Subgroups — #467×D₅₃ #20284285

Order 6784.

512 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊Q₈×D₅₃	C₂	1	2	yes	maximal
4	#16810865	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810864	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810863	C₂	1	2	yes	maximal, complemented
7	#16810862	C₂	1	2	yes	maximal, complemented
8	#16810861	C₂	1	1	yes	maximal, characteristic, complemented
9	#107×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#103×D₅₃	C₂	1	2	yes	maximal, complemented
11	C₅₃×#467	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₅₃	C₂²	1	2	yes	—
13	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic, complemented
15	C₂×#11719925	C₂²	1	2	yes	complemented
16	Q₁₆×D₅₃	—	2	4	no	—
17	C₂²×#88322	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₅₃	—	2	4	no	—
19	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
23	#13133197	C₂²	1	2	yes	—
24	#13133196	C₂²	1	2	yes	—
25	#13133195	—	2	2	no	—
26	#13133194	—	2	2	no	—
27	#13133193	C₂²	1	2	yes	—
28	C₄⋊C₄×D₅₃	—	2	2	no	—
29	C₄⋊C₄×D₅₃	C₂²	1	2	yes	—
30	C₂²⋊C₄×D₅₃	—	2	2	no	—
31	#13133151	C₂²	1	1	yes	characteristic, complemented
32	#13133150	C₂²	1	2	yes	complemented
33	#13133142	C₂²	1	2	yes	complemented
34	#13133141	C₂²	1	2	yes	complemented
35	#13133140	C₂²	1	2	yes	complemented
36	C₂×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
37	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
39	#107×C₅₃	C₂²	1	1	yes	characteristic, complemented
40	#103×C₅₃	C₂²	1	2	yes	complemented
41	C₂³×D₅₃	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
45	Q₈×D₅₃	—	2	4	no	—
46	Q₈×D₅₃	—	2	4	no	—
47	C₂×C₄×D₅₃	—	2	2	no	—
48	C₂×C₄×D₅₃	—	2	2	no	—
49	C₂×C₄×D₅₃	—	2	2	no	—
50	C₂×C₄×D₅₃	—	2	2	no	—
51	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
52	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
53	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
54	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
55	C₂×Dic₁₀₆	—	2	2	no	—
56	C₂×Dic₁₀₆	C₂³	1	2	yes	—
57	Q₁₆×C₅₃	—	2	2	no	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₅₃	—	2	2	no	—
62	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
63	C₂²⋊C₄×C₅₃	—	2	2	no	—
64	C₂²⋊Dic₅₃	—	2	2	no	—
65	#11719925	—	2	4	no	—
66	#11719921	—	2	2	no	—
67	#11719921	—	2	2	no	—
68	C₄⋊Dic₅₃	—	2	2	no	—
69	C₄⋊Dic₅₃	C₂³	1	2	yes	—
70	#11719919	—	2	2	no	—
71	#11719919	—	2	2	no	—
72	#11719919	—	2	2	no	—
73	C₂×#88322	—	2	2	no	—
74	C₂×#88322	C₂³	1	1	yes	characteristic
75	C₂×#88322	C₂³	1	1	yes	characteristic
76	Dic₂₁₂	—	2	2	no	—
77	C₈×D₅₃	—	2	2	no	—
78	C₈×D₅₃	—	2	2	no	—
79	C₈×D₅₃	—	2	2	no	—
80	C₈×D₅₃	D₄	1	4	yes	—
81	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×D₅₃	—	2	2	no	—
83	C₂²×D₅₃	—	2	2	no	—
84	C₂²×D₅₃	Q₁₆	1	4	yes	complemented
85	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
87	Q₈×C₅₃	—	2	2	no	—
88	Q₈×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
95	C₂×Dic₅₃	—	2	2	no	—
96	C₂×Dic₅₃	—	2	2	no	—
97	C₂×Dic₅₃	—	2	2	no	—
98	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
99	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
100	C₄×D₅₃	—	2	2	no	—
101	C₄×D₅₃	—	2	1	no	—
102	C₄×D₅₃	—	2	1	no	—
103	C₄×D₅₃	—	2	2	no	—
104	C₄×D₅₃	—	4	4	no	—
105	C₄×D₅₃	—	4	4	no	—
106	C₄×D₅₃	C₂×D₄	1	2	yes	—
107	C₄×D₅₃	#41	1	2	yes	—
108	Dic₁₀₆	—	2	2	no	—
109	Dic₁₀₆	—	2	2	no	—
110	Dic₁₀₆	—	4	4	no	—
111	C₈×C₅₃	—	2	1	no	—
112	C₈×C₅₃	C₂×D₄	1	2	yes	—
113	#88322	—	2	1	no	—
114	#88322	C₂×D₄	1	2	yes	—
115	C₂²×C₅₃	—	2	1	no	—
116	C₂²×C₅₃	—	2	1	no	—
117	C₂²×C₅₃	C₂×Q₁₆	1	2	yes	complemented
118	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×D₅₃	—	2	1	no	—
120	C₂×D₅₃	—	2	1	no	—
121	C₂×D₅₃	—	2	8	no	—
122	C₂×D₅₃	#121	1	2	yes	—
123	C₂×D₅₃	C₂×Q₁₆	1	2	yes	—
124	C₂×D₅₃	C₂×Q₁₆	1	2	yes	complemented
125	C₂×D₅₃	#135	1	2	yes	—
126	C₄×C₅₃	—	2	1	no	—
127	C₄×C₅₃	—	4	2	no	—
128	C₄×C₅₃	—	4	2	no	—
129	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
130	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
131	Dic₅₃	—	2	1	no	—
132	Dic₅₃	—	4	2	no	—
133	Dic₅₃	—	4	2	no	—
134	Dic₅₃	#121	1	2	yes	—
135	C₂×#467	—	53	1	no	maximal, Sylow
136	C₂×C₅₃	—	2	2	no	—
