Subgroups — #560×D₅₃ #20288828

Order 6784.

1344 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811364	C₂	1	2	yes	maximal, complemented
2	#16811363	C₂	1	3	yes	maximal, complemented
3	#16811362	C₂	1	3	yes	maximal, complemented
4	#16811361	C₂	1	6	yes	maximal, complemented
5	#16811360	C₂	1	1	yes	maximal, characteristic, complemented
6	#127×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#126×D₅₃	C₂	1	2	yes	maximal, complemented
8	#123×D₅₃	C₂	1	3	yes	maximal, complemented
9	#121×D₅₃	C₂	1	6	yes	maximal, complemented
10	#120×D₅₃	C₂	1	3	yes	maximal, complemented
11	C₅₃×#560	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂⁴×D₅₃	C₂²	1	1	yes	characteristic
13	C₂×D₄×D₅₃	C₂²	1	6	yes	—
14	C₂×D₄×D₅₃	C₂²	1	3	yes	—
15	C₂×D₄×D₅₃	C₂²	1	3	yes	complemented
16	C₂²×C₄×D₅₃	C₂²	1	3	yes	complemented
17	#13133214	C₂²	1	1	yes	characteristic, complemented
18	#13133213	C₂²	1	3	yes	complemented
19	#13133213	C₂²	1	3	yes	complemented
20	#13133212	C₂²	1	6	yes	complemented
21	#13133211	C₂²	1	2	yes	complemented
22	#13133210	C₂²	1	6	yes	complemented
23	#13133202	C₂²	1	6	yes	complemented
24	#13133201	C₂²	1	3	yes	complemented
25	#13133200	C₂²	1	3	yes	complemented
26	#13133199	C₂²	1	6	yes	complemented
27	#13133198	C₂²	1	3	yes	complemented
28	#13133192	C₂²	1	6	yes	complemented
29	#13133191	C₂²	1	6	yes	complemented
30	#13133190	C₂²	1	6	yes	complemented
31	#13133190	C₂²	1	6	yes	complemented
32	#13133189	C₂²	1	6	yes	complemented
33	#13133189	C₂²	1	6	yes	complemented
34	#13133188	C₂²	1	6	yes	complemented
35	#13133187	C₂²	1	6	yes	complemented
36	#13133187	C₂²	1	3	yes	complemented
37	#13133186	C₂²	1	3	yes	complemented
38	#13133185	C₂²	1	3	yes	complemented
39	#13133185	C₂²	1	6	yes	complemented
40	C₄⋊C₄×D₅₃	C₂²	1	6	yes	complemented
41	C₂²⋊C₄×D₅₃	C₂²	1	6	yes	—
42	C₂²⋊C₄×D₅₃	C₂²	1	3	yes	—
43	C₂²⋊C₄×D₅₃	C₂²	1	3	yes	complemented
44	C₄²×D₅₃	C₂²	1	1	yes	characteristic, complemented
45	#127×C₅₃	C₂²	1	1	yes	characteristic, complemented
46	#126×C₅₃	C₂²	1	2	yes	complemented
47	#123×C₅₃	C₂²	1	3	yes	complemented
48	#121×C₅₃	C₂²	1	6	yes	complemented
49	#120×C₅₃	C₂²	1	3	yes	complemented
50	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
51	C₂³×D₅₃	—	4	6	no	—
52	C₂³×D₅₃	C₂³	1	2	yes	—
53	C₂³×D₅₃	C₂³	1	1	yes	characteristic
54	C₂³×D₅₃	C₂³	1	3	yes	—
