Subgroups — #20286502

Order 6784.

454 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811440	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811430	C₂	1	2	yes	maximal, complemented
3	#16811428	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811007	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810898	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810893	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810887	C₂	1	2	yes	maximal, complemented
8	#16810802	C₂	1	2	yes	maximal, complemented
9	#16810797	C₂	1	2	yes	maximal, complemented
10	#16810741	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#443	C₂	1	1	yes	maximal, characteristic, complemented
12	#13133264	C₂²	1	2	yes	complemented
13	#13133263	C₂²	1	2	yes	—
14	#13133250	—	2	2	no	—
15	#13133250	C₂²	1	2	yes	complemented
16	#13133249	—	2	2	no	—
17	#13133247	C₂²	1	2	yes	—
18	#13133245	—	2	2	no	—
19	#13133245	C₂²	1	2	yes	—
20	#13133242	C₂²	1	2	yes	complemented
21	#13133241	—	2	2	no	—
22	#13133239	—	2	1	no	—
23	#13133239	—	2	1	no	—
24	#13133239	C₂²	1	1	yes	characteristic
25	#13133238	—	2	2	no	—
26	C₂×Dic₂₁₂	—	2	1	no	—
27	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic, complemented
28	C₂×#11719913	C₂²	1	2	yes	complemented
29	SD₁₆×D₅₃	—	2	2	no	—
30	C₂×D₄₂₄	—	2	1	no	—
31	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
32	#13133223	C₂²	1	2	yes	complemented
33	#13133205	C₂²	1	2	yes	complemented
34	#13133153	C₂²	1	1	yes	characteristic, complemented
35	#13133145	C₂²	1	2	yes	complemented
36	#13133135	C₂²	1	2	yes	complemented
37	#13133133	C₂²	1	2	yes	complemented
38	#135×C₅₃	C₂²	1	1	yes	characteristic
39	#131×C₅₃	C₂²	1	1	yes	characteristic
40	#131×C₅₃	C₂²	1	1	yes	characteristic, complemented
41	#108×C₅₃	C₂²	1	1	yes	characteristic, complemented
42	#104×C₅₃	C₂²	1	1	yes	characteristic
43	#104×C₅₃	C₂²	1	1	yes	characteristic, complemented
44	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
45	#41×C₅₃	C₂³	1	1	yes	characteristic
46	#41×C₅₃	C₂³	1	1	yes	characteristic
47	#11719949	—	2	2	no	—
48	Q₈×D₅₃	—	2	2	no	—
49	#11719947	—	2	2	no	—
50	#11719947	—	4	2	no	—
51	D₄×D₅₃	—	2	2	no	—
52	D₄×D₅₃	—	4	2	no	—
53	#11719945	—	2	2	no	—
54	#11719945	—	2	2	no	—
55	#11719945	C₂³	1	2	yes	—
56	C₂×D₂₁₂	—	2	2	no	—
57	C₂×D₂₁₂	C₂³	1	2	yes	—
58	C₂×Dic₁₀₆	—	2	2	no	—
59	C₂×Dic₁₀₆	C₂³	1	2	yes	—
60	Q₁₆×C₅₃	D₄	1	1	yes	characteristic
61	SD₁₆×C₅₃	D₄	1	1	yes	characteristic
62	SD₁₆×C₅₃	D₄	1	1	yes	characteristic
63	D₈×C₅₃	D₄	1	1	yes	characteristic
64	M₁₆×C₅₃	—	2	1	no	—
