Subgroups — #28292974

Order 13568.

8650 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289048	C₂	1	24	yes	maximal, complemented
2	#20288589	C₂	1	12	yes	maximal, complemented
3	#20288491	C₂	1	12	yes	maximal, complemented
4	#20288350	C₂	1	12	yes	maximal, complemented
5	#20288011	C₂	1	2	yes	maximal, complemented
6	C₂²×#127×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13133262	C₂²	1	12	yes	complemented
8	#16811360	C₂²	1	32	yes	complemented
9	#16811359	C₂²	1	24	yes	complemented
10	#16811293	C₂²	1	48	yes	complemented
11	#16811267	C₂²	1	96	yes	complemented
12	#16811258	C₂²	1	24	yes	complemented
13	D₄×C₂²⋊D₅₃	C₂²	1	96	yes	complemented
14	#16811227	C₂²	1	48	yes	complemented
15	#16811190	C₂²	1	24	yes	complemented
16	#16811186	C₂²	1	48	yes	complemented
17	#16811182	C₂²	1	48	yes	complemented
18	#16811180	C₂²	1	6	yes	complemented
19	#16811173	C₂²	1	6	yes	complemented
20	#16811171	C₂²	1	24	yes	complemented
21	#127×D₅₃	C₂²	1	8	yes	complemented
22	#16811157	C₂²	1	24	yes	complemented
23	#16811134	C₂²	1	12	yes	complemented
24	C₂×#13133186	C₂²	1	16	yes	complemented
25	#16811098	C₂²	1	12	yes	complemented
26	C₄×#11719945	C₂²	1	12	yes	complemented
27	C₂³×D₄×C₅₃	C₂²	1	6	yes	complemented
28	C₂×#127×C₅₃	C₂²	1	12	yes	complemented
29	C₂×#127×C₅₃	C₂²	1	12	yes	complemented
30	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
31	#13133262	—	4	48	no	—
32	C₂×#11719945	C₂³	1	12	yes	—
33	C₂×#11719947	C₂³	1	48	yes	complemented
34	C₂²×C₂²⋊D₅₃	C₂³	1	48	yes	—
35	C₂×D₄×D₅₃	C₂³	1	48	yes	complemented
36	#13133217	C₂³	1	6	yes	complemented
37	#13133214	C₂³	1	2	yes	complemented
38	#13133213	C₂³	1	48	yes	complemented
39	#13133211	C₂³	1	16	yes	complemented
40	#13133209	C₂³	1	12	yes	complemented
41	#13133205	C₂³	1	12	yes	complemented
42	#13133204	C₂³	1	48	yes	complemented
43	#13133200	C₂³	1	24	yes	—
44	#13133198	C₂³	1	96	yes	complemented
45	#13133196	C₂³	1	24	yes	—
46	#13133191	C₂³	1	24	yes	—
47	#13133189	C₂³	1	192	yes	complemented
48	#13133188	C₂³	1	96	yes	complemented
49	#13133186	C₂³	1	96	yes	—
50	#13133186	C₂³	1	8	yes	—
51	#13133185	C₂³	1	96	yes	complemented
52	C₄×Dic₁₀₆	C₂³	1	24	yes	complemented
53	C₄×D₂₁₂	C₂³	1	24	yes	complemented
54	D₄×Dic₅₃	C₂³	1	96	yes	complemented
55	C₄×C₂²⋊D₅₃	C₂³	1	48	yes	—
56	#13133176	C₂³	1	12	yes	—
57	#13133174	C₂³	1	24	yes	complemented
58	C₂×C₂²⋊Dic₅₃	C₂³	1	48	yes	—
59	C₄²×D₅₃	C₂³	1	8	yes	complemented
60	C₂⁵×C₅₃	C₂³	1	4	yes	—
61	C₂²×D₄×C₅₃	—	4	24	no	—
62	C₂²×D₄×C₅₃	C₂³	1	12	yes	—
63	C₂²×D₄×C₅₃	C₂³	1	48	yes	—
64	C₂²×D₄×C₅₃	C₂³	1	12	yes	—
65	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
66	#127×C₅₃	C₂³	1	24	yes	—
67	#127×C₅₃	C₂³	1	8	yes	—
68	#127×C₅₃	C₂³	1	8	yes	—
69	#127×C₅₃	C₂³	1	24	yes	—
70	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
71	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
72	C₂⁴×C₅₃	—	4	24	no	—
73	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
74	C₂⁴×C₅₃	C₂⁴	1	24	yes	—
75	C₂³×D₅₃	C₂⁴	1	16	yes	—
76	C₂×D₄×C₅₃	—	2	48	no	—
77	C₂×D₄×C₅₃	—	4	96	no	—
78	C₂×D₄×C₅₃	—	4	24	no	—
