Subgroups — C₂×#20289528 #28297992

Order 13568.

59186 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289528	C₂	1	64	yes	maximal, direct product factor
2	C₂×#16811459	C₂	1	15	yes	maximal, complemented
3	C₂×#16811458	C₂	1	20	yes	maximal, complemented
4	C₂²×#13133264	C₂	1	15	yes	maximal, complemented
5	C₂×2₋¹⁺⁴×D₅₃	C₂	1	12	yes	maximal, complemented
6	C₂×C₅₃×#585	C₂	1	1	yes	maximal, characteristic, complemented
7	#16811459	C₂²	1	480	yes	complemented
8	#16811458	C₂²	1	640	yes	complemented
9	C₂×#13133265	C₂²	1	30	yes	complemented
10	C₂×#13133264	C₂²	1	120	yes	complemented
11	C₂×#13133264	C₂²	1	60	yes	complemented
12	C₂×#13133264	C₂²	1	480	yes	complemented
13	2₋¹⁺⁴×D₅₃	C₂²	1	384	yes	complemented
14	C₂×#13133263	C₂²	1	20	yes	complemented
15	C₂×#13133262	C₂²	1	90	yes	complemented
16	C₂²×#11719945	C₂²	1	45	yes	complemented
17	C₂²×#11719947	C₂²	1	90	yes	complemented
18	C₂×#41×D₅₃	C₂²	1	120	yes	complemented
19	C₂³×Dic₁₀₆	C₂²	1	15	yes	complemented
20	C₂²×Q₈×D₅₃	C₂²	1	30	yes	complemented
21	C₅₃×#585	C₂²	1	32	yes	complemented
22	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	6	yes	complemented
23	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	10	yes	complemented
24	C₂²×#41×C₅₃	C₂²	1	15	yes	complemented
25	#13133265	C₂³	1	480	yes	complemented
26	#13133264	C₂³	1	1920	yes	complemented
27	#13133264	C₂³	1	960	yes	complemented
28	#13133263	C₂³	1	320	yes	complemented
29	#13133262	C₂³	1	1440	yes	complemented
30	C₂×#11719949	C₂³	1	40	yes	—
31	C₂×#11719945	C₂³	1	180	yes	—
32	C₂×#11719945	C₂³	1	60	yes	—
33	C₂×#11719945	C₂³	1	720	yes	complemented
34	C₂×#11719947	C₂³	1	360	yes	—
35	C₂×#11719947	C₂³	1	1440	yes	complemented
36	#41×D₅₃	C₂³	1	1920	yes	complemented
37	C₂²×Dic₁₀₆	C₂³	1	90	yes	—
38	C₂²×Dic₁₀₆	C₂³	1	45	yes	—
39	C₂²×Dic₁₀₆	C₂³	1	240	yes	complemented
40	C₂×Q₈×D₅₃	C₂³	1	120	yes	—
41	C₂×Q₈×D₅₃	C₂³	1	480	yes	complemented
42	C₂²×D₂₁₂	C₂³	1	15	yes	—
43	C₂²×C₂²⋊D₅₃	C₂³	1	90	yes	—
44	C₂×D₄×D₅₃	C₂³	1	120	yes	—
45	C₂³×Dic₅₃	C₂³	1	30	yes	—
46	C₂²×C₄×D₅₃	C₂³	1	90	yes	—
47	2₋¹⁺⁴×C₅₃	C₂³	1	96	yes	complemented
48	H(5,𝔽₂)×C₅₃	C₂³	1	160	yes	complemented
49	C₂×#41×C₅₃	C₂³	1	20	yes	—
50	C₂×#41×C₅₃	C₂³	1	60	yes	—
51	C₂×#41×C₅₃	C₂³	1	240	yes	complemented
52	C₂²×Q₈×C₅₃	C₂³	1	15	yes	—
53	C₂²×D₄×C₅₃	C₂³	1	45	yes	—
54	C₂³×C₄×C₅₃	C₂³	1	15	yes	—
55	C₂⁴×C₅₃	C₂⁴	1	15	yes	—
56	C₂³×D₅₃	C₂⁴	1	30	yes	—
57	#41×C₅₃	C₂⁴	1	160	yes	—
58	#41×C₅₃	C₂⁴	1	480	yes	—
59	C₂×Q₈×C₅₃	C₂⁴	1	20	yes	—
60	C₂×Q₈×C₅₃	C₂⁴	1	120	yes	—
61	C₂×D₄×C₅₃	C₂⁴	1	60	yes	—
62	C₂×D₄×C₅₃	C₂⁴	1	360	yes	—
63	C₂²×C₄×C₅₃	C₂⁴	1	15	yes	—
64	C₂²×C₄×C₅₃	C₂⁴	1	45	yes	—
