Subgroups — D₅₃×#2262 #27542216

Order 13568.

470 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	#508×D₅₃	C₂	1	2	yes	maximal, complemented
2	#20277025	C₂	1	1	yes	maximal, characteristic, complemented
3	#20277024	C₂	1	1	yes	maximal, characteristic, complemented
4	#20277023	C₂	1	2	yes	maximal, complemented
5	#20277022	C₂	1	2	yes	maximal, complemented
6	#20277021	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₆₄×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	SD₆₄×D₅₃	C₂	1	2	yes	maximal, complemented
9	D₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2262	C₂	1	1	yes	maximal, characteristic, complemented
12	#135×D₅₃	—	2	2	no	—
13	#16811056	C₂²	1	2	yes	complemented
14	#16811055	C₂²	1	2	yes	complemented
15	#16811054	—	2	2	no	—
16	#16811053	—	2	2	no	—
17	#16811052	C₂²	1	2	yes	complemented
18	Q₃₂×D₅₃	C₂²	1	2	yes	—
19	SD₃₂×D₅₃	—	2	2	no	—
20	D₁₆×D₅₃	C₂²	1	2	yes	—
21	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
22	Dic₈₄₈	C₂²	1	1	yes	characteristic, complemented
23	#16810463	C₂²	1	2	yes	complemented
24	#16810462	C₂²	1	2	yes	complemented
25	#16810461	C₂²	1	2	yes	complemented
26	#16810460	C₂²	1	2	yes	complemented
27	D₁₆₉₆	C₂²	1	1	yes	characteristic, complemented
28	#16810458	C₂²	1	2	yes	complemented
29	C₂×#13133123	C₂²	1	1	yes	characteristic, complemented
30	C₃₂×D₅₃	C₂²	1	1	yes	characteristic
31	C₃₂×D₅₃	C₂²	1	1	yes	characteristic
32	C₃₂×D₅₃	C₂²	1	1	yes	characteristic, complemented
33	C₃₂×D₅₃	C₂²	1	1	yes	characteristic, complemented
34	C₅₃×#508	C₂²	1	2	yes	complemented
35	C₅₃×Q₆₄	C₂²	1	1	yes	characteristic, complemented
36	C₅₃×SD₆₄	C₂²	1	2	yes	complemented
37	C₅₃×D₃₂	C₂²	1	1	yes	characteristic, complemented
38	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic, complemented
39	#41×D₅₃	—	4	2	no	—
40	#13133240	—	2	2	no	—
41	#13133239	—	2	2	no	—
42	#13133238	—	4	2	no	—
43	#13133237	—	4	2	no	—
44	#13133236	—	2	2	no	—
45	Q₁₆×D₅₃	—	2	2	no	—
46	SD₁₆×D₅₃	—	4	2	no	—
47	D₈×D₅₃	—	2	2	no	—
48	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
49	Dic₄₂₄	C₂³	1	2	yes	—
50	#13133163	—	2	2	no	—
51	#13133162	—	2	2	no	—
52	#13133161	—	2	2	no	—
53	#13133160	—	2	2	no	—
54	D₈₄₈	C₂³	1	2	yes	—
55	#13133158	—	2	2	no	—
56	C₂×#11719908	D₄	1	1	yes	characteristic
57	C₁₆×D₅₃	D₄	1	1	yes	characteristic
58	C₁₆×D₅₃	D₄	1	1	yes	characteristic
59	C₁₆×D₅₃	C₂³	1	1	yes	characteristic
60	C₁₆×D₅₃	C₂³	1	1	yes	characteristic
61	#13133123	C₂³	1	1	yes	characteristic
62	#13133123	C₂³	1	1	yes	characteristic
63	#135×C₅₃	—	2	2	no	—
64	Q₃₂×C₅₃	C₂³	1	2	yes	—
65	SD₃₂×C₅₃	—	2	2	no	—
66	D₁₆×C₅₃	C₂³	1	2	yes	—
67	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
68	C₃₂×C₅₃	C₂³	1	1	yes	characteristic
