Subgroups — #27300012

Order 13568.

566 conjugacy classes of subgroups.

The subgroup lattice has 326 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×#16810441	C₂	1	2	yes	maximal, complemented
3	C₂×#16810402	C₂	1	2	yes	maximal
4	#20272793	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1374	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	4	no	—
14	#16810441	—	2	4	no	—
15	#16810402	—	2	4	no	—
16	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₅₃	C₈	1	1	yes	characteristic
19	C₂²×C₄×D₅₃	C₈	1	1	yes	characteristic
20	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
22	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×D₅₃	—	2	2	no	—
24	C₂×C₈×D₅₃	—	2	2	no	—
25	C₂×C₈×D₅₃	—	2	2	no	—
26	C₂×C₈×D₅₃	—	2	2	no	—
27	C₂×C₈×D₅₃	—	2	2	no	—
28	C₂×C₈×D₅₃	D₄	1	2	yes	—
29	C₂×C₈×D₅₃	Q₈	1	2	yes	—
30	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
31	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
32	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄²×D₅₃	—	2	2	no	—
34	C₄²×D₅₃	—	2	2	no	—
35	C₂×#11719908	—	2	2	no	—
36	C₂×#11719908	—	2	2	no	—
37	C₂×#11719908	—	2	2	no	—
38	C₄×#88322	—	2	2	no	—
39	C₈×Dic₅₃	—	2	2	no	—
40	C₈×Dic₅₃	D₄	1	4	yes	—
41	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₁₆×C₅₃	—	2	2	no	—
45	C₂×C₁₆×C₅₃	—	2	2	no	—
46	C₂×C₁₆×C₅₃	—	2	2	no	—
47	C₄×C₈×C₅₃	—	2	2	no	—
48	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
53	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
55	C₂×C₄×D₅₃	—	2	2	no	—
56	C₂×C₄×D₅₃	—	2	2	no	—
57	C₂×C₄×D₅₃	—	2	1	no	—
58	C₂×C₄×D₅₃	—	2	1	no	—
59	C₂×C₄×D₅₃	—	2	1	no	—
60	C₂×C₄×D₅₃	—	2	1	no	—
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	C₁₆	1	4	yes	complemented
65	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
66	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
67	C₂×C₈×C₅₃	—	2	1	no	—
68	C₂×C₈×C₅₃	—	2	1	no	—
69	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
70	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₈×C₅₃	#41	1	2	yes	—
74	C₄²×C₅₃	—	2	1	no	—
75	C₄²×C₅₃	—	2	1	no	—
76	C₄×Dic₅₃	—	2	2	no	—
77	C₄×Dic₅₃	—	2	2	no	—
78	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
79	C₄×Dic₅₃	M₁₆	1	2	yes	—
80	C₂×#88322	—	2	1	no	—
81	C₂×#88322	—	2	1	no	—
82	C₂×#88322	—	2	1	no	—
83	C₂×#88322	—	2	1	no	—
84	C₂×#88322	C₂²⋊C₄	1	2	yes	—
85	C₂×#88322	C₄⋊C₄	1	2	yes	—
86	C₈×D₅₃	—	2	8	no	—
87	C₈×D₅₃	—	2	4	no	—
88	C₈×D₅₃	—	2	4	no	—
89	C₁₆×C₅₃	—	4	2	no	—
90	#11719908	—	4	2	no	—
91	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
92	C₂²×D₅₃	—	2	1	no	—
93	C₂²×D₅₃	—	2	1	no	—
94	C₂²×D₅₃	—	2	1	no	—
95	C₂²×D₅₃	—	2	1	no	—
96	C₂²×D₅₃	C₄⋊C₈	1	2	yes	—
97	C₂²×D₅₃	C₂×C₁₆	1	2	yes	—
98	C₂²×D₅₃	M₃₂	1	2	yes	—
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	2	1	no	—
101	C₂×C₄×C₅₃	—	2	1	no	—
102	C₂×C₄×C₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
104	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
105	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
111	C₂×Dic₅₃	—	2	1	no	—
112	C₂×Dic₅₃	—	2	1	no	—
113	C₂×Dic₅₃	—	2	1	no	—
114	C₂×Dic₅₃	—	2	1	no	—
115	C₂×Dic₅₃	C₂²⋊C₈	1	2	yes	—
116	C₂×Dic₅₃	C₂²⋊C₈	1	2	yes	—
117	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
118	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
119	C₂×Dic₅₃	M₃₂	1	2	yes	—
