Subgroups — #27300010

Order 13568.

566 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16810402	C₂	1	2	yes	maximal
4	#20272792	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₅₃	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₅₃	C₄	1	1	yes	characteristic
9	C₂²×#11719908	C₂²	1	2	yes	—
10	C₂×C₄×#88322	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
13	C₄×C₈×D₅₃	—	2	2	no	—
14	C₄×C₈×D₅₃	—	2	2	no	—
15	C₄⋊#11719908	—	2	2	no	—
16	#16810402	—	2	4	no	—
17	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×C₄⋊C₁₆	—	2	2	no	—
20	C₂²×C₄×D₅₃	C₈	1	1	yes	characteristic
21	C₂²×C₄×D₅₃	C₈	1	1	yes	characteristic
22	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
24	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×D₅₃	—	2	2	no	—
26	C₂×C₈×D₅₃	—	2	1	no	—
27	C₂×C₈×D₅₃	—	2	1	no	—
28	C₂×C₈×D₅₃	—	2	1	no	—
29	C₂×C₈×D₅₃	—	2	1	no	—
30	C₂×C₈×D₅₃	—	2	2	no	—
31	C₂×C₈×D₅₃	—	2	2	no	—
32	C₂×C₈×D₅₃	D₄	1	2	yes	—
33	C₂×C₈×D₅₃	Q₈	1	2	yes	—
34	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
35	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
36	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄²×D₅₃	—	2	2	no	—
38	C₄²×D₅₃	—	2	2	no	—
39	C₂×#11719908	—	2	2	no	—
40	C₂×#11719908	—	2	2	no	—
41	C₂×#11719908	—	2	2	no	—
42	C₄×#88322	—	2	2	no	—
43	C₈×Dic₅₃	—	2	2	no	—
44	C₈×Dic₅₃	—	2	1	no	—
45	C₈×Dic₅₃	—	2	1	no	—
46	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₁₆×C₅₃	—	2	2	no	—
50	C₂×C₁₆×C₅₃	—	2	2	no	—
51	C₂×C₁₆×C₅₃	—	2	2	no	—
52	C₄×C₈×C₅₃	D₄	1	2	yes	—
53	C₄×C₈×C₅₃	D₄	1	2	yes	—
54	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
59	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
60	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
61	C₂×C₄×D₅₃	—	2	2	no	—
62	C₂×C₄×D₅₃	—	2	2	no	—
63	C₂×C₄×D₅₃	—	2	2	no	—
64	C₂×C₄×D₅₃	—	2	2	no	—
65	C₂×C₄×D₅₃	—	2	1	no	—
66	C₂×C₄×D₅₃	—	2	1	no	—
67	C₂×C₄×D₅₃	—	2	1	no	—
68	C₂×C₄×D₅₃	—	2	1	no	—
69	C₂×C₄×D₅₃	—	2	2	no	—
70	C₂×C₄×D₅₃	C₁₆	1	4	yes	complemented
71	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
72	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
73	C₂×C₈×C₅₃	—	2	1	no	—
74	C₂×C₈×C₅₃	—	2	1	no	—
75	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
81	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
82	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₅₃	M₁₆	1	2	yes	—
84	C₄×Dic₅₃	—	2	2	no	—
85	C₄×Dic₅₃	—	2	2	no	—
86	C₄×Dic₅₃	—	2	1	no	—
87	C₄×Dic₅₃	—	2	1	no	—
88	C₂×#88322	—	2	1	no	—
89	C₂×#88322	—	2	1	no	—
90	C₂×#88322	—	2	1	no	—
91	C₂×#88322	—	2	1	no	—
92	C₂×#88322	C₄⋊C₄	1	2	yes	—
93	C₂×#88322	M₁₆	1	2	yes	—
94	C₈×D₅₃	—	2	8	no	—
95	C₈×D₅₃	—	2	2	no	—
96	C₈×D₅₃	—	2	2	no	—
97	C₈×D₅₃	—	2	2	no	—
