Subgroups — #27270644

Order 13568.

622 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810378	C₂	1	1	yes	maximal, characteristic
2	#20272945	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#336	C₂	1	1	yes	maximal, characteristic
4	C₂³×#88322	—	2	1	no	—
5	C₂²×C₄⋊Dic₅₃	C₄	1	2	yes	—
6	C₂×#13133124	—	2	1	no	—
7	C₂×#13133124	—	2	1	no	—
8	#16810378	—	2	1	no	—
9	#16810378	C₄	1	2	yes	complemented
10	C₂³×C₈×C₅₃	—	2	1	no	—
11	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#336	—	2	1	no	—
13	C₅₃×#336	C₄	1	2	yes	complemented
14	C₂³×Dic₅₃	—	2	2	no	—
15	C₂²×#88322	—	2	4	no	—
16	C₂²×#88322	—	2	2	no	—
17	C₂²×#88322	—	4	2	no	—
18	C₂²×#88322	—	4	1	no	—
19	C₂²×#88322	—	4	1	no	—
20	C₂×C₄⋊Dic₅₃	—	2	2	no	—
21	C₂×C₄⋊Dic₅₃	—	4	2	no	—
22	C₂×C₄⋊Dic₅₃	—	4	2	no	—
23	C₂×C₄⋊Dic₅₃	C₈	1	4	yes	complemented
24	#13133124	—	4	1	no	—
25	#13133124	—	4	1	no	—
26	#13133124	—	4	1	no	—
27	#13133124	—	4	1	no	—
28	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	—	2	4	no	—
32	C₂²×C₈×C₅₃	—	2	2	no	—
33	C₂²×C₈×C₅₃	—	4	2	no	—
34	C₂²×C₈×C₅₃	—	4	1	no	—
35	C₂²×C₈×C₅₃	—	4	1	no	—
36	C₂×C₄²×C₅₃	—	4	2	no	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
41	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	—	2	2	no	—
43	C₂²×C₄×C₅₃	—	2	2	no	—
44	C₂²×C₄×C₅₃	—	2	2	no	—
45	C₂²×C₄×C₅₃	—	2	1	no	—
46	C₂²×C₄×C₅₃	—	2	1	no	—
47	C₂²×C₄×C₅₃	—	2	2	no	—
48	C₂²×C₄×C₅₃	—	2	2	no	—
49	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
57	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
65	C₂²×Dic₅₃	—	2	4	no	—
66	C₂²×Dic₅₃	—	4	2	no	—
67	C₂²×Dic₅₃	—	4	2	no	—
68	C₂²×Dic₅₃	—	4	2	no	—
69	C₂²×Dic₅₃	—	4	2	no	—
70	C₂²×Dic₅₃	—	4	2	no	—
71	C₂²×Dic₅₃	—	4	2	no	—
72	C₂×C₈×C₅₃	—	2	4	no	—
73	C₂×C₈×C₅₃	—	4	4	no	—
74	C₂×C₈×C₅₃	—	4	4	no	—
75	C₂×C₈×C₅₃	—	4	2	no	—
76	C₂×C₈×C₅₃	—	4	2	no	—
77	C₄²×C₅₃	—	4	1	no	—
78	C₄²×C₅₃	—	4	1	no	—
79	C₄²×C₅₃	—	4	1	no	—
80	C₄²×C₅₃	—	4	1	no	—
81	C₄⋊Dic₅₃	—	4	4	no	—
82	C₄⋊Dic₅₃	—	4	4	no	—
83	C₂×#88322	—	2	4	no	—
84	C₂×#88322	—	4	4	no	—
85	C₂×#88322	—	4	4	no	—
86	C₂×#88322	—	4	2	no	—
87	C₂×#88322	—	4	2	no	—
88	C₂³×C₅₃	#95	1	1	yes	characteristic
89	C₂³×C₅₃	C₂²⋊C₈	1	2	yes	—
90	C₂³×C₅₃	#99	1	2	yes	—
91	C₂³×C₅₃	#100	1	1	yes	characteristic
92	C₂³×C₅₃	#101	1	1	yes	characteristic
93	C₂³×C₅₃	#102	1	1	yes	characteristic
94	C₂³×C₅₃	#102	1	1	yes	characteristic
95	C₂³×C₅₃	#103	1	1	yes	characteristic
96	C₂³×C₅₃	#103	1	1	yes	characteristic
97	C₂³×C₅₃	#104	1	2	yes	—
98	C₂³×C₅₃	#104	1	2	yes	—
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	—	2	2	no	—
108	C₂×C₄×C₅₃	—	2	2	no	—
109	C₂×C₄×C₅₃	—	2	2	no	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	2	1	no	—
112	C₂×C₄×C₅₃	—	2	1	no	—
113	C₂×C₄×C₅₃	—	2	1	no	—