65	M₁₆×C₅₃	—	2	1	no	—
66	M₁₆×C₅₃	C₂³	1	1	yes	characteristic
67	M₁₆×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₅₃	—	2	1	no	—
69	C₂×C₈×C₅₃	—	2	1	no	—
70	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
72	C₄²×C₅₃	C₂³	1	1	yes	characteristic
73	#11719925	—	2	2	no	—
74	#11719924	—	2	2	no	—
75	#11719923	—	2	2	no	—
76	#11719922	—	2	2	no	—
77	#11719921	—	4	2	no	—
78	#11719917	C₂³	1	2	yes	—
79	Dic₂₁₂	—	2	2	no	—
80	Dic₂₁₂	—	2	1	no	—
81	Dic₂₁₂	—	2	1	no	—
82	Dic₂₁₂	D₄	1	1	yes	characteristic
83	D₄₂₄	—	2	2	no	—
84	D₄₂₄	—	2	1	no	—
85	D₄₂₄	—	2	1	no	—
86	D₄₂₄	D₄	1	1	yes	characteristic
87	#11719913	—	2	2	no	—
88	#11719913	—	2	2	no	—
89	#11719913	—	2	2	no	—
90	#11719913	—	2	2	no	—
91	#11719913	D₄	1	2	yes	—
92	#11719912	—	2	2	no	—
93	C₈×D₅₃	—	2	2	no	—
94	C₂²×D₅₃	—	2	2	no	—
95	C₂²×D₅₃	—	4	2	no	—
96	Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
97	Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
98	D₄×C₅₃	—	2	1	no	—
99	D₄×C₅₃	—	2	1	no	—
100	D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
101	D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
106	C₂²⋊D₅₃	—	2	2	no	—
107	C₂²⋊D₅₃	—	4	2	no	—
108	C₂²⋊D₅₃	—	4	2	no	—
109	C₂×Dic₅₃	—	2	2	no	—
110	C₂×Dic₅₃	—	4	2	no	—
111	D₂₁₂	—	2	2	no	—
112	D₂₁₂	—	2	2	no	—
113	D₂₁₂	—	4	2	no	—
114	D₂₁₂	C₂×D₄	1	2	yes	—
115	C₄×D₅₃	—	2	2	no	—
116	C₄×D₅₃	—	4	2	no	—
117	C₄×D₅₃	—	4	2	no	—
118	Dic₁₀₆	—	2	2	no	—
119	Dic₁₀₆	—	2	2	no	—
120	Dic₁₀₆	—	4	2	no	—
121	Dic₁₀₆	C₂×D₄	1	2	yes	—
122	C₈×C₅₃	—	2	1	no	—
123	C₈×C₅₃	—	2	1	no	—
124	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
125	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
126	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
127	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
128	#88322	—	2	2	no	—
129	C₂²×C₅₃	—	2	1	no	—
130	C₂²×C₅₃	—	2	1	no	—
131	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×D₅₃	—	2	2	no	—
133	C₂×D₅₃	—	4	2	no	—
134	C₂×D₅₃	—	8	2	no	—
135	C₂×D₅₃	—	16	2	no	—
136	C₄×C₅₃	—	2	1	no	—
137	C₄×C₅₃	—	2	1	no	—
138	C₄×C₅₃	—	4	1	no	—
139	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
140	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
141	Dic₅₃	—	2	2	no	—
142	Dic₅₃	—	4	2	no	—
143	#3290	—	53	1	no	maximal, Sylow
144	C₂×C₅₃	—	2	1	no	—
145	C₂×C₅₃	—	4	1	no	—
146	C₂×C₅₃	—	4	1	no	—
147	C₂×C₅₃	D₄²	1	1	yes	characteristic
148	D₅₃	—	4	2	no	—
149	D₅₃	—	8	2	no	—
150	#578	—	53	1	no	—
151	#577	—	53	2	no	—
152	#576	—	53	1	no	—
153	#495	—	53	1	no	—
154	#473	—	53	1	no	—
155	#472	—	53	1	no	—