120	C₄×D₅₃	—	2	8	no	—
121	C₄×D₅₃	—	2	2	no	—
122	C₄×D₅₃	—	2	2	no	—
123	C₄×D₅₃	—	2	2	no	—
124	C₄×D₅₃	—	2	2	no	—
125	C₄×D₅₃	—	2	8	no	—
126	C₈×C₅₃	—	2	2	no	—
127	C₈×C₅₃	#121	1	2	yes	—
128	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
129	#88322	—	2	2	no	—
130	#88322	—	2	2	no	—
131	#12783	—	53	1	no	maximal, Sylow
132	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
133	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
134	C₂²×C₅₃	#431	1	1	yes	characteristic
135	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
136	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
137	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
138	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
139	C₂×D₅₃	—	2	2	no	—
140	C₂×D₅₃	—	2	2	no	—
141	C₂×D₅₃	—	2	2	no	—
142	C₂×D₅₃	—	2	2	no	—
143	C₂×D₅₃	—	2	2	no	—
144	C₂×D₅₃	—	2	2	no	—
145	C₂×D₅₃	C₄⋊C₁₆	1	4	yes	complemented
146	C₄×C₅₃	—	2	2	no	—
147	C₄×C₅₃	—	2	2	no	—
148	C₄×C₅₃	#380	1	1	yes	characteristic
149	C₄×C₅₃	#407	1	1	yes	characteristic
150	C₄×C₅₃	#423	1	1	yes	characteristic
151	C₄×C₅₃	#432	1	1	yes	characteristic
152	Dic₅₃	—	2	2	no	—
153	Dic₅₃	—	2	2	no	—
154	Dic₅₃	C₂²⋊C₁₆	1	4	yes	complemented
155	C₂²×C₄×C₈	—	53	1	no	—
156	C₂×C₄⋊C₁₆	—	53	2	no	—
157	C₂×C₂²⋊C₁₆	—	53	2	no	—
158	#1374	—	53	1	no	—
159	#1374	D₅₃	1	1	yes	characteristic, complemented
160	C₂×C₅₃	#1797	1	1	yes	characteristic
161	C₂×C₅₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
162	C₂×C₅₃	#2112	1	1	yes	characteristic
163	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
164	C₂×C₅₃	#2150	1	1	yes	characteristic
165	C₂×C₅₃	#2162	1	1	yes	characteristic
166	C₂×C₅₃	#2163	1	1	yes	characteristic
167	D₅₃	—	2	4	no	—
168	C₂³×C₈	—	53	1	no	—
169	C₂³×C₈	—	53	1	no	—
170	C₂²×C₄²	—	53	1	no	—
171	C₂²×C₁₆	—	53	2	no	—
172	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
173	C₂×C₄×C₈	—	53	1	no	—
174	C₂×C₄×C₈	—	53	1	no	—
175	C₂×C₄×C₈	—	53	1	no	—
176	C₂×C₄×C₈	—	106	4	no	—
177	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
178	C₄⋊C₁₆	—	106	4	no	—
179	C₂²⋊C₁₆	—	106	4	no	—
180	C₅₃	#12783	1	1	yes	characteristic, complemented, Sylow
181	C₂³×C₄	—	53	1	no	—
182	C₂³×C₄	—	53	1	no	—
183	C₂³×C₄	—	53	1	no	—
184	C₂²×C₈	—	53	1	no	—
185	C₂²×C₈	—	53	1	no	—
186	C₂²×C₈	—	53	2	no	—
187	C₂²×C₈	—	53	2	no	—
188	C₂²×C₈	—	106	2	no	—
189	C₂²×C₈	—	106	2	no	—
190	C₂²×C₈	—	106	2	no	—
191	C₂²×C₈	—	106	2	no	—
192	C₂²×C₈	—	106	2	no	—
193	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
195	C₂×C₄²	—	53	1	no	—
196	C₂×C₄²	—	53	1	no	—
197	C₂×C₄²	—	53	1	no	—
198	C₂×C₄²	—	106	2	no	—
199	C₂×C₄²	—	106	2	no	—
200	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
201	C₂×C₁₆	—	2	2	no	—
202	C₂×C₁₆	—	2	2	no	—
203	C₂×C₁₆	—	2	2	no	—
204	C₂×C₁₆	—	106	2	no	—
205	C₂×C₁₆	—	106	2	no	—
206	C₂×C₁₆	—	106	2	no	—
207	C₄×C₈	—	2	2	no	—
208	C₄×C₈	—	53	4	no	—
209	C₄×C₈	—	106	2	no	—
210	C₄×C₈	—	106	2	no	—
211	C₂⁴	—	53	1	no	—
212	C₂²×C₄	—	53	1	no	—
213	C₂²×C₄	—	53	1	no	—
214	C₂²×C₄	—	53	1	no	—
215	C₂²×C₄	—	53	4	no	—
216	C₂²×C₄	—	53	2	no	—
217	C₂²×C₄	—	53	2	no	—
218	C₂²×C₄	—	106	2	no	—
219	C₂²×C₄	—	106	2	no	—
220	C₂²×C₄	—	106	2	no	—
221	C₂²×C₄	—	106	2	no	—
222	C₂²×C₄	—	106	2	no	—
223	C₂²×C₄	—	106	1	no	—
224	C₂²×C₄	—	106	1	no	—
225	C₂²×C₄	—	106	1	no	—
226	C₂²×C₄	—	106	1	no	—
