Subgroups — C₅₃×#7659 #27215171

Order 13568.

566 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2858	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1270	C₂	1	4	yes	maximal, complemented
5	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
6	C₂×#127×C₅₃	C₄	1	1	yes	characteristic
7	C₂×#124×C₅₃	—	2	1	no	—
8	C₂×#124×C₅₃	—	2	1	no	—
9	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
10	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
11	C₂×C₄×C₈×C₅₃	—	2	1	no	—
12	C₅₃×#390	—	2	1	no	—
13	C₅₃×#390	—	2	1	no	—
14	C₅₃×#390	C₄	1	2	yes	—
15	C₅₃×#390	C₂²	1	2	yes	—
16	C₄³×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#336	—	2	1	no	—
18	C₅₃×#336	C₂²	1	2	yes	—
19	C₅₃×#336	C₂²	1	2	yes	—
20	C₂²×Q₈×C₅₃	—	2	1	no	—
21	C₂²×D₄×C₅₃	—	2	1	no	—
22	C₂²×D₄×C₅₃	—	2	1	no	—
23	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₅₃	—	2	2	no	—
25	C₂²×C₈×C₅₃	—	2	2	no	—
26	C₄⋊Q₈×C₅₃	—	4	2	no	—
27	#127×C₅₃	—	4	2	no	—
28	#124×C₅₃	—	4	2	no	—
29	#124×C₅₃	—	4	2	no	—
30	#124×C₅₃	—	4	2	no	—
31	#124×C₅₃	—	4	2	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
35	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
38	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₅₃	—	2	4	no	—
46	C₄⋊C₈×C₅₃	—	4	2	no	—
47	C₄×C₈×C₅₃	—	4	2	no	—
48	C₂⁴×C₅₃	—	2	2	no	—
49	C₂×Q₈×C₅₃	—	4	2	no	—
50	C₂×Q₈×C₅₃	—	4	2	no	—
51	C₂×Q₈×C₅₃	—	4	2	no	—
52	C₂×D₄×C₅₃	—	2	4	no	—
53	C₂×D₄×C₅₃	—	4	2	no	—
54	C₂×D₄×C₅₃	—	4	2	no	—
55	C₂×D₄×C₅₃	—	4	2	no	—
56	C₂×D₄×C₅₃	—	4	2	no	—
57	C₂×D₄×C₅₃	—	4	2	no	—
58	C₂×D₄×C₅₃	—	4	2	no	—
59	C₂×D₄×C₅₃	—	4	2	no	—
60	C₂×D₄×C₅₃	—	4	2	no	—
61	C₂²×C₄×C₅₃	—	2	2	no	—
62	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	—	2	4	no	—
70	C₂×C₈×C₅₃	—	4	2	no	—
71	C₂×C₈×C₅₃	—	4	2	no	—
72	C₂×C₈×C₅₃	—	4	2	no	—
73	C₂×C₈×C₅₃	—	4	2	no	—
74	C₂×C₈×C₅₃	—	4	2	no	—
75	C₄⋊C₄×C₅₃	—	4	2	no	—
76	C₄⋊C₄×C₅₃	—	4	2	no	—
77	C₂²⋊C₄×C₅₃	—	4	4	no	—
78	C₂²⋊C₄×C₅₃	—	4	4	no	—
79	C₂²⋊C₄×C₅₃	—	4	2	no	—
80	C₂²⋊C₄×C₅₃	—	4	2	no	—
81	C₂²⋊C₄×C₅₃	—	4	2	no	—
82	C₂²⋊C₄×C₅₃	—	4	2	no	—
83	C₄²×C₅₃	—	2	2	no	—
84	C₄²×C₅₃	—	2	2	no	—
85	C₄²×C₅₃	—	2	2	no	—
86	C₄²×C₅₃	—	4	1	no	—
87	C₄²×C₅₃	—	4	1	no	—
88	C₄²×C₅₃	D₈	1	2	yes	—
89	C₄²×C₅₃	D₈	1	2	yes	—
90	C₄²×C₅₃	SD₁₆	1	2	yes	—
91	C₄²×C₅₃	SD₁₆	1	2	yes	—
92	C₂³×C₅₃	—	4	2	no	—
93	C₂³×C₅₃	—	4	2	no	—
94	C₂³×C₅₃	—	4	2	no	—
95	C₂³×C₅₃	—	4	2	no	—
96	C₂³×C₅₃	—	4	2	no	—
97	C₂³×C₅₃	—	4	2	no	—
98	C₂³×C₅₃	—	4	2	no	—
99	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	Q₈×C₅₃	—	8	4	no	—
101	D₄×C₅₃	—	8	4	no	—
102	D₄×C₅₃	—	8	4	no	—
103	D₄×C₅₃	—	8	4	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	2	1	no	—
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	1	no	—
108	C₂×C₄×C₅₃	—	2	1	no	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	—	2	1	no	—
112	C₂×C₄×C₅₃	—	2	1	no	—
113	C₂×C₄×C₅₃	—	2	1	no	—
114	C₂×C₄×C₅₃	—	2	1	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	—	4	2	no	—
117	C₂×C₄×C₅₃	—	4	2	no	—
118	C₂×C₄×C₅₃	—	4	2	no	—
119	C₂×C₄×C₅₃	#100	1	2	yes	—
120	C₂×C₄×C₅₃	#102	1	2	yes	—
