Subgroups — #27276966

Order 13568.

462 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20273463	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810374	C₂	1	1	yes	maximal, characteristic, complemented
3	#20273102	C₂	1	2	yes	maximal
4	#20272566	C₂	1	2	yes	maximal, complemented
5	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	#16810376	—	2	1	no	—
9	#16810374	—	2	1	no	—
10	#16810374	—	2	1	no	—
11	#16810374	—	2	1	no	—
12	Q₈⋊#88322	—	2	2	no	—
13	D₄⋊#88322	—	2	2	no	—
14	C₂×#131×C₅₃	C₄	1	1	yes	characteristic
15	C₂×#131×C₅₃	C₄	1	1	yes	characteristic
16	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#405	—	2	1	no	—
18	C₅₃×#405	—	2	1	no	—
19	C₅₃×#405	C₄	1	1	yes	characteristic
20	C₅₃×#405	C₄	1	1	yes	characteristic
21	C₄×M₁₆×C₅₃	—	2	1	no	—
22	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₅₃	—	2	1	no	—
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×#88322	—	2	2	no	—
26	C₄⋊#88322	—	2	2	no	—
27	C₄⋊#88322	—	2	2	no	—
28	C₄⋊#88322	—	2	2	no	—
29	C₂²⋊#88322	—	2	2	no	—
30	C₄×#88322	—	2	2	no	—
31	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
32	#131×C₅₃	—	2	2	no	—
33	#131×C₅₃	—	2	2	no	—
34	#131×C₅₃	—	2	2	no	—
35	#131×C₅₃	C₈	1	4	yes	complemented
36	C₂×M₁₆×C₅₃	—	2	1	no	—
37	C₂×M₁₆×C₅₃	—	2	1	no	—
38	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	—	2	1	no	—
41	C₂²×C₈×C₅₃	—	2	1	no	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
45	C₄×Q₈×C₅₃	Q₈	1	1	yes	characteristic
46	C₄×D₄×C₅₃	—	2	1	no	—
47	C₄×D₄×C₅₃	—	2	1	no	—
48	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
49	C₄×D₄×C₅₃	Q₈	1	1	yes	characteristic
50	#117×C₅₃	—	2	1	no	—
51	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	—	2	1	no	—
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	#97×C₅₃	—	2	1	no	—
55	#97×C₅₃	—	2	1	no	—
56	#97×C₅₃	—	2	1	no	—
57	C₄×C₈×C₅₃	—	2	1	no	—
58	C₄×C₈×C₅₃	—	2	1	no	—
59	C₄×C₈×C₅₃	—	2	1	no	—
60	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
62	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
63	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
64	#41×C₅₃	—	2	1	no	—
65	#41×C₅₃	—	2	1	no	—
66	#41×C₅₃	C₂×C₈	1	2	yes	—
67	#41×C₅₃	M₁₆	1	2	yes	—
68	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
69	C₂×D₄×C₅₃	—	2	1	no	—
70	C₂×D₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	—	2	1	no	—
72	C₂²×C₄×C₅₃	—	2	1	no	—
73	C₂²×C₄×C₅₃	—	2	1	no	—
74	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
77	M₁₆×C₅₃	—	2	4	no	—
78	M₁₆×C₅₃	—	2	1	no	—
79	M₁₆×C₅₃	—	2	1	no	—
80	M₁₆×C₅₃	—	2	4	no	—
81	M₁₆×C₅₃	C₂×C₈	1	2	yes	—
82	M₁₆×C₅₃	M₁₆	1	2	yes	—
83	C₂×C₈×C₅₃	—	2	4	no	—
84	C₂×C₈×C₅₃	—	2	1	no	—
85	C₂×C₈×C₅₃	—	2	1	no	—
86	C₂×C₈×C₅₃	—	2	1	no	—
87	C₂×C₈×C₅₃	—	2	1	no	—
88	C₂×C₈×C₅₃	—	2	4	no	—
89	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
94	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
95	C₄⋊C₄×C₅₃	—	2	1	no	—
96	C₄⋊C₄×C₅₃	—	2	1	no	—
97	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
98	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
99	C₂²⋊C₄×C₅₃	—	2	1	no	—
100	C₂²⋊C₄×C₅₃	—	2	1	no	—
101	C₂²⋊C₄×C₅₃	—	2	1	no	—
102	C₄²×C₅₃	—	2	1	no	—
103	C₄²×C₅₃	—	2	1	no	—
104	C₄²×C₅₃	—	2	1	no	—
105	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
109	C₂×#88322	—	4	2	no	—
110	C₂×#88322	—	4	2	no	—
111	C₂×#88322	—	8	2	no	—
112	C₂³×C₅₃	—	2	1	no	—
113	C₂³×C₅₃	#117	1	1	yes	characteristic
114	Q₈×C₅₃	—	2	1	no	—
115	Q₈×C₅₃	C₄⋊C₈	1	2	yes	—
116	D₄×C₅₃	—	2	2	no	—
117	D₄×C₅₃	—	2	2	no	—