65	C₂²×C₄×C₅₃	C₂⁴	1	120	yes	—
66	C₂×C₂²⋊D₅₃	C₂⁴	1	120	yes	—
67	C₂×C₂²⋊D₅₃	C₂⁴	1	720	yes	—
68	C₂²×Dic₅₃	C₂⁴	1	90	yes	—
69	C₂²×Dic₅₃	C₂⁴	1	240	yes	—
70	#11719949	C₂⁴	1	320	yes	—
71	Q₈×D₅₃	C₂⁴	1	960	yes	—
72	#11719947	C₂⁴	1	2880	yes	—
73	D₄×D₅₃	C₂⁴	1	960	yes	—
74	#11719945	C₂⁴	1	1440	yes	—
75	#11719945	C₂⁴	1	480	yes	—
76	C₂×D₂₁₂	C₂⁴	1	30	yes	—
77	C₂×D₂₁₂	C₂⁴	1	6	yes	—
78	C₂×D₂₁₂	C₂⁴	1	120	yes	—
79	C₂×C₄×D₅₃	C₂⁴	1	120	yes	—
80	C₂×C₄×D₅₃	C₂⁴	1	720	yes	—
81	C₂×Dic₁₀₆	C₂⁴	1	90	yes	—
82	C₂×Dic₁₀₆	C₂⁴	1	720	yes	—
83	C₂×Dic₁₀₆	C₂⁴	1	10	yes	—
84	C₂×Dic₁₀₆	C₂⁴	1	360	yes	—
85	C₂³×C₅₃	—	4	30	no	—
86	C₂³×C₅₃	C₂⁵	1	15	yes	—
87	C₂³×C₅₃	C₂⁵	1	60	yes	—
88	C₂²×D₅₃	—	4	60	no	—
89	C₂²×D₅₃	C₂⁵	1	12	yes	—
90	C₂²×D₅₃	C₂⁵	1	120	yes	—
91	Q₈×C₅₃	C₂⁵	1	80	yes	—
92	D₄×C₅₃	C₂⁵	1	240	yes	—
93	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂⁵	1	15	yes	—
95	C₂×C₄×C₅₃	C₂⁵	1	60	yes	—
96	C₂×C₄×C₅₃	C₂⁵	1	180	yes	—
97	C₂²⋊D₅₃	C₂⁵	1	480	yes	—
98	C₂×Dic₅₃	C₂⁵	1	20	yes	—
99	C₂×Dic₅₃	C₂⁵	1	360	yes	—
100	D₂₁₂	C₂⁵	1	24	yes	—
101	D₂₁₂	C₂⁵	1	120	yes	—
102	C₄×D₅₃	C₂⁵	1	480	yes	—
103	Dic₁₀₆	C₂⁵	1	40	yes	—
104	Dic₁₀₆	C₂⁵	1	360	yes	—
105	C₂×2₋¹⁺⁶	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	—	2	30	no	—
107	C₂²×C₅₃	—	4	60	no	—
108	C₂²×C₅₃	C₂⁶	1	30	yes	—
109	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
110	C₂×D₅₃	—	2	24	no	—
111	C₂×D₅₃	—	4	120	no	—
112	C₂×D₅₃	C₂⁶	1	24	yes	—
113	C₄×C₅₃	C₂⁶	1	2	yes	—
114	C₄×C₅₃	C₂⁶	1	30	yes	—
115	Dic₅₃	C₂⁶	1	40	yes	—
116	2₋¹⁺⁶	—	53	64	no	—
117	C₂×#585	—	53	20	no	—
118	C₂×#585	—	53	15	no	—
119	C₂×#585	D₅₃	1	1	yes	characteristic, complemented
120	C₂²×2₋¹⁺⁴	—	53	12	no	—
121	C₂²×2₋¹⁺⁴	—	53	15	no	—
122	C₂×C₅₃	—	2	30	no	—
123	C₂×C₅₃	2₋¹⁺⁶	1	2	yes	complemented
124	C₂×C₅₃	C₂⁷	1	1	yes	characteristic
125	D₅₃	—	2	24	no	—
126	#585	—	53	640	no	—
127	#585	—	53	480	no	—
128	#585	C₂×D₅₃	1	32	yes	complemented
129	C₂×2₋¹⁺⁴	—	53	384	no	—
130	C₂×2₋¹⁺⁴	—	53	120	no	—
131	C₂×2₋¹⁺⁴	—	53	60	no	—
132	C₂×2₋¹⁺⁴	—	53	480	no	—
133	C₂×2₋¹⁺⁴	—	53	30	no	—
134	C₂×2₋¹⁺⁴	C₂×D₅₃	1	6	yes	complemented
135	C₂×H(5,𝔽₂)	—	53	20	no	—
136	C₂×H(5,𝔽₂)	—	53	90	no	—
137	C₂×H(5,𝔽₂)	C₂×D₅₃	1	10	yes	complemented
138	C₂²×#41	—	53	120	no	—
139	C₂²×#41	—	53	90	no	—
140	C₂²×#41	—	53	45	no	—
141	C₂²×#41	C₂×D₅₃	1	15	yes	complemented
142	C₂³×Q₈	—	53	30	no	—
143	C₂³×Q₈	—	53	15	no	—
144	C₅₃	C₂×2₋¹⁺⁶	1	1	yes	characteristic, complemented, Sylow
145	2₋¹⁺⁴	—	53	1920	no	—
146	2₋¹⁺⁴	—	53	960	no	—
147	2₋¹⁺⁴	—	53	480	no	—
148	2₋¹⁺⁴	C₂²×D₅₃	1	96	yes	complemented
149	H(5,𝔽₂)	—	53	320	no	—
150	H(5,𝔽₂)	—	53	1440	no	—
