Subgroups — #27353561

Order 13568.

678 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283794	C₂	1	2	yes	maximal
2	C₄×Q₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#20273532	C₂	1	2	yes	maximal, complemented
4	#20273531	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₁₆×#88322	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273521	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273462	C₂	1	1	yes	maximal, characteristic
9	#20273108	C₂	1	2	yes	maximal
10	#20272997	C₂	1	1	yes	maximal, characteristic
11	C₈×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₅₃	C₄	1	1	yes	characteristic
13	C₄×Q₈×D₅₃	C₂²	1	2	yes	—
14	Q₈×#88322	C₂²	1	2	yes	—
15	C₈×Dic₁₀₆	—	2	2	no	—
16	C₈×Dic₁₀₆	C₂²	1	2	yes	—
17	#16810747	—	2	2	no	—
18	#16810747	C₂²	1	2	yes	—
19	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
20	Q₁₆×Dic₅₃	C₄	1	1	yes	characteristic
21	#16810717	C₄	1	1	yes	characteristic
22	C₄×#11719925	C₂²	1	2	yes	—
23	#16810715	C₄	1	2	yes	—
24	#16810675	—	2	2	no	—
25	#16810675	C₂²	1	2	yes	—
26	#16810674	—	2	2	no	—
27	#16810674	C₂²	1	1	yes	characteristic
28	#16810674	C₂²	1	1	yes	characteristic
29	#107×D₅₃	C₄	1	1	yes	characteristic
30	#103×D₅₃	C₄	1	2	yes	—
31	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
32	#16810376	C₂²	1	1	yes	characteristic
33	#16810373	C₂²	1	2	yes	—
34	#16810355	C₂²	1	2	yes	—
35	Q₈⋊#88322	C₂²	1	2	yes	—
36	#16810353	C₂²	1	2	yes	—
37	#16810345	C₂²	1	1	yes	characteristic
38	#16810345	C₂²	1	1	yes	characteristic
39	C₈×#88322	C₂²	1	1	yes	characteristic
40	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
42	C₅₃×#335	C₂²	1	1	yes	characteristic
43	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₈²×C₅₃	C₂²	1	1	yes	characteristic
45	C₂×Q₈×D₅₃	C₂×C₄	1	2	yes	—
46	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
47	C₂×#11719925	C₂×C₄	1	2	yes	—
48	Q₁₆×D₅₃	C₈	1	8	yes	complemented
49	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
51	Q₈×Dic₅₃	C₂×C₄	1	2	yes	—
52	C₄×Dic₁₀₆	—	2	2	no	—
53	C₄×Dic₁₀₆	C₂³	1	2	yes	—
54	#13133182	—	2	2	no	—
55	#13133182	C₂×C₄	1	2	yes	—
56	C₄⋊C₄×D₅₃	—	2	2	no	—
57	C₄⋊C₄×D₅₃	C₂×C₄	1	2	yes	—
58	C₄²×D₅₃	—	2	2	no	—
59	C₄²×D₅₃	C₂³	1	1	yes	characteristic
60	#13133151	C₂×C₄	1	1	yes	characteristic
61	#13133150	C₂×C₄	1	2	yes	—
62	C₄⋊#88322	—	2	2	no	—
63	C₄⋊#88322	C₂³	1	2	yes	—
64	#13133146	—	2	2	no	—
65	#13133146	—	2	1	no	—
66	#13133146	—	2	2	no	—
67	#13133146	D₄	1	1	yes	characteristic
68	#13133146	D₄	1	1	yes	characteristic
69	#13133142	C₂×C₄	1	2	yes	—
70	#13133141	C₂×C₄	1	2	yes	—
71	#13133140	C₂×C₄	1	2	yes	—
72	#13133129	—	2	2	no	—
73	#13133129	—	2	1	no	—
74	#13133129	D₄	1	1	yes	characteristic
75	#13133129	D₄	1	1	yes	characteristic
76	C₄×#88322	—	2	2	no	—
77	C₄×#88322	C₂³	1	1	yes	characteristic
78	C₄×#88322	C₂³	1	1	yes	characteristic
79	C₄×#88322	C₂³	1	1	yes	characteristic
80	C₈×Dic₅₃	—	2	2	no	—
81	C₈×Dic₅₃	—	2	2	no	—
82	C₈×Dic₅₃	—	2	2	no	—
83	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
84	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