121	C₂×C₄×C₅₃	#102	1	2	yes	—
122	C₂×C₄×C₅₃	#102	1	2	yes	—
123	C₂×C₄×C₅₃	#102	1	2	yes	—
124	C₂×C₄×C₅₃	#104	1	2	yes	—
125	C₂×C₄×C₅₃	#131	1	2	yes	—
126	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
130	C₈×C₅₃	—	4	4	no	—
131	C₈×C₅₃	—	8	2	no	—
132	#7659	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂²×C₅₃	—	8	2	no	—
134	C₂²×C₅₃	—	8	2	no	—
135	C₂²×C₅₃	—	8	2	no	—
136	C₂²×C₅₃	—	8	2	no	—
137	C₂²×C₅₃	—	8	2	no	—
138	C₂²×C₅₃	—	8	2	no	—
139	C₂²×C₅₃	—	8	2	no	—
140	C₂²×C₅₃	#408	1	1	yes	characteristic
141	C₂²×C₅₃	#410	1	1	yes	characteristic
142	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
143	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
144	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
145	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
146	C₂²×C₅₃	#421	1	1	yes	characteristic
147	C₄×C₅₃	—	2	2	no	—
148	C₄×C₅₃	—	2	2	no	—
149	C₄×C₅₃	—	2	2	no	—
150	C₄×C₅₃	—	2	1	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	—	8	2	no	—
156	C₄×C₅₃	#331	1	4	yes	—
157	#2858	C₂×C₅₃	1	1	yes	characteristic
158	#1831	C₂×C₅₃	1	1	yes	characteristic, complemented
159	C₄×C₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
160	#1270	C₂×C₅₃	1	4	yes	complemented
161	C₂×C₅₃	—	16	2	no	—
162	C₂×C₅₃	#1469	1	1	yes	characteristic
163	C₂×C₅₃	#1480	1	1	yes	characteristic
164	C₂×C₅₃	#1492	1	1	yes	characteristic
165	C₂×C₅₃	#1502	1	1	yes	characteristic
166	C₂×C₅₃	#1517	1	1	yes	characteristic
167	C₂×C₅₃	#1527	1	1	yes	characteristic
168	C₂×C₅₃	C₂×#331	1	1	yes	characteristic
169	C₂×C₄⋊Q₈	C₄×C₅₃	1	1	yes	characteristic
170	C₂×#127	C₄×C₅₃	1	1	yes	characteristic
171	C₂×#124	—	2	1	no	—
172	C₂×#124	—	2	1	no	—
173	C₂×C₄⋊C₈	—	2	1	no	—
174	C₂×C₄⋊C₈	—	2	1	no	—
175	C₂×C₄×C₈	—	2	1	no	—
176	#390	—	2	1	no	—
177	#390	—	2	1	no	—
178	#390	C₄×C₅₃	1	2	yes	—
179	#390	C₂²×C₅₃	1	2	yes	—
180	C₄³	C₂²×C₅₃	1	1	yes	characteristic
181	#336	—	2	1	no	—
182	#336	C₂²×C₅₃	1	2	yes	—
183	#336	C₂²×C₅₃	1	2	yes	—
184	C₅₃	#7659	1	1	yes	characteristic, direct product factor, Sylow
185	C₂²×Q₈	—	2	1	no	—
186	C₂²×D₄	—	2	1	no	—
187	C₂²×D₄	—	2	1	no	—
188	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	2	2	no	—
191	C₄⋊Q₈	—	4	2	no	—
192	#127	—	4	2	no	—
193	#124	—	4	2	no	—
194	#124	—	4	2	no	—
195	#124	—	4	2	no	—
196	#124	—	4	2	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
199	C₂×C₂²⋊C₄	—	2	2	no	—
200	C₂×C₂²⋊C₄	—	2	2	no	—
201	C₂×C₂²⋊C₄	—	2	2	no	—
202	C₂×C₂²⋊C₄	—	2	2	no	—
203	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
204	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
205	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
206	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
207	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
208	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
209	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
210	C₄⋊C₈	—	2	4	no	—
211	C₄⋊C₈	—	4	2	no	—
212	C₄×C₈	—	4	2	no	—
213	C₂⁴	—	2	2	no	—
214	C₂×Q₈	—	4	2	no	—
215	C₂×Q₈	—	4	2	no	—
216	C₂×Q₈	—	4	2	no	—
217	C₂×D₄	—	2	4	no	—
