Subgroups — C₅₃×#8932 #27216444

Order 13568.

426 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2935	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1950	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1570	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1540	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1492	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1278	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131×C₅₃	—	2	1	no	—
9	C₅₃×#532	C₂²	1	1	yes	characteristic
10	C₅₃×#436	—	2	1	no	—
11	C₅₃×#435	—	2	2	no	—
12	C₈×D₄×C₅₃	—	2	1	no	—
13	C₅₃×#431	C₂²	1	1	yes	characteristic
14	C₂×#108×C₅₃	—	2	1	no	—
15	C₅₃×#423	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#408	—	2	1	no	—
19	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#341	—	2	1	no	—
22	C₅₃×#334	—	2	1	no	—
23	C₅₃×#334	—	2	1	no	—
24	C₅₃×#332	C₄	1	1	yes	characteristic
25	C₅₃×#332	C₄	1	1	yes	characteristic
26	C₅₃×#331	C₄	1	1	yes	characteristic
27	C₅₃×#331	C₄	1	1	yes	characteristic
28	C₅₃×D₄⋊C₈	—	2	2	no	—
29	C₂×#41×C₅₃	—	2	1	no	—
30	#131×C₅₃	—	4	2	no	—
31	#131×C₅₃	—	4	2	no	—
32	C₂×M₁₆×C₅₃	—	2	2	no	—
33	C₂×M₁₆×C₅₃	—	2	1	no	—
34	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
35	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	—	2	2	no	—
37	C₂²×C₈×C₅₃	—	2	1	no	—
38	C₂²×C₈×C₅₃	—	2	1	no	—
39	C₂²×C₈×C₅₃	—	2	1	no	—
40	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	#127×C₅₃	C₂×C₄	1	1	yes	characteristic
42	#124×C₅₃	—	2	1	no	—
43	#121×C₅₃	—	2	1	no	—
44	#121×C₅₃	Q₈	1	2	yes	—
45	C₄×D₄×C₅₃	—	2	1	no	—
46	C₄×D₄×C₅₃	D₄	1	2	yes	—
47	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
48	#108×C₅₃	—	4	2	no	—
49	C₄⋊C₈×C₅₃	—	2	1	no	—
50	C₄⋊C₈×C₅₃	—	2	1	no	—
51	C₄⋊C₈×C₅₃	—	2	1	no	—
52	C₄⋊C₈×C₅₃	—	2	1	no	—
53	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
58	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
59	C₂²⋊C₈×C₅₃	—	2	2	no	—
60	C₂²⋊C₈×C₅₃	—	2	2	no	—
61	C₂²⋊C₈×C₅₃	—	2	1	no	—
62	#97×C₅₃	—	2	1	no	—
63	C₄×C₈×C₅₃	—	2	1	no	—
64	C₄×C₈×C₅₃	—	2	1	no	—
65	C₄×C₈×C₅₃	—	2	1	no	—
66	C₄×C₈×C₅₃	—	2	1	no	—
67	#41×C₅₃	—	4	1	no	—
68	#41×C₅₃	—	4	1	no	—
69	#41×C₅₃	—	4	1	no	—
70	#41×C₅₃	—	4	1	no	—
71	C₂×Q₈×C₅₃	—	2	1	no	—
72	C₂×D₄×C₅₃	—	2	2	no	—
73	C₂×D₄×C₅₃	—	2	1	no	—
74	C₂×D₄×C₅₃	—	2	1	no	—
75	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₅₃	—	2	2	no	—
77	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
80	M₁₆×C₅₃	—	2	1	no	—
81	M₁₆×C₅₃	—	2	1	no	—
82	M₁₆×C₅₃	—	2	1	no	—
83	M₁₆×C₅₃	—	4	4	no	—
84	M₁₆×C₅₃	—	4	1	no	—
85	M₁₆×C₅₃	—	4	1	no	—
86	M₁₆×C₅₃	SD₁₆	1	2	yes	—
87	C₂×C₈×C₅₃	—	2	1	no	—
88	C₂×C₈×C₅₃	—	2	2	no	—
89	C₂×C₈×C₅₃	—	2	1	no	—
90	C₂×C₈×C₅₃	—	2	1	no	—
91	C₂×C₈×C₅₃	—	2	1	no	—
92	C₂×C₈×C₅₃	—	2	1	no	—
93	C₂×C₈×C₅₃	—	2	1	no	—
94	C₂×C₈×C₅₃	—	2	1	no	—
95	C₂×C₈×C₅₃	—	2	1	no	—
96	C₂×C₈×C₅₃	—	4	4	no	—
97	C₂×C₈×C₅₃	—	4	1	no	—
98	C₂×C₈×C₅₃	—	4	1	no	—
99	C₂×C₈×C₅₃	—	4	1	no	—
100	C₂×C₈×C₅₃	—	4	1	no	—
101	C₂×C₈×C₅₃	—	4	1	no	—
102	C₂×C₈×C₅₃	—	4	1	no	—
103	C₂×C₈×C₅₃	—	4	1	no	—
104	C₄⋊C₄×C₅₃	—	2	1	no	—
105	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
106	C₂²⋊C₄×C₅₃	—	2	2	no	—
107	C₂²⋊C₄×C₅₃	—	2	2	no	—
108	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₅₃	#41	1	1	yes	characteristic
112	C₂³×C₅₃	—	2	2	no	—
113	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
114	Q₈×C₅₃	—	4	1	no	—
115	Q₈×C₅₃	—	4	1	no	—
116	D₄×C₅₃	—	4	2	no	—
117	D₄×C₅₃	—	4	2	no	—
118	D₄×C₅₃	—	4	2	no	—
119	D₄×C₅₃	—	4	1	no	—
120	D₄×C₅₃	—	4	1	no	—
121	D₄×C₅₃	—	4	1	no	—
122	D₄×C₅₃	—	4	1	no	—
123	C₂×C₄×C₅₃	—	2	1	no	—
