Subgroups — #20288872

Order 6784.

1336 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	#16811362	C₂	1	3	yes	maximal, complemented
2	#16811303	C₂	1	3	yes	maximal, complemented
3	#16811266	C₂	1	6	yes	maximal, complemented
4	#16811255	C₂	1	3	yes	maximal, complemented
5	#16811207	C₂	1	6	yes	maximal, complemented
6	#16811191	C₂	1	3	yes	maximal, complemented
7	#16811183	C₂	1	3	yes	maximal, complemented
8	#120×D₅₃	C₂	1	3	yes	maximal, complemented
9	C₅₃×H(3,𝔽₄)	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₅₃	C₂²	1	1	yes	characteristic, complemented
11	C₂×#11719949	C₂²	1	3	yes	complemented
12	C₂²×C₂²⋊D₅₃	C₂²	1	3	yes	complemented
13	C₂×D₄×D₅₃	C₂²	1	6	yes	complemented
14	C₂×D₄×D₅₃	C₂²	1	3	yes	complemented
15	#13133214	C₂²	1	1	yes	characteristic, complemented
16	#13133213	C₂²	1	3	yes	complemented
17	#13133211	C₂²	1	2	yes	complemented
18	#13133210	C₂²	1	6	yes	complemented
19	#13133206	C₂²	1	3	yes	complemented
20	#13133201	C₂²	1	3	yes	complemented
21	#13133199	C₂²	1	6	yes	complemented
22	#13133199	C₂²	1	6	yes	complemented
23	#13133198	C₂²	1	3	yes	complemented
24	#13133193	C₂²	1	6	yes	complemented
25	#13133190	C₂²	1	6	yes	complemented
26	#13133190	C₂²	1	6	yes	complemented
27	#13133190	C₂²	1	6	yes	complemented
28	#13133189	C₂²	1	6	yes	complemented
29	#13133188	C₂²	1	6	yes	complemented
30	#13133187	C₂²	1	3	yes	complemented
31	#13133187	C₂²	1	6	yes	complemented
32	#13133187	C₂²	1	6	yes	complemented
33	#13133186	C₂²	1	3	yes	complemented
34	#13133186	C₂²	1	1	yes	characteristic, complemented
35	#13133185	C₂²	1	3	yes	—
36	#13133185	C₂²	1	6	yes	complemented
37	C₄×D₂₁₂	C₂²	1	6	yes	complemented
38	#13133177	C₂²	1	6	yes	complemented
39	#13133174	C₂²	1	3	yes	complemented
40	C₂²⋊C₄×D₅₃	C₂²	1	6	yes	complemented
41	C₂²⋊C₄×D₅₃	C₂²	1	3	yes	complemented
42	C₂²⋊C₄×D₅₃	C₂²	1	3	yes	complemented
43	#124×C₅₃	C₂²	1	6	yes	complemented
44	#124×C₅₃	C₂²	1	3	yes	complemented
45	#120×C₅₃	C₂²	1	3	yes	complemented
46	#120×C₅₃	C₂²	1	3	yes	complemented
47	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
48	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic, complemented
49	C₂³×D₅₃	—	4	6	no	—
50	C₂³×D₅₃	C₂³	1	3	yes	—
