Subgroups — D₅₃×#2273 #27542295

Order 13568.

5790 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊C₄×D₅₃	C₂	1	3	yes	maximal
2	#20277068	C₂	1	3	yes	maximal, complemented
3	#20277067	C₂	1	12	yes	maximal, complemented
4	C₄×C₂²⋊C₄×D₅₃	C₂	1	12	yes	maximal, complemented
5	C₅₃×#2273	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
7	C₂³×C₄×D₅₃	C₂²	1	6	yes	—
8	C₂²×C₂²⋊Dic₅₃	C₂²	1	3	yes	—
9	C₂²×#11719921	C₂²	1	6	yes	—
10	C₂×C₂²⋊C₄×D₅₃	C₄	1	48	yes	complemented
11	C₂×C₂²⋊C₄×D₅₃	C₂²	1	12	yes	—
12	C₂×C₂²⋊C₄×D₅₃	C₂²	1	12	yes	—
13	C₂×C₄²×D₅₃	C₂²	1	8	yes	complemented
14	C₂²⋊C₄×Dic₅₃	C₂²	1	24	yes	complemented
15	C₄×#11719921	C₂²	1	24	yes	complemented
16	C₄×C₂²⋊Dic₅₃	C₂²	1	12	yes	complemented
17	#16810470	C₂²	1	24	yes	complemented
18	#95×D₅₃	C₂²	1	8	yes	complemented
19	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	3	yes	—
20	C₄×C₂²⋊C₄×C₅₃	C₂²	1	12	yes	complemented
21	C₂⁴×D₅₃	—	4	8	no	—
22	C₂⁴×D₅₃	C₂×C₄	1	24	yes	—
23	C₂⁴×D₅₃	C₂³	1	3	yes	—
24	C₂⁴×D₅₃	C₂³	1	3	yes	—
25	C₂³×Dic₅₃	C₂³	1	6	yes	—
26	C₂²×C₄×D₅₃	—	4	24	no	—
27	C₂²×C₄×D₅₃	C₂×C₄	1	48	yes	complemented
28	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
29	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
30	C₂×C₂²⋊Dic₅₃	C₂×C₄	1	24	yes	complemented
31	C₂×C₂²⋊Dic₅₃	C₂³	1	12	yes	—
32	C₂×#11719921	C₂×C₄	1	48	yes	complemented
33	C₂×#11719921	C₂³	1	24	yes	—
34	C₂²⋊C₄×D₅₃	C₂×C₄	1	96	yes	complemented
35	C₂²⋊C₄×D₅₃	C₂×C₄	1	96	yes	complemented
36	C₂×C₄×Dic₅₃	C₂³	1	24	yes	complemented
37	C₄²×D₅₃	—	4	16	no	—
38	#13133124	C₂³	1	24	yes	complemented
39	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
41	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	24	yes	complemented
42	C₂×C₂²⋊C₄×C₅₃	C₂³	1	12	yes	—
43	C₂×C₄²×C₅₃	C₂³	1	8	yes	complemented
44	#95×C₅₃	C₂³	1	8	yes	complemented
45	C₂⁴×C₅₃	—	4	4	no	—
46	C₂⁴×C₅₃	C₂²×C₄	1	12	yes	—
47	C₂⁴×C₅₃	C₂⁴	1	3	yes	—
48	C₂³×D₅₃	—	2	24	no	—
49	C₂³×D₅₃	—	4	96	no	—
50	C₂³×D₅₃	—	4	12	no	—
51	C₂³×D₅₃	C₄²	1	64	yes	complemented
52	C₂³×D₅₃	C₂²×C₄	1	24	yes	—
53	C₂³×D₅₃	C₂²×C₄	1	12	yes	—
54	C₂³×D₅₃	C₂²×C₄	1	24	yes	—
55	C₂³×D₅₃	C₂²×C₄	1	12	yes	—
56	C₂³×D₅₃	C₂⁴	1	3	yes	—
57	C₂³×D₅₃	C₂⁴	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	—	4	12	no	—
59	C₂²×C₄×C₅₃	C₂²×C₄	1	24	yes	complemented
60	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
