Subgroups — #28297959

Order 13568.

10796 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289528	C₂	1	1	yes	maximal, characteristic, complemented
2	#20289501	C₂	1	15	yes	maximal, complemented
3	#20289495	C₂	1	15	yes	maximal, complemented
4	#20289486	C₂	1	15	yes	maximal, complemented
5	C₂×#16811434	C₂	1	15	yes	maximal, complemented
6	#20289427	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×H(7,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
8	#16811459	C₂²	1	15	yes	complemented
9	#16811458	—	2	10	no	—
10	C₂×#13133264	C₂²	1	15	yes	complemented
11	2₋¹⁺⁴×D₅₃	—	2	6	no	—
12	#16811438	C₂²	1	60	yes	complemented
13	#16811434	C₂²	1	60	yes	complemented
14	#16811434	C₂²	1	120	yes	complemented
15	#16811431	C₂²	1	20	yes	complemented
16	#16811426	C₂²	1	60	yes	complemented
17	#16811421	C₂²	1	90	yes	complemented
18	C₂×#13133247	C₂²	1	45	yes	complemented
19	C₂×#13133243	C₂²	1	45	yes	complemented
20	C₂×#13133237	C₂²	1	45	yes	complemented
21	C₂²×#11719923	C₂²	1	15	yes	complemented
22	#16811390	C₂²	1	15	yes	complemented
23	C₂×#13133224	C₂²	1	15	yes	complemented
24	C₅₃×#585	—	2	6	no	—
25	C₅₃×#585	C₂²	1	1	yes	characteristic
26	C₅₃×#585	C₂²	1	15	yes	complemented
27	C₂×H(5,𝔽₂)×C₅₃	—	2	10	no	—
28	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	15	yes	complemented
29	#13133265	—	2	15	no	—
30	#13133264	—	2	60	no	—
31	#13133264	C₂³	1	60	yes	complemented
32	#13133263	C₂³	1	20	yes	complemented
33	#13133262	—	2	45	no	—
34	C₂×#11719945	C₂³	1	45	yes	complemented
35	C₂×#11719947	—	2	45	no	—
36	#41×D₅₃	—	2	60	no	—
37	C₂²×Dic₁₀₆	C₂³	1	15	yes	complemented
38	C₂×Q₈×D₅₃	—	2	15	no	—
39	#13133248	C₂³	1	60	yes	complemented
40	#13133247	C₂³	1	180	yes	complemented
41	#13133243	C₂³	1	180	yes	complemented
42	#13133242	C₂³	1	60	yes	complemented
43	#13133237	C₂³	1	60	yes	complemented
44	#13133237	C₂³	1	180	yes	complemented
45	C₂×#11719925	C₂³	1	45	yes	complemented
46	C₂×#11719923	C₂³	1	45	yes	complemented
47	C₂×#11719923	C₂³	1	90	yes	complemented
48	C₂×#11719924	C₂³	1	15	yes	complemented
