Subgroups — #27541747

Order 13568.

486 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	#20287583	C₂	1	1	yes	maximal, characteristic, complemented
2	#20276946	C₂	1	2	yes	maximal, complemented
3	#20276942	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276879	C₂	1	2	yes	maximal, complemented
5	#20276862	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276767	C₂	1	2	yes	maximal
7	#20276757	C₂	1	1	yes	maximal, characteristic, complemented
8	Q₈×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13133160	C₂	1	2	yes	maximal, complemented
10	C₄×#13133158	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#3393	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811333	—	2	1	no	—
13	C₂×#13133160	C₂²	1	2	yes	—
14	C₂×#13133158	C₂²	1	1	yes	characteristic
15	#16811028	C₂²	1	1	yes	characteristic
16	#16811025	C₂²	1	2	yes	—
17	#16810924	—	2	2	no	—
18	#16810914	—	2	1	no	—
19	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
20	C₄×D₄₂₄	C₂²	1	2	yes	—
21	#16810450	C₂²	1	2	yes	—
22	#16810448	C₂²	1	1	yes	characteristic
23	C₄⋊#11719908	C₂²	1	1	yes	characteristic
24	C₄⋊#11719908	C₂²	1	2	yes	—
25	#16810427	C₂²	1	2	yes	—
26	#16810427	C₂²	1	2	yes	—
27	#16810427	C₂²	1	2	yes	—
28	#16810424	C₂²	1	2	yes	—
29	#16810424	C₂²	1	1	yes	characteristic
30	#16810423	C₂²	1	1	yes	characteristic
31	C₄×#11719908	C₂²	1	1	yes	characteristic
32	C₄×#11719908	C₂²	1	2	yes	—
33	C₅₃×#556	—	2	1	no	—
34	C₅₃×#495	C₂²	1	2	yes	—
35	C₅₃×#494	C₂²	1	1	yes	characteristic
36	C₅₃×#464	—	2	2	no	—
37	C₅₃×#461	—	2	1	no	—
38	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
40	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×D₄₂₄	C₂³	1	1	yes	characteristic
42	#13133217	—	2	1	no	—
43	#13133209	—	2	2	no	—
44	#13133195	—	4	2	no	—
45	#13133193	—	4	1	no	—
46	C₄×D₂₁₂	—	2	1	no	—
47	C₄×D₂₁₂	—	2	2	no	—
48	C₄⋊C₄×D₅₃	—	4	1	no	—
49	#13133160	—	2	4	no	—
50	#13133158	—	2	2	no	—
51	C₂×#11719908	C₂³	1	1	yes	characteristic
52	C₂×#11719908	C₂³	1	2	yes	—
53	C₂×#11719908	C₂³	1	1	yes	characteristic
54	#13133151	—	2	1	no	—
55	#13133151	C₂³	1	1	yes	characteristic
56	#13133151	C₂³	1	2	yes	—
57	#13133149	—	2	2	no	—
58	#13133139	—	2	2	no	—
59	#13133139	—	2	1	no	—
60	C₂×Q₁₆×C₅₃	—	2	1	no	—
61	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
62	C₂×SD₁₆×C₅₃	—	2	2	no	—
63	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₅₃	—	2	1	no	—
65	#126×C₅₃	—	4	2	no	—
66	#124×C₅₃	—	2	2	no	—
67	C₂²⋊Q₈×C₅₃	—	4	1	no	—
68	C₄×Q₈×C₅₃	—	2	2	no	—
69	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
70	C₄×D₄×C₅₃	—	2	1	no	—
71	#117×C₅₃	—	4	1	no	—
72	#107×C₅₃	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