98	C₈×D₅₃	—	2	2	no	—
99	C₁₆×C₅₃	—	4	2	no	—
100	#11719908	—	4	2	no	—
101	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
102	C₂²×D₅₃	—	2	1	no	—
103	C₂²×D₅₃	—	2	1	no	—
104	C₂²×D₅₃	—	2	1	no	—
105	C₂²×D₅₃	—	2	1	no	—
106	C₂²×D₅₃	C₄⋊C₈	1	2	yes	—
107	C₂²×D₅₃	C₂×C₁₆	1	2	yes	—
108	C₂²×D₅₃	M₃₂	1	2	yes	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
112	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
114	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
115	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
121	C₂×Dic₅₃	—	2	1	no	—
122	C₂×Dic₅₃	—	2	1	no	—
123	C₂×Dic₅₃	—	2	1	no	—
124	C₂×Dic₅₃	—	2	1	no	—
125	C₂×Dic₅₃	—	2	1	no	—
126	C₂×Dic₅₃	—	2	1	no	—
127	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
128	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
129	C₂×Dic₅₃	M₃₂	1	2	yes	—
130	C₄×D₅₃	—	2	8	no	—
131	C₄×D₅₃	—	2	8	no	—
132	C₄×D₅₃	—	2	2	no	—
133	C₄×D₅₃	—	2	2	no	—
134	C₄×D₅₃	—	2	2	no	—
135	C₄×D₅₃	—	2	2	no	—
136	C₈×C₅₃	—	2	2	no	—
137	C₈×C₅₃	#121	1	1	yes	characteristic
138	C₈×C₅₃	#121	1	1	yes	characteristic
139	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
140	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
141	#88322	—	2	2	no	—
142	#88322	—	2	2	no	—
143	#12783	—	53	1	no	maximal, Sylow
144	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
146	C₂²×C₅₃	#431	1	1	yes	characteristic
147	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
148	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
149	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
150	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
151	C₂×D₅₃	—	2	2	no	—
152	C₂×D₅₃	—	2	2	no	—
153	C₂×D₅₃	—	2	2	no	—
154	C₂×D₅₃	—	2	2	no	—
155	C₂×D₅₃	—	2	2	no	—
156	C₂×D₅₃	—	2	2	no	—
157	C₂×D₅₃	C₄⋊C₁₆	1	4	yes	complemented
158	C₄×C₅₃	—	2	2	no	—
159	C₄×C₅₃	C₂²⋊C₁₆	1	4	yes	complemented
160	C₄×C₅₃	#380	1	1	yes	characteristic
161	C₄×C₅₃	#407	1	1	yes	characteristic
162	C₄×C₅₃	#423	1	1	yes	characteristic
163	C₄×C₅₃	#432	1	1	yes	characteristic
164	Dic₅₃	—	2	2	no	—
165	Dic₅₃	—	2	2	no	—
166	Dic₅₃	—	2	2	no	—
167	C₂²×C₄×C₈	—	53	1	no	—
168	C₂×C₄⋊C₁₆	—	53	1	no	—
169	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic, complemented
170	C₂×C₂²⋊C₁₆	—	53	2	no	—
171	#1374	—	53	2	no	—
172	C₂×C₅₃	#1797	1	1	yes	characteristic
173	C₂×C₅₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
174	C₂×C₅₃	#2112	1	1	yes	characteristic
175	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
176	C₂×C₅₃	#2150	1	1	yes	characteristic
177	C₂×C₅₃	#2162	1	1	yes	characteristic
178	C₂×C₅₃	#2163	1	1	yes	characteristic
179	D₅₃	—	2	4	no	—
180	C₂³×C₈	—	53	1	no	—
181	C₂³×C₈	—	53	1	no	—
182	C₂²×C₄²	—	53	1	no	—
183	C₂²×C₁₆	—	53	2	no	—