124	C₂×C₄×C₅₃	—	2	1	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	4	1	no	—
127	C₂×C₄×C₅₃	—	4	2	no	—
128	C₂×C₄×C₅₃	—	4	2	no	—
129	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
137	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₈×C₅₃	—	2	2	no	—
140	C₈×C₅₃	—	4	2	no	—
141	C₈×C₅₃	—	4	1	no	—
142	C₈×C₅₃	—	4	1	no	—
143	C₈×C₅₃	—	4	1	no	—
144	C₈×C₅₃	—	4	1	no	—
145	C₈×C₅₃	—	4	1	no	—
146	C₈×C₅₃	—	4	1	no	—
147	C₈×C₅₃	—	4	1	no	—
148	C₈×C₅₃	—	8	1	no	—
149	#8932	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂²×C₅₃	—	2	1	no	—
151	C₂²×C₅₃	—	2	1	no	—
152	C₂²×C₅₃	—	4	2	no	—
153	C₂²×C₅₃	—	4	2	no	—
154	C₂²×C₅₃	—	4	2	no	—
155	C₂²×C₅₃	#385	1	1	yes	characteristic
156	C₂²×C₅₃	#476	1	1	yes	characteristic
157	C₂²×C₅₃	#478	1	1	yes	characteristic
158	C₄×C₅₃	—	2	1	no	—
159	C₄×C₅₃	—	2	1	no	—
160	C₄×C₅₃	—	2	1	no	—
161	C₄×C₅₃	—	4	2	no	—
162	C₄×C₅₃	C₄×SD₁₆	1	1	yes	characteristic
163	C₄×C₅₃	#442	1	1	yes	characteristic
164	C₄×C₅₃	#443	1	1	yes	characteristic
165	C₄×C₅₃	#444	1	1	yes	characteristic
166	C₄×C₅₃	#449	1	1	yes	characteristic
167	C₄×C₅₃	#450	1	1	yes	characteristic
168	#2935	C₂×C₅₃	1	1	yes	characteristic
169	#1950	C₂×C₅₃	1	1	yes	characteristic, complemented
170	#1570	C₂×C₅₃	1	1	yes	characteristic, complemented
171	#1540	C₂×C₅₃	1	1	yes	characteristic
172	#1492	C₂×C₅₃	1	1	yes	characteristic
173	#1278	C₂×C₅₃	1	1	yes	characteristic, complemented
174	#1277	C₂×C₅₃	1	1	yes	characteristic, complemented
175	C₂×C₅₃	—	2	1	no	—
176	C₂×C₅₃	—	2	1	no	—
177	C₂×C₅₃	—	8	2	no	—
178	C₂×C₅₃	#1858	1	1	yes	characteristic
179	C₂×C₅₃	#1864	1	1	yes	characteristic
180	C₂×C₅₃	#1866	1	1	yes	characteristic
181	C₂×#131	—	2	1	no	—
182	#532	C₂²×C₅₃	1	1	yes	characteristic
183	#436	—	2	1	no	—
184	#435	—	2	2	no	—
185	C₈×D₄	—	2	1	no	—
186	#431	C₂²×C₅₃	1	1	yes	characteristic
187	C₂×#108	—	2	1	no	—
188	#423	C₂²×C₅₃	1	1	yes	characteristic
189	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
190	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
191	#408	—	2	1	no	—
192	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
193	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
194	#341	—	2	1	no	—
195	#334	—	2	1	no	—
196	#334	—	2	1	no	—
197	#332	C₄×C₅₃	1	1	yes	characteristic
198	#332	C₄×C₅₃	1	1	yes	characteristic
199	#331	C₄×C₅₃	1	1	yes	characteristic
200	#331	C₄×C₅₃	1	1	yes	characteristic
201	D₄⋊C₈	—	2	2	no	—
202	C₅₃	#8932	1	1	yes	characteristic, direct product factor, Sylow
203	C₂×#41	—	2	1	no	—
204	#131	—	4	2	no	—
205	#131	—	4	2	no	—
206	C₂×M₁₆	—	2	2	no	—
207	C₂×M₁₆	—	2	1	no	—
208	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
209	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
210	C₂²×C₈	—	2	2	no	—
211	C₂²×C₈	—	2	1	no	—
212	C₂²×C₈	—	2	1	no	—
213	C₂²×C₈	—	2	1	no	—
214	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
215	#127	C₂×C₄×C₅₃	1	1	yes	characteristic
216	#124	—	2	1	no	—
217	#121	—	2	1	no	—
218	#121	Q₈×C₅₃	1	2	yes	—
219	C₄×D₄	—	2	1	no	—
220	C₄×D₄	D₄×C₅₃	1	2	yes	—
221	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
222	#108	—	4	2	no	—
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
228	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
229	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
230	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
231	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
232	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
233	C₂²⋊C₈	—	2	2	no	—
234	C₂²⋊C₈	—	2	2	no	—
235	C₂²⋊C₈	—	2	1	no	—
