Subgroups — C₅₃×#17824 #27225336

Order 13568.

572 conjugacy classes of subgroups.

The subgroup lattice has 324 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
2	C₅₃×#2922	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1574	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1573	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1572	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1570	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1569	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1566	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#446	—	2	2	no	—
12	C₅₃×#436	—	2	2	no	—
13	C₅₃×#435	—	2	2	no	—
14	C₅₃×#435	—	2	2	no	—
15	C₅₃×#433	C₄	1	4	yes	—
16	C₅₃×#432	C₂²	1	2	yes	—
17	C₅₃×#432	C₂²	1	2	yes	—
18	C₅₃×#431	C₂²	1	2	yes	—
19	C₅₃×#431	C₂²	1	2	yes	—
20	C₅₃×#431	C₂²	1	2	yes	—
21	C₅₃×#424	C₄	1	2	yes	—
22	C₅₃×#423	C₂²	1	2	yes	—
23	C₅₃×#423	C₂²	1	2	yes	—
24	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
25	C₅₃×#408	C₄	1	2	yes	—
26	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#335	C₂²	1	2	yes	—
29	C₅₃×#335	C₂²	1	2	yes	—
30	C₅₃×#335	C₂²	1	2	yes	—
31	C₅₃×#334	C₂²	1	2	yes	—
32	C₅₃×#334	C₂²	1	1	yes	characteristic
33	C₅₃×#334	C₂²	1	1	yes	characteristic
34	C₅₃×#334	C₂²	1	1	yes	characteristic
35	C₅₃×#334	C₂²	1	1	yes	characteristic
36	C₅₃×#334	C₂²	1	2	yes	—
37	C₅₃×#334	C₂²	1	2	yes	—
38	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₅₃	—	2	2	no	—
40	C₂×M₁₆×C₅₃	—	2	2	no	—
41	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₅₃	—	2	2	no	—
43	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
44	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
45	C₄×Q₈×C₅₃	Q₈	1	1	yes	characteristic
46	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
47	C₄×D₄×C₅₃	D₄	1	2	yes	—
48	C₄×D₄×C₅₃	Q₈	1	2	yes	—
49	C₄×D₄×C₅₃	Q₈	1	1	yes	characteristic
50	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
51	#117×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
54	C₄⋊C₈×C₅₃	—	2	2	no	—
55	C₄⋊C₈×C₅₃	—	2	2	no	—
56	C₄⋊C₈×C₅₃	—	2	2	no	—
57	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
58	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
59	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
60	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
61	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
62	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
63	C₂²⋊C₈×C₅₃	—	2	2	no	—
64	C₂²⋊C₈×C₅₃	—	2	2	no	—
65	C₂²⋊C₈×C₅₃	—	2	2	no	—
66	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
67	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
68	#97×C₅₃	—	2	2	no	—
69	#97×C₅₃	—	2	2	no	—
70	#97×C₅₃	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₅₃	—	2	2	no	—
72	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
73	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₅₃	C₂³	1	2	yes	—
76	#41×C₅₃	—	4	2	no	—
77	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂×D₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
80	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
86	M₁₆×C₅₃	—	4	2	no	—
87	M₁₆×C₅₃	—	8	2	no	—
88	M₁₆×C₅₃	—	8	2	no	—
89	C₂×C₈×C₅₃	—	2	2	no	—
90	C₂×C₈×C₅₃	—	2	2	no	—
91	C₂×C₈×C₅₃	—	2	2	no	—
92	C₂×C₈×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	—	4	2	no	—
94	C₂×C₈×C₅₃	—	8	2	no	—
95	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
98	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
99	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
100	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
101	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
102	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
103	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
104	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
105	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
107	C₄²×C₅₃	C₂×D₄	1	2	yes	—
