Subgroups — #27541797

Order 13568.

484 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	#20287725	C₂	1	1	yes	maximal, characteristic
2	#20287608	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276948	C₂	1	2	yes	maximal, complemented
4	#20276940	C₂	1	1	yes	maximal, characteristic
5	#20276871	C₂	1	2	yes	maximal, complemented
6	#20276780	C₂	1	2	yes	maximal, complemented
7	#20276756	C₂	1	1	yes	maximal, characteristic
8	#20276617	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13133163	C₂	1	1	yes	maximal, characteristic, complemented
10	#20276566	C₂	1	2	yes	maximal, complemented
11	C₅₃×#2225	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811358	—	2	1	no	—
13	#16811348	—	2	1	no	—
14	C₂×#13133163	C₂²	1	1	yes	characteristic
15	C₂×#13133162	C₂²	1	2	yes	—
16	#16811024	C₂²	1	2	yes	—
17	#16811008	C₂²	1	2	yes	—
18	#16811000	C₂²	1	1	yes	characteristic
19	#16810929	—	2	2	no	—
20	#16810905	—	2	1	no	—
21	#16810841	—	2	2	no	—
22	#16810823	—	2	1	no	—
23	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
24	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
25	#16810717	C₂²	1	2	yes	—
26	#16810705	C₂²	1	2	yes	—
27	#16810448	C₂²	1	1	yes	characteristic
28	#16810441	C₂²	1	2	yes	—
29	#16810427	C₂²	1	1	yes	characteristic
30	#16810426	C₂²	1	1	yes	characteristic
31	#16810425	C₂²	1	2	yes	—
32	#16810423	C₂²	1	2	yes	—
33	#16810423	C₂²	1	2	yes	—
34	C₄×#11719908	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₅₃	C₂²	1	2	yes	—
36	C₅₃×#500	C₂²	1	1	yes	characteristic
37	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#367	C₂²	1	2	yes	—
39	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
40	C₅₃×#358	C₂²	1	2	yes	—
41	C₂×Dic₂₁₂	C₂³	1	1	yes	characteristic
42	C₂×#11719924	—	2	2	no	—
43	C₂×D₄₂₄	—	2	1	no	—
44	C₂×D₄₂₄	C₂³	1	2	yes	—
45	#13133215	—	4	1	no	—
46	#13133214	—	2	1	no	—
47	#13133208	—	2	2	no	—
48	#13133207	—	4	2	no	—
49	#13133206	—	2	2	no	—
50	#13133205	—	4	1	no	—
51	#13133202	—	4	2	no	—
52	Q₈×Dic₅₃	—	4	1	no	—
53	C₄×Dic₁₀₆	—	2	1	no	—
54	C₄×Dic₁₀₆	D₄	1	1	yes	characteristic
55	#13133182	—	2	2	no	—
56	C₄×D₂₁₂	—	4	1	no	—
57	#13133179	—	2	2	no	—
58	#13133163	—	2	2	no	—
59	#13133162	—	2	4	no	—
60	C₂×#11719908	C₂³	1	1	yes	characteristic
61	C₂×#11719908	C₂³	1	1	yes	characteristic
62	#13133151	C₂³	1	1	yes	characteristic
63	#13133148	—	2	2	no	—
64	C₄⋊#88322	D₄	1	1	yes	characteristic
65	#13133146	D₄	1	2	yes	—
66	#13133142	—	2	1	no	—
67	#13133140	—	2	2	no	—
68	#13133137	—	2	2	no	—
69	C₈×Dic₅₃	C₂³	1	2	yes	—