85	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
86	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
87	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
88	C₄⋊C₈×C₅₃	—	2	2	no	—
89	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
90	#103×C₅₃	C₂×C₄	1	2	yes	—
91	C₄×C₈×C₅₃	—	2	2	no	—
92	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
96	Q₈×D₅₃	—	2	4	no	—
97	Q₈×D₅₃	C₂×C₈	1	8	yes	—
98	C₂×C₄×D₅₃	—	2	2	no	—
99	C₂×C₄×D₅₃	—	2	2	no	—
100	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
103	C₂×Dic₁₀₆	—	2	2	no	—
104	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
105	Q₁₆×C₅₃	C₂×C₈	1	4	yes	—
106	C₂×C₈×C₅₃	—	2	2	no	—
107	C₂×C₈×C₅₃	—	2	1	no	—
108	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₄⋊C₄×C₅₃	—	2	2	no	—
113	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
114	C₄²×C₅₃	—	2	2	no	—
115	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
116	#11719925	C₂×C₈	1	8	yes	—
117	C₄⋊Dic₅₃	—	2	2	no	—
118	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
119	#11719919	—	2	2	no	—
120	#11719919	—	2	2	no	—
121	#11719919	—	2	2	no	—
122	C₄×Dic₅₃	—	2	2	no	—
123	C₄×Dic₅₃	—	2	2	no	—
124	C₄×Dic₅₃	—	2	2	no	—
125	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
126	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
127	C₄×Dic₅₃	#41	1	1	yes	characteristic
128	C₂×#88322	—	2	2	no	—
129	C₂×#88322	—	2	1	no	—
130	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
131	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
132	C₂×#88322	C₂×D₄	1	1	yes	characteristic
133	C₂×#88322	C₂×D₄	1	1	yes	characteristic
134	Dic₂₁₂	C₂×C₈	1	4	yes	—
135	C₈×D₅₃	—	2	2	no	—
136	C₈×D₅₃	C₂×C₈	1	4	yes	—
137	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
138	Q₈×C₅₃	—	2	2	no	—
139	Q₈×C₅₃	C₂²×C₈	1	4	yes	—
140	C₂×C₄×C₅₃	—	2	2	no	—
141	C₂×C₄×C₅₃	—	2	2	no	—
142	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
145	C₂×Dic₅₃	—	2	2	no	—
146	C₂×Dic₅₃	—	2	2	no	—
147	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
148	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
149	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
150	C₄×D₅₃	—	2	8	no	—
151	C₄×D₅₃	—	2	1	no	—
152	C₄×D₅₃	—	2	1	no	—
153	C₄×D₅₃	—	4	1	no	—
154	C₄×D₅₃	—	8	2	no	—
155	C₄×D₅₃	C₂²×C₈	1	2	yes	—
156	C₄×D₅₃	#131	1	2	yes	—
157	Dic₁₀₆	—	2	2	no	—
158	Dic₁₀₆	—	2	8	no	—
159	Dic₁₀₆	C₂²×C₈	1	4	yes	—
160	C₈×C₅₃	—	2	1	no	—
161	C₈×C₅₃	—	2	1	no	—
162	C₈×C₅₃	—	2	1	no	—
163	C₈×C₅₃	—	4	1	no	—
164	C₈×C₅₃	—	4	2	no	—
165	C₈×C₅₃	C₂²×C₈	1	2	yes	—
166	#88322	—	2	1	no	—
167	#88322	—	2	1	no	—
168	#88322	—	2	1	no	—
169	#88322	—	4	1	no	—
170	#88322	—	4	2	no	—
171	#88322	C₂²×C₈	1	2	yes	—
172	#17856	—	53	1	no	maximal, Sylow
173	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
174	C₂×D₅₃	—	2	1	no	—
175	C₂×D₅₃	—	2	1	no	—
176	C₂×D₅₃	C₈×D₄	1	2	yes	—
177	C₄×C₅₃	—	2	4	no	—
178	C₄×C₅₃	—	2	1	no	—
179	C₄×C₅₃	—	4	2	no	—
180	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
181	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
182	C₄×C₅₃	#578	1	1	yes	characteristic
183	C₄×C₅₃	#578	1	1	yes	characteristic
184	Dic₅₃	—	2	4	no	—
