Subgroups — #20283302

Order 6784.

492 conjugacy classes of subgroups.

The subgroup lattice has 290 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	1	yes	maximal, characteristic
2	C₂×#13133175	C₂	1	1	yes	maximal, characteristic
3	#117×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#16810666	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810665	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810664	C₂	1	2	yes	maximal, complemented
7	#16810663	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810662	C₂	1	2	yes	maximal, complemented
9	#16810660	C₂	1	2	yes	maximal, complemented
10	#16810658	C₂	1	2	yes	maximal, complemented
11	C₅₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
13	C₂×#11719917	C₂²	1	1	yes	characteristic
14	C₂×#11719912	C₂²	1	2	yes	—
15	M₁₆×D₅₃	C₄	1	8	yes	complemented
16	C₂×C₈×D₅₃	C₂²	1	2	yes	—
17	#13133176	C₂²	1	1	yes	characteristic
18	#13133175	—	2	1	no	—
19	#13133175	—	2	1	no	—
20	#13133175	C₂²	1	1	yes	characteristic
21	#13133175	C₂²	1	1	yes	characteristic
22	#13133175	C₂²	1	2	yes	—
23	#13133175	C₂²	1	2	yes	—
24	#13133174	—	2	1	no	—
25	#13133173	—	2	1	no	—
26	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×D₅₃	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×D₅₃	C₂²	1	1	yes	characteristic
29	C₂×#11719921	—	2	1	no	—
30	C₂²⋊C₄×D₅₃	—	2	1	no	—
31	C₂×C₄×Dic₅₃	—	2	1	no	—
32	C₄²×D₅₃	—	2	1	no	—
33	#13133151	C₂²	1	2	yes	complemented
34	#13133150	C₂²	1	2	yes	complemented
35	#13133150	C₂²	1	2	yes	complemented
36	#13133149	C₂²	1	2	yes	complemented
37	#13133148	C₂²	1	2	yes	complemented
38	#13133148	C₂²	1	2	yes	complemented
39	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
40	#117×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
42	#107×C₅₃	C₂²	1	2	yes	complemented
43	#106×C₅₃	C₂²	1	2	yes	complemented
44	C₂³×D₅₃	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	—	2	1	no	—
46	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²×Dic₅₃	—	2	1	no	—
48	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
49	C₂×C₄×D₅₃	—	2	1	no	—
50	C₂×C₄×D₅₃	—	2	1	no	—
51	C₂×C₄×D₅₃	C₂×C₄	1	4	yes	—
52	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
53	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
54	C₂×C₄×D₅₃	Q₈	1	4	yes	—
55	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
56	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
57	M₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
58	C₂×C₈×C₅₃	C₂³	1	2	yes	—
59	C₄⋊C₄×C₅₃	—	2	1	no	—
60	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
62	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
63	C₂²⋊C₄×C₅₃	—	2	1	no	—
64	C₄²×C₅₃	—	2	1	no	—
65	C₂²⋊Dic₅₃	—	2	1	no	—
66	#11719921	—	2	1	no	—
67	#11719921	—	2	1	no	—
68	#11719921	—	2	2	no	—
69	#11719921	—	2	2	no	—
70	C₄⋊Dic₅₃	—	2	1	no	—
71	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₅₃	C₂³	1	2	yes	—
73	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
74	#11719919	—	2	1	no	—
75	#11719919	—	2	1	no	—
76	C₄×Dic₅₃	—	2	1	no	—
77	C₄×Dic₅₃	—	2	1	no	—
78	C₄×Dic₅₃	—	2	1	no	—
79	C₄×Dic₅₃	—	2	2	no	—
80	C₄×Dic₅₃	—	2	2	no	—
81	#11719917	C₂×C₄	1	4	yes	complemented
82	C₂×#88322	C₂³	1	2	yes	—
83	#11719912	C₂×C₄	1	8	yes	complemented
84	C₈×D₅₃	C₂×C₄	1	8	yes	complemented
85	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×D₅₃	—	4	1	no	—
