Subgroups — #27372043

Order 13568.

750 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285862	C₂	1	2	yes	maximal
2	#20285302	C₂	1	1	yes	maximal, characteristic
3	#20285256	C₂	1	2	yes	maximal
4	#494×D₅₃	C₂	1	1	yes	maximal, characteristic
5	#20274395	C₂	1	2	yes	maximal, complemented
6	#20274389	C₂	1	2	yes	maximal, complemented
7	#20274333	C₂	1	1	yes	maximal, characteristic, complemented
8	#20274331	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274313	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#11719912	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#1707	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₅₃	—	2	2	no	—
13	#16811178	C₂²	1	2	yes	—
14	C₄⋊Q₈×D₅₃	C₂²	1	2	yes	—
15	#16811030	—	2	4	no	—
16	#16811028	C₂²	1	2	yes	—
17	#16811028	C₂²	1	2	yes	—
18	#16811026	C₂²	1	2	yes	—
19	#16811026	C₂²	1	2	yes	—
20	#16811019	C₂²	1	2	yes	—
21	#16811003	C₂²	1	1	yes	characteristic
22	#16811003	C₂²	1	1	yes	characteristic
23	#16811002	C₂²	1	2	yes	—
24	#16811001	—	2	2	no	—
25	#16810975	C₂²	1	2	yes	—
26	#16810973	C₂²	1	2	yes	—
27	#16810957	—	2	4	no	—
28	#16810916	—	2	4	no	—
29	#16810865	—	2	2	no	—
30	#16810717	—	2	2	no	—
31	#16810674	C₂²	1	1	yes	characteristic
32	#16810662	—	2	4	no	—
33	C₄×#11719912	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
35	#16810345	C₂²	1	1	yes	characteristic
36	#16810344	C₂²	1	1	yes	characteristic
37	C₈×#88322	C₂²	1	1	yes	characteristic
38	C₅₃×#500	C₂²	1	1	yes	characteristic
39	C₅₃×#498	C₂²	1	2	yes	—
40	C₅₃×#494	C₂²	1	1	yes	characteristic
41	C₅₃×#487	C₂²	1	2	yes	—
42	C₈²×C₅₃	C₂²	1	1	yes	characteristic
43	C₂×Q₈×D₅₃	—	2	2	no	—
44	C₂×Dic₂₁₂	—	2	2	no	—
45	C₂×Dic₂₁₂	—	2	2	no	—
46	C₂×#11719925	—	2	4	no	—
47	Q₁₆×D₅₃	—	4	8	no	—
48	C₂×#11719912	D₄	1	2	yes	—
49	C₂×C₈×D₅₃	D₄	1	2	yes	—
50	#13133217	C₂³	1	2	yes	—
51	#13133217	C₂³	1	2	yes	—
52	#13133216	—	2	2	no	—
53	#13133216	—	2	2	no	—
54	#13133215	—	2	2	no	—
55	#13133208	—	2	2	no	—
56	#13133197	—	2	2	no	—
57	#13133193	—	2	2	no	—
58	Q₈×Dic₅₃	—	2	2	no	—
59	#13133182	—	2	2	no	—
60	#13133175	—	2	2	no	—
61	C₄⋊C₄×D₅₃	—	2	2	no	—
62	C₄⋊C₄×D₅₃	—	2	2	no	—
63	C₄²×D₅₃	C₂³	1	1	yes	characteristic
64	#13133151	—	2	4	no	—
65	#13133151	—	2	4	no	—
66	#13133150	—	2	4	no	—
67	#13133150	—	2	4	no	—
68	#13133148	—	2	4	no	—
69	#13133146	D₄	1	2	yes	—
70	#13133141	—	2	4	no	—
71	#13133140	—	2	4	no	—
72	#13133129	D₄	1	2	yes	—
73	#13133127	D₄	1	2	yes	—
74	C₄×#88322	C₂³	1	1	yes	characteristic
75	C₄×#88322	C₂³	1	1	yes	characteristic
76	C₄×#88322	C₂³	1	1	yes	characteristic
77	C₈×Dic₅₃	D₄	1	2	yes	—
78	C₂×Q₁₆×C₅₃	—	2	2	no	—