185	Dic₅₃	—	2	1	no	—
186	Dic₅₃	—	2	1	no	—
187	Dic₅₃	—	4	2	no	—
188	Dic₅₃	C₈×D₄	1	2	yes	—
189	#2982	—	53	2	no	—
190	C₂×C₄×Q₁₆	—	53	1	no	—
191	#1579	—	53	2	no	—
192	#1579	—	53	2	no	—
193	C₈×Q₁₆	—	53	1	no	—
194	C₈×Q₁₆	—	53	1	no	—
195	C₈×Q₁₆	—	53	1	no	—
196	C₈×Q₁₆	D₅₃	1	1	yes	characteristic, complemented
197	#1564	—	53	1	no	—
198	#1479	—	53	2	no	—
199	#1450	—	53	1	no	—
200	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
201	C₂×C₅₃	#2950	1	1	yes	characteristic
202	C₂×C₅₃	#2957	1	1	yes	characteristic
203	D₅₃	—	2	2	no	—
204	C₂²×Q₁₆	—	53	1	no	—
205	C₂×C₄×Q₈	—	53	2	no	—
206	C₈×Q₈	—	53	2	no	—
207	C₈×Q₈	—	53	2	no	—
208	C₈×Q₈	—	53	2	no	—
209	C₈×Q₈	—	106	2	no	—
210	C₈×Q₈	—	106	2	no	—
211	C₈×Q₈	C₂×D₅₃	1	2	yes	—
212	C₄×Q₁₆	—	53	1	no	—
213	C₄×Q₁₆	—	53	2	no	—
214	C₄×Q₁₆	—	53	1	no	—
215	C₄×Q₁₆	—	53	2	no	—
216	C₄×Q₁₆	—	53	1	no	—
217	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
218	#431	—	53	1	no	—
219	#431	—	53	1	no	—
220	#431	—	53	2	no	—
221	#431	—	106	2	no	—
222	#431	—	106	2	no	—
223	C₂×#107	—	53	1	no	—
224	C₂×#103	—	53	2	no	—
225	C₂×C₄×C₈	—	53	1	no	—
226	#335	—	53	1	no	—
227	#335	C₂×D₅₃	1	1	yes	characteristic
228	#334	—	53	2	no	—
229	Q₈⋊C₈	—	53	2	no	—
230	Q₈⋊C₈	—	53	2	no	—
231	Q₈⋊C₈	—	53	2	no	—
232	Q₈⋊C₈	C₂×D₅₃	1	2	yes	—
233	#322	—	53	1	no	—
234	#322	—	53	1	no	—
235	C₈²	—	53	1	no	—
236	C₈²	C₂×D₅₃	1	1	yes	characteristic
237	C₅₃	#17856	1	1	yes	characteristic, complemented, Sylow
238	C₂²×Q₈	—	53	2	no	—
239	C₂×Q₁₆	—	53	2	no	—
240	C₂×Q₁₆	—	53	8	no	—
241	C₂×Q₁₆	—	53	1	no	—
242	C₂×Q₁₆	C₄×D₅₃	1	1	yes	characteristic
243	C₂²×C₈	—	53	1	no	—
244	C₂²×C₈	—	53	1	no	—
245	C₄×Q₈	—	53	2	no	—
246	C₄×Q₈	—	53	2	no	—
247	C₄×Q₈	—	53	2	no	—
248	C₄×Q₈	—	106	2	no	—
249	C₄×Q₈	—	106	2	no	—
250	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
251	C₂×C₄⋊C₄	—	53	2	no	—
252	C₂×C₄⋊C₄	—	106	2	no	—
253	C₂×C₄²	—	53	1	no	—
254	C₂×C₄²	—	106	2	no	—
255	#107	—	53	1	no	—
256	#107	—	53	2	no	—
257	#107	C₄×D₅₃	1	1	yes	characteristic
258	C₄⋊C₈	—	2	2	no	—
259	C₄⋊C₈	—	53	1	no	—
260	C₄⋊C₈	—	53	1	no	—
261	C₄⋊C₈	—	53	2	no	—
262	C₄⋊C₈	—	106	2	no	—
263	C₄⋊C₈	—	106	1	no	—
264	C₄⋊C₈	—	106	2	no	—
265	C₄⋊C₈	—	106	2	no	—
266	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
267	#103	—	53	2	no	—
268	#103	—	53	2	no	—
269	#103	—	53	2	no	—
270	#103	C₄×D₅₃	1	2	yes	—
271	C₂²⋊C₈	—	53	1	no	—
272	C₂²⋊C₈	—	53	1	no	—
273	C₂²⋊C₈	—	106	1	no	—
274	C₂²⋊C₈	—	106	2	no	—
275	C₄×C₈	—	2	2	no	—
276	C₄×C₈	—	53	1	no	—
277	C₄×C₈	—	53	1	no	—
278	C₄×C₈	—	53	1	no	—
279	C₄×C₈	—	53	1	no	—
280	C₄×C₈	—	53	1	no	—
281	C₄×C₈	—	106	2	no	—
282	C₄×C₈	—	106	2	no	—
283	C₄×C₈	—	106	2	no	—
284	C₄×C₈	—	106	2	no	—
285	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
286	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
287	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
288	C₂×Q₈	—	53	8	no	—
289	C₂×Q₈	—	53	2	no	—
290	C₂×Q₈	—	106	2	no	—
291	C₂×Q₈	—	106	4	no	—
292	C₂×Q₈	C₂×C₄×D₅₃	1	2	yes	—
293	C₂²×C₄	—	53	1	no	—
294	C₂²×C₄	—	53	1	no	—
295	C₂²×C₄	—	53	1	no	—
296	C₂²×C₄	—	106	2	no	—
297	C₂²×C₄	—	106	2	no	—