55	C₂³×D₅₃	C₂³	1	1	yes	characteristic
56	C₂³×D₅₃	C₂³	1	2	yes	—
57	C₂×D₄×C₅₃	C₂³	1	6	yes	—
58	C₂×D₄×C₅₃	C₂³	1	3	yes	—
59	C₂×D₄×C₅₃	C₂³	1	3	yes	complemented
60	C₂²×C₄×C₅₃	C₂³	1	3	yes	complemented
61	C₂×C₂²⋊D₅₃	C₂³	1	6	yes	—
62	C₂×C₂²⋊D₅₃	C₂³	1	3	yes	—
63	C₂×C₂²⋊D₅₃	C₂³	1	6	yes	—
64	C₂×C₂²⋊D₅₃	C₂³	1	3	yes	—
65	C₂×C₂²⋊D₅₃	C₂³	1	6	yes	complemented
66	C₂²×Dic₅₃	C₂³	1	3	yes	complemented
67	D₄×D₅₃	—	4	12	no	—
68	D₄×D₅₃	—	4	6	no	—
69	D₄×D₅₃	—	4	6	no	—
70	C₂×D₂₁₂	C₂³	1	6	yes	—
71	C₂×D₂₁₂	C₂³	1	3	yes	—
72	C₂×D₂₁₂	C₂³	1	3	yes	complemented
73	C₂×C₄×D₅₃	—	4	6	no	—
74	C₂×C₄×D₅₃	C₂³	1	3	yes	—
75	C₂×C₄×D₅₃	C₂³	1	6	yes	—
76	C₂×C₄×D₅₃	C₂³	1	6	yes	complemented
77	C₄⋊C₄×C₅₃	C₂³	1	6	yes	complemented
78	C₂²⋊C₄×C₅₃	C₂³	1	6	yes	—
79	C₂²⋊C₄×C₅₃	C₂³	1	3	yes	—
80	C₂²⋊C₄×C₅₃	C₂³	1	3	yes	complemented
81	C₄²×C₅₃	C₂³	1	1	yes	characteristic, complemented
82	C₂²⋊Dic₅₃	C₂³	1	6	yes	—
83	C₂²⋊Dic₅₃	C₂³	1	3	yes	—
84	C₂²⋊Dic₅₃	C₂³	1	3	yes	complemented
85	#11719921	C₂³	1	6	yes	—
86	#11719921	C₂³	1	6	yes	—
87	#11719921	C₂³	1	6	yes	—
88	#11719921	C₂³	1	6	yes	complemented
89	C₄⋊Dic₅₃	C₂³	1	6	yes	complemented
90	#11719919	C₂³	1	6	yes	complemented
91	#11719919	C₂³	1	6	yes	complemented
92	C₄×Dic₅₃	C₂³	1	3	yes	complemented
93	C₂³×C₅₃	—	4	3	no	—
94	C₂³×C₅₃	C₂⁴	1	3	yes	—
95	C₂³×C₅₃	C₂⁴	1	2	yes	—
96	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
97	C₂²×D₅₃	—	2	6	no	—
98	C₂²×D₅₃	—	2	12	no	—
99	C₂²×D₅₃	—	2	12	no	—
100	C₂²×D₅₃	—	2	6	no	—
101	C₂²×D₅₃	—	4	3	no	—
102	C₂²×D₅₃	—	4	6	no	—
103	C₂²×D₅₃	—	4	16	no	—
104	C₂²×D₅₃	C₂⁴	1	3	yes	—
105	C₂²×D₅₃	C₂⁴	1	2	yes	—
106	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
107	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
108	D₄×C₅₃	—	4	6	no	—
109	D₄×C₅₃	—	4	3	no	—
110	D₄×C₅₃	—	4	3	no	—
111	C₂×C₄×C₅₃	—	4	3	no	—
112	C₂×C₄×C₅₃	C₂⁴	1	6	yes	—
113	C₂×C₄×C₅₃	C₂⁴	1	3	yes	—
114	C₂²⋊D₅₃	—	4	6	no	—
115	C₂²⋊D₅₃	—	4	6	no	—
116	C₂²⋊D₅₃	—	4	3	no	—
117	C₂²⋊D₅₃	—	4	3	no	—
118	C₂²⋊D₅₃	—	4	6	no	—
119	C₂×Dic₅₃	—	4	3	no	—
120	C₂×Dic₅₃	C₂⁴	1	6	yes	—
121	C₂×Dic₅₃	C₂⁴	1	3	yes	—
122	D₂₁₂	—	4	6	no	—
123	D₂₁₂	—	4	3	no	—
124	D₂₁₂	—	4	3	no	—
125	C₄×D₅₃	—	2	12	no	—
126	C₄×D₅₃	—	2	6	no	—
127	C₄×D₅₃	—	4	6	no	—
128	C₂²×C₅₃	—	2	3	no	—
129	C₂²×C₅₃	—	2	6	no	—
130	C₂²×C₅₃	—	2	6	no	—
131	C₂²×C₅₃	—	2	3	no	—
132	C₂²×C₅₃	—	4	4	no	—
