Subgroups — C₅₃×#19188 #27226700

Order 13568.

714 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3226	C₂	1	2	yes	maximal
2	C₂×C₅₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2931	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1689	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1669	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1546	C₂	1	2	yes	maximal, complemented
8	C₂²×Q₁₆×C₅₃	—	2	1	no	—
9	C₂²×Q₁₆×C₅₃	—	2	1	no	—
10	C₂×#131×C₅₃	—	2	1	no	—
11	C₅₃×#532	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#500	—	2	2	no	—
14	C₅₃×#494	—	2	1	no	—
15	C₅₃×#494	—	2	1	no	—
16	C₅₃×#494	C₂²	1	2	yes	—
17	C₅₃×#494	C₂²	1	2	yes	—
18	C₅₃×#486	—	2	2	no	—
19	C₅₃×#480	—	2	4	no	—
20	C₅₃×#474	C₂²	1	8	yes	—
21	C₅₃×#436	—	2	1	no	—
22	C₅₃×#436	C₂²	1	2	yes	—
23	C₈×D₄×C₅₃	—	2	1	no	—
24	C₈×D₄×C₅₃	C₂²	1	2	yes	—
25	C₅₃×#431	C₂²	1	2	yes	—
26	C₅₃×#423	C₂²	1	1	yes	characteristic
27	C₅₃×#408	—	2	1	no	—
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#335	C₂²	1	2	yes	—
30	C₅₃×#334	C₂²	1	2	yes	—
31	C₅₃×#332	C₂²	1	8	yes	—
32	C₂×#41×C₅₃	—	2	1	no	—
33	C₂²×Q₈×C₅₃	—	2	2	no	—
34	C₂×Q₁₆×C₅₃	—	2	2	no	—
35	C₂×Q₁₆×C₅₃	—	2	2	no	—
36	C₂×Q₁₆×C₅₃	—	2	2	no	—
37	C₂×Q₁₆×C₅₃	—	2	2	no	—
38	C₂×Q₁₆×C₅₃	—	4	4	no	—
39	C₂×Q₁₆×C₅₃	—	4	4	no	—
40	#131×C₅₃	—	4	4	no	—
41	C₂×M₁₆×C₅₃	—	2	2	no	—
42	C₂×M₁₆×C₅₃	—	2	1	no	—
43	C₂²×C₈×C₅₃	—	2	2	no	—
44	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
46	C₄⋊Q₈×C₅₃	—	2	2	no	—
47	C₄⋊Q₈×C₅₃	—	2	2	no	—
48	C₄⋊Q₈×C₅₃	—	2	2	no	—
49	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
50	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
51	#127×C₅₃	D₄	1	1	yes	characteristic
52	#124×C₅₃	—	2	1	no	—
53	#121×C₅₃	—	2	1	no	—
54	#121×C₅₃	D₄	1	2	yes	—
55	C₄×D₄×C₅₃	—	2	1	no	—
56	C₄×D₄×C₅₃	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
58	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
59	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
60	#107×C₅₃	—	2	8	no	—
61	C₄⋊C₈×C₅₃	—	2	1	no	—
62	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
63	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
64	#102×C₅₃	—	2	8	no	—
65	C₂²⋊C₈×C₅₃	—	2	2	no	—
66	C₂²⋊C₈×C₅₃	—	2	2	no	—
67	C₂²⋊C₈×C₅₃	—	2	2	no	—
68	C₄×C₈×C₅₃	—	2	2	no	—
69	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
71	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
73	#41×C₅₃	—	4	2	no	—
74	#41×C₅₃	—	4	2	no	—
75	C₂×Q₈×C₅₃	—	2	1	no	—
76	C₂×Q₈×C₅₃	—	2	4	no	—
77	C₂×Q₈×C₅₃	—	4	2	no	—
78	C₂×Q₈×C₅₃	—	4	4	no	—
79	C₂×Q₈×C₅₃	—	4	4	no	—
80	C₂×D₄×C₅₃	—	2	2	no	—
81	C₂×D₄×C₅₃	—	2	1	no	—
82	C₂×D₄×C₅₃	—	2	1	no	—
83	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₅₃	—	2	2	no	—
85	C₂²×C₄×C₅₃	—	4	2	no	—
86	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
89	Q₁₆×C₅₃	—	4	4	no	—
90	Q₁₆×C₅₃	—	4	4	no	—
91	Q₁₆×C₅₃	—	4	4	no	—
92	Q₁₆×C₅₃	—	4	4	no	—
93	M₁₆×C₅₃	—	4	2	no	—
94	M₁₆×C₅₃	—	4	4	no	—
95	C₂×C₈×C₅₃	—	2	2	no	—
96	C₂×C₈×C₅₃	—	2	4	no	—
97	C₂×C₈×C₅₃	—	2	2	no	—
98	C₂×C₈×C₅₃	—	4	4	no	—
99	C₂×C₈×C₅₃	D₈	1	4	yes	—
100	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₄⋊C₄×C₅₃	—	2	1	no	—
105	C₄⋊C₄×C₅₃	—	2	4	no	—
106	C₄⋊C₄×C₅₃	—	2	4	no	—
107	C₄⋊C₄×C₅₃	—	4	2	no	—
108	C₄⋊C₄×C₅₃	—	4	2	no	—
109	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
110	C₂²⋊C₄×C₅₃	—	2	2	no	—
111	C₂²⋊C₄×C₅₃	—	2	2	no	—
112	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₅₃	#41	1	1	yes	characteristic
115	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
