Subgroups — #27538775

Order 13568.

556 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283568	C₂	1	1	yes	maximal, characteristic
2	#20276869	C₂	1	2	yes	maximal, complemented
3	#20276869	C₂	1	2	yes	maximal, complemented
4	#20276859	C₂	1	2	yes	maximal, complemented
5	#20276849	C₂	1	2	yes	maximal, complemented
6	#20276501	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276497	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276474	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276430	C₂	1	1	yes	maximal, characteristic
10	C₂×#16810423	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#3164	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄₂₄	C₂²	1	1	yes	characteristic
13	#16811097	—	2	1	no	—
14	C₂×#13133157	C₂²	1	1	yes	characteristic
15	C₂²×#11719908	C₂²	1	1	yes	characteristic
16	#16810736	—	2	2	no	—
17	C₄×D₄₂₄	C₂²	1	2	yes	—
18	C₄×D₄₂₄	C₂²	1	2	yes	—
19	#16810661	C₂²	1	1	yes	characteristic
20	#16810629	—	2	1	no	—
21	#16810450	C₂²	1	1	yes	characteristic, complemented
22	#16810450	C₂²	1	1	yes	characteristic, complemented
23	#16810450	C₂²	1	2	yes	complemented
24	#16810448	C₂²	1	1	yes	characteristic, complemented
25	#16810448	C₂²	1	1	yes	characteristic, complemented
26	#16810448	C₂²	1	2	yes	complemented
27	C₄⋊#11719908	C₂²	1	2	yes	complemented
28	C₄⋊#11719908	C₂²	1	2	yes	complemented
29	#16810423	C₂²	1	2	yes	—
30	#16810423	C₂²	1	2	yes	—
31	#16810423	C₂²	1	2	yes	—
32	#16810423	C₂²	1	2	yes	—
33	C₅₃×#544	—	2	1	no	—
34	C₅₃×#501	—	2	2	no	—
35	C₅₃×#500	C₂²	1	2	yes	complemented
36	C₅₃×#499	C₂²	1	2	yes	complemented
37	C₅₃×#428	—	2	1	no	—
38	C₅₃×#427	C₂²	1	1	yes	characteristic
39	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#405	C₂²	1	1	yes	characteristic
41	C₂²×D₂₁₂	—	2	1	no	—
42	C₂×D₄₂₄	—	2	1	no	—
43	C₂×D₄₂₄	—	2	2	no	—
44	C₂×D₄₂₄	Q₈	1	4	yes	complemented
45	C₂×D₄₂₄	C₂³	1	2	yes	—
46	C₄×D₂₁₂	—	2	2	no	—
47	C₄×D₂₁₂	—	2	2	no	—
48	#13133179	—	4	1	no	—
49	#13133179	—	4	1	no	—
50	#13133176	—	2	1	no	—
51	C₂²⋊C₄×D₅₃	—	4	2	no	—
52	#13133157	—	2	2	no	—
53	C₂×#11719908	—	2	2	no	—
54	C₂×#11719908	C₂³	1	1	yes	characteristic
55	C₂×#11719908	C₂³	1	1	yes	characteristic
56	C₂×#11719908	C₂³	1	2	yes	—
57	#13133151	C₂³	1	1	yes	characteristic
58	#13133151	C₂³	1	1	yes	characteristic
59	#13133149	—	2	1	no	—
60	#13133139	—	2	2	no	—
61	#13133139	—	2	2	no	—
62	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₅₃	—	2	2	no	—
65	C₄⋊Q₈×C₅₃	—	4	1	no	—
66	#125×C₅₃	—	2	2	no	—
67	#125×C₅₃	—	4	1	no	—
68	C₂²⋊Q₈×C₅₃	—	4	2	no	—
69	#117×C₅₃	—	2	1	no	—
70	#117×C₅₃	D₄	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
72	#107×C₅₃	—	2	2	no	—
73	#107×C₅₃	—	2	1	no	—
74	#107×C₅₃	C₂³	1	1	yes	characteristic
75	#107×C₅₃	C₂³	1	2	yes	—
76	#107×C₅₃	C₂³	1	1	yes	characteristic
77	#106×C₅₃	—	2	2	no	—
78	#106×C₅₃	—	2	1	no	—
79	#97×C₅₃	D₄	1	2	yes	—
80	C₄×C₈×C₅₃	C₂³	1	2	yes	—
81	C₂³×D₅₃	—	4	1	no	—
82	C₂×Q₈×C₅₃	—	4	2	no	—
83	C₂²×C₄×C₅₃	—	4	1	no	—
84	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂×D₂₁₂	—	2	4	no	—
86	C₂×D₂₁₂	—	2	2	no	—
87	C₂×D₂₁₂	—	4	1	no	—
88	C₂×D₂₁₂	—	4	2	no	—
89	C₂×C₄×D₅₃	—	4	2	no	—
90	C₂×C₄×D₅₃	—	4	2	no	—
91	M₁₆×C₅₃	—	2	2	no	—
92	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
95	C₂×C₈×C₅₃	#41	1	2	yes	—
96	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
97	C₄⋊C₄×C₅₃	—	2	1	no	—
98	C₄⋊C₄×C₅₃	—	2	2	no	—
99	C₄⋊C₄×C₅₃	—	2	1	no	—
100	C₄⋊C₄×C₅₃	—	4	1	no	—
101	C₄⋊C₄×C₅₃	—	4	2	no	—
102	C₄⋊C₄×C₅₃	—	4	2	no	—
