Subgroups — #444×D₅₃ #20283607

Order 6784.

464 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#131×D₅₃	C₂	1	2	yes	maximal, complemented
3	#16810746	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810745	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810744	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810743	C₂	1	2	yes	maximal, complemented
7	#16810742	C₂	1	2	yes	maximal, complemented
8	#108×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#104×D₅₃	C₂	1	2	yes	maximal, complemented
10	#97×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#444	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₅₃	C₂²	1	2	yes	—
13	#13133240	C₄	1	1	yes	characteristic, complemented
14	#13133239	C₂²	1	1	yes	characteristic
15	#13133238	C₄	1	2	yes	complemented
16	#13133237	C₂²	1	2	yes	—
17	#13133236	C₄	1	1	yes	characteristic, complemented
18	Q₁₆×D₅₃	C₄	1	1	yes	characteristic, complemented
19	SD₁₆×D₅₃	C₄	1	2	yes	complemented
20	D₈×D₅₃	C₄	1	1	yes	characteristic, complemented
21	#13133225	—	2	2	no	—
22	#13133225	C₂²	1	2	yes	complemented
23	#13133224	C₂²	1	2	yes	complemented
24	#13133223	—	2	2	no	—
25	#13133223	C₂²	1	2	yes	complemented
26	M₁₆×D₅₃	—	2	2	no	—
27	M₁₆×D₅₃	C₂²	1	2	yes	—
28	C₂×C₈×D₅₃	—	2	2	no	—
29	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
31	C₄²×D₅₃	C₂²	1	1	yes	characteristic
32	#13133153	C₂²	1	1	yes	characteristic, complemented
33	#13133152	C₂²	1	2	yes	complemented
34	#13133145	C₂²	1	2	yes	complemented
35	#13133144	C₂²	1	2	yes	complemented
36	#13133143	C₂²	1	2	yes	complemented
37	#13133128	C₂²	1	1	yes	characteristic, complemented
38	#13133127	C₂²	1	1	yes	characteristic, complemented
39	#13133127	C₂²	1	1	yes	characteristic, complemented
40	#135×C₅₃	C₂²	1	1	yes	characteristic
41	#131×C₅₃	C₂²	1	2	yes	complemented
42	#108×C₅₃	C₂²	1	1	yes	characteristic, complemented
43	#104×C₅₃	C₂²	1	2	yes	complemented
44	#97×C₅₃	C₂²	1	1	yes	characteristic, complemented
45	#41×C₅₃	C₂³	1	2	yes	—
46	#11719949	C₂×C₄	1	2	yes	—
47	Q₈×D₅₃	C₂×C₄	1	2	yes	—
48	#11719947	—	2	2	no	—
49	#11719947	C₂×C₄	1	2	yes	—
50	D₄×D₅₃	—	2	2	no	—
51	D₄×D₅₃	C₂×C₄	1	2	yes	—
52	#11719945	—	2	2	no	—
53	#11719945	C₂³	1	2	yes	—
54	C₂×C₄×D₅₃	—	2	2	no	—
55	C₂×C₄×D₅₃	—	2	1	no	—
56	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
57	Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
58	SD₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
59	D₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
60	M₁₆×C₅₃	—	2	2	no	—
61	M₁₆×C₅₃	C₂³	1	2	yes	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
65	C₄²×C₅₃	C₂³	1	1	yes	characteristic
66	#11719925	C₂×C₄	1	2	yes	complemented
67	#11719924	C₂×C₄	1	2	yes	complemented
68	#11719923	C₂×C₄	1	2	yes	complemented
69	#11719922	C₂×C₄	1	2	yes	complemented
70	C₄×Dic₅₃	—	2	1	no	—
71	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
72	#11719917	—	2	2	no	—
73	#11719917	C₂³	1	2	yes	—
74	C₂×#88322	—	2	2	no	—
75	C₂×#88322	C₂³	1	1	yes	characteristic
76	C₂×#88322	C₂³	1	1	yes	characteristic
77	Dic₂₁₂	C₂×C₄	1	1	yes	characteristic, complemented