184	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
185	C₂×C₄×C₈	—	53	1	no	—
186	C₂×C₄×C₈	—	53	1	no	—
187	C₂×C₄×C₈	—	53	1	no	—
188	C₂×C₄×C₈	—	106	2	no	—
189	C₂×C₄×C₈	—	106	2	no	—
190	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
191	C₄⋊C₁₆	—	2	2	no	—
192	C₄⋊C₁₆	—	106	2	no	—
193	C₂²⋊C₁₆	—	106	4	no	—
194	C₅₃	#12783	1	1	yes	characteristic, complemented, Sylow
195	C₂³×C₄	—	53	1	no	—
196	C₂³×C₄	—	53	1	no	—
197	C₂³×C₄	—	53	1	no	—
198	C₂²×C₈	—	53	1	no	—
199	C₂²×C₈	—	53	1	no	—
200	C₂²×C₈	—	53	2	no	—
201	C₂²×C₈	—	53	2	no	—
202	C₂²×C₈	—	106	2	no	—
203	C₂²×C₈	—	106	2	no	—
204	C₂²×C₈	—	106	1	no	—
205	C₂²×C₈	—	106	1	no	—
206	C₂²×C₈	—	106	1	no	—
207	C₂²×C₈	—	106	1	no	—
208	C₂²×C₈	—	106	2	no	—
209	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
210	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
211	C₂×C₄²	—	53	1	no	—
212	C₂×C₄²	—	53	1	no	—
213	C₂×C₄²	—	53	1	no	—
214	C₂×C₄²	—	106	2	no	—
215	C₂×C₄²	—	106	2	no	—
216	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
217	C₂×C₁₆	—	2	2	no	—
218	C₂×C₁₆	—	2	2	no	—
219	C₂×C₁₆	—	2	2	no	—
220	C₂×C₁₆	—	106	2	no	—
221	C₂×C₁₆	—	106	2	no	—
222	C₂×C₁₆	—	106	2	no	—
223	C₄×C₈	—	106	1	no	—
224	C₄×C₈	—	106	1	no	—
225	C₄×C₈	—	106	2	no	—
226	C₄×C₈	—	106	2	no	—
227	C₄×C₈	D₂₁₂	1	2	yes	—
228	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
229	C₂⁴	—	53	1	no	—
230	C₂²×C₄	—	53	1	no	—
231	C₂²×C₄	—	53	1	no	—
232	C₂²×C₄	—	53	1	no	—
233	C₂²×C₄	—	53	4	no	—
234	C₂²×C₄	—	53	2	no	—
235	C₂²×C₄	—	53	2	no	—
236	C₂²×C₄	—	106	2	no	—
237	C₂²×C₄	—	106	2	no	—
238	C₂²×C₄	—	106	2	no	—
239	C₂²×C₄	—	106	2	no	—
240	C₂²×C₄	—	106	2	no	—
241	C₂²×C₄	—	106	1	no	—
242	C₂²×C₄	—	106	1	no	—
243	C₂²×C₄	—	106	1	no	—
244	C₂²×C₄	—	106	1	no	—
245	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
246	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	53	2	no	—
251	C₂×C₈	—	53	2	no	—
252	C₂×C₈	—	106	8	no	—
253	C₂×C₈	—	106	1	no	—
254	C₂×C₈	—	106	1	no	—
255	C₂×C₈	—	106	1	no	—
256	C₂×C₈	—	106	1	no	—
257	C₂×C₈	—	106	2	no	—
258	C₂×C₈	—	106	2	no	—
259	C₂×C₈	—	106	2	no	—
260	C₂×C₈	—	106	2	no	—
261	C₂×C₈	#11719921	1	2	yes	—
262	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
263	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
264	C₂×C₈	#11719947	1	1	yes	characteristic
265	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
266	C₂×C₈	#11719949	1	1	yes	characteristic
267	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
268	C₄²	—	106	2	no	—
269	C₄²	—	106	2	no	—
270	C₄²	—	106	1	no	—
271	C₄²	—	106	1	no	—
272	C₄²	#11719912	1	2	yes	—
273	C₄²	#11719921	1	2	yes	—
274	C₁₆	—	4	2	no	—
275	C₁₆	—	212	2	no	—
276	C₂³	—	53	2	no	—
277	C₂³	—	53	2	no	—
