Subgroups — #20283627

Order 6784.

452 conjugacy classes of subgroups.

The subgroup lattice has 302 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	#16810743	C₂	1	2	yes	maximal, complemented
4	#16810742	C₂	1	2	yes	maximal, complemented
5	#16810741	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810740	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810739	C₂	1	1	yes	maximal, characteristic, complemented
8	#108×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#16810638	C₂	1	2	yes	maximal, complemented
10	#16810577	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#443	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	#13133128	C₂²	1	1	yes	characteristic, complemented
38	#13133127	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	C₄×C₈×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	1	yes	characteristic
84	#11719912	D₄	1	1	yes	characteristic
85	#11719912	D₄	1	1	yes	characteristic
86	#11719912	D₄	1	1	yes	characteristic
87	C₈×D₅₃	—	2	2	no	—
88	C₈×D₅₃	—	2	2	no	—
89	C₈×D₅₃	—	2	2	no	—
90	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
91	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
92	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
93	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
94	C₂²×D₅₃	—	2	2	no	—
95	C₂²×D₅₃	#41	1	1	yes	characteristic
96	Q₈×C₅₃	C₂²×C₄	1	2	yes	—
97	D₄×C₅₃	—	2	2	no	—
98	D₄×C₅₃	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	2	1	no	—
101	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
102	C₂²⋊D₅₃	—	2	2	no	—
103	C₂²⋊D₅₃	—	2	2	no	—
104	C₂²⋊D₅₃	—	2	2	no	—
105	C₂×Dic₅₃	—	2	2	no	—
106	C₂×Dic₅₃	—	2	1	no	—
107	C₂×Dic₅₃	#41	1	1	yes	characteristic
108	D₂₁₂	—	2	2	no	—
109	D₂₁₂	C₂²×C₄	1	2	yes	—
110	C₄×D₅₃	—	2	2	no	—
111	C₄×D₅₃	—	2	2	no	—
112	C₄×D₅₃	—	2	2	no	—
113	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
116	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
117	Dic₁₀₆	—	2	2	no	—
118	Dic₁₀₆	C₂²×C₄	1	2	yes	—
119	C₈×C₅₃	—	2	2	no	—
120	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
122	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
123	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
124	#88322	—	2	2	no	—
125	#88322	C₂²×C₄	1	1	yes	characteristic
126	#88322	C₂²×C₄	1	1	yes	characteristic
127	#88322	C₂×D₄	1	1	yes	characteristic
128	#88322	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₅₃	—	2	2	no	—
130	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
131	C₂×D₅₃	—	2	2	no	—
132	C₂×D₅₃	—	4	1	no	—
133	C₂×D₅₃	—	8	2	no	—
134	C₂×D₅₃	C₄×D₄	1	1	yes	characteristic
135	C₂×D₅₃	C₄×D₄	1	1	yes	characteristic
136	C₄×C₅₃	—	2	2	no	—
137	C₄×C₅₃	—	4	1	no	—
138	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
139	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
140	Dic₅₃	—	2	2	no	—
141	Dic₅₃	—	4	1	no	—
142	Dic₅₃	C₄×D₄	1	1	yes	characteristic
143	Dic₅₃	C₄×D₄	1	1	yes	characteristic
144	#2957	—	53	1	no	maximal, Sylow
145	C₂×C₅₃	—	2	1	no	—
146	C₂×C₅₃	—	4	2	no	—
147	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
148	D₅₃	—	2	1	no	—
149	D₅₃	—	2	1	no	—
150	D₅₃	—	4	2	no	—
151	C₂×#135	—	53	1	no	—
152	#568	—	53	2	no	—
153	#444	—	53	2	no	—
154	#444	—	53	2	no	—
155	#443	—	53	1	no	—
156	#443	—	53	1	no	—
157	#443	—	53	1	no	—
158	#443	D₅₃	1	1	yes	characteristic, complemented
159	C₂×#108	—	53	1	no	—
160	#421	—	53	2	no	—
161	#405	—	53	1	no	—
162	C₅₃	#2957	1	1	yes	characteristic, complemented, Sylow
163	C₂×#41	—	53	2	no	—
164	#135	—	53	2	no	—
165	#135	—	53	1	no	—
166	#135	—	53	2	no	—
167	#135	—	53	1	no	—
168	#135	—	53	1	no	—
169	#135	C₂×D₅₃	1	1	yes	characteristic
170	C₂×Q₁₆	—	53	1	no	—
171	C₂×SD₁₆	—	53	2	no	—
172	C₂×D₈	—	53	1	no	—
173	#131	—	53	2	no	—
174	#131	—	53	2	no	—
175	#131	—	53	2	no	—
176	#131	—	106	2	no	—
177	#131	—	106	2	no	—
178	#131	C₂×D₅₃	1	2	yes	complemented
179	C₂×M₁₆	—	53	1	no	—
180	C₂×M₁₆	—	53	2	no	—
