Subgroups — #28293109

Order 13568.

7130 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20288517	C₂	1	24	yes	maximal, complemented
2	#20288516	C₂	1	24	yes	maximal, complemented
3	#20288120	C₂	1	3	yes	maximal, complemented
4	#543×D₅₃	C₂	1	8	yes	maximal, complemented
5	C₂×#16811097	C₂	1	3	yes	maximal
6	C₅₃×#3458	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic
8	#16811334	C₂²	1	48	yes	complemented
9	#16811333	C₂²	1	48	yes	complemented
10	#16811329	C₂²	1	96	yes	complemented
11	Q₈×D₂₁₂	C₂²	1	16	yes	complemented
12	#16811288	C₂²	1	48	yes	complemented
13	#16811254	C₂²	1	12	yes	complemented
14	#16811251	C₂²	1	24	yes	complemented
15	#16811249	C₂²	1	12	yes	—
16	#16811248	C₂²	1	96	yes	complemented
17	#16811180	C₂²	1	12	yes	complemented
18	#16811176	C₂²	1	24	yes	complemented
19	C₂²⋊Q₈×D₅₃	C₂²	1	96	yes	complemented
20	#16811097	C₂²	1	24	yes	—
21	#16811097	C₂²	1	24	yes	—
22	C₂×C₄×D₂₁₂	C₂²	1	12	yes	—
23	C₂×#13133179	C₂²	1	12	yes	—
24	C₂×#13133176	C₂²	1	3	yes	—
25	C₂×C₂²⋊C₄×D₅₃	C₂²	1	12	yes	—
26	C₅₃×#544	C₂²	1	12	yes	complemented
27	C₅₃×#543	C₂²	1	4	yes	—
28	C₅₃×#533	C₂²	1	12	yes	complemented
29	C₂×#117×C₅₃	C₂²	1	3	yes	—
30	C₂⁴×D₅₃	C₂³	1	2	yes	—
31	C₂×Q₈×D₅₃	C₂³	1	32	yes	complemented
32	C₂²×D₂₁₂	Q₈	1	16	yes	complemented
33	C₂²×D₂₁₂	C₂³	1	6	yes	—
34	C₂²×D₂₁₂	C₂³	1	6	yes	—
35	C₂²×C₄×D₅₃	C₂³	1	24	yes	—
36	#13133217	C₂³	1	24	yes	complemented
37	#13133216	C₂³	1	48	yes	complemented
38	#13133209	C₂³	1	24	yes	complemented
39	#13133208	C₂³	1	48	yes	complemented
40	#13133197	C₂³	1	96	yes	complemented
41	#13133196	C₂³	1	48	yes	—
42	#13133195	C₂³	1	192	yes	complemented
43	#13133194	C₂³	1	96	yes	—
44	#13133193	C₂³	1	96	yes	complemented
45	C₄×Dic₁₀₆	C₂³	1	48	yes	complemented
46	C₄×D₂₁₂	C₂³	1	48	yes	—
47	C₄×D₂₁₂	C₂³	1	48	yes	—
48	#13133179	C₂³	1	96	yes	—
49	#13133176	C₂³	1	12	yes	—
50	#13133176	C₂³	1	12	yes	—
51	C₂×C₄⋊Dic₅₃	C₂³	1	6	yes	—
52	C₄⋊C₄×D₅₃	C₂³	1	96	yes	complemented
53	C₂×C₂²⋊Dic₅₃	C₂³	1	6	yes	—
54	C₂×#11719921	C₂³	1	24	yes	—
55	C₂²⋊C₄×D₅₃	C₂³	1	96	yes	—
56	C₂×C₄×Dic₅₃	C₂³	1	6	yes	—
57	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
58	C₄⋊Q₈×C₅₃	C₂³	1	24	yes	complemented
59	#125×C₅₃	C₂³	1	24	yes	complemented
60	C₂²⋊Q₈×C₅₃	C₂³	1	48	yes	—
61	C₄×Q₈×C₅₃	C₂³	1	16	yes	complemented
62	#117×C₅₃	C₂³	1	12	yes	—
63	#117×C₅₃	C₂³	1	12	yes	—
64	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
65	C₂×C₂²⋊C₄×C₅₃	C₂³	1	6	yes	—
66	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
