Subgroups — #20288954

Order 6784.

1270 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811363	C₂	1	4	yes	maximal, complemented
2	#16811319	C₂	1	4	yes	maximal, complemented
3	D₄×D₂₁₂	C₂	1	4	yes	maximal, complemented
4	#16811255	C₂	1	4	yes	maximal, complemented
5	#16811228	C₂	1	2	yes	maximal, complemented
6	#16811213	C₂	1	4	yes	maximal, complemented
7	#16811187	C₂	1	2	yes	maximal, complemented
8	#16811181	C₂	1	4	yes	maximal, complemented
9	C₂×#13133191	C₂	1	2	yes	maximal, complemented
10	C₅₃×#525	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×D₂₁₂	C₂²	1	2	yes	complemented
12	C₂²×D₂₁₂	C₂²	1	2	yes	complemented
13	C₂²×C₂²⋊D₅₃	C₂²	1	4	yes	complemented
14	C₂×D₄×D₅₃	C₂²	1	8	yes	complemented
15	#13133214	C₂²	1	2	yes	complemented
16	#13133213	C₂²	1	4	yes	complemented
17	#13133212	C₂²	1	8	yes	complemented
18	#13133209	C₂²	1	2	yes	complemented
19	#13133206	C₂²	1	4	yes	complemented
20	#13133202	C₂²	1	8	yes	complemented
21	#13133199	C₂²	1	8	yes	complemented
22	#13133192	C₂²	1	8	yes	complemented
23	#13133191	C₂²	1	4	yes	complemented
24	#13133191	C₂²	1	4	yes	complemented
25	#13133191	C₂²	1	4	yes	complemented
26	#13133191	C₂²	1	4	yes	complemented
27	#13133191	C₂²	1	4	yes	complemented
28	#13133190	C₂²	1	8	yes	complemented
29	#13133189	C₂²	1	8	yes	complemented
30	#13133187	C₂²	1	8	yes	complemented
31	#13133187	C₂²	1	8	yes	complemented
32	#13133185	C₂²	1	4	yes	complemented
33	#13133185	C₂²	1	4	yes	complemented
34	C₄×D₂₁₂	C₂²	1	4	yes	complemented
35	#13133177	C₂²	1	8	yes	complemented
36	#13133176	C₂²	1	2	yes	complemented
37	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	complemented
38	C₂×#11719921	C₂²	1	4	yes	complemented
39	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic, complemented
40	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
41	#123×C₅₃	C₂²	1	2	yes	complemented
42	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
43	#121×C₅₃	C₂²	1	2	yes	complemented
44	#120×C₅₃	C₂²	1	2	yes	complemented
45	C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
46	#117×C₅₃	C₂²	1	1	yes	characteristic, complemented
47	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
48	C₂³×D₅₃	C₂³	1	4	yes	—
49	C₂³×D₅₃	C₂³	1	4	yes	—
50	#41×C₅₃	—	4	2	no	—
51	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
52	C₂×D₄×C₅₃	C₂³	1	4	yes	—
53	C₂×D₄×C₅₃	C₂³	1	2	yes	—
54	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	—	4	2	no	—
56	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₅₃	—	4	8	no	—
62	C₂×C₂²⋊D₅₃	C₂³	1	8	yes	—
63	C₂×C₂²⋊D₅₃	C₂³	1	8	yes	—
64	C₂×C₂²⋊D₅₃	C₂³	1	8	yes	—
65	C₂²×Dic₅₃	C₂³	1	4	yes	—
66	D₄×D₅₃	—	4	16	no	—
67	C₂×D₂₁₂	—	4	4	no	—
68	C₂×D₂₁₂	—	4	4	no	—
69	C₂×D₂₁₂	C₂³	1	4	yes	—
70	C₂×D₂₁₂	C₂³	1	8	yes	—
71	C₂×D₂₁₂	C₂³	1	4	yes	—