61	C₂²×Dic₅₃	—	4	12	no	—
62	C₂²×Dic₅₃	C₂²×C₄	1	24	yes	complemented
63	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
64	C₂×C₄×D₅₃	—	2	48	no	—
65	C₂×C₄×D₅₃	—	4	96	no	—
66	C₂×C₄×D₅₃	—	4	24	no	—
67	C₂×C₄×D₅₃	C₂²×C₄	1	48	yes	—
68	C₂×C₄×D₅₃	C₂²×C₄	1	48	yes	complemented
69	C₂²⋊C₄×C₅₃	C₂²×C₄	1	48	yes	complemented
70	C₄²×C₅₃	—	4	8	no	—
71	C₂²⋊Dic₅₃	C₂²×C₄	1	48	yes	complemented
72	#11719921	C₂²×C₄	1	96	yes	complemented
73	C₄×Dic₅₃	—	4	24	no	—
74	C₂³×C₅₃	—	2	12	no	—
75	C₂³×C₅₃	—	4	24	no	—
76	C₂³×C₅₃	C₂×C₄²	1	16	yes	complemented
77	C₂³×C₅₃	C₂³×C₄	1	6	yes	—
78	C₂³×C₅₃	C₂³×C₄	1	12	yes	—
79	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
80	C₂²×D₅₃	—	2	12	no	—
81	C₂²×D₅₃	—	2	48	no	—
82	C₂²×D₅₃	—	2	96	no	—
83	C₂²×D₅₃	—	4	128	no	—
84	C₂²×D₅₃	—	4	48	no	—
85	C₂²×D₅₃	—	4	24	no	—
86	C₂²×D₅₃	C₂×C₄²	1	48	yes	—
87	C₂²×D₅₃	C₂×C₄²	1	48	yes	complemented
88	C₂²×D₅₃	C₂³×C₄	1	6	yes	—
89	C₂²×D₅₃	C₂³×C₄	1	12	yes	—
90	C₂²×D₅₃	C₂³×C₄	1	6	yes	—
91	C₂²×D₅₃	H(5,𝔽₂)	1	8	yes	—
92	C₂×C₄×C₅₃	—	2	24	no	—
93	C₂×C₄×C₅₃	—	4	24	no	—
94	C₂×C₄×C₅₃	C₂³×C₄	1	24	yes	—
95	C₂×Dic₅₃	—	2	24	no	—
96	C₂×Dic₅₃	—	4	24	no	—
97	C₂×Dic₅₃	C₂³×C₄	1	24	yes	—
98	C₄×D₅₃	—	2	96	no	—
99	C₄×D₅₃	—	4	48	no	—
100	C₂×#2273	—	53	1	no	maximal, Sylow
101	C₂²×C₅₃	—	2	12	no	—
102	C₂²×C₅₃	—	2	24	no	—
103	C₂²×C₅₃	—	4	16	no	—
104	C₂²×C₅₃	C₂²×C₄²	1	12	yes	—
105	C₂²×C₅₃	C₂⁴×C₄	1	3	yes	—
106	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
107	C₂×D₅₃	—	2	12	no	—
108	C₂×D₅₃	—	2	24	no	—
109	C₂×D₅₃	—	2	96	no	—
110	C₂×D₅₃	—	4	48	no	—
111	C₂×D₅₃	C₂²×C₄²	1	12	yes	—
112	C₂×D₅₃	C₂²×C₄²	1	4	yes	—
113	C₂×D₅₃	#518	1	24	yes	—
114	C₄×C₅₃	—	2	24	no	—
115	Dic₅₃	—	2	24	no	—
116	C₂³×C₂²⋊C₄	—	53	3	no	—
117	#2273	—	53	12	no	—
118	#2273	—	53	3	no	—
119	#2273	D₅₃	1	1	yes	characteristic, direct product factor
120	C₂×C₄×C₂²⋊C₄	—	53	12	no	—
121	C₂×C₅₃	—	2	12	no	—
122	C₂×C₅₃	C₂³×C₄²	1	1	yes	characteristic
123	C₂×C₅₃	C₂×#518	1	6	yes	—
124	D₅₃	—	2	12	no	—
125	D₅₃	#2273	1	8	yes	direct product factor
126	C₂⁶	—	53	1	no	—
127	C₂⁴×C₄	—	53	6	no	—
128	C₂²×C₂²⋊C₄	—	53	3	no	—
129	C₂²×C₂²⋊C₄	—	53	6	no	—
130	C₂²×C₂²⋊C₄	—	53	48	no	—
131	C₂²×C₂²⋊C₄	—	53	12	no	—
132	C₂²×C₂²⋊C₄	—	53	12	no	—
133	C₂²×C₂²⋊C₄	C₂×D₅₃	1	3	yes	—
134	C₂²×C₄²	—	53	8	no	—
135	C₄×C₂²⋊C₄	—	53	12	no	—
136	C₄×C₂²⋊C₄	—	53	24	no	—
137	C₄×C₂²⋊C₄	—	53	24	no	—
