Subgroups — #27371756

Order 13568.

814 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	#20285277	C₂	1	2	yes	maximal, complemented
2	#20285276	C₂	1	2	yes	maximal, complemented
3	#20285226	C₂	1	1	yes	maximal, characteristic, complemented
4	#20285144	C₂	1	1	yes	maximal, characteristic
5	#20274318	C₂	1	2	yes	maximal, complemented
6	#20274309	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274308	C₂	1	1	yes	maximal, characteristic, complemented
8	#20274307	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274303	C₂	1	2	yes	maximal, complemented
10	C₈²×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1707	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133240	—	2	2	no	—
13	#16811178	C₂²	1	2	yes	—
14	#16811174	C₂²	1	2	yes	—
15	#16811028	C₂²	1	1	yes	characteristic
16	#16811024	—	2	2	no	—
17	#16811019	C₂²	1	2	yes	—
18	#16811019	C₂²	1	2	yes	—
19	#16811008	—	2	2	no	—
20	#16811000	C₂²	1	1	yes	characteristic
21	#16810997	C₂²	1	2	yes	—
22	#16810997	C₂²	1	2	yes	—
23	#16810996	C₂²	1	2	yes	—
24	#16810975	C₂²	1	2	yes	—
25	#16810971	C₂²	1	2	yes	—
26	#16810970	C₂²	1	2	yes	—
27	#16810855	—	2	4	no	—
28	#16810853	—	2	4	no	—
29	#16810710	—	2	4	no	—
30	#16810708	—	2	4	no	—
31	#16810660	—	2	2	no	—
32	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
35	C₈×#88322	C₂²	1	1	yes	characteristic
36	C₈×#88322	C₂²	1	1	yes	characteristic
37	C₈×#88322	C₂²	1	1	yes	characteristic
38	C₅₃×#500	C₂²	1	1	yes	characteristic
39	C₅₃×#498	C₂²	1	2	yes	—
40	C₅₃×#494	C₂²	1	1	yes	characteristic
41	C₅₃×#487	C₂²	1	2	yes	—
42	C₈²×C₅₃	C₂²	1	1	yes	characteristic
43	C₂×#11719949	—	2	2	no	—
44	#13133240	—	4	8	no	—
45	C₂×#11719913	—	2	4	no	—
46	C₂×#11719924	—	2	4	no	—
47	C₂×#11719924	—	2	4	no	—
48	C₂×D₄₂₄	—	2	2	no	—
49	C₂×C₈×D₅₃	D₄	1	2	yes	—
50	C₂×C₈×D₅₃	D₄	1	2	yes	—
51	C₂×C₈×D₅₃	D₄	1	2	yes	—
52	#13133217	C₂³	1	2	yes	—
53	#13133214	C₂³	1	2	yes	—
54	#13133208	—	2	2	no	—
55	#13133206	—	2	2	no	—
56	#13133197	—	2	2	no	—
57	#13133193	—	2	2	no	—
58	#13133192	—	2	2	no	—
59	#13133192	—	2	2	no	—
60	Q₈×Dic₅₃	—	2	2	no	—
61	#13133182	—	2	2	no	—
62	#13133179	—	2	2	no	—
63	#13133179	—	2	2	no	—
64	#13133175	—	2	2	no	—
65	C₄²×D₅₃	C₂³	1	1	yes	characteristic
66	#13133151	—	2	2	no	—
67	#13133148	—	2	4	no	—
68	#13133148	—	2	4	no	—
69	#13133140	—	2	4	no	—
70	#13133139	—	2	4	no	—
71	#13133137	—	2	4	no	—
72	C₄×#88322	C₂³	1	1	yes	characteristic
73	C₄×#88322	C₂³	1	1	yes	characteristic
74	C₄×#88322	C₂³	1	1	yes	characteristic
75	C₈×Dic₅₃	D₄	1	2	yes	—
76	C₈×Dic₅₃	D₄	1	2	yes	—
77	C₈×Dic₅₃	D₄	1	2	yes	—
78	C₂×Q₁₆×C₅₃	—	2	2	no	—
79	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
80	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
