Subgroups — #27366358

Order 13568.

694 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20284719	C₂	1	2	yes	maximal
2	C₂×C₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic
3	#20274111	C₂	1	2	yes	maximal, complemented
4	#20274101	C₂	1	2	yes	maximal, complemented
5	#20274087	C₂	1	4	yes	maximal, complemented
6	#20273463	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273372	C₂	1	2	yes	maximal, complemented
8	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₅₃	—	2	1	no	—
10	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
11	#16810951	—	2	2	no	—
12	#16810910	C₂²	1	4	yes	—
13	#16810825	C₂²	1	4	yes	—
14	#16810773	—	2	2	no	—
15	C₄×#11719924	—	2	4	no	—
16	C₈×D₂₁₂	—	2	1	no	—
17	C₈×D₂₁₂	—	2	1	no	—
18	C₈×D₂₁₂	C₂²	1	2	yes	—
19	C₈×D₂₁₂	C₂²	1	2	yes	—
20	#16810676	C₂²	1	2	yes	—
21	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
22	C₂×#13133129	—	2	1	no	—
23	#16810376	C₂²	1	2	yes	complemented
24	#16810374	C₂²	1	2	yes	complemented
25	#16810372	C₂²	1	4	yes	complemented
26	#16810369	C₂²	1	4	yes	complemented
27	C₅₃×#533	C₂²	1	2	yes	—
28	C₅₃×#500	C₂²	1	1	yes	characteristic, complemented
29	C₅₃×#500	C₂²	1	1	yes	characteristic, complemented
30	C₅₃×#499	—	2	1	no	—
31	C₅₃×#499	—	2	1	no	—
32	C₅₃×#498	C₂²	1	2	yes	complemented
33	C₅₃×#427	—	2	1	no	—
34	C₅₃×#427	—	2	1	no	—
35	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₂²×D₂₁₂	D₄	1	1	yes	characteristic
37	C₂²×C₄×D₅₃	—	2	1	no	—
38	C₂×#11719924	—	2	4	no	—
39	C₂²×#88322	—	2	1	no	—
40	C₂×C₈×D₅₃	—	2	2	no	—
41	C₂×C₈×D₅₃	—	2	2	no	—
42	C₂×C₈×D₅₃	—	4	4	no	—
43	C₄×D₂₁₂	—	2	1	no	—
44	C₄×D₂₁₂	—	2	1	no	—
45	C₄×D₂₁₂	D₄	1	2	yes	—
46	C₄×D₂₁₂	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
48	C₂×#11719921	—	2	1	no	—
49	#13133148	—	2	4	no	—
50	C₄⋊#88322	—	2	1	no	—
51	C₄⋊#88322	C₂³	1	4	yes	—
52	C₄⋊#88322	C₂³	1	2	yes	—
53	#13133141	—	2	4	no	—
54	#13133137	—	2	4	no	—
55	C₂²⋊#88322	—	2	2	no	—
56	#13133129	—	2	2	no	—
57	#13133129	—	2	2	no	—
58	C₄×#88322	—	2	2	no	—
59	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
61	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
62	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
63	#125×C₅₃	—	2	2	no	—
64	C₂²⋊Q₈×C₅₃	—	2	2	no	—
65	C₂²⋊Q₈×C₅₃	—	2	2	no	—
66	C₄×Q₈×C₅₃	—	2	2	no	—
67	C₄×Q₈×C₅₃	—	2	2	no	—
68	#117×C₅₃	—	2	2	no	—
69	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
70	#107×C₅₃	—	2	1	no	—
71	#107×C₅₃	—	2	1	no	—
72	#107×C₅₃	—	2	1	no	—
73	#107×C₅₃	—	2	1	no	—
74	#106×C₅₃	—	2	2	no	—
75	#106×C₅₃	—	2	2	no	—
76	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
80	C₂³×D₅₃	—	2	1	no	—
81	C₂×Q₈×C₅₃	—	2	2	no	—
82	C₂×Q₈×C₅₃	—	2	2	no	—
83	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
86	C₂²×Dic₅₃	—	2	1	no	—
87	C₂×D₂₁₂	—	2	1	no	—
88	C₂×D₂₁₂	—	2	4	no	—
89	C₂×D₂₁₂	C₂×D₄	1	2	yes	—
90	C₂×C₄×D₅₃	—	2	2	no	—
91	C₂×C₄×D₅₃	—	2	2	no	—
