Subgroups — C₂²×#126×D₅₃ #28290761

Order 13568.

9486 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133212	C₂	1	3	yes	maximal, complemented
2	C₂²×#13133211	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#13133210	C₂	1	3	yes	maximal, complemented
4	C₂×#126×D₅₃	C₂	1	48	yes	maximal, direct product factor
5	C₂²×C₄⋊C₄×D₅₃	C₂	1	3	yes	maximal, complemented
6	C₂²×C₂²⋊C₄×D₅₃	C₂	1	3	yes	maximal
7	C₂²×C₄²×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#126×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×D₅₃	C₂²	1	3	yes	—
11	C₂³×C₄×D₅₃	C₂²	1	3	yes	—
12	C₂×#13133212	C₂²	1	72	yes	complemented
13	C₂×#13133211	C₂²	1	24	yes	complemented
14	C₂×#13133210	C₂²	1	72	yes	complemented
15	#126×D₅₃	C₂²	1	256	yes	direct product factor
16	C₂²×C₄⋊Dic₅₃	C₂²	1	3	yes	complemented
17	C₂²×#11719919	C₂²	1	3	yes	complemented
18	C₂²×#11719919	C₂²	1	3	yes	complemented
19	C₂×C₄⋊C₄×D₅₃	C₂²	1	72	yes	complemented
20	C₂²×C₂²⋊Dic₅₃	C₂²	1	3	yes	—
21	C₂²×#11719921	C₂²	1	3	yes	—
22	C₂²×#11719921	C₂²	1	3	yes	—
23	C₂×C₂²⋊C₄×D₅₃	C₂²	1	72	yes	—
24	C₂²×C₄×Dic₅₃	C₂²	1	3	yes	complemented
25	C₂×C₄²×D₅₃	C₂²	1	24	yes	complemented
26	C₂×#126×C₅₃	C₂²	1	24	yes	complemented
27	C₂²×C₄⋊C₄×C₅₃	C₂²	1	3	yes	complemented
28	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	3	yes	—
29	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
30	C₂⁴×D₅₃	—	2	24	no	—
31	C₂⁴×D₅₃	C₂³	1	12	yes	—
32	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
33	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
34	C₂³×Dic₅₃	C₂³	1	3	yes	—
35	C₂³×Dic₅₃	C₂³	1	3	yes	—
36	C₂²×C₄×D₅₃	—	2	24	no	—
37	C₂²×C₄×D₅₃	—	2	24	no	—
38	C₂²×C₄×D₅₃	C₂³	1	36	yes	—
39	C₂²×C₄×D₅₃	C₂³	1	36	yes	—
40	#13133212	C₂³	1	192	yes	complemented
41	#13133211	C₂³	1	64	yes	complemented
42	#13133210	C₂³	1	192	yes	complemented
43	C₂×C₄⋊Dic₅₃	C₂³	1	36	yes	complemented
44	C₂×#11719919	C₂³	1	36	yes	complemented
45	C₂×#11719919	C₂³	1	36	yes	complemented
46	C₄⋊C₄×D₅₃	C₂³	1	192	yes	complemented
47	C₂×C₂²⋊Dic₅₃	C₂³	1	36	yes	—
48	C₂×#11719921	C₂³	1	36	yes	—
49	C₂×#11719921	C₂³	1	36	yes	—
50	C₂²⋊C₄×D₅₃	C₂³	1	192	yes	—
51	C₂×C₄×Dic₅₃	C₂³	1	36	yes	complemented
52	C₄²×D₅₃	C₂³	1	64	yes	complemented
53	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
