Subgroups — #27277253

Order 13568.

454 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11719945	C₂	1	1	yes	maximal, characteristic
2	#20274732	C₂	1	2	yes	maximal
3	C₄×#13133149	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272616	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1566	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#11719945	C₂²	1	1	yes	characteristic
7	C₈×Dic₁₀₆	C₄	1	1	yes	characteristic, complemented
8	C₈×Dic₁₀₆	C₄	1	1	yes	characteristic, complemented
9	C₈×D₂₁₂	C₄	1	1	yes	characteristic, complemented
10	C₈×D₂₁₂	C₄	1	1	yes	characteristic, complemented
11	C₈×C₂²⋊D₅₃	—	2	1	no	—
12	C₈×C₂²⋊D₅₃	—	2	1	no	—
13	#16810676	C₂²	1	1	yes	characteristic
14	#16810674	—	2	1	no	—
15	C₂×#13133149	—	2	1	no	—
16	C₂×#13133143	—	2	2	no	—
17	C₄×C₈×D₅₃	—	2	1	no	—
18	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
19	#16810386	—	2	1	no	—
20	#16810384	—	2	2	no	—
21	C₅₃×#423	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#334	—	2	1	no	—
24	C₅₃×#334	—	2	1	no	—
25	C₂×#11719945	D₄	1	1	yes	characteristic
26	C₂×C₈×D₅₃	—	2	1	no	—
27	C₂×C₈×D₅₃	—	2	1	no	—
28	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₁₀₆	Q₈	1	1	yes	characteristic
30	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
31	C₄×D₂₁₂	Q₈	1	1	yes	characteristic
32	C₄×C₂²⋊D₅₃	—	2	1	no	—
33	C₄×C₂²⋊D₅₃	—	2	1	no	—
34	#13133176	D₄	1	1	yes	characteristic
35	#13133174	—	2	1	no	—
36	C₂×C₄⋊Dic₅₃	—	2	2	no	—
37	C₂×C₄×Dic₅₃	—	2	2	no	—
38	C₄²×D₅₃	—	2	1	no	—
39	#13133149	—	2	4	no	—
40	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
41	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
42	#13133146	—	2	1	no	—
43	#13133146	—	2	1	no	—
44	#13133143	—	4	4	no	—
45	C₂²⋊#88322	—	2	1	no	—
46	#13133129	—	2	1	no	—
47	#13133129	—	2	1	no	—
48	C₄×#88322	—	2	1	no	—
49	C₈×Dic₅₃	—	2	1	no	—
50	C₈×Dic₅₃	—	2	1	no	—
51	#13133124	—	2	2	no	—
52	C₂×M₁₆×C₅₃	—	2	2	no	—
53	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₅₃	—	2	2	no	—
57	C₄⋊C₈×C₅₃	—	2	2	no	—
58	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
65	C₂×C₂²⋊D₅₃	—	2	1	no	—
66	C₂²×Dic₅₃	—	4	2	no	—
67	#11719945	—	2	2	no	—
68	#11719945	SD₁₆	1	4	yes	—
69	C₂×D₂₁₂	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×D₅₃	—	2	1	no	—
71	C₂×C₄×D₅₃	—	2	1	no	—
72	C₂×C₄×D₅₃	—	2	1	no	—
73	C₂×Dic₁₀₆	C₄⋊C₄	1	1	yes	characteristic
74	M₁₆×C₅₃	—	4	2	no	—
75	M₁₆×C₅₃	—	4	2	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	4	1	no	—
78	C₂×C₈×C₅₃	—	4	2	no	—
79	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
86	C₄²×C₅₃	#41	1	1	yes	characteristic
87	C₂²⋊Dic₅₃	—	2	1	no	—
88	#11719921	—	2	1	no	—
89	#11719921	—	2	1	no	—
90	C₄⋊Dic₅₃	—	2	4	no	—
91	C₄⋊Dic₅₃	—	4	2	no	—
92	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
93	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
94	#11719919	—	2	1	no	—
95	#11719919	—	2	1	no	—
96	C₄×Dic₅₃	—	2	1	no	—
97	C₄×Dic₅₃	—	2	1	no	—
98	C₄×Dic₅₃	—	2	1	no	—
99	C₄×Dic₅₃	—	4	2	no	—
100	C₄×Dic₅₃	—	4	2	no	—
101	C₂×#88322	—	2	1	no	—
102	C₂×#88322	—	2	1	no	—
103	C₈×D₅₃	—	4	2	no	—
104	C₈×D₅₃	—	4	2	no	—
105	C₂³×C₅₃	#121	1	1	yes	characteristic
106	C₂²×D₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	1	no	—
108	C₂×C₄×C₅₃	—	2	1	no	—
109	C₂×C₄×C₅₃	—	4	1	no	—
110	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
117	C₂×C₄×C₅₃	#135	1	2	yes	—
118	C₂²⋊D₅₃	—	2	4	no	—
119	C₂×Dic₅₃	—	2	1	no	—
