Subgroups — #27539647

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	#20285288	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	#20276814	C₂	1	2	yes	maximal, complemented
6	#20276803	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276802	C₂	1	1	yes	maximal, characteristic
8	C₄×#13133160	C₂	1	2	yes	maximal, complemented
9	C₄×D₈₄₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Q₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811160	—	2	2	no	—
13	C₂×#13133160	C₂²	1	2	yes	—
14	C₂×D₈₄₈	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
17	#16811024	C₂²	1	2	yes	—
18	#16811008	C₂²	1	2	yes	—
19	#16810979	—	2	2	no	—
20	#16810861	C₂²	1	2	yes	—
21	#16810860	C₂²	1	2	yes	—
22	#16810855	—	2	2	no	—
23	C₄×#11719924	—	2	2	no	—
24	C₄×D₄₂₄	C₂²	1	2	yes	—
25	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
26	#16810450	C₂²	1	2	yes	—
27	#16810449	C₂²	1	1	yes	characteristic
28	#16810427	C₂²	1	2	yes	—
29	#16810425	C₂²	1	2	yes	—
30	#16810423	C₂²	1	2	yes	—
31	C₄×#11719908	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	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
36	C₅₃×#366	C₂²	1	1	yes	characteristic
37	C₅₃×#358	C₂²	1	2	yes	—
38	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
39	C₂×#11719924	—	2	2	no	—
40	C₂×D₄₂₄	C₂³	1	2	yes	—
41	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
42	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
43	#13133212	—	4	2	no	—
44	#13133208	—	2	2	no	—
45	#13133206	—	2	2	no	—
46	#13133202	—	4	2	no	—
47	#13133193	—	2	2	no	—
48	#13133192	—	2	2	no	—
49	C₄×D₂₁₂	—	2	2	no	—
50	C₄×D₂₁₂	—	4	2	no	—
51	#13133174	—	4	2	no	—
52	C₄²×D₅₃	D₄	1	1	yes	characteristic
53	#13133160	—	2	4	no	—
54	D₈₄₈	—	2	2	no	—
55	C₂×#11719908	C₂³	1	1	yes	characteristic
56	C₂×#11719908	C₂³	1	1	yes	characteristic
57	C₁₆×D₅₃	—	2	2	no	—
58	C₁₆×D₅₃	—	2	2	no	—
59	#13133151	C₂³	1	2	yes	—
60	#13133148	—	2	2	no	—
61	#13133141	—	2	2	no	—
62	#13133139	—	2	2	no	—
63	#13133137	—	2	2	no	—
64	C₄×#88322	D₄	1	1	yes	characteristic
65	C₈×Dic₅₃	D₄	1	1	yes	characteristic
66	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
67	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
68	C₄×Q₈×C₅₃	—	2	2	no	—
69	Q₃₂×C₅₃	—	2	2	no	—
70	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
71	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
72	#107×C₅₃	C₂³	1	2	yes	—
73	#103×C₅₃	—	2	2	no	—
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₂×Q₈×C₅₃	—	2	2	no	—
76	C₂×D₂₁₂	—	2	2	no	—
77	C₂×D₂₁₂	—	4	2	no	—
78	C₂×C₄×D₅₃	—	4	2	no	—
79	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
80	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
81	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
82	Q₁₆×C₅₃	—	2	2	no	—
83	Q₁₆×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
86	C₄⋊C₄×C₅₃	—	2	2	no	—
87	C₄⋊C₄×C₅₃	—	4	2	no	—
88	C₄²×C₅₃	—	4	2	no	—
89	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
90	#11719924	—	4	4	no	—
91	#11719921	—	4	2	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	#11719919	—	4	2	no	—
99	#11719919	—	4	2	no	—
100	C₄×Dic₅₃	D₈	1	1	yes	characteristic
101	C₄×Dic₅₃	D₈	1	1	yes	characteristic
102	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
103	C₂×#88322	C₂×D₄	1	1	yes	characteristic
104	C₂×#88322	C₂×D₄	1	1	yes	characteristic
105	D₄₂₄	—	2	2	no	—
106	D₄₂₄	—	2	2	no	—
107	C₈×D₅₃	—	2	1	no	—
108	C₈×D₅₃	—	2	1	no	—
109	C₈×D₅₃	—	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	C₂²×D₅₃	—	4	2	no	—
115	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
116	Q₈×C₅₃	—	4	2	no	—
117	Q₈×C₅₃	—	4	2	no	—
118	C₂×C₄×C₅₃	—	4	2	no	—
119	C₂×C₄×C₅₃	—	4	2	no	—
120	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×Dic₅₃	—	4	2	no	—
124	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
125	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
126	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
127	D₂₁₂	—	4	2	no	—
128	D₂₁₂	—	4	2	no	—
129	D₂₁₂	—	8	4	no	—
130	C₄×D₅₃	—	2	2	no	—
131	C₄×D₅₃	—	2	1	no	—