151	H(5,𝔽₂)	C₂²×D₅₃	1	160	yes	complemented
152	C₂×#41	—	53	1920	no	—
153	C₂×#41	—	53	40	no	—
154	C₂×#41	—	53	360	no	—
155	C₂×#41	—	53	1440	no	—
156	C₂×#41	—	53	180	no	—
157	C₂×#41	—	53	60	no	—
158	C₂×#41	—	53	720	no	—
159	C₂×#41	C₂²×D₅₃	1	60	yes	—
160	C₂×#41	C₂²×D₅₃	1	20	yes	—
161	C₂×#41	C₂²×D₅₃	1	240	yes	complemented
162	C₂²×Q₈	—	53	120	no	—
163	C₂²×Q₈	—	53	480	no	—
164	C₂²×Q₈	—	53	90	no	—
165	C₂²×Q₈	—	53	45	no	—
166	C₂²×Q₈	—	53	240	no	—
167	C₂²×Q₈	C₂²×D₅₃	1	15	yes	—
168	C₂²×D₄	—	53	90	no	—
169	C₂²×D₄	—	53	120	no	—
170	C₂²×D₄	—	53	15	no	—
171	C₂²×D₄	C₂²×D₅₃	1	45	yes	—
172	C₂³×C₄	—	53	30	no	—
173	C₂³×C₄	—	53	90	no	—
174	C₂³×C₄	C₂²×D₅₃	1	15	yes	—
175	C₂⁴	—	53	30	no	—
176	C₂⁴	C₂³×D₅₃	1	15	yes	—
177	#41	—	53	320	no	—
178	#41	—	53	2880	no	—
179	#41	—	53	1440	no	—
180	#41	—	53	480	no	—
181	#41	C₂³×D₅₃	1	480	yes	—
182	#41	C₂³×D₅₃	1	160	yes	—
183	C₂×Q₈	—	53	960	no	—
184	C₂×Q₈	—	53	90	no	—
185	C₂×Q₈	—	53	10	no	—
186	C₂×Q₈	—	53	720	no	—
187	C₂×Q₈	—	53	360	no	—
188	C₂×Q₈	C₂³×D₅₃	1	20	yes	—
189	C₂×Q₈	C₂³×D₅₃	1	120	yes	—
190	C₂×D₄	—	53	120	no	—
191	C₂×D₄	—	53	720	no	—
192	C₂×D₄	—	53	960	no	—
193	C₂×D₄	—	53	30	no	—
194	C₂×D₄	—	53	6	no	—
195	C₂×D₄	—	53	120	no	—
196	C₂×D₄	C₂³×D₅₃	1	60	yes	—
197	C₂×D₄	C₂³×D₅₃	1	360	yes	—
198	C₂²×C₄	—	53	90	no	—
199	C₂²×C₄	—	53	240	no	—
200	C₂²×C₄	—	53	120	no	—
201	C₂²×C₄	—	53	720	no	—
202	C₂²×C₄	C₂³×D₅₃	1	45	yes	—
203	C₂²×C₄	C₂³×D₅₃	1	15	yes	—
204	C₂²×C₄	C₂³×D₅₃	1	120	yes	—
205	C₂³	—	4	30	no	—
206	C₂³	—	53	12	no	—
207	C₂³	—	53	120	no	—
208	C₂³	—	212	60	no	—
209	C₂³	C₂⁴×D₅₃	1	15	yes	—
210	C₂³	C₂⁴×D₅₃	1	60	yes	—
211	Q₈	—	53	360	no	—
212	Q₈	—	53	40	no	—
213	Q₈	C₂⁴×D₅₃	1	80	yes	—
214	D₄	—	53	480	no	—
215	D₄	—	53	120	no	—
216	D₄	—	53	24	no	—
217	D₄	C₂⁴×D₅₃	1	240	yes	—
218	C₂×C₄	—	53	360	no	—
219	C₂×C₄	—	53	20	no	—
220	C₂×C₄	—	53	480	no	—
221	C₂×C₄	C₂⁴×D₅₃	1	180	yes	—
222	C₂×C₄	C₂⁴×D₅₃	1	60	yes	—
223	C₂×C₄	C₂⁴×D₅₃	1	15	yes	—
224	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
225	C₂²	—	2	30	no	—
226	C₂²	—	4	60	no	—
227	C₂²	—	53	24	no	—
228	C₂²	—	106	24	no	—
229	C₂²	—	212	120	no	—
230	C₂²	C₂⁵×D₅₃	1	30	yes	—
231	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
232	C₄	—	53	40	no	—
233	C₄	C₂⁵×D₅₃	1	30	yes	—
234	C₄	C₂⁵×D₅₃	1	2	yes	—
235	C₂	—	2	30	no	—
236	C₂	—	106	24	no	—
237	C₂	#20289528	1	2	yes	direct product factor
238	C₂	C₂⁶×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂⁵×D₅₃
commutator, cosocle	C₂×C₅₃	C₂⁷
Frattini	C₂	C₂⁶×D₅₃
Fitting	C₂×C₅₃×#585	C₂
socle	C₂²×C₅₃	C₂⁶
hypercenter	C₂×#585	D₅₃
hypocenter	C₅₃	C₂×2₋¹⁺⁶
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×#585