138	C₄×C₂²⋊C₄	—	53	24	no	—
139	C₄×C₂²⋊C₄	C₂×D₅₃	1	12	yes	complemented
140	C₂×#95	—	53	8	no	—
141	C₅₃	C₂×#2273	1	1	yes	characteristic, complemented, Sylow
142	C₂⁵	—	53	3	no	—
143	C₂⁵	—	53	24	no	—
144	C₂⁵	—	53	3	no	—
145	C₂⁵	—	212	8	no	—
146	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
147	C₂³×C₄	—	53	6	no	—
148	C₂³×C₄	—	53	12	no	—
149	C₂³×C₄	—	53	48	no	—
150	C₂³×C₄	—	53	12	no	—
151	C₂³×C₄	—	212	24	no	—
152	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
153	C₂×C₂²⋊C₄	—	53	24	no	—
154	C₂×C₂²⋊C₄	—	53	12	no	—
155	C₂×C₂²⋊C₄	—	53	48	no	—
156	C₂×C₂²⋊C₄	—	53	24	no	—
157	C₂×C₂²⋊C₄	—	53	96	no	—
158	C₂×C₂²⋊C₄	—	53	96	no	—
159	C₂×C₂²⋊C₄	C₄×D₅₃	1	24	yes	complemented
160	C₂×C₂²⋊C₄	C₂²×D₅₃	1	12	yes	—
161	C₂×C₄²	—	53	24	no	—
162	C₂×C₄²	—	212	16	no	—
163	C₂×C₄²	C₂²×D₅₃	1	8	yes	complemented
164	#95	—	53	24	no	—
165	#95	C₂²×D₅₃	1	8	yes	complemented
166	C₂⁴	—	4	4	no	—
167	C₂⁴	—	53	1	no	—
168	C₂⁴	—	53	24	no	—
169	C₂⁴	—	53	12	no	—
170	C₂⁴	—	53	3	no	—
171	C₂⁴	—	53	64	no	—
172	C₂⁴	—	53	12	no	—
173	C₂⁴	—	53	24	no	—
174	C₂⁴	—	106	24	no	—
175	C₂⁴	—	212	12	no	—
176	C₂⁴	—	212	96	no	—
177	C₂⁴	C₂×C₄×D₅₃	1	12	yes	—
178	C₂⁴	C₂³×D₅₃	1	3	yes	—
179	C₂²×C₄	—	4	12	no	—
180	C₂²×C₄	—	53	12	no	—
181	C₂²×C₄	—	53	24	no	—
182	C₂²×C₄	—	53	48	no	—
183	C₂²×C₄	—	53	48	no	—
184	C₂²×C₄	—	106	48	no	—
185	C₂²×C₄	—	212	24	no	—
186	C₂²×C₄	—	212	96	no	—
187	C₂²×C₄	—	212	12	no	—
188	C₂²×C₄	C₂×C₄×D₅₃	1	24	yes	complemented
189	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
190	C₂²⋊C₄	—	53	48	no	—
191	C₂²⋊C₄	—	53	96	no	—
192	C₂²⋊C₄	C₂×C₄×D₅₃	1	48	yes	complemented
193	C₄²	—	4	8	no	—
194	C₄²	—	212	24	no	—
195	C₂³	—	2	12	no	—
196	C₂³	—	4	24	no	—
197	C₂³	—	53	6	no	—
198	C₂³	—	53	8	no	—
199	C₂³	—	53	48	no	—
200	C₂³	—	53	12	no	—
201	C₂³	—	53	6	no	—
202	C₂³	—	53	48	no	—
203	C₂³	—	106	48	no	—
204	C₂³	—	106	96	no	—
205	C₂³	—	106	12	no	—
206	C₂³	—	212	24	no	—
207	C₂³	—	212	48	no	—
208	C₂³	—	212	128	no	—
209	C₂³	C₄²×D₅₃	1	16	yes	complemented
210	C₂³	C₂²×C₄×D₅₃	1	12	yes	—
211	C₂³	C₂²×C₄×D₅₃	1	6	yes	—
212	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
213	C₂×C₄	—	2	24	no	—
214	C₂×C₄	—	4	24	no	—
215	C₂×C₄	—	53	24	no	—
216	C₂×C₄	—	106	96	no	—
217	C₂×C₄	—	106	24	no	—
218	C₂×C₄	—	212	48	no	—
219	C₂×C₄	—	212	24	no	—
220	C₂×C₄	C₂²×C₄×D₅₃	1	24	yes	—