132	C₄×D₅₃	—	2	1	no	—
133	C₄×D₅₃	—	2	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	C₂²×D₈	1	1	yes	characteristic
142	C₂×D₅₃	—	2	1	no	—
143	C₂×D₅₃	—	2	1	no	—
144	C₂×D₅₃	—	8	2	no	—
145	C₂×D₅₃	—	8	2	no	—
146	C₂×D₅₃	—	8	2	no	—
147	C₂×D₅₃	#508	1	2	yes	—
148	C₄×C₅₃	—	2	1	no	—
149	C₄×C₅₃	—	8	2	no	—
150	C₄×C₅₃	—	8	2	no	—
151	C₄×C₅₃	#508	1	2	yes	—
152	C₄×C₅₃	#532	1	1	yes	characteristic
153	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
154	Dic₅₃	—	2	1	no	—
155	Dic₅₃	—	2	1	no	—
156	Dic₅₃	—	2	1	no	—
157	Dic₅₃	—	8	2	no	—
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	—	53	1	no	—
164	#2215	—	53	1	no	—
165	C₄×Q₃₂	D₅₃	1	1	yes	characteristic, complemented
166	C₄×SD₃₂	—	53	2	no	—
167	C₄×D₁₆	—	53	1	no	—
168	C₂×C₄×C₁₆	—	53	1	no	—
169	C₂×C₅₃	#3169	1	1	yes	characteristic
170	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
171	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
172	D₅₃	—	2	2	no	—
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	—	53	1	no	—
180	#499	—	53	2	no	—
181	#495	—	53	2	no	—
182	#488	—	106	2	no	—
183	#467	—	53	2	no	—
184	#466	—	53	2	no	—
185	#465	—	106	2	no	—
186	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
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	C₂×D₅₃	1	1	yes	characteristic
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	—	53	1	no	—
207	#126	—	212	2	no	—
208	#125	—	106	2	no	—
209	#124	—	106	2	no	—
210	#123	—	212	2	no	—
211	C₂²⋊Q₈	—	106	2	no	—
212	#121	—	106	2	no	—
213	C₄×Q₈	—	2	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	—	53	1	no	—
222	C₂×C₁₆	—	53	1	no	—
223	C₂×C₁₆	—	106	2	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	C₂²×D₅₃	1	1	yes	characteristic
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	—	53	1	no	—
244	C₂²×C₄	—	212	2	no	—
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	—	53	1	no	—
251	C₂×C₈	—	53	1	no	—
252	C₂×C₈	—	106	2	no	—
253	C₂×C₈	—	106	1	no	—
254	C₂×C₈	—	106	1	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	—	212	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	—	4	2	no	—
269	C₄²	—	53	1	no	—
270	C₄²	—	53	1	no	—
271	C₄²	—	53	1	no	—
272	C₄²	D₄×D₅₃	1	1	yes	characteristic
273	C₁₆	—	2	1	no	—
274	C₁₆	—	53	2	no	—
275	C₁₆	—	106	1	no	—
276	C₁₆	#11719945	1	2	yes	—
277	C₂³	—	53	1	no	—
278	C₂³	—	212	2	no	—
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	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	53	1	no	—
287	C₂×C₄	—	53	1	no	—
288	C₂×C₄	—	53	1	no	—
289	C₂×C₄	—	106	2	no	—
290	C₂×C₄	—	106	2	no	—
291	C₂×C₄	—	106	1	no	—
292	C₂×C₄	—	106	1	no	—
293	C₂×C₄	—	424	4	no	—
294	C₂×C₄	—	212	2	no	—
295	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
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×#11719945	1	1	yes	characteristic
303	C₂²	—	53	2	no	—
304	C₂²	—	106	1	no	—
305	C₂²	—	106	1	no	—
306	C₂²	—	424	2	no	—
307	C₂²	—	424	2	no	—
308	C₂²	—	424	2	no	—
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	—	106	1	no	—
314	C₄	—	106	1	no	—
315	C₄	—	106	1	no	—
316	C₄	—	424	2	no	—
317	C₄	#16811056	1	2	yes	—
318	C₄	#16811170	1	1	yes	characteristic
319	C₄	#135×D₅₃	1	1	yes	characteristic
320	C₂	—	848	2	no	—
321	C₂	—	106	2	no	—
322	C₂	#20285291	1	1	yes	characteristic
323	C₂	#508×D₅₃	1	1	yes	characteristic
324	C₂	C₂×#16811056	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₄×Q₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#20276802	C₂
hypercenter	C₄×Q₃₂	D₅₃
hypocenter	C₅₃	#33735
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
1. #20285288	C₂	semidirect
2. #20276992	C₂	semidirect
3. #20276977	C₂	semidirect
4. #20276948	C₂	semidirect
5. #20276814	C₂	semidirect
6. #20276803	C₂	semidirect
8. C₄×#13133160	C₂	semidirect
9. C₄×D₈₄₈	C₂	semidirect
10. C₄×C₁₆×D₅₃	C₂	semidirect
11. C₄×Q₃₂×C₅₃	C₂	semidirect
165. C₄×Q₃₂	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
#20285288	2
#20276992	2
#20276977	2
#20276948	2
#20276814	2
#20276803	2
#20276802	2
C₄×#13133160	2
C₄×D₈₄₈	2
C₄×C₁₆×D₅₃	2
C₄×Q₃₂×C₅₃	2
#33735	53