Subgroups — #474×D₅₃ #20284456

Order 6784.

480 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₄×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#16810904	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810903	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810902	C₂	1	2	yes	maximal, complemented
6	#16810901	C₂	1	2	yes	maximal, complemented
7	#16810900	C₂	1	1	yes	maximal, characteristic, complemented
8	#107×D₅₃	C₂	1	2	yes	maximal, complemented
9	C₄⋊C₈×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#102×D₅₃	C₂	1	2	yes	maximal
11	C₅₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₅₃	—	2	1	no	—
13	C₂×D₄×D₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₅₃	—	2	1	no	—
15	C₂×C₈×D₅₃	C₂²	1	2	yes	—
16	#13133217	C₂²	1	1	yes	characteristic, complemented
17	#13133216	—	2	1	no	—
18	#13133216	C₂²	1	2	yes	complemented
19	#13133215	—	2	1	no	—
20	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
21	D₄×Dic₅₃	C₂²	1	1	yes	characteristic
22	#13133179	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₅₃	—	2	1	no	—
24	#13133177	—	2	1	no	—
25	C₄⋊C₄×D₅₃	—	2	1	no	—
26	C₄⋊C₄×D₅₃	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×D₅₃	C₂²	1	2	yes	—
28	C₂²⋊C₄×D₅₃	—	2	1	no	—
29	C₄²×D₅₃	C₂²	1	1	yes	characteristic
30	#13133151	C₂²	1	2	yes	complemented
31	#13133150	C₂²	1	2	yes	complemented
32	#13133150	C₂²	1	2	yes	complemented
33	C₄⋊#88322	C₂²	1	1	yes	characteristic, complemented
34	#13133146	C₂²	1	2	yes	complemented
35	#13133139	C₂²	1	2	yes	—
36	#13133138	C₂²	1	2	yes	—
37	#13133137	C₂²	1	2	yes	—
38	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
39	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
40	#107×C₅₃	C₂²	1	2	yes	complemented
41	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
42	#102×C₅₃	C₂²	1	2	yes	—
43	C₂³×D₅₃	—	2	1	no	—
44	C₂×Q₈×C₅₃	—	2	1	no	—
45	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	—	2	1	no	—
47	C₂×C₂²⋊D₅₃	—	2	1	no	—
48	C₂²×Dic₅₃	—	2	1	no	—
49	Q₈×D₅₃	—	4	4	no	—
50	D₄×D₅₃	—	2	2	no	—
51	D₄×D₅₃	Q₈	1	4	yes	complemented
52	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₅₃	—	2	1	no	—
54	C₂×C₄×D₅₃	—	2	2	no	—
55	C₂×C₄×D₅₃	—	2	1	no	—
56	C₂×C₄×D₅₃	—	2	1	no	—
57	C₂×C₄×D₅₃	—	4	4	no	—
58	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
59	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
60	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
61	C₂×Dic₁₀₆	—	2	1	no	—
62	C₂×Dic₁₀₆	—	2	2	no	—
63	C₂×C₈×C₅₃	C₂³	1	2	yes	—
64	C₄⋊C₄×C₅₃	—	2	1	no	—
65	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
66	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
67	C₂²⋊C₄×C₅₃	—	2	1	no	—
68	C₄²×C₅₃	C₂³	1	1	yes	characteristic
69	C₂²⋊Dic₅₃	—	2	1	no	—
70	#11719921	—	2	1	no	—
71	#11719921	—	2	1	no	—
72	C₄⋊Dic₅₃	—	2	1	no	—
73	C₄⋊Dic₅₃	C₂³	1	2	yes	—
74	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
75	#11719919	—	2	2	no	—
76	#11719919	—	2	1	no	—
77	#11719919	—	2	2	no	—
78	C₄×Dic₅₃	D₄	1	1	yes	characteristic
79	C₄×Dic₅₃	D₄	1	1	yes	characteristic
80	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
81	C₂×#88322	C₂³	1	2	yes	—
82	C₈×D₅₃	—	2	4	no	—
83	C₂³×C₅₃	—	2	1	no	—
84	C₂²×D₅₃	—	2	1	no	—
85	C₂²×D₅₃	—	2	4	no	—
86	C₂²×D₅₃	—	4	2	no	—
87	C₂²×D₅₃	—	4	2	no	—
88	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
89	Q₈×C₅₃	—	4	2	no	—
90	D₄×C₅₃	—	2	1	no	—
91	D₄×C₅₃	C₂×Q₈	1	2	yes	complemented
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	4	2	no	—
95	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
98	C₂²⋊D₅₃	—	2	4	no	—
99	C₂×Dic₅₃	—	2	2	no	—
100	C₂×Dic₅₃	—	2	1	no	—
101	C₂×Dic₅₃	—	4	2	no	—
102	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
103	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
104	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
105	D₂₁₂	—	2	1	no	—
106	D₂₁₂	C₂×Q₈	1	2	yes	complemented
107	C₄×D₅₃	—	2	2	no	—
108	C₄×D₅₃	—	4	2	no	—
109	C₄×D₅₃	—	4	4	no	—
110	C₄×D₅₃	—	4	2	no	—
111	C₄×D₅₃	—	4	2	no	—
112	C₄×D₅₃	D₈	1	4	yes	—
113	C₄×D₅₃	C₂×Q₈	1	2	yes	—
114	C₄×D₅₃	#41	1	2	yes	—
115	Dic₁₀₆	—	4	2	no	—
116	Dic₁₀₆	—	4	4	no	—
117	C₈×C₅₃	—	2	2	no	—
118	#88322	—	2	2	no	—
119	C₂²×C₅₃	—	2	2	no	—
120	C₂²×C₅₃	—	4	1	no	—
121	C₂²×C₅₃	—	4	1	no	—
122	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×D₅₃	—	2	2	no	—
124	C₂×D₅₃	—	4	1	no	—
125	C₂×D₅₃	—	4	1	no	—
126	C₂×D₅₃	—	4	8	no	—
127	C₂×D₅₃	C₂²⋊Q₈	1	2	yes	—
128	C₂×D₅₃	C₂×D₈	1	2	yes	—
129	C₂×D₅₃	#137	1	2	yes	—
130	C₄×C₅₃	—	2	1	no	—
131	C₄×C₅₃	—	4	2	no	—
132	C₄×C₅₃	—	4	1	no	—
133	C₄×C₅₃	C₂×D₈	1	2	yes	—
