₄	—	2	2	no	—
317	C₂²⋊C₄	—	2	2	no	—
318	C₄²	—	2	2	no	—
319	C₄²	D₄×D₅₃	1	1	yes	characteristic
320	C₄²	#11719947	1	1	yes	characteristic
321	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
322	C₄²	C₂³×D₅₃	1	1	yes	characteristic
323	C₂³	—	2	2	no	—
324	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
325	Q₈	—	2	2	no	—
326	Q₈	—	2	2	no	—
327	D₄	—	2	2	no	—
328	D₄	—	2	2	no	—
329	D₄	—	4	4	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
340	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
341	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
342	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
343	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
344	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	2	1	no	—
347	C₈	—	2	1	no	—
348	C₈	—	106	2	no	—
349	C₈	—	106	2	no	—
350	C₈	—	212	1	no	—
351	C₈	—	212	1	no	—
352	C₈	—	212	2	no	—
353	C₈	—	212	2	no	—
354	C₈	#13133224	1	2	yes	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	4	2	no	—
358	C₂²	—	4	2	no	—
359	C₂²	—	4	2	no	—
360	C₂²	#16810745	1	2	yes	—
361	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	4	2	no	—
367	C₄	—	4	2	no	—
368	C₄	—	4	2	no	—
369	C₄	C₂×#13133224	1	1	yes	characteristic
370	C₄	#16811390	1	1	yes	characteristic
371	C₄	#16811430	1	2	yes	—
372	C₂	—	2	2	no	—
373	C₂	—	8	2	no	—
374	C₂	#20283459	1	1	yes	characteristic
375	C₂	#20283599	1	1	yes	characteristic
376	C₂	C₂×#16810745	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3168	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₅₃×#466	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#3168	D₅₃
hypocenter	C₅₃	#18093
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3168

Complements

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

N	H ≅ G/N	type
1. #20283875	C₂	semidirect
2. #20273629	C₂	semidirect
3. #20273624	C₂	semidirect
4. #20273559	C₂	semidirect
5. #20273552	C₂	semidirect
6. #20273539	C₂	semidirect
7. #20273534	C₂	semidirect
8. C₂×#16810374	C₂	semidirect
9. #20273128	C₂	semidirect
10. #20273004	C₂	semidirect
233. C₅₃	#18093	semidirect

Maximal subgroups

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

subgroup	index
#20283875	2
#20273629	2
#20273624	2
#20273559	2
#20273552	2
#20273539	2
#20273534	2
C₂×#16810374	2
#20273128	2
#20273004	2
C₅₃×#3168	2
#18093	53