79	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
80	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
81	#107×C₅₃	—	2	2	no	—
82	#106×C₅₃	—	2	4	no	—
83	#103×C₅₃	—	2	4	no	—
84	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
87	C₂×Q₈×C₅₃	—	2	2	no	—
88	C₂×Q₈×C₅₃	—	2	2	no	—
89	Q₈×D₅₃	—	4	8	no	—
90	C₂×C₄×D₅₃	—	4	2	no	—
91	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂×Dic₁₀₆	—	2	2	no	—
95	C₂×Dic₁₀₆	—	2	2	no	—
96	C₂×Dic₁₀₆	—	4	2	no	—
97	Q₁₆×C₅₃	—	4	4	no	—
98	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
101	C₄⋊C₄×C₅₃	—	2	2	no	—
102	C₄⋊C₄×C₅₃	—	2	2	no	—
103	C₄⋊C₄×C₅₃	—	2	2	no	—
104	C₄⋊C₄×C₅₃	—	2	2	no	—
105	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
106	#11719925	—	4	8	no	—
107	#11719921	—	4	2	no	—
108	C₄⋊Dic₅₃	—	2	2	no	—
109	C₄⋊Dic₅₃	—	2	2	no	—
110	C₄⋊Dic₅₃	—	2	2	no	—
111	C₄⋊Dic₅₃	—	2	2	no	—
112	C₄⋊Dic₅₃	—	4	2	no	—
113	#11719919	—	4	2	no	—
114	#11719919	—	4	2	no	—
115	#11719919	—	4	2	no	—
116	C₄×Dic₅₃	—	4	2	no	—
117	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
118	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
119	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
120	C₂×#88322	C₂×D₄	1	2	yes	—
121	C₂×#88322	C₂×D₄	1	2	yes	—
122	C₂×#88322	C₂×D₄	1	2	yes	—
123	Dic₂₁₂	—	4	4	no	—
124	Dic₂₁₂	—	4	4	no	—
125	#11719912	Q₁₆	1	8	yes	complemented
126	C₈×D₅₃	—	2	4	no	—
127	C₂²×D₅₃	#127	1	1	yes	characteristic
128	Q₈×C₅₃	—	4	4	no	—
129	Q₈×C₅₃	—	4	4	no	—
130	C₂×C₄×C₅₃	—	4	2	no	—
131	C₂×C₄×C₅₃	—	4	2	no	—
132	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×Dic₅₃	—	4	2	no	—
136	C₂×Dic₅₃	—	4	2	no	—
137	C₂×Dic₅₃	#127	1	1	yes	characteristic
138	C₂×Dic₅₃	#127	1	1	yes	characteristic
139	C₂×Dic₅₃	#127	1	1	yes	characteristic
140	C₄×D₅₃	—	2	2	no	—
141	C₄×D₅₃	—	2	2	no	—
142	C₄×D₅₃	—	8	4	no	—
143	C₄×D₅₃	C₂×Q₁₆	1	4	yes	—
144	Dic₁₀₆	—	4	4	no	—
145	Dic₁₀₆	—	4	4	no	—
146	Dic₁₀₆	—	8	4	no	—
147	C₈×C₅₃	—	2	2	no	—
148	C₈×C₅₃	C₂×Q₁₆	1	4	yes	complemented
149	C₈×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
150	#88322	—	2	2	no	—
151	#88322	—	2	2	no	—
152	#88322	C₂×Q₁₆	1	4	yes	complemented
153	#19632	—	53	1	no	maximal, Sylow
154	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
155	C₂×D₅₃	—	2	1	no	—
156	C₂×D₅₃	—	2	1	no	—
157	C₂×D₅₃	#494	1	2	yes	—
158	C₄×C₅₃	—	8	2	no	—
159	C₄×C₅₃	—	8	2	no	—
160	C₄×C₅₃	C₂²×Q₁₆	1	2	yes	—
161	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
162	C₄×C₅₃	#578	1	2	yes	—
163	Dic₅₃	—	2	1	no	—
164	Dic₅₃	—	2	1	no	—
165	Dic₅₃	—	8	2	no	—
166	Dic₅₃	—	8	2	no	—
167	Dic₅₃	#494	1	2	yes	—
168	#3230	—	53	2	no	—
169	#3170	—	53	1	no	—
