Subgroups — #27541685

Order 13568.

502 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287145	C₂	1	1	yes	maximal, characteristic, complemented
2	#20287062	C₂	1	1	yes	maximal, characteristic
3	#20276919	C₂	1	2	yes	maximal, complemented
4	#20276909	C₂	1	2	yes	maximal, complemented
5	#20276854	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276849	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276829	C₂	1	2	yes	maximal, complemented
8	C₄×#13133162	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276548	C₂	1	1	yes	maximal, characteristic, complemented
10	#20276500	C₂	1	2	yes	maximal, complemented
11	C₅₃×#2252	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811347	—	2	1	no	—
13	#16811276	—	2	1	no	—
14	C₂×#13133162	C₂²	1	1	yes	characteristic
15	C₂×#13133156	C₂²	1	2	yes	—
16	#16810966	—	2	2	no	—
17	#16810938	—	2	2	no	—
18	#16810905	—	2	1	no	—
19	#16810900	—	2	1	no	—
20	#16810865	C₂²	1	2	yes	—
21	#16810860	C₂²	1	2	yes	—
22	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
23	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
24	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
25	#16810660	C₂²	1	2	yes	—
26	#107×D₅₃	C₂²	1	2	yes	—
27	#16810451	C₂²	1	1	yes	characteristic
28	#16810450	C₂²	1	2	yes	—
29	#16810448	C₂²	1	2	yes	—
30	C₄⋊#11719908	C₂²	1	1	yes	characteristic
31	#16810428	C₂²	1	1	yes	characteristic
32	#16810426	C₂²	1	2	yes	—
33	#16810425	C₂²	1	1	yes	characteristic
34	#16810423	C₂²	1	2	yes	—
35	#16810402	C₂²	1	2	yes	—
36	C₅₃×#500	C₂²	1	1	yes	characteristic
37	C₅₃×#500	C₂²	1	1	yes	characteristic
38	C₅₃×#366	C₂²	1	2	yes	—
39	C₅₃×#366	C₂²	1	2	yes	—
40	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×#11719949	—	4	1	no	—
42	C₂×Dic₂₁₂	C₂³	1	1	yes	characteristic
43	C₂×D₄₂₄	C₂³	1	1	yes	characteristic
44	C₂×C₈×D₅₃	C₂³	1	2	yes	—
45	#13133212	—	4	2	no	—
46	#13133206	—	4	1	no	—
47	#13133202	—	4	2	no	—
48	#13133197	—	2	2	no	—
49	#13133193	—	4	1	no	—
50	#13133192	—	2	2	no	—
51	Q₈×Dic₅₃	—	4	1	no	—
52	C₄×Dic₁₀₆	—	2	1	no	—
53	C₄×D₂₁₂	—	2	1	no	—
54	C₄×D₂₁₂	D₄	1	1	yes	characteristic
55	#13133175	—	2	2	no	—
56	C₄⋊C₄×D₅₃	—	2	2	no	—
57	#13133162	—	2	2	no	—
58	#13133156	—	2	4	no	—
59	C₂×#11719908	C₂³	1	2	yes	—
60	#13133151	C₂³	1	1	yes	characteristic
61	#13133151	C₂³	1	1	yes	characteristic
62	#13133150	—	2	2	no	—
63	#13133148	—	2	2	no	—
64	C₄⋊#88322	D₄	1	1	yes	characteristic
65	#13133142	—	2	1	no	—
66	#13133140	—	2	2	no	—
67	#13133139	—	2	1	no	—
68	#13133137	—	2	2	no	—
69	#13133129	D₄	1	2	yes	—
70	C₄⋊Q₈×C₅₃	—	2	1	no	—
71	C₄⋊Q₈×C₅₃	—	2	1	no	—
72	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