81	#107×C₅₃	—	2	2	no	—
82	#106×C₅₃	—	2	4	no	—
83	#103×C₅₃	—	2	4	no	—
84	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
87	C₂×Q₈×C₅₃	—	2	2	no	—
88	C₂×Q₈×C₅₃	—	2	2	no	—
89	#11719949	—	4	8	no	—
90	C₂×D₂₁₂	—	2	2	no	—
91	C₂×D₂₁₂	—	2	2	no	—
92	C₂×D₂₁₂	—	2	2	no	—
93	C₂×D₂₁₂	—	4	2	no	—
94	C₂×C₄×D₅₃	—	4	2	no	—
95	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
98	C₂×Dic₁₀₆	—	2	2	no	—
99	Q₁₆×C₅₃	—	4	4	no	—
100	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
103	C₄⋊C₄×C₅₃	—	2	2	no	—
104	C₄⋊C₄×C₅₃	—	2	2	no	—
105	C₄⋊C₄×C₅₃	—	2	2	no	—
106	C₄⋊C₄×C₅₃	—	2	2	no	—
107	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
108	#11719924	—	4	8	no	—
109	#11719924	—	4	8	no	—
110	#11719921	—	4	2	no	—
111	#11719921	—	4	2	no	—
112	#11719921	—	4	2	no	—
113	C₄⋊Dic₅₃	—	2	2	no	—
114	C₄⋊Dic₅₃	—	2	2	no	—
115	C₄⋊Dic₅₃	—	4	2	no	—
116	#11719919	—	4	2	no	—
117	C₄×Dic₅₃	—	4	2	no	—
118	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
119	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
120	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
121	C₂×#88322	C₂×D₄	1	2	yes	—
122	C₂×#88322	C₂×D₄	1	2	yes	—
123	C₂×#88322	C₂×D₄	1	2	yes	—
124	D₄₂₄	—	4	4	no	—
125	#11719913	—	4	8	no	—
126	C₈×D₅₃	—	2	4	no	—
127	C₈×D₅₃	—	2	4	no	—
128	C₈×D₅₃	SD₁₆	1	8	yes	complemented
129	C₂²×D₅₃	—	4	2	no	—
130	C₂²×D₅₃	#127	1	1	yes	characteristic
131	Q₈×C₅₃	—	4	4	no	—
132	Q₈×C₅₃	—	4	4	no	—
133	C₂×C₄×C₅₃	—	4	2	no	—
134	C₂×C₄×C₅₃	—	4	2	no	—
135	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×Dic₅₃	—	4	2	no	—
139	C₂×Dic₅₃	#127	1	1	yes	characteristic
140	C₂×Dic₅₃	#127	1	1	yes	characteristic
141	C₂×Dic₅₃	#127	1	1	yes	characteristic
142	D₂₁₂	—	4	4	no	—
143	D₂₁₂	—	4	4	no	—
144	D₂₁₂	—	4	4	no	—
145	D₂₁₂	—	8	4	no	—
146	C₄×D₅₃	—	2	2	no	—
147	C₄×D₅₃	—	2	2	no	—
148	C₄×D₅₃	—	8	4	no	—
149	C₄×D₅₃	C₂×SD₁₆	1	4	yes	—
150	Dic₁₀₆	—	4	4	no	—
151	C₈×C₅₃	—	2	2	no	—
152	C₈×C₅₃	C₂×SD₁₆	1	4	yes	—
153	C₈×C₅₃	#135	1	4	yes	complemented
154	#88322	—	2	2	no	—
155	#88322	C₂×SD₁₆	1	4	yes	—
156	#88322	#135	1	4	yes	complemented
157	#19596	—	53	1	no	maximal, Sylow
158	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
159	C₂×D₅₃	—	2	1	no	—
160	C₂×D₅₃	—	2	1	no	—
161	C₂×D₅₃	—	8	2	no	—
162	C₂×D₅₃	—	8	2	no	—
163	C₂×D₅₃	—	8	2	no	—
164	C₂×D₅₃	#492	1	2	yes	—
165	C₄×C₅₃	—	8	2	no	—
166	C₄×C₅₃	—	8	2	no	—
167	C₄×C₅₃	C₂²×SD₁₆	1	2	yes	—
168	C₄×C₅₃	C₂×#135	1	2	yes	—
169	C₄×C₅₃	C₂×#135	1	2	yes	—
170	Dic₅₃	—	2	1	no	—
171	Dic₅₃	—	2	1	no	—
172	Dic₅₃	—	8	2	no	—
173	Dic₅₃	#492	1	2	yes	—
174	#3167	—	53	2	no	—
175	#3167	—	53	2	no	—
176	#3160	—	53	1	no	—
177	#3147	—	53	1	no	—
178	#1708	—	53	2	no	—
179	#1707	—	53	1	no	—
180	#1707	D₅₃	1	1	yes	characteristic, complemented