118	D₄×C₅₃	—	2	1	no	—
119	D₄×C₅₃	C₄⋊C₈	1	2	yes	—
120	C₂×C₄×C₅₃	—	2	1	no	—
121	C₂×C₄×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	1	no	—
125	C₂×C₄×C₅₃	—	2	1	no	—
126	C₂×C₄×C₅₃	—	4	1	no	—
127	C₂×C₄×C₅₃	—	4	2	no	—
128	C₂×C₄×C₅₃	—	4	2	no	—
129	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
132	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
139	C₈×C₅₃	—	2	2	no	—
140	C₈×C₅₃	—	2	1	no	—
141	C₈×C₅₃	—	2	2	no	—
142	C₈×C₅₃	—	2	1	no	—
143	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
144	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
145	#88322	—	8	2	no	—
146	#88322	—	8	2	no	—
147	#8604	—	53	1	no	maximal, Sylow
148	C₂²×C₅₃	—	2	1	no	—
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	—	2	2	no	—
151	C₂²×C₅₃	—	4	1	no	—
152	C₂²×C₅₃	—	4	1	no	—
153	C₂²×C₅₃	#380	1	1	yes	characteristic
154	C₂²×C₅₃	#431	1	1	yes	characteristic
155	C₂²×C₅₃	#432	1	1	yes	characteristic
156	C₄×C₅₃	—	2	2	no	—
157	C₄×C₅₃	—	2	1	no	—
158	C₄×C₅₃	—	2	1	no	—
159	C₄×C₅₃	—	2	1	no	—
160	C₄×C₅₃	—	4	1	no	—
161	C₄×C₅₃	—	4	1	no	—
162	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
163	C₄×C₅₃	#407	1	1	yes	characteristic
164	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
165	C₄×C₅₃	#423	1	1	yes	characteristic
166	C₄×C₅₃	#471	1	2	yes	—
167	C₄×#131	D₅₃	1	1	yes	characteristic
168	#1564	—	53	1	no	—
169	C₂×#334	—	53	1	no	—
170	#1477	—	53	2	no	—
171	#1283	—	53	2	no	—
172	C₂×C₅₃	—	2	1	no	—
173	C₂×C₅₃	—	2	1	no	—
174	C₂×C₅₃	—	4	2	no	—
175	C₂×C₅₃	#1797	1	1	yes	characteristic
176	C₂×C₅₃	#1814	1	1	yes	characteristic
177	C₂×C₅₃	#1815	1	1	yes	characteristic
178	C₂×#131	Dic₅₃	1	1	yes	characteristic
179	C₂×#131	Dic₅₃	1	1	yes	characteristic
180	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
181	#431	—	53	2	no	—
182	C₂×C₄⋊C₈	—	53	2	no	—
183	#405	—	2	1	no	—
184	#405	—	2	1	no	—
185	#405	Dic₅₃	1	1	yes	characteristic
186	#405	Dic₅₃	1	1	yes	characteristic
187	C₄×M₁₆	—	2	1	no	—
188	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
189	C₂×C₄×C₈	—	2	1	no	—
190	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
191	#335	—	106	1	no	—
192	#334	—	106	1	no	—
193	#334	—	106	1	no	—
194	#334	—	106	1	no	—
195	Q₈⋊C₈	—	106	2	no	—
196	D₄⋊C₈	—	106	2	no	—
197	C₅₃	#8604	1	1	yes	characteristic, complemented, Sylow
198	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
199	#131	—	2	2	no	—
200	#131	—	2	2	no	—
201	#131	—	2	2	no	—
202	#131	#88322	1	4	yes	complemented
203	C₂×M₁₆	—	2	1	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
206	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
207	C₂²×C₈	—	2	1	no	—
208	C₂²×C₈	—	2	1	no	—
209	C₂²×C₈	—	106	2	no	—
210	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
211	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
212	C₄×Q₈	Dic₁₀₆	1	1	yes	characteristic
213	C₄×Q₈	D₂₁₂	1	1	yes	characteristic
214	C₄×D₄	—	2	1	no	—
215	C₄×D₄	—	2	1	no	—
216	C₄×D₄	Dic₁₀₆	1	1	yes	characteristic
217	C₄×D₄	D₂₁₂	1	1	yes	characteristic
218	#117	—	2	1	no	—
219	#117	C₂×Dic₅₃	1	1	yes	characteristic
220	C₂×C₄²	—	2	1	no	—
221	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
222	C₄⋊C₈	—	106	2	no	—
223	C₄⋊C₈	—	106	2	no	—
224	C₄⋊C₈	—	106	2	no	—
225	C₂²⋊C₈	—	106	2	no	—
226	#97	—	2	1	no	—
227	#97	—	2	1	no	—
228	#97	—	2	1	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	2	1	no	—
232	C₄×C₈	—	106	2	no	—
233	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
234	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