78	D₄₂₄	C₂×C₄	1	1	yes	characteristic, complemented
79	#11719913	C₂×C₄	1	2	yes	complemented
80	#11719912	—	2	2	no	—
81	#11719912	—	2	2	no	—
82	#11719912	—	2	2	no	—
83	C₈×D₅₃	—	2	2	no	—
84	C₈×D₅₃	—	2	2	no	—
85	C₈×D₅₃	—	2	2	no	—
86	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
87	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
88	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
89	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
90	C₈×D₅₃	D₄	1	2	yes	—
91	C₈×D₅₃	D₄	1	2	yes	—
92	C₂²×D₅₃	—	2	2	no	—
93	C₂²×D₅₃	#41	1	1	yes	characteristic
94	Q₈×C₅₃	C₂²×C₄	1	2	yes	—
95	D₄×C₅₃	—	2	2	no	—
96	D₄×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	1	no	—
99	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
100	C₂²⋊D₅₃	—	2	2	no	—
101	C₂²⋊D₅₃	—	2	2	no	—
102	C₂²⋊D₅₃	—	2	2	no	—
103	C₂×Dic₅₃	—	2	2	no	—
104	C₂×Dic₅₃	—	2	1	no	—
105	C₂×Dic₅₃	#41	1	1	yes	characteristic
106	D₂₁₂	—	2	2	no	—
107	D₂₁₂	C₂²×C₄	1	2	yes	—
108	C₄×D₅₃	—	2	2	no	—
109	C₄×D₅₃	—	2	2	no	—
110	C₄×D₅₃	—	2	2	no	—
111	C₄×D₅₃	—	4	2	no	—
112	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
115	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
116	Dic₁₀₆	—	2	2	no	—
117	Dic₁₀₆	C₂²×C₄	1	2	yes	—
118	C₈×C₅₃	—	2	2	no	—
119	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₈×C₅₃	C₂×D₄	1	2	yes	—
122	#88322	—	2	2	no	—
123	#88322	C₂²×C₄	1	1	yes	characteristic
124	#88322	C₂²×C₄	1	1	yes	characteristic
125	#88322	C₂×D₄	1	2	yes	—
126	C₂²×C₅₃	—	2	2	no	—
127	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
128	C₂×D₅₃	—	2	2	no	—
129	C₂×D₅₃	—	2	2	no	—
130	C₂×D₅₃	—	4	4	no	—
131	C₂×D₅₃	C₄×D₄	1	1	yes	characteristic
132	C₂×D₅₃	C₄×D₄	1	1	yes	characteristic
133	C₄×C₅₃	—	2	2	no	—
134	C₄×C₅₃	—	4	1	no	—
135	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
136	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
137	Dic₅₃	—	2	2	no	—
138	Dic₅₃	—	4	1	no	—
139	Dic₅₃	C₄×D₄	1	1	yes	characteristic
140	Dic₅₃	C₄×D₄	1	1	yes	characteristic
141	C₂×#444	—	53	1	no	maximal, Sylow
142	C₂×C₅₃	—	2	1	no	—
143	C₂×C₅₃	—	4	2	no	—
144	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
145	D₅₃	—	2	1	no	—
146	D₅₃	—	4	2	no	—
147	D₅₃	#444	1	2	yes	direct product factor
148	C₂×#135	—	53	1	no	—
149	C₂×#131	—	53	2	no	—
150	#444	—	53	1	no	—
151	#444	—	53	2	no	—
152	#444	—	53	1	no	—
153	#444	—	53	2	no	—
154	#444	—	53	1	no	—
155	#444	D₅₃	1	1	yes	characteristic, direct product factor
156	C₂×#108	—	53	1	no	—
157	C₂×#104	—	53	2	no	—
158	C₂×#97	—	53	1	no	—
159	C₅₃	C₂×#444	1	1	yes	characteristic, complemented, Sylow
160	C₂×#41	—	53	2	no	—
161	#135	—	53	2	no	—
162	#135	—	53	1	no	—
163	#135	—	53	2	no	—
164	#135	—	53	1	no	—
165	#135	—	53	1	no	—
166	#135	C₂×D₅₃	1	1	yes	characteristic
167	C₂×Q₁₆	—	53	1	no	—
168	C₂×SD₁₆	—	53	2	no	—
169	C₂×D₈	—	53	1	no	—
170	#131	—	53	2	no	—
171	#131	—	53	2	no	—
172	#131	—	53	2	no	—
173	#131	—	106	2	no	—
174	#131	—	106	2	no	—
175	#131	C₂×D₅₃	1	2	yes	complemented
176	C₂×M₁₆	—	53	2	no	—
177	C₂×M₁₆	—	106	2	no	—
178	C₂²×C₈	—	53	1	no	—
179	C₂²×C₈	—	53	1	no	—
180	C₂²×C₈	—	106	2	no	—
181	C₂×C₄²	—	53	1	no	—