120	C₂×Dic₅₃	—	2	1	no	—
121	C₂×Dic₅₃	—	2	1	no	—
122	C₂×Dic₅₃	—	4	4	no	—
123	C₂×Dic₅₃	—	8	2	no	—
124	C₂×Dic₅₃	—	8	2	no	—
125	D₂₁₂	—	2	1	no	—
126	D₂₁₂	#106	1	2	yes	complemented
127	C₄×D₅₃	—	2	4	no	—
128	C₄×D₅₃	—	4	2	no	—
129	C₄×D₅₃	—	4	1	no	—
130	C₄×D₅₃	—	4	1	no	—
131	Dic₁₀₆	—	2	1	no	—
132	Dic₁₀₆	#106	1	2	yes	complemented
133	C₈×C₅₃	—	2	1	no	—
134	C₈×C₅₃	—	2	1	no	—
135	C₈×C₅₃	—	2	1	no	—
136	C₈×C₅₃	—	8	2	no	—
137	C₈×C₅₃	#104	1	2	yes	complemented
138	#88322	—	4	1	no	—
139	#88322	—	4	1	no	—
140	#8666	—	53	1	no	maximal, Sylow
141	C₂²×C₅₃	—	2	1	no	—
142	C₂²×C₅₃	—	2	1	no	—
143	C₂²×C₅₃	#380	1	1	yes	characteristic
144	C₂²×C₅₃	#465	1	2	yes	—
145	C₂×D₅₃	—	2	2	no	—
146	C₂×D₅₃	—	4	1	no	—
147	C₂×D₅₃	—	4	1	no	—
148	C₄×C₅₃	—	2	1	no	—
149	C₄×C₅₃	—	2	1	no	—
150	C₄×C₅₃	—	2	1	no	—
151	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
152	C₄×C₅₃	#421	1	1	yes	characteristic
153	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
154	C₄×C₅₃	#427	1	1	yes	characteristic
155	C₄×C₅₃	#465	1	2	yes	—
156	Dic₅₃	—	2	2	no	—
157	Dic₅₃	—	4	1	no	—
158	Dic₅₃	—	4	1	no	—
159	Dic₅₃	—	8	4	no	—
160	C₈×#41	—	53	1	no	—
161	#1806	—	53	2	no	—
162	C₄×#106	—	53	1	no	—
163	#1566	D₅₃	1	1	yes	characteristic, complemented
164	#1303	—	53	2	no	—
165	C₂×C₅₃	—	2	2	no	—
166	C₂×C₅₃	#1806	1	1	yes	characteristic
167	C₂×C₅₃	#1807	1	1	yes	characteristic
168	C₂×C₅₃	#1814	1	1	yes	characteristic
169	D₅₃	—	4	2	no	—
170	C₄×#41	—	53	1	no	—
171	C₈×Q₈	—	53	1	no	—
172	C₈×Q₈	—	53	1	no	—
173	C₈×D₄	—	53	1	no	—
174	C₈×D₄	—	53	1	no	—
175	C₈×D₄	—	106	1	no	—
176	C₈×D₄	—	106	1	no	—
177	#431	—	53	1	no	—
178	#431	—	106	1	no	—
179	C₂×#106	—	106	1	no	—
180	#423	C₂×D₅₃	1	2	yes	—
181	C₂×#104	—	106	2	no	—
182	C₂×C₄×C₈	—	106	1	no	—
183	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
184	C₄×C₄⋊C₄	—	53	2	no	—
185	#340	—	106	1	no	—
186	#339	—	106	2	no	—
187	#334	—	2	1	no	—
188	#334	—	2	1	no	—
189	C₅₃	#8666	1	1	yes	characteristic, complemented, Sylow
190	C₂×#41	—	53	1	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂²×C₈	—	106	1	no	—
193	C₂²×C₈	—	106	1	no	—
194	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
195	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
196	C₄×Q₈	—	53	1	no	—
197	C₄×Q₈	—	53	1	no	—
198	C₄×D₄	—	53	1	no	—
199	C₄×D₄	—	53	1	no	—
200	C₄×D₄	—	106	1	no	—
201	C₄×D₄	—	106	1	no	—
202	#117	—	53	1	no	—
203	#117	—	106	1	no	—
204	C₂×C₄⋊C₄	—	106	2	no	—
205	C₂×C₄²	—	106	1	no	—
206	C₂×C₄²	—	106	2	no	—
207	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
208	#106	—	106	4	no	—
209	C₄⋊C₈	—	2	2	no	—
210	C₄⋊C₈	—	2	2	no	—
211	C₄⋊C₈	—	53	1	no	—
212	C₄⋊C₈	—	53	1	no	—
213	C₄⋊C₈	—	106	1	no	—
214	C₄⋊C₈	—	106	1	no	—
215	#104	—	212	4	no	—
216	C₂²⋊C₈	—	106	1	no	—
217	C₂²⋊C₈	—	106	1	no	—
218	C₂²⋊C₈	—	106	1	no	—
219	C₄×C₈	—	106	1	no	—
220	C₄×C₈	—	106	1	no	—
221	C₄×C₈	—	106	1	no	—
222	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
223	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
224	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
225	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
226	#95	—	106	2	no	—
227	#41	—	53	4	no	—
228	#41	—	106	2	no	—
229	C₂×Q₈	—	53	1	no	—
230	C₂×D₄	—	53	1	no	—
231	C₂×D₄	—	106	1	no	—
232	C₂²×C₄	—	106	1	no	—
233	C₂²×C₄	—	106	1	no	—
234	C₂²×C₄	—	106	1	no	—
235	C₂²×C₄	—	212	2	no	—
236	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