103	C₄⋊C₄×C₅₃	—	4	2	no	—
104	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
105	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
106	C₂²⋊C₄×C₅₃	—	4	1	no	—
107	C₂²⋊C₄×C₅₃	D₈	1	2	yes	—
108	C₄²×C₅₃	—	4	1	no	—
109	C₄²×C₅₃	C₂×D₄	1	2	yes	—
110	C₂²⋊Dic₅₃	—	4	1	no	—
111	#11719921	—	4	2	no	—
112	#11719921	—	4	2	no	—
113	C₄⋊Dic₅₃	—	2	1	no	—
114	C₄⋊Dic₅₃	—	2	1	no	—
115	C₄×Dic₅₃	—	4	1	no	—
116	D₄₂₄	—	2	4	no	—
117	D₄₂₄	—	2	2	no	—
118	D₄₂₄	C₂×Q₈	1	4	yes	—
119	#11719908	—	2	1	no	—
120	#11719908	—	2	1	no	—
121	#11719908	—	2	2	no	—
122	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
123	C₂²×D₅₃	—	4	2	no	—
124	C₂²×D₅₃	—	4	4	no	—
125	C₂²×D₅₃	—	16	1	no	—
126	C₂²×D₅₃	—	16	1	no	—
127	Q₈×C₅₃	—	16	2	no	—
128	C₂×C₄×C₅₃	—	4	2	no	—
129	C₂×C₄×C₅₃	—	4	1	no	—
130	C₂×C₄×C₅₃	—	4	1	no	—
131	C₂×C₄×C₅₃	—	16	1	no	—
132	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
133	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₅₃	#137	1	2	yes	—
137	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×Dic₅₃	—	4	1	no	—
139	C₂×Dic₅₃	—	4	1	no	—
140	D₂₁₂	—	2	4	no	—
141	D₂₁₂	—	4	2	no	—
142	D₂₁₂	—	4	4	no	—
143	C₄×D₅₃	—	16	2	no	—
144	C₄×D₅₃	—	16	2	no	—
145	C₈×C₅₃	—	2	2	no	—
146	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
147	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
148	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
149	#33631	—	53	1	no	maximal, Sylow
150	C₂²×C₅₃	—	2	1	no	—
151	C₂²×C₅₃	—	2	1	no	—
152	C₂²×C₅₃	#474	1	2	yes	—
153	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
154	C₂×D₅₃	—	4	4	no	—
155	C₂×D₅₃	—	8	2	no	—
156	C₂×D₅₃	—	8	2	no	—
157	C₂×D₅₃	—	16	4	no	—
158	C₄×C₅₃	—	2	2	no	—
159	C₄×C₅₃	—	2	2	no	—
160	C₄×C₅₃	—	8	2	no	—
161	C₄×C₅₃	—	8	1	no	—
162	C₄×C₅₃	—	8	1	no	—
163	C₄×C₅₃	#474	1	2	yes	—
164	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
165	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
166	Dic₅₃	—	8	1	no	—
167	Dic₅₃	—	8	1	no	—
168	#3164	D₅₃	1	1	yes	characteristic, complemented
169	#2948	—	53	1	no	—
170	#2228	—	53	2	no	—
171	#2228	—	53	2	no	—
172	#2226	—	53	2	no	—
173	#2224	—	53	2	no	—
174	#2159	—	53	1	no	—
175	#2158	—	53	1	no	—
176	#2151	—	53	1	no	—
177	#2140	—	53	1	no	—
178	C₂×#357	—	53	1	no	—
179	C₂×C₅₃	—	2	2	no	—
180	C₂×C₅₃	C₂×#474	1	1	yes	characteristic
181	C₂×C₅₃	#3415	1	1	yes	characteristic
182	C₂×C₅₃	#3416	1	1	yes	characteristic
183	D₅₃	—	8	4	no	—
184	C₂²×D₈	—	53	1	no	—
185	#544	—	2	1	no	—
186	#518	—	106	1	no	—
187	C₂×M₃₂	—	53	1	no	—
188	C₂²×C₁₆	—	53	1	no	—
189	#501	—	2	2	no	—
190	#500	C₂×D₅₃	1	2	yes	complemented
191	#499	C₂×D₅₃	1	2	yes	complemented
192	#442	—	106	2	no	—
193	C₄×D₈	—	53	2	no	—
194	C₄×D₈	—	53	2	no	—
195	#428	—	2	1	no	—
196	#427	—	53	1	no	—
197	#427	C₂×D₅₃	1	1	yes	characteristic
198	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
199	#418	—	106	1	no	—
200	#405	C₂×D₅₃	1	1	yes	characteristic
201	#367	—	53	1	no	—
202	#367	—	53	1	no	—
203	#367	—	53	2	no	—
204	#366	—	53	1	no	—
205	#366	—	53	1	no	—
206	#366	—	53	2	no	—
207	C₄⋊C₁₆	—	53	2	no	—
208	C₄⋊C₁₆	—	53	2	no	—
209	#357	—	53	2	no	—
210	#357	—	53	2	no	—
211	#357	—	53	2	no	—
212	#357	—	53	2	no	—
213	C₅₃	#33631	1	1	yes	characteristic, complemented, Sylow
214	C₂²×D₄	—	106	1	no	—
215	C₂×D₈	—	53	4	no	—
216	C₂×D₈	—	53	2	no	—
217	C₂×D₈	—	106	2	no	—
218	C₂×D₈	—	106	1	no	—
219	C₂×M₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
220	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
221	C₄⋊Q₈	—	2	2	no	—