235	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
236	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
237	#41	—	2	1	no	—
238	#41	—	2	1	no	—
239	#41	C₂×#88322	1	2	yes	—
240	#41	#11719917	1	2	yes	—
241	C₂×Q₈	C₄⋊Dic₅₃	1	1	yes	characteristic
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	C₄⋊Dic₅₃	1	1	yes	characteristic
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	#11719945	1	1	yes	characteristic
248	C₂²×C₄	#11719945	1	1	yes	characteristic
249	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
250	M₁₆	—	2	4	no	—
251	M₁₆	—	2	1	no	—
252	M₁₆	—	2	1	no	—
253	M₁₆	—	2	4	no	—
254	M₁₆	C₂×#88322	1	2	yes	—
255	M₁₆	#11719917	1	2	yes	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	424	2	no	—
263	C₂×C₈	—	212	2	no	—
264	C₂×C₈	—	212	2	no	—
265	C₂×C₈	C₂×#88322	1	2	yes	—
266	C₂×C₈	#11719917	1	2	yes	—
267	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
268	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
269	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
270	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
274	C₄⋊C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	2	1	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
282	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
283	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
284	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
285	C₂³	—	2	1	no	—
286	C₂³	#13133176	1	1	yes	characteristic
287	Q₈	—	2	1	no	—
288	Q₈	C₄⋊#88322	1	2	yes	—
289	D₄	—	2	2	no	—
290	D₄	—	2	2	no	—
291	D₄	—	2	1	no	—
292	D₄	C₄⋊#88322	1	2	yes	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
303	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
304	C₂×C₄	#13133191	1	1	yes	characteristic
305	C₂×C₄	#13133191	1	1	yes	characteristic
306	C₂×C₄	#13133196	1	1	yes	characteristic
307	C₂×C₄	#13133196	1	1	yes	characteristic
308	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
309	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
310	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
311	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
312	C₈	—	2	2	no	—
313	C₈	—	2	1	no	—
314	C₈	—	2	2	no	—
315	C₈	—	2	1	no	—
316	C₈	—	424	2	no	—
317	C₈	—	424	2	no	—
318	C₈	C₂²⋊#88322	1	2	yes	—
319	C₈	C₂²⋊#88322	1	2	yes	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	#16810483	1	1	yes	characteristic
326	C₂²	#16810676	1	1	yes	characteristic
327	C₂²	#16810680	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	C₂×C₂²⋊#88322	1	1	yes	characteristic
335	C₄	#16810587	1	1	yes	characteristic
336	C₄	C₂×C₄⋊#88322	1	1	yes	characteristic
337	C₄	#16810647	1	1	yes	characteristic
338	C₄	#16810886	1	2	yes	—
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	4	2	no	—
342	C₂	#20274695	1	1	yes	characteristic
343	C₂	#20274769	1	1	yes	characteristic
344	C₂	#20274774	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133191
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×#131×C₅₃	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#131	D₅₃
hypocenter	C₅₃	#8604
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. #20273463	C₂	semidirect
2. C₂×#16810374	C₂	semidirect
4. #20272566	C₂	semidirect
35. #131×C₅₃	C₈	semidirect
197. C₅₃	#8604	semidirect
202. #131	#88322	semidirect

Maximal subgroups

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

subgroup	index
#20273463	2
C₂×#16810374	2
#20273102	2
#20272566	2
C₄×#131×C₅₃	2
#8604	53