79	C₂×D₄×C₅₃	C₂⁴	1	48	yes	—
80	C₂×D₄×C₅₃	C₂⁴	1	48	yes	—
81	C₂²×C₄×C₅₃	—	2	12	no	—
82	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
83	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
84	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
85	C₂×C₂²⋊D₅₃	—	4	96	no	—
86	C₂×C₂²⋊D₅₃	C₂⁴	1	24	yes	—
87	C₂×C₂²⋊D₅₃	C₂⁴	1	96	yes	—
88	C₂²×Dic₅₃	C₂⁴	1	48	yes	—
89	#11719947	—	4	96	no	—
90	D₄×D₅₃	—	4	96	no	—
91	#11719945	—	2	48	no	—
92	C₂×D₂₁₂	C₂⁴	1	12	yes	—
93	C₂×C₄×D₅₃	C₂⁴	1	24	yes	—
94	C₂×Dic₁₀₆	C₂⁴	1	12	yes	—
95	C₄²×C₅₃	C₂⁴	1	2	yes	—
96	C₄²×C₅₃	C₂⁴	1	6	yes	—
97	C₄²×C₅₃	C₂⁴	1	6	yes	—
98	C₄²×C₅₃	C₂⁴	1	2	yes	—
99	C₂²⋊Dic₅₃	C₂⁴	1	24	yes	—
100	C₂²⋊Dic₅₃	C₂⁴	1	96	yes	—
101	#11719921	C₂⁴	1	48	yes	—
102	C₄⋊Dic₅₃	C₂⁴	1	24	yes	—
103	#11719919	C₂⁴	1	48	yes	—
104	C₄×Dic₅₃	C₂⁴	1	24	yes	—
105	C₂³×C₅₃	—	2	6	no	—
106	C₂³×C₅₃	—	2	48	no	—
107	C₂³×C₅₃	—	4	48	no	—
108	C₂³×C₅₃	—	8	32	no	—
109	C₂³×C₅₃	C₂⁵	1	16	yes	—
110	C₂³×C₅₃	C₂⁵	1	3	yes	—
111	C₂²×D₅₃	—	4	48	no	—
112	C₂²×D₅₃	C₂⁵	1	8	yes	—
113	D₄×C₅₃	—	2	96	no	—
114	D₄×C₅₃	—	4	48	no	—
115	C₂×C₄×C₅₃	—	2	6	no	—
116	C₂×C₄×C₅₃	—	2	24	no	—
117	C₂×C₄×C₅₃	H(5,𝔽₂)	1	12	yes	complemented
118	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
119	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
120	C₂²⋊D₅₃	—	2	48	no	—
121	C₂²⋊D₅₃	—	4	96	no	—
122	C₂×Dic₅₃	—	4	48	no	—
123	C₂×Dic₅₃	C₂⁵	1	24	yes	—
124	D₂₁₂	—	2	24	no	—
125	C₄×D₅₃	—	2	48	no	—
126	Dic₁₀₆	—	2	24	no	—
127	#62832	—	53	1	no	maximal, Sylow
128	C₂²×C₅₃	—	2	8	no	—
129	C₂²×C₅₃	—	2	6	no	—
130	C₂²×C₅₃	—	2	48	no	—
131	C₂²×C₅₃	—	8	48	no	—
132	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
133	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
134	C₂×D₅₃	—	2	24	no	—
135	C₂×D₅₃	—	16	16	no	—
136	C₄×C₅₃	—	2	6	no	—
137	C₄×C₅₃	C₂×H(5,𝔽₂)	1	12	yes	—
138	Dic₅₃	—	2	24	no	—
139	#3543	—	53	24	no	—
140	#3508	—	53	12	no	—
141	#3498	—	53	12	no	—
142	#3484	—	53	12	no	—
143	#3444	—	53	2	no	—
144	C₂²×#127	D₅₃	1	1	yes	characteristic, complemented
145	C₂×C₅₃	—	2	6	no	—
146	C₂×C₅₃	—	4	16	no	—
147	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	3	yes	—
148	D₅₃	—	4	8	no	—
149	C₂×H(5,𝔽₂)	—	53	12	no	—
150	C₂³×D₄	C₂×D₅₃	1	6	yes	complemented
151	#560	—	53	32	no	—
152	#559	—	53	24	no	—
153	#550	—	53	48	no	—
154	#546	—	53	96	no	—
155	D₄²	—	53	96	no	—
156	D₄²	—	53	24	no	—
157	#540	—	53	48	no	—
158	#535	—	53	24	no	—
159	#535	—	53	48	no	—
160	#534	—	53	48	no	—
161	#533	—	53	6	no	—
162	#532	—	53	24	no	—
163	#532	—	53	6	no	—
164	C₂×#127	—	53	8	no	—
165	C₂×#127	C₂×D₅₃	1	12	yes	complemented
166	C₂×#127	C₂×D₅₃	1	12	yes	complemented
167	#529	—	53	24	no	—
168	#525	—	53	12	no	—
169	C₂×#120	—	53	16	no	—
170	#518	—	53	12	no	—
171	C₄×#41	—	53	12	no	—
172	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic, complemented
173	C₅₃	#62832	1	1	yes	characteristic, complemented, Sylow
174	C₂⁵	C₂²×D₅₃	1	4	yes	—
175	H(5,𝔽₂)	—	212	48	no	—
176	C₂×#41	—	53	48	no	—
177	C₂×#41	—	53	12	no	—
178	C₂²×D₄	—	4	24	no	—
179	C₂²×D₄	—	53	48	no	—
180	C₂²×D₄	—	53	48	no	—
181	C₂²×D₄	C₂²×D₅₃	1	12	yes	—
