Subgroups — C₅₃×#18066 #27225578

Order 13568.

722 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3169	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2935	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1589	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1581	C₂	1	4	yes	maximal, complemented
5	C₈×D₈×C₅₃	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1480	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#131×C₅₃	—	2	2	no	—
10	C₅₃×#532	C₂²	1	2	yes	—
11	C₅₃×#500	C₄	1	1	yes	characteristic
12	C₅₃×#493	C₄	1	1	yes	characteristic
13	C₅₃×#486	C₄	1	2	yes	—
14	C₅₃×#466	C₄	1	2	yes	—
15	C₅₃×#466	C₄	1	2	yes	—
16	C₄×D₈×C₅₃	C₂²	1	2	yes	—
17	C₄×D₈×C₅₃	C₂²	1	2	yes	—
18	C₅₃×#436	C₂²	1	4	yes	—
19	C₅₃×#435	—	2	4	no	—
20	C₈×D₄×C₅₃	C₂²	1	4	yes	—
21	C₅₃×#431	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
23	C₅₃×#408	—	2	2	no	—
24	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#335	C₂²	1	1	yes	characteristic
27	C₅₃×#335	C₂²	1	2	yes	—
28	C₅₃×#335	C₂²	1	1	yes	characteristic
29	C₅₃×D₄⋊C₈	C₂²	1	4	yes	—
30	C₅₃×D₄⋊C₈	C₂²	1	4	yes	—
31	C₅₃×#322	C₂²	1	2	yes	—
32	C₈²×C₅₃	C₂²	1	2	yes	—
33	C₂×#41×C₅₃	—	2	2	no	—
34	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
36	#131×C₅₃	—	4	8	no	—
37	C₂×M₁₆×C₅₃	—	2	4	no	—
38	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
39	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	—	2	4	no	—
41	C₂²×C₈×C₅₃	—	2	2	no	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
45	#127×C₅₃	C₂×C₄	1	2	yes	—
46	#124×C₅₃	—	2	2	no	—
47	#121×C₅₃	—	2	2	no	—
48	#121×C₅₃	C₂×C₄	1	4	yes	—
49	C₄×D₄×C₅₃	—	2	2	no	—
50	C₄×D₄×C₅₃	C₂³	1	4	yes	—
51	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
52	#107×C₅₃	C₂×C₄	1	2	yes	—
53	#107×C₅₃	C₂×C₄	1	2	yes	—
54	C₄⋊C₈×C₅₃	—	2	2	no	—
55	C₄⋊C₈×C₅₃	—	2	1	no	—
56	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
57	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
58	#102×C₅₃	C₂×C₄	1	4	yes	—
59	#102×C₅₃	C₂×C₄	1	4	yes	—
60	C₂²⋊C₈×C₅₃	—	2	4	no	—
61	C₂²⋊C₈×C₅₃	—	2	4	no	—
62	C₂²⋊C₈×C₅₃	—	2	1	no	—
63	#97×C₅₃	D₄	1	2	yes	—
64	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₅₃	C₂³	1	2	yes	—
67	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₅₃	C₂³	1	2	yes	—
70	#41×C₅₃	—	4	4	no	—
71	#41×C₅₃	—	4	4	no	—
72	C₂×Q₈×C₅₃	—	2	2	no	—
73	C₂×D₄×C₅₃	—	2	4	no	—
74	C₂×D₄×C₅₃	—	2	2	no	—
75	C₂×D₄×C₅₃	—	2	2	no	—
76	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₅₃	—	2	4	no	—
78	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
81	D₈×C₅₃	—	2	4	no	—
82	D₈×C₅₃	—	4	2	no	—
83	M₁₆×C₅₃	—	2	2	no	—
84	M₁₆×C₅₃	—	4	8	no	—
85	M₁₆×C₅₃	D₈	1	4	yes	complemented
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	1	no	—
91	C₂×C₈×C₅₃	—	4	4	no	—
92	C₂×C₈×C₅₃	—	4	8	no	—
93	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
99	C₄⋊C₄×C₅₃	—	2	2	no	—
100	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
101	C₂²⋊C₄×C₅₃	—	2	4	no	—
102	C₂²⋊C₄×C₅₃	—	2	4	no	—
103	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₅₃	#41	1	1	yes	characteristic
106	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
107	C₂³×C₅₃	—	2	4	no	—
108	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
109	Q₈×C₅₃	—	4	4	no	—
110	D₄×C₅₃	—	2	4	no	—
111	D₄×C₅₃	—	2	4	no	—
112	D₄×C₅₃	—	4	8	no	—
113	D₄×C₅₃	—	4	4	no	—
114	D₄×C₅₃	—	8	2	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	—	2	1	no	—
117	C₂×C₄×C₅₃	—	2	4	no	—
118	C₂×C₄×C₅₃	—	4	1	no	—
119	C₂×C₄×C₅₃	—	4	4	no	—
120	C₂×C₄×C₅₃	—	4	4	no	—
