Subgroups — #27541916

Order 13568.

486 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287728	C₂	1	1	yes	maximal, characteristic
2	#20276954	C₂	1	2	yes	maximal, complemented
3	#20276940	C₂	1	1	yes	maximal, characteristic
4	#20276864	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276849	C₂	1	2	yes	maximal, complemented
6	#20276777	C₂	1	2	yes	maximal, complemented
7	#20276755	C₂	1	1	yes	maximal, characteristic
8	#20276624	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276577	C₂	1	2	yes	maximal, complemented
10	C₄×#13133163	C₂	1	1	yes	maximal, characteristic, complemented
11	Q₈×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811358	—	2	1	no	—
13	C₂×#13133163	C₂²	1	1	yes	characteristic
14	C₂×#13133160	C₂²	1	2	yes	—
15	#16811007	C₂²	1	2	yes	—
16	#16811000	C₂²	1	1	yes	characteristic
17	#16810839	—	2	2	no	—
18	#16810824	—	2	1	no	—
19	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
20	C₄×D₄₂₄	C₂²	1	2	yes	—
21	#16810448	C₂²	1	1	yes	characteristic
22	#16810448	C₂²	1	2	yes	—
23	C₄⋊#11719908	C₂²	1	1	yes	characteristic
24	C₄⋊#11719908	C₂²	1	2	yes	—
25	#16810427	C₂²	1	1	yes	characteristic
26	#16810426	C₂²	1	2	yes	—
27	#16810426	C₂²	1	1	yes	characteristic
28	#16810423	C₂²	1	2	yes	—
29	#16810423	C₂²	1	2	yes	—
30	#16810423	C₂²	1	2	yes	—
31	#16810399	C₂²	1	2	yes	—
32	C₄×#11719908	C₂²	1	1	yes	characteristic
33	Q₈²×C₅₃	—	2	1	no	—
34	C₅₃×#500	C₂²	1	2	yes	—
35	C₅₃×#500	C₂²	1	1	yes	characteristic
36	C₅₃×#477	—	2	1	no	—
37	C₅₃×#477	—	2	2	no	—
38	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
41	C₂×Dic₂₁₂	C₂³	1	1	yes	characteristic
42	C₂×#11719913	—	2	2	no	—
43	C₂×D₄₂₄	—	2	1	no	—
44	C₂×D₄₂₄	C₂³	1	2	yes	—
45	#13133214	—	2	1	no	—
46	#13133206	—	4	1	no	—
47	#13133205	—	2	2	no	—
48	#13133202	—	4	2	no	—
49	C₄×Dic₁₀₆	—	2	1	no	—
50	C₄×D₂₁₂	—	2	2	no	—
51	#13133179	—	4	1	no	—
52	#13133163	—	2	2	no	—
53	#13133160	—	2	4	no	—
54	C₂×#11719908	C₂³	1	2	yes	—
55	C₂×#11719908	C₂³	1	1	yes	characteristic
56	C₂×#11719908	C₂³	1	1	yes	characteristic
57	#13133151	C₂³	1	1	yes	characteristic
58	#13133142	—	2	1	no	—
59	#13133139	—	2	2	no	—
60	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₅₃	—	2	2	no	—
62	C₄⋊Q₈×C₅₃	—	2	1	no	—
63	C₄⋊Q₈×C₅₃	—	4	1	no	—
64	C₄⋊Q₈×C₅₃	—	4	2	no	—
65	C₄×Q₈×C₅₃	—	2	2	no	—
66	C₄×Q₈×C₅₃	—	2	1	no	—
67	C₄×Q₈×C₅₃	—	4	1	no	—
68	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
69	#107×C₅₃	—	2	2	no	—
70	#107×C₅₃	—	2	1	no	—
71	#107×C₅₃	C₂³	1	1	yes	characteristic
72	#107×C₅₃	C₂³	1	2	yes	—