70	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
71	C₄⋊Q₈×C₅₃	—	2	1	no	—
72	C₄×Q₈×C₅₃	—	2	1	no	—
73	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
74	#107×C₅₃	—	2	1	no	—
75	#107×C₅₃	C₂³	1	2	yes	—
76	#107×C₅₃	C₂³	1	1	yes	characteristic
77	#103×C₅₃	—	2	1	no	—
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₅₃	—	2	1	no	—
80	C₂×Q₈×C₅₃	—	4	1	no	—
81	C₂×D₂₁₂	—	2	2	no	—
82	C₂×D₂₁₂	—	4	1	no	—
83	C₂×D₂₁₂	—	4	1	no	—
84	C₂×C₄×D₅₃	—	4	2	no	—
85	C₂×Dic₁₀₆	—	2	1	no	—
86	C₂×Dic₁₀₆	D₈	1	1	yes	characteristic
87	Q₁₆×C₅₃	—	2	1	no	—
88	Q₁₆×C₅₃	—	2	1	no	—
89	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₅₃	—	2	1	no	—
92	C₄⋊C₄×C₅₃	—	2	2	no	—
93	C₄⋊C₄×C₅₃	—	4	1	no	—
94	C₄⋊C₄×C₅₃	—	4	1	no	—
95	C₄²×C₅₃	—	4	1	no	—
96	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
97	#11719924	—	8	2	no	—
98	#11719921	—	4	2	no	—
99	#11719921	—	4	2	no	—
100	#11719921	—	4	2	no	—
101	C₄⋊Dic₅₃	—	2	1	no	—
102	C₄⋊Dic₅₃	—	4	2	no	—
103	C₄⋊Dic₅₃	D₈	1	1	yes	characteristic
104	#11719919	—	4	1	no	—
105	#11719919	—	4	2	no	—
106	#11719919	—	4	2	no	—
107	#11719919	D₈	1	2	yes	—
108	C₄×Dic₅₃	—	4	1	no	—
109	C₄×Dic₅₃	—	4	2	no	—
110	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
111	C₂×#88322	C₂×D₄	1	2	yes	—
112	Dic₂₁₂	—	2	1	no	—
113	Dic₂₁₂	#41	1	2	yes	—
114	D₄₂₄	—	2	2	no	—
115	D₄₂₄	—	2	2	no	—
116	D₄₂₄	—	8	1	no	—
117	C₁₆×C₅₃	—	2	2	no	—
118	#11719908	—	2	1	no	—
119	#11719908	—	2	1	no	—
120	C₂²×D₅₃	—	4	2	no	—
121	Q₈×C₅₃	—	4	1	no	—
122	Q₈×C₅₃	—	4	1	no	—
123	Q₈×C₅₃	—	8	2	no	—
124	C₂×C₄×C₅₃	—	4	1	no	—
125	C₂×C₄×C₅₃	—	4	1	no	—
126	C₂×C₄×C₅₃	—	4	2	no	—
127	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×Dic₅₃	—	4	1	no	—
131	C₂×Dic₅₃	—	4	1	no	—
132	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
133	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
134	D₂₁₂	—	4	2	no	—
135	D₂₁₂	—	4	2	no	—
136	D₂₁₂	—	8	2	no	—
137	D₂₁₂	—	16	1	no	—
138	C₄×D₅₃	—	16	2	no	—
139	Dic₁₀₆	—	2	2	no	—
140	Dic₁₀₆	—	2	2	no	—
141	Dic₁₀₆	—	4	1	no	—
142	C₈×C₅₃	—	2	1	no	—
143	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
144	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
145	#88322	—	2	2	no	—
146	#33932	—	53	1	no	maximal, Sylow
147	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
148	C₂×D₅₃	—	8	2	no	—
149	C₂×D₅₃	—	8	2	no	—
150	C₂×D₅₃	—	8	2	no	—
151	C₄×C₅₃	—	2	1	no	—
152	C₄×C₅₃	—	8	1	no	—
153	C₄×C₅₃	—	8	2	no	—
154	C₄×C₅₃	—	8	1	no	—
155	C₄×C₅₃	#508	1	2	yes	—
156	C₄×C₅₃	#548	1	1	yes	characteristic
157	C₄×C₅₃	#577	1	1	yes	characteristic
158	Dic₅₃	—	2	2	no	—
