Subgroups — C₅₃×#17899 #27225411

Order 13568.

646 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2989	C₂	1	2	yes	maximal, complemented
2	C₄×#135×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1594	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1593	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1590	C₂	1	4	yes	maximal, complemented
6	C₂×C₅₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1477	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₅₃	C₄	1	1	yes	characteristic
10	C₄×#41×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#446	C₂²	1	4	yes	—
12	C₄×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
13	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×SD₁₆×C₅₃	C₄	1	2	yes	—
15	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	—
16	C₄×D₈×C₅₃	C₄	1	1	yes	characteristic
17	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#436	C₂²	1	4	yes	—
19	C₅₃×#435	—	2	4	no	—
20	C₅₃×#432	—	2	2	no	—
21	C₅₃×#431	C₂²	1	1	yes	characteristic
22	C₅₃×#431	C₂²	1	1	yes	characteristic
23	C₅₃×#427	C₄	1	1	yes	characteristic
24	C₅₃×#423	—	2	2	no	—
25	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
26	C₅₃×#416	C₄	1	2	yes	—
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#334	C₂²	1	2	yes	—
29	C₅₃×#334	C₂²	1	2	yes	—
30	C₅₃×Q₈⋊C₈	C₂²	1	4	yes	—
31	C₅₃×D₄⋊C₈	C₂²	1	4	yes	—
32	C₅₃×#322	C₂²	1	2	yes	—
33	C₈²×C₅₃	C₂²	1	2	yes	—
34	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
35	#135×C₅₃	—	2	4	no	—
36	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₅₃	—	2	4	no	—
40	C₂²×C₈×C₅₃	—	2	2	no	—
41	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
44	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
45	C₄×D₄×C₅₃	—	2	2	no	—
46	C₄×D₄×C₅₃	—	2	2	no	—
47	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
48	C₄×D₄×C₅₃	C₂³	1	2	yes	—
49	#117×C₅₃	—	2	2	no	—
50	#117×C₅₃	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₅₃	—	2	2	no	—
52	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
53	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
54	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
55	#106×C₅₃	C₂×C₄	1	2	yes	—
56	C₄⋊C₈×C₅₃	—	2	4	no	—
57	C₄⋊C₈×C₅₃	—	2	2	no	—
58	C₄⋊C₈×C₅₃	—	2	1	no	—
59	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
60	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
61	#103×C₅₃	C₂×C₄	1	2	yes	—
62	#103×C₅₃	C₂×C₄	1	2	yes	—
63	#102×C₅₃	C₂×C₄	1	2	yes	—
64	#102×C₅₃	C₂×C₄	1	2	yes	—
65	C₂²⋊C₈×C₅₃	—	2	4	no	—
66	C₂²⋊C₈×C₅₃	—	2	1	no	—
67	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
68	#97×C₅₃	—	2	4	no	—
69	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₅₃	C₂³	1	2	yes	—
72	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₅₃	C₂³	1	2	yes	—
75	#41×C₅₃	—	2	4	no	—
76	#41×C₅₃	—	2	4	no	—
77	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
78	C₂×D₄×C₅₃	—	2	2	no	—
79	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₅₃	—	2	2	no	—
81	C₂²×C₄×C₅₃	—	2	2	no	—
82	C₂²×C₄×C₅₃	—	2	2	no	—
83	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
86	Q₁₆×C₅₃	—	2	2	no	—
87	SD₁₆×C₅₃	—	2	4	no	—
88	D₈×C₅₃	—	2	2	no	—
89	M₁₆×C₅₃	—	8	4	no	—
90	C₂×C₈×C₅₃	—	2	2	no	—
91	C₂×C₈×C₅₃	—	2	4	no	—
92	C₂×C₈×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	—	8	2	no	—
94	C₂×C₈×C₅₃	M₁₆	1	4	yes	complemented
95	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
100	C₄⋊C₄×C₅₃	—	2	2	no	—
101	C₄⋊C₄×C₅₃	—	2	2	no	—
102	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
103	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
104	C₂²⋊C₄×C₅₃	—	2	2	no	—
105	C₂²⋊C₄×C₅₃	—	2	2	no	—
106	C₂²⋊C₄×C₅₃	—	2	2	no	—
107	C₄²×C₅₃	—	2	2	no	—
108	C₄²×C₅₃	—	2	2	no	—
109	C₄²×C₅₃	—	2	2	no	—
110	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₅₃	#41	1	1	yes	characteristic