108	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
110	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
111	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
112	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	Q₈×C₅₃	—	4	1	no	—
115	D₄×C₅₃	—	4	2	no	—
116	D₄×C₅₃	—	4	1	no	—
117	C₂×C₄×C₅₃	—	2	4	no	—
118	C₂×C₄×C₅₃	—	2	2	no	—
119	C₂×C₄×C₅₃	—	2	2	no	—
120	C₂×C₄×C₅₃	—	2	2	no	—
121	C₂×C₄×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	—	4	1	no	—
123	C₂×C₄×C₅₃	—	4	2	no	—
124	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
133	C₈×C₅₃	—	2	2	no	—
134	C₈×C₅₃	—	2	2	no	—
135	C₈×C₅₃	—	2	4	no	—
136	C₈×C₅₃	—	4	2	no	—
137	C₈×C₅₃	—	4	2	no	—
138	C₈×C₅₃	—	4	2	no	—
139	C₈×C₅₃	—	4	2	no	—
140	#17824	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂²×C₅₃	—	2	2	no	—
142	C₂²×C₅₃	—	2	1	no	—
143	C₂²×C₅₃	—	2	2	no	—
144	C₂²×C₅₃	—	2	1	no	—
145	C₂²×C₅₃	—	2	2	no	—
146	C₂²×C₅₃	—	2	1	no	—
147	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
148	C₄×C₅₃	—	2	2	no	—
149	C₄×C₅₃	—	2	1	no	—
150	C₄×C₅₃	—	2	2	no	—
151	C₄×C₅₃	—	2	1	no	—
152	C₄×C₅₃	—	2	1	no	—
153	C₄×C₅₃	—	2	1	no	—
154	C₄×C₅₃	—	2	2	no	—
155	C₄×C₅₃	#568	1	2	yes	—
156	C₄×C₅₃	#577	1	2	yes	—
157	#2989	C₂×C₅₃	1	2	yes	—
158	#2922	C₂×C₅₃	1	1	yes	characteristic
159	#1574	C₂×C₅₃	1	2	yes	complemented
160	#1573	C₂×C₅₃	1	2	yes	complemented
161	#1572	C₂×C₅₃	1	2	yes	complemented
162	#1570	C₂×C₅₃	1	2	yes	complemented
163	#1569	C₂×C₅₃	1	2	yes	complemented
164	#1568	C₂×C₅₃	1	1	yes	characteristic, complemented
165	#1566	C₂×C₅₃	1	1	yes	characteristic, complemented
166	C₂×C₅₃	—	4	2	no	—
167	C₂×C₅₃	—	4	1	no	—
168	C₂×C₅₃	#2905	1	1	yes	characteristic
169	C₂×C₅₃	#2912	1	1	yes	characteristic
170	C₂×C₅₃	#2916	1	1	yes	characteristic
171	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
172	#446	—	2	2	no	—
173	#436	—	2	2	no	—
174	#435	—	2	2	no	—
175	#435	—	2	2	no	—
176	#433	C₄×C₅₃	1	4	yes	—
177	#432	C₂²×C₅₃	1	2	yes	—
178	#432	C₂²×C₅₃	1	2	yes	—
179	#431	C₂²×C₅₃	1	2	yes	—
180	#431	C₂²×C₅₃	1	2	yes	—
181	#431	C₂²×C₅₃	1	2	yes	—
182	#424	C₄×C₅₃	1	2	yes	—
183	#423	C₂²×C₅₃	1	2	yes	—
184	#423	C₂²×C₅₃	1	2	yes	—
185	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
186	#408	C₄×C₅₃	1	2	yes	—
187	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
188	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
189	#335	C₂²×C₅₃	1	2	yes	—
190	#335	C₂²×C₅₃	1	2	yes	—
191	#335	C₂²×C₅₃	1	2	yes	—
192	#334	C₂²×C₅₃	1	2	yes	—
193	#334	C₂²×C₅₃	1	1	yes	characteristic
194	#334	C₂²×C₅₃	1	1	yes	characteristic
195	#334	C₂²×C₅₃	1	1	yes	characteristic
196	#334	C₂²×C₅₃	1	1	yes	characteristic
197	#334	C₂²×C₅₃	1	2	yes	—
198	#334	C₂²×C₅₃	1	2	yes	—
199	C₅₃	#17824	1	1	yes	characteristic, direct product factor, Sylow
200	C₂×#41	C₂×C₄×C₅₃	1	1	yes	characteristic
201	C₂×M₁₆	—	2	2	no	—
202	C₂×M₁₆	—	2	2	no	—
203	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
206	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
207	C₄×Q₈	Q₈×C₅₃	1	1	yes	characteristic
208	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
209	C₄×D₄	D₄×C₅₃	1	2	yes	—
210	C₄×D₄	Q₈×C₅₃	1	1	yes	characteristic
211	C₄×D₄	Q₈×C₅₃	1	2	yes	—
212	#117	C₂×C₄×C₅₃	1	1	yes	characteristic
213	#117	C₂×C₄×C₅₃	1	2	yes	—
214	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
215	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
216	C₄⋊C₈	—	2	2	no	—
217	C₄⋊C₈	—	2	2	no	—
218	C₄⋊C₈	—	2	2	no	—
219	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
220	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
221	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
222	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
223	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
224	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
225	C₂²⋊C₈	—	2	2	no	—
226	C₂²⋊C₈	—	2	2	no	—
227	C₂²⋊C₈	—	2	2	no	—
