Subgroups — #20283629

Order 6784.

452 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#16811388	C₂	1	2	yes	maximal, complemented
3	#16810746	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810745	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810744	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810738	C₂	1	2	yes	maximal, complemented
7	#16810737	C₂	1	2	yes	maximal, complemented
8	#108×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#16810638	C₂	1	2	yes	maximal, complemented
10	#16810580	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#444	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₅₃	C₂²	1	2	yes	—
13	#13133240	C₄	1	1	yes	characteristic, complemented
14	#13133239	C₂²	1	1	yes	characteristic
15	#13133238	C₄	1	2	yes	complemented
16	#13133237	C₂²	1	2	yes	—
17	#13133236	C₄	1	1	yes	characteristic, complemented
18	Q₁₆×D₅₃	C₄	1	1	yes	characteristic, complemented
19	SD₁₆×D₅₃	C₄	1	2	yes	complemented
20	D₈×D₅₃	C₄	1	1	yes	characteristic, complemented
21	#13133225	—	2	2	no	—
22	#13133225	C₂²	1	2	yes	complemented
23	#13133224	C₂²	1	2	yes	complemented
24	#13133223	—	2	2	no	—
25	#13133223	C₂²	1	2	yes	complemented
26	C₂×#11719912	—	2	2	no	—
27	C₂×#11719912	C₂²	1	1	yes	characteristic
28	M₁₆×D₅₃	—	2	2	no	—
29	M₁₆×D₅₃	C₂²	1	2	yes	—
30	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
31	#13133174	C₂²	1	1	yes	characteristic
32	#13133153	C₂²	1	1	yes	characteristic, complemented
33	#13133152	C₂²	1	2	yes	complemented
34	#13133145	C₂²	1	2	yes	complemented
35	#13133144	C₂²	1	2	yes	complemented
36	#13133143	C₂²	1	2	yes	complemented
37	#13133127	C₂²	1	1	yes	characteristic, complemented
38	C₄×#88322	C₂²	1	1	yes	characteristic, complemented
39	C₈×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
40	#135×C₅₃	C₂²	1	1	yes	characteristic
41	#131×C₅₃	C₂²	1	2	yes	complemented
42	#108×C₅₃	C₂²	1	1	yes	characteristic, complemented
43	#104×C₅₃	C₂²	1	2	yes	complemented
44	#97×C₅₃	C₂²	1	1	yes	characteristic, complemented
45	#41×C₅₃	C₂³	1	2	yes	—
46	#11719949	C₂×C₄	1	2	yes	—
47	Q₈×D₅₃	C₂×C₄	1	2	yes	—
48	#11719947	—	2	2	no	—
49	#11719947	C₂×C₄	1	2	yes	—
50	D₄×D₅₃	—	2	2	no	—
51	D₄×D₅₃	C₂×C₄	1	2	yes	—
52	#11719945	—	2	2	no	—
53	#11719945	C₂³	1	2	yes	—
54	C₂×C₄×D₅₃	—	2	2	no	—
55	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
56	Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
57	SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
58	D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
59	M₁₆×C₅₃	—	2	2	no	—
60	M₁₆×C₅₃	C₂³	1	2	yes	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
64	C₄²×C₅₃	C₂³	1	1	yes	characteristic
65	#11719925	C₂×C₄	1	2	yes	—
66	#11719924	C₂×C₄	1	2	yes	—
67	#11719923	C₂×C₄	1	2	yes	—
68	#11719922	C₂×C₄	1	2	yes	—
69	#11719921	—	2	1	no	—
70	#11719919	—	2	1	no	—
71	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
72	#11719917	—	2	2	no	—
73	#11719917	C₂³	1	2	yes	—
74	C₂×#88322	—	2	2	no	—
75	C₂×#88322	C₂³	1	1	yes	characteristic
76	C₂×#88322	C₂³	1	1	yes	characteristic
77	Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
78	D₄₂₄	C₂×C₄	1	1	yes	characteristic
79	#11719913	C₂×C₄	1	2	yes	—
80	#11719912	—	2	2	no	—
81	#11719912	—	2	2	no	—
82	#11719912	—	2	2	no	—
83	#11719912	D₄	1	2	yes	—
84	#11719912	D₄	1	2	yes	—
85	C₈×D₅₃	—	2	2	no	—
86	C₈×D₅₃	—	2	2	no	—
87	C₈×D₅₃	—	2	2	no	—
88	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
89	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
90	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
91	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
92	C₂²×D₅₃	—	2	2	no	—
93	C₂²×D₅₃	#41	1	1	yes	characteristic
94	Q₈×C₅₃	C₂²×C₄	1	2	yes	—
95	D₄×C₅₃	—	2	2	no	—
96	D₄×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	1	no	—
99	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
100	C₂²⋊D₅₃	—	2	2	no	—
101	C₂²⋊D₅₃	—	2	2	no	—
102	C₂²⋊D₅₃	—	2	2	no	—
103	C₂×Dic₅₃	—	2	2	no	—
104	C₂×Dic₅₃	—	2	1	no	—
105	C₂×Dic₅₃	#41	1	1	yes	characteristic
106	D₂₁₂	—	2	2	no	—
107	D₂₁₂	C₂²×C₄	1	2	yes	—
108	C₄×D₅₃	—	2	2	no	—
109	C₄×D₅₃	—	2	2	no	—
110	C₄×D₅₃	—	2	2	no	—
111	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
114	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
115	Dic₁₀₆	—	2	2	no	—
116	Dic₁₀₆	C₂²×C₄	1	2	yes	—
117	C₈×C₅₃	—	2	2	no	—
118	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₈×C₅₃	C₂×D₄	1	2	yes	—
121	#88322	—	2	2	no	—
122	#88322	C₂²×C₄	1	1	yes	characteristic
123	#88322	C₂²×C₄	1	1	yes	characteristic
124	#88322	C₂×D₄	1	2	yes	—
125	C₂²×C₅₃	—	2	2	no	—
