Subgroups — C₂×#20289527 #28297991

Order 13568.

59186 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289527	C₂	1	64	yes	maximal, direct product factor
2	C₂×#16811460	C₂	1	16	yes	maximal, complemented
3	C₂×#16811459	C₂	1	20	yes	maximal, complemented
4	C₂²×#13133265	C₂	1	10	yes	maximal, complemented
5	C₂×2₋¹⁺⁴×D₅₃	C₂	1	16	yes	maximal, complemented
6	C₂²×2₋¹⁺⁴×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#16811460	C₂²	1	512	yes	complemented
8	#16811459	C₂²	1	640	yes	complemented
9	C₂×#13133265	C₂²	1	80	yes	complemented
10	C₂×#13133265	C₂²	1	40	yes	complemented
11	C₂×#13133265	C₂²	1	320	yes	complemented
12	C₂×#13133264	C₂²	1	80	yes	complemented
13	2₋¹⁺⁴×D₅₃	C₂²	1	512	yes	complemented
14	C₂×#13133263	C₂²	1	80	yes	complemented
15	C₂×#13133262	C₂²	1	40	yes	complemented
16	C₂²×#11719949	C₂²	1	40	yes	complemented
17	C₂²×#11719945	C₂²	1	60	yes	complemented
18	C₂×#41×D₅₃	C₂²	1	160	yes	complemented
19	C₂²×Q₈×D₅₃	C₂²	1	40	yes	complemented
20	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	16	yes	complemented
21	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	32	yes	complemented
22	C₂²×#41×C₅₃	C₂²	1	10	yes	complemented
23	C₂³×Q₈×C₅₃	C₂²	1	5	yes	complemented
24	#13133265	C₂³	1	1280	yes	complemented
25	#13133265	C₂³	1	640	yes	complemented
26	#13133264	C₂³	1	1280	yes	complemented
27	#13133263	C₂³	1	1280	yes	complemented
28	#13133262	C₂³	1	640	yes	complemented
29	C₂×#11719949	C₂³	1	160	yes	—
30	C₂×#11719949	C₂³	1	640	yes	complemented
31	C₂×#11719945	C₂³	1	240	yes	—
32	C₂×#11719945	C₂³	1	80	yes	—
33	C₂×#11719945	C₂³	1	960	yes	complemented
34	C₂×#11719947	C₂³	1	160	yes	—
35	#41×D₅₃	C₂³	1	2560	yes	complemented
36	C₂²×Dic₁₀₆	C₂³	1	60	yes	—
37	C₂×Q₈×D₅₃	C₂³	1	160	yes	—
38	C₂×Q₈×D₅₃	C₂³	1	640	yes	complemented
39	C₂²×D₂₁₂	C₂³	1	60	yes	—
40	C₂²×C₂²⋊D₅₃	C₂³	1	40	yes	—
41	C₂×D₄×D₅₃	C₂³	1	160	yes	—
42	C₂²×C₄×D₅₃	C₂³	1	120	yes	—
43	2₋¹⁺⁴×C₅₃	C₂³	1	256	yes	complemented
44	C₂×#41×C₅₃	C₂³	1	80	yes	—
45	C₂×#41×C₅₃	C₂³	1	160	yes	complemented
46	C₂²×Q₈×C₅₃	C₂³	1	10	yes	—
47	C₂²×Q₈×C₅₃	C₂³	1	40	yes	—
48	C₂²×Q₈×C₅₃	C₂³	1	80	yes	complemented
49	C₂²×D₄×C₅₃	C₂³	1	10	yes	—
50	C₂³×C₄×C₅₃	C₂³	1	15	yes	—
51	C₂⁴×C₅₃	C₂⁴	1	5	yes	—
52	C₂³×D₅₃	C₂⁴	1	40	yes	—
53	#41×C₅₃	C₂⁴	1	640	yes	—
