Subgroups — #27539263

Order 13568.

556 conjugacy classes of subgroups.

The subgroup lattice has 316 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	#20276920	C₂	1	2	yes	maximal, complemented
3	#20276869	C₂	1	2	yes	maximal, complemented
4	#20276757	C₂	1	2	yes	maximal, complemented
5	#20276643	C₂	1	2	yes	maximal
6	#20276577	C₂	1	2	yes	maximal, complemented
7	C₄×#13133160	C₂	1	2	yes	maximal, complemented
8	#20276522	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#3177	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄₂₄	C₂²	1	1	yes	characteristic
11	#16811097	—	2	1	no	—
12	C₂×#13133160	C₂²	1	2	yes	—
13	C₂×#13133160	C₂²	1	2	yes	—
14	#16810736	—	2	1	no	—
15	#16810736	—	2	1	no	—
16	C₄×D₄₂₄	C₂²	1	2	yes	—
17	C₄×D₄₂₄	C₂²	1	2	yes	—
18	#16810661	C₂²	1	1	yes	characteristic
19	#16810629	—	2	1	no	—
20	#16810450	C₂²	1	2	yes	complemented
21	#16810448	C₂²	1	2	yes	complemented
22	C₄⋊#11719908	C₂²	1	2	yes	complemented
23	#16810427	C₂²	1	2	yes	complemented
24	#16810427	C₂²	1	2	yes	complemented
25	#16810423	C₂²	1	2	yes	—
26	#16810423	C₂²	1	2	yes	—
27	C₂²⋊#11719908	C₂²	1	2	yes	—
28	#16810399	C₂²	1	1	yes	characteristic, complemented
29	C₄×#11719908	C₂²	1	1	yes	characteristic, complemented
30	C₅₃×#541	—	2	1	no	—
31	C₅₃×#499	C₂²	1	1	yes	characteristic, complemented
32	C₅₃×#494	C₂²	1	1	yes	characteristic, complemented
33	C₅₃×#491	—	2	1	no	—
34	C₅₃×#470	—	2	1	no	—
35	C₅₃×#467	C₂²	1	2	yes	—
36	C₅₃×#441	—	2	1	no	—
37	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	complemented
38	C₅₃×#405	C₂²	1	1	yes	characteristic
39	C₂²×D₂₁₂	—	2	1	no	—
40	C₂×D₄₂₄	—	2	1	no	—
41	C₂×D₄₂₄	—	2	4	no	—
42	C₂×D₄₂₄	C₂³	1	2	yes	—
43	C₄×D₂₁₂	—	2	2	no	—
44	C₄×D₂₁₂	—	2	2	no	—
45	#13133179	—	4	2	no	—
46	#13133176	—	2	1	no	—
47	C₂²⋊C₄×D₅₃	—	4	2	no	—
48	#13133160	—	2	4	no	—
49	#13133160	—	2	4	no	—
50	C₂×#11719908	C₂³	1	2	yes	—
51	C₂×#11719908	C₂³	1	2	yes	—
52	#13133151	C₂³	1	2	yes	—
53	#13133149	—	2	1	no	—
54	#13133139	—	2	2	no	—
55	#13133139	—	2	2	no	—
56	C₂×Q₁₆×C₅₃	—	2	1	no	—
57	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
58	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
60	C₄⋊Q₈×C₅₃	—	2	1	no	—
61	#126×C₅₃	—	4	2	no	—
62	#125×C₅₃	—	2	1	no	—
63	#125×C₅₃	—	4	2	no	—
64	C₂²⋊Q₈×C₅₃	—	2	2	no	—
65	C₄×Q₈×C₅₃	—	2	2	no	—
66	#117×C₅₃	D₄	1	1	yes	characteristic
67	#107×C₅₃	C₂³	1	2	yes	—
68	#106×C₅₃	—	2	1	no	—
69	#103×C₅₃	—	2	2	no	—
70	#103×C₅₃	—	2	2	no	—
71	#97×C₅₃	D₄	1	1	yes	characteristic
72	#97×C₅₃	D₄	1	1	yes	characteristic
73	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₂³×D₅₃	—	4	1	no	—
76	C₂×Q₈×C₅₃	—	2	2	no	—
77	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂×D₂₁₂	—	2	4	no	—
79	C₂×D₂₁₂	—	2	2	no	—
80	C₂×D₂₁₂	—	4	2	no	—
81	C₂×D₂₁₂	—	4	1	no	—
82	C₂×C₄×D₅₃	—	4	2	no	—
83	C₂×C₄×D₅₃	—	4	2	no	—
84	Q₁₆×C₅₃	—	2	2	no	—
85	Q₁₆×C₅₃	—	2	2	no	—
86	Q₁₆×C₅₃	—	4	2	no	—
87	M₁₆×C₅₃	—	4	1	no	—
88	C₂×C₈×C₅₃	—	2	2	no	—
89	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
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	—	4	2	no	—