181	#1706	—	53	1	no	—
182	#1706	—	53	1	no	—
183	#1705	—	53	2	no	—
184	C₂×C₈²	—	53	1	no	—
185	C₂×C₅₃	C₂×#492	1	1	yes	characteristic
186	C₂×C₅₃	#3147	1	1	yes	characteristic
187	C₂×C₅₃	#3147	1	1	yes	characteristic
188	D₅₃	—	2	2	no	—
189	D₅₃	—	16	2	no	—
190	C₂×#135	—	106	2	no	—
191	#533	—	53	2	no	—
192	#532	—	53	2	no	—
193	#500	—	53	1	no	—
194	#500	C₂×D₅₃	1	1	yes	characteristic
195	#499	—	106	2	no	—
196	#498	—	53	2	no	—
197	#498	—	53	2	no	—
198	#498	C₂×D₅₃	1	2	yes	—
199	#495	—	106	2	no	—
200	#494	C₂×D₅₃	1	1	yes	characteristic
201	#493	—	53	1	no	—
202	#492	—	53	2	no	—
203	#492	—	53	2	no	—
204	#492	—	53	2	no	—
205	#487	—	53	2	no	—
206	#487	C₂×D₅₃	1	2	yes	—
207	#486	—	53	2	no	—
208	#486	—	53	2	no	—
209	#465	—	106	4	no	—
210	#465	—	106	4	no	—
211	C₄×SD₁₆	—	106	4	no	—
212	C₄×SD₁₆	—	106	4	no	—
213	#427	—	106	2	no	—
214	C₂×C₄×C₈	—	53	1	no	—
215	C₂×C₄×C₈	—	53	1	no	—
216	C₂×C₄×C₈	—	53	1	no	—
217	C₈²	—	53	1	no	—
218	C₈²	—	53	1	no	—
219	C₈²	—	53	1	no	—
220	C₈²	C₂×D₅₃	1	1	yes	characteristic
221	C₅₃	#19596	1	1	yes	characteristic, complemented, Sylow
222	C₂×#41	—	106	2	no	—
223	#135	—	212	8	no	—
224	C₂×Q₁₆	—	2	2	no	—
225	C₂×SD₁₆	—	106	4	no	—
226	C₂×SD₁₆	—	106	4	no	—
227	C₂×SD₁₆	—	106	4	no	—
228	C₂×D₈	—	106	2	no	—
229	C₂²×C₈	—	53	2	no	—
230	C₂²×C₈	—	53	2	no	—
231	C₂²×C₈	—	53	2	no	—
232	C₄⋊Q₈	—	53	2	no	—
233	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
234	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
235	#127	—	53	2	no	—
236	#125	—	106	2	no	—
237	#124	—	106	2	no	—
238	C₂²⋊Q₈	—	106	2	no	—
239	C₂²⋊Q₈	—	106	2	no	—
240	#121	—	106	2	no	—
241	#121	—	106	2	no	—
242	C₄×Q₈	—	106	2	no	—
243	C₄×Q₈	—	106	2	no	—
244	C₄×D₄	—	106	2	no	—
245	C₄×D₄	—	106	2	no	—
246	#117	—	106	2	no	—
247	C₂×C₄²	—	53	1	no	—
248	#107	—	2	2	no	—
249	#107	—	106	2	no	—
250	#106	—	2	4	no	—
251	#106	—	106	4	no	—
252	#106	—	106	4	no	—
253	#103	—	2	4	no	—
254	#103	—	106	4	no	—
255	#102	—	106	4	no	—
256	#102	—	106	4	no	—
257	C₄×C₈	—	53	2	no	—
258	C₄×C₈	—	53	2	no	—
259	C₄×C₈	—	53	2	no	—
260	C₄×C₈	—	53	1	no	—
261	C₄×C₈	—	53	1	no	—
262	C₄×C₈	—	53	1	no	—
263	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
265	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
266	#41	—	212	8	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	2	2	no	—
269	C₂×Q₈	—	106	2	no	—
270	C₂×D₄	—	106	2	no	—
271	C₂×D₄	—	106	2	no	—
272	C₂×D₄	—	106	2	no	—
273	C₂×D₄	—	212	2	no	—
274	C₂²×C₄	—	53	1	no	—
275	C₂²×C₄	—	53	1	no	—
276	C₂²×C₄	—	53	1	no	—
277	C₂²×C₄	—	212	2	no	—
278	Q₁₆	—	4	4	no	—
279	SD₁₆	—	212	8	no	—
280	SD₁₆	—	212	8	no	—
281	SD₁₆	—	212	8	no	—
282	D₈	—	212	4	no	—
283	C₂×C₈	—	53	2	no	—
284	C₂×C₈	—	53	2	no	—
285	C₂×C₈	—	53	8	no	—
286	C₂×C₈	—	53	2	no	—