54	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
55	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
56	#126×C₅₃	C₂³	1	64	yes	complemented
57	C₂×C₄⋊C₄×C₅₃	C₂³	1	36	yes	complemented
58	C₂×C₂²⋊C₄×C₅₃	C₂³	1	36	yes	—
59	C₂×C₄²×C₅₃	C₂³	1	12	yes	complemented
60	C₂⁴×C₅₃	—	2	12	no	—
61	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
62	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
63	C₂³×D₅₃	—	2	12	no	—
64	C₂³×D₅₃	—	2	144	no	—
65	C₂³×D₅₃	—	4	64	no	—
66	C₂³×D₅₃	#41	1	24	yes	—
67	C₂³×D₅₃	C₂⁴	1	16	yes	—
68	C₂³×D₅₃	C₂⁴	1	6	yes	—
69	C₂³×D₅₃	C₂⁴	1	6	yes	—
70	C₂²×C₄×C₅₃	—	2	12	no	—
71	C₂²×C₄×C₅₃	—	2	12	no	—
72	C₂²×C₄×C₅₃	C₂⁴	1	18	yes	—
73	C₂²×C₄×C₅₃	C₂⁴	1	18	yes	—
74	C₂²×Dic₅₃	—	2	12	no	—
75	C₂²×Dic₅₃	—	2	12	no	—
76	C₂²×Dic₅₃	C₂⁴	1	18	yes	—
77	C₂²×Dic₅₃	C₂⁴	1	18	yes	—
78	C₂×C₄×D₅₃	—	2	144	no	—
79	C₂×C₄×D₅₃	—	2	144	no	—
80	C₂×C₄×D₅₃	C₂⁴	1	48	yes	—
81	C₂×C₄×D₅₃	C₂⁴	1	48	yes	—
82	C₄⋊C₄×C₅₃	C₂⁴	1	48	yes	complemented
83	C₂²⋊C₄×C₅₃	C₂⁴	1	48	yes	—
84	C₄²×C₅₃	C₂⁴	1	16	yes	complemented
85	C₂²⋊Dic₅₃	C₂⁴	1	48	yes	—
86	#11719921	C₂⁴	1	48	yes	—
87	#11719921	C₂⁴	1	48	yes	—
88	C₄⋊Dic₅₃	C₂⁴	1	48	yes	complemented
89	#11719919	C₂⁴	1	48	yes	complemented
90	#11719919	C₂⁴	1	48	yes	complemented
91	C₄×Dic₅₃	C₂⁴	1	48	yes	complemented
92	C₂³×C₅₃	—	2	36	no	—
93	C₂³×C₅₃	—	4	16	no	—
94	C₂³×C₅₃	C₂×#41	1	12	yes	—
95	C₂³×C₅₃	C₂⁵	1	4	yes	—
96	C₂³×C₅₃	C₂⁵	1	3	yes	—
97	C₂²×D₅₃	—	2	36	no	—
98	C₂²×D₅₃	—	2	96	no	—
99	C₂²×D₅₃	—	4	16	no	—
100	C₂²×D₅₃	—	4	192	no	—
101	C₂²×D₅₃	#126	1	64	yes	direct product factor
102	C₂²×D₅₃	C₂×#41	1	72	yes	—
103	C₂²×D₅₃	C₂⁵	1	4	yes	—
104	C₂²×D₅₃	C₂⁵	1	4	yes	—
105	C₂×C₄×C₅₃	—	2	36	no	—
106	C₂×C₄×C₅₃	—	2	36	no	—
107	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
108	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
109	C₂×Dic₅₃	—	2	36	no	—
110	C₂×Dic₅₃	—	2	36	no	—
111	C₂×Dic₅₃	C₂⁵	1	12	yes	—
112	C₂×Dic₅₃	C₂⁵	1	12	yes	—
113	C₄×D₅₃	—	2	96	no	—
114	C₄×D₅₃	—	2	96	no	—
115	C₂³×#126	—	53	1	no	maximal, Sylow
116	C₂²×C₅₃	—	2	12	no	—
117	C₂²×C₅₃	—	4	24	no	—
118	C₂²×C₅₃	C₂×#126	1	16	yes	complemented
119	C₂²×C₅₃	C₂²×#41	1	18	yes	—
120	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
121	C₂×D₅₃	—	2	12	no	—