182	C₂²×D₄	C₂²×D₅₃	1	12	yes	—
183	C₂²×D₄	C₂²×D₅₃	1	48	yes	—
184	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
185	C₄⋊Q₈	—	53	6	no	—
186	#127	—	53	48	no	—
187	#127	—	53	2	no	—
188	#127	C₂²×D₅₃	1	8	yes	—
189	#127	C₂²×D₅₃	1	24	yes	—
190	#127	C₂²×D₅₃	1	24	yes	—
191	#127	C₂²×D₅₃	1	8	yes	—
192	#126	—	53	16	no	—
193	#125	—	53	12	no	—
194	#124	—	53	48	no	—
195	#124	—	53	12	no	—
196	#123	—	53	96	no	—
197	#123	—	53	24	no	—
198	C₂²⋊Q₈	—	53	24	no	—
199	#121	—	53	96	no	—
200	#121	—	53	192	no	—
201	#121	—	53	24	no	—
202	#120	—	53	8	no	—
203	#120	—	53	96	no	—
204	#120	—	53	96	no	—
205	C₄×Q₈	—	53	24	no	—
206	C₄×D₄	—	53	96	no	—
207	C₄×D₄	—	53	48	no	—
208	C₄×D₄	—	53	24	no	—
209	#117	—	53	12	no	—
210	#117	—	53	24	no	—
211	C₂×C₂²⋊C₄	—	53	48	no	—
212	C₂×C₄²	—	53	8	no	—
213	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
214	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
215	C₂⁴	—	4	24	no	—
216	C₂⁴	—	53	16	no	—
217	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
218	C₂⁴	C₂³×D₅₃	1	24	yes	—
219	#41	—	106	48	no	—
220	#41	—	212	96	no	—
221	C₂×Q₈	—	53	12	no	—
222	C₂×D₄	—	2	48	no	—
223	C₂×D₄	—	4	96	no	—
224	C₂×D₄	—	4	24	no	—
225	C₂×D₄	—	53	24	no	—
226	C₂×D₄	—	53	96	no	—
227	C₂×D₄	—	53	12	no	—
228	C₂×D₄	—	212	96	no	—
229	C₂×D₄	—	212	96	no	—
230	C₂×D₄	C₂³×D₅₃	1	48	yes	—
231	C₂×D₄	C₂³×D₅₃	1	48	yes	—
232	C₂²×C₄	—	2	12	no	—
233	C₂²×C₄	—	53	48	no	—
234	C₂²×C₄	—	53	24	no	—
235	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
236	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
237	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
238	C₄⋊C₄	—	53	24	no	—
239	C₄⋊C₄	—	53	48	no	—
240	C₂²⋊C₄	—	53	24	no	—
241	C₂²⋊C₄	—	53	96	no	—
242	C₂²⋊C₄	—	53	48	no	—
243	C₄²	—	53	24	no	—
244	C₄²	C₂³×D₅₃	1	2	yes	—
245	C₄²	C₂³×D₅₃	1	6	yes	—
246	C₄²	C₂³×D₅₃	1	6	yes	—
247	C₄²	C₂³×D₅₃	1	2	yes	—
248	C₂³	—	2	6	no	—
249	C₂³	—	2	48	no	—
250	C₂³	—	4	48	no	—
251	C₂³	—	8	32	no	—
252	C₂³	—	53	8	no	—
253	C₂³	—	212	48	no	—
254	C₂³	C₂⁴×D₅₃	1	16	yes	—
255	C₂³	C₂⁴×D₅₃	1	3	yes	—
256	Q₈	—	106	24	no	—
257	D₄	—	2	96	no	—
258	D₄	—	4	48	no	—
259	D₄	—	106	48	no	—
260	D₄	—	106	24	no	—
261	D₄	—	212	96	no	—
262	C₂×C₄	—	2	6	no	—
263	C₂×C₄	—	2	24	no	—
264	C₂×C₄	—	53	24	no	—
265	C₂×C₄	—	106	48	no	—
266	C₂×C₄	—	212	48	no	—
267	C₂×C₄	#13133262	1	12	yes	complemented
268	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
269	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
270	C₂²	—	2	8	no	—
271	C₂²	—	2	6	no	—
272	C₂²	—	2	48	no	—
273	C₂²	—	8	48	no	—
274	C₂²	—	848	16	no	—
275	C₂²	—	106	24	no	—
276	C₂²	C₂×#13133262	1	6	yes	—
277	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
278	C₄	—	2	6	no	—
279	C₄	—	106	24	no	—
280	C₄	C₂×#13133262	1	12	yes	—
281	C₂	—	2	6	no	—
282	C₂	—	4	16	no	—
283	C₂	—	212	8	no	—
284	C₂	C₂²×#13133262	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁵×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁶
Fitting	C₂²×#127×C₅₃	C₂
hypercenter	C₂²×#127	D₅₃
hypocenter	C₅₃	#62832
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×#127