126	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
127	C₂×D₅₃	—	2	2	no	—
128	C₂×D₅₃	—	4	1	no	—
129	C₂×D₅₃	—	8	2	no	—
130	C₂×D₅₃	C₄×D₄	1	1	yes	characteristic
131	C₂×D₅₃	C₄×D₄	1	1	yes	characteristic
132	C₄×C₅₃	—	2	2	no	—
133	C₄×C₅₃	—	4	1	no	—
134	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
135	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
136	Dic₅₃	—	2	2	no	—
137	Dic₅₃	—	4	1	no	—
138	Dic₅₃	C₄×D₄	1	1	yes	characteristic
139	Dic₅₃	C₄×D₄	1	1	yes	characteristic
140	#2957	—	53	1	no	maximal, Sylow
141	C₂×C₅₃	—	2	1	no	—
142	C₂×C₅₃	—	4	2	no	—
143	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
144	D₅₃	—	2	1	no	—
145	D₅₃	—	2	1	no	—
146	D₅₃	—	4	2	no	—
147	C₂×#135	—	53	1	no	—
148	#568	—	53	2	no	—
149	#444	—	53	1	no	—
150	#444	—	53	1	no	—
151	#444	—	53	1	no	—
152	#444	D₅₃	1	1	yes	characteristic, complemented
153	#443	—	53	2	no	—
154	#443	—	53	2	no	—
155	C₂×#108	—	53	1	no	—
156	#421	—	53	2	no	—
157	#405	—	53	1	no	—
158	C₅₃	#2957	1	1	yes	characteristic, complemented, Sylow
159	C₂×#41	—	53	2	no	—
160	#135	—	53	2	no	—
161	#135	—	53	1	no	—
162	#135	—	53	2	no	—
163	#135	—	53	1	no	—
164	#135	—	53	1	no	—
165	#135	C₂×D₅₃	1	1	yes	characteristic
166	C₂×Q₁₆	—	53	1	no	—
167	C₂×SD₁₆	—	53	2	no	—
168	C₂×D₈	—	53	1	no	—
169	#131	—	53	2	no	—
170	#131	—	53	2	no	—
171	#131	—	53	2	no	—
172	#131	—	106	2	no	—
173	#131	—	106	2	no	—
174	#131	C₂×D₅₃	1	2	yes	complemented
175	C₂×M₁₆	—	53	1	no	—
176	C₂×M₁₆	—	53	2	no	—
177	C₂×M₁₆	—	106	2	no	—
178	C₂×M₁₆	—	106	2	no	—
179	C₂²×C₈	—	53	1	no	—
180	#117	—	53	1	no	—
181	#108	—	53	2	no	—
182	#108	—	53	1	no	—
183	#108	C₂×D₅₃	1	1	yes	characteristic, complemented
184	#104	—	53	2	no	—
185	#104	—	53	2	no	—
186	#104	—	53	2	no	—
187	#104	C₂×D₅₃	1	2	yes	complemented
188	#97	—	53	1	no	—
189	#97	C₂×D₅₃	1	1	yes	characteristic, complemented
190	C₄×C₈	—	53	1	no	—
191	C₄×C₈	—	53	1	no	—
192	#41	—	53	2	no	—
193	#41	—	53	2	no	—
194	#41	—	53	2	no	—
195	#41	—	106	2	no	—
196	#41	—	106	2	no	—
197	#41	C₂²×D₅₃	1	2	yes	—
198	C₂×Q₈	—	53	2	no	—
199	C₂×D₄	—	53	2	no	—
200	C₂×D₄	—	106	2	no	—
201	C₂²×C₄	—	53	1	no	—
202	C₂²×C₄	—	106	2	no	—
203	Q₁₆	—	53	2	no	—
204	Q₁₆	—	53	1	no	—
205	Q₁₆	C₄×D₅₃	1	1	yes	characteristic
206	SD₁₆	—	53	2	no	—
207	SD₁₆	—	53	2	no	—
208	SD₁₆	—	53	2	no	—
209	SD₁₆	C₄×D₅₃	1	2	yes	—
210	D₈	—	53	2	no	—
211	D₈	—	53	1	no	—
212	D₈	C₄×D₅₃	1	1	yes	characteristic
213	M₁₆	—	2	2	no	—
214	M₁₆	—	53	2	no	—
215	M₁₆	—	53	2	no	—
216	M₁₆	—	53	2	no	—
217	M₁₆	—	106	2	no	—
218	M₁₆	—	106	2	no	—
219	M₁₆	—	106	2	no	—
220	M₁₆	—	106	2	no	—
221	M₁₆	C₂²×D₅₃	1	2	yes	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	53	1	no	—
224	C₂×C₈	—	53	1	no	—
225	C₂×C₈	—	53	1	no	—
226	C₂×C₈	—	53	1	no	—
227	C₂×C₈	—	53	1	no	—
228	C₂×C₈	—	53	1	no	—
229	C₂×C₈	—	106	2	no	—
230	C₂×C₈	—	106	2	no	—
231	C₂×C₈	—	106	2	no	—
232	C₂×C₈	—	106	2	no	—
233	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
234	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
235	C₄⋊C₄	—	106	1	no	—
236	C₂²⋊C₄	—	106	1	no	—
237	C₄²	—	53	1	no	—
238	C₄²	C₂²×D₅₃	1	1	yes	characteristic
239	C₂³	—	53	1	no	—
240	C₂³	—	106	2	no	—
241	Q₈	—	53	2	no	—
242	Q₈	—	106	2	no	—
243	Q₈	C₂×C₄×D₅₃	1	2	yes	—
244	D₄	—	2	2	no	—
245	D₄	—	53	2	no	—
246	D₄	—	106	2	no	—
247	D₄	—	106	2	no	—
248	D₄	—	106	2	no	—
249	D₄	—	106	2	no	—
250	D₄	C₂×C₄×D₅₃	1	2	yes	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	53	1	no	—
254	C₂×C₄	—	53	1	no	—
255	C₂×C₄	—	53	1	no	—
256	C₂×C₄	—	53	1	no	—
257	C₂×C₄	—	53	1	no	—
258	C₂×C₄	—	106	2	no	—
259	C₂×C₄	—	106	2	no	—
260	C₂×C₄	—	106	2	no	—
261	C₂×C₄	—	106	2	no	—
262	C₂×C₄	—	106	1	no	—
263	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	53	1	no	—
266	C₈	—	53	1	no	—
267	C₈	—	53	2	no	—
268	C₈	—	106	2	no	—
269	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
270	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
271	C₈	D₄×D₅₃	1	2	yes	—
272	C₂²	—	2	2	no	—
273	C₂²	—	53	1	no	—
274	C₂²	—	53	1	no	—
275	C₂²	—	106	2	no	—
276	C₂²	—	424	2	no	—
277	C₂²	—	212	1	no	—
278	C₂²	#41×D₅₃	1	1	yes	characteristic
279	C₄	—	2	2	no	—
280	C₄	—	4	1	no	—
281	C₄	—	53	1	no	—
282	C₄	—	53	1	no	—
283	C₄	—	106	2	no	—
284	C₄	—	212	1	no	—
285	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
286	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
287	C₂	—	2	1	no	—
288	C₂	—	4	2	no	—
289	C₂	—	106	1	no	—
290	C₂	—	106	1	no	—
291	C₂	—	212	2	no	—
292	C₂	C₄×D₄×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#444	C₂
socle	C₂×C₅₃	C₂×C₄×D₄
hypercenter	#444	D₅₃
hypocenter	C₅₃	#2957
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #444

