Subgroups — #27539646

Order 13568.

526 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285286	C₂	1	2	yes	maximal, complemented
2	#20276992	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276977	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276948	C₂	1	2	yes	maximal, complemented
5	#20276813	C₂	1	2	yes	maximal, complemented
6	#20276801	C₂	1	1	yes	maximal, characteristic
7	Q₃₂×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276564	C₂	1	2	yes	maximal, complemented
9	#20276561	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2216	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811159	—	2	2	no	—
13	C₂×#13133160	C₂²	1	2	yes	—
14	C₂×D₈₄₈	C₂²	1	1	yes	characteristic
15	C₂²×#11719908	C₂²	1	1	yes	characteristic
16	#16811024	C₂²	1	2	yes	—
17	#16811008	C₂²	1	2	yes	—
18	#16810979	—	2	2	no	—
19	#16810859	C₂²	1	2	yes	—
20	#16810854	—	2	2	no	—
21	Q₁₆×Dic₅₃	C₂²	1	2	yes	—
22	#16810708	—	2	2	no	—
23	#16810705	C₂²	1	2	yes	—
24	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
25	#16810450	C₂²	1	2	yes	—
26	#16810449	C₂²	1	1	yes	characteristic
27	#16810427	C₂²	1	2	yes	—
28	#16810425	C₂²	1	2	yes	—
29	#16810423	C₂²	1	2	yes	—
30	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
31	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
32	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
34	C₂×Q₃₂×C₅₃	C₂²	1	1	yes	characteristic
35	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
36	C₅₃×#467	C₂²	1	2	yes	—
37	C₅₃×#366	C₂²	1	1	yes	characteristic
38	C₅₃×#358	C₂²	1	2	yes	—
39	C₂×#11719924	—	2	2	no	—
40	C₂×D₄₂₄	C₂³	1	2	yes	—
41	C₂²×#88322	D₄	1	1	yes	characteristic
42	#13133212	—	4	2	no	—
43	#13133208	—	2	2	no	—
44	#13133206	—	2	2	no	—
45	#13133201	—	4	2	no	—
46	#13133191	—	2	2	no	—
47	Q₈×Dic₅₃	—	2	2	no	—
48	#13133179	—	2	2	no	—
49	#13133177	—	4	2	no	—
50	#13133175	—	4	2	no	—
51	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
52	#13133160	—	2	4	no	—
53	D₈₄₈	—	2	2	no	—
54	C₂×#11719908	—	2	2	no	—
55	C₂×#11719908	C₂³	1	1	yes	characteristic
56	C₂×#11719908	C₂³	1	1	yes	characteristic
57	#13133151	C₂³	1	2	yes	—
58	#13133148	—	2	2	no	—
59	#13133141	—	2	2	no	—
60	#13133139	—	2	2	no	—
61	#13133137	—	2	2	no	—
62	C₈×Dic₅₃	D₄	1	1	yes	characteristic
63	C₈×Dic₅₃	D₄	1	1	yes	characteristic
64	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
65	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
66	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
67	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₂²⋊Q₈×C₅₃	—	2	2	no	—
69	Q₃₂×C₅₃	—	2	2	no	—
70	C₂×C₁₆×C₅₃	—	2	2	no	—
71	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
72	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
73	#107×C₅₃	C₂³	1	2	yes	—
74	#103×C₅₃	—	2	2	no	—
75	C₂×Q₈×C₅₃	—	2	2	no	—
76	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₂²⋊D₅₃	—	4	2	no	—
78	C₂²×Dic₅₃	D₈	1	1	yes	characteristic
79	C₂²×Dic₅₃	D₈	1	1	yes	characteristic
80	C₂×D₂₁₂	—	2	2	no	—
81	C₂×C₄×D₅₃	—	4	2	no	—
82	Q₁₆×C₅₃	—	2	2	no	—
83	Q₁₆×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	—	2	1	no	—
85	C₂×C₈×C₅₃	—	2	1	no	—
86	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₅₃	—	2	2	no	—
89	C₄⋊C₄×C₅₃	—	4	2	no	—
90	C₂²⋊C₄×C₅₃	—	4	2	no	—
91	#11719924	—	4	4	no	—
92	#11719921	—	4	2	no	—
93	#11719921	—	4	2	no	—
94	#11719921	—	4	2	no	—
95	#11719921	—	4	2	no	—
96	C₄⋊Dic₅₃	—	2	2	no	—
97	C₄⋊Dic₅₃	—	4	2	no	—
98	C₄⋊Dic₅₃	—	4	2	no	—
99	#11719919	—	4	2	no	—
100	C₄×Dic₅₃	—	4	2	no	—