72	C₂×D₂₁₂	C₂³	1	4	yes	—
73	C₂×C₄×D₅₃	C₂³	1	8	yes	—
74	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
75	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
76	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
77	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
78	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
79	C₄²×C₅₃	C₂³	1	2	yes	—
80	C₂²⋊Dic₅₃	C₂³	1	4	yes	—
81	#11719921	C₂³	1	8	yes	—
82	#11719921	C₂³	1	8	yes	—
83	#11719921	C₂³	1	8	yes	—
84	#11719921	C₂³	1	8	yes	—
85	C₄⋊Dic₅₃	C₂³	1	4	yes	—
86	C₄⋊Dic₅₃	C₂³	1	4	yes	—
87	#11719919	C₂³	1	8	yes	—
88	C₄×Dic₅₃	C₂³	1	4	yes	—
89	C₂³×C₅₃	—	2	4	no	—
90	C₂³×C₅₃	—	4	1	no	—
91	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
92	C₂³×C₅₃	C₂⁴	1	2	yes	—
93	C₂³×C₅₃	C₂⁴	1	2	yes	—
94	C₂²×D₅₃	—	2	8	no	—
95	C₂²×D₅₃	—	4	4	no	—
96	C₂²×D₅₃	—	4	4	no	—
97	C₂²×D₅₃	—	4	8	no	—
98	C₂²×D₅₃	—	4	4	no	—
99	C₂²×D₅₃	C₂⁴	1	8	yes	—
100	Q₈×C₅₃	—	4	1	no	—
101	D₄×C₅₃	—	2	8	no	—
102	D₄×C₅₃	—	4	2	no	—
103	D₄×C₅₃	—	4	1	no	—
104	C₂×C₄×C₅₃	—	2	4	no	—
105	C₂×C₄×C₅₃	—	2	4	no	—
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	—	4	1	no	—
108	C₂×C₄×C₅₃	—	4	2	no	—
109	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
110	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
111	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
112	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
114	C₂²⋊D₅₃	—	2	16	no	—
115	C₂²⋊D₅₃	—	4	8	no	—
116	C₂²⋊D₅₃	—	4	8	no	—
117	C₂×Dic₅₃	—	4	4	no	—
118	C₂×Dic₅₃	C₂⁴	1	8	yes	—
119	D₂₁₂	—	2	8	no	—
120	D₂₁₂	—	4	4	no	—
121	D₂₁₂	—	4	4	no	—
122	D₂₁₂	—	4	8	no	—
123	C₄×D₅₃	—	4	8	no	—
124	C₂²×C₅₃	—	2	2	no	—
125	C₂²×C₅₃	—	2	2	no	—
126	C₂²×C₅₃	—	2	2	no	—
127	C₂²×C₅₃	—	2	4	no	—
128	C₂²×C₅₃	—	2	1	no	—
129	C₂²×C₅₃	—	2	2	no	—
130	C₂²×C₅₃	—	4	4	no	—
131	C₂²×C₅₃	H(5,𝔽₂)	1	4	yes	—
132	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
133	C₂×D₅₃	—	2	8	no	—
134	C₂×D₅₃	—	2	8	no	—
135	C₂×D₅₃	—	2	8	no	—
136	C₂×D₅₃	—	8	8	no	—
137	C₂×D₅₃	—	8	8	no	—
138	C₄×C₅₃	—	2	4	no	—
139	C₄×C₅₃	—	2	2	no	—
140	C₄×C₅₃	—	2	2	no	—
141	C₄×C₅₃	—	2	1	no	—
142	C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
143	Dic₅₃	—	2	8	no	—
144	#3537	—	53	1	no	maximal, Sylow
145	C₂×C₅₃	—	2	4	no	—
146	C₂×C₅₃	—	4	2	no	—
147	C₂×C₅₃	—	4	1	no	—
148	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
149	C₂×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
150	D₅₃	—	4	8	no	—
151	#560	—	53	4	no	—
152	#553	—	53	4	no	—
153	D₄²	—	53	4	no	—
154	D₄²	—	53	4	no	—
155	#540	—	53	2	no	—
156	#538	—	53	4	no	—
157	#535	—	53	2	no	—
158	#534	—	53	4	no	—
159	#525	D₅₃	1	1	yes	characteristic, complemented
160	C₂×#121	—	53	2	no	—
161	C₅₃	#3537	1	1	yes	characteristic, complemented, Sylow
162	C₂×#41	C₂×D₅₃	1	1	yes	characteristic, complemented