95	C₄⋊C₄×C₅₃	—	4	2	no	—
96	C₄⋊C₄×C₅₃	—	4	2	no	—
97	C₄⋊C₄×C₅₃	—	4	2	no	—
98	C₄⋊C₄×C₅₃	D₈	1	2	yes	—
99	C₂²⋊C₄×C₅₃	—	4	2	no	—
100	C₂²⋊C₄×C₅₃	D₈	1	2	yes	—
101	C₄²×C₅₃	—	4	2	no	—
102	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₂²⋊Dic₅₃	—	4	1	no	—
105	#11719921	—	4	2	no	—
106	#11719921	—	4	2	no	—
107	C₄⋊Dic₅₃	—	2	2	no	—
108	C₄×Dic₅₃	—	4	1	no	—
109	D₄₂₄	—	2	4	no	—
110	D₄₂₄	—	2	2	no	—
111	D₄₂₄	#41	1	4	yes	—
112	#11719908	—	2	2	no	—
113	#11719908	—	2	2	no	—
114	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
115	C₂²×D₅₃	—	4	2	no	—
116	C₂²×D₅₃	—	4	4	no	—
117	C₂²×D₅₃	—	16	2	no	—
118	Q₈×C₅₃	—	4	2	no	—
119	Q₈×C₅₃	—	4	2	no	—
120	C₂×C₄×C₅₃	—	2	1	no	—
121	C₂×C₄×C₅₃	—	2	1	no	—
122	C₂×C₄×C₅₃	—	4	2	no	—
123	C₂×C₄×C₅₃	—	4	2	no	—
124	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
126	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×Dic₅₃	—	4	2	no	—
129	D₂₁₂	—	2	4	no	—
130	D₂₁₂	—	4	4	no	—
131	D₂₁₂	—	4	2	no	—
132	C₄×D₅₃	—	16	2	no	—
133	C₄×D₅₃	—	16	2	no	—
134	C₈×C₅₃	—	2	1	no	—
135	C₈×C₅₃	—	2	1	no	—
136	C₈×C₅₃	—	2	1	no	—
137	C₈×C₅₃	C₂×#41	1	2	yes	—
138	#33683	—	53	1	no	maximal, Sylow
139	C₂²×C₅₃	—	2	1	no	—
140	C₂²×C₅₃	—	2	2	no	—
141	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
142	C₂×D₅₃	—	4	4	no	—
143	C₂×D₅₃	—	8	2	no	—
144	C₂×D₅₃	—	8	2	no	—
145	C₂×D₅₃	—	16	4	no	—
146	C₄×C₅₃	—	2	1	no	—
147	C₄×C₅₃	—	2	2	no	—
148	C₄×C₅₃	—	2	2	no	—
149	C₄×C₅₃	—	8	2	no	—
150	C₄×C₅₃	—	8	2	no	—
151	C₄×C₅₃	#525	1	1	yes	characteristic
152	C₄×C₅₃	#575	1	1	yes	characteristic
153	Dic₅₃	—	8	2	no	—
154	#3177	D₅₃	1	1	yes	characteristic, complemented
155	#2948	—	53	1	no	—
156	#2237	—	53	2	no	—
157	#2228	—	53	2	no	—
158	#2208	—	53	2	no	—
159	#2187	—	53	2	no	—
160	#2175	—	53	2	no	—
161	C₄×SD₃₂	—	53	2	no	—
162	#2164	—	53	1	no	—
163	C₂×C₅₃	—	4	1	no	—
164	C₂×C₅₃	#3100	1	1	yes	characteristic
165	C₂×C₅₃	#3416	1	2	yes	—
166	D₅₃	—	8	4	no	—
167	C₂²×D₈	—	53	1	no	—
168	#541	—	2	1	no	—
169	#518	—	106	1	no	—
170	C₂×SD₃₂	—	53	2	no	—
171	C₂×SD₃₂	—	53	2	no	—
172	#499	C₂×D₅₃	1	1	yes	characteristic, complemented
173	#494	C₂×D₅₃	1	1	yes	characteristic, complemented
174	#491	—	2	1	no	—
175	#470	—	2	1	no	—
176	#467	C₂×D₅₃	1	2	yes	—
177	#442	—	106	1	no	—
178	#442	—	106	1	no	—
179	#441	—	2	1	no	—
180	C₄×Q₁₆	C₂×D₅₃	1	2	yes	complemented
181	C₄×D₈	—	53	2	no	—
182	C₄×D₈	—	53	2	no	—
183	#427	—	53	1	no	—
184	#418	—	106	1	no	—
185	#405	C₂×D₅₃	1	1	yes	characteristic
186	#367	—	53	2	no	—
187	#366	—	53	2	no	—
188	C₄⋊C₁₆	—	53	2	no	—
189	#358	—	53	2	no	—
190	#358	—	53	2	no	—
191	#357	—	53	2	no	—
192	#357	—	53	2	no	—
193	C₂²⋊C₁₆	—	53	2	no	—
194	#346	—	53	1	no	—
195	C₄×C₁₆	—	53	1	no	—
196	C₅₃	#33683	1	1	yes	characteristic, complemented, Sylow
197	C₂²×D₄	—	106	1	no	—
198	C₂×Q₁₆	—	2	1	no	—
199	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
200	C₂×D₈	—	53	2	no	—
201	C₂×D₈	—	106	4	no	—
202	C₂×D₈	—	106	1	no	—
203	C₂×M₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
204	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