222	C₄⋊Q₈	—	4	1	no	—
223	#125	—	2	2	no	—
224	#125	—	4	1	no	—
225	C₂²⋊Q₈	—	4	2	no	—
226	C₄×D₄	—	106	2	no	—
227	C₄×D₄	—	106	2	no	—
228	C₄×D₄	—	212	1	no	—
229	C₄×D₄	—	212	1	no	—
230	#117	—	2	1	no	—
231	#117	—	106	1	no	—
232	#117	C₂²⋊D₅₃	1	1	yes	characteristic
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₂²⋊C₄	—	212	2	no	—
235	M₃₂	—	106	2	no	—
236	C₂×C₁₆	—	53	1	no	—
237	C₂×C₁₆	—	53	2	no	—
238	C₂×C₁₆	—	53	1	no	—
239	C₂×C₁₆	—	106	2	no	—
240	#107	—	2	1	no	—
241	#107	—	2	2	no	—
242	#107	—	53	1	no	—
243	#107	—	53	1	no	—
244	#107	C₂²×D₅₃	1	2	yes	—
245	#107	C₂²×D₅₃	1	1	yes	characteristic
246	#107	C₂²×D₅₃	1	1	yes	characteristic
247	#106	—	2	1	no	—
248	#106	—	2	2	no	—
249	#106	—	106	1	no	—
250	#102	—	106	2	no	—
251	#102	—	106	2	no	—
252	#97	C₂²⋊D₅₃	1	2	yes	—
253	C₄×C₈	C₂²×D₅₃	1	2	yes	—
254	C₂⁴	—	212	1	no	—
255	C₂×Q₈	—	4	2	no	—
256	C₂×D₄	—	106	4	no	—
257	C₂×D₄	—	106	2	no	—
258	C₂×D₄	—	212	1	no	—
259	C₂×D₄	—	212	2	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	—	212	2	no	—
262	C₂²×C₄	—	212	2	no	—
263	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
264	D₈	—	53	4	no	—
265	D₈	—	106	4	no	—
266	D₈	—	106	2	no	—
267	M₁₆	—	2	2	no	—
268	C₂×C₈	Q₈×D₅₃	1	2	yes	—
269	C₂×C₈	#11719949	1	2	yes	—
270	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
271	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
272	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	—	106	1	no	—
281	C₄⋊C₄	—	106	1	no	—
282	C₄⋊C₄	#11719922	1	1	yes	characteristic
283	C₄⋊C₄	#11719922	1	1	yes	characteristic
284	C₂²⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	212	1	no	—
286	C₂²⋊C₄	—	212	2	no	—
287	C₂²⋊C₄	—	212	2	no	—
288	C₂²⋊C₄	#11719922	1	2	yes	—
289	C₄²	—	4	1	no	—
290	C₄²	—	212	1	no	—
291	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
292	C₁₆	—	106	1	no	—
293	C₁₆	—	106	1	no	—
294	C₁₆	—	106	2	no	—
295	C₂³	—	848	1	no	—
296	C₂³	—	848	1	no	—
297	C₂³	—	212	4	no	—
298	C₂³	—	212	2	no	—
299	C₂³	C₂×#11719922	1	1	yes	characteristic
300	Q₈	—	16	2	no	—
301	D₄	—	106	4	no	—
302	D₄	—	212	4	no	—
303	D₄	—	212	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	16	1	no	—
308	C₂×C₄	—	848	2	no	—
309	C₂×C₄	—	848	2	no	—
310	C₂×C₄	—	212	1	no	—
311	C₂×C₄	—	212	1	no	—
312	C₂×C₄	#13133193	1	2	yes	—
313	C₂×C₄	C₂×#11719922	1	2	yes	—
314	C₂×C₄	C₂×#11719922	1	2	yes	—
315	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
316	C₂×C₄	#13133248	1	2	yes	—
317	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
318	C₈	—	2	2	no	—
319	C₈	#13133193	1	2	yes	—
320	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
321	C₈	C₂×#11719949	1	1	yes	characteristic
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	848	4	no	—
325	C₂²	—	424	2	no	—
326	C₂²	—	424	2	no	—
327	C₂²	—	212	4	no	—
328	C₂²	#16810900	1	2	yes	—
329	C₂²	C₂²×#11719922	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	8	2	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	1	no	—
335	C₄	—	424	1	no	—
336	C₄	—	424	1	no	—
337	C₄	#16810900	1	2	yes	—
338	C₄	C₂×#13133193	1	1	yes	characteristic
339	C₄	C₂×#13133248	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	424	4	no	—
342	C₂	C₂×#16810900	1	1	yes	characteristic
343	C₂	#20287854	1	1	yes	characteristic
344	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×#3164	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂²×D₄₂₄	C₂²
hypercenter	#3164	D₅₃
hypocenter	C₅₃	#33631
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3164

