Subgroups — C₂²×2₋¹⁺⁴×D₅₃ #28297980

Order 13568.

59806 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133265	C₂	1	5	yes	maximal, complemented
2	C₂²×#13133264	C₂	1	10	yes	maximal, complemented
3	C₂×2₋¹⁺⁴×D₅₃	C₂	1	96	yes	maximal, direct product factor
4	C₂²×#41×D₅₃	C₂	1	10	yes	maximal, complemented
5	C₂³×Q₈×D₅₃	C₂	1	5	yes	maximal, complemented
6	C₂²×2₋¹⁺⁴×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13133265	C₂²	1	240	yes	complemented
8	C₂×#13133264	C₂²	1	480	yes	complemented
9	2₋¹⁺⁴×D₅₃	C₂²	1	1024	yes	direct product factor
10	C₂²×#11719949	C₂²	1	10	yes	complemented
11	C₂²×#11719945	C₂²	1	30	yes	complemented
12	C₂²×#11719947	C₂²	1	30	yes	complemented
13	C₂×#41×D₅₃	C₂²	1	480	yes	complemented
14	C₂³×Dic₁₀₆	C₂²	1	15	yes	complemented
15	C₂²×Q₈×D₅₃	C₂²	1	10	yes	—
16	C₂²×Q₈×D₅₃	C₂²	1	20	yes	complemented
17	C₂²×Q₈×D₅₃	C₂²	1	240	yes	complemented
18	C₂²×D₄×D₅₃	C₂²	1	10	yes	—
19	C₂³×C₄×D₅₃	C₂²	1	15	yes	—
20	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	48	yes	complemented
21	C₂²×#41×C₅₃	C₂²	1	10	yes	complemented
22	C₂³×Q₈×C₅₃	C₂²	1	5	yes	complemented
23	C₂⁴×D₅₃	C₂³	1	5	yes	—
24	#13133265	C₂³	1	1280	yes	complemented
25	#13133264	C₂³	1	2560	yes	complemented
26	C₂×#11719949	C₂³	1	240	yes	complemented
27	C₂×#11719945	C₂³	1	720	yes	complemented
28	C₂×#11719947	C₂³	1	720	yes	complemented
29	#41×D₅₃	C₂³	1	2560	yes	complemented
30	C₂²×Dic₁₀₆	C₂³	1	30	yes	—
31	C₂²×Dic₁₀₆	C₂³	1	360	yes	complemented
32	C₂×Q₈×D₅₃	C₂³	1	240	yes	—
33	C₂×Q₈×D₅₃	C₂³	1	480	yes	complemented
34	C₂×Q₈×D₅₃	C₂³	1	1280	yes	complemented
35	C₂²×D₂₁₂	C₂³	1	10	yes	—
36	C₂²×C₂²⋊D₅₃	C₂³	1	20	yes	—
37	C₂×D₄×D₅₃	C₂³	1	240	yes	—
38	C₂³×Dic₅₃	C₂³	1	15	yes	—
39	C₂²×C₄×D₅₃	C₂³	1	10	yes	—
40	C₂²×C₄×D₅₃	C₂³	1	360	yes	—
41	C₂²×C₄×D₅₃	C₂³	1	30	yes	—
42	2₋¹⁺⁴×C₅₃	C₂³	1	256	yes	complemented
43	C₂×#41×C₅₃	C₂³	1	240	yes	complemented
44	C₂²×Q₈×C₅₃	C₂³	1	10	yes	—
45	C₂²×Q₈×C₅₃	C₂³	1	120	yes	complemented
46	C₂²×D₄×C₅₃	C₂³	1	10	yes	—
47	C₂³×C₄×C₅₃	C₂³	1	15	yes	—
48	C₂⁴×C₅₃	C₂⁴	1	5	yes	—
49	C₂³×D₅₃	—	2	40	no	—
50	C₂³×D₅₃	C₂⁴	1	60	yes	—
51	C₂³×D₅₃	C₂⁴	1	5	yes	—
52	C₂³×D₅₃	C₂⁴	1	1	yes	characteristic
53	#41×C₅₃	C₂⁴	1	640	yes	complemented
54	C₂×Q₈×C₅₃	C₂⁴	1	120	yes	—
55	C₂×Q₈×C₅₃	C₂⁴	1	320	yes	complemented
56	C₂×D₄×C₅₃	C₂⁴	1	120	yes	—
57	C₂²×C₄×C₅₃	C₂⁴	1	10	yes	—
58	C₂²×C₄×C₅₃	C₂⁴	1	180	yes	—