49	C₂×#11719922	C₂³	1	45	yes	complemented
50	#13133224	C₂³	1	20	yes	complemented
51	#13133224	C₂³	1	60	yes	complemented
52	C₂×#11719917	C₂³	1	45	yes	complemented
53	C₂²×#88322	C₂³	1	15	yes	complemented
54	2₋¹⁺⁴×C₅₃	—	2	15	no	—
55	2₋¹⁺⁴×C₅₃	D₄	1	6	yes	—
56	2₋¹⁺⁴×C₅₃	C₂³	1	20	yes	—
57	H(5,𝔽₂)×C₅₃	—	2	60	no	—
58	H(5,𝔽₂)×C₅₃	—	2	45	no	—
59	H(5,𝔽₂)×C₅₃	D₄	1	10	yes	—
60	H(5,𝔽₂)×C₅₃	C₂³	1	60	yes	—
61	C₂×#41×C₅₃	—	2	60	no	—
62	C₂×#41×C₅₃	—	2	15	no	—
63	C₂×#41×C₅₃	C₂³	1	45	yes	—
64	C₂×#41×C₅₃	C₂³	1	15	yes	—
65	C₂²×D₄×C₅₃	—	2	45	no	—
66	C₂²×D₄×C₅₃	C₂³	1	15	yes	—
67	C₂⁴×C₅₃	—	2	15	no	—
68	#41×C₅₃	—	2	60	no	—
69	#41×C₅₃	—	2	60	no	—
70	#41×C₅₃	—	2	90	no	—
71	#41×C₅₃	C₂×D₄	1	60	yes	—
72	#41×C₅₃	C₂⁴	1	60	yes	—
73	#41×C₅₃	C₂⁴	1	20	yes	—
74	C₂×Q₈×C₅₃	—	2	20	no	—
75	C₂×Q₈×C₅₃	C₂×D₄	1	15	yes	—
76	C₂×Q₈×C₅₃	C₂⁴	1	15	yes	—
77	C₂×D₄×C₅₃	—	2	45	no	—
78	C₂×D₄×C₅₃	—	2	180	no	—
79	C₂×D₄×C₅₃	—	2	45	no	—
80	C₂×D₄×C₅₃	C₂×D₄	1	45	yes	—
81	C₂×D₄×C₅₃	C₂⁴	1	45	yes	—
82	C₂²×C₄×C₅₃	—	2	45	no	—
83	C₂²×C₄×C₅₃	C₂⁴	1	15	yes	—
84	C₂×C₂²⋊D₅₃	—	2	45	no	—
85	C₂²×Dic₅₃	—	2	15	no	—
86	#11719949	—	2	20	no	—
87	Q₈×D₅₃	—	2	60	no	—
88	#11719947	—	2	180	no	—
89	D₄×D₅₃	—	2	60	no	—
90	#11719945	—	2	90	no	—
91	#11719945	C₂⁴	1	60	yes	—
92	C₂×D₂₁₂	C₂⁴	1	15	yes	—
93	C₂×C₄×D₅₃	—	2	45	no	—
94	C₂×Dic₁₀₆	—	2	45	no	—
95	C₂×Dic₁₀₆	C₂⁴	1	45	yes	—
96	#11719925	C₂⁴	1	60	yes	—
97	#11719924	C₂⁴	1	6	yes	—
98	#11719924	C₂⁴	1	30	yes	—
99	#11719923	C₂⁴	1	10	yes	—
100	#11719923	C₂⁴	1	90	yes	—
101	#11719922	C₂⁴	1	60	yes	—
102	#11719917	C₂⁴	1	60	yes	—
103	C₂×#88322	C₂⁴	1	15	yes	—
104	C₂×#88322	C₂⁴	1	45	yes	—
105	C₂³×C₅₃	—	2	45	no	—
106	C₂³×C₅₃	—	16	15	no	—
107	C₂³×C₅₃	C₂²×D₄	1	15	yes	—
108	C₂²×D₅₃	—	2	15	no	—
109	Q₈×C₅₃	—	2	15	no	—
110	Q₈×C₅₃	C₂²×D₄	1	20	yes	—
111	Q₈×C₅₃	C₂⁵	1	6	yes	—
112	D₄×C₅₃	—	2	60	no	—
113	D₄×C₅₃	—	2	45	no	—
114	D₄×C₅₃	C₂²×D₄	1	60	yes	—
115	D₄×C₅₃	C₂⁵	1	10	yes	—
116	C₂×C₄×C₅₃	—	2	15	no	—
117	C₂×C₄×C₅₃	—	2	60	no	—
118	C₂×C₄×C₅₃	C₂²×D₄	1	45	yes	—
119	C₂×C₄×C₅₃	C₂⁵	1	15	yes	—
120	C₂²⋊D₅₃	—	2	60	no	—
121	C₂×Dic₅₃	—	2	45	no	—