236	#97	—	2	1	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	—	2	1	no	—
239	C₄×C₈	—	2	1	no	—
240	C₄×C₈	—	2	1	no	—
241	#41	—	4	1	no	—
242	#41	—	4	1	no	—
243	#41	—	4	1	no	—
244	#41	—	4	1	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	C₄⋊C₄×C₅₃	1	2	yes	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
252	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
253	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
254	M₁₆	—	2	1	no	—
255	M₁₆	—	2	1	no	—
256	M₁₆	—	2	1	no	—
257	M₁₆	—	4	4	no	—
258	M₁₆	—	4	1	no	—
259	M₁₆	—	4	1	no	—
260	M₁₆	SD₁₆×C₅₃	1	2	yes	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	4	4	no	—
271	C₂×C₈	—	4	1	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	4	1	no	—
275	C₂×C₈	—	4	1	no	—
276	C₂×C₈	—	4	1	no	—
277	C₂×C₈	—	4	1	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	C₄⋊C₄×C₅₃	1	2	yes	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₂²⋊C₄	—	2	2	no	—
282	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
283	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
284	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
285	C₄²	#41×C₅₃	1	1	yes	characteristic
286	C₂³	—	2	2	no	—
287	C₂³	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
288	Q₈	—	4	1	no	—
289	Q₈	—	4	1	no	—
290	D₄	—	4	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	1	no	—
294	D₄	—	4	1	no	—
295	D₄	—	4	1	no	—
296	D₄	—	4	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
304	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
305	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
306	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
307	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
308	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
309	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
310	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
311	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
312	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
313	C₈	—	2	2	no	—
314	C₈	—	4	2	no	—
315	C₈	—	4	1	no	—
316	C₈	—	4	1	no	—
317	C₈	—	4	1	no	—
318	C₈	—	4	1	no	—
319	C₈	—	4	1	no	—
320	C₈	—	4	1	no	—
321	C₈	—	4	1	no	—
322	C₈	—	8	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	4	2	no	—
326	C₂²	—	4	2	no	—
327	C₂²	—	4	2	no	—
328	C₂²	C₅₃×#385	1	1	yes	characteristic
329	C₂²	C₅₃×#476	1	1	yes	characteristic
330	C₂²	C₅₃×#478	1	1	yes	characteristic
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	4	2	no	—
335	C₄	C₄×SD₁₆×C₅₃	1	1	yes	characteristic
336	C₄	C₅₃×#442	1	1	yes	characteristic
337	C₄	C₅₃×#443	1	1	yes	characteristic
338	C₄	C₅₃×#444	1	1	yes	characteristic
339	C₄	C₅₃×#449	1	1	yes	characteristic
340	C₄	C₅₃×#450	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	8	2	no	—
344	C₂	C₅₃×#1858	1	1	yes	characteristic
345	C₂	C₅₃×#1864	1	1	yes	characteristic
346	C₂	C₅₃×#1866	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#120
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#385
prime radical	C₅₃×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#1950	C₂	semidirect
3. C₅₃×#1570	C₂	semidirect
6. C₅₃×#1278	C₂	semidirect
7. C₅₃×#1277	C₂	semidirect
149. #8932	C₅₃	direct (N × H)
169. #1950	C₂×C₅₃	semidirect
170. #1570	C₂×C₅₃	semidirect
173. #1278	C₂×C₅₃	semidirect
174. #1277	C₂×C₅₃	semidirect
202. C₅₃	#8932	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×#2935	2
C₅₃×#1950	2
C₅₃×#1570	2
C₅₃×#1540	2
C₅₃×#1492	2
C₅₃×#1278	2
C₅₃×#1277	2
#8932	53

Subgroups — C53×#8932 #27216444

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8932 #27216444