227	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
228	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	53	2	no	—
233	C₂×C₈	—	53	2	no	—
234	C₂×C₈	—	106	8	no	—
235	C₂×C₈	—	106	1	no	—
236	C₂×C₈	—	106	1	no	—
237	C₂×C₈	—	106	1	no	—
238	C₂×C₈	—	106	1	no	—
239	C₂×C₈	—	106	4	no	—
240	C₂×C₈	—	106	4	no	—
241	C₂×C₈	#11719912	1	2	yes	—
242	C₂×C₈	#11719945	1	2	yes	—
243	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
244	C₂×C₈	D₄×D₅₃	1	2	yes	—
245	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
246	C₄²	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	53	2	no	—
249	C₄²	—	53	2	no	—
250	C₄²	—	106	2	no	—
251	C₄²	—	106	2	no	—
252	C₁₆	—	4	2	no	—
253	C₁₆	—	212	2	no	—
254	C₂³	—	53	2	no	—
255	C₂³	—	53	2	no	—
256	C₂³	—	53	2	no	—
257	C₂³	—	106	1	no	—
258	C₂³	—	106	1	no	—
259	C₂³	—	106	1	no	—
260	C₂³	—	106	1	no	—
261	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	53	2	no	—
267	C₂×C₄	—	53	2	no	—
268	C₂×C₄	—	53	2	no	—
269	C₂×C₄	—	53	2	no	—
270	C₂×C₄	—	53	2	no	—
271	C₂×C₄	—	106	8	no	—
272	C₂×C₄	—	106	8	no	—
273	C₂×C₄	—	106	1	no	—
274	C₂×C₄	—	106	1	no	—
275	C₂×C₄	—	106	1	no	—
276	C₂×C₄	—	106	1	no	—
277	C₂×C₄	—	106	2	no	—
278	C₂×C₄	—	106	2	no	—
279	C₂×C₄	—	106	2	no	—
280	C₂×C₄	—	106	2	no	—
281	C₂×C₄	C₁₆×D₅₃	1	2	yes	—
282	C₂×C₄	#13133156	1	2	yes	—
283	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
284	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
285	C₂×C₄	#13133174	1	1	yes	characteristic
286	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
287	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
288	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	106	2	no	—
291	C₈	—	106	2	no	—
292	C₈	#13133190	1	2	yes	—
293	C₈	#13133195	1	2	yes	—
294	C₂²	—	53	4	no	—
295	C₂²	—	106	2	no	—
296	C₂²	—	106	2	no	—
297	C₂²	—	106	2	no	—
298	C₂²	—	106	2	no	—
299	C₂²	—	106	2	no	—
300	C₂²	—	106	2	no	—
301	C₂²	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
302	C₂²	C₄⋊C₈×D₅₃	1	1	yes	characteristic
303	C₂²	#16810674	1	1	yes	characteristic
304	C₂²	C₂×C₁₆×D₅₃	1	1	yes	characteristic
305	C₂²	M₃₂×D₅₃	1	1	yes	characteristic
306	C₂²	M₃₂×D₅₃	1	1	yes	characteristic
307	C₂²	C₂×#13133156	1	1	yes	characteristic
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	53	4	no	—
311	C₄	—	106	2	no	—
312	C₄	—	106	2	no	—
313	C₄	#16810482	1	1	yes	characteristic
314	C₄	#16810585	1	1	yes	characteristic
315	C₄	#16810645	1	1	yes	characteristic
316	C₄	#16810678	1	1	yes	characteristic
317	C₂	—	106	4	no	—
318	C₂	#20274694	1	1	yes	characteristic
319	C₂	C₂²⋊C₁₆×D₅₃	1	1	yes	characteristic
320	C₂	#20276324	1	1	yes	characteristic
321	C₂	C₄⋊C₁₆×D₅₃	1	1	yes	characteristic
322	C₂	#20276469	1	1	yes	characteristic
323	C₂	#20276512	1	1	yes	characteristic
324	C₂	#20276516	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₅₃×#1374	C₂
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1374	D₅₃
hypocenter	C₅₃	#12783
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
2. C₂×#16810441	C₂	semidirect
4. #20272793	C₂	semidirect
5. C₅₃×#1374	C₂	semidirect
64. C₂×C₄×D₅₃	C₁₆	semidirect
145. C₂×D₅₃	C₄⋊C₁₆	semidirect
154. Dic₅₃	C₂²⋊C₁₆	semidirect
159. #1374	D₅₃	semidirect
180. C₅₃	#12783	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×#16810441	2
C₂×#16810402	2
#20272793	2
C₅₃×#1374	2
#12783	53