Complements

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

N	H ≅ G/N	type
2. #20276869	C₂	semidirect
3. #20276869	C₂	semidirect
4. #20276859	C₂	semidirect
5. #20276849	C₂	semidirect
6. #20276501	C₂	semidirect
7. #20276497	C₂	semidirect
8. #20276474	C₂	semidirect
11. C₅₃×#3164	C₂	semidirect
21. #16810450	C₂²	semidirect
22. #16810450	C₂²	semidirect
23. #16810450	C₂²	semidirect
24. #16810448	C₂²	semidirect
25. #16810448	C₂²	semidirect
26. #16810448	C₂²	semidirect
27. C₄⋊#11719908	C₂²	semidirect
28. C₄⋊#11719908	C₂²	semidirect
35. C₅₃×#500	C₂²	semidirect
36. C₅₃×#499	C₂²	semidirect
44. C₂×D₄₂₄	Q₈	semidirect
168. #3164	D₅₃	semidirect
190. #500	C₂×D₅₃	semidirect
191. #499	C₂×D₅₃	semidirect
213. C₅₃	#33631	semidirect

Maximal subgroups

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

subgroup	index
#20283568	2
#20276869	2
#20276859	2
#20276849	2
#20276501	2
#20276497	2
#20276474	2
#20276430	2
C₂×#16810423	2
C₅₃×#3164	2
#33631	53