113	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
114	C₂³×C₅₃	—	2	2	no	—
115	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
116	Q₈×C₅₃	—	2	2	no	—
117	Q₈×C₅₃	—	2	2	no	—
118	D₄×C₅₃	—	2	2	no	—
119	D₄×C₅₃	—	2	2	no	—
120	D₄×C₅₃	—	4	4	no	—
121	C₂×C₄×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	4	1	no	—
126	C₂×C₄×C₅₃	—	4	2	no	—
127	C₂×C₄×C₅₃	—	4	2	no	—
128	C₂×C₄×C₅₃	—	4	4	no	—
129	C₂×C₄×C₅₃	—	8	2	no	—
130	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
134	C₂×C₄×C₅₃	#131	1	2	yes	—
135	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
138	C₈×C₅₃	—	2	1	no	—
139	C₈×C₅₃	—	2	1	no	—
140	C₈×C₅₃	—	2	4	no	—
141	C₈×C₅₃	—	4	2	no	—
142	C₈×C₅₃	—	4	4	no	—
143	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
144	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
145	#17899	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂²×C₅₃	—	2	1	no	—
147	C₂²×C₅₃	—	2	1	no	—
148	C₂²×C₅₃	—	4	2	no	—
149	C₂²×C₅₃	—	4	2	no	—
150	C₂²×C₅₃	—	4	2	no	—
151	C₂²×C₅₃	#436	1	2	yes	—
152	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
153	C₄×C₅₃	—	2	1	no	—
154	C₄×C₅₃	—	2	1	no	—
155	C₄×C₅₃	—	2	1	no	—
156	C₄×C₅₃	—	4	2	no	—
157	C₄×C₅₃	—	4	2	no	—
158	C₄×C₅₃	—	4	2	no	—
159	C₄×C₅₃	#436	1	2	yes	—
160	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
161	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
162	C₄×C₅₃	#577	1	2	yes	—
163	#2989	C₂×C₅₃	1	2	yes	complemented
164	C₄×#135	C₂×C₅₃	1	1	yes	characteristic
165	#1594	C₂×C₅₃	1	2	yes	complemented
166	#1593	C₂×C₅₃	1	2	yes	complemented
167	#1590	C₂×C₅₃	1	4	yes	complemented
168	C₂×#334	C₂×C₅₃	1	1	yes	characteristic, complemented
169	#1477	C₂×C₅₃	1	2	yes	complemented
170	#1450	C₂×C₅₃	1	1	yes	characteristic, complemented
171	C₂×C₅₃	—	2	2	no	—
172	C₂×C₅₃	—	8	2	no	—
173	C₂×C₅₃	C₂×#436	1	1	yes	characteristic
174	C₂×C₅₃	#2952	1	1	yes	characteristic
175	C₂×C₅₃	#2957	1	1	yes	characteristic
176	C₂×#135	C₄×C₅₃	1	1	yes	characteristic
177	C₄×#41	C₂²×C₅₃	1	2	yes	—
178	#446	C₂²×C₅₃	1	4	yes	—
179	C₄×Q₁₆	C₄×C₅₃	1	1	yes	characteristic
180	C₄×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
181	C₄×SD₁₆	C₄×C₅₃	1	2	yes	—
182	C₄×SD₁₆	C₂²×C₅₃	1	2	yes	—
183	C₄×D₈	C₄×C₅₃	1	1	yes	characteristic
184	C₄×D₈	C₂²×C₅₃	1	1	yes	characteristic
185	#436	C₂²×C₅₃	1	4	yes	—
186	#435	—	2	4	no	—
187	#432	—	2	2	no	—
188	#431	C₂²×C₅₃	1	1	yes	characteristic
189	#431	C₂²×C₅₃	1	1	yes	characteristic
190	#427	C₄×C₅₃	1	1	yes	characteristic
191	#423	—	2	2	no	—
192	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
193	#416	C₄×C₅₃	1	2	yes	—
194	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
195	#334	C₂²×C₅₃	1	2	yes	—
196	#334	C₂²×C₅₃	1	2	yes	—
197	Q₈⋊C₈	C₂²×C₅₃	1	4	yes	—
198	D₄⋊C₈	C₂²×C₅₃	1	4	yes	—
199	#322	C₂²×C₅₃	1	2	yes	—
200	C₈²	C₂²×C₅₃	1	2	yes	—
201	C₅₃	#17899	1	1	yes	characteristic, direct product factor, Sylow
202	C₂×#41	C₂×C₄×C₅₃	1	2	yes	—
203	#135	—	2	4	no	—
204	C₂×Q₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
205	C₂×SD₁₆	C₂×C₄×C₅₃	1	2	yes	—
206	C₂×D₈	C₂×C₄×C₅₃	1	1	yes	characteristic
207	C₂×M₁₆	—	2	4	no	—
208	C₂²×C₈	—	2	2	no	—
209	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
210	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
211	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	—
212	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
213	C₄×D₄	—	2	2	no	—
214	C₄×D₄	—	2	2	no	—
215	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
216	C₄×D₄	C₂³×C₅₃	1	2	yes	—
217	#117	—	2	2	no	—
218	#117	C₂×C₄×C₅₃	1	2	yes	—
219	C₂×C₄²	—	2	2	no	—
220	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
221	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
222	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
223	#106	C₂×C₄×C₅₃	1	2	yes	—
224	C₄⋊C₈	—	2	4	no	—
225	C₄⋊C₈	—	2	2	no	—
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