69	C₃₂×C₅₃	C₂³	1	1	yes	characteristic
70	#41×C₅₃	—	4	2	no	—
71	#11719949	—	4	2	no	—
72	Q₈×D₅₃	—	4	2	no	—
73	#11719947	—	4	2	no	—
74	#11719947	—	8	2	no	—
75	D₄×D₅₃	—	4	2	no	—
76	D₄×D₅₃	—	8	2	no	—
77	#11719945	—	4	2	no	—
78	#11719945	—	8	2	no	—
79	C₂×C₄×D₅₃	—	8	2	no	—
80	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
81	Q₁₆×C₅₃	—	2	2	no	—
82	SD₁₆×C₅₃	—	4	2	no	—
83	D₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
85	#11719925	—	4	2	no	—
86	#11719924	—	4	2	no	—
87	#11719923	—	4	2	no	—
88	#11719922	—	4	2	no	—
89	C₂×#88322	D₈	1	1	yes	characteristic
90	Dic₂₁₂	—	2	2	no	—
91	D₄₂₄	—	2	2	no	—
92	#11719913	—	4	2	no	—
93	C₈×D₅₃	D₈	1	1	yes	characteristic
94	C₈×D₅₃	D₈	1	1	yes	characteristic
95	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
96	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
97	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₁₆×C₅₃	C₂⁴	1	1	yes	characteristic
99	#11719908	C₂×D₄	1	1	yes	characteristic
100	#11719908	C₂×D₄	1	1	yes	characteristic
101	C₂²×D₅₃	—	8	2	no	—
102	C₂²×D₅₃	D₁₆	1	1	yes	characteristic
103	Q₈×C₅₃	—	4	2	no	—
104	D₄×C₅₃	—	4	2	no	—
105	D₄×C₅₃	—	8	2	no	—
106	C₂×C₄×C₅₃	—	8	2	no	—
107	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
108	C₂²⋊D₅₃	—	8	2	no	—
109	C₂²⋊D₅₃	—	8	2	no	—
110	C₂²⋊D₅₃	—	8	2	no	—
111	C₂×Dic₅₃	—	8	2	no	—
112	C₂×Dic₅₃	D₁₆	1	1	yes	characteristic
113	D₂₁₂	—	4	2	no	—
114	D₂₁₂	—	8	2	no	—
115	C₄×D₅₃	—	8	2	no	—
116	C₄×D₅₃	—	8	2	no	—
117	C₄×D₅₃	—	8	2	no	—
118	C₄×D₅₃	D₁₆	1	1	yes	characteristic
119	C₄×D₅₃	D₁₆	1	1	yes	characteristic
120	C₄×D₅₃	C₂×D₈	1	1	yes	characteristic
121	C₄×D₅₃	C₂×D₈	1	1	yes	characteristic
122	Dic₁₀₆	—	4	2	no	—
123	Dic₁₀₆	—	8	2	no	—
124	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
125	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
126	#88322	C₂×D₈	1	1	yes	characteristic
127	#88322	C₂×D₈	1	1	yes	characteristic
128	C₂×#2262	—	53	1	no	maximal, Sylow
129	C₂²×C₅₃	—	8	2	no	—
130	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
131	C₂×D₅₃	—	2	2	no	—
132	C₂×D₅₃	—	8	2	no	—
133	C₂×D₅₃	—	16	4	no	—
134	C₂×D₅₃	C₂×D₁₆	1	1	yes	characteristic
135	C₂×D₅₃	C₂×D₁₆	1	1	yes	characteristic
136	C₄×C₅₃	—	8	2	no	—
137	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
138	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
139	Dic₅₃	—	8	2	no	—
140	Dic₅₃	C₂×D₁₆	1	1	yes	characteristic
141	Dic₅₃	C₂×D₁₆	1	1	yes	characteristic
142	C₂×#508	—	53	2	no	—
143	#2262	—	53	2	no	—
144	#2262	—	53	1	no	—
145	#2262	—	53	2	no	—
146	#2262	—	53	1	no	—
147	#2262	—	53	1	no	—
148	#2262	D₅₃	1	1	yes	characteristic, direct product factor
149	C₂×Q₆₄	—	53	1	no	—
150	C₂×SD₆₄	—	53	2	no	—
151	C₂×D₃₂	—	53	1	no	—
