Subgroups — #20283562

Order 6784.

528 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#16811095	C₂	1	2	yes	maximal, complemented
3	#16810733	C₂	1	2	yes	maximal, complemented
4	#16810732	C₂	1	2	yes	maximal, complemented
5	C₄×D₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
6	D₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810705	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810660	C₂	1	1	yes	maximal, characteristic, complemented
9	#16810626	C₂	1	2	yes	maximal, complemented
10	#16810577	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₅₃	C₂²	1	2	yes	—
13	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
14	C₂×#11719922	C₂²	1	2	yes	—
15	D₈×D₅₃	C₄	1	8	yes	complemented
16	C₂×#11719912	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
18	C₄×D₂₁₂	C₂²	1	2	yes	complemented
19	D₄×Dic₅₃	C₂²	1	2	yes	complemented
20	#13133179	C₂²	1	2	yes	complemented
21	C₄×C₂²⋊D₅₃	—	2	2	no	—
22	#13133177	—	2	2	no	—
23	#13133175	C₂²	1	2	yes	—
24	#13133174	C₂²	1	1	yes	characteristic
25	C₂×#11719921	—	2	2	no	—
26	C₂²⋊C₄×D₅₃	—	2	2	no	—
27	#13133151	C₂²	1	1	yes	characteristic, complemented
28	#13133148	C₂²	1	2	yes	complemented
29	#13133139	C₂²	1	2	yes	complemented
30	#13133138	C₂²	1	2	yes	complemented
31	#13133137	C₂²	1	2	yes	complemented
32	#13133128	C₂²	1	1	yes	characteristic, complemented
33	#13133127	C₂²	1	1	yes	characteristic, complemented
34	C₈×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
35	C₂×D₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
37	#107×C₅₃	C₂²	1	1	yes	characteristic, complemented
38	#102×C₅₃	C₂²	1	2	yes	complemented
39	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
40	C₂³×D₅₃	—	2	2	no	—
41	C₂×D₄×C₅₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₅₃	—	2	2	no	—
43	C₂×C₂²⋊D₅₃	—	2	2	no	—
44	C₂²×Dic₅₃	—	2	2	no	—
45	D₄×D₅₃	—	2	4	no	—
46	D₄×D₅₃	C₂×C₄	1	8	yes	complemented
47	C₂×D₂₁₂	C₂³	1	2	yes	—
48	C₂×C₄×D₅₃	—	2	2	no	—
49	C₂×C₄×D₅₃	—	2	2	no	—
50	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
51	D₈×C₅₃	C₂×C₄	1	4	yes	complemented
52	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
55	C₂²⋊C₄×C₅₃	—	2	2	no	—
56	C₄²×C₅₃	C₂³	1	1	yes	characteristic
57	C₂²⋊Dic₅₃	—	2	2	no	—
58	#11719922	C₂×C₄	1	8	yes	complemented
59	#11719921	—	2	2	no	—
60	#11719921	—	2	2	no	—
61	#11719921	—	2	2	no	—
62	#11719921	D₄	1	1	yes	characteristic
63	#11719921	D₄	1	1	yes	characteristic
64	C₄⋊Dic₅₃	C₂³	1	2	yes	—
65	#11719919	D₄	1	1	yes	characteristic
66	#11719919	D₄	1	1	yes	characteristic
67	C₄×Dic₅₃	—	2	2	no	—
68	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
69	C₂×#88322	C₂³	1	1	yes	characteristic
70	C₂×#88322	C₂³	1	1	yes	characteristic
71	D₄₂₄	C₂×C₄	1	4	yes	complemented
72	#11719912	—	2	2	no	—
73	C₈×D₅₃	C₂×C₄	1	4	yes	complemented
74	C₂³×C₅₃	—	2	2	no	—
75	C₂²×D₅₃	—	2	2	no	—
76	C₂²×D₅₃	—	2	8	no	—
77	C₂²×D₅₃	—	8	2	no	—
78	C₂²×D₅₃	—	8	2	no	—
79	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
80	D₄×C₅₃	—	2	2	no	—