228	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
229	#103	C₂×C₄×C₅₃	1	2	yes	—
230	#103	C₂×C₄×C₅₃	1	2	yes	—
231	#102	C₂×C₄×C₅₃	1	2	yes	—
232	#102	C₂×C₄×C₅₃	1	2	yes	—
233	C₂²⋊C₈	—	2	1	no	—
234	C₂²⋊C₈	—	2	4	no	—
235	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
236	#97	—	2	4	no	—
237	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
238	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
239	C₄×C₈	C₂³×C₅₃	1	2	yes	—
240	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
241	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
242	C₄×C₈	C₂³×C₅₃	1	2	yes	—
243	#41	—	2	4	no	—
244	#41	—	2	4	no	—
245	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
253	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
254	Q₁₆	—	2	2	no	—
255	SD₁₆	—	2	4	no	—
256	D₈	—	2	2	no	—
257	M₁₆	—	8	4	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	8	2	no	—
262	C₂×C₈	M₁₆×C₅₃	1	4	yes	complemented
263	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
264	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
265	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
266	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
267	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
271	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
272	C₂²⋊C₄	—	2	2	no	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₂²⋊C₄	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
279	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
280	C₄²	#41×C₅₃	1	1	yes	characteristic
281	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
282	C₂³	—	2	2	no	—
283	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
284	Q₈	—	2	2	no	—
285	Q₈	—	2	2	no	—
286	D₄	—	2	2	no	—
287	D₄	—	2	2	no	—
288	D₄	—	4	4	no	—
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	—	4	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	8	2	no	—
298	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
299	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
300	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
301	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
302	C₂×C₄	#131×C₅₃	1	2	yes	—
303	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
305	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	2	4	no	—
309	C₈	—	4	2	no	—
310	C₈	—	4	4	no	—
311	C₈	C₂×M₁₆×C₅₃	1	2	yes	—
312	C₈	C₂×M₁₆×C₅₃	1	2	yes	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	C₅₃×#436	1	2	yes	—
319	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	2	no	—
325	C₄	—	4	2	no	—
326	C₄	C₅₃×#436	1	2	yes	—
327	C₄	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
328	C₄	C₂×#131×C₅₃	1	1	yes	characteristic
329	C₄	C₅₃×#577	1	2	yes	—
330	C₂	—	2	2	no	—
331	C₂	—	8	2	no	—
332	C₂	C₂×C₅₃×#436	1	1	yes	characteristic
333	C₂	C₅₃×#2952	1	1	yes	characteristic
334	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	C₂×#135×C₅₃	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₅₃×#431 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#2989	C₂	semidirect
3. C₅₃×#1594	C₂	semidirect
4. C₅₃×#1593	C₂	semidirect
5. C₅₃×#1590	C₂	semidirect
6. C₂×C₅₃×#334	C₂	semidirect
7. C₅₃×#1477	C₂	semidirect
8. C₅₃×#1450	C₂	semidirect
94. C₂×C₈×C₅₃	M₁₆	semidirect
145. #17899	C₅₃	direct (N × H)
163. #2989	C₂×C₅₃	semidirect
165. #1594	C₂×C₅₃	semidirect
166. #1593	C₂×C₅₃	semidirect
167. #1590	C₂×C₅₃	semidirect
168. C₂×#334	C₂×C₅₃	semidirect
169. #1477	C₂×C₅₃	semidirect
170. #1450	C₂×C₅₃	semidirect
201. C₅₃	#17899	direct (N × H)
262. C₂×C₈	M₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2989	2
C₄×#135×C₅₃	2
C₅₃×#1594	2
C₅₃×#1593	2
C₅₃×#1590	2
C₂×C₅₃×#334	2
C₅₃×#1477	2
C₅₃×#1450	2
#17899	53

Subgroups — C53×#17899 #27225411

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17899 #27225411