51	C₂³×D₅₃	C₂³	1	3	yes	—
52	C₂³×D₅₃	C₂³	1	3	yes	—
53	C₂×Q₈×C₅₃	C₂³	1	3	yes	complemented
54	C₂×D₄×C₅₃	C₂³	1	3	yes	—
55	C₂×D₄×C₅₃	C₂³	1	6	yes	—
56	C₂×C₂²⋊D₅₃	—	4	6	no	—
57	C₂×C₂²⋊D₅₃	C₂³	1	6	yes	—
58	C₂×C₂²⋊D₅₃	C₂³	1	3	yes	—
59	C₂×C₂²⋊D₅₃	C₂³	1	3	yes	—
60	C₂×C₂²⋊D₅₃	C₂³	1	6	yes	—
61	C₂×C₂²⋊D₅₃	C₂³	1	6	yes	complemented
62	C₂²×Dic₅₃	C₂³	1	3	yes	complemented
63	#11719949	—	4	6	no	—
64	D₄×D₅₃	—	4	6	no	—
65	D₄×D₅₃	—	4	12	no	—
66	C₂×D₂₁₂	C₂³	1	6	yes	—
67	C₂×D₂₁₂	C₂³	1	3	yes	—
68	C₂×D₂₁₂	C₂³	1	3	yes	complemented
69	C₂×C₄×D₅₃	C₂³	1	3	yes	—
70	C₂×C₄×D₅₃	C₂³	1	6	yes	—
71	C₂×C₄×D₅₃	C₂³	1	6	yes	complemented
72	C₂²⋊C₄×C₅₃	C₂³	1	6	yes	—
73	C₂²⋊C₄×C₅₃	C₂³	1	3	yes	—
74	C₂²⋊C₄×C₅₃	C₂³	1	3	yes	complemented
75	C₂²⋊C₄×C₅₃	C₂³	1	6	yes	complemented
76	C₄²×C₅₃	C₂³	1	1	yes	characteristic, complemented
77	C₄²×C₅₃	C₂³	1	2	yes	complemented
78	C₂²⋊Dic₅₃	C₂³	1	3	yes	—
79	C₂²⋊Dic₅₃	C₂³	1	6	yes	—
80	C₂²⋊Dic₅₃	C₂³	1	3	yes	complemented
81	#11719921	C₂³	1	6	yes	—
82	#11719921	C₂³	1	6	yes	—
83	#11719921	C₂³	1	6	yes	—
84	#11719921	C₂³	1	6	yes	complemented
85	C₄⋊Dic₅₃	C₂³	1	6	yes	complemented
86	#11719919	C₂³	1	6	yes	complemented
87	#11719919	C₂³	1	6	yes	complemented
88	C₄×Dic₅₃	C₂³	1	3	yes	complemented
89	C₂³×C₅₃	—	4	3	no	—
90	C₂³×C₅₃	—	4	3	no	—
91	C₂³×C₅₃	C₂⁴	1	3	yes	—
92	C₂³×C₅₃	C₂⁴	1	3	yes	—
93	C₂²×D₅₃	—	2	12	no	—
94	C₂²×D₅₃	—	2	6	no	—
95	C₂²×D₅₃	—	2	6	no	—
96	C₂²×D₅₃	—	4	3	no	—
97	C₂²×D₅₃	—	4	6	no	—
98	C₂²×D₅₃	—	4	6	no	—
99	C₂²×D₅₃	—	4	16	no	—
100	C₂²×D₅₃	C₂⁴	1	3	yes	—
101	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
102	C₂²×D₅₃	C₂⁴	1	3	yes	—
103	Q₈×C₅₃	—	4	3	no	—
104	D₄×C₅₃	—	4	3	no	—
105	D₄×C₅₃	—	4	6	no	—
106	C₂×C₄×C₅₃	C₂⁴	1	6	yes	—
107	C₂×C₄×C₅₃	C₂⁴	1	3	yes	—
108	C₂²⋊D₅₃	—	2	12	no	—
109	C₂²⋊D₅₃	—	4	3	no	—
110	C₂²⋊D₅₃	—	4	3	no	—
111	C₂²⋊D₅₃	—	4	6	no	—
112	C₂²⋊D₅₃	—	4	6	no	—
113	C₂×Dic₅₃	—	4	3	no	—
114	C₂×Dic₅₃	C₂⁴	1	3	yes	—
115	C₂×Dic₅₃	C₂⁴	1	6	yes	—
116	D₂₁₂	—	2	12	no	—
117	D₂₁₂	—	4	3	no	—
118	D₂₁₂	—	4	3	no	—
119	C₄×D₅₃	—	2	6	no	—
120	C₄×D₅₃	—	4	6	no	—
121	C₄×D₅₃	—	4	6	no	—
122	C₂²×C₅₃	—	2	3	no	—
123	C₂²×C₅₃	—	2	6	no	—
124	C₂²×C₅₃	—	2	3	no	—
125	C₂²×C₅₃	—	2	6	no	—
126	C₂²×C₅₃	—	4	4	no	—
127	C₂²×C₅₃	—	8	2	no	—
