Subgroups — #27537477

Order 13568.

674 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	C₂×#16811056	C₂	1	1	yes	maximal, characteristic
2	#20283590	C₂	1	2	yes	maximal, complemented
3	#20276582	C₂	1	2	yes	maximal, complemented
4	#20276577	C₂	1	2	yes	maximal, complemented
5	Q₃₂×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₈₄₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276561	C₂	1	1	yes	maximal, characteristic, complemented
8	#366×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276426	C₂	1	2	yes	maximal, complemented
10	#20276310	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Q₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133240	C₂²	1	2	yes	—
13	#16811105	—	2	2	no	—
14	#16811056	C₄	1	8	yes	complemented
15	C₂×#13133160	C₂²	1	2	yes	—
16	C₂×D₈₄₈	C₂²	1	1	yes	characteristic
17	C₂×#13133156	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
19	#16810726	—	2	2	no	—
20	#16810721	—	2	2	no	—
21	Q₁₆×Dic₅₃	C₂²	1	2	yes	—
22	C₄×D₄₂₄	C₂²	1	2	yes	—
23	#16810705	C₂²	1	2	yes	—
24	#107×D₅₃	C₂²	1	2	yes	—
25	#16810623	—	2	2	no	—
26	#16810577	C₂²	1	1	yes	characteristic
27	#16810449	C₂²	1	1	yes	characteristic
28	#16810448	C₂²	1	2	yes	—
29	#16810427	C₂²	1	2	yes	—
30	#16810425	C₂²	1	2	yes	—
31	#16810423	C₂²	1	2	yes	—
32	#16810399	C₂²	1	1	yes	characteristic
33	#16810398	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
35	C₂×Q₃₂×C₅₃	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
37	C₅₃×#366	C₂²	1	1	yes	characteristic
38	C₅₃×#358	C₂²	1	2	yes	—
39	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
40	C₂×#11719949	—	2	2	no	—
41	#13133240	—	2	4	no	—
42	#13133240	C₂×C₄	1	8	yes	complemented
43	C₂×#11719924	—	2	2	no	—
44	C₂×D₄₂₄	C₂³	1	2	yes	—
45	C₂×#11719912	D₄	1	1	yes	characteristic
46	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
47	Q₈×Dic₅₃	—	2	2	no	—
48	C₄×D₂₁₂	—	2	2	no	—
49	C₄×D₂₁₂	—	4	2	no	—
50	#13133179	—	2	2	no	—
51	#13133179	—	4	2	no	—
52	#13133174	—	4	2	no	—
53	#13133174	D₄	1	1	yes	characteristic
54	C₄⋊C₄×D₅₃	—	2	2	no	—
55	C₄⋊C₄×D₅₃	—	4	2	no	—
56	#13133160	C₂×C₄	1	8	yes	complemented
57	D₈₄₈	C₂×C₄	1	4	yes	complemented
58	#13133156	—	2	2	no	—
59	C₂×#11719908	C₂³	1	1	yes	characteristic
60	C₂×#11719908	C₂³	1	1	yes	characteristic
61	C₁₆×D₅₃	C₂×C₄	1	4	yes	complemented
62	#13133151	C₂³	1	2	yes	—
63	#13133150	—	2	2	no	—
64	#13133141	—	2	2	no	—
65	#13133139	—	2	2	no	—
66	#13133137	—	2	2	no	—
67	#13133128	D₄	1	1	yes	characteristic
68	#13133127	D₄	1	1	yes	characteristic
69	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
70	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
71	C₄×Q₈×C₅₃	—	2	2	no	—
72	Q₃₂×C₅₃	C₂×C₄	1	4	yes	complemented