114	C₂×C₄×C₅₃	—	2	1	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	—	2	1	no	—
117	C₂×C₄×C₅₃	—	2	1	no	—
118	C₂×C₄×C₅₃	—	2	1	no	—
119	C₂×C₄×C₅₃	—	2	1	no	—
120	C₂×C₄×C₅₃	—	2	1	no	—
121	C₂×C₄×C₅₃	—	2	1	no	—
122	C₂×C₄×C₅₃	—	2	1	no	—
123	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
124	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
125	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
126	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
127	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
128	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
130	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
131	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
132	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
133	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
134	C₂×Dic₅₃	—	4	4	no	—
135	C₂×Dic₅₃	—	4	4	no	—
136	C₂×Dic₅₃	—	4	4	no	—
137	C₂×Dic₅₃	—	8	2	no	—
138	C₂×Dic₅₃	—	8	2	no	—
139	C₂×Dic₅₃	—	8	2	no	—
140	C₂×Dic₅₃	—	8	2	no	—
141	C₈×C₅₃	—	4	4	no	—
142	#88322	—	4	4	no	—
143	#7025	—	53	1	no	maximal, Sylow
144	C₂²×C₅₃	#323	1	2	yes	—
145	C₂²×C₅₃	#324	1	2	yes	—
146	C₂²×C₅₃	D₄⋊C₈	1	2	yes	—
147	C₂²×C₅₃	D₄⋊C₈	1	2	yes	—
148	C₂²×C₅₃	Q₈⋊C₈	1	2	yes	—
149	C₂²×C₅₃	Q₈⋊C₈	1	2	yes	—
150	C₂²×C₅₃	#327	1	2	yes	—
151	C₂²×C₅₃	#327	1	2	yes	—
152	C₂²×C₅₃	#328	1	2	yes	—
153	C₂²×C₅₃	#328	1	2	yes	—
154	C₂²×C₅₃	#329	1	2	yes	—
155	C₂²×C₅₃	#330	1	2	yes	—
156	C₂²×C₅₃	#331	1	1	yes	characteristic
157	C₂²×C₅₃	#332	1	1	yes	characteristic
158	C₂²×C₅₃	#332	1	1	yes	characteristic
159	C₂²×C₅₃	#333	1	1	yes	characteristic
160	C₂²×C₅₃	#336	1	1	yes	characteristic
161	C₂²×C₅₃	#339	1	1	yes	characteristic
162	C₂²×C₅₃	#339	1	1	yes	characteristic
163	C₂²×C₅₃	#340	1	1	yes	characteristic
164	C₂²×C₅₃	#340	1	1	yes	characteristic
165	C₂²×C₅₃	#342	1	1	yes	characteristic
166	C₂²×C₅₃	#343	1	1	yes	characteristic
167	C₄×C₅₃	—	2	1	no	—
168	C₄×C₅₃	—	2	1	no	—
169	C₄×C₅₃	—	2	2	no	—
170	C₄×C₅₃	—	2	1	no	—
171	C₄×C₅₃	—	2	1	no	—
172	C₄×C₅₃	—	2	1	no	—
173	C₄×C₅₃	—	2	1	no	—
174	C₄×C₅₃	—	2	1	no	—
175	C₄×C₅₃	—	2	1	no	—
176	C₄×C₅₃	—	2	1	no	—
177	C₄×C₅₃	—	2	1	no	—
178	Dic₅₃	—	8	4	no	—
179	C₂×#336	—	53	1	no	—
180	C₂×#336	D₅₃	1	1	yes	characteristic
181	C₄⋊#95	—	53	1	no	—
182	C₂×C₅₃	#1270	1	2	yes	—
183	C₂×C₅₃	#1271	1	2	yes	—
184	C₂×C₅₃	#1271	1	2	yes	—
185	C₂×C₅₃	#1272	1	2	yes	—
186	C₂×C₅₃	#1279	1	1	yes	characteristic
187	C₂×C₅₃	#1279	1	1	yes	characteristic
188	C₂×C₅₃	#1280	1	1	yes	characteristic
189	C₂×C₅₃	#1294	1	1	yes	characteristic
190	C₂×C₅₃	#1294	1	1	yes	characteristic
191	C₂×C₅₃	#1295	1	1	yes	characteristic
192	C₂×C₅₃	#1296	1	1	yes	characteristic
193	C₂³×C₈	—	2	1	no	—
194	C₂³×C₈	—	106	1	no	—
195	C₂²×C₄⋊C₄	—	53	2	no	—
196	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