Complements

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

N	H ≅ G/N	type
1. #20289528	C₂	direct (N × H)
2. C₂×#16811459	C₂	semidirect
3. C₂×#16811458	C₂	semidirect
4. C₂²×#13133264	C₂	semidirect
5. C₂×2₋¹⁺⁴×D₅₃	C₂	semidirect
6. C₂×C₅₃×#585	C₂	semidirect
7. #16811459	C₂²	semidirect
8. #16811458	C₂²	semidirect
9. C₂×#13133265	C₂²	semidirect
10. C₂×#13133264	C₂²	semidirect
11. C₂×#13133264	C₂²	semidirect
12. C₂×#13133264	C₂²	semidirect
13. 2₋¹⁺⁴×D₅₃	C₂²	semidirect
14. C₂×#13133263	C₂²	semidirect
15. C₂×#13133262	C₂²	semidirect
16. C₂²×#11719945	C₂²	semidirect
17. C₂²×#11719947	C₂²	semidirect
18. C₂×#41×D₅₃	C₂²	semidirect
19. C₂³×Dic₁₀₆	C₂²	semidirect
20. C₂²×Q₈×D₅₃	C₂²	semidirect
21. C₅₃×#585	C₂²	semidirect
22. C₂×2₋¹⁺⁴×C₅₃	C₂²	semidirect
23. C₂×H(5,𝔽₂)×C₅₃	C₂²	semidirect
24. C₂²×#41×C₅₃	C₂²	semidirect
25. #13133265	C₂³	semidirect
26. #13133264	C₂³	semidirect
27. #13133264	C₂³	semidirect
28. #13133263	C₂³	semidirect
29. #13133262	C₂³	semidirect
33. C₂×#11719945	C₂³	semidirect
35. C₂×#11719947	C₂³	semidirect
36. #41×D₅₃	C₂³	semidirect
39. C₂²×Dic₁₀₆	C₂³	semidirect
41. C₂×Q₈×D₅₃	C₂³	semidirect
47. 2₋¹⁺⁴×C₅₃	C₂³	semidirect
48. H(5,𝔽₂)×C₅₃	C₂³	semidirect
51. C₂×#41×C₅₃	C₂³	semidirect
119. C₂×#585	D₅₃	semidirect
123. C₂×C₅₃	2₋¹⁺⁶	semidirect
128. #585	C₂×D₅₃	semidirect
134. C₂×2₋¹⁺⁴	C₂×D₅₃	semidirect
137. C₂×H(5,𝔽₂)	C₂×D₅₃	semidirect
141. C₂²×#41	C₂×D₅₃	semidirect
144. C₅₃	C₂×2₋¹⁺⁶	semidirect
148. 2₋¹⁺⁴	C₂²×D₅₃	semidirect
151. H(5,𝔽₂)	C₂²×D₅₃	semidirect
161. C₂×#41	C₂²×D₅₃	semidirect
237. C₂	#20289528	direct (N × H)

Maximal subgroups

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

subgroup	index
#20289528	2
C₂×#16811459	2
C₂×#16811458	2
C₂²×#13133264	2
C₂×2₋¹⁺⁴×D₅₃	2
C₂×C₅₃×#585	2
C₂×2₋¹⁺⁶	53

Subgroups — C2×#20289528 #28297992

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20289528 #28297992