159	Dic₅₃	—	2	2	no	—
160	Dic₅₃	—	4	2	no	—
161	Dic₅₃	—	8	1	no	—
162	#3396	—	53	1	no	—
163	#3383	—	53	1	no	—
164	#2242	—	53	2	no	—
165	#2240	—	53	1	no	—
166	#2228	—	53	2	no	—
167	#2225	D₅₃	1	1	yes	characteristic, complemented
168	#2211	—	53	2	no	—
169	#2207	—	53	1	no	—
170	Q₈×C₁₆	—	53	1	no	—
171	C₄×Q₃₂	—	53	1	no	—
172	C₄×SD₃₂	—	53	2	no	—
173	C₂×C₅₃	#3358	1	1	yes	characteristic
174	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
175	C₂×C₅₃	#3416	1	1	yes	characteristic
176	D₅₃	—	16	2	no	—
177	#559	—	106	1	no	—
178	#557	—	106	1	no	—
179	C₂×Q₃₂	—	53	1	no	—
180	C₂×SD₃₂	—	53	2	no	—
181	#500	C₂×D₅₃	1	1	yes	characteristic
182	#499	—	53	2	no	—
183	#495	—	53	2	no	—
184	#493	—	53	1	no	—
185	#479	—	106	2	no	—
186	#475	—	106	1	no	—
187	#463	—	106	2	no	—
188	#460	—	106	1	no	—
189	C₈×Q₈	—	53	1	no	—
190	C₄×Q₁₆	—	53	2	no	—
191	C₄×Q₁₆	—	53	1	no	—
192	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
193	C₄×D₈	—	53	2	no	—
194	#367	C₂×D₅₃	1	2	yes	—
195	#366	—	53	1	no	—
196	C₄⋊C₁₆	—	53	2	no	—
197	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
198	#358	—	53	1	no	—
199	#358	—	53	1	no	—
200	#358	C₂×D₅₃	1	2	yes	—
201	#357	—	53	2	no	—
202	#357	—	53	2	no	—
203	#357	—	53	2	no	—
204	C₄×C₁₆	—	53	2	no	—
205	C₄×C₁₆	—	53	1	no	—
206	C₅₃	#33932	1	1	yes	characteristic, complemented, Sylow
207	C₂×Q₁₆	—	53	1	no	—
208	C₂×Q₁₆	C₂²×D₅₃	1	1	yes	characteristic
209	C₂×SD₁₆	—	106	2	no	—
210	C₂×D₈	—	53	2	no	—
211	C₂×D₈	—	106	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	212	1	no	—
214	#127	—	106	1	no	—
215	#125	—	106	2	no	—
216	#125	—	212	2	no	—
217	#124	—	106	2	no	—
218	#124	—	212	1	no	—
219	#123	—	212	2	no	—
220	C₄×Q₈	—	2	1	no	—
221	C₄×Q₈	—	53	1	no	—
222	C₄×Q₈	—	106	2	no	—
223	C₄×Q₈	—	106	1	no	—
224	C₄×Q₈	—	212	1	no	—
225	C₄×D₄	—	106	2	no	—
226	C₄×D₄	—	212	1	no	—
227	Q₃₂	—	106	2	no	—
228	SD₃₂	—	106	4	no	—
229	C₂×C₁₆	—	53	1	no	—
230	C₂×C₁₆	—	53	1	no	—
231	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
232	#107	—	2	1	no	—
233	#107	—	53	1	no	—
234	#107	C₂²×D₅₃	1	2	yes	—
235	#107	C₂²×D₅₃	1	1	yes	characteristic
236	#106	—	106	2	no	—
237	C₄⋊C₈	—	53	2	no	—
238	C₄⋊C₈	—	53	1	no	—
239	#103	—	2	1	no	—
240	#103	—	106	2	no	—
241	#103	—	106	1	no	—
242	#102	—	106	2	no	—
243	C₄×C₈	—	53	2	no	—
244	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	53	1	no	—
248	C₂×Q₈	—	106	1	no	—
249	C₂×D₄	—	106	2	no	—
250	C₂×D₄	—	212	1	no	—
251	C₂×D₄	—	212	1	no	—
252	C₂²×C₄	—	212	2	no	—
253	Q₁₆	—	2	1	no	—
254	Q₁₆	—	2	1	no	—
255	Q₁₆	—	53	2	no	—
256	Q₁₆	—	106	1	no	—
