Subgroups — #27539656

Order 13568.

558 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285302	C₂	1	1	yes	maximal, characteristic, complemented
2	#20285295	C₂	1	1	yes	maximal, characteristic
3	#20276971	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276944	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276940	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276812	C₂	1	4	yes	maximal, complemented
7	#20276566	C₂	1	4	yes	maximal, complemented
8	C₄×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#2254	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811178	—	2	1	no	—
11	#16811174	—	2	1	no	—
12	C₂×#13133162	C₂²	1	4	yes	—
13	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
14	#16811003	C₂²	1	1	yes	characteristic
15	#16811000	C₂²	1	1	yes	characteristic
16	#16810975	—	2	1	no	—
17	#16810971	—	2	1	no	—
18	#16810865	—	2	1	no	—
19	#16810865	C₂²	1	2	yes	—
20	#16810860	—	2	1	no	—
21	#16810860	C₂²	1	2	yes	—
22	#16810717	—	2	1	no	—
23	#16810717	C₂²	1	2	yes	—
24	#16810705	—	2	1	no	—
25	#16810705	C₂²	1	2	yes	—
26	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
27	#16810426	C₂²	1	4	yes	—
28	#16810423	C₂²	1	4	yes	—
29	C₄×#11719908	C₂²	1	1	yes	characteristic
30	C₁₆×Dic₅₃	C₂²	1	2	yes	—
31	C₅₃×#500	C₂²	1	1	yes	characteristic
32	C₅₃×#500	C₂²	1	1	yes	characteristic
33	C₅₃×#367	C₂²	1	4	yes	—
34	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×#11719949	—	4	1	no	—
36	C₂×Dic₂₁₂	—	2	1	no	—
37	C₂×Dic₂₁₂	C₂³	1	2	yes	—
38	C₂×D₄₂₄	—	2	1	no	—
39	C₂×D₄₂₄	C₂³	1	2	yes	—
40	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
41	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
42	#13133217	—	2	1	no	—
43	#13133214	—	2	1	no	—
44	#13133208	—	4	1	no	—
45	#13133206	—	4	1	no	—
46	#13133197	—	2	2	no	—
47	#13133193	—	4	1	no	—
48	#13133192	—	2	2	no	—
49	#13133192	—	4	1	no	—
50	Q₈×Dic₅₃	—	4	1	no	—
51	#13133182	—	2	2	no	—
52	#13133179	—	2	2	no	—
53	#13133179	—	4	1	no	—
54	#13133175	—	4	1	no	—
55	C₄²×D₅₃	D₄	1	1	yes	characteristic
56	#13133162	—	2	8	no	—
57	C₂×#11719908	C₂³	1	2	yes	—
58	C₁₆×D₅₃	—	2	4	no	—
59	#13133140	—	2	2	no	—
60	#13133140	—	2	2	no	—
61	#13133137	—	2	2	no	—
62	#13133137	—	2	2	no	—
63	C₄×#88322	D₄	1	1	yes	characteristic
64	C₈×Dic₅₃	D₄	1	1	yes	characteristic
65	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
66	C₄⋊Q₈×C₅₃	—	2	1	no	—
67	C₄⋊Q₈×C₅₃	—	2	1	no	—
68	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
69	#107×C₅₃	—	2	1	no	—
70	#107×C₅₃	—	2	1	no	—
71	#107×C₅₃	C₂³	1	2	yes	—
72	#107×C₅₃	C₂³	1	2	yes	—
73	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₂×Q₈×C₅₃	—	4	1	no	—
75	C₂×Q₈×C₅₃	—	4	1	no	—
76	#11719949	—	8	4	no	—
77	C₂×D₂₁₂	—	2	2	no	—
78	C₂×D₂₁₂	—	4	2	no	—
79	C₂×D₂₁₂	—	4	1	no	—
80	C₂×D₂₁₂	—	4	1	no	—
81	C₂×C₄×D₅₃	—	4	2	no	—
82	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
83	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
84	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂×Dic₁₀₆	—	2	2	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	—	2	2	no	—
90	C₄⋊C₄×C₅₃	—	4	1	no	—
91	C₄⋊C₄×C₅₃	—	4	1	no	—
92	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
93	#11719921	—	4	2	no	—
94	#11719921	—	4	2	no	—
95	#11719921	—	4	2	no	—
96	C₄⋊Dic₅₃	—	4	1	no	—
