Subgroups — #27367194

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285845	C₂	1	1	yes	maximal, characteristic
2	#20285642	C₂	1	1	yes	maximal, characteristic
3	C₂×#16811021	C₂	1	1	yes	maximal, characteristic, complemented
4	#20283428	C₂	1	1	yes	maximal, characteristic
5	#20274215	C₂	1	4	yes	maximal, complemented
6	#20274091	C₂	1	4	yes	maximal, complemented
7	#20273399	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273395	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13133223	—	2	1	no	—
11	#16811173	C₂²	1	1	yes	characteristic
12	C₂×#13133217	C₂²	1	1	yes	characteristic
13	C₂×#13133217	C₂²	1	1	yes	characteristic
14	#16811021	—	2	1	no	—
15	#16811021	—	2	1	no	—
16	#16811021	—	2	1	no	—
17	#16811021	C₂²	1	2	yes	—
18	#16811021	C₂²	1	2	yes	—
19	#16810996	—	2	1	no	—
20	#16810974	—	2	1	no	—
21	#16810970	—	2	1	no	—
22	#16810942	—	2	2	no	—
23	#16810914	C₂²	1	4	yes	—
24	#16810829	C₂²	1	4	yes	—
25	#16810764	—	2	2	no	—
26	#16810702	—	2	1	no	—
27	#16810702	C₂²	1	2	yes	—
28	C₈×D₂₁₂	—	2	1	no	—
29	C₈×D₂₁₂	C₂²	1	2	yes	—
30	C₂×#13133149	—	2	1	no	—
31	C₂×#13133149	—	2	1	no	—
32	#16810647	C₂²	1	1	yes	characteristic
33	#16810589	—	2	1	no	—
34	#16810371	C₂²	1	4	yes	—
35	#16810370	C₂²	1	4	yes	—
36	C₅₃×#431	C₂²	1	1	yes	characteristic
37	C₅₃×#431	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₅₃×#335	C₂²	1	2	yes	—
40	C₅₃×#334	C₂²	1	2	yes	—
41	C₂×#11719945	—	2	1	no	—
42	C₂²×Dic₁₀₆	—	2	1	no	—
43	C₂²×Dic₁₀₆	—	2	1	no	—
44	C₂×#11719913	—	2	4	no	—
45	#13133223	—	4	4	no	—
46	C₂×#11719917	—	2	1	no	—
47	C₂×#11719912	—	2	2	no	—
48	C₂×C₈×D₅₃	—	2	2	no	—
49	#13133217	—	2	1	no	—
50	#13133217	—	2	1	no	—
51	#13133217	—	2	1	no	—
52	#13133217	—	2	1	no	—
53	#13133217	—	2	1	no	—
54	#13133217	—	2	1	no	—
55	#13133217	D₄	1	1	yes	characteristic
56	#13133217	C₂³	1	2	yes	—
57	#13133217	C₂³	1	2	yes	—
58	#13133214	D₄	1	1	yes	characteristic
59	#13133205	—	2	1	no	—
60	#13133191	—	2	1	no	—
61	#13133191	D₄	1	2	yes	—
62	C₄×D₂₁₂	—	2	1	no	—
63	C₄×D₂₁₂	C₂³	1	2	yes	—
64	C₂×C₄⋊Dic₅₃	—	2	1	no	—
65	C₂×C₄⋊Dic₅₃	—	2	1	no	—
66	C₂×C₄⋊Dic₅₃	—	2	1	no	—
67	C₂×C₄⋊Dic₅₃	—	2	1	no	—
68	#13133149	—	2	2	no	—
69	#13133149	—	2	2	no	—
70	#13133149	—	2	2	no	—
71	#13133149	—	2	2	no	—
72	#13133149	—	2	4	no	—
73	C₄⋊#88322	—	2	1	no	—
74	C₄⋊#88322	C₂³	1	2	yes	—
75	#13133142	—	2	4	no	—
76	#13133139	—	2	4	no	—
77	#13133129	—	2	2	no	—
78	#13133129	—	2	2	no	—
79	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
80	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
81	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
82	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
83	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
84	C₂²⋊C₈×C₅₃	—	2	1	no	—
85	C₂²⋊C₈×C₅₃	—	2	1	no	—
86	C₄×C₈×C₅₃	—	2	1	no	—
87	C₄×C₈×C₅₃	—	2	1	no	—
88	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
89	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
90	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
95	C₂×C₂²⋊D₅₃	—	2	2	no	—
96	C₂²×Dic₅₃	—	4	1	no	—