59	C₂×C₂²⋊D₅₃	C₂⁴	1	240	yes	—
60	C₂²×Dic₅₃	C₂⁴	1	10	yes	—
61	C₂²×Dic₅₃	C₂⁴	1	180	yes	—
62	#11719949	C₂⁴	1	640	yes	complemented
63	Q₈×D₅₃	C₂⁴	1	640	yes	—
64	Q₈×D₅₃	C₂⁴	1	1280	yes	complemented
65	#11719947	C₂⁴	1	1920	yes	complemented
66	D₄×D₅₃	C₂⁴	1	640	yes	—
67	#11719945	C₂⁴	1	1920	yes	complemented
68	C₂×D₂₁₂	C₂⁴	1	120	yes	—
69	C₂×C₄×D₅₃	C₂⁴	1	120	yes	—
70	C₂×C₄×D₅₃	C₂⁴	1	960	yes	—
71	C₂×C₄×D₅₃	C₂⁴	1	360	yes	—
72	C₂×Dic₁₀₆	C₂⁴	1	360	yes	—
73	C₂×Dic₁₀₆	C₂⁴	1	960	yes	complemented
74	C₂³×C₅₃	—	2	20	no	—
75	C₂³×C₅₃	C₂⁵	1	30	yes	—
76	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
77	C₂²×D₅₃	—	2	20	no	—
78	C₂²×D₅₃	—	2	240	no	—
79	C₂²×D₅₃	2₋¹⁺⁴	1	8	yes	direct product factor
80	C₂²×D₅₃	C₂⁵	1	30	yes	—
81	C₂²×D₅₃	C₂⁵	1	80	yes	—
82	C₂²×D₅₃	C₂⁵	1	6	yes	—
83	Q₈×C₅₃	C₂⁵	1	160	yes	—
84	D₄×C₅₃	C₂⁵	1	160	yes	—
85	C₂×C₄×C₅₃	C₂⁵	1	60	yes	—
86	C₂×C₄×C₅₃	C₂⁵	1	240	yes	—
87	C₂²⋊D₅₃	C₂⁵	1	320	yes	—
88	C₂×Dic₅₃	C₂⁵	1	60	yes	—
89	C₂×Dic₅₃	C₂⁵	1	240	yes	—
90	D₂₁₂	C₂⁵	1	160	yes	—
91	C₄×D₅₃	C₂⁵	1	160	yes	—
92	C₄×D₅₃	C₂⁵	1	480	yes	—
93	Dic₁₀₆	C₂⁵	1	480	yes	—
94	C₂³×2₋¹⁺⁴	—	53	1	no	maximal, Sylow
95	C₂²×C₅₃	—	2	60	no	—
96	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	4	yes	complemented
97	C₂²×C₅₃	C₂⁶	1	20	yes	—
98	C₂²×C₅₃	C₂⁶	1	3	yes	—
99	C₂×D₅₃	—	2	60	no	—
100	C₂×D₅₃	—	2	160	no	—
101	C₂×D₅₃	C₂×2₋¹⁺⁴	1	24	yes	direct product factor
102	C₂×D₅₃	C₂⁶	1	20	yes	—
103	C₂×D₅₃	C₂⁶	1	4	yes	—
104	C₄×C₅₃	C₂⁶	1	40	yes	—
105	Dic₅₃	C₂⁶	1	40	yes	—
106	C₂²×2₋¹⁺⁴	—	53	96	no	—
107	C₂²×2₋¹⁺⁴	—	53	10	no	—
108	C₂²×2₋¹⁺⁴	—	53	5	no	—
109	C₂²×2₋¹⁺⁴	D₅₃	1	1	yes	characteristic, direct product factor
110	C₂³×#41	—	53	10	no	—
111	C₂⁴×Q₈	—	53	5	no	—
112	C₂×C₅₃	—	2	20	no	—
113	C₂×C₅₃	C₂²×2₋¹⁺⁴	1	6	yes	complemented
114	C₂×C₅₃	C₂⁷	1	1	yes	characteristic
115	D₅₃	—	2	20	no	—
116	D₅₃	C₂²×2₋¹⁺⁴	1	8	yes	direct product factor
117	C₂×2₋¹⁺⁴	—	53	1024	no	—
118	C₂×2₋¹⁺⁴	—	53	480	no	—
119	C₂×2₋¹⁺⁴	—	53	240	no	—
120	C₂×2₋¹⁺⁴	C₂×D₅₃	1	48	yes	direct product factor
121	C₂²×#41	—	53	480	no	—
122	C₂²×#41	—	53	10	no	—
123	C₂²×#41	—	53	30	no	—
124	C₂²×#41	—	53	30	no	—
125	C₂²×#41	C₂×D₅₃	1	10	yes	complemented
126	C₂³×Q₈	—	53	10	no	—
127	C₂³×Q₈	—	53	20	no	—
128	C₂³×Q₈	—	53	240	no	—
129	C₂³×Q₈	—	53	15	no	—
130	C₂³×Q₈	C₂×D₅₃	1	5	yes	complemented
131	C₂³×D₄	—	53	10	no	—
132	C₂⁴×C₄	—	53	15	no	—