122	C₂×D₅₃	—	4	24	no	—
123	C₂×D₅₃	—	4	64	no	—
124	C₂×D₅₃	C₂×#126	1	96	yes	direct product factor
125	C₂×D₅₃	C₂²×#41	1	24	yes	—
126	C₄×C₅₃	—	2	12	no	—
127	C₄×C₅₃	—	2	12	no	—
128	Dic₅₃	—	2	12	no	—
129	Dic₅₃	—	2	12	no	—
130	C₂²×#126	—	53	1	no	—
131	C₂²×#126	—	53	48	no	—
132	C₂²×#126	—	53	3	no	—
133	C₂²×#126	—	53	3	no	—
134	C₂²×#126	D₅₃	1	1	yes	characteristic, direct product factor
135	C₂³×C₄⋊C₄	—	53	3	no	—
136	C₂³×C₂²⋊C₄	—	53	3	no	—
137	C₂³×C₄²	—	53	1	no	—
138	C₂×C₅₃	—	4	4	no	—
139	C₂×C₅₃	C₂²×#126	1	12	yes	complemented
140	C₂×C₅₃	C₂³×#41	1	3	yes	—
141	D₅₃	—	4	4	no	—
142	D₅₃	C₂²×#126	1	16	yes	direct product factor
143	C₂⁶	—	53	1	no	—
144	C₂⁴×C₄	—	53	3	no	—
145	C₂⁴×C₄	—	53	3	no	—
146	C₂×#126	—	53	256	no	—
147	C₂×#126	—	53	72	no	—
148	C₂×#126	—	53	72	no	—
149	C₂×#126	—	53	24	no	—
150	C₂×#126	C₂×D₅₃	1	24	yes	direct product factor
151	C₂²×C₄⋊C₄	—	53	3	no	—
152	C₂²×C₄⋊C₄	—	53	3	no	—
153	C₂²×C₄⋊C₄	—	53	3	no	—
154	C₂²×C₄⋊C₄	—	53	72	no	—
155	C₂²×C₄⋊C₄	C₂×D₅₃	1	3	yes	complemented
156	C₂²×C₂²⋊C₄	—	53	3	no	—
157	C₂²×C₂²⋊C₄	—	53	3	no	—
158	C₂²×C₂²⋊C₄	—	53	3	no	—
159	C₂²×C₂²⋊C₄	—	53	72	no	—
160	C₂²×C₂²⋊C₄	C₂×D₅₃	1	3	yes	—
161	C₂²×C₄²	—	53	3	no	—
162	C₂²×C₄²	—	53	24	no	—
163	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic, complemented
164	C₅₃	C₂³×#126	1	1	yes	characteristic, complemented, Sylow
165	C₂⁵	—	53	1	no	—
166	C₂⁵	—	53	12	no	—
167	C₂⁵	—	53	1	no	—
168	C₂⁵	—	106	24	no	—
169	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
170	C₂³×C₄	—	53	3	no	—
171	C₂³×C₄	—	53	3	no	—
172	C₂³×C₄	—	53	36	no	—
173	C₂³×C₄	—	53	36	no	—
174	C₂³×C₄	—	106	24	no	—
175	C₂³×C₄	—	106	24	no	—
176	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
177	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
178	#126	—	53	192	no	—
179	#126	—	53	192	no	—
180	#126	—	53	64	no	—
181	#126	C₂²×D₅₃	1	64	yes	direct product factor
182	C₂×C₄⋊C₄	—	53	36	no	—
183	C₂×C₄⋊C₄	—	53	36	no	—
184	C₂×C₄⋊C₄	—	53	36	no	—
185	C₂×C₄⋊C₄	—	53	192	no	—
186	C₂×C₄⋊C₄	C₂²×D₅₃	1	36	yes	complemented
187	C₂×C₂²⋊C₄	—	53	36	no	—
188	C₂×C₂²⋊C₄	—	53	36	no	—
189	C₂×C₂²⋊C₄	—	53	36	no	—
190	C₂×C₂²⋊C₄	—	53	192	no	—
191	C₂×C₂²⋊C₄	C₂²×D₅₃	1	36	yes	—
192	C₂×C₄²	—	53	36	no	—
193	C₂×C₄²	—	53	64	no	—
