Subgroups — #27541829

Order 13568.

518 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287689	C₂	1	1	yes	maximal, characteristic
2	#20287604	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276955	C₂	1	2	yes	maximal, complemented
4	#20276939	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276871	C₂	1	2	yes	maximal, complemented
6	#20276852	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276755	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276753	C₂	1	2	yes	maximal, complemented
9	#20276623	C₂	1	1	yes	maximal, characteristic, complemented
10	#20276582	C₂	1	2	yes	maximal, complemented
11	C₄×SD₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811352	—	2	1	no	—
13	#16811294	—	2	1	no	—
14	C₂×#13133160	C₂²	1	2	yes	—
15	#16811028	C₂²	1	1	yes	characteristic
16	#16811024	C₂²	1	2	yes	—
17	#16811000	C₂²	1	1	yes	characteristic
18	#16810998	C₂²	1	2	yes	—
19	#16810932	—	2	2	no	—
20	#16810908	—	2	1	no	—
21	#16810818	—	2	1	no	—
22	#16810817	—	2	2	no	—
23	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
24	Q₁₆×Dic₅₃	C₂²	1	2	yes	—
25	#16810705	C₂²	1	2	yes	—
26	#16810448	C₂²	1	2	yes	—
27	C₄⋊#11719908	C₂²	1	1	yes	characteristic
28	#16810441	C₂²	1	2	yes	—
29	#16810425	C₂²	1	2	yes	—
30	#16810425	C₂²	1	2	yes	—
31	#16810424	C₂²	1	2	yes	—
32	#16810423	C₂²	1	2	yes	—
33	#16810398	C₂²	1	2	yes	—
34	C₂×SD₃₂×C₅₃	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#367	C₂²	1	1	yes	characteristic
38	C₅₃×#358	C₂²	1	1	yes	characteristic
39	C₅₃×#357	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×#11719945	—	4	1	no	—
42	C₂×D₄₂₄	—	2	1	no	—
43	C₂×D₄₂₄	C₂³	1	2	yes	—
44	C₂×#11719922	—	2	2	no	—
45	#13133217	—	2	1	no	—
46	#13133217	—	4	1	no	—
47	#13133214	—	2	1	no	—
48	#13133213	—	2	2	no	—
49	#13133208	—	2	2	no	—
50	#13133208	—	4	2	no	—
51	#13133191	—	4	1	no	—
52	#13133190	—	4	2	no	—
53	Q₈×Dic₅₃	—	2	2	no	—
54	C₄×Dic₁₀₆	—	4	1	no	—
55	C₄×Dic₁₀₆	D₄	1	1	yes	characteristic
56	#13133179	—	2	2	no	—
57	#13133160	—	2	4	no	—
58	C₂×#11719908	C₂³	1	2	yes	—
59	#13133151	—	2	1	no	—
60	#13133151	C₂³	1	2	yes	—
61	#13133148	—	2	2	no	—
62	C₄⋊#88322	D₄	1	1	yes	characteristic
63	#13133146	D₄	1	2	yes	—
64	#13133141	—	2	2	no	—
65	#13133137	—	2	2	no	—
66	C₈×Dic₅₃	C₂³	1	2	yes	—
67	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₄×Q₈×C₅₃	—	2	1	no	—
70	C₄×D₄×C₅₃	—	2	1	no	—
71	SD₃₂×C₅₃	—	2	2	no	—
72	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
74	#107×C₅₃	C₂³	1	1	yes	characteristic
75	#107×C₅₃	C₂³	1	1	yes	characteristic
76	#103×C₅₃	—	2	1	no	—
77	#102×C₅₃	—	2	1	no	—
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₅₃	—	2	1	no	—
80	C₂×D₄×C₅₃	—	2	1	no	—
