Subgroups — #28297960

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	#20289526	C₂	1	1	yes	maximal, characteristic, complemented
3	#20289503	C₂	1	15	yes	maximal, complemented
4	#20289496	C₂	1	15	yes	maximal, complemented
5	#20289489	C₂	1	15	yes	maximal, complemented
6	C₂×#16811435	C₂	1	15	yes	maximal, complemented
7	C₅₃×#3573	C₂	1	1	yes	maximal, characteristic, complemented
8	#16811460	—	2	6	no	—
9	#16811459	C₂²	1	15	yes	complemented
10	#16811459	C₂²	1	15	yes	complemented
11	#16811458	—	2	10	no	—
12	C₂×#13133264	C₂²	1	15	yes	complemented
13	2₋¹⁺⁴×D₅₃	—	2	6	no	—
14	C₂×#13133263	C₂²	1	15	yes	complemented
15	H(5,𝔽₂)×D₅₃	—	2	10	no	—
16	#16811440	C₂²	1	60	yes	complemented
17	#16811435	C₂²	1	60	yes	complemented
18	#16811435	C₂²	1	120	yes	complemented
19	#16811435	C₂²	1	20	yes	complemented
20	#16811428	C₂²	1	60	yes	complemented
21	#16811423	C₂²	1	90	yes	complemented
22	C₂×#13133250	C₂²	1	45	yes	complemented
23	C₂×#13133245	C₂²	1	45	yes	complemented
24	C₂×#13133239	C₂²	1	45	yes	complemented
25	C₂²×#11719913	C₂²	1	15	yes	complemented
26	C₅₃×#585	C₂²	1	1	yes	characteristic
27	C₅₃×#568	C₂²	1	15	yes	complemented
28	C₂×#131×C₅₃	C₂²	1	15	yes	complemented
29	#13133265	—	2	15	no	—
30	#13133265	—	2	15	no	—
31	#13133264	—	2	60	no	—
32	#13133264	C₂³	1	20	yes	—
33	#13133264	C₂³	1	60	yes	complemented
34	#13133263	—	2	60	no	—
35	#13133263	C₂³	1	60	yes	—
36	#13133263	C₂³	1	20	yes	complemented
37	#13133262	—	2	45	no	—
38	#13133262	—	2	45	no	—
39	C₂×#11719949	—	2	15	no	—
40	C₂×#11719945	C₂³	1	45	yes	—
41	C₂×#11719945	C₂³	1	45	yes	complemented
42	C₂×#11719947	—	2	45	no	—
43	#41×D₅₃	—	2	60	no	—
44	#41×D₅₃	—	2	60	no	—
45	C₂²×Dic₁₀₆	C₂³	1	15	yes	complemented
46	C₂×Q₈×D₅₃	—	2	15	no	—
47	C₂²×D₂₁₂	C₂³	1	15	yes	—
48	C₂×D₄×D₅₃	—	2	45	no	—
49	#13133250	C₂³	1	180	yes	complemented
50	#13133250	C₂³	1	60	yes	complemented
51	#13133245	C₂³	1	180	yes	complemented
52	#13133245	C₂³	1	60	yes	complemented
53	#13133239	C₂³	1	60	yes	complemented
54	#13133239	C₂³	1	180	yes	complemented
55	C₂×Dic₂₁₂	C₂³	1	45	yes	complemented
56	C₂×#11719913	C₂³	1	45	yes	complemented
57	C₂×#11719913	C₂³	1	90	yes	complemented
58	C₂×#11719913	C₂³	1	15	yes	complemented
59	C₂×D₄₂₄	C₂³	1	45	yes	complemented
60	2₋¹⁺⁴×C₅₃	D₄	1	6	yes	—
61	H(5,𝔽₂)×C₅₃	D₄	1	10	yes	—
62	C₂×#41×C₅₃	C₂³	1	15	yes	—
63	#131×C₅₃	C₂³	1	20	yes	complemented
64	#131×C₅₃	C₂³	1	60	yes	complemented
65	C₂×M₁₆×C₅₃	C₂³	1	45	yes	complemented
66	C₂²×C₈×C₅₃	C₂³	1	15	yes	complemented
67	C₂³×D₅₃	—	2	15	no	—