134	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
135	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
136	Dic₅₃	—	2	1	no	—
137	Dic₅₃	—	4	2	no	—
138	Dic₅₃	—	4	1	no	—
139	Dic₅₃	C₂²⋊Q₈	1	2	yes	—
140	Dic₅₃	C₂×D₈	1	2	yes	—
141	C₂×#474	—	53	1	no	maximal, Sylow
142	C₂×C₅₃	—	4	2	no	—
143	C₂×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
144	C₂×C₅₃	C₂²×D₈	1	1	yes	characteristic
145	C₂×C₅₃	C₂×#137	1	1	yes	characteristic
146	D₅₃	—	4	2	no	—
147	D₅₃	#474	1	4	yes	direct product factor
148	C₂×C₄⋊Q₈	—	53	1	no	—
149	C₂×C₄×D₄	—	53	1	no	—
150	#474	—	53	1	no	—
151	#474	—	53	1	no	—
152	#474	—	53	2	no	—
153	#474	—	53	1	no	—
154	#474	—	53	2	no	—
155	#474	D₅₃	1	1	yes	characteristic, direct product factor
156	C₂×#107	—	53	2	no	—
157	C₂×C₄⋊C₈	—	53	1	no	—
158	C₂×#102	—	53	2	no	—
159	C₅₃	C₂×#474	1	1	yes	characteristic, complemented, Sylow
160	C₂²×Q₈	—	106	1	no	—
161	C₂²×D₄	—	53	1	no	—
162	C₂³×C₄	—	106	1	no	—
163	C₂²×C₈	—	53	2	no	—
164	C₄⋊Q₈	—	53	2	no	—
165	C₄⋊Q₈	—	53	1	no	—
166	C₄⋊Q₈	—	106	1	no	—
167	C₄⋊Q₈	—	106	1	no	—
168	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
169	C₄×D₄	—	53	1	no	—
170	C₄×D₄	—	53	1	no	—
171	C₄×D₄	—	53	1	no	—
172	C₄×D₄	—	106	1	no	—
173	C₄×D₄	—	106	1	no	—
174	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
175	C₂×C₄⋊C₄	—	53	1	no	—
176	C₂×C₄⋊C₄	—	53	2	no	—
177	C₂×C₄⋊C₄	—	106	1	no	—
178	C₂×C₂²⋊C₄	—	106	1	no	—
179	C₂×C₄²	—	53	1	no	—
180	#107	—	53	2	no	—
181	#107	—	53	2	no	—
182	#107	—	53	2	no	—
183	#107	C₂×D₅₃	1	2	yes	complemented
184	C₄⋊C₈	—	53	2	no	—
185	C₄⋊C₈	—	53	1	no	—
186	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic, complemented
187	#102	—	53	2	no	—
188	#102	—	53	2	no	—
189	#102	—	53	2	no	—
190	#102	C₂×D₅₃	1	2	yes	—
191	C₂⁴	—	106	1	no	—
192	C₂×Q₈	—	2	1	no	—
193	C₂×Q₈	—	106	2	no	—
194	C₂×Q₈	—	106	1	no	—
195	C₂×Q₈	—	212	4	no	—
196	C₂×D₄	—	53	4	no	—
197	C₂×D₄	—	53	1	no	—
198	C₂×D₄	—	106	2	no	—
199	C₂×D₄	—	106	1	no	—
200	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	53	1	no	—
203	C₂²×C₄	—	53	1	no	—
204	C₂²×C₄	—	53	1	no	—
205	C₂²×C₄	—	106	1	no	—
206	C₂²×C₄	—	106	1	no	—
207	C₂²×C₄	—	106	2	no	—
208	C₂²×C₄	—	106	1	no	—
209	C₂²×C₄	—	106	1	no	—
210	C₂²×C₄	—	212	4	no	—
211	C₂×C₈	—	53	2	no	—
212	C₂×C₈	—	106	4	no	—
213	C₂×C₈	C₂²×D₅₃	1	2	yes	—
214	C₄⋊C₄	—	2	1	no	—
215	C₄⋊C₄	—	53	2	no	—
216	C₄⋊C₄	—	53	1	no	—
217	C₄⋊C₄	—	106	1	no	—
218	C₄⋊C₄	—	106	2	no	—
219	C₄⋊C₄	—	106	2	no	—
220	C₄⋊C₄	—	106	1	no	—
221	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
222	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
223	C₂²⋊C₄	—	2	1	no	—
224	C₂²⋊C₄	—	106	1	no	—
225	C₂²⋊C₄	—	106	1	no	—
226	C₂²⋊C₄	—	106	1	no	—
227	C₄²	—	53	1	no	—
228	C₄²	—	53	1	no	—
229	C₄²	—	53	1	no	—
230	C₄²	C₂²×D₅₃	1	1	yes	characteristic
231	C₂³	—	2	1	no	—
232	C₂³	—	53	1	no	—
233	C₂³	—	106	4	no	—
234	C₂³	—	106	1	no	—
235	C₂³	—	212	2	no	—
236	C₂³	—	212	2	no	—
237	Q₈	—	4	2	no	—
238	Q₈	—	212	4	no	—
239	Q₈	—	212	2	no	—
240	D₄	—	2	1	no	—
241	D₄	—	53	2	no	—
242	D₄	—	106	4	no	—
243	D₄	—	106	1	no	—
244	D₄	Q₈×D₅₃	1	2	yes	complemented
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	53	1	no	—
249	C₂×C₄	—	53	1	no	—
250	C₂×C₄	—	53	4	no	—
251	C₂×C₄	—	53	2	no	—
252	C₂×C₄	—	53	2	no	—
253	C₂×C₄	—	53	1	no	—
254	C₂×C₄	—	106	2	no	—
255	C₂×C₄	—	106	1	no	—
256	C₂×C₄	—	106	2	no	—
257	C₂×C₄	—	212	2	no	—
258	C₂×C₄	—	212	2	no	—
259	C₂×C₄	—	212	4	no	—
260	C₂×C₄	—	212	2	no	—
261	C₂×C₄	—	212	2	no	—
262	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
263	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
264	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	106	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	1	no	—
270	C₂²	—	53	2	no	—
271	C₂²	—	53	2	no	—
272	C₂²	—	53	2	no	—
273	C₂²	—	106	2	no	—
274	C₂²	—	212	8	no	—
275	C₂²	—	212	1	no	—
276	C₂²	—	212	1	no	—
277	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	4	1	no	—
280	C₄	—	4	2	no	—
281	C₄	—	53	2	no	—
282	C₄	—	53	2	no	—
283	C₄	—	106	1	no	—
284	C₄	—	212	1	no	—
285	C₄	—	212	2	no	—
286	C₄	D₈×D₅₃	1	2	yes	—
287	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
288	C₄	#41×D₅₃	1	1	yes	characteristic
289	C₂	—	4	2	no	—
290	C₂	—	53	4	no	—
291	C₂	—	212	2	no	—
292	C₂	C₂²⋊Q₈×D₅₃	1	1	yes	characteristic
293	C₂	C₂×D₈×D₅₃	1	1	yes	characteristic
294	C₂	#137×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#474	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#474	D₅₃
hypocenter	C₅₃	C₂×#474
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #474