92	C₂×C₄×D₅₃	—	4	2	no	—
93	C₂×C₄×D₅₃	—	4	2	no	—
94	C₂×C₈×C₅₃	—	2	2	no	—
95	C₂×C₈×C₅₃	SD₁₆	1	4	yes	complemented
96	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
100	C₄⋊C₄×C₅₃	—	2	2	no	—
101	C₄⋊C₄×C₅₃	—	2	2	no	—
102	C₄⋊C₄×C₅₃	—	2	2	no	—
103	C₄⋊C₄×C₅₃	—	2	2	no	—
104	C₄⋊C₄×C₅₃	—	4	2	no	—
105	C₄⋊C₄×C₅₃	—	4	2	no	—
106	C₂²⋊C₄×C₅₃	—	4	2	no	—
107	C₄²×C₅₃	—	4	2	no	—
108	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₅₃	#41	1	1	yes	characteristic
111	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
112	#11719924	—	4	8	no	—
113	#11719921	—	2	2	no	—
114	#11719921	—	2	2	no	—
115	C₄⋊Dic₅₃	—	2	1	no	—
116	C₄⋊Dic₅₃	C₂×D₄	1	2	yes	—
117	C₂×#88322	—	2	4	no	—
118	C₂×#88322	—	2	2	no	—
119	C₂×#88322	—	4	2	no	—
120	C₈×D₅₃	—	4	4	no	—
121	C₈×D₅₃	—	4	4	no	—
122	C₂³×C₅₃	#121	1	1	yes	characteristic
123	C₂²×D₅₃	—	2	4	no	—
124	C₂²×D₅₃	—	2	2	no	—
125	C₂²×D₅₃	—	4	2	no	—
126	C₂²×D₅₃	—	4	2	no	—
127	Q₈×C₅₃	—	4	4	no	—
128	Q₈×C₅₃	—	8	2	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	4	1	no	—
131	C₂×C₄×C₅₃	—	4	2	no	—
132	C₂×C₄×C₅₃	—	4	2	no	—
133	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
134	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
135	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
137	C₂×C₄×C₅₃	#137	1	2	yes	—
138	C₂×C₄×C₅₃	C₂²×D₄	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	—	4	2	no	—
143	D₂₁₂	—	2	4	no	—
144	D₂₁₂	—	2	2	no	—
145	D₂₁₂	#135	1	4	yes	—
146	C₄×D₅₃	—	4	2	no	—
147	C₄×D₅₃	—	4	2	no	—
148	C₄×D₅₃	—	4	4	no	—
149	C₈×C₅₃	—	2	1	no	—
150	C₈×C₅₃	—	2	1	no	—
151	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
152	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
153	#88322	—	4	2	no	—
154	#88322	—	4	4	no	—
155	#19105	—	53	1	no	maximal, Sylow
156	C₂²×C₅₃	—	2	1	no	—
157	C₂²×C₅₃	—	2	1	no	—
158	C₂²×C₅₃	#461	1	2	yes	—
159	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
160	C₂×D₅₃	—	2	4	no	—
161	C₂×D₅₃	—	4	8	no	—
162	C₂×D₅₃	—	4	2	no	—
163	C₂×D₅₃	—	4	2	no	—
164	C₄×C₅₃	—	2	1	no	—
165	C₄×C₅₃	—	2	1	no	—
166	C₄×C₅₃	—	2	1	no	—
167	C₄×C₅₃	—	8	2	no	—
168	C₄×C₅₃	—	8	2	no	—
169	C₄×C₅₃	#461	1	2	yes	—
170	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
171	C₄×C₅₃	C₂×#135	1	2	yes	—
172	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
173	Dic₅₃	—	4	2	no	—
174	#3160	D₅₃	1	1	yes	characteristic, complemented
175	#3101	—	53	2	no	—
176	C₂×C₈×D₄	—	53	1	no	—
177	#1675	—	53	2	no	—
178	#1673	—	53	2	no	—
179	#1671	—	53	4	no	—
180	#1564	—	53	1	no	—
181	#1542	—	53	2	no	—
182	C₂×C₅₃	—	2	2	no	—
183	C₂×C₅₃	C₂×#461	1	1	yes	characteristic
184	C₂×C₅₃	#3050	1	1	yes	characteristic
185	C₂×C₅₃	#3067	1	1	yes	characteristic
186	D₅₃	—	4	4	no	—
187	C₂³×C₈	—	106	1	no	—
188	#533	C₂×D₅₃	1	2	yes	—
189	C₂×C₄×D₄	—	53	1	no	—
190	#500	C₂×D₅₃	1	1	yes	characteristic, complemented
191	#500	C₂×D₅₃	1	1	yes	characteristic, complemented
192	#499	—	2	1	no	—