218	C₂×D₄	—	4	2	no	—
219	C₂×D₄	—	4	2	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂×D₄	—	4	2	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂×D₄	—	4	2	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
228	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
229	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
230	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
231	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₂²⋊C₄	—	4	4	no	—
243	C₂²⋊C₄	—	4	4	no	—
244	C₂²⋊C₄	—	4	2	no	—
245	C₂²⋊C₄	—	4	2	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₂²⋊C₄	—	4	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	4	1	no	—
252	C₄²	—	4	1	no	—
253	C₄²	D₈×C₅₃	1	2	yes	—
254	C₄²	D₈×C₅₃	1	2	yes	—
255	C₄²	SD₁₆×C₅₃	1	2	yes	—
256	C₄²	SD₁₆×C₅₃	1	2	yes	—
257	C₂³	—	4	2	no	—
258	C₂³	—	4	2	no	—
259	C₂³	—	4	2	no	—
260	C₂³	—	4	2	no	—
261	C₂³	—	4	2	no	—
262	C₂³	—	4	2	no	—
263	C₂³	—	4	2	no	—
264	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
265	Q₈	—	8	4	no	—
266	D₄	—	8	4	no	—
267	D₄	—	8	4	no	—
268	D₄	—	8	4	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	#100×C₅₃	1	2	yes	—
285	C₂×C₄	#102×C₅₃	1	2	yes	—
286	C₂×C₄	#102×C₅₃	1	2	yes	—
287	C₂×C₄	#102×C₅₃	1	2	yes	—
288	C₂×C₄	#102×C₅₃	1	2	yes	—
289	C₂×C₄	#104×C₅₃	1	2	yes	—
290	C₂×C₄	#131×C₅₃	1	2	yes	—
291	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
292	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
293	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
294	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
295	C₈	—	4	4	no	—
296	C₈	—	8	2	no	—
297	C₂²	—	8	2	no	—
298	C₂²	—	8	2	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	8	2	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	C₅₃×#408	1	1	yes	characteristic
305	C₂²	C₅₃×#410	1	1	yes	characteristic
306	C₂²	C₂×#100×C₅₃	1	1	yes	characteristic
307	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
308	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
309	C₂²	C₂×#104×C₅₃	1	1	yes	characteristic
310	C₂²	C₅₃×#421	1	1	yes	characteristic
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	2	no	—
319	C₄	—	8	2	no	—
320	C₄	C₅₃×#331	1	4	yes	—
321	C₂	—	16	2	no	—
322	C₂	C₅₃×#1469	1	1	yes	characteristic
323	C₂	C₅₃×#1480	1	1	yes	characteristic
324	C₂	C₅₃×#1492	1	1	yes	characteristic
325	C₂	C₅₃×#1502	1	1	yes	characteristic
326	C₂	C₅₃×#1517	1	1	yes	characteristic
327	C₂	C₅₃×#1527	1	1	yes	characteristic
328	C₂	C₂×C₅₃×#331	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂×#127×C₅₃	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#1831	C₂	semidirect
3. C₄×C₄⋊C₈×C₅₃	C₂	semidirect
4. C₅₃×#1270	C₂	semidirect
132. #7659	C₅₃	direct (N × H)
158. #1831	C₂×C₅₃	semidirect
159. C₄×C₄⋊C₈	C₂×C₅₃	semidirect
160. #1270	C₂×C₅₃	semidirect
184. C₅₃	#7659	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×#2858	2
C₅₃×#1831	2
C₄×C₄⋊C₈×C₅₃	2
C₅₃×#1270	2
#7659	53

Subgroups — C53×#7659 #27215171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7659 #27215171