73	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
75	#107×C₅₃	C₂³	1	2	yes	—
76	#103×C₅₃	—	2	2	no	—
77	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₂×Q₈×C₅₃	—	2	2	no	—
79	#11719949	—	2	8	no	—
80	#11719949	—	4	4	no	—
81	C₂×D₂₁₂	—	2	2	no	—
82	C₂×D₂₁₂	—	4	2	no	—
83	C₂×C₄×D₅₃	—	4	2	no	—
84	C₂×C₄×D₅₃	—	4	2	no	—
85	C₂×C₄×D₅₃	—	4	2	no	—
86	C₂×C₄×D₅₃	—	4	2	no	—
87	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
88	Q₁₆×C₅₃	—	2	2	no	—
89	Q₁₆×C₅₃	C₂²×C₄	1	4	yes	—
90	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₅₃	—	2	2	no	—
93	C₄⋊C₄×C₅₃	—	4	2	no	—
94	C₄²×C₅₃	—	4	2	no	—
95	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
96	#11719924	—	2	8	no	—
97	#11719921	—	4	2	no	—
98	#11719921	—	4	2	no	—
99	#11719921	D₈	1	1	yes	characteristic
100	#11719921	D₈	1	1	yes	characteristic
101	C₄⋊Dic₅₃	—	2	2	no	—
102	C₄⋊Dic₅₃	—	4	2	no	—
103	#11719919	—	4	2	no	—
104	#11719919	D₈	1	1	yes	characteristic
105	#11719919	D₈	1	1	yes	characteristic
106	C₄×Dic₅₃	—	4	2	no	—
107	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
108	C₂×#88322	C₂×D₄	1	1	yes	characteristic
109	C₂×#88322	C₂×D₄	1	1	yes	characteristic
110	D₄₂₄	—	2	2	no	—
111	D₄₂₄	C₂²×C₄	1	4	yes	—
112	#11719912	—	2	2	no	—
113	C₈×D₅₃	C₂²×C₄	1	2	yes	—
114	C₈×D₅₃	#41	1	2	yes	—
115	C₁₆×C₅₃	—	2	1	no	—
116	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
117	#11719908	—	2	1	no	—
118	#11719908	C₂²×C₄	1	2	yes	—
119	C₂²×D₅₃	—	4	2	no	—
120	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
121	Q₈×C₅₃	—	2	4	no	—
122	Q₈×C₅₃	—	4	2	no	—
123	C₂×C₄×C₅₃	—	4	2	no	—
124	C₂×C₄×C₅₃	—	4	2	no	—
125	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
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	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
132	D₂₁₂	—	2	4	no	—
133	D₂₁₂	—	4	8	no	—
134	D₂₁₂	—	4	2	no	—
135	C₄×D₅₃	—	4	8	no	—
136	C₄×D₅₃	—	16	2	no	—
137	C₄×D₅₃	—	16	2	no	—
138	C₄×D₅₃	—	16	2	no	—
139	C₄×D₅₃	C₄×D₄	1	2	yes	—
140	C₄×D₅₃	#135	1	2	yes	—
141	C₈×C₅₃	—	2	1	no	—
142	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
143	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
144	#88322	—	2	1	no	—
145	#88322	C₄×D₄	1	2	yes	—
146	#33459	—	53	1	no	maximal, Sylow
147	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
148	C₂×D₅₃	—	2	1	no	—
149	C₂×D₅₃	—	2	1	no	—
150	C₂×D₅₃	—	4	4	no	—
151	C₂×D₅₃	—	8	2	no	—
152	C₂×D₅₃	—	8	2	no	—
153	C₂×D₅₃	C₄×D₈	1	2	yes	—
154	C₄×C₅₃	—	2	1	no	—
155	C₄×C₅₃	—	2	1	no	—
156	C₄×C₅₃	—	4	4	no	—
157	C₄×C₅₃	—	8	2	no	—
158	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
159	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
160	Dic₅₃	—	2	1	no	—
161	Dic₅₃	—	2	1	no	—
162	Dic₅₃	—	8	2	no	—
163	Dic₅₃	C₄×D₈	1	2	yes	—
164	C₂×#508	—	53	1	no	—