67	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
68	C₂³×D₅₃	—	8	4	no	—
69	C₂³×D₅₃	C₂×Q₈	1	16	yes	complemented
70	C₂³×D₅₃	C₂⁴	1	12	yes	—
71	C₂×Q₈×C₅₃	C₂⁴	1	16	yes	—
72	C₂²×C₄×C₅₃	C₂×Q₈	1	8	yes	complemented
73	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
74	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
75	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
76	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
77	Q₈×D₅₃	—	8	32	no	—
78	C₂×D₂₁₂	C₂×Q₈	1	48	yes	complemented
79	C₂×D₂₁₂	C₂×Q₈	1	48	yes	complemented
80	C₂×D₂₁₂	C₂⁴	1	12	yes	—
81	C₂×D₂₁₂	C₂⁴	1	4	yes	—
82	C₂×C₄×D₅₃	—	8	24	no	—
83	C₂×C₄×D₅₃	C₂⁴	1	96	yes	—
84	C₂×Dic₁₀₆	C₂⁴	1	48	yes	—
85	C₄⋊C₄×C₅₃	C₂⁴	1	24	yes	—
86	C₄⋊C₄×C₅₃	C₂⁴	1	48	yes	—
87	C₂²⋊C₄×C₅₃	C₂⁴	1	24	yes	—
88	C₄²×C₅₃	C₂⁴	1	12	yes	—
89	C₄²×C₅₃	C₂⁴	1	12	yes	—
90	C₂²⋊Dic₅₃	C₂⁴	1	24	yes	—
91	#11719921	C₂⁴	1	96	yes	—
92	C₄⋊Dic₅₃	C₂⁴	1	48	yes	—
93	C₄⋊Dic₅₃	C₂⁴	1	12	yes	—
94	C₄⋊Dic₅₃	C₂⁴	1	12	yes	—
95	#11719919	C₂⁴	1	96	yes	—
96	C₄×Dic₅₃	C₂⁴	1	24	yes	—
97	C₂³×C₅₃	—	4	2	no	—
98	C₂³×C₅₃	C₂²×Q₈	1	4	yes	—
99	C₂³×C₅₃	C₂⁵	1	3	yes	—
100	C₂²×D₅₃	—	4	24	no	—
101	C₂²×D₅₃	—	8	16	no	—
102	C₂²×D₅₃	C₂²×Q₈	1	48	yes	—
103	C₂²×D₅₃	C₂⁵	1	8	yes	—
104	Q₈×C₅₃	—	4	16	no	—
105	C₂×C₄×C₅₃	—	4	12	no	—
106	C₂×C₄×C₅₃	C₂²×Q₈	1	12	yes	—
107	C₂×C₄×C₅₃	C₂²×Q₈	1	12	yes	—
108	C₂×C₄×C₅₃	C₂⁵	1	24	yes	—
109	C₂×C₄×C₅₃	C₂⁵	1	3	yes	—
110	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
111	C₂×Dic₅₃	—	4	12	no	—
112	C₂×Dic₅₃	C₂⁵	1	24	yes	—
113	D₂₁₂	C₂²×Q₈	1	48	yes	—
114	D₂₁₂	C₂²×Q₈	1	16	yes	—
115	C₄×D₅₃	—	4	96	no	—
116	Dic₁₀₆	—	4	48	no	—
117	#62843	—	53	1	no	maximal, Sylow
118	C₂²×C₅₃	—	2	6	no	—
119	C₂²×C₅₃	—	4	4	no	—
120	C₂²×C₅₃	C₂³×Q₈	1	6	yes	—
121	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
122	C₂×D₅₃	—	2	16	no	—
123	C₂×D₅₃	—	4	48	no	—
124	C₂×D₅₃	C₂³×Q₈	1	16	yes	—
125	C₄×C₅₃	—	2	24	no	—
126	C₄×C₅₃	C₂³×Q₈	1	6	yes	—
127	C₄×C₅₃	C₂³×Q₈	1	2	yes	—
128	Dic₅₃	—	2	24	no	—
129	#3501	—	53	24	no	—
130	#3501	—	53	24	no	—
131	#3458	—	53	3	no	—
132	#3458	D₅₃	1	1	yes	characteristic, complemented
133	C₂×#543	—	53	8	no	—
134	C₂×#518	—	53	3	no	—
135	C₂×C₅₃	—	2	6	no	—
136	C₂×C₅₃	C₂⁴×Q₈	1	1	yes	characteristic
137	C₂×C₅₃	H(7,𝔽₂)	1	2	yes	—
138	D₅₃	—	2	16	no	—
139	C₂³×D₄	—	53	1	no	—
140	#554	—	53	48	no	—
141	#554	—	53	96	no	—
142	#554	—	53	48	no	—
143	D₄×Q₈	—	53	48	no	—
144	D₄×Q₈	—	53	16	no	—
145	#544	—	53	24	no	—
146	#544	—	53	12	no	—
147	#544	C₂×D₅₃	1	12	yes	complemented
148	#543	—	53	96	no	—
149	#543	—	53	12	no	—