298	Q₁₆	—	53	8	no	—
299	Q₁₆	—	53	4	no	—
300	Q₁₆	C₈×D₅₃	1	4	yes	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	1	no	—
303	C₂×C₈	—	53	1	no	—
304	C₂×C₈	—	53	4	no	—
305	C₂×C₈	—	53	1	no	—
306	C₂×C₈	—	53	1	no	—
307	C₂×C₈	—	53	1	no	—
308	C₂×C₈	—	106	2	no	—
309	C₂×C₈	—	106	2	no	—
310	C₂×C₈	—	106	1	no	—
311	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
312	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
313	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
314	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	53	2	no	—
317	C₄⋊C₄	—	106	2	no	—
318	C₄⋊C₄	—	106	2	no	—
319	C₄⋊C₄	—	106	2	no	—
320	C₄⋊C₄	—	106	2	no	—
321	C₄⋊C₄	C₂×C₄×D₅₃	1	2	yes	—
322	C₄²	—	2	2	no	—
323	C₄²	—	53	1	no	—
324	C₄²	—	53	1	no	—
325	C₄²	—	53	1	no	—
326	C₄²	—	106	2	no	—
327	C₄²	—	106	2	no	—
328	C₄²	—	106	2	no	—
329	C₄²	C₂³×D₅₃	1	1	yes	characteristic
330	C₂³	—	53	1	no	—
331	Q₈	—	2	2	no	—
332	Q₈	—	53	4	no	—
333	Q₈	—	106	8	no	—
334	Q₈	—	106	2	no	—
335	Q₈	C₂×C₈×D₅₃	1	4	yes	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	53	1	no	—
339	C₂×C₄	—	53	2	no	—
340	C₂×C₄	—	53	2	no	—
341	C₂×C₄	—	53	1	no	—
342	C₂×C₄	—	53	1	no	—
343	C₂×C₄	—	106	8	no	—
344	C₂×C₄	—	106	2	no	—
345	C₂×C₄	—	106	1	no	—
346	C₂×C₄	—	106	1	no	—
347	C₂×C₄	—	106	2	no	—
348	C₂×C₄	—	424	2	no	—
349	C₂×C₄	—	212	1	no	—
350	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
351	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
352	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
353	C₈	—	2	1	no	—
354	C₈	—	2	1	no	—
355	C₈	—	2	1	no	—
356	C₈	—	4	1	no	—
357	C₈	—	4	2	no	—
358	C₈	—	53	2	no	—
359	C₈	—	106	1	no	—
360	C₈	—	106	1	no	—
361	C₈	—	106	1	no	—
362	C₈	—	212	1	no	—
363	C₈	—	212	2	no	—
364	C₈	C₂×C₈×D₅₃	1	2	yes	—
365	C₂²	—	53	2	no	—
366	C₂²	—	106	1	no	—
367	C₂²	—	106	1	no	—
368	C₂²	C₄×D₄×D₅₃	1	1	yes	characteristic
369	C₄	—	2	1	no	—
370	C₄	—	2	4	no	—
371	C₄	—	4	2	no	—
372	C₄	—	53	2	no	—
373	C₄	—	106	1	no	—
374	C₄	—	106	1	no	—
375	C₄	—	106	4	no	—
376	C₄	—	212	2	no	—
377	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
378	C₄	#131×D₅₃	1	1	yes	characteristic
379	C₄	#16811436	1	1	yes	characteristic
380	C₄	#16811437	1	1	yes	characteristic
381	C₂	—	106	2	no	—
382	C₂	C₈×D₄×D₅₃	1	1	yes	characteristic
383	C₂	#20283578	1	1	yes	characteristic
384	C₂	#20283627	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₄²	C₂³×D₅₃
Fitting	C₈×Q₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂²×D₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₈×Q₁₆	D₅₃
hypocenter	C₅₃	#17856
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #20273532	C₂	semidirect
4. #20273531	C₂	semidirect
5. C₈×Dic₂₁₂	C₂	semidirect
6. Q₁₆×#88322	C₂	semidirect
7. #20273521	C₂	semidirect
11. C₈×Q₁₆×C₅₃	C₂	semidirect
48. Q₁₆×D₅₃	C₈	semidirect
196. C₈×Q₁₆	D₅₃	semidirect
237. C₅₃	#17856	semidirect

Maximal subgroups

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

subgroup	index
#20283794	2
C₄×Q₁₆×D₅₃	2
#20273532	2
#20273531	2
C₈×Dic₂₁₂	2
Q₁₆×#88322	2
#20273521	2
#20273462	2
#20273108	2
#20272997	2
C₈×Q₁₆×C₅₃	2
#17856	53