128	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
129	C₂×D₅₃	—	2	6	no	—
130	C₂×D₅₃	—	2	6	no	—
131	C₂×D₅₃	—	2	3	no	—
132	C₂×D₅₃	—	2	3	no	—
133	C₂×D₅₃	—	4	12	no	—
134	C₂×D₅₃	—	4	12	no	—
135	C₂×D₅₃	—	8	6	no	—
136	C₂×D₅₃	H(5,𝔽₂)	1	6	yes	—
137	C₄×C₅₃	—	2	3	no	—
138	C₄×C₅₃	—	2	6	no	—
139	Dic₅₃	—	2	3	no	—
140	Dic₅₃	—	2	6	no	—
141	#3532	—	53	1	no	maximal, Sylow
142	C₂×C₅₃	—	4	3	no	—
143	C₂×C₅₃	—	4	3	no	—
144	C₂×C₅₃	C₂×H(5,𝔽₂)	1	3	yes	—
145	D₅₃	—	2	6	no	—
146	D₅₃	—	4	1	no	—
147	D₅₃	—	4	3	no	—
148	H(3,𝔽₄)	D₅₃	1	1	yes	characteristic, complemented
149	#560	—	53	3	no	—
150	#551	—	53	3	no	—
151	#546	—	53	6	no	—
152	D₄²	—	53	3	no	—
153	#538	—	53	6	no	—
154	#535	—	53	3	no	—
155	#534	—	53	3	no	—
156	C₂×#120	—	53	3	no	—
157	C₅₃	#3532	1	1	yes	characteristic, complemented, Sylow
158	C₂⁵	—	53	1	no	—
159	C₂×#41	—	53	3	no	—
160	C₂²×D₄	—	53	3	no	—
161	C₂²×D₄	—	53	3	no	—
162	C₂²×D₄	—	53	6	no	—
163	#127	—	53	3	no	—
164	#127	—	53	1	no	—
165	#126	—	53	6	no	—
166	#126	—	53	2	no	—
167	#124	—	53	3	no	—
168	#124	C₂×D₅₃	1	3	yes	complemented
169	#124	C₂×D₅₃	1	6	yes	complemented
170	#123	—	53	3	no	—
171	#123	—	53	6	no	—
172	#123	—	53	3	no	—
173	#123	—	53	6	no	—
174	C₂²⋊Q₈	—	53	6	no	—
175	#121	—	53	6	no	—
176	#121	—	53	6	no	—
177	#121	—	53	6	no	—
178	#121	—	53	6	no	—
179	#121	—	53	6	no	—
180	#120	—	53	3	no	—
181	#120	—	53	1	no	—
182	#120	—	53	6	no	—
183	#120	—	53	3	no	—
184	#120	—	53	6	no	—
185	#120	—	53	3	no	—
186	#120	—	53	6	no	—
187	#120	C₂×D₅₃	1	3	yes	complemented
188	#120	C₂×D₅₃	1	3	yes	complemented
189	C₄×D₄	—	53	6	no	—
190	C₄×D₄	—	53	6	no	—
191	#117	—	53	3	no	—
192	C₂×C₂²⋊C₄	—	53	3	no	—
193	C₂×C₂²⋊C₄	—	53	6	no	—
194	C₂×C₂²⋊C₄	—	53	3	no	—
195	C₂⁴	—	53	3	no	—
196	C₂⁴	—	53	3	no	—
197	C₂⁴	—	53	3	no	—
198	C₂⁴	—	212	6	no	—
199	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
200	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic, complemented
201	#41	—	212	6	no	—
202	C₂×Q₈	C₂²×D₅₃	1	3	yes	complemented
203	C₂×D₄	—	53	6	no	—
204	C₂×D₄	—	53	3	no	—
205	C₂×D₄	—	53	3	no	—
206	C₂×D₄	—	53	6	no	—
207	C₂×D₄	—	53	6	no	—
208	C₂×D₄	—	53	6	no	—
209	C₂×D₄	—	53	3	no	—
210	C₂×D₄	—	53	3	no	—
211	C₂×D₄	—	212	6	no	—
212	C₂×D₄	—	212	12	no	—