73	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
74	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
75	#103×C₅₃	—	2	2	no	—
76	#103×C₅₃	—	2	1	no	—
77	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₅₃	C₂³	1	2	yes	—
79	C₂×Q₈×C₅₃	—	2	1	no	—
80	C₂×Q₈×C₅₃	—	4	2	no	—
81	C₂×Q₈×C₅₃	—	4	1	no	—
82	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
83	C₂×D₂₁₂	—	2	2	no	—
84	C₂×D₂₁₂	—	4	1	no	—
85	C₂×D₂₁₂	—	4	1	no	—
86	C₂×C₄×D₅₃	—	4	2	no	—
87	C₂×Dic₁₀₆	—	2	1	no	—
88	Q₁₆×C₅₃	—	2	1	no	—
89	Q₁₆×C₅₃	#41	1	2	yes	—
90	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₅₃	—	2	2	no	—
94	C₄⋊C₄×C₅₃	—	2	1	no	—
95	C₄⋊C₄×C₅₃	—	4	2	no	—
96	C₄⋊C₄×C₅₃	—	4	1	no	—
97	C₄⋊C₄×C₅₃	—	4	2	no	—
98	C₄⋊C₄×C₅₃	—	4	1	no	—
99	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
100	C₄⋊C₄×C₅₃	D₈	1	2	yes	—
101	C₄²×C₅₃	—	4	2	no	—
102	C₄²×C₅₃	—	4	1	no	—
103	C₄²×C₅₃	C₂×D₄	1	2	yes	—
104	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
105	#11719921	—	4	2	no	—
106	#11719921	—	4	2	no	—
107	#11719921	—	4	2	no	—
108	C₄⋊Dic₅₃	—	2	1	no	—
109	#11719919	—	4	1	no	—
110	C₄×Dic₅₃	—	4	1	no	—
111	Dic₂₁₂	—	2	1	no	—
112	Dic₂₁₂	—	2	1	no	—
113	D₄₂₄	—	2	2	no	—
114	D₄₂₄	—	2	2	no	—
115	D₄₂₄	—	8	1	no	—
116	#11719913	—	8	2	no	—
117	#11719908	—	2	2	no	—
118	#11719908	—	2	1	no	—
119	#11719908	—	2	1	no	—
120	C₂²×D₅₃	—	4	2	no	—
121	Q₈×C₅₃	—	2	2	no	—
122	Q₈×C₅₃	—	2	2	no	—
123	Q₈×C₅₃	—	4	1	no	—
124	Q₈×C₅₃	—	16	2	no	—
125	Q₈×C₅₃	—	16	1	no	—
126	C₂×C₄×C₅₃	—	4	1	no	—
127	C₂×C₄×C₅₃	—	4	1	no	—
128	C₂×C₄×C₅₃	—	4	2	no	—
129	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×Dic₅₃	—	4	1	no	—
135	C₂×Dic₅₃	—	4	1	no	—
136	D₂₁₂	—	4	2	no	—
137	D₂₁₂	—	4	2	no	—
138	D₂₁₂	—	8	2	no	—
139	D₂₁₂	—	16	1	no	—
140	C₄×D₅₃	—	16	2	no	—
141	Dic₁₀₆	—	4	1	no	—
142	Dic₁₀₆	—	4	1	no	—
143	C₈×C₅₃	—	2	2	no	—
144	C₈×C₅₃	—	2	1	no	—
145	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
146	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
147	#33942	—	53	1	no	maximal, Sylow
148	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
149	C₂×D₅₃	—	8	2	no	—
150	C₂×D₅₃	—	8	2	no	—
151	C₂×D₅₃	—	8	2	no	—
152	C₄×C₅₃	—	2	2	no	—
153	C₄×C₅₃	—	2	2	no	—
154	C₄×C₅₃	—	2	1	no	—
155	C₄×C₅₃	—	4	2	no	—
156	C₄×C₅₃	—	8	1	no	—
157	C₄×C₅₃	—	8	2	no	—
158	C₄×C₅₃	#510	1	2	yes	—
159	C₄×C₅₃	#548	1	1	yes	characteristic
160	C₄×C₅₃	#577	1	1	yes	characteristic
161	Dic₅₃	—	8	1	no	—
162	Dic₅₃	—	8	1	no	—
163	#3396	—	53	1	no	—
164	Q₈×Q₁₆	D₅₃	1	1	yes	characteristic, complemented
165	#2243	—	53	2	no	—
166	#2240	—	53	1	no	—
167	#2227	—	53	1	no	—
168	#2224	—	53	2	no	—