Complements

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

N	H ≅ G/N	type
1. C₄⋊Q₈×D₅₃	C₂	semidirect
3. #16810904	C₂	semidirect
4. #16810903	C₂	semidirect
5. #16810902	C₂	semidirect
6. #16810901	C₂	semidirect
7. #16810900	C₂	semidirect
8. #107×D₅₃	C₂	semidirect
9. C₄⋊C₈×D₅₃	C₂	semidirect
11. C₅₃×#474	C₂	semidirect
16. #13133217	C₂²	semidirect
18. #13133216	C₂²	semidirect
30. #13133151	C₂²	semidirect
31. #13133150	C₂²	semidirect
32. #13133150	C₂²	semidirect
33. C₄⋊#88322	C₂²	semidirect
34. #13133146	C₂²	semidirect
38. C₄⋊Q₈×C₅₃	C₂²	semidirect
40. #107×C₅₃	C₂²	semidirect
41. C₄⋊C₈×C₅₃	C₂²	semidirect
51. D₄×D₅₃	Q₈	semidirect
91. D₄×C₅₃	C₂×Q₈	semidirect
106. D₂₁₂	C₂×Q₈	semidirect
147. D₅₃	#474	direct (N × H)
155. #474	D₅₃	direct (N × H)
159. C₅₃	C₂×#474	semidirect
168. C₄⋊Q₈	C₂×D₅₃	semidirect
183. #107	C₂×D₅₃	semidirect
186. C₄⋊C₈	C₂×D₅₃	semidirect
244. D₄	Q₈×D₅₃	semidirect

Subgroups — #474×D53 #20284456

Conjugacy classes of subgroups

Complements

Subgroups — #474×D₅₃ #20284456