182	#108	—	53	2	no	—
183	#108	—	53	1	no	—
184	#108	C₂×D₅₃	1	1	yes	characteristic, complemented
185	#104	—	53	2	no	—
186	#104	—	53	2	no	—
187	#104	—	53	2	no	—
188	#104	C₂×D₅₃	1	2	yes	complemented
189	#97	—	53	1	no	—
190	#97	—	53	1	no	—
191	#97	—	53	1	no	—
192	#97	C₂×D₅₃	1	1	yes	characteristic, complemented
193	#41	—	53	2	no	—
194	#41	—	53	2	no	—
195	#41	—	53	2	no	—
196	#41	—	106	2	no	—
197	#41	—	106	2	no	—
198	#41	C₂²×D₅₃	1	2	yes	—
199	C₂×Q₈	—	53	2	no	—
200	C₂×D₄	—	53	2	no	—
201	C₂×D₄	—	106	2	no	—
202	C₂²×C₄	—	53	1	no	—
203	C₂²×C₄	—	106	2	no	—
204	C₂²×C₄	—	106	1	no	—
205	Q₁₆	—	53	2	no	—
206	Q₁₆	—	53	1	no	—
207	Q₁₆	C₄×D₅₃	1	1	yes	characteristic, complemented
208	SD₁₆	—	53	2	no	—
209	SD₁₆	—	53	2	no	—
210	SD₁₆	—	53	2	no	—
211	SD₁₆	C₄×D₅₃	1	2	yes	complemented
212	D₈	—	53	2	no	—
213	D₈	—	53	1	no	—
214	D₈	C₄×D₅₃	1	1	yes	characteristic, complemented
215	M₁₆	—	2	2	no	—
216	M₁₆	—	53	2	no	—
217	M₁₆	—	106	2	no	—
218	M₁₆	—	106	2	no	—
219	M₁₆	—	106	2	no	—
220	M₁₆	—	106	2	no	—
221	M₁₆	C₂²×D₅₃	1	2	yes	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	53	1	no	—
224	C₂×C₈	—	53	1	no	—
225	C₂×C₈	—	53	1	no	—
226	C₂×C₈	—	53	1	no	—
227	C₂×C₈	—	53	1	no	—
228	C₂×C₈	—	53	1	no	—
229	C₂×C₈	—	53	2	no	—
230	C₂×C₈	—	53	2	no	—
231	C₂×C₈	—	106	2	no	—
232	C₂×C₈	—	106	2	no	—
233	C₂×C₈	—	106	2	no	—
234	C₂×C₈	—	106	2	no	—
235	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
236	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
237	C₄²	—	53	1	no	—
238	C₄²	—	106	1	no	—
239	C₄²	C₂²×D₅₃	1	1	yes	characteristic
240	C₂³	—	53	1	no	—
241	C₂³	—	106	2	no	—
242	Q₈	—	53	2	no	—
243	Q₈	—	106	2	no	—
244	Q₈	C₂×C₄×D₅₃	1	2	yes	—
245	D₄	—	2	2	no	—
246	D₄	—	53	2	no	—
247	D₄	—	106	2	no	—
248	D₄	—	106	2	no	—
249	D₄	—	106	2	no	—
250	D₄	—	106	2	no	—
251	D₄	C₂×C₄×D₅₃	1	2	yes	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	53	1	no	—
255	C₂×C₄	—	53	1	no	—
256	C₂×C₄	—	53	1	no	—
257	C₂×C₄	—	53	1	no	—
258	C₂×C₄	—	53	1	no	—
259	C₂×C₄	—	106	2	no	—
260	C₂×C₄	—	106	2	no	—
261	C₂×C₄	—	106	2	no	—
262	C₂×C₄	—	106	2	no	—
263	C₂×C₄	—	106	1	no	—
264	C₂×C₄	—	212	2	no	—
265	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	—	53	1	no	—
268	C₈	—	53	1	no	—
269	C₈	—	53	2	no	—
270	C₈	—	106	2	no	—
271	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
272	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
273	C₈	D₄×D₅₃	1	2	yes	—
274	C₂²	—	2	2	no	—
275	C₂²	—	53	1	no	—
276	C₂²	—	53	1	no	—
277	C₂²	—	106	2	no	—
278	C₂²	—	106	2	no	—
279	C₂²	—	212	4	no	—
280	C₂²	#41×D₅₃	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	4	1	no	—
283	C₄	—	53	1	no	—
284	C₄	—	53	1	no	—
285	C₄	—	106	2	no	—
286	C₄	—	212	1	no	—
287	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
288	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
289	C₂	—	2	1	no	—
290	C₂	—	4	2	no	—
291	C₂	—	53	2	no	—
292	C₂	—	106	1	no	—
293	C₂	—	212	2	no	—
294	C₂	C₄×D₄×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#444	C₂
socle	C₂×C₅₃	C₂×C₄×D₄
hypercenter	#444	D₅₃
hypocenter	C₅₃	C₂×#444
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #444