257	SD₁₆	—	424	2	no	—
258	D₈	—	106	2	no	—
259	D₈	—	106	2	no	—
260	D₈	—	424	1	no	—
261	C₂×C₈	—	53	2	no	—
262	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
263	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	53	1	no	—
269	C₄⋊C₄	—	53	2	no	—
270	C₄⋊C₄	—	106	1	no	—
271	C₄⋊C₄	—	212	2	no	—
272	C₄⋊C₄	—	212	2	no	—
273	C₄⋊C₄	—	212	2	no	—
274	C₄⋊C₄	—	212	1	no	—
275	C₂²⋊C₄	—	212	2	no	—
276	C₂²⋊C₄	—	212	2	no	—
277	C₂²⋊C₄	—	212	2	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	53	2	no	—
280	C₄²	—	212	2	no	—
281	C₄²	—	212	1	no	—
282	C₄²	D₄×D₅₃	1	1	yes	characteristic
283	C₁₆	—	2	2	no	—
284	C₁₆	—	106	1	no	—
285	C₁₆	—	106	1	no	—
286	C₂³	—	212	2	no	—
287	Q₈	—	4	1	no	—
288	Q₈	—	4	1	no	—
289	Q₈	—	8	2	no	—
290	Q₈	—	106	2	no	—
291	Q₈	—	106	2	no	—
292	Q₈	—	212	1	no	—
293	D₄	—	848	1	no	—
294	D₄	—	424	2	no	—
295	D₄	—	212	2	no	—
296	D₄	—	212	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	53	2	no	—
301	C₂×C₄	—	53	2	no	—
302	C₂×C₄	—	848	2	no	—
303	C₂×C₄	—	212	1	no	—
304	C₂×C₄	—	212	1	no	—
305	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
306	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	106	2	no	—
310	C₈	#41×D₅₃	1	1	yes	characteristic
311	C₈	#13133265	1	1	yes	characteristic
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	—	2	1	no	—
317	C₄	—	8	2	no	—
318	C₄	—	8	1	no	—
319	C₄	—	8	1	no	—
320	C₄	—	106	2	no	—
321	C₄	—	106	2	no	—
322	C₄	—	424	1	no	—
323	C₄	—	212	2	no	—
324	C₄	#16811054	1	2	yes	—
325	C₄	#16811283	1	1	yes	characteristic
326	C₄	#16811432	1	1	yes	characteristic
327	C₂	—	848	2	no	—
328	C₂	#20287349	1	1	yes	characteristic
329	C₂	C₂×#16811054	1	1	yes	characteristic
330	C₂	#20287860	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×#2225	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#16811000	C₂²
hypercenter	#2225	D₅₃
hypocenter	C₅₃	#33932
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2225

Complements

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

N	H ≅ G/N	type
2. #20287608	C₂	semidirect
3. #20276948	C₂	semidirect
5. #20276871	C₂	semidirect
6. #20276780	C₂	semidirect
8. #20276617	C₂	semidirect
9. C₄×#13133163	C₂	semidirect
10. #20276566	C₂	semidirect
11. C₅₃×#2225	C₂	semidirect
167. #2225	D₅₃	semidirect
206. C₅₃	#33932	semidirect

Maximal subgroups

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

subgroup	index
#20287725	2
#20287608	2
#20276948	2
#20276940	2
#20276871	2
#20276780	2
#20276756	2
#20276617	2
C₄×#13133163	2
#20276566	2
C₅₃×#2225	2
#33932	53