213	C₂×D₄	—	212	6	no	—
214	C₂×D₄	C₂²×D₅₃	1	6	yes	—
215	C₂×D₄	C₂²×D₅₃	1	3	yes	—
216	C₂²×C₄	—	53	3	no	—
217	C₂²×C₄	—	53	3	no	—
218	C₂²×C₄	—	53	6	no	—
219	C₂²×C₄	—	53	6	no	—
220	C₄⋊C₄	—	53	6	no	—
221	C₄⋊C₄	—	53	6	no	—
222	C₄⋊C₄	—	53	6	no	—
223	C₂²⋊C₄	—	53	3	no	—
224	C₂²⋊C₄	—	53	6	no	—
225	C₂²⋊C₄	—	53	3	no	—
226	C₂²⋊C₄	—	53	6	no	—
227	C₂²⋊C₄	—	53	6	no	—
228	C₂²⋊C₄	—	53	6	no	—
229	C₂²⋊C₄	—	53	6	no	—
230	C₂²⋊C₄	C₂²×D₅₃	1	6	yes	—
231	C₂²⋊C₄	C₂²×D₅₃	1	3	yes	—
232	C₂²⋊C₄	C₂²×D₅₃	1	3	yes	complemented
233	C₂²⋊C₄	C₂²×D₅₃	1	6	yes	complemented
234	C₄²	—	53	3	no	—
235	C₄²	C₂²×D₅₃	1	1	yes	characteristic, complemented
236	C₄²	C₂²×D₅₃	1	2	yes	complemented
237	C₂³	—	4	3	no	—
238	C₂³	—	4	3	no	—
239	C₂³	—	53	3	no	—
240	C₂³	—	53	1	no	—
241	C₂³	—	53	3	no	—
242	C₂³	—	106	6	no	—
243	C₂³	—	106	12	no	—
244	C₂³	—	106	6	no	—
245	C₂³	—	212	6	no	—
246	C₂³	—	212	3	no	—
247	C₂³	—	212	6	no	—
248	C₂³	—	212	16	no	—
249	C₂³	C₂³×D₅₃	1	3	yes	—
250	C₂³	C₂³×D₅₃	1	3	yes	—
251	Q₈	—	4	3	no	—
252	D₄	—	4	3	no	—
253	D₄	—	4	6	no	—
254	D₄	—	106	12	no	—
255	D₄	—	106	12	no	—
256	D₄	—	212	3	no	—
257	D₄	—	212	3	no	—
258	D₄	—	212	3	no	—
259	D₄	—	212	3	no	—
260	D₄	—	212	6	no	—
261	D₄	—	212	6	no	—
262	C₂×C₄	—	53	6	no	—
263	C₂×C₄	—	53	3	no	—
264	C₂×C₄	—	106	6	no	—
265	C₂×C₄	—	212	6	no	—
266	C₂×C₄	—	212	6	no	—
267	C₂×C₄	—	212	3	no	—
268	C₂×C₄	C₂³×D₅₃	1	3	yes	—
269	C₂×C₄	C₂³×D₅₃	1	6	yes	—
270	C₂²	—	2	3	no	—
271	C₂²	—	2	6	no	—
272	C₂²	—	2	3	no	—
273	C₂²	—	2	6	no	—
274	C₂²	—	4	4	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	53	6	no	—
277	C₂²	—	106	6	no	—
278	C₂²	—	106	6	no	—
279	C₂²	—	106	3	no	—
280	C₂²	—	106	3	no	—
281	C₂²	—	424	6	no	—
282	C₂²	—	212	12	no	—
283	C₂²	—	212	12	no	—
284	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
285	C₄	—	2	3	no	—
286	C₄	—	2	6	no	—
287	C₄	—	106	3	no	—
288	C₄	—	106	6	no	—
289	C₂	—	4	3	no	—
290	C₂	—	4	3	no	—
291	C₂	—	106	6	no	—
292	C₂	—	212	3	no	—
293	C₂	—	212	1	no	—
294	C₂	H(5,𝔽₂)×D₅₃	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	C₅₃×H(3,𝔽₄)	C₂
hypercenter	H(3,𝔽₄)	D₅₃
hypocenter	C₅₃	#3532
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ H(3,𝔽₄)

Complements

