Subgroups — #27371780

Order 13568.

750 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285294	C₂	1	1	yes	maximal, characteristic
2	#20285275	C₂	1	2	yes	maximal, complemented
3	#20285040	C₂	1	2	yes	maximal, complemented
4	#20274328	C₂	1	2	yes	maximal, complemented
5	#20274324	C₂	1	1	yes	maximal, characteristic, complemented
6	#20274322	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274318	C₂	1	2	yes	maximal, complemented
8	#20274309	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274308	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811173	C₂²	1	2	yes	—
13	#16811028	C₂²	1	1	yes	characteristic
14	#16811019	C₂²	1	2	yes	—
15	#16811000	C₂²	1	1	yes	characteristic
16	#16810997	C₂²	1	2	yes	—
17	#16810980	—	2	4	no	—
18	#16810975	C₂²	1	2	yes	—
19	#16810975	C₂²	1	2	yes	—
20	#16810973	C₂²	1	2	yes	—
21	#16810971	C₂²	1	2	yes	—
22	#16810971	C₂²	1	2	yes	—
23	#16810971	C₂²	1	2	yes	—
24	#16810970	C₂²	1	2	yes	—
25	#16810859	—	2	2	no	—
26	#16810854	—	2	4	no	—
27	C₄×#11719924	—	2	4	no	—
28	C₄×D₄₂₄	—	2	2	no	—
29	#16810617	—	2	4	no	—
30	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
31	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
32	C₈×#88322	C₂²	1	1	yes	characteristic
33	C₈×#88322	C₂²	1	1	yes	characteristic
34	C₈×#88322	C₂²	1	1	yes	characteristic
35	C₈×#88322	C₂²	1	1	yes	characteristic
36	C₅₃×#533	C₂²	1	2	yes	—
37	C₅₃×#500	C₂²	1	1	yes	characteristic
38	C₅₃×#494	C₂²	1	1	yes	characteristic
39	C₅₃×#487	C₂²	1	2	yes	—
40	C₅₃×#467	—	2	2	no	—
41	C₄×Q₁₆×C₅₃	—	2	2	no	—
42	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
43	C₂×#11719945	—	2	2	no	—
44	C₂×#11719924	—	2	4	no	—
45	C₂×D₄₂₄	—	2	2	no	—
46	C₂²×#88322	D₄	1	2	yes	—
47	C₂²×#88322	D₄	1	2	yes	—
48	#13133217	C₂³	1	2	yes	—
49	#13133214	C₂³	1	2	yes	—
50	#13133205	—	2	2	no	—
51	#13133191	—	2	2	no	—
52	#13133191	—	2	2	no	—
53	C₄×D₂₁₂	—	2	2	no	—
54	C₄×D₂₁₂	—	2	2	no	—
55	#13133151	—	2	2	no	—
56	#13133148	—	2	4	no	—
57	#13133141	—	2	4	no	—
58	#13133140	—	2	4	no	—
59	#13133140	—	2	4	no	—
60	#13133139	—	2	4	no	—
61	#13133137	—	2	4	no	—
62	#13133137	—	2	4	no	—
63	#13133137	—	2	4	no	—
64	C₄×#88322	D₄	1	2	yes	—
65	C₄×#88322	D₄	1	2	yes	—
66	C₄×#88322	C₂³	1	1	yes	characteristic
67	C₄×#88322	C₂³	1	1	yes	characteristic
68	C₄×#88322	C₂³	1	1	yes	characteristic
69	C₄×#88322	C₂³	1	1	yes	characteristic
70	C₂×Q₁₆×C₅₃	—	2	2	no	—
71	C₂²×C₈×C₅₃	D₄	1	2	yes	—
72	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
73	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
74	#125×C₅₃	—	2	2	no	—
75	C₂²⋊Q₈×C₅₃	—	2	2	no	—
76	C₂²⋊Q₈×C₅₃	—	2	2	no	—
77	C₄×Q₈×C₅₃	—	2	2	no	—
78	C₄×Q₈×C₅₃	—	2	2	no	—