165	#2951	—	53	2	no	—
166	#2175	—	53	2	no	—
167	#2175	—	53	2	no	—
168	C₄×Q₃₂	—	53	1	no	—
169	C₄×Q₃₂	D₅₃	1	1	yes	characteristic, complemented
170	C₄×D₁₆	—	53	1	no	—
171	C₄×D₁₆	—	53	1	no	—
172	C₂×#366	—	53	1	no	—
173	#2139	—	53	2	no	—
174	#2108	—	53	1	no	—
175	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
176	C₂×C₅₃	#3415	1	1	yes	characteristic
177	C₂×C₅₃	#3417	1	1	yes	characteristic
178	D₅₃	—	2	2	no	—
179	D₅₃	—	8	4	no	—
180	C₂×#135	—	53	2	no	—
181	#520	—	106	2	no	—
182	#508	—	53	8	no	—
183	C₂×Q₃₂	C₂×D₅₃	1	1	yes	characteristic
184	C₂×SD₃₂	—	53	2	no	—
185	C₂×D₁₆	—	53	1	no	—
186	C₂×M₃₂	—	53	1	no	—
187	C₂²×C₁₆	—	53	1	no	—
188	#440	—	106	2	no	—
189	#440	—	106	2	no	—
190	C₄×Q₁₆	—	53	2	no	—
191	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
192	C₄×D₈	—	53	2	no	—
193	C₄×D₈	—	53	2	no	—
194	C₂×#107	—	53	2	no	—
195	#417	—	106	2	no	—
196	#405	—	53	1	no	—
197	#366	—	53	1	no	—
198	#366	—	53	2	no	—
199	#366	C₂×D₅₃	1	1	yes	characteristic
200	#358	—	53	2	no	—
201	#358	C₂×D₅₃	1	2	yes	—
202	#357	—	53	2	no	—
203	#357	—	53	2	no	—
204	#346	—	53	1	no	—
205	#346	—	53	1	no	—
206	C₄×C₁₆	—	53	1	no	—
207	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
208	C₅₃	#33459	1	1	yes	characteristic, complemented, Sylow
209	C₂×#41	—	106	2	no	—
210	#135	—	53	8	no	—
211	#135	—	106	4	no	—
212	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
213	C₂×SD₁₆	—	106	2	no	—
214	C₂×D₈	—	53	2	no	—
215	C₂×M₁₆	—	53	1	no	—
216	C₂²×C₈	—	53	1	no	—
217	C₄×Q₈	—	2	2	no	—
218	C₄×Q₈	—	106	2	no	—
219	C₄×D₄	—	106	2	no	—
220	C₄×D₄	—	106	2	no	—
221	C₄×D₄	—	212	2	no	—
222	C₄×D₄	—	212	2	no	—
223	#117	—	53	1	no	—
224	#117	—	212	2	no	—
225	C₂×C₄⋊C₄	—	106	2	no	—
226	C₂×C₄⋊C₄	—	212	2	no	—
227	Q₃₂	C₄×D₅₃	1	4	yes	complemented
228	SD₃₂	—	53	8	no	—
229	D₁₆	—	53	4	no	—
230	M₃₂	—	106	2	no	—
231	C₂×C₁₆	—	53	1	no	—
232	C₂×C₁₆	—	53	4	no	—
233	C₂×C₁₆	—	53	1	no	—
234	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
235	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
236	#107	—	53	2	no	—
237	#107	—	106	2	no	—
238	#107	C₂²×D₅₃	1	2	yes	—
239	#103	—	2	2	no	—
240	#103	—	106	2	no	—
241	#102	—	106	2	no	—
242	#102	—	106	2	no	—
243	#97	—	53	1	no	—
244	#97	—	53	1	no	—
245	C₄×C₈	—	53	1	no	—
246	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
247	#41	—	106	8	no	—
248	#41	—	212	4	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×D₄	—	106	2	no	—
251	C₂×D₄	—	212	2	no	—
252	C₂²×C₄	—	53	1	no	—
253	C₂²×C₄	—	212	2	no	—
254	C₂²×C₄	—	212	2	no	—
255	C₂²×C₄	—	212	2	no	—
256	C₂²×C₄	—	212	2	no	—
257	Q₁₆	—	2	2	no	—
258	Q₁₆	C₂×C₄×D₅₃	1	4	yes	—
259	SD₁₆	—	106	8	no	—
260	D₈	—	53	4	no	—
261	D₈	—	106	2	no	—
262	M₁₆	—	106	2	no	—
263	C₂×C₈	—	53	1	no	—
264	C₂×C₈	—	53	2	no	—
