Subgroups — C₄×D₈×D₅₃ #20283462

Order 6784.

616 conjugacy classes of subgroups.

The subgroup lattice has 268 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	C₄×D₄×D₅₃	C₂	1	2	yes	maximal, complemented
3	C₄×D₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
4	D₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810705	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#11719922	C₂	1	2	yes	maximal, complemented
7	#16810703	C₂	1	2	yes	maximal, complemented
8	#107×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#102×D₅₃	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₅₃	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	C₂²×C₄×D₅₃	—	2	2	no	—
14	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
15	C₂×#11719922	C₂²	1	2	yes	—
16	D₈×D₅₃	C₄	1	8	yes	direct product factor
17	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
19	C₄×D₂₁₂	C₂²	1	2	yes	complemented
20	D₄×Dic₅₃	C₂²	1	2	yes	complemented
21	#13133179	C₂²	1	2	yes	complemented
22	C₄×C₂²⋊D₅₃	—	2	2	no	—
23	#13133177	—	2	2	no	—
24	C₄⋊C₄×D₅₃	C₂²	1	2	yes	—
25	C₂²⋊C₄×D₅₃	—	2	2	no	—
26	C₄²×D₅₃	C₂²	1	1	yes	characteristic
27	#13133151	C₂²	1	1	yes	characteristic, complemented
28	#13133150	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	C₄×#88322	C₂²	1	1	yes	characteristic, complemented
33	C₈×Dic₅₃	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	—	2	2	no	—
51	C₂×C₄×D₅₃	—	4	8	no	—
52	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
53	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
54	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
55	D₈×C₅₃	C₂×C₄	1	4	yes	complemented
56	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
59	C₂²⋊C₄×C₅₃	—	2	2	no	—
60	C₄²×C₅₃	C₂³	1	1	yes	characteristic
61	C₂²⋊Dic₅₃	—	2	2	no	—
62	#11719922	C₂×C₄	1	8	yes	complemented
63	#11719921	—	2	2	no	—
64	#11719921	—	2	2	no	—
65	C₄⋊Dic₅₃	C₂³	1	2	yes	—
66	#11719919	—	2	2	no	—
67	C₄×Dic₅₃	D₄	1	1	yes	characteristic
68	C₄×Dic₅₃	D₄	1	1	yes	characteristic
69	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
70	C₂×#88322	C₂³	1	1	yes	characteristic
71	C₂×#88322	C₂³	1	1	yes	characteristic
72	D₄₂₄	C₂×C₄	1	4	yes	complemented
73	C₈×D₅₃	—	2	2	no	—
74	C₈×D₅₃	C₂×C₄	1	4	yes	complemented
75	C₂³×C₅₃	—	2	2	no	—
76	C₂²×D₅₃	—	2	2	no	—
77	C₂²×D₅₃	—	2	8	no	—
78	C₂²×D₅₃	—	4	4	no	—
79	C₂²×D₅₃	—	4	4	no	—
80	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
81	D₄×C₅₃	—	2	2	no	—
82	D₄×C₅₃	C₂²×C₄	1	4	yes	complemented
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	4	4	no	—
85	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
88	C₂²⋊D₅₃	—	2	8	no	—
89	C₂×Dic₅₃	—	2	2	no	—
90	C₂×Dic₅₃	—	4	4	no	—
91	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
94	D₂₁₂	—	2	2	no	—
95	D₂₁₂	C₂²×C₄	1	4	yes	complemented
96	C₄×D₅₃	—	2	2	no	—
97	C₄×D₅₃	—	4	4	no	—
98	C₄×D₅₃	—	4	4	no	—
99	C₄×D₅₃	—	4	4	no	—
100	C₄×D₅₃	D₈	1	4	yes	direct product factor
101	C₄×D₅₃	C₂²×C₄	1	2	yes	—
102	C₄×D₅₃	#41	1	2	yes	—
103	C₈×C₅₃	—	2	1	no	—
104	C₈×C₅₃	C₂²×C₄	1	2	yes	complemented
105	#88322	—	2	1	no	—