74	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
75	#103×C₅₃	—	2	2	no	—
76	#102×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	2	no	—
80	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
81	C₂×D₄×C₅₃	—	2	1	no	—
82	C₂²×C₄×C₅₃	—	4	1	no	—
83	C₂×D₂₁₂	—	2	1	no	—
84	C₂×C₄×D₅₃	—	4	1	no	—
85	C₂×C₄×D₅₃	—	4	2	no	—
86	C₂×Dic₁₀₆	—	4	1	no	—
87	Q₁₆×C₅₃	—	2	2	no	—
88	Q₁₆×C₅₃	—	2	2	no	—
89	Q₁₆×C₅₃	—	8	1	no	—
90	SD₁₆×C₅₃	—	8	2	no	—
91	D₈×C₅₃	—	2	1	no	—
92	D₈×C₅₃	—	2	1	no	—
93	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₅₃	—	2	1	no	—
97	C₄⋊C₄×C₅₃	—	4	1	no	—
98	C₄⋊C₄×C₅₃	—	4	2	no	—
99	C₄⋊C₄×C₅₃	—	4	1	no	—
100	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
101	C₄⋊C₄×C₅₃	D₈	1	2	yes	—
102	C₂²⋊C₄×C₅₃	—	4	2	no	—
103	C₂²⋊C₄×C₅₃	—	4	1	no	—
104	C₂²⋊C₄×C₅₃	—	4	2	no	—
105	C₄²×C₅₃	—	4	2	no	—
106	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₂×D₄	1	2	yes	—
108	#11719921	—	4	2	no	—
109	#11719921	—	4	1	no	—
110	C₄⋊Dic₅₃	—	2	1	no	—
111	C₄⋊Dic₅₃	—	2	2	no	—
112	C₄⋊Dic₅₃	—	4	1	no	—
113	#11719919	—	4	2	no	—
114	#11719919	—	4	2	no	—
115	D₄₂₄	—	2	1	no	—
116	D₄₂₄	#41	1	2	yes	—
117	#11719908	—	2	1	no	—
118	#11719908	—	2	2	no	—
119	#11719908	—	2	1	no	—
120	C₂³×C₅₃	—	4	1	no	—
121	C₂²×D₅₃	—	4	1	no	—
122	Q₈×C₅₃	—	2	2	no	—
123	Q₈×C₅₃	—	4	2	no	—
124	Q₈×C₅₃	—	4	2	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	2	no	—
129	C₂×C₄×C₅₃	—	8	2	no	—
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	2	yes	—
133	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×Dic₅₃	—	4	2	no	—
136	C₂×Dic₅₃	—	4	1	no	—
137	D₂₁₂	—	2	2	no	—
138	D₂₁₂	—	4	1	no	—
139	C₄×D₅₃	—	8	2	no	—
140	C₄×D₅₃	—	16	2	no	—
141	Dic₁₀₆	—	16	1	no	—
142	C₈×C₅₃	—	2	2	no	—
143	C₈×C₅₃	—	2	1	no	—
144	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
145	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
146	#33927	—	53	1	no	maximal, Sylow
147	C₂²×C₅₃	—	8	1	no	—
148	C₂²×C₅₃	—	8	1	no	—
149	C₂²×C₅₃	—	8	1	no	—
150	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
151	C₂×D₅₃	—	4	2	no	—
152	C₂×D₅₃	—	8	1	no	—
153	C₂×D₅₃	—	8	1	no	—
154	C₄×C₅₃	—	2	2	no	—
155	C₄×C₅₃	—	2	1	no	—
156	C₄×C₅₃	—	2	2	no	—
157	C₄×C₅₃	—	8	2	no	—
158	C₄×C₅₃	—	8	1	no	—
159	C₄×C₅₃	#508	1	2	yes	—
160	C₄×C₅₃	#548	1	1	yes	characteristic
161	C₄×C₅₃	#577	1	1	yes	characteristic
162	Dic₅₃	—	8	1	no	—
163	Dic₅₃	—	8	2	no	—
164	#3393	D₅₃	1	1	yes	characteristic, complemented
165	#3380	—	53	1	no	—
166	#2242	—	53	2	no	—
167	#2241	—	53	1	no	—
168	#2229	—	53	2	no	—
169	#2226	—	53	1	no	—
170	#2210	—	53	2	no	—
171	#2208	—	53	1	no	—
172	Q₈×C₁₆	—	53	1	no	—
173	C₄×SD₃₂	—	53	2	no	—
174	C₄×D₁₆	—	53	1	no	—