81	D₄×C₅₃	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	8	2	no	—
84	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
87	C₂²⋊D₅₃	—	2	8	no	—
88	C₂×Dic₅₃	—	2	2	no	—
89	C₂×Dic₅₃	—	8	2	no	—
90	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
93	D₂₁₂	—	2	2	no	—
94	D₂₁₂	C₂²×C₄	1	4	yes	—
95	C₄×D₅₃	—	8	2	no	—
96	C₄×D₅₃	—	8	2	no	—
97	C₄×D₅₃	C₂²×C₄	1	2	yes	—
98	C₄×D₅₃	#41	1	2	yes	—
99	C₈×C₅₃	—	2	1	no	—
100	C₈×C₅₃	C₂²×C₄	1	2	yes	—
101	#88322	—	2	1	no	—
102	#88322	C₂²×C₄	1	2	yes	—
103	C₂²×C₅₃	—	2	4	no	—
104	C₂²×C₅₃	—	4	2	no	—
105	C₂²×C₅₃	—	4	2	no	—
106	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
107	C₂×D₅₃	—	2	1	no	—
108	C₂×D₅₃	—	2	1	no	—
109	C₂×D₅₃	—	2	4	no	—
110	C₂×D₅₃	—	4	2	no	—
111	C₂×D₅₃	—	4	2	no	—
112	C₂×D₅₃	—	8	8	no	—
113	C₂×D₅₃	C₄×D₄	1	2	yes	—
114	C₄×C₅₃	—	2	1	no	—
115	C₄×C₅₃	—	2	1	no	—
116	C₄×C₅₃	—	4	2	no	—
117	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
118	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
119	Dic₅₃	—	2	1	no	—
120	Dic₅₃	—	2	1	no	—
121	Dic₅₃	—	4	2	no	—
122	Dic₅₃	C₄×D₄	1	2	yes	—
123	#2948	—	53	1	no	maximal, Sylow
124	C₂×C₅₃	—	4	4	no	—
125	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
126	C₂×C₅₃	#576	1	1	yes	characteristic
127	C₂×C₅₃	#576	1	1	yes	characteristic
128	D₅₃	—	2	2	no	—
129	D₅₃	—	4	4	no	—
130	C₂²×D₈	—	53	1	no	—
131	#518	—	53	2	no	—
132	#442	—	53	2	no	—
133	#442	—	53	2	no	—
134	C₄×D₈	—	53	1	no	—
135	C₄×D₈	—	53	1	no	—
136	C₄×D₈	—	53	1	no	—
137	C₄×D₈	D₅₃	1	1	yes	characteristic, complemented
138	#427	—	53	1	no	—
139	#418	—	53	2	no	—
140	#405	—	53	1	no	—
141	C₅₃	#2948	1	1	yes	characteristic, complemented, Sylow
142	C₂²×D₄	—	53	2	no	—
143	C₂×D₈	—	53	2	no	—
144	C₂×D₈	—	53	8	no	—
145	C₂×D₈	—	53	1	no	—
146	C₂×D₈	C₂×D₅₃	1	1	yes	characteristic
147	C₂×M₁₆	—	53	1	no	—
148	C₂²×C₈	—	53	1	no	—
149	C₄×D₄	—	53	2	no	—
150	C₄×D₄	—	53	2	no	—
151	C₄×D₄	—	53	2	no	—
152	C₄×D₄	—	106	2	no	—
153	C₄×D₄	—	106	2	no	—
154	C₄×D₄	C₂×D₅₃	1	2	yes	complemented
155	#117	—	53	1	no	—
156	#117	—	53	2	no	—
157	C₂×C₂²⋊C₄	—	106	2	no	—
158	C₂×C₂²⋊C₄	—	106	2	no	—
159	#107	—	53	1	no	—
160	#107	C₂×D₅₃	1	1	yes	characteristic, complemented
161	#106	—	53	2	no	—
162	#102	—	53	2	no	—
163	#102	—	53	2	no	—
164	#102	—	53	2	no	—
165	#102	C₂×D₅₃	1	2	yes	complemented
166	#97	—	53	1	no	—
167	#97	—	53	1	no	—
168	C₄×C₈	—	53	1	no	—
169	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic, complemented
170	C₂⁴	—	106	2	no	—
171	C₂×D₄	—	53	8	no	—
172	C₂×D₄	—	53	2	no	—
173	C₂×D₄	—	106	4	no	—
174	C₂×D₄	—	106	2	no	—
175	C₂×D₄	C₂²×D₅₃	1	2	yes	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	53	1	no	—
178	C₂²×C₄	—	106	2	no	—
179	C₂²×C₄	—	106	2	no	—
180	C₂²×C₄	—	106	2	no	—
181	D₈	—	53	8	no	—
182	D₈	—	53	4	no	—
183	D₈	C₄×D₅₃	1	4	yes	complemented
184	M₁₆	—	106	2	no	—
185	C₂×C₈	—	53	1	no	—
186	C₂×C₈	—	53	4	no	—
187	C₂×C₈	—	53	1	no	—
188	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
189	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
190	C₄⋊C₄	—	53	1	no	—