265	C₂×C₈	—	53	2	no	—
266	C₂×C₈	—	53	1	no	—
267	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
268	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	53	1	no	—
272	C₄⋊C₄	—	53	1	no	—
273	C₄⋊C₄	—	106	2	no	—
274	C₄⋊C₄	—	212	2	no	—
275	C₄⋊C₄	—	212	2	no	—
276	C₂²⋊C₄	—	53	1	no	—
277	C₂²⋊C₄	—	53	1	no	—
278	C₂²⋊C₄	—	212	2	no	—
279	C₂²⋊C₄	—	212	2	no	—
280	C₄²	—	4	2	no	—
281	C₄²	—	53	1	no	—
282	C₄²	—	212	2	no	—
283	C₄²	D₄×D₅₃	1	1	yes	characteristic
284	C₁₆	—	2	1	no	—
285	C₁₆	—	53	2	no	—
286	C₁₆	—	106	1	no	—
287	C₁₆	C₂×C₄×D₅₃	1	2	yes	—
288	C₂³	—	53	1	no	—
289	C₂³	—	212	2	no	—
290	Q₈	—	2	4	no	—
291	Q₈	—	4	2	no	—
292	D₄	—	106	4	no	—
293	D₄	—	212	8	no	—
294	D₄	—	212	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	53	1	no	—
298	C₂×C₄	—	53	2	no	—
299	C₂×C₄	—	53	2	no	—
300	C₂×C₄	—	53	1	no	—
301	C₂×C₄	—	53	1	no	—
302	C₂×C₄	—	848	2	no	—
303	C₂×C₄	—	848	2	no	—
304	C₂×C₄	—	848	2	no	—
305	C₂×C₄	—	212	8	no	—
306	C₂×C₄	—	212	2	no	—
307	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
308	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	53	2	no	—
312	C₈	—	106	1	no	—
313	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
314	C₈	#41×D₅₃	1	1	yes	characteristic
315	C₂²	—	53	2	no	—
316	C₂²	—	106	1	no	—
317	C₂²	—	106	1	no	—
318	C₂²	—	424	2	no	—
319	C₂²	—	424	2	no	—
320	C₂²	—	212	4	no	—
321	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	4	no	—
325	C₄	—	8	2	no	—
326	C₄	—	53	2	no	—
327	C₄	—	106	1	no	—
328	C₄	—	106	1	no	—
329	C₄	—	424	2	no	—
330	C₄	C₄×D₄×D₅₃	1	1	yes	characteristic
331	C₄	#135×D₅₃	1	1	yes	characteristic
332	C₂	—	106	2	no	—
333	C₂	—	424	4	no	—
334	C₂	C₄×D₈×D₅₃	1	1	yes	characteristic
335	C₂	#20287853	1	1	yes	characteristic
336	C₂	#20287864	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₄×Q₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×#16811056	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₄×Q₃₂	D₅₃
hypocenter	C₅₃	#33459
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
2. #20283590	C₂	semidirect
3. #20276582	C₂	semidirect
4. #20276577	C₂	semidirect
5. Q₃₂×Dic₅₃	C₂	semidirect
6. C₄×D₈₄₈	C₂	semidirect
7. #20276561	C₂	semidirect
8. #366×D₅₃	C₂	semidirect
9. #20276426	C₂	semidirect
10. #20276310	C₂	semidirect
11. C₄×Q₃₂×C₅₃	C₂	semidirect
14. #16811056	C₄	semidirect
42. #13133240	C₂×C₄	semidirect
56. #13133160	C₂×C₄	semidirect
57. D₈₄₈	C₂×C₄	semidirect
61. C₁₆×D₅₃	C₂×C₄	semidirect
72. Q₃₂×C₅₃	C₂×C₄	semidirect
169. C₄×Q₃₂	D₅₃	semidirect
208. C₅₃	#33459	semidirect
227. Q₃₂	C₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811056	2
#20283590	2
#20276582	2
#20276577	2
Q₃₂×Dic₅₃	2
C₄×D₈₄₈	2
#20276561	2
#366×D₅₃	2
#20276426	2
#20276310	2
C₄×Q₃₂×C₅₃	2
#33459	53