175	C₂×C₅₃	—	16	1	no	—
176	C₂×C₅₃	#3358	1	1	yes	characteristic
177	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
178	C₂×C₅₃	#3416	1	1	yes	characteristic
179	D₅₃	—	8	2	no	—
180	#556	—	2	1	no	—
181	#554	—	106	1	no	—
182	C₂×SD₃₂	—	53	2	no	—
183	C₂×D₁₆	—	53	1	no	—
184	#500	—	53	1	no	—
185	#499	—	53	2	no	—
186	#495	C₂×D₅₃	1	2	yes	—
187	#494	C₂×D₅₃	1	1	yes	characteristic
188	#478	—	106	2	no	—
189	#476	—	106	1	no	—
190	#464	—	2	2	no	—
191	#461	—	2	1	no	—
192	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
193	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
194	C₄×D₈	—	53	1	no	—
195	C₄×D₈	—	53	2	no	—
196	C₄×D₈	C₂×D₅₃	1	1	yes	characteristic
197	#367	—	53	2	no	—
198	#366	—	53	1	no	—
199	C₄⋊C₁₆	—	53	1	no	—
200	C₄⋊C₁₆	—	53	2	no	—
201	#358	—	53	2	no	—
202	#358	—	53	2	no	—
203	#358	—	53	2	no	—
204	#357	—	53	1	no	—
205	#357	—	53	2	no	—
206	#357	—	53	1	no	—
207	C₄×C₁₆	—	53	1	no	—
208	C₄×C₁₆	—	53	2	no	—
209	C₅₃	#33927	1	1	yes	characteristic, complemented, Sylow
210	C₂×Q₁₆	—	2	1	no	—
211	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
212	C₂×SD₁₆	—	2	2	no	—
213	C₂×D₈	—	53	1	no	—
214	C₂×D₈	C₂²×D₅₃	1	1	yes	characteristic
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	106	1	no	—
217	#126	—	4	2	no	—
218	#125	—	106	2	no	—
219	#124	—	2	2	no	—
220	C₂²⋊Q₈	—	4	1	no	—
221	C₂²⋊Q₈	—	212	1	no	—
222	C₂²⋊Q₈	—	212	2	no	—
223	C₄×Q₈	—	2	2	no	—
224	C₄×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
225	C₄×D₄	—	2	1	no	—
226	C₄×D₄	—	106	1	no	—
227	C₄×D₄	—	106	2	no	—
228	#117	—	4	1	no	—
229	C₂×C₄⋊C₄	—	212	1	no	—
230	SD₃₂	—	106	4	no	—
231	D₁₆	—	106	2	no	—
232	C₂×C₁₆	—	53	1	no	—
233	C₂×C₁₆	—	53	2	no	—
234	C₂×C₁₆	—	53	1	no	—
235	#107	—	53	1	no	—
236	#107	—	53	2	no	—
237	#107	—	106	1	no	—
238	#107	C₂²×D₅₃	1	1	yes	characteristic
239	#106	—	106	2	no	—
240	C₄⋊C₈	C₂²⋊D₅₃	1	1	yes	characteristic
241	C₄⋊C₈	C₂²⋊D₅₃	1	2	yes	—
242	#103	—	2	2	no	—
243	#102	—	2	1	no	—
244	#102	—	106	2	no	—
245	#102	—	106	1	no	—
246	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₅₃	1	2	yes	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	212	1	no	—
250	C₂×Q₈	#11719922	1	1	yes	characteristic
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	106	1	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	212	2	no	—
255	C₂²×C₄	—	212	1	no	—
256	Q₁₆	—	2	2	no	—
257	Q₁₆	—	2	2	no	—
258	Q₁₆	—	8	1	no	—
259	SD₁₆	—	8	2	no	—
260	D₈	—	2	1	no	—
261	D₈	—	2	1	no	—
262	D₈	—	53	2	no	—
263	D₈	—	106	1	no	—
264	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
265	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
266	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	106	1	no	—