152	C₂²×C₃₂	—	53	1	no	—
153	C₂×C₅₃	—	2	1	no	—
154	C₂×C₅₃	—	16	2	no	—
155	C₂×C₅₃	C₂²×D₁₆	1	1	yes	characteristic
156	D₅₃	—	2	1	no	—
157	D₅₃	—	16	2	no	—
158	D₅₃	#2262	1	2	yes	direct product factor
159	C₂×#135	—	106	2	no	—
160	#508	—	53	2	no	—
161	#508	—	53	2	no	—
162	#508	—	53	2	no	—
163	#508	—	106	2	no	—
164	#508	—	106	2	no	—
165	#508	C₂×D₅₃	1	2	yes	complemented
166	C₂×Q₃₂	—	53	2	no	—
167	C₂×SD₃₂	—	106	2	no	—
168	C₂×D₁₆	—	53	2	no	—
169	C₂²×C₁₆	—	53	1	no	—
170	Q₆₄	—	53	2	no	—
171	Q₆₄	—	53	1	no	—
172	Q₆₄	C₂×D₅₃	1	1	yes	characteristic, complemented
173	SD₆₄	—	53	2	no	—
174	SD₆₄	—	53	2	no	—
175	SD₆₄	—	53	2	no	—
176	SD₆₄	C₂×D₅₃	1	2	yes	complemented
177	D₃₂	—	53	2	no	—
178	D₃₂	—	53	1	no	—
179	D₃₂	C₂×D₅₃	1	1	yes	characteristic, complemented
180	C₂×C₃₂	—	53	1	no	—
181	C₂×C₃₂	—	53	1	no	—
182	C₂×C₃₂	—	53	1	no	—
183	C₂×C₃₂	—	53	1	no	—
184	C₂×C₃₂	—	53	1	no	—
185	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic, complemented
186	C₅₃	C₂×#2262	1	1	yes	characteristic, complemented, Sylow
187	C₂×#41	—	212	2	no	—
188	#135	—	2	2	no	—
189	#135	—	106	2	no	—
190	#135	—	106	2	no	—
191	#135	—	106	2	no	—
192	#135	—	212	2	no	—
193	#135	—	212	2	no	—
194	C₂×Q₁₆	—	106	2	no	—
195	C₂×SD₁₆	—	212	2	no	—
196	C₂×D₈	—	106	2	no	—
197	C₂²×C₈	—	53	1	no	—
198	Q₃₂	—	53	2	no	—
199	Q₃₂	—	106	2	no	—
200	Q₃₂	C₂²×D₅₃	1	2	yes	—
201	SD₃₂	—	2	2	no	—
202	SD₃₂	—	106	2	no	—
203	SD₃₂	—	106	2	no	—
204	SD₃₂	—	106	2	no	—
205	D₁₆	—	53	2	no	—
206	D₁₆	—	106	2	no	—
207	D₁₆	C₂²×D₅₃	1	2	yes	—
208	C₂×C₁₆	—	53	1	no	—
209	C₂×C₁₆	—	53	1	no	—
210	C₂×C₁₆	—	53	1	no	—
211	C₂×C₁₆	—	53	1	no	—
212	C₂×C₁₆	—	53	1	no	—
213	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
214	C₃₂	—	53	1	no	—
215	C₃₂	—	53	1	no	—
216	C₃₂	C₂²×D₅₃	1	1	yes	characteristic
217	C₃₂	C₂²×D₅₃	1	1	yes	characteristic
218	#41	—	4	2	no	—
219	#41	—	424	2	no	—
220	#41	—	424	2	no	—
221	#41	—	212	2	no	—
222	#41	—	212	2	no	—
223	#41	—	212	2	no	—
224	C₂×Q₈	—	212	2	no	—
225	C₂×D₄	—	424	2	no	—
226	C₂×D₄	—	212	2	no	—
227	C₂²×C₄	—	53	1	no	—
228	C₂²×C₄	—	424	2	no	—
229	Q₁₆	—	2	2	no	—
230	Q₁₆	—	106	2	no	—
231	Q₁₆	—	212	2	no	—
232	SD₁₆	—	4	2	no	—
233	SD₁₆	—	212	2	no	—
234	SD₁₆	—	212	2	no	—
235	SD₁₆	—	212	2	no	—
236	D₈	—	2	2	no	—
237	D₈	—	106	2	no	—
238	D₈	—	212	2	no	—
239	C₂×C₈	—	53	1	no	—
240	C₂×C₈	—	53	1	no	—
241	C₂×C₈	—	53	1	no	—
242	C₂×C₈	—	53	1	no	—
243	C₂×C₈	—	53	1	no	—
244	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
245	C₁₆	—	53	1	no	—
246	C₁₆	—	53	1	no	—
247	C₁₆	D₄×D₅₃	1	1	yes	characteristic
248	C₁₆	C₂³×D₅₃	1	1	yes	characteristic
249	C₂³	—	53	1	no	—
250	C₂³	—	424	2	no	—
251	Q₈	—	4	2	no	—
252	Q₈	—	424	2	no	—
253	Q₈	—	212	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	8	2	no	—