68	#41×C₅₃	C₂×D₄	1	60	yes	—
69	#41×C₅₃	C₂⁴	1	20	yes	—
70	C₂×Q₈×C₅₃	C₂×D₄	1	15	yes	—
71	C₂×D₄×C₅₃	C₂×D₄	1	45	yes	—
72	C₂²×C₄×C₅₃	C₂⁴	1	15	yes	—
73	C₂×C₂²⋊D₅₃	—	2	45	no	—
74	C₂×C₂²⋊D₅₃	—	2	45	no	—
75	C₂²×Dic₅₃	—	2	15	no	—
76	#11719949	—	2	20	no	—
77	#11719949	—	2	60	no	—
78	Q₈×D₅₃	—	2	60	no	—
79	Q₈×D₅₃	—	2	20	no	—
80	#11719947	—	2	60	no	—
81	#11719947	—	2	180	no	—
82	D₄×D₅₃	—	2	180	no	—
83	D₄×D₅₃	—	2	60	no	—
84	#11719945	—	2	90	no	—
85	#11719945	—	2	90	no	—
86	#11719945	C₂⁴	1	60	yes	—
87	#11719945	C₂⁴	1	60	yes	—
88	C₂×D₂₁₂	—	2	45	no	—
89	C₂×D₂₁₂	C₂⁴	1	15	yes	—
90	C₂×D₂₁₂	C₂⁴	1	45	yes	—
91	C₂×C₄×D₅₃	—	2	45	no	—
92	C₂×C₄×D₅₃	—	2	45	no	—
93	C₂×Dic₁₀₆	—	2	45	no	—
94	C₂×Dic₁₀₆	C₂⁴	1	45	yes	—
95	C₂×Dic₁₀₆	C₂⁴	1	15	yes	—
96	M₁₆×C₅₃	C₂⁴	1	60	yes	—
97	C₂×C₈×C₅₃	C₂⁴	1	15	yes	—
98	C₂×C₈×C₅₃	C₂⁴	1	45	yes	—
99	Dic₂₁₂	C₂⁴	1	60	yes	—
100	D₄₂₄	C₂⁴	1	60	yes	—
101	#11719913	C₂⁴	1	10	yes	—
102	#11719913	C₂⁴	1	6	yes	—
103	#11719913	C₂⁴	1	90	yes	—
104	#11719913	C₂⁴	1	30	yes	—
105	C₂³×C₅₃	C₂²×D₄	1	15	yes	—
106	C₂²×D₅₃	—	2	15	no	—
107	C₂²×D₅₃	—	2	45	no	—
108	C₂²×D₅₃	—	16	15	no	—
109	Q₈×C₅₃	C₂²×D₄	1	20	yes	—
110	D₄×C₅₃	C₂²×D₄	1	60	yes	—
111	C₂×C₄×C₅₃	C₂²×D₄	1	45	yes	—
112	C₂×C₄×C₅₃	C₂⁵	1	15	yes	—
113	C₂²⋊D₅₃	—	2	60	no	—
114	C₂²⋊D₅₃	—	2	60	no	—
115	C₂×Dic₅₃	—	2	45	no	—
116	C₂×Dic₅₃	—	2	15	no	—
117	D₂₁₂	—	2	45	no	—
118	D₂₁₂	—	2	15	no	—
119	D₂₁₂	C₂⁵	1	10	yes	—
120	D₂₁₂	C₂⁵	1	6	yes	—
121	C₄×D₅₃	—	2	60	no	—
122	C₄×D₅₃	—	2	60	no	—
123	Dic₁₀₆	—	2	45	no	—
124	Dic₁₀₆	—	2	15	no	—
125	Dic₁₀₆	C₂⁵	1	10	yes	—
126	Dic₁₀₆	C₂⁵	1	6	yes	—
127	C₈×C₅₃	C₂⁵	1	1	yes	characteristic
128	C₈×C₅₃	C₂⁵	1	15	yes	—
129	#63092	—	53	1	no	maximal, Sylow
130	C₂²×C₅₃	—	4	15	no	—
131	C₂²×C₅₃	C₂³×D₄	1	15	yes	—
132	C₂×D₅₃	—	2	6	no	—
133	C₂×D₅₃	—	2	10	no	—
134	C₂×D₅₃	—	8	15	no	—
135	C₂×D₅₃	—	8	45	no	—
136	C₄×C₅₃	C₂³×D₄	1	15	yes	—
137	C₄×C₅₃	C₂⁶	1	1	yes	characteristic
138	Dic₅₃	—	2	10	no	—
139	Dic₅₃	—	2	6	no	—
140	2₋¹⁺⁶	—	53	1	no	—
141	H(7,𝔽₂)	—	53	1	no	—
142	#3586	—	53	15	no	—
143	#3585	—	53	15	no	—
144	#3584	—	53	15	no	—
145	C₂×#577	—	53	15	no	—
146	#3573	D₅₃	1	1	yes	characteristic, complemented
147	C₂×C₅₃	—	2	15	no	—
148	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
149	D₅₃	—	4	6	no	—
150	D₅₃	—	4	10	no	—
151	#585	—	53	15	no	—
152	#585	—	53	15	no	—
153	#585	—	106	6	no	—
154	#585	—	106	10	no	—
155	#585	C₂×D₅₃	1	1	yes	characteristic
156	C₂×2₋¹⁺⁴	—	53	15	no	—
157	C₂×2₋¹⁺⁴	—	106	6	no	—
158	C₂×H(5,𝔽₂)	—	53	15	no	—
159	C₂×H(5,𝔽₂)	—	106	10	no	—