133	C₅₃	C₂³×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
134	C₂⁵	—	53	5	no	—
135	2₋¹⁺⁴	—	53	2560	no	—
136	2₋¹⁺⁴	—	53	1280	no	—
137	2₋¹⁺⁴	C₂²×D₅₃	1	256	yes	direct product factor
138	C₂×#41	—	53	2560	no	—
139	C₂×#41	—	53	240	no	—
140	C₂×#41	—	53	720	no	—
141	C₂×#41	—	53	720	no	—
142	C₂×#41	C₂²×D₅₃	1	240	yes	complemented
143	C₂²×Q₈	—	53	240	no	—
144	C₂²×Q₈	—	53	480	no	—
145	C₂²×Q₈	—	53	1280	no	—
146	C₂²×Q₈	—	53	30	no	—
147	C₂²×Q₈	—	53	360	no	—
148	C₂²×Q₈	C₂²×D₅₃	1	10	yes	—
149	C₂²×Q₈	C₂²×D₅₃	1	120	yes	complemented
150	C₂²×D₄	—	53	20	no	—
151	C₂²×D₄	—	53	240	no	—
152	C₂²×D₄	—	53	10	no	—
153	C₂²×D₄	C₂²×D₅₃	1	10	yes	—
154	C₂³×C₄	—	53	15	no	—
155	C₂³×C₄	—	53	10	no	—
156	C₂³×C₄	—	53	360	no	—
157	C₂³×C₄	—	53	30	no	—
158	C₂³×C₄	C₂²×D₅₃	1	15	yes	—
159	C₂⁴	—	53	1	no	—
160	C₂⁴	—	53	60	no	—
161	C₂⁴	—	53	5	no	—
162	C₂⁴	—	106	40	no	—
163	C₂⁴	C₂³×D₅₃	1	5	yes	—
164	#41	—	53	640	no	—
165	#41	—	53	1920	no	—
166	#41	—	53	1920	no	—
167	#41	C₂³×D₅₃	1	640	yes	complemented
168	C₂×Q₈	—	53	640	no	—
169	C₂×Q₈	—	53	1280	no	—
170	C₂×Q₈	—	53	360	no	—
171	C₂×Q₈	—	53	960	no	—
172	C₂×Q₈	C₂³×D₅₃	1	120	yes	—
173	C₂×Q₈	C₂³×D₅₃	1	320	yes	complemented
174	C₂×D₄	—	53	240	no	—
175	C₂×D₄	—	53	640	no	—
176	C₂×D₄	—	53	120	no	—
177	C₂×D₄	C₂³×D₅₃	1	120	yes	—
178	C₂²×C₄	—	53	180	no	—
179	C₂²×C₄	—	53	10	no	—
180	C₂²×C₄	—	53	120	no	—
181	C₂²×C₄	—	53	360	no	—
182	C₂²×C₄	—	53	960	no	—
183	C₂²×C₄	C₂³×D₅₃	1	180	yes	—
184	C₂²×C₄	C₂³×D₅₃	1	10	yes	—
185	C₂³	—	2	20	no	—
186	C₂³	—	53	8	no	—
187	C₂³	—	53	6	no	—
188	C₂³	—	53	80	no	—
189	C₂³	—	53	30	no	—
190	C₂³	—	106	240	no	—
191	C₂³	—	106	20	no	—
192	C₂³	C₂⁴×D₅₃	1	30	yes	—
193	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
194	Q₈	—	53	480	no	—
195	Q₈	C₂⁴×D₅₃	1	160	yes	—
196	D₄	—	53	320	no	—
197	D₄	—	53	160	no	—
198	D₄	C₂⁴×D₅₃	1	160	yes	—
199	C₂×C₄	—	53	60	no	—
200	C₂×C₄	—	53	240	no	—
201	C₂×C₄	—	53	160	no	—
202	C₂×C₄	—	53	480	no	—
203	C₂×C₄	C₂⁴×D₅₃	1	60	yes	—
204	C₂×C₄	C₂⁴×D₅₃	1	240	yes	—
205	C₂²	—	2	60	no	—
206	C₂²	—	53	24	no	—
207	C₂²	—	53	4	no	—
208	C₂²	—	53	20	no	—
209	C₂²	—	106	160	no	—
210	C₂²	—	106	60	no	—
211	C₂²	2₋¹⁺⁴×D₅₃	1	4	yes	direct product factor
212	C₂²	C₂⁵×D₅₃	1	20	yes	—
213	C₂²	C₂⁵×D₅₃	1	3	yes	—
214	C₄	—	53	40	no	—
215	C₄	C₂⁵×D₅₃	1	40	yes	—
216	C₂	—	2	20	no	—
217	C₂	—	53	8	no	—
218	C₂	—	106	20	no	—