163	C₂²×D₄	—	53	4	no	—
164	C₂²×D₄	—	53	8	no	—
165	C₂²×D₄	—	53	2	no	—
166	C₂²×D₄	—	53	2	no	—
167	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
168	#127	—	53	4	no	—
169	#127	—	53	2	no	—
170	#126	—	53	8	no	—
171	#125	—	53	2	no	—
172	#124	—	53	4	no	—
173	#123	—	53	8	no	—
174	#123	—	53	8	no	—
175	#123	C₂×D₅₃	1	2	yes	complemented
176	C₂²⋊Q₈	C₂×D₅₃	1	2	yes	complemented
177	#121	—	53	8	no	—
178	#121	—	53	4	no	—
179	#121	—	53	4	no	—
180	#121	—	53	4	no	—
181	#121	—	53	4	no	—
182	#121	—	53	4	no	—
183	#121	—	53	8	no	—
184	#121	—	53	8	no	—
185	#121	C₂×D₅₃	1	2	yes	complemented
186	#120	—	53	4	no	—
187	#120	—	53	4	no	—
188	#120	—	53	8	no	—
189	#120	—	53	8	no	—
190	#120	C₂×D₅₃	1	2	yes	complemented
191	C₄×D₄	—	53	8	no	—
192	C₄×D₄	—	53	4	no	—
193	C₄×D₄	C₂×D₅₃	1	4	yes	complemented
194	#117	—	53	2	no	—
195	#117	C₂×D₅₃	1	1	yes	characteristic, complemented
196	C₂×C₄⋊C₄	—	53	2	no	—
197	C₂×C₂²⋊C₄	—	53	4	no	—
198	C₂⁴	—	53	4	no	—
199	C₂⁴	—	53	4	no	—
200	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
201	#41	—	4	2	no	—
202	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
203	C₂×D₄	—	53	8	no	—
204	C₂×D₄	—	53	8	no	—
205	C₂×D₄	—	53	8	no	—
206	C₂×D₄	—	53	4	no	—
207	C₂×D₄	—	53	4	no	—
208	C₂×D₄	—	53	8	no	—
209	C₂×D₄	—	53	4	no	—
210	C₂×D₄	—	212	4	no	—
211	C₂×D₄	—	212	4	no	—
212	C₂×D₄	—	212	16	no	—
213	C₂×D₄	—	212	8	no	—
214	C₂×D₄	C₂²×D₅₃	1	4	yes	—
215	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
216	C₂×D₄	C₂²×D₅₃	1	2	yes	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	—	53	4	no	—
219	C₂²×C₄	—	53	8	no	—
220	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
221	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
222	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
223	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
224	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
225	C₄⋊C₄	—	53	4	no	—
226	C₄⋊C₄	—	53	4	no	—
227	C₄⋊C₄	—	53	8	no	—
228	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
229	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
230	C₂²⋊C₄	—	53	4	no	—
231	C₂²⋊C₄	—	53	8	no	—
232	C₂²⋊C₄	—	53	8	no	—
233	C₂²⋊C₄	—	53	8	no	—
234	C₂²⋊C₄	—	53	8	no	—
235	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
236	C₂²⋊C₄	C₂²×D₅₃	1	4	yes	—
237	C₂²⋊C₄	C₂²×D₅₃	1	4	yes	—
238	C₄²	—	53	4	no	—
239	C₄²	C₂²×D₅₃	1	2	yes	—
240	C₂³	—	2	4	no	—
241	C₂³	—	4	1	no	—
242	C₂³	—	53	8	no	—
243	C₂³	—	106	8	no	—
244	C₂³	—	212	4	no	—
245	C₂³	—	212	4	no	—
246	C₂³	—	212	4	no	—
247	C₂³	—	212	8	no	—
248	C₂³	C₂³×D₅₃	1	2	yes	—
249	C₂³	C₂³×D₅₃	1	1	yes	characteristic
250	C₂³	C₂³×D₅₃	1	2	yes	—
251	Q₈	—	4	1	no	—
252	D₄	—	2	8	no	—
253	D₄	—	4	1	no	—
254	D₄	—	4	2	no	—
255	D₄	—	106	16	no	—
256	D₄	—	106	8	no	—
257	D₄	—	212	4	no	—