205	C₄⋊Q₈	—	2	1	no	—
206	#126	—	4	2	no	—
207	#125	—	2	1	no	—
208	#125	—	4	2	no	—
209	C₂²⋊Q₈	—	2	2	no	—
210	C₄×Q₈	—	2	2	no	—
211	C₄×D₄	—	106	2	no	—
212	C₄×D₄	—	106	2	no	—
213	C₄×D₄	—	212	2	no	—
214	#117	—	106	1	no	—
215	#117	C₂²⋊D₅₃	1	1	yes	characteristic
216	C₂×C₂²⋊C₄	—	212	2	no	—
217	SD₃₂	—	106	4	no	—
218	SD₃₂	—	106	4	no	—
219	C₂×C₁₆	—	53	2	no	—
220	C₂×C₁₆	—	53	2	no	—
221	#107	—	53	2	no	—
222	#107	C₂²×D₅₃	1	2	yes	—
223	#106	—	2	1	no	—
224	#106	—	106	1	no	—
225	#103	—	2	2	no	—
226	#103	—	2	2	no	—
227	#102	—	106	2	no	—
228	#102	—	106	2	no	—
229	#97	C₂²⋊D₅₃	1	1	yes	characteristic
230	#97	C₂²⋊D₅₃	1	1	yes	characteristic
231	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
232	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
233	C₂⁴	—	212	1	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×D₄	—	106	4	no	—
236	C₂×D₄	—	106	2	no	—
237	C₂×D₄	—	212	2	no	—
238	C₂×D₄	—	212	1	no	—
239	C₂²×C₄	—	212	2	no	—
240	C₂²×C₄	—	212	2	no	—
241	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
242	Q₁₆	—	2	2	no	—
243	Q₁₆	—	2	2	no	—
244	Q₁₆	—	4	2	no	—
245	D₈	—	53	4	no	—
246	D₈	—	106	4	no	—
247	D₈	—	106	2	no	—
248	M₁₆	—	4	1	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
251	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
252	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
253	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	106	2	no	—
260	C₄⋊C₄	#11719922	1	2	yes	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	212	1	no	—
263	C₂²⋊C₄	—	212	2	no	—
264	C₂²⋊C₄	—	212	2	no	—
265	C₂²⋊C₄	#11719922	1	2	yes	—
266	C₄²	—	4	2	no	—
267	C₄²	—	212	1	no	—
268	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
269	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
270	C₁₆	—	106	2	no	—
271	C₁₆	—	106	2	no	—
272	C₂³	—	848	2	no	—
273	C₂³	—	212	4	no	—
274	C₂³	—	212	2	no	—
275	C₂³	C₂×#11719922	1	1	yes	characteristic
276	Q₈	—	4	2	no	—
277	Q₈	—	4	2	no	—
278	D₄	—	106	4	no	—
279	D₄	—	212	4	no	—
280	D₄	—	212	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	848	2	no	—
286	C₂×C₄	—	848	2	no	—
287	C₂×C₄	—	212	2	no	—
288	C₂×C₄	C₂×#11719922	1	2	yes	—
289	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
290	C₂×C₄	C₂×#11719922	1	2	yes	—
291	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	#41×D₅₃	1	2	yes	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	848	4	no	—
299	C₂²	—	424	2	no	—
300	C₂²	—	424	2	no	—
301	C₂²	—	212	4	no	—
302	C₂²	C₂²×#11719922	1	1	yes	characteristic
303	C₄	—	2	1	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	8	2	no	—
307	C₄	—	8	2	no	—
308	C₄	—	424	2	no	—
309	C₄	#16811130	1	1	yes	characteristic
310	C₄	#16811421	1	1	yes	characteristic
311	C₂	—	4	1	no	—
312	C₂	—	424	4	no	—
313	C₂	#20284711	1	1	yes	characteristic
314	C₂	#20287859	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#3177	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂²×D₄₂₄	C₂²
hypercenter	#3177	D₅₃
hypocenter	C₅₃	#33683
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3177