122	D₂₁₂	—	2	15	no	—
123	D₂₁₂	C₂⁵	1	6	yes	—
124	C₄×D₅₃	—	2	60	no	—
125	Dic₁₀₆	—	2	45	no	—
126	Dic₁₀₆	C₂⁵	1	10	yes	—
127	#88322	C₂⁵	1	1	yes	characteristic
128	#88322	C₂⁵	1	15	yes	—
129	#63092	—	53	1	no	maximal, Sylow
130	C₂²×C₅₃	—	2	10	no	—
131	C₂²×C₅₃	—	4	15	no	—
132	C₂²×C₅₃	—	8	45	no	—
133	C₂²×C₅₃	C₂³×D₄	1	15	yes	—
134	C₂×D₅₃	—	2	6	no	—
135	C₂×D₅₃	—	8	15	no	—
136	C₄×C₅₃	—	2	6	no	—
137	C₄×C₅₃	C₂³×D₄	1	15	yes	—
138	C₄×C₅₃	C₂⁶	1	1	yes	characteristic
139	Dic₅₃	—	2	10	no	—
140	2₋¹⁺⁶	—	53	1	no	—
141	H(7,𝔽₂)	D₅₃	1	1	yes	characteristic, complemented
142	#3586	—	53	15	no	—
143	#3585	—	53	15	no	—
144	#3584	—	53	15	no	—
145	C₂×#577	—	53	15	no	—
146	#3573	—	53	1	no	—
147	C₂×C₅₃	—	2	15	no	—
148	C₂×C₅₃	—	4	10	no	—
149	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
150	D₅₃	—	4	6	no	—
151	#585	—	2	6	no	—
152	#585	—	53	15	no	—
153	#585	—	106	10	no	—
154	#585	C₂×D₅₃	1	1	yes	characteristic
155	#585	C₂×D₅₃	1	15	yes	complemented
156	C₂×2₋¹⁺⁴	—	53	15	no	—
157	C₂×2₋¹⁺⁴	—	106	6	no	—
158	C₂×H(5,𝔽₂)	—	2	10	no	—
159	C₂×H(5,𝔽₂)	C₂×D₅₃	1	15	yes	complemented
160	#578	—	53	60	no	—
161	#577	—	53	20	no	—
162	#577	—	53	60	no	—
163	#577	—	53	120	no	—
164	#576	—	53	60	no	—
165	#575	—	53	90	no	—
166	C₂×#137	—	53	45	no	—
167	C₂×AGL(1,ℤ/8)	—	53	45	no	—
168	C₂×#135	—	53	45	no	—
169	C₂²×SD₁₆	—	53	15	no	—
170	#568	—	53	15	no	—
171	C₂×#131	—	53	15	no	—
172	C₅₃	#63092	1	1	yes	characteristic, complemented, Sylow
173	2₋¹⁺⁴	—	2	15	no	—
174	2₋¹⁺⁴	—	53	60	no	—
175	2₋¹⁺⁴	—	106	60	no	—
176	2₋¹⁺⁴	—	106	15	no	—
177	2₋¹⁺⁴	C₂²⋊D₅₃	1	6	yes	—
178	2₋¹⁺⁴	C₂²×D₅₃	1	20	yes	—
179	H(5,𝔽₂)	—	2	60	no	—
180	H(5,𝔽₂)	—	2	45	no	—
181	H(5,𝔽₂)	—	53	20	no	—
182	H(5,𝔽₂)	—	106	45	no	—
183	H(5,𝔽₂)	C₂²⋊D₅₃	1	10	yes	—
184	H(5,𝔽₂)	C₂²×D₅₃	1	60	yes	—
185	C₂×#41	—	2	60	no	—
186	C₂×#41	—	2	15	no	—
187	C₂×#41	—	53	45	no	—
188	C₂×#41	—	106	60	no	—
189	C₂×#41	—	106	45	no	—
190	C₂×#41	C₂²×D₅₃	1	45	yes	—
191	C₂×#41	C₂²×D₅₃	1	15	yes	—
192	C₂²×Q₈	—	53	15	no	—
193	C₂²×Q₈	—	106	15	no	—
194	C₂²×D₄	—	2	45	no	—
195	C₂²×D₄	C₂²×D₅₃	1	15	yes	—
