Subgroups — C₅₃×#21660 #27229172

Order 13568.

746 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3131	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2960	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2931	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1994	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1962	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1938	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1837	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1765	C₂	1	4	yes	maximal, complemented
9	C₄×#97×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#131×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#533	C₂²	1	1	yes	characteristic
12	C₅₃×#532	C₂²	1	1	yes	characteristic
13	C₅₃×#488	C₄	1	4	yes	complemented
14	C₅₃×#487	C₄	1	2	yes	complemented
15	C₅₃×#486	C₄	1	2	yes	complemented
16	C₅₃×#446	—	2	2	no	—
17	C₈×Q₈×C₅₃	—	2	2	no	—
18	C₅₃×#436	—	2	2	no	—
19	C₈×D₄×C₅₃	—	2	2	no	—
20	C₅₃×#424	C₂²	1	2	yes	—
21	C₅₃×#423	C₂²	1	1	yes	characteristic
22	C₅₃×#423	C₂²	1	1	yes	characteristic
23	C₂×#104×C₅₃	C₂²	1	4	yes	—
24	C₅₃×#416	C₂²	1	4	yes	—
25	C₅₃×#408	C₂²	1	2	yes	—
26	C₅₃×#405	C₂²	1	2	yes	—
27	C₂×#97×C₅₃	C₂²	1	2	yes	—
28	C₂×#97×C₅₃	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₄³×C₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#341	C₂²	1	4	yes	—
33	C₅₃×#339	C₂²	1	4	yes	—
34	C₂×#41×C₅₃	C₂³	1	2	yes	—
35	#131×C₅₃	—	2	8	no	—
36	C₂×M₁₆×C₅₃	—	2	2	no	—
37	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
38	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₅₃	—	2	2	no	—
40	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
44	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	#127×C₅₃	C₂×C₄	1	1	yes	characteristic
46	#125×C₅₃	C₂×C₄	1	2	yes	—
47	#124×C₅₃	C₂×C₄	1	2	yes	—
48	C₂²⋊Q₈×C₅₃	—	2	2	no	—
49	#121×C₅₃	—	2	2	no	—
50	C₄×Q₈×C₅₃	—	2	2	no	—
51	C₄×D₄×C₅₃	—	2	2	no	—
52	#117×C₅₃	C₂³	1	2	yes	—
53	C₂×C₄²×C₅₃	—	2	2	no	—
54	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
55	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₅₃	—	2	4	no	—
57	C₄⋊C₈×C₅₃	—	2	2	no	—
58	C₄⋊C₈×C₅₃	—	2	2	no	—
59	#104×C₅₃	—	2	8	no	—
60	#103×C₅₃	C₂×C₄	1	8	yes	—
61	#102×C₅₃	C₂×C₄	1	8	yes	—
62	C₂²⋊C₈×C₅₃	—	2	4	no	—
63	#97×C₅₃	—	2	2	no	—
64	#97×C₅₃	—	2	4	no	—
65	#97×C₅₃	—	2	4	no	—
66	C₄×C₈×C₅₃	—	2	2	no	—
67	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₅₃	D₄	1	2	yes	—
70	C₄×C₈×C₅₃	D₄	1	2	yes	—
71	#41×C₅₃	—	2	4	no	—
72	#41×C₅₃	—	2	4	no	—
73	C₂×Q₈×C₅₃	—	2	2	no	—
74	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
75	C₂×D₄×C₅₃	—	2	2	no	—
76	C₂×D₄×C₅₃	—	2	2	no	—
77	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₅₃	—	2	2	no	—
79	C₂²×C₄×C₅₃	—	2	2	no	—
80	C₂²×C₄×C₅₃	#41	1	2	yes	—
81	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
82	M₁₆×C₅₃	—	2	4	no	—
83	M₁₆×C₅₃	—	2	4	no	—
84	M₁₆×C₅₃	—	4	4	no	—
85	C₂×C₈×C₅₃	—	2	2	no	—
86	C₂×C₈×C₅₃	—	2	2	no	—
87	C₂×C₈×C₅₃	—	2	2	no	—
88	C₂×C₈×C₅₃	—	2	2	no	—
89	C₂×C₈×C₅₃	—	2	2	no	—
90	C₂×C₈×C₅₃	—	4	4	no	—
91	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
92	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	#41	1	4	yes	—
96	C₄⋊C₄×C₅₃	—	2	2	no	—
97	C₄⋊C₄×C₅₃	—	2	4	no	—
98	C₄⋊C₄×C₅₃	—	2	2	no	—
99	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
100	C₂²⋊C₄×C₅₃	—	2	2	no	—
101	C₂²⋊C₄×C₅₃	—	2	4	no	—
102	C₄²×C₅₃	—	2	2	no	—
103	C₄²×C₅₃	—	2	2	no	—