97	C₂²×Dic₅₃	—	4	1	no	—
98	#11719945	—	4	2	no	—
99	#11719945	—	4	2	no	—
100	C₂×D₂₁₂	—	2	1	no	—
101	C₂×D₂₁₂	—	2	1	no	—
102	C₂×D₂₁₂	C₂×D₄	1	2	yes	—
103	C₂×C₄×D₅₃	—	2	2	no	—
104	C₂×Dic₁₀₆	—	2	1	no	—
105	C₂×Dic₁₀₆	—	2	2	no	—
106	C₂×Dic₁₀₆	—	2	2	no	—
107	C₂×Dic₁₀₆	—	4	1	no	—
108	C₂×Dic₁₀₆	—	4	1	no	—
109	C₂×Dic₁₀₆	—	8	2	no	—
110	C₂×Dic₁₀₆	—	8	2	no	—
111	C₂×C₈×C₅₃	—	2	2	no	—
112	C₂×C₈×C₅₃	—	2	2	no	—
113	C₂×C₈×C₅₃	—	2	2	no	—
114	C₂×C₈×C₅₃	SD₁₆	1	4	yes	complemented
115	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₅₃	#41	1	1	yes	characteristic
122	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
123	#11719921	—	2	2	no	—
124	#11719921	—	2	2	no	—
125	C₄⋊Dic₅₃	—	2	1	no	—
126	C₄⋊Dic₅₃	—	2	2	no	—
127	C₄⋊Dic₅₃	—	2	2	no	—
128	C₄⋊Dic₅₃	—	2	2	no	—
129	C₄⋊Dic₅₃	—	2	2	no	—
130	C₄⋊Dic₅₃	—	4	1	no	—
131	C₄⋊Dic₅₃	—	4	1	no	—
132	C₄⋊Dic₅₃	—	4	1	no	—
133	C₄⋊Dic₅₃	—	4	1	no	—
134	C₄⋊Dic₅₃	C₂×D₄	1	2	yes	—
135	#11719917	—	4	2	no	—
136	C₂×#88322	—	2	2	no	—
137	#11719913	—	8	4	no	—
138	#11719912	—	4	4	no	—
139	C₈×D₅₃	—	4	4	no	—
140	C₂³×C₅₃	#121	1	1	yes	characteristic
141	C₂²×D₅₃	—	2	2	no	—
142	C₂×C₄×C₅₃	—	2	2	no	—
143	C₂×C₄×C₅₃	—	2	2	no	—
144	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
145	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
146	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
148	C₂×C₄×C₅₃	#137	1	2	yes	—
149	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
152	C₂²⋊D₅₃	—	4	4	no	—
153	C₂×Dic₅₃	—	2	2	no	—
154	C₂×Dic₅₃	—	4	2	no	—
155	C₂×Dic₅₃	—	4	2	no	—
156	C₂×Dic₅₃	—	8	2	no	—
157	C₂×Dic₅₃	—	8	2	no	—
158	D₂₁₂	—	2	4	no	—
159	D₂₁₂	—	4	2	no	—
160	D₂₁₂	—	4	2	no	—
161	C₄×D₅₃	—	4	2	no	—
162	C₄×D₅₃	—	4	2	no	—
163	Dic₁₀₆	—	4	2	no	—
164	Dic₁₀₆	—	4	4	no	—
165	Dic₁₀₆	—	8	2	no	—
166	Dic₁₀₆	—	8	2	no	—
167	Dic₁₀₆	—	8	2	no	—
168	C₈×C₅₃	—	2	1	no	—
169	C₈×C₅₃	—	2	1	no	—
170	C₈×C₅₃	—	4	2	no	—
171	C₈×C₅₃	—	4	2	no	—
172	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
173	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
174	#88322	—	4	2	no	—
175	#19191	—	53	1	no	maximal, Sylow
176	C₂²×C₅₃	—	2	1	no	—
177	C₂²×C₅₃	—	2	1	no	—
178	C₂²×C₅₃	#461	1	2	yes	—
179	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
180	C₂×D₅₃	—	4	2	no	—
181	C₂×D₅₃	—	4	2	no	—
182	C₂×D₅₃	—	4	2	no	—
183	C₄×C₅₃	—	2	1	no	—
184	C₄×C₅₃	—	2	1	no	—
185	C₄×C₅₃	#461	1	2	yes	—
186	C₄×C₅₃	#471	1	2	yes	—
187	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
188	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
189	C₄×C₅₃	#577	1	2	yes	—
190	Dic₅₃	—	4	2	no	—
191	Dic₅₃	—	8	2	no	—
192	Dic₅₃	—	8	2	no	—
193	#3227	—	53	1	no	—
194	#3206	—	53	1	no	—
195	C₂×#498	—	53	1	no	—
196	#2931	—	53	1	no	—
197	#1691	—	53	4	no	—
198	#1671	—	53	4	no	—
199	#1564	D₅₃	1	1	yes	characteristic, complemented
200	#1549	—	53	1	no	—
201	#1548	—	53	1	no	—
202	C₂×C₅₃	—	2	2	no	—
203	C₂×C₅₃	C₂×#461	1	1	yes	characteristic
204	C₂×C₅₃	#3057	1	1	yes	characteristic
205	C₂×C₅₃	C₂×#471	1	1	yes	characteristic
206	D₅₃	—	8	2	no	—
207	C₂×#131	—	106	1	no	—