196	#137	—	53	180	no	—
197	#137	—	53	60	no	—
198	AGL(1,ℤ/8)	—	53	60	no	—
199	AGL(1,ℤ/8)	—	53	180	no	—
200	#135	—	53	60	no	—
201	#135	—	53	180	no	—
202	C₂×Q₁₆	—	53	45	no	—
203	C₂×SD₁₆	—	53	15	no	—
204	C₂×SD₁₆	—	53	45	no	—
205	C₂×SD₁₆	—	53	90	no	—
206	C₂×D₈	—	53	45	no	—
207	#131	—	53	20	no	—
208	#131	—	53	60	no	—
209	C₂×M₁₆	—	53	45	no	—
210	C₂²×C₈	—	53	15	no	—
211	C₂⁴	—	2	15	no	—
212	#41	—	2	60	no	—
213	#41	—	2	60	no	—
214	#41	—	2	90	no	—
215	#41	—	53	60	no	—
216	#41	—	106	20	no	—
217	#41	—	106	180	no	—
218	#41	—	106	90	no	—
219	#41	C₂×C₂²⋊D₅₃	1	60	yes	—
220	#41	C₂³×D₅₃	1	60	yes	—
221	#41	C₂³×D₅₃	1	20	yes	—
222	C₂×Q₈	—	2	20	no	—
223	C₂×Q₈	—	53	45	no	—
224	C₂×Q₈	—	106	60	no	—
225	C₂×Q₈	—	106	45	no	—
226	C₂×Q₈	C₂×C₂²⋊D₅₃	1	15	yes	—
227	C₂×Q₈	C₂³×D₅₃	1	15	yes	—
228	C₂×D₄	—	2	45	no	—
229	C₂×D₄	—	2	180	no	—
230	C₂×D₄	—	2	45	no	—
231	C₂×D₄	—	53	15	no	—
232	C₂×D₄	—	106	45	no	—
233	C₂×D₄	—	106	60	no	—
234	C₂×D₄	C₂×C₂²⋊D₅₃	1	45	yes	—
235	C₂×D₄	C₂³×D₅₃	1	45	yes	—
236	C₂²×C₄	—	2	45	no	—
237	C₂²×C₄	—	106	15	no	—
238	C₂²×C₄	—	106	45	no	—
239	C₂²×C₄	C₂³×D₅₃	1	15	yes	—
240	Q₁₆	—	53	60	no	—
241	SD₁₆	—	53	6	no	—
242	SD₁₆	—	53	30	no	—
243	SD₁₆	—	53	10	no	—
244	SD₁₆	—	53	90	no	—
245	D₈	—	53	60	no	—
246	M₁₆	—	53	60	no	—
247	C₂×C₈	—	53	15	no	—
248	C₂×C₈	—	53	45	no	—
249	C₂³	—	2	45	no	—
250	C₂³	—	16	15	no	—
251	C₂³	—	106	15	no	—
252	C₂³	C₂²×C₂²⋊D₅₃	1	15	yes	—
253	Q₈	—	2	15	no	—
254	Q₈	—	53	10	no	—
255	Q₈	—	106	45	no	—
256	Q₈	C₂²×C₂²⋊D₅₃	1	20	yes	—
257	Q₈	C₂⁴×D₅₃	1	6	yes	—
258	D₄	—	2	60	no	—
259	D₄	—	2	45	no	—
260	D₄	—	53	6	no	—
261	D₄	—	106	60	no	—
262	D₄	—	106	15	no	—
263	D₄	C₂²×C₂²⋊D₅₃	1	60	yes	—
264	D₄	C₂⁴×D₅₃	1	10	yes	—
265	C₂×C₄	—	2	15	no	—
266	C₂×C₄	—	2	60	no	—
267	C₂×C₄	—	106	45	no	—
268	C₂×C₄	—	106	60	no	—
269	C₂×C₄	C₂²×C₂²⋊D₅₃	1	45	yes	—
270	C₂×C₄	C₂⁴×D₅₃	1	15	yes	—
271	C₈	—	53	1	no	—
272	C₈	—	53	15	no	—
273	C₂²	—	2	10	no	—
274	C₂²	—	4	15	no	—
275	C₂²	—	8	45	no	—
276	C₂²	—	106	6	no	—
277	C₂²	—	424	15	no	—
278	C₂²	C₂³×C₂²⋊D₅₃	1	15	yes	—
279	C₄	—	2	6	no	—
280	C₄	—	106	10	no	—
281	C₄	C₂³×C₂²⋊D₅₃	1	15	yes	—