87	C₂²×D₅₃	—	4	1	no	—
88	C₂²×D₅₃	C₄⋊C₄	1	4	yes	—
89	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	4	2	no	—
95	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
98	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
99	C₂×Dic₅₃	—	2	1	no	—
100	C₂×Dic₅₃	—	2	1	no	—
101	C₂×Dic₅₃	—	2	2	no	—
102	C₂×Dic₅₃	—	4	2	no	—
103	C₂×Dic₅₃	C₄⋊C₄	1	4	yes	—
104	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
105	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
106	C₄×D₅₃	—	4	2	no	—
107	C₄×D₅₃	—	4	2	no	—
108	C₄×D₅₃	C₄⋊C₄	1	4	yes	—
109	C₄×D₅₃	C₄⋊C₄	1	4	yes	—
110	C₄×D₅₃	C₂²×C₄	1	2	yes	—
111	C₄×D₅₃	C₂²×C₄	1	2	yes	—
112	C₄×D₅₃	C₂×Q₈	1	2	yes	—
113	C₄×D₅₃	C₂×Q₈	1	2	yes	—
114	C₈×C₅₃	C₂²×C₄	1	4	yes	—
115	#88322	C₂²×C₄	1	4	yes	—
116	C₂²×C₅₃	—	2	1	no	—
117	C₂²×C₅₃	—	2	1	no	—
118	C₂²×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
119	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×D₅₃	—	2	1	no	—
121	C₂×D₅₃	—	2	1	no	—
122	C₂×D₅₃	—	2	1	no	—
123	C₂×D₅₃	—	2	1	no	—
124	C₂×D₅₃	—	4	4	no	—
125	C₂×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
127	C₄×C₅₃	—	4	1	no	—
128	C₄×C₅₃	—	4	1	no	—
129	C₄×C₅₃	—	4	2	no	—
130	C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
131	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
132	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
133	Dic₅₃	—	4	1	no	—
134	Dic₅₃	—	4	1	no	—
135	Dic₅₃	—	4	2	no	—
136	Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
137	Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
138	#2911	—	53	1	no	maximal, Sylow
139	C₂×C₅₃	—	2	2	no	—
140	C₂×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₅₃	#575	1	1	yes	characteristic
142	C₂×C₅₃	#575	1	1	yes	characteristic
143	D₅₃	—	2	2	no	—
144	D₅₃	—	2	2	no	—
145	C₂²×M₁₆	—	53	1	no	—
146	C₂×#117	—	53	1	no	—
147	C₂×#117	—	53	1	no	—
148	#428	—	53	1	no	—
149	#428	—	53	1	no	—
150	#428	—	53	1	no	—
151	#428	—	53	2	no	—
152	#428	—	53	2	no	—
153	#428	D₅₃	1	1	yes	characteristic, complemented
154	#427	—	53	2	no	—
155	#427	—	53	2	no	—
156	C₅₃	#2911	1	1	yes	characteristic, complemented, Sylow
157	C₂³×C₄	—	53	1	no	—
158	C₂×M₁₆	—	53	1	no	—
159	C₂×M₁₆	—	53	8	no	—
160	C₂×M₁₆	—	53	2	no	—
161	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
162	C₂²×C₈	—	53	2	no	—
163	#117	—	53	1	no	—
164	#117	—	53	1	no	—
165	#117	—	53	2	no	—
166	#117	—	53	1	no	—
167	#117	—	53	2	no	—
168	#117	—	106	1	no	—
169	#117	—	106	1	no	—
170	#117	—	106	1	no	—
171	#117	—	106	1	no	—
172	#117	C₂×D₅₃	1	1	yes	characteristic
173	C₂×C₄⋊C₄	—	53	1	no	—
174	C₂×C₄⋊C₄	—	53	1	no	—
175	C₂×C₄⋊C₄	—	53	1	no	—
176	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
177	C₂×C₂²⋊C₄	—	106	1	no	—
178	C₂×C₂²⋊C₄	—	106	1	no	—
179	C₂×C₄²	—	106	1	no	—
180	C₂×C₄²	—	106	1	no	—
181	#107	—	53	2	no	—
182	#107	—	53	2	no	—
183	#107	—	53	2	no	—
184	#107	C₂×D₅₃	1	2	yes	complemented
185	#106	—	53	2	no	—
186	#106	—	53	2	no	—
187	#106	—	53	2	no	—
188	#106	C₂×D₅₃	1	2	yes	complemented
189	C₂⁴	—	53	1	no	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	53	1	no	—
192	C₂²×C₄	—	53	4	no	—
193	C₂²×C₄	—	53	1	no	—
194	C₂²×C₄	—	53	1	no	—
195	C₂²×C₄	—	53	4	no	—
196	C₂²×C₄	—	53	1	no	—
197	C₂²×C₄	—	53	1	no	—
198	C₂²×C₄	—	106	1	no	—
199	C₂²×C₄	—	106	1	no	—
200	C₂²×C₄	—	106	1	no	—
201	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
202	M₁₆	—	53	4	no	—
