Subgroups — #27300563

Order 13568.

420 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810444	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16810443	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276325	C₂	1	1	yes	maximal, characteristic
5	C₈×#11719921	C₂	1	1	yes	maximal, characteristic
6	#20272796	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
9	C₂×#13133157	C₂²	1	1	yes	characteristic
10	C₂×#13133157	C₂²	1	1	yes	characteristic
11	C₂×#13133156	—	2	1	no	—
12	M₃₂×D₅₃	—	2	2	no	—
13	M₃₂×D₅₃	—	2	2	no	—
14	C₂×C₁₆×D₅₃	—	2	1	no	—
15	C₂×#13133129	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
17	C₄×#11719921	C₄	1	1	yes	characteristic
18	#16810444	—	2	2	no	—
19	#16810443	—	2	2	no	—
20	#16810402	—	2	1	no	—
21	#16810402	—	2	1	no	—
22	#16810378	C₄	1	1	yes	characteristic
23	#16810377	C₄	1	1	yes	characteristic
24	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
29	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×D₅₃	—	2	1	no	—
31	C₂×C₈×D₅₃	—	2	1	no	—
32	C₂×C₈×D₅₃	—	2	1	no	—
33	C₂×C₈×D₅₃	—	2	1	no	—
34	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
35	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
36	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
37	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
38	C₂×#11719921	—	2	1	no	—
39	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
40	#13133157	—	2	1	no	—
41	#13133157	—	2	1	no	—
42	#13133157	—	2	1	no	—
43	#13133157	D₄	1	2	yes	—
44	#13133156	—	2	4	no	—
45	C₂×#11719908	—	2	1	no	—
46	C₂×#11719908	—	2	1	no	—
47	C₁₆×D₅₃	—	2	4	no	—
48	#13133129	C₈	1	2	yes	—
49	#13133129	C₈	1	2	yes	—
50	C₈×Dic₅₃	—	2	1	no	—
51	C₈×Dic₅₃	—	2	1	no	—
52	#13133124	—	2	1	no	—
53	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
56	M₃₂×C₅₃	—	2	1	no	—
57	M₃₂×C₅₃	—	2	1	no	—
58	M₃₂×C₅₃	—	2	1	no	—
59	M₃₂×C₅₃	D₄	1	2	yes	—
60	C₂×C₁₆×C₅₃	—	2	1	no	—
61	C₂×C₁₆×C₅₃	—	2	1	no	—
62	C₄×C₈×C₅₃	—	2	1	no	—
63	C₄×C₈×C₅₃	—	2	1	no	—
64	C₂³×D₅₃	M₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	—	2	1	no	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×Dic₅₃	—	2	1	no	—
68	C₂²×Dic₅₃	M₁₆	1	1	yes	characteristic
69	C₂×C₄×D₅₃	—	2	1	no	—
70	C₂×C₄×D₅₃	—	2	1	no	—
71	C₂×C₄×D₅₃	—	2	1	no	—
72	C₂×C₄×D₅₃	—	2	1	no	—
73	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
76	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
77	C₂×C₈×C₅₃	—	2	1	no	—
78	C₂×C₈×C₅₃	—	2	1	no	—
79	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
80	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
84	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
85	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
86	C₄²×C₅₃	—	2	1	no	—
87	C₄²×C₅₃	—	2	1	no	—
88	#11719921	—	2	4	no	—