Complements

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

N	H ≅ G/N	type
1. #20289048	C₂	semidirect
2. #20288589	C₂	semidirect
3. #20288491	C₂	semidirect
4. #20288350	C₂	semidirect
5. #20288011	C₂	semidirect
6. C₂²×#127×C₅₃	C₂	semidirect
7. C₂×#13133262	C₂²	semidirect
8. #16811360	C₂²	semidirect
9. #16811359	C₂²	semidirect
10. #16811293	C₂²	semidirect
11. #16811267	C₂²	semidirect
12. #16811258	C₂²	semidirect
13. D₄×C₂²⋊D₅₃	C₂²	semidirect
14. #16811227	C₂²	semidirect
15. #16811190	C₂²	semidirect
16. #16811186	C₂²	semidirect
17. #16811182	C₂²	semidirect
18. #16811180	C₂²	semidirect
19. #16811173	C₂²	semidirect
20. #16811171	C₂²	semidirect
21. #127×D₅₃	C₂²	semidirect
22. #16811157	C₂²	semidirect
23. #16811134	C₂²	semidirect
24. C₂×#13133186	C₂²	semidirect
25. #16811098	C₂²	semidirect
26. C₄×#11719945	C₂²	semidirect
27. C₂³×D₄×C₅₃	C₂²	semidirect
28. C₂×#127×C₅₃	C₂²	semidirect
29. C₂×#127×C₅₃	C₂²	semidirect
30. C₂²×C₄²×C₅₃	C₂²	semidirect
33. C₂×#11719947	C₂³	semidirect
35. C₂×D₄×D₅₃	C₂³	semidirect
36. #13133217	C₂³	semidirect
37. #13133214	C₂³	semidirect
38. #13133213	C₂³	semidirect
39. #13133211	C₂³	semidirect
40. #13133209	C₂³	semidirect
41. #13133205	C₂³	semidirect
42. #13133204	C₂³	semidirect
44. #13133198	C₂³	semidirect
47. #13133189	C₂³	semidirect
48. #13133188	C₂³	semidirect
51. #13133185	C₂³	semidirect
52. C₄×Dic₁₀₆	C₂³	semidirect
53. C₄×D₂₁₂	C₂³	semidirect
54. D₄×Dic₅₃	C₂³	semidirect
57. #13133174	C₂³	semidirect
59. C₄²×D₅₃	C₂³	semidirect
117. C₂×C₄×C₅₃	H(5,𝔽₂)	semidirect
144. C₂²×#127	D₅₃	semidirect
150. C₂³×D₄	C₂×D₅₃	semidirect
165. C₂×#127	C₂×D₅₃	semidirect
166. C₂×#127	C₂×D₅₃	semidirect
172. C₂²×C₄²	C₂×D₅₃	semidirect
173. C₅₃	#62832	semidirect
267. C₂×C₄	#13133262	semidirect

Maximal subgroups

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

subgroup	index
#20289048	2
#20288589	2
#20288491	2
#20288350	2
#20288011	2
C₂²×#127×C₅₃	2
#62832	53