Subgroups — #27355852

Order 13568.

722 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285294	C₂	1	1	yes	maximal, characteristic
2	#20283461	C₂	1	2	yes	maximal, complemented
3	#20273601	C₂	1	2	yes	maximal, complemented
4	#20273546	C₂	1	4	yes	maximal, complemented
5	C₈×D₄₂₄	C₂	1	2	yes	maximal, complemented
6	C₂×#16810376	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273115	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133225	—	2	2	no	—
10	#16811173	C₂²	1	2	yes	—
11	#16811028	C₄	1	1	yes	characteristic
12	#16811000	C₄	1	1	yes	characteristic
13	#16810970	C₄	1	2	yes	—
14	#16810859	C₄	1	2	yes	—
15	#16810859	C₄	1	2	yes	—
16	C₄×D₄₂₄	C₂²	1	2	yes	—
17	C₄×D₄₂₄	C₂²	1	2	yes	—
18	#16810702	C₂²	1	4	yes	—
19	#16810696	—	2	4	no	—
20	C₈×D₂₁₂	C₂²	1	4	yes	—
21	C₂×C₄⋊#88322	C₂²	1	2	yes	—
22	#16810591	—	2	2	no	—
23	#16810376	C₂²	1	1	yes	characteristic
24	#16810376	C₂²	1	2	yes	—
25	#16810376	C₂²	1	1	yes	characteristic
26	#16810352	C₂²	1	4	yes	—
27	#16810352	C₂²	1	4	yes	—
28	C₅₃×#431	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#322	C₂²	1	2	yes	—
32	C₈²×C₅₃	C₂²	1	2	yes	—
33	C₂×#11719945	—	2	2	no	—
34	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
35	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
36	#13133225	—	4	8	no	—
37	C₂×#11719912	—	2	4	no	—
38	C₂²×#88322	—	2	2	no	—
39	C₂×C₈×D₅₃	—	2	4	no	—
40	#13133217	C₂×C₄	1	2	yes	—
41	#13133214	C₂×C₄	1	2	yes	—
42	#13133205	—	2	2	no	—
43	#13133191	—	2	2	no	—
44	#13133191	C₂×C₄	1	4	yes	—
45	C₄×D₂₁₂	—	2	2	no	—
46	C₄×D₂₁₂	C₂³	1	4	yes	—
47	#13133151	C₂×C₄	1	2	yes	—
48	#13133151	C₂×C₄	1	2	yes	—
49	C₄⋊#88322	—	2	2	no	—
50	C₄⋊#88322	C₂³	1	2	yes	—
51	C₄⋊#88322	C₂³	1	2	yes	—
52	#13133139	C₂×C₄	1	4	yes	—
53	#13133139	C₂×C₄	1	4	yes	—
54	#13133129	—	2	4	no	—
55	#13133129	—	2	4	no	—
56	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
57	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
58	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₅₃	—	2	1	no	—
62	C₂²⋊C₈×C₅₃	—	2	1	no	—
63	#97×C₅₃	D₄	1	2	yes	—
64	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₅₃	C₂³	1	2	yes	—
67	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₅₃	C₂³	1	2	yes	—
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
73	C₂×C₂²⋊D₅₃	—	2	4	no	—
74	#11719945	—	4	4	no	—
75	#11719945	—	4	4	no	—
76	C₂×D₂₁₂	—	2	2	no	—
77	C₂×D₂₁₂	—	2	2	no	—
78	C₂×D₂₁₂	C₂²×C₄	1	4	yes	—
79	C₂×C₄×D₅₃	—	2	4	no	—
80	C₂×Dic₁₀₆	—	2	2	no	—
81	M₁₆×C₅₃	—	2	2	no	—
82	M₁₆×C₅₃	D₈	1	4	yes	complemented
83	C₂×C₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	—	2	2	no	—
85	C₂×C₈×C₅₃	—	4	1	no	—
86	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
92	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₅₃	#41	1	1	yes	characteristic
95	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
96	#11719921	—	2	4	no	—
97	#11719921	—	2	4	no	—
98	C₄⋊Dic₅₃	—	2	2	no	—
99	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
100	C₂×#88322	—	2	2	no	—
101	C₂×#88322	—	2	2	no	—
102	C₂×#88322	—	4	4	no	—
103	D₄₂₄	—	2	4	no	—
104	D₄₂₄	—	4	2	no	—
105	#11719912	—	4	8	no	—
106	C₈×D₅₃	—	4	8	no	—
107	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
108	C₂²×D₅₃	—	2	4	no	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	—	4	1	no	—
112	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₅₃	#135	1	2	yes	—