121	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₅₃	#135	1	2	yes	—
126	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
129	C₈×C₅₃	—	2	1	no	—
130	C₈×C₅₃	—	2	1	no	—
131	C₈×C₅₃	—	2	1	no	—
132	C₈×C₅₃	—	2	2	no	—
133	C₈×C₅₃	—	4	2	no	—
134	C₈×C₅₃	—	4	2	no	—
135	C₈×C₅₃	—	4	2	no	—
136	C₈×C₅₃	#131	1	2	yes	complemented
137	C₈×C₅₃	C₂×D₈	1	4	yes	—
138	#18066	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂²×C₅₃	—	2	1	no	—
140	C₂²×C₅₃	—	2	1	no	—
141	C₂²×C₅₃	—	4	4	no	—
142	C₂²×C₅₃	—	4	4	no	—
143	C₂²×C₅₃	—	4	4	no	—
144	C₂²×C₅₃	C₄×D₈	1	2	yes	—
145	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
146	C₄×C₅₃	—	2	1	no	—
147	C₄×C₅₃	—	2	1	no	—
148	C₄×C₅₃	—	2	1	no	—
149	C₄×C₅₃	—	4	4	no	—
150	C₄×C₅₃	C₄×D₈	1	2	yes	—
151	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
152	C₄×C₅₃	#568	1	1	yes	characteristic
153	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
154	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
155	#3169	C₂×C₅₃	1	1	yes	characteristic
156	#2935	C₂×C₅₃	1	2	yes	complemented
157	#1589	C₂×C₅₃	1	2	yes	complemented
158	#1581	C₂×C₅₃	1	4	yes	complemented
159	C₈×D₈	C₂×C₅₃	1	2	yes	complemented
160	C₂×#335	C₂×C₅₃	1	1	yes	characteristic, complemented
161	#1480	C₂×C₅₃	1	2	yes	complemented
162	C₈×M₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
163	C₂×C₅₃	—	2	2	no	—
164	C₂×C₅₃	—	8	4	no	—
165	C₂×C₅₃	#2935	1	1	yes	characteristic
166	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
167	C₂×C₅₃	#2957	1	1	yes	characteristic
168	C₂×#131	—	2	2	no	—
169	#532	C₂²×C₅₃	1	2	yes	—
170	#500	C₄×C₅₃	1	1	yes	characteristic
171	#493	C₄×C₅₃	1	1	yes	characteristic
172	#486	C₄×C₅₃	1	2	yes	—
173	#466	C₄×C₅₃	1	2	yes	—
174	#466	C₄×C₅₃	1	2	yes	—
175	C₄×D₈	C₂²×C₅₃	1	2	yes	—
176	C₄×D₈	C₂²×C₅₃	1	2	yes	—
177	#436	C₂²×C₅₃	1	4	yes	—
178	#435	—	2	4	no	—
179	C₈×D₄	C₂²×C₅₃	1	4	yes	—
180	#431	C₂²×C₅₃	1	1	yes	characteristic
181	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
182	#408	—	2	2	no	—
183	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
184	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
185	#335	C₂²×C₅₃	1	1	yes	characteristic
186	#335	C₂²×C₅₃	1	2	yes	—
187	#335	C₂²×C₅₃	1	1	yes	characteristic
188	D₄⋊C₈	C₂²×C₅₃	1	4	yes	—
189	D₄⋊C₈	C₂²×C₅₃	1	4	yes	—
190	#322	C₂²×C₅₃	1	2	yes	—
191	C₈²	C₂²×C₅₃	1	2	yes	—
192	C₅₃	#18066	1	1	yes	characteristic, direct product factor, Sylow
193	C₂×#41	—	2	2	no	—
194	C₂×D₈	C₂×C₄×C₅₃	1	2	yes	—
195	C₂×D₈	C₂×C₄×C₅₃	1	2	yes	—
196	#131	—	4	8	no	—
197	C₂×M₁₆	—	2	4	no	—
198	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
199	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	—	2	4	no	—
202	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
203	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
204	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
205	#127	C₂×C₄×C₅₃	1	2	yes	—
206	#124	—	2	2	no	—
207	#121	—	2	2	no	—
208	#121	C₂×C₄×C₅₃	1	4	yes	—
209	C₄×D₄	—	2	2	no	—
210	C₄×D₄	C₂³×C₅₃	1	4	yes	—
211	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
212	#107	C₂×C₄×C₅₃	1	2	yes	—
213	#107	C₂×C₄×C₅₃	1	2	yes	—
214	C₄⋊C₈	—	2	2	no	—
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
217	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
218	#102	C₂×C₄×C₅₃	1	4	yes	—
219	#102	C₂×C₄×C₅₃	1	4	yes	—
220	C₂²⋊C₈	—	2	1	no	—
221	C₂²⋊C₈	—	2	4	no	—
222	C₂²⋊C₈	—	2	4	no	—
223	#97	D₄×C₅₃	1	2	yes	—
224	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
