Subgroups — #20283613

Order 6784.

456 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811420	C₂	1	1	yes	maximal, characteristic
2	#131×D₅₃	C₂	1	2	yes	maximal, complemented
3	#16810745	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810744	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810742	C₂	1	2	yes	maximal, complemented
6	#16810741	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810738	C₂	1	2	yes	maximal, complemented
8	#16810670	C₂	1	1	yes	maximal, characteristic, complemented
9	#16810638	C₂	1	2	yes	maximal, complemented
10	#16810577	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#443	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₅₃	C₂²	1	2	yes	—
13	#13133249	C₄	1	2	yes	complemented
14	#13133246	C₄	1	2	yes	complemented
15	#13133244	C₄	1	2	yes	complemented
16	#13133241	C₄	1	2	yes	complemented
17	#13133239	C₂²	1	1	yes	characteristic
18	#13133237	C₂²	1	2	yes	—
19	#13133225	—	2	2	no	—
20	#13133225	C₂²	1	2	yes	complemented
21	#13133224	C₂²	1	2	yes	complemented
22	#13133223	—	2	2	no	—
23	#13133223	C₂²	1	2	yes	complemented
24	C₂×#11719912	C₂²	1	1	yes	characteristic
25	M₁₆×D₅₃	—	2	2	no	—
26	M₁₆×D₅₃	C₂²	1	2	yes	—
27	C₂×C₈×D₅₃	—	2	2	no	—
28	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
29	#13133174	C₂²	1	1	yes	characteristic
30	#13133153	C₂²	1	1	yes	characteristic, complemented
31	#13133152	C₂²	1	2	yes	complemented
32	#13133145	C₂²	1	2	yes	complemented
33	#13133144	C₂²	1	2	yes	complemented
34	#13133143	C₂²	1	2	yes	complemented
35	#13133128	C₂²	1	1	yes	characteristic, complemented
36	#13133127	C₂²	1	1	yes	characteristic, complemented
37	C₈×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
38	#135×C₅₃	C₂²	1	1	yes	characteristic
39	#131×C₅₃	C₂²	1	2	yes	complemented
40	#108×C₅₃	C₂²	1	1	yes	characteristic, complemented
41	#104×C₅₃	C₂²	1	2	yes	complemented
42	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
43	#41×C₅₃	C₂³	1	2	yes	—
44	#11719949	C₂×C₄	1	2	yes	—
45	Q₈×D₅₃	C₂×C₄	1	2	yes	—
46	#11719947	—	2	2	no	—
47	#11719947	C₂×C₄	1	2	yes	—
48	D₄×D₅₃	—	2	2	no	—
49	D₄×D₅₃	C₂×C₄	1	2	yes	—
50	#11719945	—	2	2	no	—
51	#11719945	C₂³	1	2	yes	—
52	C₂×C₄×D₅₃	—	2	2	no	—
53	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
54	Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
55	SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
56	D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
57	M₁₆×C₅₃	—	2	2	no	—
58	M₁₆×C₅₃	C₂³	1	2	yes	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₄²×C₅₃	C₂³	1	1	yes	characteristic
63	#11719925	C₂×C₄	1	2	yes	—
64	#11719924	C₂×C₄	1	2	yes	—
65	#11719923	C₂×C₄	1	2	yes	—
66	#11719922	C₂×C₄	1	2	yes	—
67	#11719921	—	2	1	no	—
68	#11719919	—	2	1	no	—
69	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
70	#11719917	—	2	2	no	—
71	#11719917	C₂³	1	2	yes	—
72	C₂×#88322	—	2	2	no	—
73	C₂×#88322	C₂³	1	1	yes	characteristic
74	C₂×#88322	C₂³	1	1	yes	characteristic
75	Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
76	D₄₂₄	C₂×C₄	1	1	yes	characteristic
77	#11719913	C₂×C₄	1	2	yes	—
78	#11719912	—	2	2	no	—