133	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
134	C₂×D₅₃	—	2	3	no	—
135	C₂×D₅₃	—	2	6	no	—
136	C₂×D₅₃	—	2	6	no	—
137	C₂×D₅₃	—	2	3	no	—
138	C₂×D₅₃	—	4	4	no	—
139	C₂×D₅₃	—	4	8	no	—
140	C₂×D₅₃	—	4	12	no	—
141	C₂×D₅₃	—	4	8	no	—
142	C₂×D₅₃	—	4	4	no	—
143	C₂×D₅₃	H(5,𝔽₂)	1	6	yes	—
144	C₄×C₅₃	—	2	6	no	—
145	C₄×C₅₃	—	2	3	no	—
146	Dic₅₃	—	2	6	no	—
147	Dic₅₃	—	2	3	no	—
148	C₂×#560	—	53	1	no	maximal, Sylow
149	C₂×C₅₃	—	4	2	no	—
150	C₂×C₅₃	—	4	1	no	—
151	C₂×C₅₃	—	4	3	no	—
152	C₂×C₅₃	C₂×H(5,𝔽₂)	1	3	yes	—
153	D₅₃	—	4	3	no	—
154	D₅₃	—	4	2	no	—
155	D₅₃	—	4	1	no	—
156	D₅₃	#560	1	4	yes	direct product factor
157	#560	—	53	1	no	—
158	#560	—	53	2	no	—
159	#560	—	53	3	no	—
160	#560	—	53	6	no	—
161	#560	—	53	3	no	—
162	#560	D₅₃	1	1	yes	characteristic, direct product factor
163	C₂×#127	—	53	1	no	—
164	C₂×#126	—	53	2	no	—
165	C₂×#123	—	53	3	no	—
166	C₂×#121	—	53	6	no	—
167	C₂×#120	—	53	3	no	—
168	C₅₃	C₂×#560	1	1	yes	characteristic, complemented, Sylow
169	C₂⁵	—	53	1	no	—
170	C₂²×D₄	—	53	6	no	—
171	C₂²×D₄	—	53	3	no	—
172	C₂²×D₄	—	53	3	no	—
173	C₂³×C₄	—	53	3	no	—
174	#127	—	53	3	no	—
175	#127	—	53	3	no	—
176	#127	—	53	1	no	—
177	#127	C₂×D₅₃	1	1	yes	characteristic, complemented
178	#126	—	53	6	no	—
179	#126	—	53	6	no	—
180	#126	—	53	2	no	—
181	#126	C₂×D₅₃	1	2	yes	complemented
182	#123	—	53	3	no	—
183	#123	—	53	3	no	—
184	#123	—	53	6	no	—
185	#123	—	53	6	no	—
186	#123	—	53	3	no	—
187	#123	C₂×D₅₃	1	3	yes	complemented
188	#121	—	53	6	no	—
189	#121	—	53	6	no	—
190	#121	—	53	6	no	—
191	#121	—	53	6	no	—
192	#121	—	53	6	no	—
193	#121	—	53	6	no	—
194	#121	—	53	6	no	—
195	#121	C₂×D₅₃	1	6	yes	complemented
196	#120	—	53	3	no	—
197	#120	—	53	6	no	—
198	#120	—	53	3	no	—
199	#120	—	53	3	no	—
200	#120	—	53	6	no	—
201	#120	C₂×D₅₃	1	3	yes	complemented
202	C₂×C₄⋊C₄	—	53	6	no	—
203	C₂×C₂²⋊C₄	—	53	6	no	—
204	C₂×C₂²⋊C₄	—	53	3	no	—
205	C₂×C₂²⋊C₄	—	53	3	no	—
206	C₂×C₄²	—	53	1	no	—
207	C₂⁴	—	53	2	no	—
208	C₂⁴	—	53	1	no	—
209	C₂⁴	—	53	3	no	—
210	C₂⁴	—	53	1	no	—
211	C₂⁴	—	53	2	no	—
212	C₂⁴	—	212	6	no	—
213	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
214	C₂×D₄	—	53	6	no	—
215	C₂×D₄	—	53	6	no	—
216	C₂×D₄	—	53	3	no	—
217	C₂×D₄	—	53	3	no	—
218	C₂×D₄	—	53	6	no	—
219	C₂×D₄	—	53	6	no	—
220	C₂×D₄	—	53	3	no	—
221	C₂×D₄	—	53	3	no	—
222	C₂×D₄	—	212	12	no	—
223	C₂×D₄	—	212	6	no	—