74	#107×C₅₃	—	2	1	no	—
75	#107×C₅₃	—	2	1	no	—
76	#107×C₅₃	C₂³	1	2	yes	—
77	#107×C₅₃	C₂³	1	2	yes	—
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₅₃	—	4	1	no	—
80	C₂×Q₈×C₅₃	—	4	1	no	—
81	#11719949	—	8	4	no	—
82	C₂×D₂₁₂	—	2	1	no	—
83	C₂×D₂₁₂	—	4	2	no	—
84	C₂×D₂₁₂	D₈	1	1	yes	characteristic
85	C₂×C₄×D₅₃	—	4	2	no	—
86	C₂×C₄×D₅₃	—	4	1	no	—
87	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
88	C₂×Dic₁₀₆	—	2	1	no	—
89	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₅₃	—	2	2	no	—
92	C₄⋊C₄×C₅₃	—	2	2	no	—
93	C₄⋊C₄×C₅₃	—	4	1	no	—
94	C₄⋊C₄×C₅₃	—	4	1	no	—
95	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
96	#11719921	—	4	2	no	—
97	#11719921	—	4	2	no	—
98	#11719921	—	4	1	no	—
99	#11719921	—	4	2	no	—
100	#11719921	D₈	1	2	yes	—
101	C₄⋊Dic₅₃	—	2	1	no	—
102	C₄⋊Dic₅₃	—	2	1	no	—
103	C₄⋊Dic₅₃	—	4	2	no	—
104	C₄⋊Dic₅₃	D₈	1	1	yes	characteristic
105	#11719919	—	4	1	no	—
106	#11719919	—	4	2	no	—
107	C₄×Dic₅₃	—	4	1	no	—
108	C₄×Dic₅₃	—	4	2	no	—
109	C₂×#88322	C₂×D₄	1	2	yes	—
110	Dic₂₁₂	—	2	1	no	—
111	Dic₂₁₂	—	2	1	no	—
112	D₄₂₄	—	2	1	no	—
113	D₄₂₄	—	2	1	no	—
114	C₈×D₅₃	—	2	2	no	—
115	C₈×D₅₃	—	2	2	no	—
116	C₁₆×C₅₃	—	2	1	no	—
117	C₁₆×C₅₃	#41	1	2	yes	complemented
118	#11719908	—	2	2	no	—
119	C₂²×D₅₃	—	4	1	no	—
120	C₂²×D₅₃	C₂×D₈	1	2	yes	—
121	Q₈×C₅₃	—	8	2	no	—
122	Q₈×C₅₃	—	8	2	no	—
123	C₂×C₄×C₅₃	—	4	2	no	—
124	C₂×C₄×C₅₃	—	4	2	no	—
125	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×Dic₅₃	—	4	1	no	—
129	C₂×Dic₅₃	—	4	1	no	—
130	C₂×Dic₅₃	—	4	1	no	—
131	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
132	D₂₁₂	—	2	2	no	—
133	D₂₁₂	—	4	1	no	—
134	D₂₁₂	—	4	1	no	—
135	D₂₁₂	—	8	4	no	—
136	C₄×D₅₃	—	2	2	no	—
137	C₄×D₅₃	—	2	2	no	—
138	C₄×D₅₃	—	8	4	no	—
139	C₄×D₅₃	—	8	2	no	—
140	Dic₁₀₆	—	4	1	no	—
141	Dic₁₀₆	—	4	1	no	—
142	C₈×C₅₃	—	2	1	no	—
143	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
144	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
145	#88322	—	2	2	no	—
146	#33920	—	53	1	no	maximal, Sylow
147	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
148	C₂×D₅₃	—	2	2	no	—
149	C₂×D₅₃	—	2	2	no	—
150	C₂×D₅₃	—	2	2	no	—
151	C₂×D₅₃	—	8	1	no	—
152	C₂×D₅₃	—	8	1	no	—
153	C₂×D₅₃	—	8	1	no	—
154	C₄×C₅₃	—	2	1	no	—
155	C₄×C₅₃	—	8	2	no	—
156	C₄×C₅₃	—	8	2	no	—
157	C₄×C₅₃	#509	1	2	yes	—
158	C₄×C₅₃	#547	1	1	yes	characteristic
159	C₄×C₅₃	#578	1	1	yes	characteristic
160	Dic₅₃	—	2	2	no	—
161	Dic₅₃	—	8	1	no	—
162	Dic₅₃	—	8	1	no	—
163	Dic₅₃	—	8	1	no	—
164	#3342	—	53	1	no	—
165	#3334	—	53	1	no	—
166	#2252	D₅₃	1	1	yes	characteristic, complemented
167	#2236	—	53	2	no	—
168	#2234	—	53	2	no	—