54	C₂×Q₈×C₅₃	C₂⁴	1	40	yes	—
55	C₂×Q₈×C₅₃	C₂⁴	1	80	yes	—
56	C₂×Q₈×C₅₃	C₂⁴	1	320	yes	—
57	C₂×D₄×C₅₃	C₂⁴	1	40	yes	—
58	C₂×D₄×C₅₃	C₂⁴	1	80	yes	—
59	C₂²×C₄×C₅₃	C₂⁴	1	10	yes	—
60	C₂²×C₄×C₅₃	C₂⁴	1	60	yes	—
61	C₂²×C₄×C₅₃	C₂⁴	1	120	yes	—
62	C₂×C₂²⋊D₅₃	C₂⁴	1	80	yes	—
63	C₂×C₂²⋊D₅₃	C₂⁴	1	16	yes	—
64	C₂×C₂²⋊D₅₃	C₂⁴	1	320	yes	—
65	C₂²×Dic₅₃	C₂⁴	1	40	yes	—
66	#11719949	C₂⁴	1	1280	yes	—
67	Q₈×D₅₃	C₂⁴	1	1280	yes	—
68	#11719947	C₂⁴	1	1280	yes	—
69	D₄×D₅₃	C₂⁴	1	1280	yes	—
70	#11719945	C₂⁴	1	1920	yes	—
71	#11719945	C₂⁴	1	640	yes	—
72	C₂×D₂₁₂	C₂⁴	1	80	yes	—
73	C₂×D₂₁₂	C₂⁴	1	480	yes	—
74	C₂×C₄×D₅₃	C₂⁴	1	160	yes	—
75	C₂×C₄×D₅₃	C₂⁴	1	960	yes	—
76	C₂×Dic₁₀₆	C₂⁴	1	80	yes	—
77	C₂×Dic₁₀₆	C₂⁴	1	480	yes	—
78	C₂³×C₅₃	—	4	10	no	—
79	C₂³×C₅₃	C₂⁵	1	10	yes	—
80	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
81	C₂³×C₅₃	C₂⁵	1	20	yes	—
82	C₂²×D₅₃	—	4	80	no	—
83	C₂²×D₅₃	C₂⁵	1	16	yes	—
84	C₂²×D₅₃	C₂⁵	1	160	yes	—
85	Q₈×C₅₃	C₂⁵	1	160	yes	—
86	D₄×C₅₃	C₂⁵	1	160	yes	—
87	C₂×C₄×C₅₃	C₂⁵	1	20	yes	—
88	C₂×C₄×C₅₃	C₂⁵	1	240	yes	—
89	C₂×C₄×C₅₃	C₂⁵	1	40	yes	—
90	C₂²⋊D₅₃	C₂⁵	1	64	yes	—
91	C₂²⋊D₅₃	C₂⁵	1	320	yes	—
92	C₂×Dic₅₃	C₂⁵	1	16	yes	—
93	C₂×Dic₅₃	C₂⁵	1	160	yes	—
94	D₂₁₂	C₂⁵	1	320	yes	—
95	C₄×D₅₃	C₂⁵	1	640	yes	—
96	Dic₁₀₆	C₂⁵	1	320	yes	—
97	C₂×2₋¹⁺⁶	—	53	1	no	maximal, Sylow
98	C₂²×C₅₃	—	2	20	no	—
99	C₂²×C₅₃	—	2	2	no	—
100	C₂²×C₅₃	—	4	20	no	—
101	C₂²×C₅₃	C₂⁶	1	20	yes	—
102	C₂²×C₅₃	C₂⁶	1	2	yes	—
103	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
104	C₂×D₅₃	—	2	32	no	—
105	C₂×D₅₃	—	4	160	no	—
106	C₂×D₅₃	C₂⁶	1	32	yes	—
107	C₄×C₅₃	C₂⁶	1	40	yes	—
108	Dic₅₃	C₂⁶	1	32	yes	—
109	2₋¹⁺⁶	—	53	64	no	—
110	C₂×#585	—	53	16	no	—
111	C₂×#585	—	53	20	no	—
112	C₂²×2₋¹⁺⁴	—	53	16	no	—
113	C₂²×2₋¹⁺⁴	—	53	10	no	—
114	C₂²×2₋¹⁺⁴	D₅₃	1	1	yes	characteristic, complemented
115	C₂×C₅₃	—	2	20	no	—
116	C₂×C₅₃	—	2	2	no	—
117	C₂×C₅₃	2₋¹⁺⁶	1	2	yes	complemented
118	C₂×C₅₃	C₂⁷	1	1	yes	characteristic
119	D₅₃	—	2	32	no	—
120	#585	—	53	512	no	—
121	#585	—	53	640	no	—
122	C₂×2₋¹⁺⁴	—	53	512	no	—
123	C₂×2₋¹⁺⁴	—	53	80	no	—
124	C₂×2₋¹⁺⁴	—	53	80	no	—
125	C₂×2₋¹⁺⁴	—	53	40	no	—
126	C₂×2₋¹⁺⁴	—	53	320	no	—
127	C₂×2₋¹⁺⁴	C₂×D₅₃	1	16	yes	complemented
128	C₂×2₋¹⁺⁴	C₂×D₅₃	1	32	yes	complemented
129	C₂×H(5,𝔽₂)	—	53	80	no	—
130	C₂×H(5,𝔽₂)	—	53	40	no	—
131	C₂²×#41	—	53	160	no	—