224	C₂×D₄	—	212	6	no	—
225	C₂×D₄	C₂²×D₅₃	1	6	yes	—
226	C₂×D₄	C₂²×D₅₃	1	3	yes	—
227	C₂×D₄	C₂²×D₅₃	1	3	yes	complemented
228	C₂²×C₄	—	53	3	no	—
229	C₂²×C₄	—	53	3	no	—
230	C₂²×C₄	—	53	6	no	—
231	C₂²×C₄	—	53	6	no	—
232	C₂²×C₄	—	212	6	no	—
233	C₂²×C₄	C₂²×D₅₃	1	3	yes	complemented
234	C₄⋊C₄	—	53	6	no	—
235	C₄⋊C₄	—	53	6	no	—
236	C₄⋊C₄	—	53	6	no	—
237	C₄⋊C₄	C₂²×D₅₃	1	6	yes	complemented
238	C₂²⋊C₄	—	53	6	no	—
239	C₂²⋊C₄	—	53	3	no	—
240	C₂²⋊C₄	—	53	3	no	—
241	C₂²⋊C₄	—	53	6	no	—
242	C₂²⋊C₄	—	53	6	no	—
243	C₂²⋊C₄	—	53	6	no	—
244	C₂²⋊C₄	—	53	6	no	—
245	C₂²⋊C₄	C₂²×D₅₃	1	6	yes	—
246	C₂²⋊C₄	C₂²×D₅₃	1	3	yes	—
247	C₂²⋊C₄	C₂²×D₅₃	1	3	yes	complemented
248	C₄²	—	53	3	no	—
249	C₄²	C₂²×D₅₃	1	1	yes	characteristic, complemented
250	C₂³	—	4	3	no	—
251	C₂³	—	53	1	no	—
252	C₂³	—	53	2	no	—
253	C₂³	—	53	1	no	—
254	C₂³	—	53	3	no	—
255	C₂³	—	106	6	no	—
256	C₂³	—	106	12	no	—
257	C₂³	—	106	12	no	—
258	C₂³	—	106	6	no	—
259	C₂³	—	212	6	no	—
260	C₂³	—	212	3	no	—
261	C₂³	—	212	16	no	—
262	C₂³	C₂³×D₅₃	1	2	yes	—
263	C₂³	C₂³×D₅₃	1	1	yes	characteristic
264	C₂³	C₂³×D₅₃	1	3	yes	—
265	D₄	—	4	6	no	—
266	D₄	—	4	3	no	—
267	D₄	—	4	3	no	—
268	D₄	—	212	6	no	—
269	D₄	—	212	3	no	—
270	D₄	—	212	3	no	—
271	D₄	—	212	6	no	—
272	D₄	—	212	6	no	—
273	D₄	—	212	3	no	—
274	D₄	—	212	6	no	—
275	D₄	—	212	3	no	—
276	C₂×C₄	—	4	3	no	—
277	C₂×C₄	—	53	3	no	—
278	C₂×C₄	—	53	6	no	—
279	C₂×C₄	—	106	12	no	—
280	C₂×C₄	—	106	6	no	—
281	C₂×C₄	—	212	6	no	—
282	C₂×C₄	—	212	3	no	—
283	C₂×C₄	C₂³×D₅₃	1	3	yes	—
284	C₂×C₄	C₂³×D₅₃	1	6	yes	—
285	C₂²	—	2	3	no	—
286	C₂²	—	2	6	no	—
287	C₂²	—	2	6	no	—
288	C₂²	—	2	3	no	—
289	C₂²	—	4	4	no	—
290	C₂²	—	53	6	no	—
291	C₂²	—	106	3	no	—
292	C₂²	—	106	6	no	—
293	C₂²	—	106	6	no	—
294	C₂²	—	106	3	no	—
295	C₂²	—	212	8	no	—
296	C₂²	—	212	4	no	—
297	C₂²	—	212	12	no	—
298	C₂²	—	212	8	no	—
299	C₂²	—	212	4	no	—
300	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
301	C₄	—	2	6	no	—
302	C₄	—	2	3	no	—
303	C₄	—	106	6	no	—
304	C₄	—	106	3	no	—
305	C₂	—	4	2	no	—
306	C₂	—	4	1	no	—
307	C₂	—	4	3	no	—
308	C₂	—	53	4	no	—
309	C₂	—	212	2	no	—
310	C₂	—	212	1	no	—
311	C₂	—	212	3	no	—
312	C₂	H(5,𝔽₂)×D₅₃	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	C₅₃×#560	C₂
hypercenter	#560	D₅₃
hypocenter	C₅₃	C₂×#560
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #560