89	C₄×Dic₅₃	—	2	1	no	—
90	C₄×Dic₅₃	—	2	1	no	—
91	C₂×#88322	—	2	1	no	—
92	C₂×#88322	—	2	1	no	—
93	C₂×#88322	—	2	1	no	—
94	C₂×#88322	—	2	1	no	—
95	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
96	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
97	C₂×#88322	C₂×C₈	1	2	yes	—
98	C₈×D₅₃	—	2	1	no	—
99	C₈×D₅₃	—	2	1	no	—
100	C₈×D₅₃	—	2	1	no	—
101	C₈×D₅₃	—	2	1	no	—
102	C₈×D₅₃	—	2	1	no	—
103	C₈×D₅₃	—	2	1	no	—
104	C₈×D₅₃	—	2	1	no	—
105	C₈×D₅₃	—	2	1	no	—
106	C₁₆×C₅₃	—	2	2	no	—
107	C₁₆×C₅₃	—	4	1	no	—
108	#11719908	—	2	2	no	—
109	#11719908	—	4	1	no	—
110	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
111	C₂²×D₅₃	—	2	1	no	—
112	C₂²×D₅₃	—	2	1	no	—
113	C₂²×D₅₃	—	2	2	no	—
114	C₂²×D₅₃	—	4	1	no	—
115	C₂²×D₅₃	C₄⋊C₈	1	1	yes	characteristic
116	C₂²×D₅₃	C₄⋊C₈	1	1	yes	characteristic
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	4	1	no	—
119	C₂×C₄×C₅₃	—	4	1	no	—
120	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
125	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
126	C₂×Dic₅₃	—	2	1	no	—
127	C₂×Dic₅₃	—	2	1	no	—
128	C₂×Dic₅₃	—	2	2	no	—
129	C₂×Dic₅₃	—	4	1	no	—
130	C₂×Dic₅₃	—	4	1	no	—
131	C₂×Dic₅₃	C₄⋊C₈	1	1	yes	characteristic
132	C₂×Dic₅₃	C₄⋊C₈	1	1	yes	characteristic
133	C₄×D₅₃	—	2	1	no	—
134	C₄×D₅₃	—	2	1	no	—
135	C₄×D₅₃	—	2	1	no	—
136	C₄×D₅₃	—	2	1	no	—
137	C₄×D₅₃	—	2	1	no	—
138	C₄×D₅₃	—	2	1	no	—
139	C₄×D₅₃	—	2	1	no	—
140	C₄×D₅₃	—	2	1	no	—
141	C₈×C₅₃	—	2	2	no	—
142	C₈×C₅₃	#120	1	1	yes	characteristic
143	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
144	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
145	C₈×C₅₃	#123	1	1	yes	characteristic
146	#88322	—	2	1	no	—
147	#88322	—	2	1	no	—
148	#88322	—	2	2	no	—
149	#12914	—	53	1	no	maximal, Sylow
150	C₂²×C₅₃	—	2	1	no	—
151	C₂²×C₅₃	—	2	1	no	—
152	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
153	C₂²×C₅₃	#435	1	1	yes	characteristic
154	C₂²×C₅₃	#435	1	1	yes	characteristic
155	C₂×D₅₃	—	2	1	no	—
156	C₂×D₅₃	—	2	1	no	—
157	C₂×D₅₃	—	2	1	no	—
158	C₂×D₅₃	—	2	1	no	—
159	C₂×D₅₃	—	2	1	no	—
160	C₂×D₅₃	—	4	2	no	—
161	C₂×D₅₃	—	4	2	no	—
162	C₂×D₅₃	#364	1	2	yes	complemented
163	C₄×C₅₃	—	4	2	no	—
164	C₄×C₅₃	#385	1	1	yes	characteristic
165	C₄×C₅₃	#424	1	1	yes	characteristic
166	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
167	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
168	Dic₅₃	—	2	1	no	—
169	Dic₅₃	—	2	1	no	—
170	Dic₅₃	—	4	2	no	—
171	C₂²×M₃₂	—	53	1	no	—
172	C₂×#364	—	53	1	no	—
173	C₂×#364	—	53	1	no	—
174	#2112	—	53	1	no	—
175	C₈×C₂²⋊C₄	—	53	1	no	—
176	#1376	—	53	1	no	—
177	#1376	D₅₃	1	1	yes	characteristic, complemented
178	C₂×C₅₃	—	2	1	no	—
179	C₂×C₅₃	—	2	1	no	—
180	C₂×C₅₃	#1850	1	1	yes	characteristic
181	C₂×C₅₃	C₂×#364	1	1	yes	characteristic