272	C₄⋊C₄	—	106	2	no	—
273	C₄⋊C₄	—	212	2	no	—
274	C₄⋊C₄	—	212	1	no	—
275	C₄⋊C₄	—	212	2	no	—
276	C₄⋊C₄	#11719922	1	1	yes	characteristic
277	C₄⋊C₄	#11719922	1	2	yes	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₂²⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	212	1	no	—
282	C₂²⋊C₄	—	212	2	no	—
283	C₄²	—	4	2	no	—
284	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
285	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
286	C₁₆	—	106	1	no	—
287	C₁₆	—	106	2	no	—
288	C₁₆	—	106	1	no	—
289	C₂³	—	4	1	no	—
290	C₂³	—	212	1	no	—
291	Q₈	—	2	2	no	—
292	Q₈	—	4	2	no	—
293	Q₈	—	4	2	no	—
294	Q₈	—	848	1	no	—
295	D₄	—	4	1	no	—
296	D₄	—	4	1	no	—
297	D₄	—	106	2	no	—
298	D₄	—	212	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	—	848	2	no	—
303	C₂×C₄	—	424	2	no	—
304	C₂×C₄	—	212	1	no	—
305	C₂×C₄	—	212	2	no	—
306	C₂×C₄	C₂×#11719922	1	2	yes	—
307	C₂×C₄	C₂×#11719922	1	2	yes	—
308	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
309	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
310	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	2	1	no	—
313	C₈	#41×D₅₃	1	1	yes	characteristic
314	C₈	#13133264	1	1	yes	characteristic
315	C₂²	—	8	1	no	—
316	C₂²	—	8	1	no	—
317	C₂²	—	8	1	no	—
318	C₂²	—	424	1	no	—
319	C₂²	—	424	1	no	—
320	C₂²	—	212	2	no	—
321	C₂²	C₂²×#11719922	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	1	no	—
325	C₄	—	8	1	no	—
326	C₄	—	8	2	no	—
327	C₄	—	424	1	no	—
328	C₄	—	424	2	no	—
329	C₄	#16811053	1	2	yes	—
330	C₄	#16811280	1	1	yes	characteristic
331	C₄	#16811434	1	1	yes	characteristic
332	C₂	—	16	1	no	—
333	C₂	—	424	2	no	—
334	C₂	#20287346	1	1	yes	characteristic
335	C₂	C₂×#16811053	1	1	yes	characteristic
336	C₂	#20287859	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#3393	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×#13133158	C₂²
hypercenter	#3393	D₅₃
hypocenter	C₅₃	#33927
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3393

Complements

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

N	H ≅ G/N	type
1. #20287583	C₂	semidirect
2. #20276946	C₂	semidirect
3. #20276942	C₂	semidirect
4. #20276879	C₂	semidirect
5. #20276862	C₂	semidirect
7. #20276757	C₂	semidirect
8. Q₈×#11719908	C₂	semidirect
9. C₄×#13133160	C₂	semidirect
11. C₅₃×#3393	C₂	semidirect
164. #3393	D₅₃	semidirect
209. C₅₃	#33927	semidirect

Maximal subgroups

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

subgroup	index
#20287583	2
#20276946	2
#20276942	2
#20276879	2
#20276862	2
#20276767	2
#20276757	2
Q₈×#11719908	2
C₄×#13133160	2
C₄×#13133158	2
C₅₃×#3393	2
#33927	53