81	C₂²×C₄×C₅₃	—	4	1	no	—
82	C₂×C₂²⋊D₅₃	—	4	2	no	—
83	C₂×C₂²⋊D₅₃	—	4	2	no	—
84	#11719945	—	8	4	no	—
85	C₂×D₂₁₂	—	2	2	no	—
86	C₂×D₂₁₂	—	4	1	no	—
87	C₂×D₂₁₂	—	4	1	no	—
88	C₂×C₄×D₅₃	—	4	2	no	—
89	C₂×Dic₁₀₆	—	4	1	no	—
90	C₂×Dic₁₀₆	D₈	1	1	yes	characteristic
91	Q₁₆×C₅₃	—	2	1	no	—
92	Q₁₆×C₅₃	#41	1	2	yes	complemented
93	D₈×C₅₃	—	2	1	no	—
94	D₈×C₅₃	—	2	1	no	—
95	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
97	C₄⋊C₄×C₅₃	—	2	1	no	—
98	C₄⋊C₄×C₅₃	—	2	1	no	—
99	C₄⋊C₄×C₅₃	—	4	1	no	—
100	C₂²⋊C₄×C₅₃	—	4	1	no	—
101	C₄²×C₅₃	—	4	1	no	—
102	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
103	#11719922	—	8	2	no	—
104	#11719921	—	4	2	no	—
105	C₄⋊Dic₅₃	—	2	2	no	—
106	C₄⋊Dic₅₃	—	4	1	no	—
107	C₄⋊Dic₅₃	—	4	2	no	—
108	C₄⋊Dic₅₃	—	4	2	no	—
109	C₄⋊Dic₅₃	D₈	1	1	yes	characteristic
110	#11719919	—	4	2	no	—
111	#11719919	D₈	1	2	yes	—
112	C₄×Dic₅₃	—	4	2	no	—
113	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
114	C₂×#88322	C₂×D₄	1	2	yes	—
115	D₄₂₄	—	2	2	no	—
116	D₄₂₄	—	2	2	no	—
117	D₄₂₄	—	8	1	no	—
118	C₁₆×C₅₃	—	2	1	no	—
119	C₁₆×C₅₃	—	2	1	no	—
120	#11719908	—	2	2	no	—
121	C₂³×C₅₃	—	4	1	no	—
122	C₂²×D₅₃	—	4	2	no	—
123	Q₈×C₅₃	—	2	2	no	—
124	Q₈×C₅₃	—	4	1	no	—
125	D₄×C₅₃	—	4	1	no	—
126	D₄×C₅₃	—	4	1	no	—
127	C₂×C₄×C₅₃	—	4	1	no	—
128	C₂×C₄×C₅₃	—	4	1	no	—
129	C₂×C₄×C₅₃	—	4	1	no	—
130	C₂×C₄×C₅₃	—	8	2	no	—
131	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
134	C₂²⋊D₅₃	—	8	4	no	—
135	C₂²⋊D₅₃	—	16	2	no	—
136	C₂×Dic₅₃	—	4	2	no	—
137	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
138	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
139	D₂₁₂	—	4	2	no	—
140	D₂₁₂	—	4	2	no	—
141	D₂₁₂	—	8	2	no	—
142	D₂₁₂	—	16	1	no	—
143	C₄×D₅₃	—	8	4	no	—
144	Dic₁₀₆	—	2	2	no	—
145	Dic₁₀₆	—	8	2	no	—
146	C₈×C₅₃	—	2	1	no	—
147	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
148	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
149	#88322	—	2	2	no	—
150	#33935	—	53	1	no	maximal, Sylow
151	C₂²×C₅₃	—	8	1	no	—
152	C₂²×C₅₃	—	8	1	no	—
153	C₂²×C₅₃	—	8	1	no	—
154	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
155	C₂×D₅₃	—	8	2	no	—
156	C₂×D₅₃	—	8	2	no	—
157	C₂×D₅₃	—	8	2	no	—
158	C₄×C₅₃	—	2	1	no	—
159	C₄×C₅₃	—	4	2	no	—
160	C₄×C₅₃	—	8	1	no	—
161	C₄×C₅₃	—	8	1	no	—
162	C₄×C₅₃	#509	1	2	yes	—
163	C₄×C₅₃	#548	1	1	yes	characteristic
164	C₄×C₅₃	#577	1	1	yes	characteristic
165	Dic₅₃	—	2	2	no	—
166	Dic₅₃	—	2	2	no	—
167	Dic₅₃	—	8	2	no	—
168	#3391	—	53	1	no	—
169	#3383	—	53	1	no	—
170	#2244	—	53	2	no	—
171	#2240	—	53	1	no	—
172	#2228	—	53	2	no	—
173	#2224	—	53	1	no	—
174	#2207	—	53	2	no	—
175	#2207	—	53	1	no	—
176	#2183	—	53	1	no	—
177	#2175	—	53	2	no	—
178	C₄×SD₃₂	D₅₃	1	1	yes	characteristic, complemented