258	D₄	—	212	4	no	—
259	D₄	—	212	8	no	—
260	D₄	—	212	8	no	—
261	D₄	—	212	8	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	53	8	no	—
268	C₂×C₄	—	212	8	no	—
269	C₂×C₄	—	212	4	no	—
270	C₂×C₄	C₂³×D₅₃	1	2	yes	—
271	C₂×C₄	C₂³×D₅₃	1	4	yes	—
272	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
273	C₂×C₄	C₂³×D₅₃	1	2	yes	—
274	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	4	4	no	—
282	C₂²	—	106	8	no	—
283	C₂²	—	106	8	no	—
284	C₂²	—	106	8	no	—
285	C₂²	—	424	8	no	—
286	C₂²	—	424	8	no	—
287	C₂²	#13133263	1	4	yes	—
288	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
289	C₄	—	2	1	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	4	no	—
293	C₄	—	106	8	no	—
294	C₄	#13133262	1	2	yes	—
295	C₂	—	2	4	no	—
296	C₂	—	4	1	no	—
297	C₂	—	4	2	no	—
298	C₂	—	212	8	no	—
299	C₂	C₂×#13133262	1	1	yes	characteristic
300	C₂	C₂×#13133263	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	C₅₃×#525	C₂
hypercenter	#525	D₅₃
hypocenter	C₅₃	#3537
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #525

Complements

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

N	H ≅ G/N	type
1. #16811363	C₂	semidirect
2. #16811319	C₂	semidirect
3. D₄×D₂₁₂	C₂	semidirect
4. #16811255	C₂	semidirect
5. #16811228	C₂	semidirect
6. #16811213	C₂	semidirect
7. #16811187	C₂	semidirect
8. #16811181	C₂	semidirect
9. C₂×#13133191	C₂	semidirect
10. C₅₃×#525	C₂	semidirect
11. C₂²×D₂₁₂	C₂²	semidirect
12. C₂²×D₂₁₂	C₂²	semidirect
13. C₂²×C₂²⋊D₅₃	C₂²	semidirect
14. C₂×D₄×D₅₃	C₂²	semidirect
15. #13133214	C₂²	semidirect
16. #13133213	C₂²	semidirect
17. #13133212	C₂²	semidirect
18. #13133209	C₂²	semidirect
19. #13133206	C₂²	semidirect
20. #13133202	C₂²	semidirect
21. #13133199	C₂²	semidirect
22. #13133192	C₂²	semidirect
23. #13133191	C₂²	semidirect
24. #13133191	C₂²	semidirect
25. #13133191	C₂²	semidirect
26. #13133191	C₂²	semidirect
27. #13133191	C₂²	semidirect
28. #13133190	C₂²	semidirect
29. #13133189	C₂²	semidirect
30. #13133187	C₂²	semidirect
31. #13133187	C₂²	semidirect
32. #13133185	C₂²	semidirect
33. #13133185	C₂²	semidirect
34. C₄×D₂₁₂	C₂²	semidirect
35. #13133177	C₂²	semidirect
36. #13133176	C₂²	semidirect
37. C₂×C₄⋊Dic₅₃	C₂²	semidirect
38. C₂×#11719921	C₂²	semidirect
39. C₂×#41×C₅₃	C₂²	semidirect
40. C₂³×C₄×C₅₃	C₂²	semidirect
41. #123×C₅₃	C₂²	semidirect
42. C₂²⋊Q₈×C₅₃	C₂²	semidirect
43. #121×C₅₃	C₂²	semidirect
44. #120×C₅₃	C₂²	semidirect
45. C₄×D₄×C₅₃	C₂²	semidirect
46. #117×C₅₃	C₂²	semidirect
159. #525	D₅₃	semidirect
161. C₅₃	#3537	semidirect
162. C₂×#41	C₂×D₅₃	semidirect
167. C₂³×C₄	C₂×D₅₃	semidirect
175. #123	C₂×D₅₃	semidirect
176. C₂²⋊Q₈	C₂×D₅₃	semidirect
185. #121	C₂×D₅₃	semidirect
190. #120	C₂×D₅₃	semidirect
193. C₄×D₄	C₂×D₅₃	semidirect
195. #117	C₂×D₅₃	semidirect