101	C₄×Dic₅₃	D₈	1	1	yes	characteristic
102	C₄×Dic₅₃	D₈	1	1	yes	characteristic
103	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
104	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
105	C₂×#88322	—	2	2	no	—
106	C₂×#88322	C₂×D₄	1	1	yes	characteristic
107	C₂×#88322	C₂×D₄	1	1	yes	characteristic
108	D₄₂₄	—	2	2	no	—
109	D₄₂₄	—	2	2	no	—
110	C₁₆×C₅₃	—	2	1	no	—
111	C₁₆×C₅₃	#41	1	2	yes	—
112	#11719908	—	2	1	no	—
113	#11719908	#41	1	2	yes	—
114	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
115	C₂²×D₅₃	—	4	2	no	—
116	Q₈×C₅₃	—	4	2	no	—
117	Q₈×C₅₃	—	4	2	no	—
118	C₂×C₄×C₅₃	—	2	1	no	—
119	C₂×C₄×C₅₃	—	2	1	no	—
120	C₂×C₄×C₅₃	—	4	2	no	—
121	C₂×C₄×C₅₃	—	4	2	no	—
122	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
124	C₂²⋊D₅₃	—	8	4	no	—
125	C₂×Dic₅₃	—	2	2	no	—
126	C₂×Dic₅₃	—	2	2	no	—
127	C₂×Dic₅₃	—	4	2	no	—
128	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
129	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
132	D₂₁₂	—	4	2	no	—
133	D₂₁₂	—	4	2	no	—
134	C₄×D₅₃	—	8	4	no	—
135	C₈×C₅₃	—	2	1	no	—
136	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
137	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
138	#88322	—	2	1	no	—
139	#88322	#135	1	2	yes	—
140	#33735	—	53	1	no	maximal, Sylow
141	C₂²×C₅₃	—	2	1	no	—
142	C₂²×C₅₃	—	2	1	no	—
143	C₂²×C₅₃	#508	1	2	yes	—
144	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
145	C₂×D₅₃	—	8	2	no	—
146	C₂×D₅₃	—	8	2	no	—
147	C₂×D₅₃	—	8	2	no	—
148	C₄×C₅₃	—	2	1	no	—
149	C₄×C₅₃	—	8	2	no	—
150	C₄×C₅₃	—	8	2	no	—
151	C₄×C₅₃	#532	1	1	yes	characteristic
152	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
153	Dic₅₃	—	2	1	no	—
154	Dic₅₃	—	2	1	no	—
155	Dic₅₃	—	2	1	no	—
156	Dic₅₃	—	8	2	no	—
157	Dic₅₃	#508	1	2	yes	—
158	#3168	—	53	2	no	—
159	#2253	—	53	1	no	—
160	#2249	—	53	1	no	—
161	#2242	—	53	2	no	—
162	#2217	—	53	2	no	—
163	#2216	D₅₃	1	1	yes	characteristic, complemented
164	#2215	—	53	1	no	—
165	C₄×Q₃₂	—	53	1	no	—
166	C₄×SD₃₂	—	53	2	no	—
167	C₄×D₁₆	—	53	1	no	—
168	C₂×C₄×C₁₆	—	53	1	no	—
169	C₂×C₅₃	—	2	2	no	—
170	C₂×C₅₃	#3169	1	1	yes	characteristic
171	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
172	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
173	D₅₃	—	16	2	no	—
174	#529	—	106	2	no	—
175	C₂×Q₃₂	C₂×D₅₃	1	1	yes	characteristic
176	C₂×SD₃₂	—	53	2	no	—
177	C₂×D₁₆	—	53	1	no	—
178	C₂²×C₁₆	—	53	1	no	—
179	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
180	#499	—	53	2	no	—
181	#495	—	53	2	no	—
182	#488	—	106	2	no	—
183	#467	C₂×D₅₃	1	2	yes	—
184	#466	—	53	2	no	—
185	#465	—	106	2	no	—
186	C₄×Q₁₆	—	53	2	no	—
187	C₄×SD₁₆	—	106	2	no	—
188	C₄×D₈	—	53	2	no	—
189	C₂×C₄×C₈	—	53	1	no	—
190	#367	—	53	2	no	—
191	#366	—	53	1	no	—
192	#366	C₂×D₅₃	1	1	yes	characteristic
193	#358	—	53	2	no	—
194	#358	C₂×D₅₃	1	2	yes	—
195	#357	—	53	2	no	—
196	#357	—	53	2	no	—
197	C₄×C₁₆	—	53	1	no	—
198	C₄×C₁₆	—	53	1	no	—
199	C₄×C₁₆	—	53	1	no	—
200	C₄×C₁₆	—	53	1	no	—
201	C₅₃	#33735	1	1	yes	characteristic, complemented, Sylow
202	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
203	C₂×SD₁₆	—	106	2	no	—
204	C₂×D₈	—	53	2	no	—
205	C₂²×C₈	—	53	1	no	—
206	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
207	#126	—	212	2	no	—
208	#125	—	106	2	no	—
209	#124	—	106	2	no	—
210	#123	—	212	2	no	—
211	C₂²⋊Q₈	—	2	2	no	—
212	#121	—	106	2	no	—
213	C₄×Q₈	—	106	2	no	—
214	C₄×D₄	—	106	2	no	—
215	C₄×D₄	—	212	2	no	—
216	#117	—	212	2	no	—
217	C₂×C₄²	—	53	1	no	—
218	Q₃₂	—	2	2	no	—
219	SD₃₂	—	106	4	no	—