150	#543	C₂×D₅₃	1	4	yes	—
151	#533	—	53	24	no	—
152	#533	—	53	12	no	—
153	#533	C₂×D₅₃	1	12	yes	complemented
154	C₂×C₂²⋊Q₈	—	53	96	no	—
155	#518	—	53	24	no	—
156	#518	—	53	24	no	—
157	C₂×C₄×D₄	—	53	12	no	—
158	C₂×C₄×D₄	—	53	12	no	—
159	C₂×#117	—	53	3	no	—
160	C₂×#117	C₂×D₅₃	1	3	yes	—
161	C₂²×C₂²⋊C₄	—	53	12	no	—
162	C₅₃	#62843	1	1	yes	characteristic, complemented, Sylow
163	C₂⁵	—	53	2	no	—
164	C₂²×Q₈	—	53	32	no	—
165	C₂²×D₄	—	53	16	no	—
166	C₂²×D₄	—	53	6	no	—
167	C₂²×D₄	—	53	6	no	—
168	C₂³×C₄	—	53	24	no	—
169	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
170	C₄⋊Q₈	—	53	48	no	—
171	C₄⋊Q₈	—	53	24	no	—
172	C₄⋊Q₈	C₂²×D₅₃	1	24	yes	complemented
173	#125	—	53	48	no	—
174	#125	—	53	24	no	—
175	#125	C₂²×D₅₃	1	24	yes	complemented
176	C₂²⋊Q₈	—	53	96	no	—
177	C₂²⋊Q₈	—	53	48	no	—
178	C₂²⋊Q₈	—	53	192	no	—
179	C₂²⋊Q₈	—	53	96	no	—
180	C₂²⋊Q₈	—	53	96	no	—
181	C₂²⋊Q₈	C₂²×D₅₃	1	48	yes	—
182	C₄×Q₈	—	53	48	no	—
183	C₄×Q₈	C₂²×D₅₃	1	16	yes	complemented
184	C₄×D₄	—	53	96	no	—
185	C₄×D₄	—	53	48	no	—
186	C₄×D₄	—	53	48	no	—
187	#117	—	53	12	no	—
188	#117	—	53	12	no	—
189	#117	C₂²×D₅₃	1	12	yes	—
190	#117	C₂²×D₅₃	1	12	yes	—
191	C₂×C₄⋊C₄	—	53	6	no	—
192	C₂×C₄⋊C₄	—	53	96	no	—
193	C₂×C₄⋊C₄	C₂²×D₅₃	1	6	yes	—
194	C₂×C₂²⋊C₄	—	53	6	no	—
195	C₂×C₂²⋊C₄	—	53	24	no	—
196	C₂×C₂²⋊C₄	—	53	96	no	—
197	C₂×C₂²⋊C₄	C₂²×D₅₃	1	6	yes	—
198	C₂×C₄²	—	53	6	no	—
199	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
200	C₂⁴	—	53	16	no	—
201	C₂⁴	—	53	12	no	—
202	C₂⁴	—	424	4	no	—
203	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
204	C₂×Q₈	—	53	48	no	—
205	C₂×Q₈	—	424	32	no	—
206	C₂×Q₈	C₂³×D₅₃	1	16	yes	—
207	C₂×D₄	—	53	48	no	—
208	C₂×D₄	—	53	48	no	—
209	C₂×D₄	—	53	12	no	—
210	C₂×D₄	—	53	4	no	—
211	C₂²×C₄	—	53	12	no	—
212	C₂²×C₄	—	53	96	no	—
213	C₂²×C₄	—	424	24	no	—
214	C₂²×C₄	Q₈×D₅₃	1	8	yes	complemented
215	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
216	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
217	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
218	C₄⋊C₄	—	53	96	no	—
219	C₄⋊C₄	—	53	48	no	—
220	C₄⋊C₄	—	53	12	no	—
221	C₄⋊C₄	—	53	12	no	—
222	C₄⋊C₄	C₂³×D₅₃	1	48	yes	—
223	C₄⋊C₄	C₂³×D₅₃	1	24	yes	—
224	C₂²⋊C₄	—	53	24	no	—
225	C₂²⋊C₄	—	53	96	no	—
226	C₂²⋊C₄	C₂³×D₅₃	1	24	yes	—
227	C₄²	—	53	24	no	—
228	C₄²	C₂³×D₅₃	1	12	yes	—
229	C₄²	C₂³×D₅₃	1	12	yes	—
230	C₂³	—	4	2	no	—
231	C₂³	—	53	8	no	—
232	C₂³	—	53	48	no	—
233	C₂³	—	424	16	no	—
234	C₂³	—	212	24	no	—
235	C₂³	C₂×Q₈×D₅₃	1	4	yes	—
236	C₂³	C₂⁴×D₅₃	1	3	yes	—
237	Q₈	—	4	16	no	—