116	C₂³×C₅₃	—	2	2	no	—
117	C₂³×C₅₃	#121	1	1	yes	characteristic
118	Q₈×C₅₃	—	4	2	no	—
119	Q₈×C₅₃	—	4	4	no	—
120	Q₈×C₅₃	—	4	4	no	—
121	Q₈×C₅₃	—	8	4	no	—
122	D₄×C₅₃	—	2	4	no	—
123	D₄×C₅₃	—	4	4	no	—
124	D₄×C₅₃	—	4	2	no	—
125	D₄×C₅₃	—	4	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	4	2	no	—
130	C₂×C₄×C₅₃	—	4	2	no	—
131	C₂×C₄×C₅₃	—	4	4	no	—
132	C₂×C₄×C₅₃	—	8	4	no	—
133	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
134	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
135	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
138	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
141	C₈×C₅₃	—	2	1	no	—
142	C₈×C₅₃	—	2	1	no	—
143	C₈×C₅₃	—	4	2	no	—
144	C₈×C₅₃	—	4	4	no	—
145	C₈×C₅₃	C₂×D₈	1	2	yes	—
146	C₈×C₅₃	C₂×D₈	1	2	yes	—
147	#19188	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂²×C₅₃	—	2	1	no	—
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	—	4	2	no	—
151	C₂²×C₅₃	—	4	2	no	—
152	C₂²×C₅₃	—	4	2	no	—
153	C₂²×C₅₃	#459	1	2	yes	—
154	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
155	C₄×C₅₃	—	2	1	no	—
156	C₄×C₅₃	—	2	1	no	—
157	C₄×C₅₃	—	4	2	no	—
158	C₄×C₅₃	—	8	4	no	—
159	C₄×C₅₃	#459	1	2	yes	—
160	C₄×C₅₃	#473	1	2	yes	—
161	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
162	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
163	C₄×C₅₃	#578	1	2	yes	—
164	#3226	C₂×C₅₃	1	2	yes	—
165	C₂×#494	C₂×C₅₃	1	1	yes	characteristic, complemented
166	#2931	C₂×C₅₃	1	1	yes	characteristic
167	#1689	C₂×C₅₃	1	4	yes	complemented
168	#1669	C₂×C₅₃	1	4	yes	complemented
169	#1564	C₂×C₅₃	1	1	yes	characteristic, complemented
170	#1546	C₂×C₅₃	1	2	yes	complemented
171	C₂×C₅₃	—	2	2	no	—
172	C₂×C₅₃	—	8	2	no	—
173	C₂×C₅₃	C₂×#459	1	1	yes	characteristic
174	C₂×C₅₃	#3059	1	1	yes	characteristic
175	C₂×C₅₃	C₂×#473	1	1	yes	characteristic
176	C₂²×Q₁₆	—	2	1	no	—
177	C₂²×Q₁₆	—	2	1	no	—
178	C₂×#131	—	2	1	no	—
179	#532	C₂²×C₅₃	1	1	yes	characteristic
180	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
181	#500	—	2	2	no	—
182	#494	—	2	1	no	—
183	#494	—	2	1	no	—
184	#494	C₂²×C₅₃	1	2	yes	—
185	#494	C₂²×C₅₃	1	2	yes	—
186	#486	—	2	2	no	—
187	#480	—	2	4	no	—
188	#474	C₂²×C₅₃	1	8	yes	—
189	#436	—	2	1	no	—
190	#436	C₂²×C₅₃	1	2	yes	—
191	C₈×D₄	—	2	1	no	—
192	C₈×D₄	C₂²×C₅₃	1	2	yes	—
193	#431	C₂²×C₅₃	1	2	yes	—
194	#423	C₂²×C₅₃	1	1	yes	characteristic
195	#408	—	2	1	no	—
196	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
197	#335	C₂²×C₅₃	1	2	yes	—
198	#334	C₂²×C₅₃	1	2	yes	—
199	#332	C₂²×C₅₃	1	8	yes	—
200	C₅₃	#19188	1	1	yes	characteristic, direct product factor, Sylow
201	C₂×#41	—	2	1	no	—
202	C₂²×Q₈	—	2	2	no	—
203	C₂×Q₁₆	—	2	2	no	—
204	C₂×Q₁₆	—	2	2	no	—
205	C₂×Q₁₆	—	2	2	no	—
206	C₂×Q₁₆	—	2	2	no	—
207	C₂×Q₁₆	—	4	4	no	—
208	C₂×Q₁₆	—	4	4	no	—
209	#131	—	4	4	no	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂×M₁₆	—	2	2	no	—
212	C₂²×C₈	—	2	2	no	—
213	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
214	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
215	C₄⋊Q₈	—	2	2	no	—
216	C₄⋊Q₈	—	2	2	no	—
217	C₄⋊Q₈	—	2	2	no	—
218	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
219	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
220	#127	D₄×C₅₃	1	1	yes	characteristic
221	#124	—	2	1	no	—
222	#121	—	2	1	no	—
223	#121	D₄×C₅₃	1	2	yes	—
224	C₄×D₄	—	2	1	no	—
225	C₄×D₄	C₂³×C₅₃	1	2	yes	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
229	#107	—	2	8	no	—
230	C₄⋊C₈	—	2	1	no	—
231	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