132	C₂²×#41	—	53	40	no	—
133	C₂²×#41	—	53	60	no	—
134	C₂²×#41	C₂×D₅₃	1	10	yes	complemented
135	C₂³×Q₈	—	53	40	no	—
136	C₂³×Q₈	C₂×D₅₃	1	5	yes	complemented
137	C₅₃	C₂×2₋¹⁺⁶	1	1	yes	characteristic, complemented, Sylow
138	2₋¹⁺⁴	—	53	1280	no	—
139	2₋¹⁺⁴	—	53	1280	no	—
140	2₋¹⁺⁴	—	53	640	no	—
141	2₋¹⁺⁴	C₂²×D₅₃	1	256	yes	complemented
142	H(5,𝔽₂)	—	53	1280	no	—
143	H(5,𝔽₂)	—	53	640	no	—
144	C₂×#41	—	53	2560	no	—
145	C₂×#41	—	53	160	no	—
146	C₂×#41	—	53	640	no	—
147	C₂×#41	—	53	160	no	—
148	C₂×#41	—	53	240	no	—
149	C₂×#41	—	53	80	no	—
150	C₂×#41	—	53	960	no	—
151	C₂×#41	C₂²×D₅₃	1	80	yes	—
152	C₂×#41	C₂²×D₅₃	1	160	yes	complemented
153	C₂²×Q₈	—	53	160	no	—
154	C₂²×Q₈	—	53	640	no	—
155	C₂²×Q₈	—	53	60	no	—
156	C₂²×Q₈	C₂²×D₅₃	1	40	yes	—
157	C₂²×Q₈	C₂²×D₅₃	1	10	yes	—
158	C₂²×Q₈	C₂²×D₅₃	1	80	yes	complemented
159	C₂²×D₄	—	53	40	no	—
160	C₂²×D₄	—	53	160	no	—
161	C₂²×D₄	—	53	60	no	—
162	C₂²×D₄	C₂²×D₅₃	1	10	yes	—
163	C₂³×C₄	—	53	120	no	—
164	C₂³×C₄	C₂²×D₅₃	1	15	yes	—
165	C₂⁴	—	53	40	no	—
166	C₂⁴	C₂³×D₅₃	1	5	yes	—
167	#41	—	53	1280	no	—
168	#41	—	53	1280	no	—
169	#41	—	53	1920	no	—
170	#41	—	53	640	no	—
171	#41	C₂³×D₅₃	1	640	yes	—
172	C₂×Q₈	—	53	1280	no	—
173	C₂×Q₈	—	53	80	no	—
174	C₂×Q₈	—	53	480	no	—
175	C₂×Q₈	C₂³×D₅₃	1	40	yes	—
176	C₂×Q₈	C₂³×D₅₃	1	320	yes	—
177	C₂×Q₈	C₂³×D₅₃	1	80	yes	—
178	C₂×D₄	—	53	80	no	—
179	C₂×D₄	—	53	16	no	—
180	C₂×D₄	—	53	320	no	—
181	C₂×D₄	—	53	1280	no	—
182	C₂×D₄	—	53	80	no	—
183	C₂×D₄	—	53	480	no	—
184	C₂×D₄	C₂³×D₅₃	1	40	yes	—
185	C₂×D₄	C₂³×D₅₃	1	80	yes	—
186	C₂²×C₄	—	53	40	no	—
187	C₂²×C₄	—	53	160	no	—
188	C₂²×C₄	—	53	960	no	—
189	C₂²×C₄	C₂³×D₅₃	1	60	yes	—
190	C₂²×C₄	C₂³×D₅₃	1	10	yes	—
191	C₂²×C₄	C₂³×D₅₃	1	120	yes	—
192	C₂³	—	4	10	no	—
193	C₂³	—	53	16	no	—
194	C₂³	—	53	160	no	—
195	C₂³	—	212	80	no	—
196	C₂³	C₂⁴×D₅₃	1	10	yes	—
197	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
198	C₂³	C₂⁴×D₅₃	1	20	yes	—
199	Q₈	—	53	320	no	—
200	Q₈	C₂⁴×D₅₃	1	160	yes	—
201	D₄	—	53	320	no	—
202	D₄	—	53	64	no	—
203	D₄	—	53	320	no	—
204	D₄	C₂⁴×D₅₃	1	160	yes	—
205	C₂×C₄	—	53	160	no	—
206	C₂×C₄	—	53	16	no	—
207	C₂×C₄	—	53	640	no	—
208	C₂×C₄	C₂⁴×D₅₃	1	240	yes	—
209	C₂×C₄	C₂⁴×D₅₃	1	40	yes	—
210	C₂×C₄	C₂⁴×D₅₃	1	20	yes	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	20	no	—