193	#499	—	2	1	no	—
194	#498	C₂×D₅₃	1	2	yes	complemented
195	#483	—	106	2	no	—
196	#476	—	53	4	no	—
197	#461	—	53	4	no	—
198	#450	—	106	2	no	—
199	C₄×SD₁₆	—	106	4	no	—
200	C₈×D₄	—	53	2	no	—
201	C₈×D₄	—	53	2	no	—
202	C₈×D₄	—	106	1	no	—
203	C₈×D₄	—	106	1	no	—
204	#431	—	53	2	no	—
205	#427	—	2	1	no	—
206	#427	—	2	1	no	—
207	C₂×C₄⋊C₈	—	53	1	no	—
208	C₂×C₂²⋊C₈	—	106	1	no	—
209	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
210	#335	—	53	2	no	—
211	#334	—	53	2	no	—
212	#333	—	53	4	no	—
213	#332	—	53	4	no	—
214	C₅₃	#19105	1	1	yes	characteristic, complemented, Sylow
215	C₂²×D₄	—	53	1	no	—
216	C₂³×C₄	—	106	1	no	—
217	C₂×SD₁₆	—	106	4	no	—
218	C₂²×C₈	—	106	1	no	—
219	C₂²×C₈	—	106	2	no	—
220	C₂²×C₈	—	106	2	no	—
221	C₂²×C₈	—	212	4	no	—
222	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
223	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
224	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
225	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
226	#125	—	2	2	no	—
227	C₂²⋊Q₈	—	2	2	no	—
228	C₂²⋊Q₈	—	2	2	no	—
229	C₄×Q₈	—	2	2	no	—
230	C₄×Q₈	—	2	2	no	—
231	C₄×D₄	—	53	2	no	—
232	C₄×D₄	—	53	2	no	—
233	C₄×D₄	—	106	1	no	—
234	C₄×D₄	—	106	1	no	—
235	#117	—	2	2	no	—
236	C₂×C₄⋊C₄	—	53	1	no	—
237	C₂×C₂²⋊C₄	—	106	1	no	—
238	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
239	#107	—	2	1	no	—
240	#107	—	2	1	no	—
241	#107	—	2	1	no	—
242	#107	—	2	1	no	—
243	#106	—	2	2	no	—
244	#106	—	2	2	no	—
245	#106	—	106	4	no	—
246	C₄⋊C₈	—	53	2	no	—
247	C₄⋊C₈	—	53	4	no	—
248	C₄⋊C₈	—	106	1	no	—
249	#103	—	106	4	no	—
250	#102	—	106	4	no	—
251	C₂²⋊C₈	—	106	2	no	—
252	C₂²⋊C₈	—	106	2	no	—
253	C₂²⋊C₈	—	106	2	no	—
254	C₄×C₈	—	106	2	no	—
255	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
256	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
257	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
258	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
259	C₂⁴	—	106	1	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×D₄	—	53	2	no	—
263	C₂×D₄	—	106	4	no	—
264	C₂×D₄	—	106	1	no	—
265	C₂²×C₄	—	106	1	no	—
266	C₂²×C₄	—	106	2	no	—
267	C₂²×C₄	—	106	2	no	—
268	C₂²×C₄	—	212	2	no	—
269	C₂²×C₄	—	212	2	no	—
270	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
271	C₂²×C₄	#11719947	1	1	yes	characteristic
272	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
273	SD₁₆	—	212	8	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	106	4	no	—
276	C₂×C₈	—	106	2	no	—
277	C₂×C₈	—	212	2	no	—
278	C₂×C₈	—	212	4	no	—
279	C₂×C₈	—	212	4	no	—
280	C₂×C₈	#11719924	1	4	yes	complemented
281	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
282	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
283	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
284	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	4	2	no	—
290	C₄⋊C₄	—	4	2	no	—
291	C₄⋊C₄	—	53	2	no	—