203	M₁₆	—	53	8	no	—
204	M₁₆	C₄×D₅₃	1	4	yes	complemented
205	C₂×C₈	—	53	2	no	—
206	C₂×C₈	—	53	8	no	—
207	C₂×C₈	C₂²×D₅₃	1	2	yes	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	53	1	no	—
210	C₄⋊C₄	—	53	2	no	—
211	C₄⋊C₄	—	53	1	no	—
212	C₄⋊C₄	—	106	1	no	—
213	C₄⋊C₄	—	106	1	no	—
214	C₄⋊C₄	—	106	1	no	—
215	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
216	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
217	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
218	C₂²⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	106	1	no	—
220	C₂²⋊C₄	—	106	1	no	—
221	C₂²⋊C₄	—	106	2	no	—
222	C₂²⋊C₄	—	106	2	no	—
223	C₂²⋊C₄	—	106	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	106	1	no	—
226	C₄²	—	106	1	no	—
227	C₄²	—	106	2	no	—
228	C₄²	—	106	2	no	—
229	C₄²	—	106	1	no	—
230	C₂³	—	53	1	no	—
231	C₂³	—	53	1	no	—
232	C₂³	—	53	4	no	—
233	C₂³	—	212	1	no	—
234	C₂³	—	212	1	no	—
235	C₂³	D₄×D₅₃	1	1	yes	characteristic
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	53	4	no	—
241	C₂×C₄	—	53	1	no	—
242	C₂×C₄	—	53	1	no	—
243	C₂×C₄	—	53	4	no	—
244	C₂×C₄	—	53	4	no	—
245	C₂×C₄	—	53	2	no	—
246	C₂×C₄	—	53	2	no	—
247	C₂×C₄	—	53	2	no	—
248	C₂×C₄	—	53	2	no	—
249	C₂×C₄	—	106	1	no	—
250	C₂×C₄	—	106	1	no	—
251	C₂×C₄	—	106	2	no	—
252	C₂×C₄	—	212	2	no	—
253	C₂×C₄	—	212	2	no	—
254	C₂×C₄	—	212	2	no	—
255	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
256	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
257	C₂×C₄	Q₈×D₅₃	1	2	yes	—
258	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
259	C₈	—	53	4	no	—
260	C₈	C₂×C₄×D₅₃	1	4	yes	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	53	2	no	—
264	C₂²	—	53	2	no	—
265	C₂²	—	106	1	no	—
266	C₂²	—	106	1	no	—
267	C₂²	—	106	1	no	—
268	C₂²	—	106	1	no	—
269	C₂²	—	212	4	no	—
270	C₂²	C₄⋊C₄×D₅₃	1	2	yes	—
271	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
272	C₄	—	4	2	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	1	no	—
275	C₄	—	53	2	no	—
276	C₄	—	53	2	no	—
277	C₄	—	212	1	no	—
278	C₄	—	212	1	no	—
279	C₄	—	212	2	no	—
280	C₄	C₄⋊C₄×D₅₃	1	2	yes	—
281	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
282	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
283	C₂	—	2	2	no	—
284	C₂	—	106	2	no	—
285	C₂	—	106	2	no	—
286	C₂	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
287	C₂	#16811419	1	1	yes	characteristic
288	C₂	#16811422	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#428	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#428	D₅₃
hypocenter	C₅₃	#2911
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
4. #16810666	C₂	semidirect
5. #16810665	C₂	semidirect
6. #16810664	C₂	semidirect
7. #16810663	C₂	semidirect
8. #16810662	C₂	semidirect
9. #16810660	C₂	semidirect
10. #16810658	C₂	semidirect
11. C₅₃×#428	C₂	semidirect
15. M₁₆×D₅₃	C₄	semidirect
33. #13133151	C₂²	semidirect
34. #13133150	C₂²	semidirect
35. #13133150	C₂²	semidirect
36. #13133149	C₂²	semidirect
37. #13133148	C₂²	semidirect
38. #13133148	C₂²	semidirect
42. #107×C₅₃	C₂²	semidirect
43. #106×C₅₃	C₂²	semidirect
57. M₁₆×C₅₃	C₂×C₄	semidirect
81. #11719917	C₂×C₄	semidirect
83. #11719912	C₂×C₄	semidirect
84. C₈×D₅₃	C₂×C₄	semidirect
153. #428	D₅₃	semidirect
156. C₅₃	#2911	semidirect
184. #107	C₂×D₅₃	semidirect
188. #106	C₂×D₅₃	semidirect
204. M₁₆	C₄×D₅₃	semidirect