Complements

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

N	H ≅ G/N	type
2. #131×D₅₃	C₂	semidirect
3. #16810746	C₂	semidirect
4. #16810745	C₂	semidirect
5. #16810744	C₂	semidirect
6. #16810743	C₂	semidirect
7. #16810742	C₂	semidirect
8. #108×D₅₃	C₂	semidirect
9. #104×D₅₃	C₂	semidirect
10. #97×D₅₃	C₂	semidirect
11. C₅₃×#444	C₂	semidirect
13. #13133240	C₄	semidirect
15. #13133238	C₄	semidirect
17. #13133236	C₄	semidirect
18. Q₁₆×D₅₃	C₄	semidirect
19. SD₁₆×D₅₃	C₄	semidirect
20. D₈×D₅₃	C₄	semidirect
22. #13133225	C₂²	semidirect
23. #13133224	C₂²	semidirect
25. #13133223	C₂²	semidirect
32. #13133153	C₂²	semidirect
33. #13133152	C₂²	semidirect
34. #13133145	C₂²	semidirect
35. #13133144	C₂²	semidirect
36. #13133143	C₂²	semidirect
37. #13133128	C₂²	semidirect
38. #13133127	C₂²	semidirect
39. #13133127	C₂²	semidirect
41. #131×C₅₃	C₂²	semidirect
42. #108×C₅₃	C₂²	semidirect
43. #104×C₅₃	C₂²	semidirect
44. #97×C₅₃	C₂²	semidirect
57. Q₁₆×C₅₃	C₂×C₄	semidirect
58. SD₁₆×C₅₃	C₂×C₄	semidirect
59. D₈×C₅₃	C₂×C₄	semidirect
66. #11719925	C₂×C₄	semidirect
67. #11719924	C₂×C₄	semidirect
68. #11719923	C₂×C₄	semidirect
69. #11719922	C₂×C₄	semidirect
77. Dic₂₁₂	C₂×C₄	semidirect
78. D₄₂₄	C₂×C₄	semidirect
79. #11719913	C₂×C₄	semidirect
147. D₅₃	#444	direct (N × H)
155. #444	D₅₃	direct (N × H)
159. C₅₃	C₂×#444	semidirect
175. #131	C₂×D₅₃	semidirect
184. #108	C₂×D₅₃	semidirect
188. #104	C₂×D₅₃	semidirect
192. #97	C₂×D₅₃	semidirect
207. Q₁₆	C₄×D₅₃	semidirect
211. SD₁₆	C₄×D₅₃	semidirect
214. D₈	C₄×D₅₃	semidirect

Subgroups — #444×D53 #20283607

Conjugacy classes of subgroups

Complements

Subgroups — #444×D₅₃ #20283607