97	C₄⋊Dic₅₃	—	4	1	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	C₂×#88322	C₂×D₄	1	1	yes	characteristic
105	C₂×#88322	C₂×D₄	1	1	yes	characteristic
106	Dic₂₁₂	—	2	4	no	—
107	Dic₂₁₂	—	4	2	no	—
108	D₄₂₄	—	2	4	no	—
109	D₄₂₄	—	4	2	no	—
110	C₈×D₅₃	—	2	2	no	—
111	C₈×D₅₃	—	2	2	no	—
112	C₁₆×C₅₃	#41	1	4	yes	complemented
113	#11719908	—	2	2	no	—
114	C₂²×D₅₃	—	4	2	no	—
115	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
116	Q₈×C₅₃	—	8	2	no	—
117	Q₈×C₅₃	—	8	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	D₂₁₂	—	8	2	no	—
131	D₂₁₂	—	8	2	no	—
132	C₄×D₅₃	—	2	1	no	—
133	C₄×D₅₃	—	2	1	no	—
134	C₄×D₅₃	—	2	2	no	—
135	C₄×D₅₃	—	8	4	no	—
136	C₄×D₅₃	SD₃₂	1	4	yes	complemented
137	Dic₁₀₆	—	4	2	no	—
138	Dic₁₀₆	—	4	2	no	—
139	C₈×C₅₃	#135	1	2	yes	—
140	C₈×C₅₃	C₂×#41	1	2	yes	—
141	#88322	—	2	1	no	—
142	#88322	—	2	1	no	—
143	#33736	—	53	1	no	maximal, Sylow
144	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
145	C₂×D₅₃	—	2	1	no	—
146	C₂×D₅₃	—	2	1	no	—
147	C₂×D₅₃	—	8	2	no	—
148	C₂×D₅₃	—	8	2	no	—
149	C₂×D₅₃	—	8	2	no	—
150	C₂×D₅₃	C₂×SD₃₂	1	2	yes	—
151	C₄×C₅₃	—	8	2	no	—
152	C₄×C₅₃	—	8	2	no	—
153	C₄×C₅₃	C₂×SD₃₂	1	2	yes	—
154	C₄×C₅₃	#508	1	2	yes	—
155	C₄×C₅₃	#532	1	1	yes	characteristic
156	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
157	Dic₅₃	—	2	1	no	—
158	Dic₅₃	—	2	1	no	—
159	Dic₅₃	—	8	2	no	—
160	Dic₅₃	C₂×SD₃₂	1	2	yes	—
161	#3170	—	53	1	no	—
162	#3169	—	53	1	no	—
163	#2254	D₅₃	1	1	yes	characteristic, complemented
164	#2248	—	53	1	no	—
165	#2241	—	53	1	no	—
166	#2240	—	53	1	no	—
167	#2217	—	53	4	no	—
168	C₄×SD₃₂	—	53	4	no	—
169	C₂×C₄×C₁₆	—	53	1	no	—
170	C₂×C₅₃	#3169	1	1	yes	characteristic
171	C₂×C₅₃	C₂²×SD₃₂	1	1	yes	characteristic
172	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
173	D₅₃	—	2	2	no	—
174	D₅₃	—	16	2	no	—
175	#533	—	106	1	no	—
176	#532	—	106	1	no	—
177	C₂×SD₃₂	—	53	4	no	—
178	C₂²×C₁₆	—	53	2	no	—
179	#500	C₂×D₅₃	1	1	yes	characteristic
180	#500	C₂×D₅₃	1	1	yes	characteristic
181	#494	—	53	1	no	—
182	#493	—	53	1	no	—
183	#487	—	106	1	no	—
184	#486	—	106	1	no	—
185	#467	—	53	2	no	—
186	#467	—	106	1	no	—
187	#466	—	53	2	no	—
188	#466	—	106	1	no	—
189	C₄×Q₁₆	—	53	2	no	—
190	C₄×Q₁₆	—	106	1	no	—
191	C₄×D₈	—	53	2	no	—
192	C₄×D₈	—	106	1	no	—
193	C₂×C₄×C₈	—	53	1	no	—
194	#367	C₂×D₅₃	1	4	yes	—
195	#358	—	53	4	no	—
196	#357	—	53	4	no	—
197	C₄×C₁₆	—	53	2	no	—
198	C₄×C₁₆	—	53	1	no	—
199	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
200	C₅₃	#33736	1	1	yes	characteristic, complemented, Sylow
201	C₂×#41	—	212	1	no	—
202	C₂×Q₁₆	—	53	2	no	—
203	C₂×Q₁₆	—	106	1	no	—
204	C₂×D₈	—	53	2	no	—
205	C₂×D₈	—	106	1	no	—
206	C₂²×C₈	—	53	1	no	—
207	C₂²×C₈	—	53	1	no	—
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	106	1	no	—
211	#127	—	106	1	no	—
212	#125	—	212	1	no	—
213	#124	—	212	1	no	—
214	C₂²⋊Q₈	—	106	2	no	—
215	C₂²⋊Q₈	—	212	1	no	—
216	#121	—	106	2	no	—
217	#121	—	212	1	no	—
218	C₄×Q₈	—	106	2	no	—
219	C₄×Q₈	—	212	1	no	—
220	C₄×D₄	—	106	2	no	—
221	C₄×D₄	—	212	1	no	—
222	#117	—	212	1	no	—
223	C₂×C₄²	—	53	1	no	—
224	SD₃₂	—	106	8	no	—
225	C₂×C₁₆	—	53	2	no	—
226	C₂×C₁₆	—	106	4	no	—
227	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
228	#107	—	2	1	no	—
229	#107	—	2	1	no	—
230	#107	C₂²×D₅₃	1	2	yes	—
231	#107	C₂²×D₅₃	1	2	yes	—
232	#103	—	106	2	no	—
233	#103	—	106	2	no	—
234	#102	—	106	2	no	—