287	C₂×C₈	—	106	4	no	—
288	C₂×C₈	—	106	4	no	—
289	C₂×C₈	D₄×D₅₃	1	2	yes	—
290	C₂×C₈	D₄×D₅₃	1	2	yes	—
291	C₂×C₈	D₄×D₅₃	1	2	yes	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	106	2	no	—
297	C₄⋊C₄	—	106	2	no	—
298	C₄⋊C₄	—	212	2	no	—
299	C₄⋊C₄	—	212	2	no	—
300	C₂²⋊C₄	—	212	2	no	—
301	C₂²⋊C₄	—	212	2	no	—
302	C₂²⋊C₄	—	212	2	no	—
303	C₄²	—	53	1	no	—
304	C₄²	—	53	1	no	—
305	C₄²	—	53	1	no	—
306	C₄²	—	212	2	no	—
307	C₄²	C₂³×D₅₃	1	1	yes	characteristic
308	C₂³	—	53	1	no	—
309	C₂³	—	212	2	no	—
310	Q₈	—	4	4	no	—
311	Q₈	—	4	4	no	—
312	Q₈	—	212	4	no	—
313	D₄	—	424	4	no	—
314	D₄	—	212	4	no	—
315	D₄	—	212	4	no	—
316	D₄	—	212	4	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	53	1	no	—
320	C₂×C₄	—	53	4	no	—
321	C₂×C₄	—	53	1	no	—
322	C₂×C₄	—	53	1	no	—
323	C₂×C₄	—	106	2	no	—
324	C₂×C₄	—	106	2	no	—
325	C₂×C₄	—	424	4	no	—
326	C₂×C₄	—	212	2	no	—
327	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
328	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
329	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
330	C₈	—	2	2	no	—
331	C₈	—	53	4	no	—
332	C₈	—	53	4	no	—
333	C₈	—	106	2	no	—
334	C₈	SD₁₆×D₅₃	1	4	yes	—
335	C₈	#13133240	1	4	yes	complemented
336	C₂²	—	53	2	no	—
337	C₂²	—	106	1	no	—
338	C₂²	—	106	1	no	—
339	C₂²	—	424	2	no	—
340	C₂²	—	424	2	no	—
341	C₂²	—	424	2	no	—
342	C₂²	#127×D₅₃	1	1	yes	characteristic
343	C₄	—	8	2	no	—
344	C₄	—	8	2	no	—
345	C₄	—	53	2	no	—
346	C₄	—	106	1	no	—
347	C₄	—	106	1	no	—
348	C₄	—	424	2	no	—
349	C₄	C₂×SD₁₆×D₅₃	1	2	yes	—
350	C₄	#135×D₅₃	1	2	yes	—
351	C₄	C₂×#13133240	1	2	yes	—
352	C₂	—	848	2	no	—
353	C₂	—	106	2	no	—
354	C₂	#492×D₅₃	1	1	yes	characteristic
355	C₂	#20285140	1	1	yes	characteristic
356	C₂	#20285144	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1707	C₂
socle	C₂²×C₅₃	C₂×#127
prime radical	#20285144	C₂
hypercenter	#1707	D₅₃
hypocenter	C₅₃	#19596
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1707

Complements

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

N	H ≅ G/N	type
1. #20285277	C₂	semidirect
2. #20285276	C₂	semidirect
3. #20285226	C₂	semidirect
5. #20274318	C₂	semidirect
6. #20274309	C₂	semidirect
7. #20274308	C₂	semidirect
8. #20274307	C₂	semidirect
9. #20274303	C₂	semidirect
10. C₈²×D₅₃	C₂	semidirect
11. C₅₃×#1707	C₂	semidirect
128. C₈×D₅₃	SD₁₆	semidirect
153. C₈×C₅₃	#135	semidirect
156. #88322	#135	semidirect
180. #1707	D₅₃	semidirect
221. C₅₃	#19596	semidirect
335. C₈	#13133240	semidirect

Maximal subgroups

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

subgroup	index
#20285277	2
#20285276	2
#20285226	2
#20285144	2
#20274318	2
#20274309	2
#20274308	2
#20274307	2
#20274303	2
C₈²×D₅₃	2
C₅₃×#1707	2
#19596	53