194	C₂×C₄²	C₂²×D₅₃	1	12	yes	complemented
195	C₂⁴	—	2	12	no	—
196	C₂⁴	—	53	24	no	—
197	C₂⁴	—	53	6	no	—
198	C₂⁴	—	53	16	no	—
199	C₂⁴	—	53	6	no	—
200	C₂⁴	—	106	144	no	—
201	C₂⁴	—	106	12	no	—
202	C₂⁴	—	212	64	no	—
203	C₂⁴	C₂³×D₅₃	1	6	yes	—
204	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
205	C₂²×C₄	—	2	12	no	—
206	C₂²×C₄	—	2	12	no	—
207	C₂²×C₄	—	53	18	no	—
208	C₂²×C₄	—	53	18	no	—
209	C₂²×C₄	—	53	48	no	—
210	C₂²×C₄	—	53	48	no	—
211	C₂²×C₄	—	106	144	no	—
212	C₂²×C₄	—	106	144	no	—
213	C₂²×C₄	—	106	12	no	—
214	C₂²×C₄	—	106	12	no	—
215	C₂²×C₄	C₂³×D₅₃	1	18	yes	—
216	C₂²×C₄	C₂³×D₅₃	1	18	yes	—
217	C₄⋊C₄	—	53	48	no	—
218	C₄⋊C₄	—	53	48	no	—
219	C₄⋊C₄	—	53	48	no	—
220	C₄⋊C₄	C₂³×D₅₃	1	48	yes	complemented
221	C₂²⋊C₄	—	53	48	no	—
222	C₂²⋊C₄	—	53	48	no	—
223	C₂²⋊C₄	—	53	48	no	—
224	C₂²⋊C₄	C₂³×D₅₃	1	48	yes	—
225	C₄²	—	53	48	no	—
226	C₄²	C₂³×D₅₃	1	16	yes	complemented
227	C₂³	—	2	36	no	—
228	C₂³	—	4	16	no	—
229	C₂³	—	53	64	no	—
230	C₂³	—	53	72	no	—
231	C₂³	—	53	4	no	—
232	C₂³	—	53	4	no	—
233	C₂³	—	106	96	no	—
234	C₂³	—	106	36	no	—
235	C₂³	—	212	192	no	—
236	C₂³	—	212	16	no	—
237	C₂³	#41×D₅₃	1	12	yes	—
238	C₂³	C₂⁴×D₅₃	1	4	yes	—
239	C₂³	C₂⁴×D₅₃	1	3	yes	—
240	C₂×C₄	—	2	36	no	—
241	C₂×C₄	—	2	36	no	—
242	C₂×C₄	—	53	12	no	—
243	C₂×C₄	—	53	12	no	—
244	C₂×C₄	—	106	96	no	—
245	C₂×C₄	—	106	96	no	—
246	C₂×C₄	—	106	36	no	—
247	C₂×C₄	—	106	36	no	—
248	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
249	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
250	C₂²	—	2	12	no	—
251	C₂²	—	4	24	no	—
252	C₂²	—	53	96	no	—
253	C₂²	—	53	24	no	—
254	C₂²	—	106	12	no	—
255	C₂²	—	212	64	no	—
256	C₂²	—	212	24	no	—
257	C₂²	#126×D₅₃	1	16	yes	direct product factor
258	C₂²	C₂×#41×D₅₃	1	18	yes	—
259	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
260	C₄	—	2	12	no	—
261	C₄	—	2	12	no	—
262	C₄	—	106	12	no	—
263	C₄	—	106	12	no	—
264	C₂	—	4	4	no	—
265	C₂	—	53	16	no	—
266	C₂	—	212	4	no	—
267	C₂	C₂×#126×D₅₃	1	12	yes	direct product factor
268	C₂	C₂²×#41×D₅₃	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×#126×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂⁵×D₅₃	C₂²
hypercenter	C₂²×#126	D₅₃
hypocenter	C₅₃	C₂³×#126
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#126

