Subgroups — #20283451

Order 6784.

544 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×M₁₆×D₅₃	C₂	1	2	yes	maximal, complemented
2	C₄×D₄×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#16810701	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810697	C₂	1	2	yes	maximal, complemented
5	#16810692	C₂	1	2	yes	maximal, complemented
6	C₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810688	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810645	C₂	1	1	yes	maximal, characteristic, complemented
9	#16810585	C₂	1	2	yes	maximal, complemented
10	C₄×#11719912	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₅₃	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₅₃	C₂²	1	2	yes	—
14	C₂×#11719917	C₂²	1	2	yes	complemented
15	C₂×#11719912	C₂²	1	1	yes	characteristic
16	C₂×#11719912	C₂²	1	2	yes	—
17	C₂×#11719912	C₂²	1	1	yes	characteristic
18	M₁₆×D₅₃	—	2	8	no	—
19	C₂×C₈×D₅₃	C₂²	1	2	yes	complemented
20	C₂×C₈×D₅₃	C₂²	1	2	yes	complemented
21	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
22	D₄×Dic₅₃	C₄	1	1	yes	characteristic
23	#13133179	C₄	1	1	yes	characteristic
24	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
25	#13133177	C₄	1	2	yes	—
26	C₄⋊C₄×D₅₃	C₄	1	1	yes	characteristic
27	C₂²⋊C₄×D₅₃	C₄	1	2	yes	—
28	C₄²×D₅₃	C₂²	1	1	yes	characteristic
29	C₄⋊#88322	C₂²	1	1	yes	characteristic, complemented
30	#13133146	C₂²	1	2	yes	complemented
31	C₂²⋊#88322	C₂²	1	2	yes	complemented
32	#13133129	C₂²	1	2	yes	complemented
33	#13133129	C₂²	1	2	yes	complemented
34	#13133127	C₂²	1	1	yes	characteristic, complemented
35	#13133127	C₂²	1	1	yes	characteristic, complemented
36	C₄×#88322	C₂²	1	1	yes	characteristic, complemented
37	C₂×M₁₆×C₅₃	C₂²	1	2	yes	complemented
38	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
40	C₂²⋊C₈×C₅₃	C₂²	1	2	yes	complemented
41	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
42	C₂³×D₅₃	C₂×C₄	1	2	yes	—
43	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
45	C₂×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
46	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
47	D₄×D₅₃	—	2	4	no	—
48	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₅₃	—	2	4	no	—
50	C₂×C₄×D₅₃	—	2	4	no	—
51	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
54	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₅₃	C₂³	1	2	yes	—
56	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
57	M₁₆×C₅₃	—	2	4	no	—
58	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₅₃	C₂³	1	2	yes	—
60	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
63	C₄²×C₅₃	C₂³	1	1	yes	characteristic
64	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
65	#11719921	C₂×C₄	1	2	yes	—
66	#11719921	C₂×C₄	1	2	yes	—
67	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
68	#11719919	C₂×C₄	1	2	yes	—
69	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
72	#11719917	—	2	4	no	—
73	C₂×#88322	C₂³	1	1	yes	characteristic
74	C₂×#88322	C₂³	1	2	yes	—
75	C₂×#88322	C₂³	1	1	yes	characteristic
76	#11719912	—	2	4	no	—
77	#11719912	—	2	2	no	—
78	#11719912	D₄	1	4	yes	complemented
79	C₈×D₅₃	—	2	4	no	—
80	C₈×D₅₃	—	2	4	no	—
81	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
82	C₂²×D₅₃	—	2	4	no	—
83	C₂²×D₅₃	—	2	4	no	—
84	C₂²×D₅₃	—	4	2	no	—