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

N	H ≅ G/N	type
1. #16811362	C₂	semidirect
2. #16811303	C₂	semidirect
3. #16811266	C₂	semidirect
4. #16811255	C₂	semidirect
5. #16811207	C₂	semidirect
6. #16811191	C₂	semidirect
7. #16811183	C₂	semidirect
8. #120×D₅₃	C₂	semidirect
9. C₅₃×H(3,𝔽₄)	C₂	semidirect
10. C₂⁴×D₅₃	C₂²	semidirect
11. C₂×#11719949	C₂²	semidirect
12. C₂²×C₂²⋊D₅₃	C₂²	semidirect
13. C₂×D₄×D₅₃	C₂²	semidirect
14. C₂×D₄×D₅₃	C₂²	semidirect
15. #13133214	C₂²	semidirect
16. #13133213	C₂²	semidirect
17. #13133211	C₂²	semidirect
18. #13133210	C₂²	semidirect
19. #13133206	C₂²	semidirect
20. #13133201	C₂²	semidirect
21. #13133199	C₂²	semidirect
22. #13133199	C₂²	semidirect
23. #13133198	C₂²	semidirect
24. #13133193	C₂²	semidirect
25. #13133190	C₂²	semidirect
26. #13133190	C₂²	semidirect
27. #13133190	C₂²	semidirect
28. #13133189	C₂²	semidirect
29. #13133188	C₂²	semidirect
30. #13133187	C₂²	semidirect
31. #13133187	C₂²	semidirect
32. #13133187	C₂²	semidirect
33. #13133186	C₂²	semidirect
34. #13133186	C₂²	semidirect
36. #13133185	C₂²	semidirect
37. C₄×D₂₁₂	C₂²	semidirect
38. #13133177	C₂²	semidirect
39. #13133174	C₂²	semidirect
40. C₂²⋊C₄×D₅₃	C₂²	semidirect
41. C₂²⋊C₄×D₅₃	C₂²	semidirect
42. C₂²⋊C₄×D₅₃	C₂²	semidirect
43. #124×C₅₃	C₂²	semidirect
44. #124×C₅₃	C₂²	semidirect
45. #120×C₅₃	C₂²	semidirect
46. #120×C₅₃	C₂²	semidirect
48. C₂⁴×C₅₃	C₂³	semidirect
53. C₂×Q₈×C₅₃	C₂³	semidirect
61. C₂×C₂²⋊D₅₃	C₂³	semidirect
62. C₂²×Dic₅₃	C₂³	semidirect
68. C₂×D₂₁₂	C₂³	semidirect
71. C₂×C₄×D₅₃	C₂³	semidirect
74. C₂²⋊C₄×C₅₃	C₂³	semidirect
75. C₂²⋊C₄×C₅₃	C₂³	semidirect
76. C₄²×C₅₃	C₂³	semidirect
77. C₄²×C₅₃	C₂³	semidirect
80. C₂²⋊Dic₅₃	C₂³	semidirect
84. #11719921	C₂³	semidirect
85. C₄⋊Dic₅₃	C₂³	semidirect
86. #11719919	C₂³	semidirect
87. #11719919	C₂³	semidirect
88. C₄×Dic₅₃	C₂³	semidirect
148. H(3,𝔽₄)	D₅₃	semidirect
157. C₅₃	#3532	semidirect
168. #124	C₂×D₅₃	semidirect
169. #124	C₂×D₅₃	semidirect
187. #120	C₂×D₅₃	semidirect
188. #120	C₂×D₅₃	semidirect
200. C₂⁴	C₂²×D₅₃	semidirect
202. C₂×Q₈	C₂²×D₅₃	semidirect
232. C₂²⋊C₄	C₂²×D₅₃	semidirect
233. C₂²⋊C₄	C₂²×D₅₃	semidirect
235. C₄²	C₂²×D₅₃	semidirect
236. C₄²	C₂²×D₅₃	semidirect