Subgroups — #20283543

Order 6784.

506 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133242	C₂	1	1	yes	maximal, characteristic
2	#16811097	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810733	C₂	1	2	yes	maximal, complemented
4	#16810721	C₂	1	2	yes	maximal, complemented
5	C₄×#11719924	C₂	1	2	yes	maximal, complemented
6	C₄×#11719922	C₂	1	2	yes	maximal, complemented
7	#16810663	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810619	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133137	C₂	1	1	yes	maximal, characteristic, complemented
10	#16810579	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#520	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic
13	#13133242	C₄	1	8	yes	complemented
14	C₂×#11719924	C₂²	1	2	yes	—
15	C₂×#11719922	C₂²	1	2	yes	—
16	C₂×#11719917	C₂²	1	1	yes	characteristic
17	C₂²×#88322	C₂²	1	1	yes	characteristic
18	C₄×D₂₁₂	C₂²	1	2	yes	—
19	C₄×D₂₁₂	C₂²	1	2	yes	—
20	#13133179	—	2	1	no	—
21	#13133179	—	2	1	no	—
22	#13133176	C₂²	1	1	yes	characteristic
23	C₂²⋊C₄×D₅₃	—	2	2	no	—
24	#13133150	C₂²	1	2	yes	complemented
25	#13133148	C₂²	1	2	yes	complemented
26	#13133141	C₂²	1	1	yes	characteristic, complemented
27	#13133141	C₂²	1	1	yes	characteristic, complemented
28	#13133138	C₂²	1	1	yes	characteristic, complemented
29	#13133138	C₂²	1	1	yes	characteristic, complemented
30	#13133137	C₂²	1	2	yes	—
31	#13133137	C₂²	1	2	yes	—
32	#13133128	C₂²	1	2	yes	complemented
33	C₄×#88322	C₂²	1	2	yes	complemented
34	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
36	C₄×D₄×C₅₃	—	2	2	no	—
37	C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
38	#117×C₅₃	—	2	1	no	—
39	#117×C₅₃	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
41	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
42	C₂³×D₅₃	—	2	1	no	—
43	#41×C₅₃	—	2	2	no	—
44	#41×C₅₃	C₂×C₄	1	4	yes	complemented
45	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₅₃	—	2	1	no	—
47	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	—	2	1	no	—
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	—	2	2	no	—
51	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
52	C₂×D₂₁₂	—	2	1	no	—
53	C₂×D₂₁₂	—	2	2	no	—
54	C₂×D₂₁₂	C₂×C₄	1	4	yes	complemented
55	C₂×D₂₁₂	C₂³	1	2	yes	—
56	C₂×C₄×D₅₃	—	2	2	no	—
57	C₂×C₄×D₅₃	—	2	2	no	—
58	C₄⋊C₄×C₅₃	—	2	1	no	—
59	C₄⋊C₄×C₅₃	—	2	2	no	—
60	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
63	C₂²⋊C₄×C₅₃	—	2	2	no	—
64	C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
65	C₄²×C₅₃	—	2	2	no	—
66	C₄²×C₅₃	C₂³	1	2	yes	—
67	C₂²⋊Dic₅₃	—	2	1	no	—
68	#11719924	C₂×C₄	1	8	yes	complemented
69	#11719922	C₂×C₄	1	8	yes	complemented
70	#11719921	—	2	2	no	—
71	#11719921	—	2	2	no	—
72	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
74	C₄×Dic₅₃	—	2	1	no	—
75	#11719917	C₂×C₄	1	4	yes	complemented
76	C₂×#88322	—	2	2	no	—
77	C₂×#88322	C₂³	1	1	yes	characteristic
78	C₂×#88322	C₂³	1	1	yes	characteristic
79	C₂×#88322	C₂³	1	2	yes	—
80	C₂³×C₅₃	—	2	1	no	—
81	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×D₅₃	—	2	2	no	—
83	C₂²×D₅₃	—	2	4	no	—
84	C₂²×D₅₃	—	8	1	no	—