85	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
87	D₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₅₃	—	2	4	no	—
95	C₂×Dic₅₃	—	2	4	no	—
96	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
97	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
100	D₂₁₂	M₁₆	1	4	yes	complemented
101	C₄×D₅₃	—	2	4	no	—
102	C₄×D₅₃	—	2	2	no	—
103	C₄×D₅₃	—	2	2	no	—
104	C₄×D₅₃	—	2	2	no	—
105	C₄×D₅₃	—	2	4	no	—
106	C₄×D₅₃	—	4	1	no	—
107	C₄×D₅₃	C₂×D₄	1	2	yes	—
108	C₄×D₅₃	#41	1	2	yes	—
109	C₈×C₅₃	—	2	2	no	—
110	C₈×C₅₃	—	2	1	no	—
111	C₈×C₅₃	C₂×D₄	1	2	yes	complemented
112	#88322	—	2	2	no	—
113	#88322	—	2	1	no	—
114	#88322	C₂×D₄	1	2	yes	complemented
115	C₂²×C₅₃	—	2	2	no	—
116	C₂²×C₅₃	—	2	2	no	—
117	C₂²×C₅₃	—	2	2	no	—
118	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×D₅₃	—	2	2	no	—
120	C₂×D₅₃	—	2	1	no	—
121	C₂×D₅₃	—	2	1	no	—
122	C₂×D₅₃	—	2	2	no	—
123	C₂×D₅₃	—	4	8	no	—
124	C₂×D₅₃	C₄×D₄	1	2	yes	—
125	C₂×D₅₃	C₂×M₁₆	1	4	yes	complemented
126	C₄×C₅₃	—	2	2	no	—
127	C₄×C₅₃	—	2	1	no	—
128	C₄×C₅₃	C₂×M₁₆	1	2	yes	complemented
129	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
130	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
131	Dic₅₃	—	2	2	no	—
132	Dic₅₃	—	2	1	no	—
133	Dic₅₃	—	2	1	no	—
134	Dic₅₃	C₄×D₄	1	2	yes	—
135	D₄×M₁₆	—	53	1	no	maximal, Sylow
136	C₂×C₅₃	—	4	2	no	—
137	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
139	C₂×C₅₃	#568	1	1	yes	characteristic
140	D₅₃	—	2	2	no	—
141	D₅₃	—	2	4	no	—
142	C₂²×M₁₆	—	53	2	no	—
143	C₂×C₄×D₄	—	53	1	no	—
144	#436	—	53	1	no	—
145	#436	D₅₃	1	1	yes	characteristic, complemented
146	#435	—	53	2	no	—
147	#435	—	53	2	no	—
148	C₈×D₄	—	53	1	no	—
149	C₈×D₄	—	53	1	no	—
150	#423	—	53	1	no	—
151	#407	—	53	2	no	—
152	C₄×M₁₆	—	53	1	no	—
153	C₅₃	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
154	C₂²×D₄	—	53	1	no	—
155	C₂³×C₄	—	53	2	no	—
156	C₂×M₁₆	—	53	2	no	—
157	C₂×M₁₆	—	53	1	no	—
158	C₂×M₁₆	—	53	2	no	—
159	C₂×M₁₆	—	53	1	no	—
160	C₂×M₁₆	—	106	8	no	—
161	C₂×M₁₆	C₂×D₅₃	1	2	yes	complemented
162	C₂²×C₈	—	53	2	no	—
163	C₂²×C₈	—	53	2	no	—
164	C₄×D₄	—	53	1	no	—
165	C₄×D₄	—	53	2	no	—
166	C₄×D₄	—	53	1	no	—
167	C₄×D₄	—	53	2	no	—
168	C₄×D₄	—	53	1	no	—
169	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
170	C₂×C₄⋊C₄	—	53	1	no	—
171	C₂×C₂²⋊C₄	—	53	2	no	—
172	C₂×C₄²	—	53	1	no	—
173	C₄⋊C₈	—	53	2	no	—
174	C₄⋊C₈	—	53	1	no	—
175	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic, complemented
176	C₂²⋊C₈	—	53	2	no	—
177	C₂²⋊C₈	—	53	2	no	—
178	C₂²⋊C₈	—	53	2	no	—
179	C₂²⋊C₈	C₂×D₅₃	1	2	yes	complemented
180	#97	—	53	1	no	—
181	#97	—	53	1	no	—
182	C₄×C₈	—	53	1	no	—
183	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic, complemented
184	C₂⁴	—	53	2	no	—
185	C₂×D₄	—	53	2	no	—
186	C₂×D₄	—	53	1	no	—
187	C₂×D₄	—	106	4	no	—
188	C₂×D₄	C₄×D₅₃	1	1	yes	characteristic
189	C₂²×C₄	—	53	1	no	—
190	C₂²×C₄	—	53	2	no	—
191	C₂²×C₄	—	53	1	no	—
192	C₂²×C₄	—	53	2	no	—
193	C₂²×C₄	—	53	2	no	—
194	C₂²×C₄	—	53	2	no	—
195	C₂²×C₄	—	53	1	no	—
196	C₂²×C₄	—	106	4	no	—
197	C₂²×C₄	—	106	4	no	—
198	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
199	M₁₆	—	2	4	no	—
200	M₁₆	—	53	4	no	—
201	M₁₆	—	106	4	no	—
202	M₁₆	—	106	2	no	—
203	M₁₆	—	106	4	no	—
204	C₂×C₈	—	53	1	no	—
205	C₂×C₈	—	53	1	no	—