79	#117×C₅₃	—	2	2	no	—
80	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
81	#107×C₅₃	—	2	2	no	—
82	#103×C₅₃	—	2	4	no	—
83	C₄×C₈×C₅₃	D₄	1	2	yes	—
84	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₂×Q₈×C₅₃	—	2	2	no	—
87	C₂×Q₈×C₅₃	—	2	2	no	—
88	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₂²⋊D₅₃	—	4	2	no	—
92	#11719945	—	4	8	no	—
93	C₂×D₂₁₂	—	2	2	no	—
94	C₂×D₂₁₂	—	2	2	no	—
95	C₂×D₂₁₂	—	2	2	no	—
96	C₂×C₄×D₅₃	—	4	2	no	—
97	C₂×Dic₁₀₆	—	2	2	no	—
98	Q₁₆×C₅₃	—	4	4	no	—
99	C₂×C₈×C₅₃	—	2	4	no	—
100	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
102	C₄⋊C₄×C₅₃	—	2	2	no	—
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	—	4	2	no	—
107	C₄⋊C₄×C₅₃	—	4	2	no	—
108	C₂²⋊C₄×C₅₃	—	4	2	no	—
109	C₄²×C₅₃	—	4	2	no	—
110	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
114	#11719924	—	4	8	no	—
115	#11719921	—	4	2	no	—
116	#11719921	—	4	2	no	—
117	C₄⋊Dic₅₃	—	2	2	no	—
118	C₄⋊Dic₅₃	—	2	2	no	—
119	C₂×#88322	—	2	4	no	—
120	C₂×#88322	—	2	4	no	—
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	C₂×#88322	C₂×D₄	1	2	yes	—
125	D₄₂₄	—	4	4	no	—
126	C₂³×C₅₃	#127	1	1	yes	characteristic
127	C₂²×D₅₃	—	4	2	no	—
128	Q₈×C₅₃	—	4	4	no	—
129	Q₈×C₅₃	—	4	4	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	4	2	no	—
133	C₂×C₄×C₅₃	—	4	2	no	—
134	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
135	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
136	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
137	C₂×C₄×C₅₃	#135	1	4	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	C₂²×D₄	1	1	yes	characteristic
141	C₂²⋊D₅₃	—	8	4	no	—
142	C₂×Dic₅₃	—	4	2	no	—
143	D₂₁₂	—	4	4	no	—
144	D₂₁₂	—	4	4	no	—
145	D₂₁₂	—	4	4	no	—
146	C₄×D₅₃	—	8	4	no	—
147	Dic₁₀₆	—	4	4	no	—
148	C₈×C₅₃	—	2	2	no	—
149	C₈×C₅₃	#135	1	4	yes	—
150	#88322	—	2	2	no	—
151	#88322	—	2	2	no	—
152	#88322	—	2	2	no	—
153	#88322	#135	1	4	yes	—
154	#19600	—	53	1	no	maximal, Sylow
155	C₂²×C₅₃	—	2	1	no	—
156	C₂²×C₅₃	—	2	1	no	—
157	C₂²×C₅₃	#495	1	2	yes	—
158	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
159	C₂×D₅₃	—	8	2	no	—
160	C₂×D₅₃	—	8	2	no	—
161	C₂×D₅₃	—	8	2	no	—
162	C₄×C₅₃	—	2	1	no	—
163	C₄×C₅₃	—	2	1	no	—
164	C₄×C₅₃	—	8	2	no	—
165	C₄×C₅₃	—	8	2	no	—
166	C₄×C₅₃	#495	1	2	yes	—
167	C₄×C₅₃	C₂×#135	1	2	yes	—
168	C₄×C₅₃	C₂×#135	1	2	yes	—
169	C₄×C₅₃	C₂×#135	1	2	yes	—
170	Dic₅₃	—	8	2	no	—
171	#3170	D₅₃	1	1	yes	characteristic, complemented
172	#3169	—	53	1	no	—
173	#3167	—	53	2	no	—
174	#3131	—	53	2	no	—
175	#1711	—	53	2	no	—
176	#1710	—	53	1	no	—
177	#1709	—	53	1	no	—
178	#1708	—	53	2	no	—
179	#1707	—	53	1	no	—
180	#1706	—	53	1	no	—
181	C₂×C₈²	—	53	1	no	—
182	C₂×C₅₃	—	2	2	no	—