208	#532	—	53	1	no	—
209	C₂×C₄⋊Q₈	—	53	1	no	—
210	C₂×C₄⋊Q₈	—	53	1	no	—
211	#498	—	53	2	no	—
212	#498	—	53	2	no	—
213	#498	—	106	1	no	—
214	#498	—	106	1	no	—
215	#498	—	106	1	no	—
216	#492	—	106	1	no	—
217	#487	—	106	1	no	—
218	#486	—	106	1	no	—
219	#481	—	106	2	no	—
220	#476	—	53	4	no	—
221	#461	—	53	4	no	—
222	#448	—	106	2	no	—
223	#436	—	53	2	no	—
224	#436	—	106	1	no	—
225	C₈×D₄	—	53	2	no	—
226	C₈×D₄	—	106	1	no	—
227	#431	C₂×D₅₃	1	1	yes	characteristic
228	#431	C₂×D₅₃	1	1	yes	characteristic
229	C₂×#106	—	106	1	no	—
230	C₂×#106	—	106	1	no	—
231	#423	—	53	1	no	—
232	#408	—	106	1	no	—
233	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
234	#335	C₂×D₅₃	1	2	yes	—
235	#334	C₂×D₅₃	1	2	yes	—
236	#333	—	53	4	no	—
237	#332	—	53	4	no	—
238	C₅₃	#19191	1	1	yes	characteristic, complemented, Sylow
239	C₂×#41	—	106	1	no	—
240	C₂²×Q₈	—	106	1	no	—
241	C₂²×Q₈	—	106	1	no	—
242	C₂×SD₁₆	—	106	4	no	—
243	#131	—	212	4	no	—
244	C₂×M₁₆	—	106	1	no	—
245	C₂×M₁₆	—	106	2	no	—
246	C₂²×C₈	—	106	2	no	—
247	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
248	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
249	C₄⋊Q₈	—	53	1	no	—
250	C₄⋊Q₈	—	53	2	no	—
251	C₄⋊Q₈	—	53	2	no	—
252	C₄⋊Q₈	—	106	1	no	—
253	C₄⋊Q₈	—	106	1	no	—
254	C₄⋊Q₈	—	106	1	no	—
255	C₄⋊Q₈	—	106	1	no	—
256	C₄⋊Q₈	—	106	1	no	—
257	C₄⋊Q₈	—	106	1	no	—
258	#127	—	53	1	no	—
259	#124	—	106	1	no	—
260	#121	—	53	2	no	—
261	#121	—	106	1	no	—
262	C₄×D₄	—	53	2	no	—
263	C₄×D₄	—	106	1	no	—
264	C₂×C₄⋊C₄	—	106	1	no	—
265	C₂×C₄⋊C₄	—	106	1	no	—
266	C₂×C₄⋊C₄	—	106	1	no	—
267	C₂×C₄⋊C₄	—	106	1	no	—
268	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
269	#106	—	106	4	no	—
270	#106	—	106	2	no	—
271	#106	—	106	2	no	—
272	#106	—	106	2	no	—
273	#106	—	106	2	no	—
274	C₄⋊C₈	—	53	2	no	—
275	C₄⋊C₈	—	106	1	no	—
276	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
277	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
278	#103	—	106	4	no	—
279	#102	—	106	4	no	—
280	C₂²⋊C₈	—	2	1	no	—
281	C₂²⋊C₈	—	2	1	no	—
282	C₂²⋊C₈	—	106	2	no	—
283	C₂²⋊C₈	—	106	2	no	—
284	C₄×C₈	—	2	1	no	—
285	C₄×C₈	—	2	1	no	—
286	C₄×C₈	D₂₁₂	1	1	yes	characteristic
287	C₄×C₈	D₂₁₂	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
290	#41	—	212	2	no	—
291	#41	—	212	2	no	—
292	C₂×Q₈	—	106	1	no	—
293	C₂×Q₈	—	106	2	no	—
294	C₂×Q₈	—	106	2	no	—
295	C₂×Q₈	—	424	2	no	—
296	C₂×Q₈	—	424	2	no	—
297	C₂×Q₈	—	212	1	no	—
298	C₂×Q₈	—	212	1	no	—
299	C₂×D₄	—	53	2	no	—
300	C₂×D₄	—	106	2	no	—
301	C₂×D₄	—	106	1	no	—
302	C₂×D₄	—	106	1	no	—
303	C₂²×C₄	—	106	2	no	—
304	C₂²×C₄	—	212	1	no	—
305	C₂²×C₄	—	212	1	no	—
306	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
307	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
308	C₂²×C₄	#11719949	1	1	yes	characteristic
309	SD₁₆	—	424	4	no	—
310	M₁₆	—	212	2	no	—
311	M₁₆	—	212	4	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	106	2	no	—
316	C₂×C₈	—	212	4	no	—
317	C₂×C₈	#11719913	1	4	yes	complemented
318	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