137	C₂×C₅₃	C₂×#121	1	1	yes	characteristic
138	C₂×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
139	C₂×C₅₃	C₂×#135	1	1	yes	characteristic
140	D₅₃	—	2	2	no	—
141	D₅₃	#467	1	4	yes	direct product factor
142	C₂²×Q₁₆	—	53	1	no	—
143	C₂³×C₈	—	53	1	no	—
144	C₂×C₂²⋊Q₈	—	53	2	no	—
145	#467	—	53	1	no	—
146	#467	—	53	1	no	—
147	#467	—	53	2	no	—
148	#467	—	53	1	no	—
149	#467	—	53	2	no	—
150	#467	D₅₃	1	1	yes	characteristic, direct product factor
151	C₂×#107	—	53	1	no	—
152	C₂×#103	—	53	2	no	—
153	C₅₃	C₂×#467	1	1	yes	characteristic, complemented, Sylow
154	C₂²×Q₈	—	53	2	no	—
155	C₂³×C₄	—	53	1	no	—
156	C₂×Q₁₆	—	53	2	no	—
157	C₂×Q₁₆	—	53	1	no	—
158	C₂×Q₁₆	—	106	4	no	—
159	C₂×Q₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
160	C₂²×C₈	—	53	1	no	—
161	C₂²×C₈	—	53	1	no	—
162	C₂²×C₈	—	53	1	no	—
163	C₂²×C₈	—	53	1	no	—
164	C₂²×C₈	—	53	1	no	—
165	C₂²×C₈	—	106	4	no	—
166	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
167	C₂²⋊Q₈	—	53	2	no	—
168	C₂²⋊Q₈	—	53	2	no	—
169	C₂²⋊Q₈	—	53	2	no	—
170	C₂²⋊Q₈	—	106	2	no	—
171	C₂²⋊Q₈	—	106	2	no	—
172	C₂²⋊Q₈	C₂×D₅₃	1	2	yes	—
173	C₂×C₄⋊C₄	—	53	2	no	—
174	C₂×C₄⋊C₄	—	106	2	no	—
175	C₂×C₂²⋊C₄	—	106	2	no	—
176	#107	—	53	1	no	—
177	#107	—	53	2	no	—
178	#107	C₂×D₅₃	1	1	yes	characteristic, complemented
179	#103	—	53	2	no	—
180	#103	—	53	2	no	—
181	#103	—	53	2	no	—
182	#103	C₂×D₅₃	1	2	yes	complemented
183	C₂⁴	—	53	1	no	—
184	C₂×Q₈	—	53	2	no	—
185	C₂×Q₈	—	106	2	no	—
186	C₂×Q₈	—	106	4	no	—
187	C₂×Q₈	—	106	4	no	—
188	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
189	C₂²×C₄	—	53	1	no	—
190	C₂²×C₄	—	53	1	no	—
191	C₂²×C₄	—	53	1	no	—
192	C₂²×C₄	—	53	1	no	—
193	C₂²×C₄	—	53	1	no	—
194	C₂²×C₄	—	106	2	no	—
195	C₂²×C₄	—	106	2	no	—
196	C₂²×C₄	—	106	2	no	—
197	C₂²×C₄	—	106	2	no	—
198	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
199	Q₁₆	—	2	2	no	—
200	Q₁₆	—	106	4	no	—
201	Q₁₆	—	106	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	53	1	no	—
204	C₂×C₈	—	53	4	no	—
205	C₂×C₈	—	53	1	no	—
206	C₂×C₈	—	106	2	no	—
207	C₂×C₈	—	106	2	no	—
208	C₂×C₈	—	106	2	no	—
209	C₂×C₈	—	106	2	no	—
210	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
211	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	53	2	no	—
214	C₄⋊C₄	—	106	2	no	—
215	C₄⋊C₄	—	106	2	no	—
216	C₄⋊C₄	—	106	2	no	—
217	C₄⋊C₄	—	106	2	no	—
218	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
219	C₂²⋊C₄	—	2	2	no	—
220	C₂²⋊C₄	—	106	2	no	—
221	C₂²⋊C₄	—	106	2	no	—
222	C₂²⋊C₄	—	106	2	no	—
223	C₂³	—	53	1	no	—
224	C₂³	—	53	4	no	—
225	C₂³	—	53	1	no	—
226	C₂³	—	106	2	no	—
227	C₂³	—	106	2	no	—
228	C₂³	D₄×D₅₃	1	1	yes	characteristic
229	Q₈	—	2	2	no	—
230	Q₈	—	2	2	no	—
231	Q₈	—	106	2	no	—
232	Q₈	—	106	2	no	—
233	Q₈	—	212	4	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	53	1	no	—
239	C₂×C₄	—	53	2	no	—
240	C₂×C₄	—	53	2	no	—
241	C₂×C₄	—	53	1	no	—
242	C₂×C₄	—	106	2	no	—
243	C₂×C₄	—	106	2	no	—
244	C₂×C₄	—	106	2	no	—
245	C₂×C₄	—	106	1	no	—
246	C₂×C₄	—	106	1	no	—
247	C₂×C₄	—	106	2	no	—
248	C₂×C₄	—	106	2	no	—
249	C₂×C₄	—	212	4	no	—
250	C₂×C₄	—	212	4	no	—
251	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
252	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
253	C₈	—	2	1	no	—
254	C₈	—	53	2	no	—
255	C₈	—	106	1	no	—
256	C₈	D₄×D₅₃	1	2	yes	—
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	53	2	no	—
260	C₂²	—	53	2	no	—
261	C₂²	—	53	2	no	—
262	C₂²	—	53	2	no	—
263	C₂²	—	106	8	no	—
264	C₂²	—	106	1	no	—
265	C₂²	—	106	1	no	—
266	C₂²	Q₁₆×D₅₃	1	2	yes	complemented
267	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	4	2	no	—
270	C₄	—	4	2	no	—
271	C₄	—	53	2	no	—
272	C₄	—	106	1	no	—
273	C₄	—	212	2	no	—
274	C₄	—	212	2	no	—
275	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
276	C₄	#41×D₅₃	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	53	4	no	—
279	C₂	—	106	2	no	—
280	C₂	#121×D₅₃	1	1	yes	characteristic
281	C₂	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
282	C₂	#135×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#467	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#467	D₅₃
hypocenter	C₅₃	C₂×#467
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #467