213	C₂²	—	4	20	no	—
214	C₂²	—	53	32	no	—
215	C₂²	—	106	32	no	—
216	C₂²	—	212	160	no	—
217	C₂²	C₂⁵×D₅₃	1	20	yes	—
218	C₂²	C₂⁵×D₅₃	1	2	yes	—
219	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
220	C₄	—	53	32	no	—
221	C₄	C₂⁵×D₅₃	1	40	yes	—
222	C₂	—	2	2	no	—
223	C₂	—	2	20	no	—
224	C₂	—	106	32	no	—
225	C₂	#20289527	1	2	yes	direct product factor
226	C₂	C₂⁶×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂⁵×D₅₃
commutator, cosocle	C₂×C₅₃	C₂⁷
Frattini	C₂	C₂⁶×D₅₃
Fitting	C₂²×2₋¹⁺⁴×C₅₃	C₂
socle	C₂²×C₅₃	C₂⁶
hypercenter	C₂²×2₋¹⁺⁴	D₅₃
hypocenter	C₅₃	C₂×2₋¹⁺⁶
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. #20289527	C₂	direct (N × H)
2. C₂×#16811460	C₂	semidirect
3. C₂×#16811459	C₂	semidirect
4. C₂²×#13133265	C₂	semidirect
5. C₂×2₋¹⁺⁴×D₅₃	C₂	semidirect
6. C₂²×2₋¹⁺⁴×C₅₃	C₂	semidirect
7. #16811460	C₂²	semidirect
8. #16811459	C₂²	semidirect
9. C₂×#13133265	C₂²	semidirect
10. C₂×#13133265	C₂²	semidirect
11. C₂×#13133265	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×#11719949	C₂²	semidirect
17. C₂²×#11719945	C₂²	semidirect
18. C₂×#41×D₅₃	C₂²	semidirect
19. C₂²×Q₈×D₅₃	C₂²	semidirect
20. C₂×2₋¹⁺⁴×C₅₃	C₂²	semidirect
21. C₂×2₋¹⁺⁴×C₅₃	C₂²	semidirect
22. C₂²×#41×C₅₃	C₂²	semidirect
23. C₂³×Q₈×C₅₃	C₂²	semidirect
24. #13133265	C₂³	semidirect
25. #13133265	C₂³	semidirect
26. #13133264	C₂³	semidirect
27. #13133263	C₂³	semidirect
28. #13133262	C₂³	semidirect
30. C₂×#11719949	C₂³	semidirect
33. C₂×#11719945	C₂³	semidirect
35. #41×D₅₃	C₂³	semidirect
38. C₂×Q₈×D₅₃	C₂³	semidirect
43. 2₋¹⁺⁴×C₅₃	C₂³	semidirect
45. C₂×#41×C₅₃	C₂³	semidirect
48. C₂²×Q₈×C₅₃	C₂³	semidirect
114. C₂²×2₋¹⁺⁴	D₅₃	semidirect
117. C₂×C₅₃	2₋¹⁺⁶	semidirect
127. C₂×2₋¹⁺⁴	C₂×D₅₃	semidirect
128. C₂×2₋¹⁺⁴	C₂×D₅₃	semidirect
134. C₂²×#41	C₂×D₅₃	semidirect
136. C₂³×Q₈	C₂×D₅₃	semidirect
137. C₅₃	C₂×2₋¹⁺⁶	semidirect
141. 2₋¹⁺⁴	C₂²×D₅₃	semidirect
152. C₂×#41	C₂²×D₅₃	semidirect
158. C₂²×Q₈	C₂²×D₅₃	semidirect
225. C₂	#20289527	direct (N × H)

Maximal subgroups

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

subgroup	index
#20289527	2
C₂×#16811460	2
C₂×#16811459	2
C₂²×#13133265	2
C₂×2₋¹⁺⁴×D₅₃	2
C₂²×2₋¹⁺⁴×C₅₃	2
C₂×2₋¹⁺⁶	53

Subgroups — C2×#20289527 #28297991

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20289527 #28297991