104	C₄²×C₅₃	—	4	4	no	—
105	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₂×D₄	1	2	yes	—
108	C₄²×C₅₃	#41	1	4	yes	—
109	C₂³×C₅₃	—	2	2	no	—
110	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
111	Q₈×C₅₃	—	2	4	no	—
112	Q₈×C₅₃	—	4	4	no	—
113	D₄×C₅₃	—	2	4	no	—
114	D₄×C₅₃	—	4	4	no	—
115	D₄×C₅₃	—	4	4	no	—
116	C₂×C₄×C₅₃	—	2	4	no	—
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	2	4	no	—
119	C₂×C₄×C₅₃	—	2	4	no	—
120	C₂×C₄×C₅₃	—	4	4	no	—
121	C₂×C₄×C₅₃	—	4	2	no	—
122	C₂×C₄×C₅₃	—	4	2	no	—
123	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
128	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
129	C₈×C₅₃	—	2	4	no	—
130	C₈×C₅₃	—	2	2	no	—
131	C₈×C₅₃	—	4	2	no	—
132	C₈×C₅₃	—	4	2	no	—
133	C₈×C₅₃	#124	1	2	yes	—
134	C₈×C₅₃	#124	1	2	yes	—
135	#21660	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂²×C₅₃	—	2	2	no	—
137	C₂²×C₅₃	—	4	4	no	—
138	C₂²×C₅₃	—	4	2	no	—
139	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂²×C₅₃	#529	1	2	yes	—
141	C₄×C₅₃	—	2	2	no	—
142	C₄×C₅₃	—	4	4	no	—
143	C₄×C₅₃	—	4	4	no	—
144	C₄×C₅₃	—	4	2	no	—
145	C₄×C₅₃	#444	1	4	yes	—
146	C₄×C₅₃	C₄×#41	1	2	yes	—
147	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
148	C₄×C₅₃	#532	1	1	yes	characteristic
149	#3131	C₂×C₅₃	1	1	yes	characteristic
150	#2960	C₂×C₅₃	1	1	yes	characteristic, complemented
151	#2931	C₂×C₅₃	1	1	yes	characteristic
152	#1994	C₂×C₅₃	1	1	yes	characteristic, complemented
153	#1962	C₂×C₅₃	1	1	yes	characteristic
154	#1938	C₂×C₅₃	1	4	yes	complemented
155	#1837	C₂×C₅₃	1	1	yes	characteristic, complemented
156	#1765	C₂×C₅₃	1	4	yes	complemented
157	C₄×#97	C₂×C₅₃	1	1	yes	characteristic, complemented
158	C₂×C₅₃	—	2	2	no	—
159	C₂×C₅₃	—	8	2	no	—
160	C₂×C₅₃	C₄×#124	1	1	yes	characteristic
161	C₂×C₅₃	C₂×#444	1	2	yes	—
162	C₂×#131	C₂²×C₅₃	1	2	yes	—
163	#533	C₂²×C₅₃	1	1	yes	characteristic
164	#532	C₂²×C₅₃	1	1	yes	characteristic
165	#488	C₄×C₅₃	1	4	yes	complemented
166	#487	C₄×C₅₃	1	2	yes	complemented
167	#486	C₄×C₅₃	1	2	yes	complemented
168	#446	—	2	2	no	—
169	C₈×Q₈	—	2	2	no	—
170	#436	—	2	2	no	—
171	C₈×D₄	—	2	2	no	—
172	#424	C₂²×C₅₃	1	2	yes	—
173	#423	C₂²×C₅₃	1	1	yes	characteristic
174	#423	C₂²×C₅₃	1	1	yes	characteristic
175	C₂×#104	C₂²×C₅₃	1	4	yes	—
176	#416	C₂²×C₅₃	1	4	yes	—
177	#408	C₂²×C₅₃	1	2	yes	—
178	#405	C₂²×C₅₃	1	2	yes	—
179	C₂×#97	C₂²×C₅₃	1	2	yes	—
180	C₂×#97	C₂²×C₅₃	1	2	yes	—
181	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
182	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
183	C₄³	C₂²×C₅₃	1	1	yes	characteristic
184	#341	C₂²×C₅₃	1	4	yes	—
185	#339	C₂²×C₅₃	1	4	yes	—
186	C₅₃	#21660	1	1	yes	characteristic, direct product factor, Sylow
187	C₂×#41	C₂³×C₅₃	1	2	yes	—
188	#131	—	2	8	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
191	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
194	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
195	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
196	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
197	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
198	#127	C₂×C₄×C₅₃	1	1	yes	characteristic
199	#125	C₂×C₄×C₅₃	1	2	yes	—
200	#124	C₂×C₄×C₅₃	1	2	yes	—
201	C₂²⋊Q₈	—	2	2	no	—
202	#121	—	2	2	no	—
203	C₄×Q₈	—	2	2	no	—
204	C₄×D₄	—	2	2	no	—
205	#117	C₂³×C₅₃	1	2	yes	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
208	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