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133212	C₂	semidirect
2. C₂²×#13133211	C₂	semidirect
3. C₂²×#13133210	C₂	semidirect
4. C₂×#126×D₅₃	C₂	direct (N × H)
5. C₂²×C₄⋊C₄×D₅₃	C₂	semidirect
7. C₂²×C₄²×D₅₃	C₂	semidirect
8. C₂²×#126×C₅₃	C₂	semidirect
12. C₂×#13133212	C₂²	semidirect
13. C₂×#13133211	C₂²	semidirect
14. C₂×#13133210	C₂²	semidirect
15. #126×D₅₃	C₂²	direct (N × H)
16. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
17. C₂²×#11719919	C₂²	semidirect
18. C₂²×#11719919	C₂²	semidirect
19. C₂×C₄⋊C₄×D₅₃	C₂²	semidirect
24. C₂²×C₄×Dic₅₃	C₂²	semidirect
25. C₂×C₄²×D₅₃	C₂²	semidirect
26. C₂×#126×C₅₃	C₂²	semidirect
27. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
29. C₂²×C₄²×C₅₃	C₂²	semidirect
40. #13133212	C₂³	semidirect
41. #13133211	C₂³	semidirect
42. #13133210	C₂³	semidirect
43. C₂×C₄⋊Dic₅₃	C₂³	semidirect
44. C₂×#11719919	C₂³	semidirect
45. C₂×#11719919	C₂³	semidirect
46. C₄⋊C₄×D₅₃	C₂³	semidirect
51. C₂×C₄×Dic₅₃	C₂³	semidirect
52. C₄²×D₅₃	C₂³	semidirect
56. #126×C₅₃	C₂³	semidirect
57. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
59. C₂×C₄²×C₅₃	C₂³	semidirect
82. C₄⋊C₄×C₅₃	C₂⁴	semidirect
84. C₄²×C₅₃	C₂⁴	semidirect
88. C₄⋊Dic₅₃	C₂⁴	semidirect
89. #11719919	C₂⁴	semidirect
90. #11719919	C₂⁴	semidirect
91. C₄×Dic₅₃	C₂⁴	semidirect
101. C₂²×D₅₃	#126	direct (N × H)
118. C₂²×C₅₃	C₂×#126	semidirect
124. C₂×D₅₃	C₂×#126	direct (N × H)
134. C₂²×#126	D₅₃	direct (N × H)
139. C₂×C₅₃	C₂²×#126	semidirect
142. D₅₃	C₂²×#126	direct (N × H)
150. C₂×#126	C₂×D₅₃	direct (N × H)
155. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
163. C₂²×C₄²	C₂×D₅₃	semidirect
164. C₅₃	C₂³×#126	semidirect
181. #126	C₂²×D₅₃	direct (N × H)
186. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
194. C₂×C₄²	C₂²×D₅₃	semidirect
220. C₄⋊C₄	C₂³×D₅₃	semidirect
226. C₄²	C₂³×D₅₃	semidirect
257. C₂²	#126×D₅₃	direct (N × H)
267. C₂	C₂×#126×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×#13133212	2
C₂²×#13133211	2
C₂²×#13133210	2
C₂×#126×D₅₃	2
C₂²×C₄⋊C₄×D₅₃	2
C₂²×C₂²⋊C₄×D₅₃	2
C₂²×C₄²×D₅₃	2
C₂²×#126×C₅₃	2
C₂³×#126	53

Subgroups — C22×#126×D53 #28290761

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#126×D₅₃ #28290761