156	#471	—	53	2	no	—
157	#455	—	53	2	no	—
158	#454	—	53	2	no	—
159	#443	—	53	1	no	—
160	#443	D₅₃	1	1	yes	characteristic, complemented
161	C₅₃	#3290	1	1	yes	characteristic, complemented, Sylow
162	2₋¹⁺⁴	—	53	2	no	—
163	H(5,𝔽₂)	—	53	2	no	—
164	#137	—	53	2	no	—
165	#137	—	53	2	no	—
166	#137	—	106	2	no	—
167	#137	—	106	2	no	—
168	AGL(1,ℤ/8)	—	53	2	no	—
169	AGL(1,ℤ/8)	—	53	2	no	—
170	AGL(1,ℤ/8)	—	106	2	no	—
171	AGL(1,ℤ/8)	—	106	2	no	—
172	#135	—	53	1	no	—
173	#135	—	106	1	no	—
174	#135	—	106	1	no	—
175	#135	—	106	2	no	—
176	#135	C₂×D₅₃	1	1	yes	characteristic
177	C₂×Q₁₆	—	53	1	no	—
178	C₂×Q₁₆	—	106	1	no	—
179	C₂×SD₁₆	—	53	2	no	—
180	C₂×SD₁₆	—	106	2	no	—
181	C₂×D₈	—	53	1	no	—
182	C₂×D₈	—	106	1	no	—
183	#131	—	53	2	no	—
184	#131	C₂×D₅₃	1	1	yes	characteristic
185	#131	C₂×D₅₃	1	1	yes	characteristic, complemented
186	#124	—	53	2	no	—
187	#108	—	53	1	no	—
188	#108	C₂×D₅₃	1	1	yes	characteristic, complemented
189	#104	—	53	2	no	—
190	#104	C₂×D₅₃	1	1	yes	characteristic
191	#104	C₂×D₅₃	1	1	yes	characteristic, complemented
192	#101	—	53	2	no	—
193	#100	—	53	2	no	—
194	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic, complemented
195	#41	—	53	2	no	—
196	#41	—	106	2	no	—
197	#41	—	106	2	no	—
198	#41	—	106	2	no	—
199	#41	—	106	2	no	—
200	#41	—	212	2	no	—
201	#41	C₂²×D₅₃	1	1	yes	characteristic
202	#41	C₂²×D₅₃	1	1	yes	characteristic
203	C₂×Q₈	—	53	2	no	—
204	C₂×Q₈	—	106	2	no	—
205	C₂×Q₈	—	106	2	no	—
206	C₂×D₄	—	53	2	no	—
207	C₂×D₄	—	106	2	no	—
208	C₂×D₄	—	106	2	no	—
209	C₂×D₄	—	212	2	no	—
210	Q₁₆	—	53	1	no	—
211	Q₁₆	—	106	2	no	—
212	Q₁₆	—	106	2	no	—
213	Q₁₆	—	106	1	no	—
214	Q₁₆	—	106	1	no	—
215	Q₁₆	D₂₁₂	1	1	yes	characteristic
216	SD₁₆	—	53	2	no	—
217	SD₁₆	—	106	2	no	—
218	SD₁₆	—	106	2	no	—
219	SD₁₆	—	106	2	no	—
220	SD₁₆	—	106	2	no	—
221	SD₁₆	—	106	2	no	—
222	SD₁₆	—	106	2	no	—
223	SD₁₆	D₂₁₂	1	1	yes	characteristic
224	SD₁₆	D₂₁₂	1	1	yes	characteristic
225	D₈	—	53	1	no	—
226	D₈	—	106	2	no	—
227	D₈	—	106	2	no	—
228	D₈	—	106	1	no	—
229	D₈	—	106	1	no	—
230	D₈	D₂₁₂	1	1	yes	characteristic
231	M₁₆	—	2	1	no	—
232	M₁₆	—	2	1	no	—
233	M₁₆	—	53	2	no	—
234	M₁₆	—	106	2	no	—
235	M₁₆	C₂²×D₅₃	1	1	yes	characteristic
236	M₁₆	C₂²×D₅₃	1	1	yes	characteristic
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	106	2	no	—
240	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
241	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
242	C₂²⋊C₄	—	212	2	no	—
243	C₄²	C₂²×D₅₃	1	1	yes	characteristic
244	C₂³	—	106	2	no	—
245	C₂³	—	212	2	no	—