292	C₄⋊C₄	—	106	1	no	—
293	C₂²⋊C₄	—	4	2	no	—
294	C₂²⋊C₄	—	106	2	no	—
295	C₂²⋊C₄	—	106	2	no	—
296	C₄²	—	4	2	no	—
297	C₄²	D₄×D₅₃	1	1	yes	characteristic
298	C₄²	#11719947	1	1	yes	characteristic
299	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
300	C₄²	C₂³×D₅₃	1	1	yes	characteristic
301	C₂³	—	106	4	no	—
302	C₂³	—	106	2	no	—
303	C₂³	—	212	2	no	—
304	C₂³	—	212	2	no	—
305	C₂³	#13133188	1	1	yes	characteristic
306	Q₈	—	4	4	no	—
307	Q₈	—	8	2	no	—
308	D₄	—	53	4	no	—
309	D₄	—	106	4	no	—
310	D₄	—	106	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	106	2	no	—
316	C₂×C₄	—	212	2	no	—
317	C₂×C₄	—	212	4	no	—
318	C₂×C₄	—	212	2	no	—
319	C₂×C₄	—	212	2	no	—
320	C₂×C₄	#13133188	1	1	yes	characteristic
321	C₂×C₄	#13133188	1	1	yes	characteristic
322	C₂×C₄	#13133188	1	1	yes	characteristic
323	C₂×C₄	C₂×#11719924	1	2	yes	—
324	C₂×C₄	#13133248	1	2	yes	—
325	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
326	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
327	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	212	2	no	—
331	C₈	—	212	4	no	—
332	C₈	C₂×#11719924	1	2	yes	—
333	C₈	C₂×#11719924	1	2	yes	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	106	4	no	—
337	C₂²	—	212	8	no	—
338	C₂²	—	212	2	no	—
339	C₂²	—	212	2	no	—
340	C₂²	#16810825	1	2	yes	—
341	C₂²	C₂×#13133188	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	8	2	no	—
346	C₄	—	8	2	no	—
347	C₄	—	212	2	no	—
348	C₄	#16810825	1	2	yes	—
349	C₄	C₂²×#11719924	1	1	yes	characteristic
350	C₄	#135×D₅₃	1	2	yes	—
351	C₄	C₂×#13133248	1	1	yes	characteristic
352	C₂	—	2	2	no	—
353	C₂	—	212	4	no	—
354	C₂	C₂×#16810825	1	1	yes	characteristic
355	C₂	#20284292	1	1	yes	characteristic
356	C₂	#20284431	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133188
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3160	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂²×D₂₁₂	D₄
hypercenter	#3160	D₅₃
hypocenter	C₅₃	#19105
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

Complements

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

N	H ≅ G/N	type
3. #20274111	C₂	semidirect
4. #20274101	C₂	semidirect
5. #20274087	C₂	semidirect
6. #20273463	C₂	semidirect
7. #20273372	C₂	semidirect
8. C₅₃×#3160	C₂	semidirect
23. #16810376	C₂²	semidirect
24. #16810374	C₂²	semidirect
25. #16810372	C₂²	semidirect
26. #16810369	C₂²	semidirect
28. C₅₃×#500	C₂²	semidirect
29. C₅₃×#500	C₂²	semidirect
32. C₅₃×#498	C₂²	semidirect
95. C₂×C₈×C₅₃	SD₁₆	semidirect
174. #3160	D₅₃	semidirect
190. #500	C₂×D₅₃	semidirect
191. #500	C₂×D₅₃	semidirect
194. #498	C₂×D₅₃	semidirect
214. C₅₃	#19105	semidirect
280. C₂×C₈	#11719924	semidirect

Maximal subgroups

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

subgroup	index
#20284719	2
C₂×C₈×D₂₁₂	2
#20274111	2
#20274101	2
#20274087	2
#20273463	2
#20273372	2
C₅₃×#3160	2
#19105	53