191	C₄⋊C₄	—	53	1	no	—
192	C₄⋊C₄	—	53	2	no	—
193	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₂²⋊C₄	—	53	1	no	—
196	C₂²⋊C₄	—	53	1	no	—
197	C₂²⋊C₄	—	106	2	no	—
198	C₂²⋊C₄	—	106	2	no	—
199	C₂²⋊C₄	—	106	2	no	—
200	C₂²⋊C₄	—	106	2	no	—
201	C₄²	—	53	1	no	—
202	C₄²	—	106	2	no	—
203	C₄²	C₂²×D₅₃	1	1	yes	characteristic
204	C₂³	—	2	2	no	—
205	C₂³	—	53	1	no	—
206	C₂³	—	106	8	no	—
207	C₂³	—	106	2	no	—
208	C₂³	—	424	2	no	—
209	C₂³	—	424	2	no	—
210	D₄	—	2	2	no	—
211	D₄	—	53	4	no	—
212	D₄	—	106	8	no	—
213	D₄	—	106	2	no	—
214	D₄	C₂×C₄×D₅₃	1	4	yes	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	53	1	no	—
218	C₂×C₄	—	53	2	no	—
219	C₂×C₄	—	53	2	no	—
220	C₂×C₄	—	53	1	no	—
221	C₂×C₄	—	53	1	no	—
222	C₂×C₄	—	106	2	no	—
223	C₂×C₄	—	424	2	no	—
224	C₂×C₄	—	424	2	no	—
225	C₂×C₄	—	424	2	no	—
226	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
227	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
228	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
229	C₈	—	2	1	no	—
230	C₈	—	53	2	no	—
231	C₈	—	106	1	no	—
232	C₈	C₂×C₄×D₅₃	1	2	yes	—
233	C₂²	—	2	4	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	53	2	no	—
237	C₂²	—	106	1	no	—
238	C₂²	—	106	1	no	—
239	C₂²	—	106	4	no	—
240	C₂²	—	424	8	no	—
241	C₂²	—	212	2	no	—
242	C₂²	—	212	2	no	—
243	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	4	2	no	—
247	C₄	—	53	2	no	—
248	C₄	—	106	1	no	—
249	C₄	—	106	1	no	—
250	C₄	—	212	2	no	—
251	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
252	C₄	#41×D₅₃	1	1	yes	characteristic
253	C₂	—	4	4	no	—
254	C₂	—	106	2	no	—
255	C₂	—	212	4	no	—
256	C₂	C₄×D₄×D₅₃	1	1	yes	characteristic
257	C₂	#16811424	1	1	yes	characteristic
258	C₂	#16811425	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₄×D₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	C₄×D₈	D₅₃
hypocenter	C₅₃	#2948
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
2. #16811095	C₂	semidirect
3. #16810733	C₂	semidirect
4. #16810732	C₂	semidirect
5. C₄×D₄₂₄	C₂	semidirect
6. D₈×Dic₅₃	C₂	semidirect
7. #16810705	C₂	semidirect
8. #16810660	C₂	semidirect
9. #16810626	C₂	semidirect
10. #16810577	C₂	semidirect
11. C₄×D₈×C₅₃	C₂	semidirect
15. D₈×D₅₃	C₄	semidirect
18. C₄×D₂₁₂	C₂²	semidirect
19. D₄×Dic₅₃	C₂²	semidirect
20. #13133179	C₂²	semidirect
27. #13133151	C₂²	semidirect
28. #13133148	C₂²	semidirect
29. #13133139	C₂²	semidirect
30. #13133138	C₂²	semidirect
31. #13133137	C₂²	semidirect
32. #13133128	C₂²	semidirect
33. #13133127	C₂²	semidirect
34. C₈×Dic₅₃	C₂²	semidirect
36. C₄×D₄×C₅₃	C₂²	semidirect
37. #107×C₅₃	C₂²	semidirect
38. #102×C₅₃	C₂²	semidirect
39. C₄×C₈×C₅₃	C₂²	semidirect
46. D₄×D₅₃	C₂×C₄	semidirect
51. D₈×C₅₃	C₂×C₄	semidirect
58. #11719922	C₂×C₄	semidirect
71. D₄₂₄	C₂×C₄	semidirect
73. C₈×D₅₃	C₂×C₄	semidirect
137. C₄×D₈	D₅₃	semidirect
141. C₅₃	#2948	semidirect
154. C₄×D₄	C₂×D₅₃	semidirect
160. #107	C₂×D₅₃	semidirect
165. #102	C₂×D₅₃	semidirect
169. C₄×C₈	C₂×D₅₃	semidirect
183. D₈	C₄×D₅₃	semidirect