282	C₄	C₂⁵×D₅₃	1	1	yes	characteristic
283	C₂	—	2	15	no	—
284	C₂	—	4	10	no	—
285	C₂	—	212	6	no	—
286	C₂	C₂⁴×C₂²⋊D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂⁴×C₂²⋊D₅₃
commutator, cosocle	C₄×C₅₃	C₂⁶
Frattini	C₄	C₂⁵×D₅₃
Fitting	C₅₃×H(7,𝔽₂)	C₂
socle	C₂×C₅₃	C₂⁴×D₄
hypercenter	H(7,𝔽₂)	D₅₃
hypocenter	C₅₃	#63092
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ #585 ◁ H(7,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. #20289528	C₂	semidirect
2. #20289501	C₂	semidirect
3. #20289495	C₂	semidirect
4. #20289486	C₂	semidirect
5. C₂×#16811434	C₂	semidirect
6. #20289427	C₂	semidirect
7. C₅₃×H(7,𝔽₂)	C₂	semidirect
8. #16811459	C₂²	semidirect
10. C₂×#13133264	C₂²	semidirect
12. #16811438	C₂²	semidirect
13. #16811434	C₂²	semidirect
14. #16811434	C₂²	semidirect
15. #16811431	C₂²	semidirect
16. #16811426	C₂²	semidirect
17. #16811421	C₂²	semidirect
18. C₂×#13133247	C₂²	semidirect
19. C₂×#13133243	C₂²	semidirect
20. C₂×#13133237	C₂²	semidirect
21. C₂²×#11719923	C₂²	semidirect
22. #16811390	C₂²	semidirect
23. C₂×#13133224	C₂²	semidirect
26. C₅₃×#585	C₂²	semidirect
28. C₂×H(5,𝔽₂)×C₅₃	C₂²	semidirect
31. #13133264	C₂³	semidirect
32. #13133263	C₂³	semidirect
34. C₂×#11719945	C₂³	semidirect
37. C₂²×Dic₁₀₆	C₂³	semidirect
39. #13133248	C₂³	semidirect
40. #13133247	C₂³	semidirect
41. #13133243	C₂³	semidirect
42. #13133242	C₂³	semidirect
43. #13133237	C₂³	semidirect
44. #13133237	C₂³	semidirect
45. C₂×#11719925	C₂³	semidirect
46. C₂×#11719923	C₂³	semidirect
47. C₂×#11719923	C₂³	semidirect
48. C₂×#11719924	C₂³	semidirect
49. C₂×#11719922	C₂³	semidirect
50. #13133224	C₂³	semidirect
51. #13133224	C₂³	semidirect
52. C₂×#11719917	C₂³	semidirect
53. C₂²×#88322	C₂³	semidirect
141. H(7,𝔽₂)	D₅₃	semidirect
155. #585	C₂×D₅₃	semidirect
159. C₂×H(5,𝔽₂)	C₂×D₅₃	semidirect
172. C₅₃	#63092	semidirect

Maximal subgroups

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

subgroup	index
#20289528	2
#20289501	2
#20289495	2
#20289486	2
C₂×#16811434	2
#20289427	2
C₅₃×H(7,𝔽₂)	2
#63092	53