85	C₂²×D₅₃	—	8	1	no	—
86	Q₈×C₅₃	—	2	1	no	—
87	Q₈×C₅₃	C₂²×C₄	1	2	yes	—
88	D₄×C₅₃	—	2	4	no	—
89	D₄×C₅₃	—	2	1	no	—
90	D₄×C₅₃	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	4	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	8	2	no	—
95	C₂×C₄×C₅₃	—	8	1	no	—
96	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#41	1	2	yes	—
101	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
102	C₂×Dic₅₃	—	2	1	no	—
103	C₂×Dic₅₃	—	2	1	no	—
104	D₂₁₂	—	2	4	no	—
105	D₂₁₂	—	2	2	no	—
106	D₂₁₂	C₂²×C₄	1	4	yes	—
107	C₄×D₅₃	—	8	2	no	—
108	C₄×D₅₃	—	8	2	no	—
109	#88322	—	2	1	no	—
110	#88322	—	2	1	no	—
111	#88322	C₂²×C₄	1	4	yes	—
112	C₂²×C₅₃	—	2	1	no	—
113	C₂²×C₅₃	—	2	1	no	—
114	C₂²×C₅₃	—	2	2	no	—
115	C₂²×C₅₃	—	4	1	no	—
116	C₂²×C₅₃	—	4	1	no	—
117	C₂²×C₅₃	C₄×D₄	1	2	yes	—
118	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×D₅₃	—	2	4	no	—
120	C₂×D₅₃	—	4	2	no	—
121	C₂×D₅₃	—	4	2	no	—
122	C₂×D₅₃	—	8	4	no	—
123	C₄×C₅₃	—	2	2	no	—
124	C₄×C₅₃	—	2	2	no	—
125	C₄×C₅₃	—	2	2	no	—
126	C₄×C₅₃	—	4	2	no	—
127	C₄×C₅₃	C₄×D₄	1	2	yes	—
128	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
129	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
130	Dic₅₃	—	4	1	no	—
131	Dic₅₃	—	4	1	no	—
132	#2946	—	53	1	no	maximal, Sylow
133	C₂×C₅₃	—	2	2	no	—
134	C₂×C₅₃	—	4	2	no	—
135	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂×C₅₃	#576	1	1	yes	characteristic
137	C₂×C₅₃	#577	1	1	yes	characteristic
138	D₅₃	—	4	4	no	—
139	C₂×AGL(1,ℤ/8)	—	53	1	no	—
140	#520	D₅₃	1	1	yes	characteristic, complemented
141	#518	—	53	1	no	—
142	#442	—	53	2	no	—
143	#440	—	53	2	no	—
144	C₄×SD₁₆	—	53	2	no	—
145	C₄×D₈	—	53	2	no	—
146	#428	—	53	1	no	—
147	#416	—	53	1	no	—
148	C₂×#102	—	53	1	no	—
149	#405	—	53	1	no	—
150	C₅₃	#2946	1	1	yes	characteristic, complemented, Sylow
151	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
152	C₂²×D₄	—	53	1	no	—
153	AGL(1,ℤ/8)	—	53	8	no	—
154	C₂×SD₁₆	—	53	2	no	—
155	C₂×D₈	—	53	2	no	—
156	C₂×M₁₆	—	53	1	no	—
157	C₂²×C₈	—	53	1	no	—
158	C₄×Q₈	C₂×D₅₃	1	2	yes	complemented
159	C₄×D₄	—	2	2	no	—
160	C₄×D₄	—	53	2	no	—
161	C₄×D₄	—	53	2	no	—
162	C₄×D₄	—	106	1	no	—
163	C₄×D₄	—	106	1	no	—
164	C₄×D₄	C₂×D₅₃	1	2	yes	complemented
165	#117	—	2	1	no	—
166	#117	—	53	1	no	—
167	#117	C₂×D₅₃	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
170	C₂×C₂²⋊C₄	—	106	2	no	—
171	#107	—	53	2	no	—
172	#106	—	53	2	no	—
173	#103	—	53	1	no	—
174	#103	—	53	1	no	—
175	#102	—	53	1	no	—
176	#102	—	53	1	no	—
177	#102	—	53	2	no	—
178	#102	—	53	2	no	—
179	#97	—	53	2	no	—
180	C₄×C₈	—	53	2	no	—
181	C₂⁴	—	106	1	no	—
182	#41	—	2	2	no	—
183	#41	C₄×D₅₃	1	4	yes	complemented
184	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	53	4	no	—
187	C₂×D₄	—	53	2	no	—
188	C₂×D₄	—	106	2	no	—
189	C₂×D₄	—	106	1	no	—
190	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	106	2	no	—
195	C₂²×C₄	—	106	2	no	—
196	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
197	SD₁₆	—	53	8	no	—
198	D₈	—	53	8	no	—
199	M₁₆	—	53	4	no	—
200	C₂×C₈	—	53	2	no	—
201	C₂×C₈	—	53	1	no	—
202	C₂×C₈	—	53	1	no	—
203	C₂×C₈	—	106	2	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	53	1	no	—