169	#2225	—	53	1	no	—
170	#2224	—	53	1	no	—
171	#2220	—	53	2	no	—
172	C₄×SD₃₂	—	53	1	no	—
173	#2169	—	53	1	no	—
174	#2159	—	53	2	no	—
175	C₂×C₅₃	#3334	1	1	yes	characteristic
176	C₂×C₅₃	C₂×#509	1	1	yes	characteristic
177	C₂×C₅₃	#3417	1	1	yes	characteristic
178	D₅₃	—	4	2	no	—
179	D₅₃	—	16	1	no	—
180	#556	—	106	1	no	—
181	#547	—	106	1	no	—
182	C₂×SD₃₂	—	53	1	no	—
183	C₂×M₃₂	—	53	2	no	—
184	#500	C₂×D₅₃	1	1	yes	characteristic
185	#500	C₂×D₅₃	1	1	yes	characteristic
186	#485	—	106	2	no	—
187	#480	—	106	2	no	—
188	#475	—	106	1	no	—
189	#474	—	106	1	no	—
190	#467	—	53	2	no	—
191	#466	—	53	2	no	—
192	C₄×Q₁₆	—	53	1	no	—
193	C₄×D₈	—	53	1	no	—
194	C₈×D₄	—	53	1	no	—
195	#427	—	53	2	no	—
196	C₂×#107	—	53	2	no	—
197	#367	—	53	1	no	—
198	#367	—	53	2	no	—
199	#366	—	53	2	no	—
200	#366	C₂×D₅₃	1	2	yes	—
201	#366	C₂×D₅₃	1	2	yes	—
202	C₄⋊C₁₆	—	53	1	no	—
203	#358	—	53	1	no	—
204	#358	—	53	2	no	—
205	#357	—	53	1	no	—
206	#357	—	53	2	no	—
207	C₂²⋊C₁₆	—	53	2	no	—
208	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
209	C₅₃	#33920	1	1	yes	characteristic, complemented, Sylow
210	C₂×#41	—	212	1	no	—
211	C₂×Q₁₆	—	53	1	no	—
212	C₂×D₈	—	53	1	no	—
213	C₂²×C₈	—	53	2	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	#126	—	212	2	no	—
217	#124	—	212	1	no	—
218	#123	—	212	2	no	—
219	C₂²⋊Q₈	—	106	2	no	—
220	C₂²⋊Q₈	—	212	1	no	—
221	#121	—	106	2	no	—
222	C₄×Q₈	—	106	1	no	—
223	C₄×Q₈	—	212	1	no	—
224	C₄×D₄	—	53	1	no	—
225	C₄×D₄	—	106	1	no	—
226	#117	—	106	2	no	—
227	C₂×C₄⋊C₄	—	106	2	no	—
228	SD₃₂	—	106	2	no	—
229	M₃₂	—	106	4	no	—
230	C₂×C₁₆	—	53	2	no	—
231	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
232	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
233	#107	—	2	1	no	—
234	#107	—	2	1	no	—
235	#107	—	53	1	no	—
236	#107	—	53	1	no	—
237	#107	—	106	2	no	—
238	#107	C₂²×D₅₃	1	2	yes	—
239	#107	C₂²×D₅₃	1	2	yes	—
240	#106	—	106	2	no	—
241	C₄⋊C₈	—	53	1	no	—
242	#103	—	106	2	no	—
243	#103	—	106	1	no	—
244	#102	—	106	2	no	—
245	#102	—	106	1	no	—
246	C₂²⋊C₈	—	53	2	no	—
247	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
248	#41	—	424	4	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×Q₈	—	4	1	no	—
251	C₂×Q₈	—	106	1	no	—
252	C₂×D₄	—	53	1	no	—
253	C₂×D₄	—	106	1	no	—
254	C₂×D₄	—	212	2	no	—
255	C₂²×C₄	—	53	2	no	—
256	C₂²×C₄	—	212	2	no	—
257	C₂²×C₄	—	212	1	no	—
258	Q₁₆	—	106	1	no	—
259	Q₁₆	—	106	1	no	—
260	D₈	—	106	1	no	—
261	D₈	—	106	1	no	—
262	C₂×C₈	—	53	2	no	—
263	C₂×C₈	—	106	2	no	—
264	C₂×C₈	—	106	2	no	—
265	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
266	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	53	1	no	—