106	#88322	C₂²×C₄	1	2	yes	complemented
107	C₂²×C₅₃	—	2	4	no	—
108	C₂²×C₅₃	—	4	2	no	—
109	C₂²×C₅₃	—	4	2	no	—
110	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
111	C₂×D₅₃	—	2	4	no	—
112	C₂×D₅₃	—	4	2	no	—
113	C₂×D₅₃	—	4	2	no	—
114	C₂×D₅₃	—	4	16	no	—
115	C₂×D₅₃	C₄×D₄	1	2	yes	—
116	C₂×D₅₃	C₂×D₈	1	2	yes	—
117	C₂×D₅₃	#135	1	2	yes	—
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	—	4	2	no	—
120	C₄×C₅₃	C₂×D₈	1	2	yes	complemented
121	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
122	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
123	Dic₅₃	—	2	1	no	—
124	Dic₅₃	—	4	2	no	—
125	Dic₅₃	C₄×D₄	1	2	yes	—
126	Dic₅₃	C₂×D₈	1	2	yes	complemented
127	C₂×C₄×D₈	—	53	1	no	maximal, Sylow
128	C₂×C₅₃	—	4	4	no	—
129	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
130	C₂×C₅₃	C₂²×D₈	1	1	yes	characteristic
131	C₂×C₅₃	C₂×#135	1	1	yes	characteristic
132	D₅₃	—	4	4	no	—
133	D₅₃	C₄×D₈	1	4	yes	direct product factor
134	C₂²×D₈	—	53	1	no	—
135	C₂×C₄×D₄	—	53	2	no	—
136	C₄×D₈	—	53	1	no	—
137	C₄×D₈	—	53	2	no	—
138	C₄×D₈	—	53	1	no	—
139	C₄×D₈	—	53	2	no	—
140	C₄×D₈	—	53	1	no	—
141	C₄×D₈	D₅₃	1	1	yes	characteristic, direct product factor
142	C₂×#107	—	53	1	no	—
143	C₂×#102	—	53	2	no	—
144	C₂×C₄×C₈	—	53	1	no	—
145	C₅₃	C₂×C₄×D₈	1	1	yes	characteristic, complemented, Sylow
146	C₂²×D₄	—	53	2	no	—
147	C₂³×C₄	—	106	2	no	—
148	C₂×D₈	—	53	2	no	—
149	C₂×D₈	—	53	8	no	—
150	C₂×D₈	—	53	1	no	—
151	C₂×D₈	C₂×D₅₃	1	1	yes	characteristic
152	C₂²×C₈	—	53	1	no	—
153	C₂²×C₈	—	53	1	no	—
154	C₄×D₄	—	53	2	no	—
155	C₄×D₄	—	53	2	no	—
156	C₄×D₄	—	53	2	no	—
157	C₄×D₄	—	106	2	no	—
158	C₄×D₄	—	106	2	no	—
159	C₄×D₄	C₂×D₅₃	1	2	yes	complemented
160	C₂×C₄⋊C₄	—	53	2	no	—
161	C₂×C₂²⋊C₄	—	106	2	no	—
162	C₂×C₄²	—	53	1	no	—
163	#107	—	53	1	no	—
164	#107	—	53	2	no	—
165	#107	C₂×D₅₃	1	1	yes	characteristic, complemented
166	#102	—	53	2	no	—
167	#102	—	53	2	no	—
168	#102	—	53	2	no	—
169	#102	C₂×D₅₃	1	2	yes	complemented
170	C₄×C₈	—	53	1	no	—
171	C₄×C₈	—	53	1	no	—
172	C₄×C₈	—	53	1	no	—
173	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic, complemented
174	C₂⁴	—	106	2	no	—
175	C₂×D₄	—	53	8	no	—
176	C₂×D₄	—	53	2	no	—
177	C₂×D₄	—	106	4	no	—
178	C₂×D₄	—	106	2	no	—
179	C₂×D₄	C₂²×D₅₃	1	2	yes	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	53	1	no	—
182	C₂²×C₄	—	53	1	no	—
183	C₂²×C₄	—	53	1	no	—
184	C₂²×C₄	—	106	2	no	—
185	C₂²×C₄	—	106	2	no	—
186	C₂²×C₄	—	106	2	no	—
187	C₂²×C₄	—	106	2	no	—
188	C₂²×C₄	—	212	8	no	—
189	D₈	—	53	8	no	—
190	D₈	—	53	4	no	—
191	D₈	C₄×D₅₃	1	4	yes	direct product factor
192	C₂×C₈	—	53	1	no	—
193	C₂×C₈	—	53	4	no	—
194	C₂×C₈	—	53	1	no	—
195	C₂×C₈	—	106	2	no	—
196	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
197	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
198	C₄⋊C₄	—	53	2	no	—
199	C₄⋊C₄	—	106	2	no	—
200	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₂²⋊C₄	—	106	2	no	—
203	C₂²⋊C₄	—	106	2	no	—
204	C₂²⋊C₄	—	106	2	no	—
205	C₄²	—	53	1	no	—
206	C₄²	—	53	1	no	—
207	C₄²	—	53	1	no	—
208	C₄²	C₂²×D₅₃	1	1	yes	characteristic
209	C₂³	—	2	2	no	—