Complements

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

N	H ≅ G/N	type
1. #16811364	C₂	semidirect
2. #16811363	C₂	semidirect
3. #16811362	C₂	semidirect
4. #16811361	C₂	semidirect
5. #16811360	C₂	semidirect
6. #127×D₅₃	C₂	semidirect
7. #126×D₅₃	C₂	semidirect
8. #123×D₅₃	C₂	semidirect
9. #121×D₅₃	C₂	semidirect
10. #120×D₅₃	C₂	semidirect
11. C₅₃×#560	C₂	semidirect
15. C₂×D₄×D₅₃	C₂²	semidirect
16. C₂²×C₄×D₅₃	C₂²	semidirect
17. #13133214	C₂²	semidirect
18. #13133213	C₂²	semidirect
19. #13133213	C₂²	semidirect
20. #13133212	C₂²	semidirect
21. #13133211	C₂²	semidirect
22. #13133210	C₂²	semidirect
23. #13133202	C₂²	semidirect
24. #13133201	C₂²	semidirect
25. #13133200	C₂²	semidirect
26. #13133199	C₂²	semidirect
27. #13133198	C₂²	semidirect
28. #13133192	C₂²	semidirect
29. #13133191	C₂²	semidirect
30. #13133190	C₂²	semidirect
31. #13133190	C₂²	semidirect
32. #13133189	C₂²	semidirect
33. #13133189	C₂²	semidirect
34. #13133188	C₂²	semidirect
35. #13133187	C₂²	semidirect
36. #13133187	C₂²	semidirect
37. #13133186	C₂²	semidirect
38. #13133185	C₂²	semidirect
39. #13133185	C₂²	semidirect
40. C₄⋊C₄×D₅₃	C₂²	semidirect
43. C₂²⋊C₄×D₅₃	C₂²	semidirect
44. C₄²×D₅₃	C₂²	semidirect
45. #127×C₅₃	C₂²	semidirect
46. #126×C₅₃	C₂²	semidirect
47. #123×C₅₃	C₂²	semidirect
48. #121×C₅₃	C₂²	semidirect
49. #120×C₅₃	C₂²	semidirect
59. C₂×D₄×C₅₃	C₂³	semidirect
60. C₂²×C₄×C₅₃	C₂³	semidirect
65. C₂×C₂²⋊D₅₃	C₂³	semidirect
66. C₂²×Dic₅₃	C₂³	semidirect
72. C₂×D₂₁₂	C₂³	semidirect
76. C₂×C₄×D₅₃	C₂³	semidirect
77. C₄⋊C₄×C₅₃	C₂³	semidirect
80. C₂²⋊C₄×C₅₃	C₂³	semidirect
81. C₄²×C₅₃	C₂³	semidirect
84. C₂²⋊Dic₅₃	C₂³	semidirect
88. #11719921	C₂³	semidirect
89. C₄⋊Dic₅₃	C₂³	semidirect
90. #11719919	C₂³	semidirect
91. #11719919	C₂³	semidirect
92. C₄×Dic₅₃	C₂³	semidirect
156. D₅₃	#560	direct (N × H)
162. #560	D₅₃	direct (N × H)
168. C₅₃	C₂×#560	semidirect
177. #127	C₂×D₅₃	semidirect
181. #126	C₂×D₅₃	semidirect
187. #123	C₂×D₅₃	semidirect
195. #121	C₂×D₅₃	semidirect
201. #120	C₂×D₅₃	semidirect
227. C₂×D₄	C₂²×D₅₃	semidirect
233. C₂²×C₄	C₂²×D₅₃	semidirect
237. C₄⋊C₄	C₂²×D₅₃	semidirect
247. C₂²⋊C₄	C₂²×D₅₃	semidirect
249. C₄²	C₂²×D₅₃	semidirect

Subgroups — #560×D53 #20288828

Conjugacy classes of subgroups

Complements

Subgroups — #560×D₅₃ #20288828