209	C₄⋊C₈	—	2	4	no	—
210	C₄⋊C₈	—	2	2	no	—
211	C₄⋊C₈	—	2	2	no	—
212	#104	—	2	8	no	—
213	#103	C₂×C₄×C₅₃	1	8	yes	—
214	#102	C₂×C₄×C₅₃	1	8	yes	—
215	C₂²⋊C₈	—	2	4	no	—
216	#97	—	2	2	no	—
217	#97	—	2	4	no	—
218	#97	—	2	4	no	—
219	C₄×C₈	—	2	2	no	—
220	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
221	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
222	C₄×C₈	D₄×C₅₃	1	2	yes	—
223	C₄×C₈	D₄×C₅₃	1	2	yes	—
224	#41	—	2	4	no	—
225	#41	—	2	4	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	#41×C₅₃	1	2	yes	—
234	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
235	M₁₆	—	2	4	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	4	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	4	4	no	—
244	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
245	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
246	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
247	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
248	C₂×C₈	#41×C₅₃	1	4	yes	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	—	2	4	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	4	4	no	—
258	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
259	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
260	C₄²	C₂×D₄×C₅₃	1	2	yes	—
261	C₄²	#41×C₅₃	1	4	yes	—
262	C₂³	—	2	2	no	—
263	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
264	Q₈	—	2	4	no	—
265	Q₈	—	4	4	no	—
266	D₄	—	2	4	no	—
267	D₄	—	4	4	no	—
268	D₄	—	4	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
277	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
278	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
279	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
280	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
281	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
282	C₈	—	2	4	no	—
283	C₈	—	2	2	no	—
284	C₈	—	4	2	no	—
285	C₈	—	4	2	no	—
286	C₈	#124×C₅₃	1	2	yes	—
287	C₈	#124×C₅₃	1	2	yes	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	4	no	—
290	C₂²	—	4	2	no	—
291	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
292	C₂²	C₅₃×#529	1	2	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	4	4	no	—
295	C₄	—	4	4	no	—
296	C₄	—	4	2	no	—
297	C₄	C₅₃×#444	1	4	yes	—
298	C₄	C₄×#41×C₅₃	1	2	yes	—
299	C₄	C₂×#124×C₅₃	1	1	yes	characteristic
300	C₄	C₅₃×#532	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	8	2	no	—
303	C₂	C₄×#124×C₅₃	1	1	yes	characteristic
304	C₂	C₂×C₅₃×#444	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₅₃×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2960	C₂	semidirect
4. C₅₃×#1994	C₂	semidirect
6. C₅₃×#1938	C₂	semidirect
7. C₅₃×#1837	C₂	semidirect
8. C₅₃×#1765	C₂	semidirect
9. C₄×#97×C₅₃	C₂	semidirect
13. C₅₃×#488	C₄	semidirect
14. C₅₃×#487	C₄	semidirect
15. C₅₃×#486	C₄	semidirect
135. #21660	C₅₃	direct (N × H)
150. #2960	C₂×C₅₃	semidirect
152. #1994	C₂×C₅₃	semidirect
154. #1938	C₂×C₅₃	semidirect
155. #1837	C₂×C₅₃	semidirect
156. #1765	C₂×C₅₃	semidirect
157. C₄×#97	C₂×C₅₃	semidirect
165. #488	C₄×C₅₃	semidirect
166. #487	C₄×C₅₃	semidirect
167. #486	C₄×C₅₃	semidirect
186. C₅₃	#21660	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#3131	2
C₅₃×#2960	2
C₅₃×#2931	2
C₅₃×#1994	2
C₅₃×#1962	2
C₅₃×#1938	2
C₅₃×#1837	2
C₅₃×#1765	2
C₄×#97×C₅₃	2
#21660	53

Subgroups — C53×#21660 #27229172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21660 #27229172