272	C₄⋊C₄	—	106	1	no	—
273	C₄⋊C₄	—	106	1	no	—
274	C₄⋊C₄	—	212	2	no	—
275	C₄⋊C₄	—	212	2	no	—
276	C₄⋊C₄	—	212	1	no	—
277	C₂²⋊C₄	—	53	2	no	—
278	C₂²⋊C₄	—	212	2	no	—
279	C₂²⋊C₄	—	212	2	no	—
280	C₂²⋊C₄	—	212	2	no	—
281	C₂²⋊C₄	—	212	1	no	—
282	C₄²	—	212	2	no	—
283	C₄²	—	212	1	no	—
284	C₄²	D₄×D₅₃	1	1	yes	characteristic
285	C₁₆	—	2	1	no	—
286	C₁₆	—	106	2	no	—
287	C₁₆	#11719949	1	2	yes	complemented
288	C₂³	—	53	2	no	—
289	C₂³	—	212	1	no	—
290	Q₈	—	8	2	no	—
291	Q₈	—	8	2	no	—
292	Q₈	—	212	1	no	—
293	Q₈	—	212	1	no	—
294	D₄	—	106	2	no	—
295	D₄	—	424	4	no	—
296	D₄	—	212	1	no	—
297	D₄	—	212	1	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	53	2	no	—
301	C₂×C₄	—	106	2	no	—
302	C₂×C₄	—	106	2	no	—
303	C₂×C₄	—	424	4	no	—
304	C₂×C₄	—	424	2	no	—
305	C₂×C₄	—	212	1	no	—
306	C₂×C₄	—	212	1	no	—
307	C₂×C₄	—	212	1	no	—
308	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
309	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
310	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	106	2	no	—
313	C₈	C₂×#11719949	1	1	yes	characteristic
314	C₈	#13133265	1	1	yes	characteristic
315	C₂²	—	106	2	no	—
316	C₂²	—	106	2	no	—
317	C₂²	—	106	2	no	—
318	C₂²	—	424	1	no	—
319	C₂²	—	424	1	no	—
320	C₂²	—	424	1	no	—
321	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	106	2	no	—
326	C₄	—	424	1	no	—
327	C₄	—	424	1	no	—
328	C₄	—	424	1	no	—
329	C₄	#16811057	1	2	yes	—
330	C₄	#16811276	1	1	yes	characteristic
331	C₄	#16811436	1	1	yes	characteristic
332	C₂	—	848	1	no	—
333	C₂	—	212	2	no	—
334	C₂	#20287062	1	1	yes	characteristic
335	C₂	C₂×#16811057	1	1	yes	characteristic
336	C₂	#20287864	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×#2252	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#20287062	C₂
hypercenter	#2252	D₅₃
hypocenter	C₅₃	#33920
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

Complements

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

N	H ≅ G/N	type
1. #20287145	C₂	semidirect
3. #20276919	C₂	semidirect
4. #20276909	C₂	semidirect
5. #20276854	C₂	semidirect
6. #20276849	C₂	semidirect
7. #20276829	C₂	semidirect
8. C₄×#13133162	C₂	semidirect
9. #20276548	C₂	semidirect
10. #20276500	C₂	semidirect
11. C₅₃×#2252	C₂	semidirect
117. C₁₆×C₅₃	#41	semidirect
166. #2252	D₅₃	semidirect
209. C₅₃	#33920	semidirect
287. C₁₆	#11719949	semidirect

Maximal subgroups

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

subgroup	index
#20287145	2
#20287062	2
#20276919	2
#20276909	2
#20276854	2
#20276849	2
#20276829	2
C₄×#13133162	2
#20276548	2
#20276500	2
C₅₃×#2252	2
#33920	53