319	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
320	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
321	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
322	C₄⋊C₄	—	53	2	no	—
323	C₄⋊C₄	—	106	1	no	—
324	C₄⋊C₄	—	106	2	no	—
325	C₄⋊C₄	—	106	2	no	—
326	C₄⋊C₄	—	106	2	no	—
327	C₄⋊C₄	—	106	2	no	—
328	C₄⋊C₄	—	212	1	no	—
329	C₄⋊C₄	—	212	1	no	—
330	C₄⋊C₄	—	212	1	no	—
331	C₄⋊C₄	—	212	1	no	—
332	C₂²⋊C₄	—	106	2	no	—
333	C₂²⋊C₄	—	106	2	no	—
334	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
335	C₄²	D₄×D₅₃	1	1	yes	characteristic
336	C₄²	#11719949	1	1	yes	characteristic
337	C₄²	C₂³×D₅₃	1	1	yes	characteristic
338	C₂³	—	106	2	no	—
339	C₂³	#13133192	1	1	yes	characteristic
340	Q₈	—	424	2	no	—
341	Q₈	—	424	2	no	—
342	Q₈	—	424	2	no	—
343	Q₈	—	212	4	no	—
344	Q₈	—	212	2	no	—
345	D₄	—	106	4	no	—
346	D₄	—	212	4	no	—
347	D₄	—	212	2	no	—
348	D₄	—	212	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	106	2	no	—
352	C₂×C₄	—	424	2	no	—
353	C₂×C₄	—	424	2	no	—
354	C₂×C₄	—	212	2	no	—
355	C₂×C₄	—	212	2	no	—
356	C₂×C₄	—	212	2	no	—
357	C₂×C₄	—	212	2	no	—
358	C₂×C₄	#13133192	1	1	yes	characteristic
359	C₂×C₄	#13133192	1	1	yes	characteristic
360	C₂×C₄	#13133192	1	1	yes	characteristic
361	C₂×C₄	C₂×#11719913	1	2	yes	—
362	C₂×C₄	#13133250	1	2	yes	—
363	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
364	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
365	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
366	C₈	—	2	1	no	—
367	C₈	—	2	1	no	—
368	C₈	—	4	2	no	—
369	C₈	—	4	2	no	—
370	C₈	—	212	2	no	—
371	C₈	C₂×#11719913	1	2	yes	—
372	C₈	C₂×#11719913	1	2	yes	—
373	C₂²	—	2	1	no	—
374	C₂²	—	2	1	no	—
375	C₂²	—	212	2	no	—
376	C₂²	—	212	2	no	—
377	C₂²	—	212	2	no	—
378	C₂²	#16810829	1	2	yes	—
379	C₂²	C₂×#13133192	1	1	yes	characteristic
380	C₄	—	2	1	no	—
381	C₄	—	2	1	no	—
382	C₄	—	424	2	no	—
383	C₄	—	424	2	no	—
384	C₄	—	212	2	no	—
385	C₄	#16810829	1	2	yes	—
386	C₄	#16810887	1	2	yes	—
387	C₄	C₂²×#11719913	1	1	yes	characteristic
388	C₄	C₂×#13133250	1	1	yes	characteristic
389	C₄	#16811430	1	2	yes	—
390	C₂	—	2	2	no	—
391	C₂	—	424	2	no	—
392	C₂	C₂×#16810829	1	1	yes	characteristic
393	C₂	#20284355	1	1	yes	characteristic
394	C₂	C₂×#16810887	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133192
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1564	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	#16811173	C₂²
hypercenter	#1564	D₅₃
hypocenter	C₅₃	#19191
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
3. C₂×#16811021	C₂	semidirect
5. #20274215	C₂	semidirect
6. #20274091	C₂	semidirect
7. #20273399	C₂	semidirect
8. #20273395	C₂	semidirect
9. C₅₃×#1564	C₂	semidirect
114. C₂×C₈×C₅₃	SD₁₆	semidirect
199. #1564	D₅₃	semidirect
238. C₅₃	#19191	semidirect
317. C₂×C₈	#11719913	semidirect

Maximal subgroups

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

subgroup	index
#20285845	2
#20285642	2
C₂×#16811021	2
#20283428	2
#20274215	2
#20274091	2
#20273399	2
#20273395	2
C₅₃×#1564	2
#19191	53