197	C₂×#95	—	106	1	no	—
198	C₂×#95	—	106	1	no	—
199	#336	—	2	1	no	—
200	#336	—	53	2	no	—
201	#336	—	106	1	no	—
202	#336	Dic₅₃	1	2	yes	complemented
203	C₅₃	#7025	1	1	yes	characteristic, complemented, Sylow
204	C₂³×C₄	—	106	2	no	—
205	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
206	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
207	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
208	C₂²×C₈	—	2	4	no	—
209	C₂²×C₈	—	2	2	no	—
210	C₂²×C₈	—	4	2	no	—
211	C₂²×C₈	—	4	1	no	—
212	C₂²×C₈	—	4	1	no	—
213	C₂²×C₈	—	106	4	no	—
214	C₂²×C₈	—	106	2	no	—
215	C₂²×C₈	—	212	2	no	—
216	C₂²×C₈	—	212	1	no	—
217	C₂²×C₈	—	212	1	no	—
218	C₂×C₄⋊C₄	—	53	4	no	—
219	C₂×C₄⋊C₄	—	106	2	no	—
220	C₂×C₄⋊C₄	—	212	2	no	—
221	C₂×C₄⋊C₄	—	212	2	no	—
222	C₂×C₄²	—	4	2	no	—
223	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
224	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
225	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
226	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
227	#95	—	212	1	no	—
228	#95	—	212	1	no	—
229	#95	—	212	1	no	—
230	#95	—	212	1	no	—
231	C₂⁴	#11719921	1	1	yes	characteristic
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	106	4	no	—
240	C₂²×C₄	—	212	2	no	—
241	C₂²×C₄	—	212	2	no	—
242	C₂²×C₄	—	212	2	no	—
243	C₂²×C₄	—	212	2	no	—
244	C₂²×C₄	—	212	2	no	—
245	C₂²×C₄	—	212	2	no	—
246	C₂²×C₄	C₈×D₅₃	1	2	yes	—
247	C₂²×C₄	#11719912	1	2	yes	—
248	C₂²×C₄	#11719913	1	1	yes	characteristic
249	C₂²×C₄	#11719913	1	1	yes	characteristic
250	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
251	C₂²×C₄	Dic₂₁₂	1	1	yes	characteristic
252	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
253	C₂²×C₄	#11719919	1	1	yes	characteristic
254	C₂²×C₄	#11719919	1	1	yes	characteristic
255	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
256	C₂²×C₄	#11719921	1	1	yes	characteristic
257	C₂²×C₄	#11719922	1	1	yes	characteristic
258	C₂²×C₄	#11719923	1	1	yes	characteristic
259	C₂²×C₄	#11719924	1	1	yes	characteristic
260	C₂²×C₄	#11719925	1	1	yes	characteristic
261	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	4	4	no	—
264	C₂×C₈	—	4	4	no	—
265	C₂×C₈	—	4	2	no	—
266	C₂×C₈	—	4	2	no	—
267	C₂×C₈	—	106	4	no	—
268	C₂×C₈	—	212	4	no	—
269	C₂×C₈	—	212	4	no	—
270	C₂×C₈	—	212	2	no	—
271	C₂×C₈	—	212	2	no	—
272	C₄⋊C₄	—	212	4	no	—
273	C₄⋊C₄	—	212	4	no	—
274	C₄²	—	4	1	no	—
275	C₄²	—	4	1	no	—
276	C₄²	—	4	1	no	—
277	C₄²	—	4	1	no	—
278	C₂³	#13133124	1	1	yes	characteristic
279	C₂³	#13133129	1	2	yes	—
280	C₂³	#13133131	1	2	yes	—
281	C₂³	#13133133	1	1	yes	characteristic
282	C₂³	#13133135	1	1	yes	characteristic
283	C₂³	#13133137	1	1	yes	characteristic
284	C₂³	#13133139	1	1	yes	characteristic
285	C₂³	#13133140	1	1	yes	characteristic