Complements

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

N	H ≅ G/N	type
2. #20276920	C₂	semidirect
3. #20276869	C₂	semidirect
4. #20276757	C₂	semidirect
6. #20276577	C₂	semidirect
7. C₄×#13133160	C₂	semidirect
8. #20276522	C₂	semidirect
9. C₅₃×#3177	C₂	semidirect
20. #16810450	C₂²	semidirect
21. #16810448	C₂²	semidirect
22. C₄⋊#11719908	C₂²	semidirect
23. #16810427	C₂²	semidirect
24. #16810427	C₂²	semidirect
28. #16810399	C₂²	semidirect
29. C₄×#11719908	C₂²	semidirect
31. C₅₃×#499	C₂²	semidirect
32. C₅₃×#494	C₂²	semidirect
37. C₄×Q₁₆×C₅₃	C₂²	semidirect
154. #3177	D₅₃	semidirect
172. #499	C₂×D₅₃	semidirect
173. #494	C₂×D₅₃	semidirect
180. C₄×Q₁₆	C₂×D₅₃	semidirect
196. C₅₃	#33683	semidirect

Maximal subgroups

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

subgroup	index
#20283568	2
#20276920	2
#20276869	2
#20276757	2
#20276643	2
#20276577	2
C₄×#13133160	2
#20276522	2
C₅₃×#3177	2
#33683	53