207	C₄⋊C₄	—	53	1	no	—
208	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
209	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
210	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	106	1	no	—
213	C₂²⋊C₄	—	106	2	no	—
214	C₂²⋊C₄	—	106	2	no	—
215	C₂²⋊C₄	C₂²⋊D₅₃	1	2	yes	—
216	C₄²	—	2	2	no	—
217	C₄²	—	106	1	no	—
218	C₄²	C₂²×D₅₃	1	2	yes	—
219	C₂³	—	2	1	no	—
220	C₂³	—	106	4	no	—
221	C₂³	—	106	2	no	—
222	C₂³	—	424	1	no	—
223	C₂³	—	424	1	no	—
224	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
225	Q₈	—	2	1	no	—
226	Q₈	C₂×C₄×D₅₃	1	2	yes	—
227	D₄	—	2	4	no	—
228	D₄	—	2	1	no	—
229	D₄	—	53	4	no	—
230	D₄	—	106	4	no	—
231	D₄	—	106	2	no	—
232	D₄	C₂×C₄×D₅₃	1	2	yes	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	8	1	no	—
238	C₂×C₄	—	106	1	no	—
239	C₂×C₄	—	106	1	no	—
240	C₂×C₄	—	424	2	no	—
241	C₂×C₄	—	424	2	no	—
242	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
243	C₂×C₄	#11719945	1	2	yes	—
244	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
245	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
246	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
247	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
248	C₈	—	53	4	no	—
249	C₈	—	106	1	no	—
250	C₈	—	106	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	4	1	no	—
255	C₂²	—	4	1	no	—
256	C₂²	—	106	4	no	—
257	C₂²	—	424	4	no	—
258	C₂²	—	212	2	no	—
259	C₂²	—	212	2	no	—
260	C₂²	C₄×C₂²⋊D₅₃	1	2	yes	—
261	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	4	2	no	—
266	C₄	—	212	1	no	—
267	C₄	—	212	1	no	—
268	C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
269	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
270	C₄	C₂×#11719945	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	4	2	no	—
273	C₂	—	212	4	no	—
274	C₂	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
275	C₂	#16811426	1	1	yes	characteristic
276	C₂	#16811431	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#520	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#520	D₅₃
hypocenter	C₅₃	#2946
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

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

N	H ≅ G/N	type
2. #16811097	C₂	semidirect
3. #16810733	C₂	semidirect
4. #16810721	C₂	semidirect
5. C₄×#11719924	C₂	semidirect
6. C₄×#11719922	C₂	semidirect
7. #16810663	C₂	semidirect
8. #16810619	C₂	semidirect
9. C₂×#13133137	C₂	semidirect
10. #16810579	C₂	semidirect
11. C₅₃×#520	C₂	semidirect
13. #13133242	C₄	semidirect
24. #13133150	C₂²	semidirect
25. #13133148	C₂²	semidirect
26. #13133141	C₂²	semidirect
27. #13133141	C₂²	semidirect
28. #13133138	C₂²	semidirect
29. #13133138	C₂²	semidirect
32. #13133128	C₂²	semidirect
33. C₄×#88322	C₂²	semidirect
35. C₄×Q₈×C₅₃	C₂²	semidirect
37. C₄×D₄×C₅₃	C₂²	semidirect
44. #41×C₅₃	C₂×C₄	semidirect
54. C₂×D₂₁₂	C₂×C₄	semidirect
68. #11719924	C₂×C₄	semidirect
69. #11719922	C₂×C₄	semidirect
75. #11719917	C₂×C₄	semidirect
140. #520	D₅₃	semidirect
150. C₅₃	#2946	semidirect
158. C₄×Q₈	C₂×D₅₃	semidirect
164. C₄×D₄	C₂×D₅₃	semidirect
183. #41	C₄×D₅₃	semidirect