Subgroups — #20283547

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×#13133243	C₂	1	1	yes	maximal, characteristic
2	#16811103	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810733	C₂	1	2	yes	maximal, complemented
4	#16810723	C₂	1	2	yes	maximal, complemented
5	C₄×#11719923	C₂	1	2	yes	maximal, complemented
6	C₄×#11719922	C₂	1	2	yes	maximal, complemented
7	#16810666	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810617	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133138	C₂	1	1	yes	maximal, characteristic, complemented
10	#16810579	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#518	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719945	C₂²	1	1	yes	characteristic
13	#13133243	C₄	1	8	yes	complemented
14	C₂×#11719923	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×Dic₁₀₆	C₂²	1	2	yes	complemented
19	C₄×D₂₁₂	C₂²	1	2	yes	complemented
20	#13133177	—	2	2	no	—
21	#13133175	—	2	1	no	—
22	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₄×D₅₃	—	2	1	no	—
24	#13133150	C₂²	1	2	yes	complemented
25	#13133148	C₂²	1	2	yes	complemented
26	#13133140	C₂²	1	1	yes	characteristic, complemented
27	#13133140	C₂²	1	1	yes	characteristic, complemented
28	#13133138	C₂²	1	2	yes	—
29	#13133138	C₂²	1	2	yes	—
30	#13133137	C₂²	1	1	yes	characteristic, complemented
31	#13133137	C₂²	1	1	yes	characteristic, complemented
32	#13133128	C₂²	1	2	yes	complemented
33	C₄×#88322	C₂²	1	2	yes	complemented
34	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
35	C₄×D₄×C₅₃	—	2	1	no	—
36	C₄×D₄×C₅₃	—	2	1	no	—
37	C₄×D₄×C₅₃	C₂²	1	2	yes	—
38	C₄×D₄×C₅₃	C₂²	1	2	yes	—
39	#117×C₅₃	C₂²	1	1	yes	characteristic
40	#117×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
42	C₂⁴×C₅₃	—	2	1	no	—
43	C₂×D₄×C₅₃	—	2	1	no	—
44	C₂×D₄×C₅₃	—	2	2	no	—
45	C₂×D₄×C₅₃	C₂×C₄	1	4	yes	complemented
46	C₂×D₄×C₅₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₅₃	—	2	2	no	—
48	C₂²×C₄×C₅₃	—	2	2	no	—
49	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
50	C₂×C₂²⋊D₅₃	—	2	1	no	—
51	C₂²×Dic₅₃	—	2	1	no	—
52	#11719945	—	2	2	no	—
53	#11719945	C₂×C₄	1	4	yes	complemented
54	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
55	C₂×C₄×D₅₃	—	2	1	no	—
56	C₂×C₄×D₅₃	—	2	2	no	—
57	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
59	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
60	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
62	C₂²⋊C₄×C₅₃	—	2	1	no	—
63	C₂²⋊C₄×C₅₃	—	2	2	no	—
64	C₂²⋊C₄×C₅₃	—	2	2	no	—
65	C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
66	C₄²×C₅₃	—	2	1	no	—
67	C₄²×C₅₃	C₂³	1	2	yes	—
68	#11719923	C₂×C₄	1	8	yes	complemented
69	#11719922	C₂×C₄	1	8	yes	complemented
70	#11719921	—	2	2	no	—
71	C₄⋊Dic₅₃	—	2	1	no	—
72	C₄⋊Dic₅₃	C₂³	1	2	yes	—
73	#11719919	—	2	2	no	—
74	C₄×Dic₅₃	—	2	2	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	2	no	—
81	C₂³×C₅₃	—	2	4	no	—
82	C₂³×C₅₃	—	8	1	no	—
83	C₂³×C₅₃	—	8	1	no	—
84	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×D₅₃	—	2	1	no	—
86	D₄×C₅₃	—	2	4	no	—
87	D₄×C₅₃	—	2	2	no	—
88	D₄×C₅₃	C₂²×C₄	1	4	yes	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	8	2	no	—
92	C₂×C₄×C₅₃	—	8	2	no	—
93	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#41	1	2	yes	—
98	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
99	C₂²⋊D₅₃	—	2	4	no	—
100	C₂×Dic₅₃	—	2	1	no	—
101	C₂×Dic₅₃	—	2	2	no	—
102	C₂×Dic₅₃	—	8	1	no	—
103	D₂₁₂	—	2	1	no	—
104	D₂₁₂	C₂²×C₄	1	2	yes	—
105	C₄×D₅₃	—	2	4	no	—
106	C₄×D₅₃	—	8	2	no	—
107	Dic₁₀₆	—	2	1	no	—
108	Dic₁₀₆	C₂²×C₄	1	2	yes	—
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	4	no	—
115	C₂²×C₅₃	—	4	2	no	—
116	C₂²×C₅₃	—	4	2	no	—
117	C₂²×C₅₃	—	8	4	no	—
118	C₂²×C₅₃	C₄×D₄	1	2	yes	—
119	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×D₅₃	—	2	2	no	—
121	C₂×D₅₃	—	4	1	no	—
122	C₂×D₅₃	—	4	1	no	—
123	C₄×C₅₃	—	2	2	no	—
124	C₄×C₅₃	—	2	2	no	—
125	C₄×C₅₃	—	4	1	no	—
126	C₄×C₅₃	—	4	1	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	—	2	2	no	—
131	Dic₅₃	—	4	2	no	—
132	#2946	—	53	1	no	maximal, Sylow
133	C₂×C₅₃	—	2	2	no	—
134	C₂×C₅₃	—	4	4	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	2	no	—
139	C₂×AGL(1,ℤ/8)	—	53	1	no	—
140	#520	—	53	1	no	—
141	#518	D₅₃	1	1	yes	characteristic, complemented
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	—	53	1	no	—
152	C₂²×D₄	C₂×D₅₃	1	1	yes	characteristic
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	—	53	2	no	—
159	C₄×D₄	—	2	1	no	—
160	C₄×D₄	—	2	1	no	—
161	C₄×D₄	—	53	2	no	—
162	C₄×D₄	—	106	2	no	—
163	C₄×D₄	C₂×D₅₃	1	2	yes	—
164	C₄×D₄	C₂×D₅₃	1	2	yes	—
165	#117	—	106	1	no	—
166	#117	C₂×D₅₃	1	1	yes	characteristic
167	#117	C₂×D₅₃	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	53	1	no	—
169	C₂×C₄⋊C₄	—	106	1	no	—
170	C₂×C₂²⋊C₄	—	2	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	2	no	—
176	#102	—	53	2	no	—
177	#102	—	53	1	no	—
178	#102	—	53	1	no	—
179	#97	—	53	2	no	—
180	C₄×C₈	—	53	2	no	—
181	C₂⁴	—	2	1	no	—
182	#41	—	53	4	no	—
183	#41	—	106	2	no	—
184	C₂×Q₈	—	53	1	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	—	53	1	no	—
188	C₂×D₄	—	106	1	no	—
189	C₂×D₄	C₄×D₅₃	1	4	yes	complemented
190	C₂×D₄	C₂²×D₅₃	1	2	yes	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	106	1	no	—
194	C₂²×C₄	—	106	1	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	—	53	2	no	—
205	C₄⋊C₄	—	106	1	no	—
206	C₄⋊C₄	—	106	2	no	—
207	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
208	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
209	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
210	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	106	2	no	—
215	C₂²⋊C₄	C₂²⋊D₅₃	1	2	yes	—
216	C₄²	—	2	1	no	—
217	C₄²	—	106	2	no	—
218	C₄²	C₂²×D₅₃	1	2	yes	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	4	no	—
221	C₂³	—	8	1	no	—
222	C₂³	—	8	1	no	—
223	C₂³	—	106	1	no	—
224	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
225	Q₈	—	53	2	no	—
226	Q₈	—	106	1	no	—
227	D₄	—	2	4	no	—
228	D₄	—	2	2	no	—
229	D₄	—	53	2	no	—
230	D₄	—	106	4	no	—
231	D₄	—	106	1	no	—
232	D₄	C₂×C₄×D₅₃	1	4	yes	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	106	1	no	—
238	C₂×C₄	—	106	4	no	—
239	C₂×C₄	—	106	2	no	—
240	C₂×C₄	—	424	2	no	—
241	C₂×C₄	—	424	1	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	4	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	8	4	no	—
257	C₂²	—	106	2	no	—
258	C₂²	—	212	1	no	—
259	C₂²	—	212	1	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	—	4	1	no	—
265	C₄	—	4	1	no	—
266	C₄	—	106	2	no	—
267	C₄	—	212	2	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	4	no	—
273	C₂	—	212	2	no	—
274	C₂	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
275	C₂	#16811426	1	1	yes	characteristic
276	C₂	#16811434	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#518	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#518	D₅₃
hypocenter	C₅₃	#2946
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #518