237	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
238	C₂²×C₄	#11719949	1	1	yes	characteristic
239	M₁₆	—	4	2	no	—
240	M₁₆	—	4	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	106	1	no	—
245	C₂×C₈	—	106	1	no	—
246	C₂×C₈	—	212	2	no	—
247	C₂×C₈	—	212	2	no	—
248	C₂×C₈	#11719921	1	1	yes	characteristic
249	C₂×C₈	#11719921	1	1	yes	characteristic
250	C₂×C₈	#11719921	1	1	yes	characteristic
251	C₂×C₈	#11719921	1	1	yes	characteristic
252	C₄⋊C₄	—	53	1	no	—
253	C₄⋊C₄	—	53	1	no	—
254	C₄⋊C₄	—	106	4	no	—
255	C₄⋊C₄	—	106	1	no	—
256	C₄⋊C₄	—	106	1	no	—
257	C₄⋊C₄	—	212	2	no	—
258	C₂²⋊C₄	—	106	1	no	—
259	C₂²⋊C₄	—	106	1	no	—
260	C₂²⋊C₄	—	106	1	no	—
261	C₄²	—	106	1	no	—
262	C₄²	—	106	1	no	—
263	C₄²	—	106	1	no	—
264	C₄²	—	212	2	no	—
265	C₄²	—	212	2	no	—
266	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
267	C₄²	#11719947	1	1	yes	characteristic
268	C₄²	Q₈×D₅₃	1	1	yes	characteristic
269	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
270	C₂³	—	106	1	no	—
271	C₂³	#13133192	1	1	yes	characteristic
272	Q₈	—	53	2	no	—
273	Q₈	—	106	1	no	—
274	D₄	—	53	2	no	—
275	D₄	—	106	4	no	—
276	D₄	—	106	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	106	1	no	—
281	C₂×C₄	—	106	4	no	—
282	C₂×C₄	—	106	1	no	—
283	C₂×C₄	—	106	1	no	—
284	C₂×C₄	—	424	2	no	—
285	C₂×C₄	—	424	2	no	—
286	C₂×C₄	—	212	2	no	—
287	C₂×C₄	—	212	4	no	—
288	C₂×C₄	—	212	1	no	—
289	C₂×C₄	—	212	1	no	—
290	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
291	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
292	C₂×C₄	#13133175	1	1	yes	characteristic
293	C₂×C₄	#13133188	1	1	yes	characteristic
294	C₂×C₄	#13133193	1	1	yes	characteristic
295	C₂×C₄	#13133197	1	1	yes	characteristic
296	C₂×C₄	SD₁₆×D₅₃	1	2	yes	—
297	C₂×C₄	#13133238	1	2	yes	—
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	8	2	no	—
302	C₈	—	212	1	no	—
303	C₈	—	212	1	no	—
304	C₈	#13133143	1	2	yes	complemented
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	106	2	no	—
308	C₂²	—	212	1	no	—
309	C₂²	—	212	1	no	—
310	C₂²	#16810480	1	1	yes	characteristic
311	C₂²	#16810853	1	2	yes	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	106	2	no	—
316	C₄	—	424	4	no	—
317	C₄	—	212	1	no	—
318	C₄	—	212	1	no	—
319	C₄	C₂×#13133143	1	1	yes	characteristic
320	C₄	#16810639	1	1	yes	characteristic
321	C₄	#106×D₅₃	1	1	yes	characteristic
322	C₄	#16810658	1	1	yes	characteristic
323	C₄	#16810849	1	2	yes	—
324	C₂	—	2	2	no	—
325	C₂	—	212	2	no	—
326	C₂	#20274732	1	1	yes	characteristic
327	C₂	#20274737	1	1	yes	characteristic
328	C₂	#20274770	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133188
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1566	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂×#11719945	D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1566	D₅₃
hypocenter	C₅₃	#8666
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
3. C₄×#13133149	C₂	semidirect
4. #20272616	C₂	semidirect
5. C₅₃×#1566	C₂	semidirect
7. C₈×Dic₁₀₆	C₄	semidirect
8. C₈×Dic₁₀₆	C₄	semidirect
9. C₈×D₂₁₂	C₄	semidirect
10. C₈×D₂₁₂	C₄	semidirect
126. D₂₁₂	#106	semidirect
132. Dic₁₀₆	#106	semidirect
137. C₈×C₅₃	#104	semidirect
163. #1566	D₅₃	semidirect
189. C₅₃	#8666	semidirect
304. C₈	#13133143	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11719945	2
#20274732	2
C₄×#13133149	2
#20272616	2
C₅₃×#1566	2
#8666	53