182	C₂×C₅₃	#2153	1	1	yes	characteristic
183	D₅₃	—	2	2	no	—
184	D₅₃	—	4	1	no	—
185	C₂³×C₈	—	53	1	no	—
186	C₂×M₃₂	—	53	1	no	—
187	C₂×M₃₂	—	53	1	no	—
188	C₂×M₃₂	—	106	1	no	—
189	C₂×M₃₂	—	106	2	no	—
190	C₂×M₃₂	—	106	2	no	—
191	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
192	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
193	C₂²×C₁₆	—	106	1	no	—
194	C₂×C₂²⋊C₈	—	53	1	no	—
195	C₂×C₄×C₈	—	53	1	no	—
196	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
197	C₄×C₂²⋊C₄	—	53	1	no	—
198	#364	—	106	2	no	—
199	#364	—	106	2	no	—
200	C₂²⋊C₁₆	—	106	1	no	—
201	C₂²⋊C₁₆	—	106	1	no	—
202	#336	—	53	1	no	—
203	#336	—	53	1	no	—
204	C₅₃	#12914	1	1	yes	characteristic, complemented, Sylow
205	C₂³×C₄	—	53	1	no	—
206	C₂²×C₈	—	53	1	no	—
207	C₂²×C₈	—	53	1	no	—
208	C₂²×C₈	—	53	1	no	—
209	C₂²×C₈	—	53	1	no	—
210	C₂²×C₈	—	53	1	no	—
211	C₂²×C₈	—	53	1	no	—
212	C₂²×C₈	—	106	1	no	—
213	C₂²×C₈	—	106	1	no	—
214	C₂²×C₈	—	106	1	no	—
215	C₂²×C₈	—	106	1	no	—
216	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
217	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
218	C₂×C₂²⋊C₄	—	106	1	no	—
219	C₂×C₄²	—	53	1	no	—
220	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
221	M₃₂	—	2	1	no	—
222	M₃₂	—	2	1	no	—
223	M₃₂	—	2	1	no	—
224	M₃₂	—	53	2	no	—
225	M₃₂	—	106	4	no	—
226	M₃₂	—	106	1	no	—
227	M₃₂	—	106	1	no	—
228	M₃₂	—	106	1	no	—
229	M₃₂	D₂₁₂	1	2	yes	—
230	C₂×C₁₆	—	2	1	no	—
231	C₂×C₁₆	—	2	1	no	—
232	C₂×C₁₆	—	106	4	no	—
233	C₂×C₁₆	—	106	1	no	—
234	C₂×C₁₆	—	106	1	no	—
235	C₂²⋊C₈	—	53	2	no	—
236	C₂²⋊C₈	—	53	2	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	—	2	1	no	—
239	C₄×C₈	—	106	1	no	—
240	C₄×C₈	—	106	1	no	—
241	#95	—	106	1	no	—
242	C₂⁴	—	53	1	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	53	1	no	—
245	C₂²×C₄	—	53	1	no	—
246	C₂²×C₄	—	53	1	no	—
247	C₂²×C₄	—	53	1	no	—
248	C₂²×C₄	—	53	1	no	—
249	C₂²×C₄	—	106	1	no	—
250	C₂²×C₄	—	106	1	no	—
251	C₂²×C₄	—	106	1	no	—
252	C₂²×C₄	—	106	1	no	—
253	C₂²×C₄	—	106	1	no	—
254	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	53	1	no	—
258	C₂×C₈	—	53	1	no	—
259	C₂×C₈	—	53	2	no	—
260	C₂×C₈	—	106	1	no	—
261	C₂×C₈	—	106	1	no	—
262	C₂×C₈	—	106	1	no	—
263	C₂×C₈	—	106	1	no	—
264	C₂×C₈	—	106	1	no	—
265	C₂×C₈	—	106	1	no	—
266	C₂×C₈	—	106	1	no	—
267	C₂×C₈	—	106	1	no	—
268	C₂×C₈	—	106	1	no	—
269	C₂×C₈	—	106	1	no	—
270	C₂×C₈	—	106	1	no	—
271	C₂×C₈	—	106	1	no	—
272	C₂×C₈	C₈×D₅₃	1	2	yes	—
273	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
274	C₂×C₈	#11719945	1	1	yes	characteristic
275	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
276	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
277	C₂×C₈	#11719947	1	1	yes	characteristic
278	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