235	#102	—	106	2	no	—
236	C₄×C₈	—	53	1	no	—
237	C₄×C₈	—	53	1	no	—
238	C₄×C₈	—	53	1	no	—
239	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
240	#41	—	424	4	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×Q₈	—	106	2	no	—
244	C₂×D₄	—	106	2	no	—
245	C₂×D₄	—	212	2	no	—
246	C₂×D₄	—	212	1	no	—
247	C₂×D₄	—	212	1	no	—
248	C₂²×C₄	—	53	1	no	—
249	C₂²×C₄	—	53	1	no	—
250	C₂²×C₄	—	53	1	no	—
251	C₂²×C₄	—	212	2	no	—
252	Q₁₆	—	106	4	no	—
253	Q₁₆	—	212	2	no	—
254	D₈	—	106	4	no	—
255	D₈	—	212	2	no	—
256	C₂×C₈	—	53	1	no	—
257	C₂×C₈	—	53	1	no	—
258	C₂×C₈	—	106	2	no	—
259	C₂×C₈	—	106	2	no	—
260	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
261	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	212	2	no	—
267	C₄⋊C₄	—	212	2	no	—
268	C₄⋊C₄	—	212	1	no	—
269	C₄⋊C₄	—	212	1	no	—
270	C₂²⋊C₄	—	212	2	no	—
271	C₂²⋊C₄	—	212	2	no	—
272	C₂²⋊C₄	—	212	2	no	—
273	C₄²	—	53	1	no	—
274	C₄²	—	53	1	no	—
275	C₄²	—	53	1	no	—
276	C₄²	—	212	2	no	—
277	C₄²	D₄×D₅₃	1	1	yes	characteristic
278	C₁₆	—	106	2	no	—
279	C₁₆	#11719949	1	4	yes	complemented
280	C₂³	—	53	1	no	—
281	C₂³	—	212	2	no	—
282	Q₈	—	8	2	no	—
283	Q₈	—	8	2	no	—
284	Q₈	—	212	2	no	—
285	Q₈	—	212	2	no	—
286	D₄	—	424	4	no	—
287	D₄	—	424	2	no	—
288	D₄	—	424	2	no	—
289	D₄	—	212	2	no	—
290	D₄	—	212	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	53	1	no	—
294	C₂×C₄	—	53	4	no	—
295	C₂×C₄	—	53	1	no	—
296	C₂×C₄	—	53	1	no	—
297	C₂×C₄	—	106	2	no	—
298	C₂×C₄	—	106	1	no	—
299	C₂×C₄	—	106	1	no	—
300	C₂×C₄	—	424	4	no	—
301	C₂×C₄	—	212	2	no	—
302	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
303	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
304	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
305	C₈	—	106	1	no	—
306	C₈	—	106	1	no	—
307	C₈	#13133240	1	2	yes	—
308	C₈	C₂×#11719949	1	2	yes	—
309	C₂²	—	53	2	no	—
310	C₂²	—	106	1	no	—
311	C₂²	—	106	1	no	—
312	C₂²	—	424	2	no	—
313	C₂²	—	424	2	no	—
314	C₂²	—	424	2	no	—
315	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
316	C₄	—	8	2	no	—
317	C₄	—	8	2	no	—
318	C₄	—	53	2	no	—
319	C₄	—	106	1	no	—
320	C₄	—	106	1	no	—
321	C₄	—	424	2	no	—
322	C₄	SD₃₂×D₅₃	1	2	yes	—
323	C₄	#16811054	1	2	yes	—
324	C₄	#16811174	1	1	yes	characteristic
325	C₄	C₂×#13133240	1	1	yes	characteristic
326	C₂	—	848	2	no	—
327	C₂	—	106	2	no	—
328	C₂	#20285295	1	1	yes	characteristic
329	C₂	C₂×SD₃₂×D₅₃	1	1	yes	characteristic
330	C₂	C₂×#16811054	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×#2254	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#20285295	C₂
hypercenter	#2254	D₅₃
hypocenter	C₅₃	#33736
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

Complements

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

N	H ≅ G/N	type
1. #20285302	C₂	semidirect
3. #20276971	C₂	semidirect
4. #20276944	C₂	semidirect
5. #20276940	C₂	semidirect
6. #20276812	C₂	semidirect
7. #20276566	C₂	semidirect
8. C₄×C₁₆×D₅₃	C₂	semidirect
9. C₅₃×#2254	C₂	semidirect
112. C₁₆×C₅₃	#41	semidirect
136. C₄×D₅₃	SD₃₂	semidirect
163. #2254	D₅₃	semidirect
200. C₅₃	#33736	semidirect
279. C₁₆	#11719949	semidirect

Maximal subgroups

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

subgroup	index
#20285302	2
#20285295	2
#20276971	2
#20276944	2
#20276940	2
#20276812	2
#20276566	2
C₄×C₁₆×D₅₃	2
C₅₃×#2254	2
#33736	53