221	C₂²	—	2	12	no	—
222	C₂²	—	2	24	no	—
223	C₂²	—	4	16	no	—
224	C₂²	—	53	4	no	—
225	C₂²	—	53	24	no	—
226	C₂²	—	53	12	no	—
227	C₂²	—	106	96	no	—
228	C₂²	—	106	12	no	—
229	C₂²	—	106	24	no	—
230	C₂²	—	212	48	no	—
231	C₂²	C₂×C₄²×D₅₃	1	12	yes	—
232	C₂²	C₂³×C₄×D₅₃	1	3	yes	—
233	C₂²	H(5,𝔽₂)×D₅₃	1	4	yes	—
234	C₄	—	2	24	no	—
235	C₄	—	106	24	no	—
236	C₂	—	2	12	no	—
237	C₂	—	53	8	no	—
238	C₂	—	106	12	no	—
239	C₂	C₂²×C₄²×D₅₃	1	1	yes	characteristic
240	C₂	#518×D₅₃	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator	C₂×C₅₃	C₂³×C₄²
Fitting	C₅₃×#2273	C₂
socle, cosocle	C₂³×C₅₃	C₂⁵
prime radical	C₂⁵×D₅₃	C₂²
hypercenter	#2273	D₅₃
hypocenter	C₅₃	C₂×#2273
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #2273

Complements

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

N	H ≅ G/N	type
2. #20277068	C₂	semidirect
3. #20277067	C₂	semidirect
4. C₄×C₂²⋊C₄×D₅₃	C₂	semidirect
5. C₅₃×#2273	C₂	semidirect
10. C₂×C₂²⋊C₄×D₅₃	C₄	semidirect
13. C₂×C₄²×D₅₃	C₂²	semidirect
14. C₂²⋊C₄×Dic₅₃	C₂²	semidirect
15. C₄×#11719921	C₂²	semidirect
16. C₄×C₂²⋊Dic₅₃	C₂²	semidirect
17. #16810470	C₂²	semidirect
18. #95×D₅₃	C₂²	semidirect
20. C₄×C₂²⋊C₄×C₅₃	C₂²	semidirect
27. C₂²×C₄×D₅₃	C₂×C₄	semidirect
30. C₂×C₂²⋊Dic₅₃	C₂×C₄	semidirect
32. C₂×#11719921	C₂×C₄	semidirect
34. C₂²⋊C₄×D₅₃	C₂×C₄	semidirect
35. C₂²⋊C₄×D₅₃	C₂×C₄	semidirect
36. C₂×C₄×Dic₅₃	C₂³	semidirect
38. #13133124	C₂³	semidirect
41. C₂×C₂²⋊C₄×C₅₃	C₂×C₄	semidirect
43. C₂×C₄²×C₅₃	C₂³	semidirect
44. #95×C₅₃	C₂³	semidirect
51. C₂³×D₅₃	C₄²	semidirect
59. C₂²×C₄×C₅₃	C₂²×C₄	semidirect
62. C₂²×Dic₅₃	C₂²×C₄	semidirect
68. C₂×C₄×D₅₃	C₂²×C₄	semidirect
69. C₂²⋊C₄×C₅₃	C₂²×C₄	semidirect
71. C₂²⋊Dic₅₃	C₂²×C₄	semidirect
72. #11719921	C₂²×C₄	semidirect
76. C₂³×C₅₃	C₂×C₄²	semidirect
87. C₂²×D₅₃	C₂×C₄²	semidirect
119. #2273	D₅₃	direct (N × H)
125. D₅₃	#2273	direct (N × H)
139. C₄×C₂²⋊C₄	C₂×D₅₃	semidirect
141. C₅₃	C₂×#2273	semidirect
159. C₂×C₂²⋊C₄	C₄×D₅₃	semidirect
163. C₂×C₄²	C₂²×D₅₃	semidirect
165. #95	C₂²×D₅₃	semidirect
188. C₂²×C₄	C₂×C₄×D₅₃	semidirect
192. C₂²⋊C₄	C₂×C₄×D₅₃	semidirect
209. C₂³	C₄²×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂²⋊C₄×D₅₃	2
#20277068	2
#20277067	2
C₄×C₂²⋊C₄×D₅₃	2
C₅₃×#2273	2
C₂×#2273	53

Subgroups — D53×#2273 #27542295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#2273 #27542295