179	C₂×C₅₃	—	16	1	no	—
180	C₂×C₅₃	#3358	1	1	yes	characteristic
181	C₂×C₅₃	C₂×#509	1	1	yes	characteristic
182	C₂×C₅₃	#3416	1	1	yes	characteristic
183	D₅₃	—	16	2	no	—
184	#557	—	106	1	no	—
185	#550	—	106	1	no	—
186	C₂×SD₃₂	—	53	2	no	—
187	C₂×SD₃₂	C₂×D₅₃	1	1	yes	characteristic
188	#500	—	53	1	no	—
189	#499	—	53	2	no	—
190	#493	—	53	2	no	—
191	#493	—	53	1	no	—
192	#479	—	106	2	no	—
193	#475	—	106	1	no	—
194	#459	—	106	2	no	—
195	#459	—	106	1	no	—
196	C₈×Q₈	—	53	1	no	—
197	C₄×Q₁₆	—	53	2	no	—
198	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
199	C₄×D₈	—	53	2	no	—
200	C₄×D₈	C₂×D₅₃	1	1	yes	characteristic
201	#367	C₂×D₅₃	1	1	yes	characteristic
202	#366	—	53	2	no	—
203	C₄⋊C₁₆	—	53	2	no	—
204	C₄⋊C₁₆	—	53	1	no	—
205	#358	C₂×D₅₃	1	1	yes	characteristic
206	#357	—	53	2	no	—
207	#357	—	53	2	no	—
208	#357	—	53	2	no	—
209	#357	—	53	2	no	—
210	#357	C₂×D₅₃	1	1	yes	characteristic
211	#346	—	53	2	no	—
212	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
213	C₅₃	#33935	1	1	yes	characteristic, complemented, Sylow
214	C₂×#41	—	212	1	no	—
215	C₂×Q₁₆	C₂²×D₅₃	1	1	yes	characteristic
216	C₂×D₈	—	53	2	no	—
217	C₂×D₈	—	106	2	no	—
218	C₂×D₈	—	106	1	no	—
219	C₂×D₈	C₂²×D₅₃	1	1	yes	characteristic
220	C₄⋊Q₈	—	106	1	no	—
221	C₄⋊Q₈	—	212	1	no	—
222	#127	—	106	2	no	—
223	#127	—	106	1	no	—
224	#125	—	106	2	no	—
225	#125	—	212	2	no	—
226	#121	—	212	2	no	—
227	#121	—	212	1	no	—
228	C₄×Q₈	—	2	1	no	—
229	C₄×Q₈	—	53	1	no	—
230	C₄×Q₈	—	106	2	no	—
231	C₄×Q₈	—	212	1	no	—
232	C₄×D₄	—	2	1	no	—
233	C₄×D₄	—	106	2	no	—
234	SD₃₂	—	2	2	no	—
235	SD₃₂	—	106	4	no	—
236	C₂×C₁₆	—	53	2	no	—
237	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
238	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
239	#107	—	53	2	no	—
240	#107	—	106	1	no	—
241	#107	C₂²×D₅₃	1	1	yes	characteristic
242	#107	C₂²×D₅₃	1	1	yes	characteristic
243	#106	—	106	2	no	—
244	C₄⋊C₈	—	53	2	no	—
245	C₄⋊C₈	—	53	1	no	—
246	#103	—	2	1	no	—
247	#103	—	106	2	no	—
248	#102	—	2	1	no	—
249	#102	—	106	2	no	—
250	C₄×C₈	—	53	2	no	—
251	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
252	#41	—	424	4	no	—
253	C₂×Q₈	—	2	1	no	—
254	C₂×Q₈	—	53	1	no	—
255	C₂×Q₈	—	212	1	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	106	2	no	—
258	C₂×D₄	—	212	2	no	—
259	C₂×D₄	—	212	2	no	—
260	C₂×D₄	—	212	1	no	—
261	C₂×D₄	—	212	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	212	2	no	—
264	Q₁₆	—	2	1	no	—
265	Q₁₆	#11719945	1	2	yes	complemented
266	D₈	—	2	1	no	—
267	D₈	—	2	1	no	—
268	D₈	—	106	2	no	—
269	D₈	—	106	2	no	—
270	D₈	—	424	1	no	—
271	D₈	—	424	2	no	—
272	C₂×C₈	—	53	2	no	—
273	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