246	Q₈	—	53	2	no	—
247	Q₈	—	106	2	no	—
248	Q₈	—	106	2	no	—
249	Q₈	—	212	2	no	—
250	Q₈	C₂×D₂₁₂	1	1	yes	characteristic
251	Q₈	C₂×D₂₁₂	1	1	yes	characteristic
252	D₄	—	2	1	no	—
253	D₄	—	2	1	no	—
254	D₄	—	53	2	no	—
255	D₄	—	106	2	no	—
256	D₄	—	106	2	no	—
257	D₄	—	106	2	no	—
258	D₄	—	212	2	no	—
259	D₄	—	212	2	no	—
260	D₄	—	212	2	no	—
261	D₄	C₂×D₂₁₂	1	1	yes	characteristic
262	D₄	C₂×D₂₁₂	1	1	yes	characteristic
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	106	2	no	—
267	C₂×C₄	—	106	2	no	—
268	C₂×C₄	—	212	2	no	—
269	C₂×C₄	—	212	2	no	—
270	C₂×C₄	—	212	2	no	—
271	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
272	C₈	—	2	1	no	—
273	C₈	—	2	1	no	—
274	C₈	—	106	2	no	—
275	C₈	C₂×D₂₁₂	1	1	yes	characteristic
276	C₈	C₂×D₂₁₂	1	1	yes	characteristic
277	C₈	D₄×D₅₃	1	1	yes	characteristic
278	C₈	D₄×D₅₃	1	1	yes	characteristic
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	848	2	no	—
282	C₂²	—	106	2	no	—
283	C₂²	—	424	2	no	—
284	C₂²	—	212	2	no	—
285	C₂²	#13133263	1	1	yes	characteristic
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	4	1	no	—
289	C₄	—	106	2	no	—
290	C₄	—	212	2	no	—
291	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
292	C₄	C₂²×D₂₁₂	1	1	yes	characteristic
293	C₂	—	2	1	no	—
294	C₂	—	4	1	no	—
295	C₂	—	4	1	no	—
296	C₂	—	424	2	no	—
297	C₂	—	212	2	no	—
298	C₂	D₄×D₂₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#443	C₂
socle	C₂×C₅₃	D₄²
hypercenter	#443	D₅₃
hypocenter	C₅₃	#3290
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #443

Complements

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

N	H ≅ G/N	type
1. #16811440	C₂	semidirect
2. #16811430	C₂	semidirect
3. #16811428	C₂	semidirect
4. #16811007	C₂	semidirect
5. #16810898	C₂	semidirect
6. #16810893	C₂	semidirect
7. #16810887	C₂	semidirect
8. #16810802	C₂	semidirect
9. #16810797	C₂	semidirect
10. #16810741	C₂	semidirect
11. C₅₃×#443	C₂	semidirect
12. #13133264	C₂²	semidirect
15. #13133250	C₂²	semidirect
20. #13133242	C₂²	semidirect
27. C₂×Dic₂₁₂	C₂²	semidirect
28. C₂×#11719913	C₂²	semidirect
32. #13133223	C₂²	semidirect
33. #13133205	C₂²	semidirect
34. #13133153	C₂²	semidirect
35. #13133145	C₂²	semidirect
36. #13133135	C₂²	semidirect
37. #13133133	C₂²	semidirect
40. #131×C₅₃	C₂²	semidirect
41. #108×C₅₃	C₂²	semidirect
43. #104×C₅₃	C₂²	semidirect
44. C₄×C₈×C₅₃	C₂²	semidirect
160. #443	D₅₃	semidirect
161. C₅₃	#3290	semidirect
185. #131	C₂×D₅₃	semidirect
188. #108	C₂×D₅₃	semidirect
191. #104	C₂×D₅₃	semidirect
194. C₄×C₈	C₂×D₅₃	semidirect