Complements

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

N	H ≅ G/N	type
2. #16811103	C₂	semidirect
3. #16810733	C₂	semidirect
4. #16810723	C₂	semidirect
5. C₄×#11719923	C₂	semidirect
6. C₄×#11719922	C₂	semidirect
7. #16810666	C₂	semidirect
8. #16810617	C₂	semidirect
9. C₂×#13133138	C₂	semidirect
10. #16810579	C₂	semidirect
11. C₅₃×#518	C₂	semidirect
13. #13133243	C₄	semidirect
18. C₄×Dic₁₀₆	C₂²	semidirect
19. C₄×D₂₁₂	C₂²	semidirect
24. #13133150	C₂²	semidirect
25. #13133148	C₂²	semidirect
26. #13133140	C₂²	semidirect
27. #13133140	C₂²	semidirect
30. #13133137	C₂²	semidirect
31. #13133137	C₂²	semidirect
32. #13133128	C₂²	semidirect
33. C₄×#88322	C₂²	semidirect
45. C₂×D₄×C₅₃	C₂×C₄	semidirect
53. #11719945	C₂×C₄	semidirect
68. #11719923	C₂×C₄	semidirect
69. #11719922	C₂×C₄	semidirect
75. #11719917	C₂×C₄	semidirect
141. #518	D₅₃	semidirect
150. C₅₃	#2946	semidirect
189. C₂×D₄	C₄×D₅₃	semidirect