274	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	53	1	no	—
279	C₄⋊C₄	—	53	2	no	—
280	C₄⋊C₄	—	106	2	no	—
281	C₄⋊C₄	—	212	2	no	—
282	C₄⋊C₄	—	212	2	no	—
283	C₄⋊C₄	—	212	2	no	—
284	C₄⋊C₄	—	212	1	no	—
285	C₂²⋊C₄	—	4	1	no	—
286	C₂²⋊C₄	—	212	2	no	—
287	C₄²	—	4	1	no	—
288	C₄²	—	53	2	no	—
289	C₄²	—	212	2	no	—
290	C₄²	D₄×D₅₃	1	1	yes	characteristic
291	C₁₆	—	2	1	no	—
292	C₁₆	—	2	1	no	—
293	C₁₆	—	106	2	no	—
294	C₂³	—	4	1	no	—
295	C₂³	—	212	2	no	—
296	Q₈	—	2	2	no	—
297	Q₈	—	4	1	no	—
298	Q₈	—	106	2	no	—
299	Q₈	—	424	2	no	—
300	D₄	—	4	1	no	—
301	D₄	—	4	1	no	—
302	D₄	—	848	1	no	—
303	D₄	—	848	2	no	—
304	D₄	—	424	4	no	—
305	D₄	—	424	2	no	—
306	D₄	—	212	2	no	—
307	D₄	—	212	2	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	8	2	no	—
312	C₂×C₄	—	53	2	no	—
313	C₂×C₄	—	53	2	no	—
314	C₂×C₄	—	424	4	no	—
315	C₂×C₄	—	212	2	no	—
316	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
317	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
318	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
319	C₈	—	2	1	no	—
320	C₈	—	106	2	no	—
321	C₈	C₂×#11719945	1	1	yes	characteristic
322	C₈	#13133264	1	1	yes	characteristic
323	C₂²	—	8	1	no	—
324	C₂²	—	8	1	no	—
325	C₂²	—	8	1	no	—
326	C₂²	—	424	2	no	—
327	C₂²	—	424	2	no	—
328	C₂²	—	424	2	no	—
329	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	4	2	no	—
332	C₄	—	8	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	106	2	no	—
335	C₄	—	106	2	no	—
336	C₄	—	424	2	no	—
337	C₄	#16811060	1	2	yes	—
338	C₄	#16811282	1	1	yes	characteristic
339	C₄	#16811430	1	1	yes	characteristic
340	C₂	—	16	1	no	—
341	C₂	—	848	2	no	—
342	C₂	#20287348	1	1	yes	characteristic
343	C₂	C₂×#16811060	1	1	yes	characteristic
344	C₂	#20287858	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₄×SD₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#20287689	C₂
hypercenter	C₄×SD₃₂	D₅₃
hypocenter	C₅₃	#33935
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×SD₃₂

Complements

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

N	H ≅ G/N	type
2. #20287604	C₂	semidirect
3. #20276955	C₂	semidirect
4. #20276939	C₂	semidirect
5. #20276871	C₂	semidirect
6. #20276852	C₂	semidirect
7. #20276755	C₂	semidirect
8. #20276753	C₂	semidirect
9. #20276623	C₂	semidirect
10. #20276582	C₂	semidirect
11. C₄×SD₃₂×C₅₃	C₂	semidirect
92. Q₁₆×C₅₃	#41	semidirect
178. C₄×SD₃₂	D₅₃	semidirect
213. C₅₃	#33935	semidirect
265. Q₁₆	#11719945	semidirect

Maximal subgroups

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

subgroup	index
#20287689	2
#20287604	2
#20276955	2
#20276939	2
#20276871	2
#20276852	2
#20276755	2
#20276753	2
#20276623	2
#20276582	2
C₄×SD₃₂×C₅₃	2
#33935	53