169	#2211	—	53	2	no	—
170	#2207	—	53	1	no	—
171	#2183	—	53	1	no	—
172	#2175	—	53	2	no	—
173	C₄×Q₃₂	—	53	1	no	—
174	C₂×C₅₃	#3358	1	1	yes	characteristic
175	C₂×C₅₃	C₂×#510	1	1	yes	characteristic
176	C₂×C₅₃	#3415	1	1	yes	characteristic
177	D₅₃	—	16	2	no	—
178	#559	—	106	1	no	—
179	Q₈²	—	2	1	no	—
180	C₂×Q₃₂	—	53	1	no	—
181	C₂×SD₃₂	—	53	2	no	—
182	#500	C₂×D₅₃	1	2	yes	—
183	#500	C₂×D₅₃	1	1	yes	characteristic
184	#495	—	53	2	no	—
185	#493	—	53	1	no	—
186	#477	—	2	1	no	—
187	#477	—	2	2	no	—
188	#463	—	106	2	no	—
189	#460	—	106	1	no	—
190	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
191	C₄×Q₁₆	—	53	1	no	—
192	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
193	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
194	C₄×D₈	—	53	2	no	—
195	#366	—	53	1	no	—
196	#366	—	53	2	no	—
197	C₄⋊C₁₆	—	53	1	no	—
198	C₄⋊C₁₆	—	53	2	no	—
199	#358	—	53	1	no	—
200	#358	—	53	1	no	—
201	#358	—	53	2	no	—
202	#357	—	53	2	no	—
203	#357	—	53	2	no	—
204	#357	—	53	2	no	—
205	#346	—	53	2	no	—
206	C₄×C₁₆	—	53	1	no	—
207	C₅₃	#33942	1	1	yes	characteristic, complemented, Sylow
208	C₂×Q₁₆	—	53	1	no	—
209	C₂×Q₁₆	C₂²×D₅₃	1	1	yes	characteristic
210	C₂×SD₁₆	—	106	2	no	—
211	C₂×D₈	—	53	2	no	—
212	C₂×D₈	—	106	1	no	—
213	C₄⋊Q₈	—	2	2	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	4	1	no	—
216	C₄⋊Q₈	—	4	2	no	—
217	#127	—	106	1	no	—
218	#124	—	106	2	no	—
219	#124	—	212	1	no	—
220	#123	—	212	2	no	—
221	C₄×Q₈	—	2	2	no	—
222	C₄×Q₈	—	2	1	no	—
223	C₄×Q₈	—	4	1	no	—
224	C₄×Q₈	—	106	1	no	—
225	C₄×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
226	C₄×D₄	—	106	2	no	—
227	C₄×D₄	—	212	1	no	—
228	Q₃₂	—	106	2	no	—
229	SD₃₂	—	106	4	no	—
230	C₂×C₁₆	—	53	1	no	—
231	C₂×C₁₆	—	53	1	no	—
232	C₂×C₁₆	—	53	2	no	—
233	#107	—	2	2	no	—
234	#107	—	2	1	no	—
235	#107	—	53	1	no	—
236	#107	C₂²×D₅₃	1	2	yes	—
237	#107	C₂²×D₅₃	1	1	yes	characteristic
238	C₄⋊C₈	C₂²⋊D₅₃	1	1	yes	characteristic
239	C₄⋊C₈	C₂²⋊D₅₃	1	2	yes	—
240	#103	—	2	2	no	—
241	#103	—	2	1	no	—
242	#103	—	106	1	no	—
243	#102	—	106	2	no	—
244	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₅₃	1	2	yes	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	4	2	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	106	1	no	—
250	C₂×Q₈	#11719922	1	1	yes	characteristic
251	C₂×D₄	—	106	2	no	—
252	C₂×D₄	—	212	1	no	—
253	C₂×D₄	—	212	1	no	—
254	C₂²×C₄	—	212	2	no	—
255	Q₁₆	—	2	1	no	—
256	Q₁₆	—	106	1	no	—
257	Q₁₆	—	106	1	no	—
258	Q₁₆	#11719949	1	2	yes	—
259	SD₁₆	—	424	2	no	—
260	D₈	—	106	2	no	—
261	D₈	—	106	2	no	—