278	C₂³	—	53	2	no	—
279	C₂³	—	106	1	no	—
280	C₂³	—	106	1	no	—
281	C₂³	—	106	1	no	—
282	C₂³	—	106	1	no	—
283	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	53	2	no	—
287	C₂×C₄	—	53	2	no	—
288	C₂×C₄	—	53	2	no	—
289	C₂×C₄	—	106	8	no	—
290	C₂×C₄	—	106	1	no	—
291	C₂×C₄	—	106	1	no	—
292	C₂×C₄	—	106	1	no	—
293	C₂×C₄	—	106	1	no	—
294	C₂×C₄	—	106	1	no	—
295	C₂×C₄	—	106	1	no	—
296	C₂×C₄	—	106	2	no	—
297	C₂×C₄	—	106	2	no	—
298	C₂×C₄	—	106	8	no	—
299	C₂×C₄	—	106	2	no	—
300	C₂×C₄	—	106	2	no	—
301	C₂×C₄	#13133129	1	2	yes	—
302	C₂×C₄	#13133129	1	2	yes	—
303	C₂×C₄	C₁₆×D₅₃	1	2	yes	—
304	C₂×C₄	#13133156	1	2	yes	—
305	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
306	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
307	C₂×C₄	#13133175	1	1	yes	characteristic
308	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
309	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
310	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	106	2	no	—
313	C₈	—	106	2	no	—
314	C₈	#13133188	1	1	yes	characteristic
315	C₈	#13133192	1	1	yes	characteristic
316	C₈	#13133193	1	1	yes	characteristic
317	C₈	#13133197	1	1	yes	characteristic
318	C₂²	—	53	4	no	—
319	C₂²	—	106	2	no	—
320	C₂²	—	106	2	no	—
321	C₂²	—	106	2	no	—
322	C₂²	—	106	2	no	—
323	C₂²	—	106	2	no	—
324	C₂²	—	106	2	no	—
325	C₂²	C₂×#13133129	1	1	yes	characteristic
326	C₂²	C₄⋊C₈×D₅₃	1	1	yes	characteristic
327	C₂²	#16810675	1	1	yes	characteristic
328	C₂²	C₂×C₁₆×D₅₃	1	1	yes	characteristic
329	C₂²	M₃₂×D₅₃	1	1	yes	characteristic
330	C₂²	M₃₂×D₅₃	1	1	yes	characteristic
331	C₂²	C₂×#13133156	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	106	2	no	—
334	C₄	—	106	2	no	—
335	C₄	—	106	2	no	—
336	C₄	#16810402	1	4	yes	complemented
337	C₄	#16810480	1	1	yes	characteristic
338	C₄	#16810586	1	1	yes	characteristic
339	C₄	#16810645	1	1	yes	characteristic
340	C₄	#16810679	1	1	yes	characteristic
341	C₂	—	106	4	no	—
342	C₂	#20274692	1	1	yes	characteristic
343	C₂	C₂×#16810402	1	1	yes	characteristic
344	C₂	#20276325	1	1	yes	characteristic
345	C₂	C₄⋊C₁₆×D₅₃	1	1	yes	characteristic
346	C₂	#20276469	1	1	yes	characteristic
347	C₂	#20276513	1	1	yes	characteristic
348	C₂	#20276517	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#12783
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊#11719908	C₂	semidirect
4. #20272792	C₂	semidirect
5. C₂×C₅₃×C₄⋊C₁₆	C₂	semidirect
70. C₂×C₄×D₅₃	C₁₆	semidirect
157. C₂×D₅₃	C₄⋊C₁₆	semidirect
159. C₄×C₅₃	C₂²⋊C₁₆	semidirect
169. C₂×C₄⋊C₁₆	D₅₃	semidirect
194. C₅₃	#12783	semidirect
336. C₄	#16810402	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×D₅₃	2
C₂×C₄⋊#11719908	2
C₂×#16810402	2
#20272792	2
C₂×C₅₃×C₄⋊C₁₆	2
#12783	53