228	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
229	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
230	#97	—	2	2	no	—
231	#97	—	2	2	no	—
232	#97	C₂×C₄×C₅₃	1	2	yes	—
233	C₄×C₈	—	2	2	no	—
234	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
235	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
236	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
237	C₄×C₈	C₂³×C₅₃	1	2	yes	—
238	#41	—	4	2	no	—
239	C₂×Q₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
240	C₂×D₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
241	C₂×D₄	C₄⋊C₄×C₅₃	1	2	yes	—
242	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
243	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
245	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
246	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
247	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
248	M₁₆	—	4	2	no	—
249	M₁₆	—	8	2	no	—
250	M₁₆	—	8	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	—	8	2	no	—
257	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
258	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
259	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
260	C₄⋊C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
261	C₄⋊C₄	C₄⋊C₄×C₅₃	1	2	yes	—
262	C₄⋊C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
263	C₄⋊C₄	C₄⋊C₄×C₅₃	1	2	yes	—
264	C₂²⋊C₄	C₄⋊C₄×C₅₃	1	2	yes	—
265	C₂²⋊C₄	C₄⋊C₄×C₅₃	1	2	yes	—
266	C₂²⋊C₄	C₄⋊C₄×C₅₃	1	2	yes	—
267	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
268	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
269	C₄²	C₂×D₄×C₅₃	1	2	yes	—
270	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
271	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
272	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
273	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
274	C₂³	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
275	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
276	Q₈	—	4	1	no	—
277	D₄	—	4	1	no	—
278	D₄	—	4	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
288	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
289	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
290	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
291	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
292	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
293	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
294	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	2	4	no	—
298	C₈	—	4	2	no	—
299	C₈	—	4	2	no	—
300	C₈	—	4	2	no	—
301	C₈	—	4	2	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	1	no	—
308	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	2	no	—
316	C₄	C₅₃×#568	1	2	yes	—
317	C₄	C₅₃×#577	1	2	yes	—
318	C₂	—	4	1	no	—
319	C₂	—	4	2	no	—
320	C₂	C₅₃×#2905	1	1	yes	characteristic
321	C₂	C₅₃×#2912	1	1	yes	characteristic
322	C₂	C₅₃×#2916	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	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₅₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₄×#41×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1574	C₂	semidirect
4. C₅₃×#1573	C₂	semidirect
5. C₅₃×#1572	C₂	semidirect
6. C₅₃×#1570	C₂	semidirect
7. C₅₃×#1569	C₂	semidirect
8. C₅₃×#1568	C₂	semidirect
9. C₅₃×#1566	C₂	semidirect
140. #17824	C₅₃	direct (N × H)
159. #1574	C₂×C₅₃	semidirect
160. #1573	C₂×C₅₃	semidirect
161. #1572	C₂×C₅₃	semidirect
162. #1570	C₂×C₅₃	semidirect
163. #1569	C₂×C₅₃	semidirect
164. #1568	C₂×C₅₃	semidirect
165. #1566	C₂×C₅₃	semidirect
199. C₅₃	#17824	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×#2989	2
C₅₃×#2922	2
C₅₃×#1574	2
C₅₃×#1573	2
C₅₃×#1572	2
C₅₃×#1570	2
C₅₃×#1569	2
C₅₃×#1568	2
C₅₃×#1566	2
#17824	53

Subgroups — C53×#17824 #27225336

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17824 #27225336