286	C₂³	#13133142	1	1	yes	characteristic
287	C₂³	#13133143	1	2	yes	—
288	C₂³	#13133145	1	2	yes	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	424	2	no	—
314	C₂×C₄	—	424	2	no	—
315	C₂×C₄	—	424	2	no	—
316	C₂×C₄	—	424	2	no	—
317	C₂×C₄	—	212	4	no	—
318	C₂×C₄	—	212	4	no	—
319	C₂×C₄	—	212	4	no	—
320	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
321	C₂×C₄	#13133127	1	1	yes	characteristic
322	C₂×C₄	#13133138	1	1	yes	characteristic
323	C₂×C₄	#13133139	1	1	yes	characteristic
324	C₂×C₄	#13133141	1	1	yes	characteristic
325	C₂×C₄	#13133142	1	1	yes	characteristic
326	C₂×C₄	#13133146	1	2	yes	—
327	C₂×C₄	#13133148	1	1	yes	characteristic
328	C₂×C₄	#13133149	1	1	yes	characteristic
329	C₂×C₄	#13133150	1	1	yes	characteristic
330	C₂×C₄	#13133151	1	1	yes	characteristic
331	C₈	—	4	4	no	—
332	C₈	—	212	4	no	—
333	C₂²	#16810346	1	2	yes	—
334	C₂²	#16810348	1	2	yes	—
335	C₂²	#16810350	1	2	yes	—
336	C₂²	#16810352	1	2	yes	—
337	C₂²	#16810353	1	2	yes	—
338	C₂²	#16810355	1	2	yes	—
339	C₂²	#16810356	1	2	yes	—
340	C₂²	#16810358	1	2	yes	—
341	C₂²	#16810359	1	2	yes	—
342	C₂²	#16810361	1	2	yes	—
343	C₂²	#16810362	1	2	yes	—
344	C₂²	#16810364	1	2	yes	—
345	C₂²	#16810366	1	1	yes	characteristic
346	C₂²	#16810368	1	1	yes	characteristic
347	C₂²	#16810370	1	1	yes	characteristic
348	C₂²	#16810371	1	1	yes	characteristic
349	C₂²	#16810377	1	1	yes	characteristic
350	C₂²	#16810383	1	1	yes	characteristic
351	C₂²	#16810384	1	1	yes	characteristic
352	C₂²	#16810385	1	1	yes	characteristic
353	C₂²	#16810386	1	1	yes	characteristic
354	C₂²	#16810389	1	1	yes	characteristic
355	C₂²	#16810391	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	2	no	—
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	424	4	no	—
368	C₂	#20272535	1	2	yes	—
369	C₂	#20272537	1	2	yes	—
370	C₂	#20272539	1	2	yes	—
371	C₂	#20272540	1	2	yes	—
372	C₂	#20272554	1	1	yes	characteristic
373	C₂	#20272555	1	1	yes	characteristic
374	C₂	#20272557	1	1	yes	characteristic
375	C₂	#20272594	1	1	yes	characteristic
376	C₂	#20272595	1	1	yes	characteristic
377	C₂	#20272597	1	1	yes	characteristic
378	C₂	#20272599	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	#11719921
commutator	C₂×C₄×C₅₃	C₄×C₈
Frattini	C₂²×C₄²	C₂×D₅₃
Fitting	C₂×C₅₃×#336	C₂
socle, prime radical	C₂⁴×C₅₃	C₂²⋊C₄
cosocle	C₂²×C₄²×C₅₃	C₂²
hypercenter	C₂×#336	D₅₃
hypocenter	C₅₃	#7025
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#336

Complements

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

N	H ≅ G/N	type
9. #16810378	C₄	semidirect
13. C₅₃×#336	C₄	semidirect
23. C₂×C₄⋊Dic₅₃	C₈	semidirect
202. #336	Dic₅₃	semidirect
203. C₅₃	#7025	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810378	2
#20272945	2
C₂×C₅₃×#336	2
#7025	53