279	C₂²⋊C₄	—	106	4	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	106	1	no	—
283	C₄²	—	106	1	no	—
284	C₁₆	—	2	2	no	—
285	C₁₆	—	4	1	no	—
286	C₁₆	—	106	2	no	—
287	C₁₆	—	212	1	no	—
288	C₂³	—	53	1	no	—
289	C₂³	—	53	1	no	—
290	C₂³	—	106	2	no	—
291	C₂³	—	106	1	no	—
292	C₂³	—	106	1	no	—
293	C₂³	—	212	1	no	—
294	C₂³	M₁₆×D₅₃	1	1	yes	characteristic
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	53	1	no	—
299	C₂×C₄	—	53	1	no	—
300	C₂×C₄	—	106	1	no	—
301	C₂×C₄	—	106	1	no	—
302	C₂×C₄	—	106	1	no	—
303	C₂×C₄	—	106	1	no	—
304	C₂×C₄	—	106	1	no	—
305	C₂×C₄	—	106	1	no	—
306	C₂×C₄	—	106	1	no	—
307	C₂×C₄	—	106	1	no	—
308	C₂×C₄	—	106	1	no	—
309	C₂×C₄	—	106	1	no	—
310	C₂×C₄	—	106	2	no	—
311	C₂×C₄	—	212	1	no	—
312	C₂×C₄	—	212	1	no	—
313	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
314	C₂×C₄	#13133177	1	1	yes	characteristic
315	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
316	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
317	C₂×C₄	#13133223	1	1	yes	characteristic
318	C₂×C₄	#13133225	1	1	yes	characteristic
319	C₈	—	2	2	no	—
320	C₈	—	106	1	no	—
321	C₈	—	106	1	no	—
322	C₈	—	106	2	no	—
323	C₈	#13133187	1	1	yes	characteristic
324	C₈	#13133195	1	1	yes	characteristic
325	C₈	#13133197	1	1	yes	characteristic
326	C₈	#13133199	1	1	yes	characteristic
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	53	2	no	—
330	C₂²	—	106	1	no	—
331	C₂²	—	106	1	no	—
332	C₂²	—	106	1	no	—
333	C₂²	—	106	1	no	—
334	C₂²	—	106	1	no	—
335	C₂²	—	212	2	no	—
336	C₂²	—	212	2	no	—
337	C₂²	C₄⋊C₈×D₅₃	1	1	yes	characteristic
338	C₂²	#16810696	1	1	yes	characteristic
339	C₂²	#16810697	1	1	yes	characteristic
340	C₄	—	4	2	no	—
341	C₄	—	106	1	no	—
342	C₄	—	106	1	no	—
343	C₄	—	212	2	no	—
344	C₄	#16810497	1	1	yes	characteristic
345	C₄	#16810648	1	1	yes	characteristic
346	C₄	#16810686	1	1	yes	characteristic
347	C₄	C₈×D₂₁₂	1	1	yes	characteristic
348	C₂	—	2	1	no	—
349	C₂	—	2	1	no	—
350	C₂	—	106	2	no	—
351	C₂	—	212	1	no	—
352	C₂	#20274894	1	1	yes	characteristic
353	C₂	#364×D₅₃	1	1	yes	characteristic
354	C₂	#20276479	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1376	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×D₅₃	M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1376	D₅₃
hypocenter	C₅₃	#12914
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

Complements

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

N	H ≅ G/N	type
2. C₂×#16810444	C₂	semidirect
3. C₂×#16810443	C₂	semidirect
6. #20272796	C₂	semidirect
7. C₅₃×#1376	C₂	semidirect
162. C₂×D₅₃	#364	semidirect
177. #1376	D₅₃	semidirect
204. C₅₃	#12914	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂×D₅₃	2
C₂×#16810444	2
C₂×#16810443	2
#20276325	2
C₈×#11719921	2
#20272796	2
C₅₃×#1376	2
#12914	53