225	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
226	C₄×C₈	C₂³×C₅₃	1	2	yes	—
227	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
228	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
229	C₄×C₈	C₂³×C₅₃	1	2	yes	—
230	#41	—	4	4	no	—
231	#41	—	4	4	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×D₄	—	2	4	no	—
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
240	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
241	D₈	—	2	4	no	—
242	D₈	—	4	2	no	—
243	M₁₆	—	2	2	no	—
244	M₁₆	—	4	8	no	—
245	M₁₆	D₈×C₅₃	1	4	yes	complemented
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	4	4	no	—
252	C₂×C₈	—	4	8	no	—
253	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
254	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
255	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
256	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
257	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
258	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
261	C₂²⋊C₄	—	2	4	no	—
262	C₂²⋊C₄	—	2	4	no	—
263	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
264	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
265	C₄²	#41×C₅₃	1	1	yes	characteristic
266	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
267	C₂³	—	2	4	no	—
268	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
269	Q₈	—	4	4	no	—
270	D₄	—	2	4	no	—
271	D₄	—	2	4	no	—
272	D₄	—	4	8	no	—
273	D₄	—	4	4	no	—
274	D₄	—	8	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
282	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
283	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
284	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
285	C₂×C₄	#135×C₅₃	1	2	yes	—
286	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
288	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	2	2	no	—
293	C₈	—	4	2	no	—
294	C₈	—	4	2	no	—
295	C₈	—	4	2	no	—
296	C₈	#131×C₅₃	1	2	yes	complemented
297	C₈	C₂×D₈×C₅₃	1	4	yes	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	4	4	no	—
301	C₂²	—	4	4	no	—
302	C₂²	—	4	4	no	—
303	C₂²	C₄×D₈×C₅₃	1	2	yes	—
304	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	4	4	no	—
309	C₄	C₄×D₈×C₅₃	1	2	yes	—
310	C₄	C₂×#131×C₅₃	1	1	yes	characteristic
311	C₄	C₅₃×#568	1	1	yes	characteristic
312	C₄	C₂²×D₈×C₅₃	1	1	yes	characteristic
313	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	8	4	no	—
316	C₂	C₅₃×#2935	1	1	yes	characteristic
317	C₂	C₂×C₄×D₈×C₅₃	1	1	yes	characteristic
318	C₂	C₅₃×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₅₃×#3169	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₄×M₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2935	C₂	semidirect
3. C₅₃×#1589	C₂	semidirect
4. C₅₃×#1581	C₂	semidirect
5. C₈×D₈×C₅₃	C₂	semidirect
6. C₂×C₅₃×#335	C₂	semidirect
7. C₅₃×#1480	C₂	semidirect
8. C₈×M₁₆×C₅₃	C₂	semidirect
85. M₁₆×C₅₃	D₈	semidirect
136. C₈×C₅₃	#131	semidirect
138. #18066	C₅₃	direct (N × H)
156. #2935	C₂×C₅₃	semidirect
157. #1589	C₂×C₅₃	semidirect
158. #1581	C₂×C₅₃	semidirect
159. C₈×D₈	C₂×C₅₃	semidirect
160. C₂×#335	C₂×C₅₃	semidirect
161. #1480	C₂×C₅₃	semidirect
162. C₈×M₁₆	C₂×C₅₃	semidirect
192. C₅₃	#18066	direct (N × H)
245. M₁₆	D₈×C₅₃	semidirect
296. C₈	#131×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3169	2
C₅₃×#2935	2
C₅₃×#1589	2
C₅₃×#1581	2
C₈×D₈×C₅₃	2
C₂×C₅₃×#335	2
C₅₃×#1480	2
C₈×M₁₆×C₅₃	2
#18066	53

Subgroups — C53×#18066 #27225578

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#18066 #27225578