183	C₂×C₅₃	C₂×#495	1	1	yes	characteristic
184	C₂×C₅₃	#3147	1	1	yes	characteristic
185	C₂×C₅₃	#3147	1	1	yes	characteristic
186	D₅₃	—	16	2	no	—
187	#533	C₂×D₅₃	1	2	yes	—
188	#532	—	53	2	no	—
189	#500	—	53	1	no	—
190	#500	C₂×D₅₃	1	1	yes	characteristic
191	#498	—	53	2	no	—
192	#494	C₂×D₅₃	1	1	yes	characteristic
193	#493	—	53	1	no	—
194	#492	—	53	2	no	—
195	#488	—	106	4	no	—
196	#487	—	53	2	no	—
197	#487	—	53	2	no	—
198	#487	—	53	2	no	—
199	#487	C₂×D₅₃	1	2	yes	—
200	#486	—	53	2	no	—
201	#486	—	53	2	no	—
202	#486	—	53	2	no	—
203	#486	—	53	2	no	—
204	#467	—	2	2	no	—
205	#466	—	106	2	no	—
206	#465	—	106	4	no	—
207	C₄×Q₁₆	—	2	2	no	—
208	C₄×SD₁₆	—	106	4	no	—
209	C₄×D₈	—	106	2	no	—
210	#416	—	106	4	no	—
211	C₂×C₄×C₈	—	53	1	no	—
212	C₂×C₄×C₈	—	53	1	no	—
213	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
214	C₈²	—	53	1	no	—
215	C₈²	—	53	1	no	—
216	C₈²	—	53	1	no	—
217	C₈²	—	53	1	no	—
218	C₅₃	#19600	1	1	yes	characteristic, complemented, Sylow
219	C₂×#41	—	106	2	no	—
220	C₂×Q₁₆	—	2	2	no	—
221	C₂×SD₁₆	—	106	4	no	—
222	C₂×D₈	—	106	2	no	—
223	C₂²×C₈	—	53	2	no	—
224	C₂²×C₈	—	53	2	no	—
225	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
226	C₄⋊Q₈	—	53	2	no	—
227	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
228	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
229	#127	—	53	2	no	—
230	#125	—	2	2	no	—
231	#124	—	106	2	no	—
232	C₂²⋊Q₈	—	2	2	no	—
233	C₂²⋊Q₈	—	2	2	no	—
234	#121	—	106	2	no	—
235	#121	—	106	2	no	—
236	C₄×Q₈	—	2	2	no	—
237	C₄×Q₈	—	2	2	no	—
238	C₄×D₄	—	106	2	no	—
239	C₄×D₄	—	106	2	no	—
240	#117	—	2	2	no	—
241	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
242	#107	—	2	2	no	—
243	#107	—	106	2	no	—
244	#106	—	106	4	no	—
245	#103	—	2	4	no	—
246	#103	—	106	4	no	—
247	#103	—	106	4	no	—
248	#103	—	106	4	no	—
249	#102	—	106	4	no	—
250	#102	—	106	4	no	—
251	#102	—	106	4	no	—
252	#102	—	106	4	no	—
253	C₄×C₈	—	53	2	no	—
254	C₄×C₈	—	53	2	no	—
255	C₄×C₈	—	53	1	no	—
256	C₄×C₈	—	53	1	no	—
257	C₄×C₈	—	53	1	no	—
258	C₄×C₈	—	53	1	no	—
259	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
260	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
262	#41	—	212	8	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	106	2	no	—
266	C₂×D₄	—	106	2	no	—
267	C₂×D₄	—	106	2	no	—
268	C₂×D₄	—	106	2	no	—
269	C₂×D₄	—	212	2	no	—
270	C₂²×C₄	—	212	2	no	—
271	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
272	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
274	Q₁₆	—	4	4	no	—
275	SD₁₆	—	212	8	no	—
276	D₈	—	212	4	no	—
277	C₂×C₈	—	2	4	no	—
278	C₂×C₈	—	53	2	no	—
279	C₂×C₈	—	53	2	no	—
280	C₂×C₈	—	53	2	no	—
281	C₂×C₈	—	53	2	no	—
282	C₂×C₈	—	106	4	no	—