219	C₂	C₂×2₋¹⁺⁴×D₅₃	1	6	yes	direct product factor
220	C₂	C₂⁶×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator, cosocle	C₂×C₅₃	C₂⁷
Frattini	C₂	C₂⁶×D₅₃
Fitting	C₂²×2₋¹⁺⁴×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁵
hypercenter	C₂²×2₋¹⁺⁴	D₅₃
hypocenter	C₅₃	C₂³×2₋¹⁺⁴
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133265	C₂	semidirect
2. C₂²×#13133264	C₂	semidirect
3. C₂×2₋¹⁺⁴×D₅₃	C₂	direct (N × H)
4. C₂²×#41×D₅₃	C₂	semidirect
5. C₂³×Q₈×D₅₃	C₂	semidirect
6. C₂²×2₋¹⁺⁴×C₅₃	C₂	semidirect
7. C₂×#13133265	C₂²	semidirect
8. C₂×#13133264	C₂²	semidirect
9. 2₋¹⁺⁴×D₅₃	C₂²	direct (N × H)
10. C₂²×#11719949	C₂²	semidirect
11. C₂²×#11719945	C₂²	semidirect
12. C₂²×#11719947	C₂²	semidirect
13. C₂×#41×D₅₃	C₂²	semidirect
14. C₂³×Dic₁₀₆	C₂²	semidirect
16. C₂²×Q₈×D₅₃	C₂²	semidirect
17. C₂²×Q₈×D₅₃	C₂²	semidirect
20. C₂×2₋¹⁺⁴×C₅₃	C₂²	semidirect
21. C₂²×#41×C₅₃	C₂²	semidirect
22. C₂³×Q₈×C₅₃	C₂²	semidirect
24. #13133265	C₂³	semidirect
25. #13133264	C₂³	semidirect
26. C₂×#11719949	C₂³	semidirect
27. C₂×#11719945	C₂³	semidirect
28. C₂×#11719947	C₂³	semidirect
29. #41×D₅₃	C₂³	semidirect
31. C₂²×Dic₁₀₆	C₂³	semidirect
33. C₂×Q₈×D₅₃	C₂³	semidirect
34. C₂×Q₈×D₅₃	C₂³	semidirect
42. 2₋¹⁺⁴×C₅₃	C₂³	semidirect
43. C₂×#41×C₅₃	C₂³	semidirect
45. C₂²×Q₈×C₅₃	C₂³	semidirect
53. #41×C₅₃	C₂⁴	semidirect
55. C₂×Q₈×C₅₃	C₂⁴	semidirect
62. #11719949	C₂⁴	semidirect
64. Q₈×D₅₃	C₂⁴	semidirect
65. #11719947	C₂⁴	semidirect
67. #11719945	C₂⁴	semidirect
73. C₂×Dic₁₀₆	C₂⁴	semidirect
79. C₂²×D₅₃	2₋¹⁺⁴	direct (N × H)
96. C₂²×C₅₃	C₂×2₋¹⁺⁴	semidirect
101. C₂×D₅₃	C₂×2₋¹⁺⁴	direct (N × H)
109. C₂²×2₋¹⁺⁴	D₅₃	direct (N × H)
113. C₂×C₅₃	C₂²×2₋¹⁺⁴	semidirect
116. D₅₃	C₂²×2₋¹⁺⁴	direct (N × H)
120. C₂×2₋¹⁺⁴	C₂×D₅₃	direct (N × H)
125. C₂²×#41	C₂×D₅₃	semidirect
130. C₂³×Q₈	C₂×D₅₃	semidirect
133. C₅₃	C₂³×2₋¹⁺⁴	semidirect
137. 2₋¹⁺⁴	C₂²×D₅₃	direct (N × H)
142. C₂×#41	C₂²×D₅₃	semidirect
149. C₂²×Q₈	C₂²×D₅₃	semidirect
167. #41	C₂³×D₅₃	semidirect
173. C₂×Q₈	C₂³×D₅₃	semidirect
211. C₂²	2₋¹⁺⁴×D₅₃	direct (N × H)
219. C₂	C₂×2₋¹⁺⁴×D₅₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13133265	2
C₂²×#13133264	2
C₂×2₋¹⁺⁴×D₅₃	2
C₂²×#41×D₅₃	2
C₂³×Q₈×D₅₃	2
C₂²×2₋¹⁺⁴×C₅₃	2
C₂³×2₋¹⁺⁴	53

Subgroups — C22×2−1+4×D53 #28297980

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×2₋¹⁺⁴×D₅₃ #28297980