256	D₄	—	424	2	no	—
257	D₄	—	424	2	no	—
258	D₄	—	424	2	no	—
259	D₄	—	424	2	no	—
260	D₄	—	212	2	no	—
261	C₂×C₄	—	8	2	no	—
262	C₂×C₄	—	53	1	no	—
263	C₂×C₄	—	53	1	no	—
264	C₂×C₄	—	53	1	no	—
265	C₂×C₄	—	53	1	no	—
266	C₂×C₄	—	53	1	no	—
267	C₂×C₄	—	424	2	no	—
268	C₂×C₄	—	424	2	no	—
269	C₂×C₄	—	424	2	no	—
270	C₂×C₄	—	424	2	no	—
271	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
272	C₈	—	53	1	no	—
273	C₈	—	53	1	no	—
274	C₈	D₈×D₅₃	1	1	yes	characteristic
275	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
276	C₂²	—	8	2	no	—
277	C₂²	—	53	1	no	—
278	C₂²	—	53	1	no	—
279	C₂²	—	848	4	no	—
280	C₂²	—	106	2	no	—
281	C₂²	—	424	2	no	—
282	C₂²	D₁₆×D₅₃	1	1	yes	characteristic
283	C₄	—	8	2	no	—
284	C₄	—	53	1	no	—
285	C₄	—	53	1	no	—
286	C₄	—	424	2	no	—
287	C₄	D₁₆×D₅₃	1	1	yes	characteristic
288	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
289	C₂	—	2	1	no	—
290	C₂	—	16	2	no	—
291	C₂	—	53	2	no	—
292	C₂	—	848	2	no	—
293	C₂	—	106	1	no	—
294	C₂	C₂×D₁₆×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₆×D₅₃
commutator, cosocle	C₁₆×C₅₃	C₂⁴
Frattini	C₁₆	C₂³×D₅₃
Fitting	C₅₃×#2262	C₂
socle	C₂×C₅₃	C₂²×D₁₆
hypercenter	#2262	D₅₃
hypocenter	C₅₃	C₂×#2262
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #2262

Complements

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

N	H ≅ G/N	type
1. #508×D₅₃	C₂	semidirect
2. #20277025	C₂	semidirect
3. #20277024	C₂	semidirect
4. #20277023	C₂	semidirect
5. #20277022	C₂	semidirect
6. #20277021	C₂	semidirect
7. Q₆₄×D₅₃	C₂	semidirect
8. SD₆₄×D₅₃	C₂	semidirect
9. D₃₂×D₅₃	C₂	semidirect
10. C₂×C₃₂×D₅₃	C₂	semidirect
11. C₅₃×#2262	C₂	semidirect
13. #16811056	C₂²	semidirect
14. #16811055	C₂²	semidirect
17. #16811052	C₂²	semidirect
22. Dic₈₄₈	C₂²	semidirect
23. #16810463	C₂²	semidirect
24. #16810462	C₂²	semidirect
25. #16810461	C₂²	semidirect
26. #16810460	C₂²	semidirect
27. D₁₆₉₆	C₂²	semidirect
28. #16810458	C₂²	semidirect
29. C₂×#13133123	C₂²	semidirect
32. C₃₂×D₅₃	C₂²	semidirect
33. C₃₂×D₅₃	C₂²	semidirect
34. C₅₃×#508	C₂²	semidirect
35. C₅₃×Q₆₄	C₂²	semidirect
36. C₅₃×SD₆₄	C₂²	semidirect
37. C₅₃×D₃₂	C₂²	semidirect
38. C₂×C₃₂×C₅₃	C₂²	semidirect
148. #2262	D₅₃	direct (N × H)
158. D₅₃	#2262	direct (N × H)
165. #508	C₂×D₅₃	semidirect
172. Q₆₄	C₂×D₅₃	semidirect
176. SD₆₄	C₂×D₅₃	semidirect
179. D₃₂	C₂×D₅₃	semidirect
185. C₂×C₃₂	C₂×D₅₃	semidirect
186. C₅₃	C₂×#2262	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#508×D₅₃	2
#20277025	2
#20277024	2
#20277023	2
#20277022	2
#20277021	2
Q₆₄×D₅₃	2
SD₆₄×D₅₃	2
D₃₂×D₅₃	2
C₂×C₃₂×D₅₃	2
C₅₃×#2262	2
C₂×#2262	53

Subgroups — D53×#2262 #27542216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#2262 #27542216