79	#11719912	—	2	2	no	—
80	#11719912	—	2	2	no	—
81	#11719912	C₂×C₄	1	2	yes	—
82	#11719912	C₂×C₄	1	2	yes	—
83	C₈×D₅₃	—	2	2	no	—
84	C₈×D₅₃	—	2	2	no	—
85	C₈×D₅₃	—	2	2	no	—
86	C₈×D₅₃	D₄	1	2	yes	—
87	C₈×D₅₃	D₄	1	2	yes	—
88	C₂²×D₅₃	—	2	2	no	—
89	C₂²×D₅₃	#41	1	1	yes	characteristic
90	Q₈×C₅₃	C₂²×C₄	1	2	yes	—
91	D₄×C₅₃	—	2	2	no	—
92	D₄×C₅₃	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
96	C₂²⋊D₅₃	—	2	2	no	—
97	C₂²⋊D₅₃	—	2	2	no	—
98	C₂²⋊D₅₃	—	2	2	no	—
99	C₂×Dic₅₃	—	2	2	no	—
100	C₂×Dic₅₃	—	2	1	no	—
101	C₂×Dic₅₃	#41	1	1	yes	characteristic
102	D₂₁₂	—	2	2	no	—
103	D₂₁₂	C₂²×C₄	1	2	yes	—
104	C₄×D₅₃	—	2	2	no	—
105	C₄×D₅₃	—	2	2	no	—
106	C₄×D₅₃	—	2	2	no	—
107	C₄×D₅₃	C₂²×C₄	1	2	yes	—
108	C₄×D₅₃	C₂×D₄	1	2	yes	—
109	Dic₁₀₆	—	2	2	no	—
110	Dic₁₀₆	C₂²×C₄	1	2	yes	—
111	C₈×C₅₃	—	2	2	no	—
112	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
116	#88322	—	2	2	no	—
117	#88322	C₂²×C₄	1	2	yes	—
118	#88322	C₂×D₄	1	2	yes	—
119	C₂²×C₅₃	—	2	2	no	—
120	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
121	C₂×D₅₃	—	2	2	no	—
122	C₂×D₅₃	—	4	1	no	—
123	C₂×D₅₃	—	4	4	no	—
124	C₂×D₅₃	C₄×D₄	1	2	yes	—
125	C₄×C₅₃	—	2	2	no	—
126	C₄×C₅₃	—	4	1	no	—
127	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
128	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
129	Dic₅₃	—	2	2	no	—
130	Dic₅₃	—	4	1	no	—
131	Dic₅₃	C₄×D₄	1	2	yes	—
132	#2955	—	53	1	no	maximal, Sylow
133	C₂×C₅₃	—	2	1	no	—
134	C₂×C₅₃	—	4	2	no	—
135	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
136	D₅₃	—	2	2	no	—
137	D₅₃	—	4	2	no	—
138	#575	—	53	1	no	—
139	C₂×#131	—	53	2	no	—
140	#444	—	53	1	no	—
141	#444	—	53	1	no	—
142	#444	—	53	2	no	—
143	#443	—	53	2	no	—
144	#443	—	53	1	no	—
145	#443	D₅₃	1	1	yes	characteristic, complemented
146	#430	—	53	1	no	—
147	#421	—	53	2	no	—
148	#405	—	53	1	no	—
149	C₅₃	#2955	1	1	yes	characteristic, complemented, Sylow
150	C₂×#41	—	53	2	no	—
151	#137	—	53	2	no	—
152	#137	—	53	2	no	—
153	AGL(1,ℤ/8)	—	53	2	no	—
154	AGL(1,ℤ/8)	—	53	2	no	—
155	#135	—	53	2	no	—
156	#135	—	53	1	no	—
157	#135	C₂×D₅₃	1	1	yes	characteristic
158	#131	—	53	2	no	—
159	#131	—	53	2	no	—
160	#131	—	53	2	no	—
161	#131	—	106	2	no	—
162	#131	—	106	2	no	—
163	#131	C₂×D₅₃	1	2	yes	complemented
164	C₂×M₁₆	—	53	1	no	—
165	C₂×M₁₆	—	53	2	no	—
166	C₂×M₁₆	—	106	2	no	—
167	C₂²×C₈	—	53	1	no	—
168	C₂²×C₈	—	106	2	no	—
169	#117	—	53	1	no	—
170	#108	—	53	2	no	—
171	#108	—	53	1	no	—
172	#108	C₂×D₅₃	1	1	yes	characteristic, complemented
173	#104	—	53	2	no	—
174	#104	—	53	2	no	—
175	#104	—	53	2	no	—
176	#104	C₂×D₅₃	1	2	yes	complemented
177	#97	—	53	1	no	—
178	#97	—	53	1	no	—
179	C₄×C₈	—	53	1	no	—
180	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic, complemented
181	#41	—	53	2	no	—
182	#41	—	53	2	no	—
183	#41	—	53	2	no	—
184	#41	—	106	2	no	—
185	#41	—	106	2	no	—
186	#41	C₂²×D₅₃	1	2	yes	—
187	C₂×Q₈	—	53	2	no	—
188	C₂×D₄	—	53	2	no	—
189	C₂×D₄	—	106	2	no	—
190	C₂²×C₄	—	53	1	no	—
191	C₂²×C₄	—	106	2	no	—
192	Q₁₆	—	53	2	no	—
193	Q₁₆	—	53	1	no	—
194	Q₁₆	C₄×D₅₃	1	1	yes	characteristic
195	SD₁₆	—	53	2	no	—
196	SD₁₆	—	53	2	no	—
197	SD₁₆	—	53	2	no	—
198	SD₁₆	C₄×D₅₃	1	2	yes	—