Complements

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

N	H ≅ G/N	type
1. C₂×Q₁₆×D₅₃	C₂	semidirect
4. #16810865	C₂	semidirect
5. #16810864	C₂	semidirect
6. #16810863	C₂	semidirect
7. #16810862	C₂	semidirect
8. #16810861	C₂	semidirect
9. #107×D₅₃	C₂	semidirect
10. #103×D₅₃	C₂	semidirect
11. C₅₃×#467	C₂	semidirect
14. C₂×Dic₂₁₂	C₂²	semidirect
15. C₂×#11719925	C₂²	semidirect
31. #13133151	C₂²	semidirect
32. #13133150	C₂²	semidirect
33. #13133142	C₂²	semidirect
34. #13133141	C₂²	semidirect
35. #13133140	C₂²	semidirect
36. C₂×Q₁₆×C₅₃	C₂²	semidirect
39. #107×C₅₃	C₂²	semidirect
40. #103×C₅₃	C₂²	semidirect
84. C₂²×D₅₃	Q₁₆	semidirect
117. C₂²×C₅₃	C₂×Q₁₆	semidirect
124. C₂×D₅₃	C₂×Q₁₆	semidirect
141. D₅₃	#467	direct (N × H)
150. #467	D₅₃	direct (N × H)
153. C₅₃	C₂×#467	semidirect
159. C₂×Q₁₆	C₂×D₅₃	semidirect
178. #107	C₂×D₅₃	semidirect
182. #103	C₂×D₅₃	semidirect
266. C₂²	Q₁₆×D₅₃	semidirect

Subgroups — #467×D53 #20284285

Conjugacy classes of subgroups

Complements

Subgroups — #467×D₅₃ #20284285