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	C₂×D₅₃	1	15	yes	complemented
171	C₂×#131	C₂×D₅₃	1	15	yes	complemented
172	C₅₃	#63092	1	1	yes	characteristic, complemented, Sylow
173	2₋¹⁺⁴	—	53	60	no	—
174	2₋¹⁺⁴	—	53	20	no	—
175	2₋¹⁺⁴	—	106	60	no	—
176	2₋¹⁺⁴	—	106	15	no	—
177	2₋¹⁺⁴	—	106	15	no	—
178	2₋¹⁺⁴	D₂₁₂	1	6	yes	—
179	H(5,𝔽₂)	—	53	20	no	—
180	H(5,𝔽₂)	—	53	60	no	—
181	H(5,𝔽₂)	—	106	60	no	—
182	H(5,𝔽₂)	—	106	45	no	—
183	H(5,𝔽₂)	—	106	45	no	—
184	H(5,𝔽₂)	D₂₁₂	1	10	yes	—
185	C₂×#41	—	53	45	no	—
186	C₂×#41	—	53	45	no	—
187	C₂×#41	—	106	60	no	—
188	C₂×#41	—	106	60	no	—
189	C₂×#41	—	106	15	no	—
190	C₂×#41	—	106	45	no	—
191	C₂×#41	C₂²×D₅₃	1	15	yes	—
192	C₂²×Q₈	—	53	15	no	—
193	C₂²×Q₈	—	106	15	no	—
194	C₂²×D₄	—	53	15	no	—
195	C₂²×D₄	—	106	45	no	—
196	#137	—	53	180	no	—
197	#137	—	53	60	no	—
198	AGL(1,ℤ/8)	—	53	180	no	—
199	AGL(1,ℤ/8)	—	53	60	no	—
200	#135	—	53	60	no	—
201	#135	—	53	180	no	—
202	C₂×Q₁₆	—	53	45	no	—
203	C₂×SD₁₆	—	53	45	no	—
204	C₂×SD₁₆	—	53	90	no	—
205	C₂×SD₁₆	—	53	15	no	—
206	C₂×D₈	—	53	45	no	—
207	#131	C₂²×D₅₃	1	20	yes	complemented
208	#131	C₂²×D₅₃	1	60	yes	complemented
209	C₂×M₁₆	C₂²×D₅₃	1	45	yes	complemented
210	C₂²×C₈	C₂²×D₅₃	1	15	yes	complemented
211	C₂⁴	—	106	15	no	—
212	#41	—	53	60	no	—
213	#41	—	53	60	no	—
214	#41	—	106	60	no	—
215	#41	—	106	20	no	—
216	#41	—	106	60	no	—
217	#41	—	106	180	no	—
218	#41	—	106	90	no	—
219	#41	—	106	90	no	—
220	#41	C₂×D₂₁₂	1	60	yes	—
221	#41	C₂³×D₅₃	1	20	yes	—
222	C₂×Q₈	—	53	45	no	—
223	C₂×Q₈	—	53	15	no	—
224	C₂×Q₈	—	106	60	no	—
225	C₂×Q₈	—	106	20	no	—
226	C₂×Q₈	—	106	45	no	—
227	C₂×Q₈	C₂×D₂₁₂	1	15	yes	—
228	C₂×D₄	—	53	15	no	—
229	C₂×D₄	—	53	45	no	—
230	C₂×D₄	—	106	45	no	—
231	C₂×D₄	—	106	45	no	—
232	C₂×D₄	—	106	180	no	—
233	C₂×D₄	—	106	60	no	—
234	C₂×D₄	—	106	45	no	—
235	C₂×D₄	C₂×D₂₁₂	1	45	yes	—
236	C₂²×C₄	—	106	15	no	—
237	C₂²×C₄	—	106	45	no	—
238	C₂²×C₄	—	106	45	no	—
239	C₂²×C₄	C₂³×D₅₃	1	15	yes	—
240	Q₁₆	—	53	60	no	—
241	SD₁₆	—	53	10	no	—
242	SD₁₆	—	53	6	no	—
243	SD₁₆	—	53	90	no	—
244	SD₁₆	—	53	30	no	—
245	D₈	—	53	60	no	—
246	M₁₆	C₂³×D₅₃	1	60	yes	—
247	C₂×C₈	C₂³×D₅₃	1	15	yes	—
248	C₂×C₈	C₂³×D₅₃	1	45	yes	—
249	C₂³	—	848	15	no	—
250	C₂³	—	106	15	no	—
251	C₂³	—	106	45	no	—
252	C₂³	C₂²×D₂₁₂	1	15	yes	—
253	Q₈	—	53	10	no	—
254	Q₈	—	53	6	no	—
255	Q₈	—	106	45	no	—
256	Q₈	—	106	15	no	—
257	Q₈	C₂²×D₂₁₂	1	20	yes	—
258	D₄	—	53	6	no	—
259	D₄	—	53	10	no	—
260	D₄	—	106	60	no	—
261	D₄	—	106	60	no	—
262	D₄	—	106	45	no	—
263	D₄	—	106	15	no	—
264	D₄	C₂²×D₂₁₂	1	60	yes	—
265	C₂×C₄	—	106	15	no	—