170	#3165	—	53	2	no	—
171	C₂×#494	—	53	1	no	—
172	#1723	—	53	2	no	—
173	#1723	—	53	2	no	—
174	#1713	—	53	1	no	—
175	#1713	—	53	1	no	—
176	#1707	—	53	1	no	—
177	#1707	D₅₃	1	1	yes	characteristic, complemented
178	C₈×M₁₆	—	53	1	no	—
179	C₂×C₅₃	C₂×#494	1	1	yes	characteristic
180	C₂×C₅₃	#3151	1	1	yes	characteristic
181	C₂×C₅₃	#3156	1	1	yes	characteristic
182	D₅₃	—	2	2	no	—
183	C₂²×Q₁₆	—	106	2	no	—
184	#533	—	53	2	no	—
185	C₂×C₄⋊Q₈	—	53	2	no	—
186	#501	—	106	4	no	—
187	#500	—	53	2	no	—
188	#500	—	53	2	no	—
189	#500	—	53	2	no	—
190	#500	—	53	2	no	—
191	#500	C₂×D₅₃	1	1	yes	characteristic
192	#498	—	53	2	no	—
193	#498	C₂×D₅₃	1	2	yes	—
194	#494	—	53	2	no	—
195	#494	—	53	1	no	—
196	#494	—	53	1	no	—
197	#494	—	106	2	no	—
198	#494	C₂×D₅₃	1	1	yes	characteristic
199	#487	—	53	2	no	—
200	#487	—	53	2	no	—
201	#487	C₂×D₅₃	1	2	yes	—
202	#484	—	106	4	no	—
203	#477	—	106	4	no	—
204	#467	—	106	2	no	—
205	C₄×Q₁₆	—	106	2	no	—
206	#431	—	53	1	no	—
207	#428	—	106	4	no	—
208	C₄×M₁₆	—	53	1	no	—
209	C₂×C₄×C₈	—	53	1	no	—
210	#322	—	53	1	no	—
211	#322	—	53	1	no	—
212	C₈²	—	53	1	no	—
213	C₈²	C₂×D₅₃	1	1	yes	characteristic
214	C₅₃	#19632	1	1	yes	characteristic, complemented, Sylow
215	C₂²×Q₈	—	106	2	no	—
216	C₂×Q₁₆	—	2	2	no	—
217	C₂×Q₁₆	—	106	4	no	—
218	C₂×Q₁₆	—	106	2	no	—
219	C₂×Q₁₆	—	106	2	no	—
220	C₂×Q₁₆	—	212	8	no	—
221	C₂×M₁₆	—	53	2	no	—
222	C₂²×C₈	—	53	2	no	—
223	C₄⋊Q₈	—	53	2	no	—
224	C₄⋊Q₈	—	53	2	no	—
225	C₄⋊Q₈	—	106	2	no	—
226	C₄⋊Q₈	—	106	2	no	—
227	C₄⋊Q₈	—	106	2	no	—
228	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
229	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
230	#125	—	106	2	no	—
231	C₂²⋊Q₈	—	106	2	no	—
232	C₂²⋊Q₈	—	106	2	no	—
233	C₄×Q₈	—	106	2	no	—
234	C₄×Q₈	—	106	2	no	—
235	#117	—	106	2	no	—
236	C₂×C₄⋊C₄	—	106	2	no	—
237	C₂×C₄⋊C₄	—	106	2	no	—
238	C₂×C₄²	—	53	1	no	—
239	#107	—	2	2	no	—
240	#107	—	106	4	no	—
241	#107	—	106	4	no	—
242	#107	—	106	4	no	—
243	#107	—	106	4	no	—
244	#106	—	2	4	no	—
245	#106	—	106	4	no	—
246	C₄⋊C₈	—	53	2	no	—
247	#103	—	2	4	no	—
248	#103	—	106	4	no	—
249	#103	—	106	4	no	—
250	C₂²⋊C₈	—	53	2	no	—
251	#97	—	53	2	no	—
252	C₄×C₈	—	53	2	no	—
253	C₄×C₈	—	53	1	no	—
254	C₄×C₈	—	53	1	no	—
255	C₄×C₈	—	53	1	no	—
256	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
257	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
258	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	106	2	no	—
262	C₂×Q₈	—	106	2	no	—
263	C₂×Q₈	—	212	2	no	—
264	C₂×Q₈	—	212	8	no	—
265	C₂²×C₄	—	53	1	no	—
266	C₂²×C₄	—	53	1	no	—
267	C₂²×C₄	—	53	1	no	—
268	C₂²×C₄	—	212	2	no	—
269	Q₁₆	—	4	4	no	—