283	C₂×C₈	—	106	4	no	—
284	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
285	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	4	2	no	—
291	C₄⋊C₄	—	4	2	no	—
292	C₄⋊C₄	—	106	2	no	—
293	C₄⋊C₄	—	106	2	no	—
294	C₂²⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	212	2	no	—
296	C₂²⋊C₄	—	212	2	no	—
297	C₄²	—	4	2	no	—
298	C₄²	D₄×D₅₃	1	1	yes	characteristic
299	C₄²	D₄×D₅₃	1	1	yes	characteristic
300	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
301	C₄²	C₂³×D₅₃	1	1	yes	characteristic
302	C₂³	—	212	2	no	—
303	C₂³	#13133213	1	1	yes	characteristic
304	Q₈	—	4	4	no	—
305	Q₈	—	4	4	no	—
306	Q₈	—	212	4	no	—
307	D₄	—	424	4	no	—
308	D₄	—	212	4	no	—
309	D₄	—	212	4	no	—
310	D₄	—	212	4	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	424	4	no	—
316	C₂×C₄	—	212	2	no	—
317	C₂×C₄	#13133213	1	1	yes	characteristic
318	C₂×C₄	#13133213	1	1	yes	characteristic
319	C₂×C₄	#13133213	1	1	yes	characteristic
320	C₂×C₄	#13133240	1	4	yes	—
321	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
322	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
323	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	53	4	no	—
326	C₈	—	106	2	no	—
327	C₈	—	106	2	no	—
328	C₈	—	106	2	no	—
329	C₈	#13133237	1	4	yes	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	424	2	no	—
333	C₂²	—	424	2	no	—
334	C₂²	—	424	2	no	—
335	C₂²	#16811008	1	2	yes	—
336	C₂²	C₂×#13133213	1	1	yes	characteristic
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	8	2	no	—
340	C₄	—	8	2	no	—
341	C₄	—	424	2	no	—
342	C₄	#16811008	1	2	yes	—
343	C₄	#135×D₅₃	1	2	yes	—
344	C₄	C₂×#13133237	1	2	yes	—
345	C₄	C₂×#13133240	1	2	yes	—
346	C₂	—	2	2	no	—
347	C₂	—	848	2	no	—
348	C₂	C₂×#16811008	1	1	yes	characteristic
349	C₂	#20285139	1	1	yes	characteristic
350	C₂	#20285141	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133213
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3170	C₂
socle	C₂²×C₅₃	C₂×#127
prime radical	#20285294	C₂
hypercenter	#3170	D₅₃
hypocenter	C₅₃	#19600
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #20285275	C₂	semidirect
3. #20285040	C₂	semidirect
4. #20274328	C₂	semidirect
5. #20274324	C₂	semidirect
6. #20274322	C₂	semidirect
7. #20274318	C₂	semidirect
8. #20274309	C₂	semidirect
9. #20274308	C₂	semidirect
10. C₂×C₈×#88322	C₂	semidirect
11. C₅₃×#3170	C₂	semidirect
171. #3170	D₅₃	semidirect
218. C₅₃	#19600	semidirect

Maximal subgroups

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

subgroup	index
#20285294	2
#20285275	2
#20285040	2
#20274328	2
#20274324	2
#20274322	2
#20274318	2
#20274309	2
#20274308	2
C₂×C₈×#88322	2
C₅₃×#3170	2
#19600	53