181	C₂×M₁₆	—	106	2	no	—
182	C₂×M₁₆	—	106	2	no	—
183	C₂²×C₈	—	53	1	no	—
184	#117	—	53	1	no	—
185	#108	—	53	2	no	—
186	#108	—	53	1	no	—
187	#108	C₂×D₅₃	1	1	yes	characteristic, complemented
188	#104	—	53	2	no	—
189	#104	—	53	2	no	—
190	#104	—	53	2	no	—
191	#104	C₂×D₅₃	1	2	yes	complemented
192	#97	—	53	1	no	—
193	#97	—	53	1	no	—
194	C₄×C₈	—	53	1	no	—
195	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic, complemented
196	#41	—	53	2	no	—
197	#41	—	53	2	no	—
198	#41	—	53	2	no	—
199	#41	—	106	2	no	—
200	#41	—	106	2	no	—
201	#41	C₂²×D₅₃	1	2	yes	—
202	C₂×Q₈	—	53	2	no	—
203	C₂×D₄	—	53	2	no	—
204	C₂×D₄	—	106	2	no	—
205	C₂²×C₄	—	53	1	no	—
206	C₂²×C₄	—	106	2	no	—
207	Q₁₆	—	53	2	no	—
208	Q₁₆	—	53	1	no	—
209	Q₁₆	C₄×D₅₃	1	1	yes	characteristic
210	SD₁₆	—	53	2	no	—
211	SD₁₆	—	53	2	no	—
212	SD₁₆	—	53	2	no	—
213	SD₁₆	C₄×D₅₃	1	2	yes	—
214	D₈	—	53	2	no	—
215	D₈	—	53	1	no	—
216	D₈	C₄×D₅₃	1	1	yes	characteristic
217	M₁₆	—	2	2	no	—
218	M₁₆	—	53	1	no	—
219	M₁₆	—	53	1	no	—
220	M₁₆	—	53	1	no	—
221	M₁₆	—	53	1	no	—
222	M₁₆	—	53	2	no	—
223	M₁₆	—	106	2	no	—
224	M₁₆	—	106	2	no	—
225	M₁₆	—	106	2	no	—
226	M₁₆	—	106	2	no	—
227	M₁₆	C₂²×D₅₃	1	2	yes	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	53	1	no	—
230	C₂×C₈	—	53	1	no	—
231	C₂×C₈	—	53	1	no	—
232	C₂×C₈	—	53	1	no	—
233	C₂×C₈	—	53	1	no	—
234	C₂×C₈	—	53	1	no	—
235	C₂×C₈	—	106	2	no	—
236	C₂×C₈	—	106	2	no	—
237	C₂×C₈	—	106	2	no	—
238	C₂×C₈	—	106	2	no	—
239	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
241	C₄⋊C₄	—	106	1	no	—
242	C₂²⋊C₄	—	106	1	no	—
243	C₄²	—	53	1	no	—
244	C₄²	C₂²×D₅₃	1	1	yes	characteristic
245	C₂³	—	53	1	no	—
246	C₂³	—	106	2	no	—
247	Q₈	—	53	2	no	—
248	Q₈	—	106	2	no	—
249	Q₈	C₂×C₄×D₅₃	1	2	yes	—
250	D₄	—	2	2	no	—
251	D₄	—	53	2	no	—
252	D₄	—	106	2	no	—
253	D₄	—	106	2	no	—
254	D₄	—	106	2	no	—
255	D₄	—	106	2	no	—
256	D₄	C₂×C₄×D₅₃	1	2	yes	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	53	1	no	—
260	C₂×C₄	—	53	1	no	—
261	C₂×C₄	—	53	1	no	—
262	C₂×C₄	—	53	1	no	—
263	C₂×C₄	—	53	1	no	—
264	C₂×C₄	—	106	2	no	—
265	C₂×C₄	—	106	2	no	—
266	C₂×C₄	—	106	2	no	—
267	C₂×C₄	—	106	2	no	—
268	C₂×C₄	—	106	1	no	—
269	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	53	1	no	—
272	C₈	—	53	1	no	—
273	C₈	—	53	1	no	—
274	C₈	—	53	1	no	—
275	C₈	—	106	2	no	—
276	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
277	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
278	C₈	D₄×D₅₃	1	1	yes	characteristic
279	C₈	D₄×D₅₃	1	1	yes	characteristic
280	C₂²	—	2	2	no	—
281	C₂²	—	53	1	no	—
282	C₂²	—	53	1	no	—
283	C₂²	—	106	2	no	—
284	C₂²	—	424	2	no	—
285	C₂²	—	212	1	no	—
286	C₂²	#41×D₅₃	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	4	1	no	—
289	C₄	—	53	1	no	—
290	C₄	—	53	1	no	—
291	C₄	—	106	2	no	—
292	C₄	—	212	1	no	—
293	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
294	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
295	C₂	—	2	1	no	—
296	C₂	—	4	2	no	—
297	C₂	—	106	1	no	—
298	C₂	—	106	1	no	—
299	C₂	—	212	2	no	—
300	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×#443	C₂
socle	C₂×C₅₃	C₂×C₄×D₄
hypercenter	#443	D₅₃
hypocenter	C₅₃	#2957
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
2. #16811388	C₂	semidirect
3. #16810743	C₂	semidirect
4. #16810742	C₂	semidirect
5. #16810741	C₂	semidirect
6. #16810740	C₂	semidirect
7. #16810739	C₂	semidirect
8. #108×D₅₃	C₂	semidirect
9. #16810638	C₂	semidirect
10. #16810577	C₂	semidirect
11. C₅₃×#443	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. #13133128	C₂²	semidirect
38. #13133127	C₂²	semidirect
39. C₈×Dic₅₃	C₂²	semidirect
41. #131×C₅₃	C₂²	semidirect
42. #108×C₅₃	C₂²	semidirect
43. #104×C₅₃	C₂²	semidirect
44. C₄×C₈×C₅₃	C₂²	semidirect
158. #443	D₅₃	semidirect
162. C₅₃	#2957	semidirect
178. #131	C₂×D₅₃	semidirect
187. #108	C₂×D₅₃	semidirect
191. #104	C₂×D₅₃	semidirect
195. C₄×C₈	C₂×D₅₃	semidirect