270	Q₁₆	—	212	8	no	—
271	Q₁₆	—	212	4	no	—
272	Q₁₆	—	212	4	no	—
273	M₁₆	—	53	8	no	—
274	C₂×C₈	—	53	2	no	—
275	C₂×C₈	—	53	2	no	—
276	C₂×C₈	—	53	2	no	—
277	C₂×C₈	—	106	4	no	—
278	C₂×C₈	D₄×D₅₃	1	2	yes	—
279	C₂×C₈	D₄×D₅₃	1	2	yes	—
280	C₂×C₈	D₄×D₅₃	1	2	yes	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	106	2	no	—
286	C₄⋊C₄	—	106	2	no	—
287	C₄⋊C₄	—	106	2	no	—
288	C₄⋊C₄	—	106	2	no	—
289	C₄⋊C₄	—	212	2	no	—
290	C₄⋊C₄	—	212	2	no	—
291	C₄⋊C₄	—	212	2	no	—
292	C₄⋊C₄	—	212	2	no	—
293	C₂²⋊C₄	—	212	2	no	—
294	C₄²	—	53	1	no	—
295	C₄²	—	53	1	no	—
296	C₄²	—	53	1	no	—
297	C₄²	—	212	2	no	—
298	C₄²	C₂³×D₅₃	1	1	yes	characteristic
299	C₂³	—	53	1	no	—
300	Q₈	—	4	4	no	—
301	Q₈	—	4	4	no	—
302	Q₈	—	424	4	no	—
303	Q₈	—	212	4	no	—
304	Q₈	—	212	4	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	53	1	no	—
308	C₂×C₄	—	53	4	no	—
309	C₂×C₄	—	53	1	no	—
310	C₂×C₄	—	53	1	no	—
311	C₂×C₄	—	106	2	no	—
312	C₂×C₄	—	106	2	no	—
313	C₂×C₄	—	424	4	no	—
314	C₂×C₄	—	212	2	no	—
315	C₂×C₄	—	212	2	no	—
316	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
317	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
318	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
319	C₈	—	2	2	no	—
320	C₈	—	53	4	no	—
321	C₈	—	106	2	no	—
322	C₈	—	106	2	no	—
323	C₈	Q₁₆×D₅₃	1	4	yes	complemented
324	C₈	#13133244	1	4	yes	—
325	C₂²	—	53	2	no	—
326	C₂²	—	106	1	no	—
327	C₂²	—	106	1	no	—
328	C₂²	#127×D₅₃	1	1	yes	characteristic
329	C₄	—	8	2	no	—
330	C₄	—	8	2	no	—
331	C₄	—	53	2	no	—
332	C₄	—	106	1	no	—
333	C₄	—	106	1	no	—
334	C₄	—	424	2	no	—
335	C₄	—	424	2	no	—
336	C₄	C₂×Q₁₆×D₅₃	1	2	yes	—
337	C₄	C₂×#13133244	1	2	yes	—
338	C₄	#16811437	1	2	yes	—
339	C₂	—	106	2	no	—
340	C₂	#494×D₅₃	1	1	yes	characteristic
341	C₂	#20285162	1	1	yes	characteristic
342	C₂	#20285205	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1707	C₂
socle	C₂²×C₅₃	C₂×#127
prime radical	C₂²×D₅₃	#127
hypercenter	#1707	D₅₃
hypocenter	C₅₃	#19632
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1707

Complements

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

N	H ≅ G/N	type
5. #20274395	C₂	semidirect
6. #20274389	C₂	semidirect
7. #20274333	C₂	semidirect
8. #20274331	C₂	semidirect
9. #20274313	C₂	semidirect
11. C₅₃×#1707	C₂	semidirect
125. #11719912	Q₁₆	semidirect
148. C₈×C₅₃	C₂×Q₁₆	semidirect
152. #88322	C₂×Q₁₆	semidirect
177. #1707	D₅₃	semidirect
214. C₅₃	#19632	semidirect
323. C₈	Q₁₆×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#20285862	2
#20285302	2
#20285256	2
#494×D₅₃	2
#20274395	2
#20274389	2
#20274333	2
#20274331	2
#20274313	2
C₈×#11719912	2
C₅₃×#1707	2
#19632	53