Complements

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

N	H ≅ G/N	type
2. #16811388	C₂	semidirect
3. #16810746	C₂	semidirect
4. #16810745	C₂	semidirect
5. #16810744	C₂	semidirect
6. #16810738	C₂	semidirect
7. #16810737	C₂	semidirect
8. #108×D₅₃	C₂	semidirect
9. #16810638	C₂	semidirect
10. #16810580	C₂	semidirect
11. C₅₃×#444	C₂	semidirect
13. #13133240	C₄	semidirect
15. #13133238	C₄	semidirect
17. #13133236	C₄	semidirect
18. Q₁₆×D₅₃	C₄	semidirect
19. SD₁₆×D₅₃	C₄	semidirect
20. D₈×D₅₃	C₄	semidirect
22. #13133225	C₂²	semidirect
23. #13133224	C₂²	semidirect
25. #13133223	C₂²	semidirect
32. #13133153	C₂²	semidirect
33. #13133152	C₂²	semidirect
34. #13133145	C₂²	semidirect
35. #13133144	C₂²	semidirect
36. #13133143	C₂²	semidirect
37. #13133127	C₂²	semidirect
38. C₄×#88322	C₂²	semidirect
39. C₈×Dic₅₃	C₂²	semidirect
41. #131×C₅₃	C₂²	semidirect
42. #108×C₅₃	C₂²	semidirect
43. #104×C₅₃	C₂²	semidirect
44. #97×C₅₃	C₂²	semidirect
152. #444	D₅₃	semidirect
158. C₅₃	#2957	semidirect
174. #131	C₂×D₅₃	semidirect
183. #108	C₂×D₅₃	semidirect
187. #104	C₂×D₅₃	semidirect
189. #97	C₂×D₅₃	semidirect