117	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
120	C₂²⋊D₅₃	—	4	8	no	—
121	C₂×Dic₅₃	—	2	4	no	—
122	D₂₁₂	—	2	4	no	—
123	D₂₁₂	—	2	4	no	—
124	D₂₁₂	—	4	4	no	—
125	D₂₁₂	—	8	2	no	—
126	C₄×D₅₃	—	4	4	no	—
127	C₄×D₅₃	—	4	4	no	—
128	Dic₁₀₆	—	4	4	no	—
129	C₈×C₅₃	—	2	1	no	—
130	C₈×C₅₃	—	2	1	no	—
131	C₈×C₅₃	—	2	1	no	—
132	C₈×C₅₃	—	2	2	no	—
133	C₈×C₅₃	—	4	2	no	—
134	C₈×C₅₃	#131	1	2	yes	complemented
135	C₈×C₅₃	C₂×D₈	1	4	yes	—
136	#88322	—	4	2	no	—
137	#88322	—	4	2	no	—
138	#18066	—	53	1	no	maximal, Sylow
139	C₂²×C₅₃	—	2	1	no	—
140	C₂²×C₅₃	—	2	1	no	—
141	C₂²×C₅₃	C₄×D₈	1	2	yes	—
142	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
143	C₂×D₅₃	—	4	4	no	—
144	C₂×D₅₃	—	4	4	no	—
145	C₂×D₅₃	—	4	4	no	—
146	C₄×C₅₃	—	2	1	no	—
147	C₄×C₅₃	—	2	1	no	—
148	C₄×C₅₃	—	2	1	no	—
149	C₄×C₅₃	C₄×D₈	1	2	yes	—
150	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
151	C₄×C₅₃	#568	1	1	yes	characteristic
152	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
153	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
154	Dic₅₃	—	4	4	no	—
155	#3169	—	53	1	no	—
156	#2935	—	53	2	no	—
157	#1589	—	53	2	no	—
158	#1581	—	53	4	no	—
159	C₈×D₈	—	53	2	no	—
160	C₂×#335	—	53	1	no	—
161	#1480	—	53	2	no	—
162	C₈×M₁₆	D₅₃	1	1	yes	characteristic, complemented
163	C₂×C₅₃	—	2	2	no	—
164	C₂×C₅₃	#2935	1	1	yes	characteristic
165	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
166	C₂×C₅₃	#2957	1	1	yes	characteristic
167	D₅₃	—	8	4	no	—
168	C₂×#131	—	106	2	no	—
169	#532	—	53	2	no	—
170	#500	—	53	1	no	—
171	#493	—	53	1	no	—
172	#486	—	53	2	no	—
173	#466	—	53	2	no	—
174	#466	—	53	2	no	—
175	C₄×D₈	—	53	2	no	—
176	C₄×D₈	—	53	2	no	—
177	#436	—	53	4	no	—
178	#435	—	106	4	no	—
179	C₈×D₄	—	53	4	no	—
180	#431	C₂×D₅₃	1	1	yes	characteristic
181	C₂×C₄⋊C₈	—	53	2	no	—
182	#408	—	106	2	no	—
183	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
184	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
185	#335	—	53	1	no	—
186	#335	—	53	2	no	—
187	#335	—	53	1	no	—
188	D₄⋊C₈	—	53	4	no	—
189	D₄⋊C₈	—	53	4	no	—
190	#322	C₂×D₅₃	1	2	yes	—
191	C₈²	C₂×D₅₃	1	2	yes	—
192	C₅₃	#18066	1	1	yes	characteristic, complemented, Sylow
193	C₂×#41	—	106	2	no	—
194	C₂×D₈	—	53	2	no	—
195	C₂×D₈	—	53	2	no	—
196	#131	—	212	8	no	—
197	C₂×M₁₆	—	106	4	no	—
198	C₂×M₁₆	D₂₁₂	1	1	yes	characteristic
199	C₂×M₁₆	D₂₁₂	1	1	yes	characteristic
200	C₂²×C₈	—	106	2	no	—
201	C₂²×C₈	—	106	4	no	—
202	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
203	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
204	C₄⋊Q₈	—	53	2	no	—
205	#127	—	53	2	no	—
206	#124	—	106	2	no	—
207	#121	—	53	4	no	—
208	#121	—	106	2	no	—
209	C₄×D₄	—	53	4	no	—
210	C₄×D₄	—	106	2	no	—
211	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
212	#107	—	53	2	no	—
213	#107	—	53	2	no	—
214	C₄⋊C₈	—	2	1	no	—
215	C₄⋊C₈	—	53	2	no	—
216	C₄⋊C₈	—	53	2	no	—
217	C₄⋊C₈	—	106	2	no	—
218	#102	—	53	4	no	—
219	#102	—	53	4	no	—
220	C₂²⋊C₈	—	2	1	no	—
221	C₂²⋊C₈	—	106	4	no	—
222	C₂²⋊C₈	—	106	4	no	—
223	#97	D₂₁₂	1	2	yes	—
224	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
225	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
226	C₄×C₈	C₂²×D₅₃	1	2	yes	—
227	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
228	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
229	C₄×C₈	C₂²×D₅₃	1	2	yes	—
230	#41	—	212	4	no	—
231	#41	—	212	4	no	—
232	C₂×Q₈	—	106	2	no	—