210	C₂³	—	53	1	no	—
211	C₂³	—	106	8	no	—
212	C₂³	—	106	2	no	—
213	C₂³	—	212	4	no	—
214	C₂³	—	212	4	no	—
215	D₄	—	2	2	no	—
216	D₄	—	53	4	no	—
217	D₄	—	106	8	no	—
218	D₄	—	106	2	no	—
219	D₄	C₂×C₄×D₅₃	1	4	yes	complemented
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	53	1	no	—
223	C₂×C₄	—	53	1	no	—
224	C₂×C₄	—	53	4	no	—
225	C₂×C₄	—	53	2	no	—
226	C₂×C₄	—	53	2	no	—
227	C₂×C₄	—	53	1	no	—
228	C₂×C₄	—	106	2	no	—
229	C₂×C₄	—	106	2	no	—
230	C₂×C₄	—	212	4	no	—
231	C₂×C₄	—	212	4	no	—
232	C₂×C₄	—	212	4	no	—
233	C₂×C₄	—	212	4	no	—
234	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
235	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
236	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
237	C₈	—	2	1	no	—
238	C₈	—	53	2	no	—
239	C₈	—	106	1	no	—
240	C₈	C₂×C₄×D₅₃	1	2	yes	complemented
241	C₂²	—	2	4	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	—	53	2	no	—
245	C₂²	—	53	2	no	—
246	C₂²	—	53	2	no	—
247	C₂²	—	106	4	no	—
248	C₂²	—	212	16	no	—
249	C₂²	—	212	2	no	—
250	C₂²	—	212	2	no	—
251	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	4	2	no	—
254	C₄	—	53	2	no	—
255	C₄	—	53	2	no	—
256	C₄	—	106	1	no	—
257	C₄	—	212	2	no	—
258	C₄	D₈×D₅₃	1	2	yes	direct product factor
259	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
260	C₄	#41×D₅₃	1	1	yes	characteristic
261	C₂	—	4	4	no	—
262	C₂	—	53	4	no	—
263	C₂	—	212	4	no	—
264	C₂	C₄×D₄×D₅₃	1	1	yes	characteristic
265	C₂	C₂×D₈×D₅₃	1	1	yes	characteristic
266	C₂	#135×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	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	C₂×C₄×D₈
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
2. C₄×D₄×D₅₃	C₂	semidirect
3. C₄×D₄₂₄	C₂	semidirect
4. D₈×Dic₅₃	C₂	semidirect
5. #16810705	C₂	semidirect
6. C₄×#11719922	C₂	semidirect
7. #16810703	C₂	semidirect
8. #107×D₅₃	C₂	semidirect
9. #102×D₅₃	C₂	semidirect
10. C₄×C₈×D₅₃	C₂	semidirect
11. C₄×D₈×C₅₃	C₂	semidirect
16. D₈×D₅₃	C₄	direct (N × H)
19. C₄×D₂₁₂	C₂²	semidirect
20. D₄×Dic₅₃	C₂²	semidirect
21. #13133179	C₂²	semidirect
27. #13133151	C₂²	semidirect
28. #13133150	C₂²	semidirect
29. #13133139	C₂²	semidirect
30. #13133138	C₂²	semidirect
31. #13133137	C₂²	semidirect
32. C₄×#88322	C₂²	semidirect
33. C₈×Dic₅₃	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
55. D₈×C₅₃	C₂×C₄	semidirect
62. #11719922	C₂×C₄	semidirect
72. D₄₂₄	C₂×C₄	semidirect
74. C₈×D₅₃	C₂×C₄	semidirect
82. D₄×C₅₃	C₂²×C₄	semidirect
95. D₂₁₂	C₂²×C₄	semidirect
100. C₄×D₅₃	D₈	direct (N × H)
104. C₈×C₅₃	C₂²×C₄	semidirect
106. #88322	C₂²×C₄	semidirect
120. C₄×C₅₃	C₂×D₈	semidirect
126. Dic₅₃	C₂×D₈	semidirect
133. D₅₃	C₄×D₈	direct (N × H)
141. C₄×D₈	D₅₃	direct (N × H)
145. C₅₃	C₂×C₄×D₈	semidirect
159. C₄×D₄	C₂×D₅₃	semidirect
165. #107	C₂×D₅₃	semidirect
169. #102	C₂×D₅₃	semidirect
173. C₄×C₈	C₂×D₅₃	semidirect
191. D₈	C₄×D₅₃	direct (N × H)
219. D₄	C₂×C₄×D₅₃	semidirect
240. C₈	C₂×C₄×D₅₃	semidirect
258. C₄	D₈×D₅₃	direct (N × H)

Subgroups — C4×D8×D53 #20283462

Conjugacy classes of subgroups

Complements

Subgroups — C₄×D₈×D₅₃ #20283462