206	C₂×C₈	—	53	2	no	—
207	C₂×C₈	—	106	4	no	—
208	C₂×C₈	—	106	4	no	—
209	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
210	C₂×C₈	C₂²×D₅₃	1	2	yes	—
211	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
212	C₄⋊C₄	—	53	2	no	—
213	C₄⋊C₄	—	53	1	no	—
214	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
215	C₂²⋊C₄	—	53	2	no	—
216	C₂²⋊C₄	—	53	2	no	—
217	C₂²⋊C₄	—	53	2	no	—
218	C₂²⋊C₄	C₄×D₅₃	1	2	yes	—
219	C₄²	—	53	1	no	—
220	C₄²	—	53	1	no	—
221	C₄²	—	53	1	no	—
222	C₄²	C₂²×D₅₃	1	1	yes	characteristic
223	C₂³	—	53	2	no	—
224	C₂³	—	53	1	no	—
225	C₂³	—	106	4	no	—
226	C₂³	—	106	4	no	—
227	C₂³	—	212	2	no	—
228	C₂³	C₂×C₄×D₅₃	1	2	yes	—
229	D₄	—	2	2	no	—
230	D₄	—	53	4	no	—
231	D₄	—	106	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	53	1	no	—
235	C₂×C₄	—	53	2	no	—
236	C₂×C₄	—	53	2	no	—
237	C₂×C₄	—	53	1	no	—
238	C₂×C₄	—	53	1	no	—
239	C₂×C₄	—	53	2	no	—
240	C₂×C₄	—	106	4	no	—
241	C₂×C₄	—	106	4	no	—
242	C₂×C₄	—	106	4	no	—
243	C₂×C₄	—	106	2	no	—
244	C₂×C₄	—	106	2	no	—
245	C₂×C₄	—	106	2	no	—
246	C₂×C₄	—	212	1	no	—
247	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
249	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
250	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	2	1	no	—
253	C₈	—	53	2	no	—
254	C₈	—	106	2	no	—
255	C₈	—	106	1	no	—
256	C₈	D₄×D₅₃	1	2	yes	complemented
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	53	2	no	—
261	C₂²	—	53	4	no	—
262	C₂²	—	106	2	no	—
263	C₂²	—	106	1	no	—
264	C₂²	—	106	1	no	—
265	C₂²	—	106	2	no	—
266	C₂²	—	212	8	no	—
267	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	—	53	2	no	—
271	C₄	—	106	2	no	—
272	C₄	—	106	1	no	—
273	C₄	—	106	1	no	—
274	C₄	M₁₆×D₅₃	1	2	yes	complemented
275	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
276	C₄	#41×D₅₃	1	1	yes	characteristic
277	C₂	—	4	2	no	—
278	C₂	—	106	2	no	—
279	C₂	—	106	4	no	—
280	C₂	C₄×D₄×D₅₃	1	1	yes	characteristic
281	C₂	C₂×M₁₆×D₅₃	1	1	yes	characteristic
282	C₂	#16811388	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#436	C₂
hypercenter	#436	D₅₃
hypocenter	C₅₃	D₄×M₁₆
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
1. C₂×M₁₆×D₅₃	C₂	semidirect
3. #16810701	C₂	semidirect
4. #16810697	C₂	semidirect
5. #16810692	C₂	semidirect
6. C₈×D₂₁₂	C₂	semidirect
7. #16810688	C₂	semidirect
8. #16810645	C₂	semidirect
9. #16810585	C₂	semidirect
10. C₄×#11719912	C₂	semidirect
11. C₅₃×#436	C₂	semidirect
14. C₂×#11719917	C₂²	semidirect
19. C₂×C₈×D₅₃	C₂²	semidirect
20. C₂×C₈×D₅₃	C₂²	semidirect
29. C₄⋊#88322	C₂²	semidirect
30. #13133146	C₂²	semidirect
31. C₂²⋊#88322	C₂²	semidirect
32. #13133129	C₂²	semidirect
33. #13133129	C₂²	semidirect
34. #13133127	C₂²	semidirect
35. #13133127	C₂²	semidirect
36. C₄×#88322	C₂²	semidirect
37. C₂×M₁₆×C₅₃	C₂²	semidirect
39. C₄⋊C₈×C₅₃	C₂²	semidirect
40. C₂²⋊C₈×C₅₃	C₂²	semidirect
41. C₄×C₈×C₅₃	C₂²	semidirect
78. #11719912	D₄	semidirect
100. D₂₁₂	M₁₆	semidirect
111. C₈×C₅₃	C₂×D₄	semidirect
114. #88322	C₂×D₄	semidirect
125. C₂×D₅₃	C₂×M₁₆	semidirect
128. C₄×C₅₃	C₂×M₁₆	semidirect
145. #436	D₅₃	semidirect
153. C₅₃	D₄×M₁₆	semidirect
161. C₂×M₁₆	C₂×D₅₃	semidirect
175. C₄⋊C₈	C₂×D₅₃	semidirect
179. C₂²⋊C₈	C₂×D₅₃	semidirect
183. C₄×C₈	C₂×D₅₃	semidirect
256. C₈	D₄×D₅₃	semidirect
274. C₄	M₁₆×D₅₃	semidirect