233	C₂×D₄	—	53	4	no	—
234	C₂×D₄	—	106	4	no	—
235	C₂×D₄	—	106	2	no	—
236	C₂×D₄	—	106	2	no	—
237	C₂²×C₄	—	106	4	no	—
238	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
240	C₂²×C₄	#11719945	1	1	yes	characteristic
241	D₈	—	106	4	no	—
242	D₈	—	212	2	no	—
243	M₁₆	—	2	2	no	—
244	M₁₆	—	212	8	no	—
245	M₁₆	D₄₂₄	1	4	yes	complemented
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	106	2	no	—
250	C₂×C₈	—	106	2	no	—
251	C₂×C₈	—	212	4	no	—
252	C₂×C₈	—	212	8	no	—
253	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
254	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
255	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
256	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
257	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
258	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
259	C₄⋊C₄	—	53	4	no	—
260	C₄⋊C₄	—	106	2	no	—
261	C₂²⋊C₄	—	106	4	no	—
262	C₂²⋊C₄	—	106	4	no	—
263	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
264	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
265	C₄²	#11719945	1	1	yes	characteristic
266	C₄²	C₂³×D₅₃	1	1	yes	characteristic
267	C₂³	—	106	4	no	—
268	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
269	Q₈	—	212	4	no	—
270	D₄	—	106	4	no	—
271	D₄	—	106	4	no	—
272	D₄	—	424	2	no	—
273	D₄	—	212	8	no	—
274	D₄	—	212	4	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	106	4	no	—
279	C₂×C₄	—	212	4	no	—
280	C₂×C₄	—	212	4	no	—
281	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
282	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
283	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
284	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
285	C₂×C₄	#13133239	1	2	yes	—
286	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
288	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	2	2	no	—
293	C₈	—	4	2	no	—
294	C₈	—	212	2	no	—
295	C₈	—	212	2	no	—
296	C₈	#13133225	1	2	yes	complemented
297	C₈	C₂×D₄₂₄	1	4	yes	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	212	4	no	—
301	C₂²	—	212	4	no	—
302	C₂²	—	212	4	no	—
303	C₂²	C₄×D₄₂₄	1	2	yes	—
304	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	212	4	no	—
309	C₄	C₄×D₄₂₄	1	2	yes	—
310	C₄	C₂×#13133225	1	1	yes	characteristic
311	C₄	#16811389	1	1	yes	characteristic
312	C₄	C₂²×D₄₂₄	1	1	yes	characteristic
313	C₄	C₂×#13133239	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	424	4	no	—
316	C₂	#20283461	1	1	yes	characteristic
317	C₂	C₂×C₄×D₄₂₄	1	1	yes	characteristic
318	C₂	#20283633	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₈×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#20285294	C₂
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₈×M₁₆	D₅₃
hypocenter	C₅₃	#18066
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
2. #20283461	C₂	semidirect
3. #20273601	C₂	semidirect
4. #20273546	C₂	semidirect
5. C₈×D₄₂₄	C₂	semidirect
6. C₂×#16810376	C₂	semidirect
7. #20273115	C₂	semidirect
8. C₈×M₁₆×C₅₃	C₂	semidirect
82. M₁₆×C₅₃	D₈	semidirect
134. C₈×C₅₃	#131	semidirect
162. C₈×M₁₆	D₅₃	semidirect
192. C₅₃	#18066	semidirect
245. M₁₆	D₄₂₄	semidirect
296. C₈	#13133225	semidirect

Maximal subgroups

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

subgroup	index
#20285294	2
#20283461	2
#20273601	2
#20273546	2
C₈×D₄₂₄	2
C₂×#16810376	2
#20273115	2
C₈×M₁₆×C₅₃	2
#18066	53