220	D₁₆	—	106	2	no	—
221	C₂×C₁₆	—	2	2	no	—
222	C₂×C₁₆	—	53	1	no	—
223	C₂×C₁₆	—	53	1	no	—
224	C₂×C₁₆	—	106	2	no	—
225	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
226	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
227	#107	—	53	2	no	—
228	#107	C₂²×D₅₃	1	2	yes	—
229	#106	—	106	2	no	—
230	#103	—	2	2	no	—
231	#103	—	106	2	no	—
232	#102	—	106	2	no	—
233	#102	—	106	2	no	—
234	C₄×C₈	—	53	1	no	—
235	C₄×C₈	—	53	1	no	—
236	C₄×C₈	—	53	1	no	—
237	C₄×C₈	—	53	1	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×D₄	—	106	2	no	—
240	C₂×D₄	—	212	2	no	—
241	C₂²×C₄	—	53	1	no	—
242	C₂²×C₄	—	53	1	no	—
243	C₂²×C₄	—	212	2	no	—
244	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
245	Q₁₆	—	2	2	no	—
246	Q₁₆	—	2	2	no	—
247	SD₁₆	—	212	4	no	—
248	D₈	—	106	2	no	—
249	D₈	—	106	2	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	53	1	no	—
253	C₂×C₈	—	53	1	no	—
254	C₂×C₈	—	106	2	no	—
255	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
256	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	106	2	no	—
260	C₄⋊C₄	—	212	2	no	—
261	C₄⋊C₄	—	212	2	no	—
262	C₄⋊C₄	—	212	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	212	2	no	—
265	C₂²⋊C₄	—	212	2	no	—
266	C₂²⋊C₄	—	212	2	no	—
267	C₂²⋊C₄	—	212	2	no	—
268	C₄²	—	53	1	no	—
269	C₄²	—	53	1	no	—
270	C₄²	—	53	1	no	—
271	C₄²	—	53	1	no	—
272	C₄²	—	212	2	no	—
273	C₁₆	—	2	1	no	—
274	C₁₆	—	53	2	no	—
275	C₁₆	—	106	1	no	—
276	C₁₆	#11719947	1	2	yes	—
277	C₂³	—	212	2	no	—
278	C₂³	D₈×D₅₃	1	1	yes	characteristic
279	Q₈	—	4	2	no	—
280	Q₈	—	4	2	no	—
281	D₄	—	424	4	no	—
282	D₄	—	212	2	no	—
283	D₄	—	212	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	53	1	no	—
289	C₂×C₄	—	53	1	no	—
290	C₂×C₄	—	53	1	no	—
291	C₂×C₄	—	53	1	no	—
292	C₂×C₄	—	106	2	no	—
293	C₂×C₄	—	106	2	no	—
294	C₂×C₄	—	424	4	no	—
295	C₂×C₄	—	212	2	no	—
296	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
297	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	53	2	no	—
300	C₈	—	106	1	no	—
301	C₈	#41×D₅₃	1	1	yes	characteristic
302	C₈	C₂×#11719947	1	1	yes	characteristic
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	424	2	no	—
306	C₂²	—	424	2	no	—
307	C₂²	—	424	2	no	—
308	C₂²	#16811056	1	2	yes	—
309	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	—	53	2	no	—
314	C₄	—	106	1	no	—
315	C₄	—	106	1	no	—
316	C₄	—	106	1	no	—
317	C₄	—	424	2	no	—
318	C₄	#16811168	1	1	yes	characteristic
319	C₄	#135×D₅₃	1	1	yes	characteristic
320	C₂	—	2	2	no	—
321	C₂	—	848	2	no	—
322	C₂	#20285289	1	1	yes	characteristic
323	C₂	#508×D₅₃	1	1	yes	characteristic
324	C₂	C₂×#16811056	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2216	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#20276801	C₂
hypercenter	#2216	D₅₃
hypocenter	C₅₃	#33735
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2216

Complements

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

N	H ≅ G/N	type
1. #20285286	C₂	semidirect
2. #20276992	C₂	semidirect
3. #20276977	C₂	semidirect
4. #20276948	C₂	semidirect
5. #20276813	C₂	semidirect
7. Q₃₂×Dic₅₃	C₂	semidirect
8. #20276564	C₂	semidirect
9. #20276561	C₂	semidirect
10. C₂×C₁₆×Dic₅₃	C₂	semidirect
11. C₅₃×#2216	C₂	semidirect
163. #2216	D₅₃	semidirect
201. C₅₃	#33735	semidirect

Maximal subgroups

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

subgroup	index
#20285286	2
#20276992	2
#20276977	2
#20276948	2
#20276813	2
#20276801	2
Q₃₂×Dic₅₃	2
#20276564	2
#20276561	2
C₂×C₁₆×Dic₅₃	2
C₅₃×#2216	2
#33735	53