238	Q₈	—	212	48	no	—
239	D₄	—	53	48	no	—
240	D₄	—	53	16	no	—
241	C₂×C₄	—	4	12	no	—
242	C₂×C₄	—	53	24	no	—
243	C₂×C₄	—	212	96	no	—
244	C₂×C₄	—	212	12	no	—
245	C₂×C₄	C₂×Q₈×D₅₃	1	12	yes	—
246	C₂×C₄	C₂×Q₈×D₅₃	1	12	yes	—
247	C₂×C₄	C₂⁴×D₅₃	1	24	yes	—
248	C₂×C₄	C₂⁴×D₅₃	1	3	yes	—
249	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
250	C₂²	—	2	6	no	—
251	C₂²	—	4	4	no	—
252	C₂²	—	53	16	no	—
253	C₂²	—	106	16	no	—
254	C₂²	—	212	48	no	—
255	C₂²	C₂²×Q₈×D₅₃	1	6	yes	—
256	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
257	C₄	—	2	24	no	—
258	C₄	—	106	24	no	—
259	C₄	C₂²×Q₈×D₅₃	1	6	yes	—
260	C₄	C₂²×Q₈×D₅₃	1	2	yes	—
261	C₂	—	2	6	no	—
262	C₂	—	106	16	no	—
263	C₂	C₂³×Q₈×D₅₃	1	1	yes	characteristic
264	C₂	#20289526	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁵×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁶
Fitting	C₅₃×#3458	C₂
prime radical	C₂³×D₂₁₂	C₂²
hypercenter	#3458	D₅₃
hypocenter	C₅₃	#62843
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #3458

Complements

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

N	H ≅ G/N	type
1. #20288517	C₂	semidirect
2. #20288516	C₂	semidirect
3. #20288120	C₂	semidirect
4. #543×D₅₃	C₂	semidirect
6. C₅₃×#3458	C₂	semidirect
8. #16811334	C₂²	semidirect
9. #16811333	C₂²	semidirect
10. #16811329	C₂²	semidirect
11. Q₈×D₂₁₂	C₂²	semidirect
12. #16811288	C₂²	semidirect
13. #16811254	C₂²	semidirect
14. #16811251	C₂²	semidirect
16. #16811248	C₂²	semidirect
17. #16811180	C₂²	semidirect
18. #16811176	C₂²	semidirect
19. C₂²⋊Q₈×D₅₃	C₂²	semidirect
26. C₅₃×#544	C₂²	semidirect
28. C₅₃×#533	C₂²	semidirect
31. C₂×Q₈×D₅₃	C₂³	semidirect
32. C₂²×D₂₁₂	Q₈	semidirect
36. #13133217	C₂³	semidirect
37. #13133216	C₂³	semidirect
38. #13133209	C₂³	semidirect
39. #13133208	C₂³	semidirect
40. #13133197	C₂³	semidirect
42. #13133195	C₂³	semidirect
44. #13133193	C₂³	semidirect
45. C₄×Dic₁₀₆	C₂³	semidirect
52. C₄⋊C₄×D₅₃	C₂³	semidirect
58. C₄⋊Q₈×C₅₃	C₂³	semidirect
59. #125×C₅₃	C₂³	semidirect
61. C₄×Q₈×C₅₃	C₂³	semidirect
69. C₂³×D₅₃	C₂×Q₈	semidirect
72. C₂²×C₄×C₅₃	C₂×Q₈	semidirect
78. C₂×D₂₁₂	C₂×Q₈	semidirect
79. C₂×D₂₁₂	C₂×Q₈	semidirect
132. #3458	D₅₃	semidirect
147. #544	C₂×D₅₃	semidirect
153. #533	C₂×D₅₃	semidirect
162. C₅₃	#62843	semidirect
172. C₄⋊Q₈	C₂²×D₅₃	semidirect
175. #125	C₂²×D₅₃	semidirect
183. C₄×Q₈	C₂²×D₅₃	semidirect
214. C₂²×C₄	Q₈×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#20288517	2
#20288516	2
#20288120	2
#543×D₅₃	2
C₂×#16811097	2
C₅₃×#3458	2
#62843	53