199	D₈	—	53	2	no	—
200	D₈	—	53	1	no	—
201	D₈	C₄×D₅₃	1	1	yes	characteristic
202	M₁₆	—	2	2	no	—
203	M₁₆	—	53	2	no	—
204	M₁₆	—	53	2	no	—
205	M₁₆	—	53	2	no	—
206	M₁₆	—	106	2	no	—
207	M₁₆	—	106	2	no	—
208	M₁₆	—	106	2	no	—
209	M₁₆	—	106	2	no	—
210	M₁₆	C₂²×D₅₃	1	2	yes	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	53	1	no	—
213	C₂×C₈	—	53	1	no	—
214	C₂×C₈	—	53	2	no	—
215	C₂×C₈	—	53	2	no	—
216	C₂×C₈	—	106	2	no	—
217	C₂×C₈	—	106	2	no	—
218	C₂×C₈	—	106	2	no	—
219	C₂×C₈	—	106	2	no	—
220	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
221	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
222	C₄⋊C₄	—	106	1	no	—
223	C₂²⋊C₄	—	106	1	no	—
224	C₄²	—	53	1	no	—
225	C₄²	C₂²×D₅₃	1	1	yes	characteristic
226	C₂³	—	53	1	no	—
227	C₂³	—	106	2	no	—
228	Q₈	—	53	2	no	—
229	Q₈	—	106	2	no	—
230	Q₈	C₂×C₄×D₅₃	1	2	yes	—
231	D₄	—	2	2	no	—
232	D₄	—	53	2	no	—
233	D₄	—	106	2	no	—
234	D₄	—	106	2	no	—
235	D₄	—	106	2	no	—
236	D₄	—	106	2	no	—
237	D₄	C₂×C₄×D₅₃	1	2	yes	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	53	1	no	—
241	C₂×C₄	—	53	2	no	—
242	C₂×C₄	—	53	2	no	—
243	C₂×C₄	—	106	2	no	—
244	C₂×C₄	—	106	2	no	—
245	C₂×C₄	—	106	2	no	—
246	C₂×C₄	—	106	2	no	—
247	C₂×C₄	—	106	1	no	—
248	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	53	2	no	—
251	C₈	—	53	2	no	—
252	C₈	—	106	2	no	—
253	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
254	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
255	C₈	D₄×D₅₃	1	1	yes	characteristic
256	C₈	D₄×D₅₃	1	1	yes	characteristic
257	C₂²	—	2	2	no	—
258	C₂²	—	53	2	no	—
259	C₂²	—	106	2	no	—
260	C₂²	—	212	1	no	—
261	C₂²	—	212	4	no	—
262	C₂²	#41×D₅₃	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	53	2	no	—
266	C₄	—	106	2	no	—
267	C₄	—	212	1	no	—
268	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
269	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	4	2	no	—
272	C₂	—	106	2	no	—
273	C₂	—	212	2	no	—
274	C₂	C₄×D₄×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#443	C₂
socle	C₂×C₅₃	C₂×C₄×D₄
hypercenter	#443	D₅₃
hypocenter	C₅₃	#2955
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
2. #131×D₅₃	C₂	semidirect
3. #16810745	C₂	semidirect
4. #16810744	C₂	semidirect
5. #16810742	C₂	semidirect
6. #16810741	C₂	semidirect
7. #16810738	C₂	semidirect
8. #16810670	C₂	semidirect
9. #16810638	C₂	semidirect
10. #16810577	C₂	semidirect
11. C₅₃×#443	C₂	semidirect
13. #13133249	C₄	semidirect
14. #13133246	C₄	semidirect
15. #13133244	C₄	semidirect
16. #13133241	C₄	semidirect
20. #13133225	C₂²	semidirect
21. #13133224	C₂²	semidirect
23. #13133223	C₂²	semidirect
30. #13133153	C₂²	semidirect
31. #13133152	C₂²	semidirect
32. #13133145	C₂²	semidirect
33. #13133144	C₂²	semidirect
34. #13133143	C₂²	semidirect
35. #13133128	C₂²	semidirect
36. #13133127	C₂²	semidirect
37. C₈×Dic₅₃	C₂²	semidirect
39. #131×C₅₃	C₂²	semidirect
40. #108×C₅₃	C₂²	semidirect
41. #104×C₅₃	C₂²	semidirect
42. C₄×C₈×C₅₃	C₂²	semidirect
145. #443	D₅₃	semidirect
149. C₅₃	#2955	semidirect
163. #131	C₂×D₅₃	semidirect
172. #108	C₂×D₅₃	semidirect
176. #104	C₂×D₅₃	semidirect
180. C₄×C₈	C₂×D₅₃	semidirect