262	D₈	—	424	1	no	—
263	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
264	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
265	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	106	1	no	—
273	C₄⋊C₄	—	212	1	no	—
274	C₄⋊C₄	#11719922	1	1	yes	characteristic
275	C₄⋊C₄	#11719922	1	2	yes	—
276	C₂²⋊C₄	—	212	2	no	—
277	C₂²⋊C₄	—	212	2	no	—
278	C₂²⋊C₄	—	212	2	no	—
279	C₄²	—	4	2	no	—
280	C₄²	—	4	1	no	—
281	C₄²	—	212	1	no	—
282	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
283	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
284	C₁₆	—	106	2	no	—
285	C₁₆	—	106	1	no	—
286	C₁₆	—	106	1	no	—
287	C₂³	—	212	2	no	—
288	Q₈	—	2	2	no	—
289	Q₈	—	2	2	no	—
290	Q₈	—	4	1	no	—
291	Q₈	—	16	2	no	—
292	Q₈	—	16	1	no	—
293	Q₈	—	212	1	no	—
294	Q₈	—	212	1	no	—
295	D₄	—	848	1	no	—
296	D₄	—	424	2	no	—
297	D₄	—	212	2	no	—
298	D₄	—	212	2	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	848	2	no	—
303	C₂×C₄	—	212	1	no	—
304	C₂×C₄	—	212	1	no	—
305	C₂×C₄	C₂×#11719922	1	2	yes	—
306	C₂×C₄	C₂×#11719922	1	2	yes	—
307	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
308	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
309	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	2	no	—
312	C₈	C₂×#11719949	1	1	yes	characteristic
313	C₈	#13133265	1	1	yes	characteristic
314	C₂²	—	424	2	no	—
315	C₂²	—	424	2	no	—
316	C₂²	—	424	2	no	—
317	C₂²	C₂²×#11719922	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	1	no	—
321	C₄	—	4	2	no	—
322	C₄	—	8	1	no	—
323	C₄	—	8	2	no	—
324	C₄	—	424	1	no	—
325	C₄	—	424	1	no	—
326	C₄	#16811064	1	2	yes	—
327	C₄	#16811284	1	1	yes	characteristic
328	C₄	#16811431	1	1	yes	characteristic
329	C₂	—	848	2	no	—
330	C₂	#20287350	1	1	yes	characteristic
331	C₂	C₂×#16811064	1	1	yes	characteristic
332	C₂	#20287854	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	Q₈×Q₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#16811000	C₂²
hypercenter	Q₈×Q₁₆	D₅₃
hypocenter	C₅₃	#33942
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #20276954	C₂	semidirect
4. #20276864	C₂	semidirect
5. #20276849	C₂	semidirect
6. #20276777	C₂	semidirect
8. #20276624	C₂	semidirect
9. #20276577	C₂	semidirect
10. C₄×#13133163	C₂	semidirect
11. Q₈×Q₁₆×C₅₃	C₂	semidirect
164. Q₈×Q₁₆	D₅₃	semidirect
207. C₅₃	#33942	semidirect

Maximal subgroups

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

subgroup	index
#20287728	2
#20276954	2
#20276940	2
#20276864	2
#20276849	2
#20276777	2
#20276755	2
#20276624	2
#20276577	2
C₄×#13133163	2
Q₈×Q₁₆×C₅₃	2
#33942	53