266	C₂×C₄	—	106	45	no	—
267	C₂×C₄	—	106	60	no	—
268	C₂×C₄	—	106	60	no	—
269	C₂×C₄	C₂²×D₂₁₂	1	45	yes	—
270	C₂×C₄	C₂⁴×D₅₃	1	15	yes	—
271	C₈	C₂⁴×D₅₃	1	1	yes	characteristic
272	C₈	C₂⁴×D₅₃	1	15	yes	—
273	C₂²	—	4	15	no	—
274	C₂²	—	106	6	no	—
275	C₂²	—	106	10	no	—
276	C₂²	—	424	45	no	—
277	C₂²	—	424	15	no	—
278	C₂²	C₂³×D₂₁₂	1	15	yes	—
279	C₄	—	106	6	no	—
280	C₄	—	106	10	no	—
281	C₄	C₂³×D₂₁₂	1	15	yes	—
282	C₄	C₂⁵×D₅₃	1	1	yes	characteristic
283	C₂	—	2	15	no	—
284	C₂	—	212	10	no	—
285	C₂	—	212	6	no	—
286	C₂	C₂⁴×D₂₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂⁴×D₂₁₂
commutator, cosocle	C₄×C₅₃	C₂⁶
Frattini	C₄	C₂⁵×D₅₃
Fitting	C₅₃×#3573	C₂
socle	C₂×C₅₃	C₂⁴×D₄
hypercenter	#3573	D₅₃
hypocenter	C₅₃	#63092
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ #585 ◁ #3573

Complements

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

N	H ≅ G/N	type
1. #20289528	C₂	semidirect
2. #20289526	C₂	semidirect
3. #20289503	C₂	semidirect
4. #20289496	C₂	semidirect
5. #20289489	C₂	semidirect
6. C₂×#16811435	C₂	semidirect
7. C₅₃×#3573	C₂	semidirect
9. #16811459	C₂²	semidirect
10. #16811459	C₂²	semidirect
12. C₂×#13133264	C₂²	semidirect
14. C₂×#13133263	C₂²	semidirect
16. #16811440	C₂²	semidirect
17. #16811435	C₂²	semidirect
18. #16811435	C₂²	semidirect
19. #16811435	C₂²	semidirect
20. #16811428	C₂²	semidirect
21. #16811423	C₂²	semidirect
22. C₂×#13133250	C₂²	semidirect
23. C₂×#13133245	C₂²	semidirect
24. C₂×#13133239	C₂²	semidirect
25. C₂²×#11719913	C₂²	semidirect
27. C₅₃×#568	C₂²	semidirect
28. C₂×#131×C₅₃	C₂²	semidirect
33. #13133264	C₂³	semidirect
36. #13133263	C₂³	semidirect
41. C₂×#11719945	C₂³	semidirect
45. C₂²×Dic₁₀₆	C₂³	semidirect
49. #13133250	C₂³	semidirect
50. #13133250	C₂³	semidirect
51. #13133245	C₂³	semidirect
52. #13133245	C₂³	semidirect
53. #13133239	C₂³	semidirect
54. #13133239	C₂³	semidirect
55. C₂×Dic₂₁₂	C₂³	semidirect
56. C₂×#11719913	C₂³	semidirect
57. C₂×#11719913	C₂³	semidirect
58. C₂×#11719913	C₂³	semidirect
59. C₂×D₄₂₄	C₂³	semidirect
63. #131×C₅₃	C₂³	semidirect
64. #131×C₅₃	C₂³	semidirect
65. C₂×M₁₆×C₅₃	C₂³	semidirect
66. C₂²×C₈×C₅₃	C₂³	semidirect
146. #3573	D₅₃	semidirect
170. #568	C₂×D₅₃	semidirect
171. C₂×#131	C₂×D₅₃	semidirect
172. C₅₃	#63092	semidirect
207. #131	C₂²×D₅₃	semidirect
208. #131	C₂²×D₅₃	semidirect
209. C₂×M₁₆	C₂²×D₅₃	semidirect
210. C₂²×C₈	C₂²×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#20289528	2
#20289526	2
#20289503	2
#20289496	2
#20289489	2
C₂×#16811435	2
C₅₃×#3573	2
#63092	53