232	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
233	#102	—	2	8	no	—
234	C₂²⋊C₈	—	2	2	no	—
235	C₂²⋊C₈	—	2	2	no	—
236	C₂²⋊C₈	—	2	2	no	—
237	C₄×C₈	—	2	2	no	—
238	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
239	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
240	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
241	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
242	#41	—	4	2	no	—
243	#41	—	4	2	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	2	4	no	—
246	C₂×Q₈	—	4	2	no	—
247	C₂×Q₈	—	4	4	no	—
248	C₂×Q₈	—	4	4	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	4	2	no	—
255	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
257	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
258	Q₁₆	—	4	4	no	—
259	Q₁₆	—	4	4	no	—
260	Q₁₆	—	4	4	no	—
261	Q₁₆	—	4	4	no	—
262	M₁₆	—	4	2	no	—
263	M₁₆	—	4	4	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	4	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	4	4	no	—
268	C₂×C₈	D₈×C₅₃	1	4	yes	—
269	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
270	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
271	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
272	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	4	no	—
275	C₄⋊C₄	—	2	4	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
282	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
283	C₄²	#41×C₅₃	1	1	yes	characteristic
284	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
285	C₂³	—	2	2	no	—
286	C₂³	#121×C₅₃	1	1	yes	characteristic
287	Q₈	—	4	2	no	—
288	Q₈	—	4	4	no	—
289	Q₈	—	4	4	no	—
290	Q₈	—	8	4	no	—
291	D₄	—	2	4	no	—
292	D₄	—	4	4	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	8	4	no	—
302	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
303	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
304	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
305	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
306	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
307	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
308	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
309	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	4	2	no	—
313	C₈	—	4	4	no	—
314	C₈	C₂×D₈×C₅₃	1	2	yes	—
315	C₈	C₂×D₈×C₅₃	1	2	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	C₅₃×#459	1	2	yes	—
322	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	2	no	—
326	C₄	—	8	4	no	—
327	C₄	C₅₃×#459	1	2	yes	—
328	C₄	C₅₃×#473	1	2	yes	—
329	C₄	C₂²×D₈×C₅₃	1	1	yes	characteristic
330	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
331	C₄	C₅₃×#578	1	2	yes	—
332	C₂	—	2	2	no	—
333	C₂	—	8	2	no	—
334	C₂	C₂×C₅₃×#459	1	1	yes	characteristic
335	C₂	C₅₃×#3059	1	1	yes	characteristic
336	C₂	C₂×C₅₃×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₅₃×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#494	C₂	semidirect
4. C₅₃×#1689	C₂	semidirect
5. C₅₃×#1669	C₂	semidirect
6. C₅₃×#1564	C₂	semidirect
7. C₅₃×#1546	C₂	semidirect
147. #19188	C₅₃	direct (N × H)
165. C₂×#494	C₂×C₅₃	semidirect
167. #1689	C₂×C₅₃	semidirect
168. #1669	C₂×C₅₃	semidirect
169. #1564	C₂×C₅₃	semidirect
170. #1546	C₂×C₅₃	semidirect
200. C₅₃	#19188	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×#3226	2
C₂×C₅₃×#494	2
C₅₃×#2931	2
C₅₃×#1689	2
C₅₃×#1669	2
C₅₃×#1564	2
C₅₃×#1546	2
#19188	53

Subgroups — C53×#19188 #27226700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19188 #27226700