Subgroups — C₅₃×#9830 #27217342

Order 13568.

578 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#486	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1990	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1957	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1957	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1956	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1845	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#486	—	2	2	no	—
11	C₅₃×#486	C₄	1	4	yes	complemented
12	C₂×#107×C₅₃	—	2	2	no	—
13	C₂×#103×C₅₃	—	2	2	no	—
14	C₂×#102×C₅₃	—	2	2	no	—
15	C₂×#102×C₅₃	—	2	2	no	—
16	C₂×#102×C₅₃	—	2	2	no	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#384	C₂²	1	1	yes	characteristic
21	C₅₃×#384	C₂²	1	1	yes	characteristic
22	C₅₃×#340	—	2	2	no	—
23	C₅₃×#322	—	2	2	no	—
24	C₂²×Q₈×C₅₃	—	2	1	no	—
25	C₂²×D₄×C₅₃	—	2	1	no	—
26	C₂²×D₄×C₅₃	—	2	1	no	—
27	C₂²×D₄×C₅₃	—	2	1	no	—
28	C₂²×C₈×C₅₃	D₄	1	2	yes	—
29	C₂²×C₈×C₅₃	D₄	1	2	yes	—
30	C₂²×C₈×C₅₃	D₄	1	2	yes	—
31	C₄⋊Q₈×C₅₃	—	2	1	no	—
32	C₄⋊Q₈×C₅₃	—	2	1	no	—
33	C₄⋊Q₈×C₅₃	—	2	1	no	—
34	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
35	#127×C₅₃	—	2	1	no	—
36	#127×C₅₃	—	2	1	no	—
37	#127×C₅₃	—	2	1	no	—
38	#127×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
40	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
41	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
42	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
43	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
44	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
45	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
46	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
47	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
48	#107×C₅₃	—	4	4	no	—
49	#103×C₅₃	—	4	4	no	—
50	#102×C₅₃	—	2	8	no	—
51	#102×C₅₃	—	4	4	no	—
52	#102×C₅₃	—	4	4	no	—
53	C₄×C₈×C₅₃	—	2	1	no	—
54	C₄×C₈×C₅₃	—	2	1	no	—
55	C₄×C₈×C₅₃	—	2	1	no	—
56	C₄×C₈×C₅₃	—	2	1	no	—
57	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₅₃	D₄	1	2	yes	—
59	C₂⁴×C₅₃	—	4	1	no	—
60	C₂×Q₈×C₅₃	—	2	2	no	—
61	C₂×Q₈×C₅₃	—	4	1	no	—
62	C₂×Q₈×C₅₃	—	4	2	no	—
63	C₂×Q₈×C₅₃	—	4	2	no	—
64	C₂×D₄×C₅₃	—	2	2	no	—
65	C₂×D₄×C₅₃	—	2	2	no	—
66	C₂×D₄×C₅₃	—	2	2	no	—
67	C₂×D₄×C₅₃	—	4	1	no	—
68	C₂×D₄×C₅₃	—	4	1	no	—
69	C₂×D₄×C₅₃	—	4	1	no	—
70	C₂×D₄×C₅₃	—	4	2	no	—
71	C₂×D₄×C₅₃	—	4	2	no	—
72	C₂×D₄×C₅₃	—	4	2	no	—
73	C₂×D₄×C₅₃	—	4	2	no	—
74	C₂×D₄×C₅₃	—	4	2	no	—
75	C₂×D₄×C₅₃	—	4	2	no	—
76	C₂²×C₄×C₅₃	—	4	1	no	—
77	C₂²×C₄×C₅₃	—	4	1	no	—
78	C₂²×C₄×C₅₃	—	4	1	no	—
79	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	—	2	2	no	—
83	C₂×C₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	—	2	2	no	—
85	C₂×C₈×C₅₃	—	2	2	no	—
86	C₂×C₈×C₅₃	—	2	2	no	—
87	C₂×C₈×C₅₃	—	2	2	no	—
88	C₂×C₈×C₅₃	—	2	2	no	—
89	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
90	C₂×C₈×C₅₃	D₈	1	4	yes	—
91	C₄⋊C₄×C₅₃	—	2	2	no	—
92	C₄⋊C₄×C₅₃	—	2	2	no	—
93	C₄⋊C₄×C₅₃	—	4	1	no	—
94	C₄⋊C₄×C₅₃	—	4	1	no	—
95	C₄⋊C₄×C₅₃	—	4	2	no	—
96	C₄⋊C₄×C₅₃	—	4	2	no	—
97	C₄⋊C₄×C₅₃	—	4	2	no	—
98	C₄⋊C₄×C₅₃	—	4	2	no	—
99	C₄⋊C₄×C₅₃	—	4	2	no	—
100	C₄⋊C₄×C₅₃	—	4	2	no	—
101	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₅₃	#41	1	1	yes	characteristic
104	C₄²×C₅₃	#41	1	1	yes	characteristic
105	C₂³×C₅₃	—	4	2	no	—
106	C₂³×C₅₃	—	8	1	no	—
107	C₂³×C₅₃	—	8	1	no	—
108	C₂³×C₅₃	—	8	1	no	—
109	C₂³×C₅₃	—	8	1	no	—
110	C₂³×C₅₃	—	8	1	no	—
111	C₂³×C₅₃	—	8	1	no	—
112	C₂³×C₅₃	#127	1	1	yes	characteristic
113	Q₈×C₅₃	—	4	4	no	—
114	Q₈×C₅₃	—	8	2	no	—
115	D₄×C₅₃	—	4	4	no	—
116	D₄×C₅₃	—	4	4	no	—
117	D₄×C₅₃	—	4	4	no	—
118	D₄×C₅₃	—	8	2	no	—
119	D₄×C₅₃	—	8	2	no	—
120	D₄×C₅₃	—	8	2	no	—
121	C₂×C₄×C₅₃	—	4	2	no	—
122	C₂×C₄×C₅₃	—	8	1	no	—
123	C₂×C₄×C₅₃	—	8	1	no	—
124	C₂×C₄×C₅₃	—	8	1	no	—
125	C₂×C₄×C₅₃	—	8	1	no	—
126	C₂×C₄×C₅₃	—	8	1	no	—
127	C₂×C₄×C₅₃	—	8	1	no	—
128	C₂×C₄×C₅₃	—	8	1	no	—
129	C₂×C₄×C₅₃	—	8	1	no	—
130	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	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	#121	1	1	yes	characteristic
134	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
135	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₅₃	#135	1	2	yes	—
138	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
139	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
140	C₂×C₄×C₅₃	#137	1	2	yes	—
141	C₂×C₄×C₅₃	#137	1	2	yes	—
142	C₈×C₅₃	—	2	4	no	—
143	C₈×C₅₃	—	2	4	no	—
144	C₈×C₅₃	—	4	2	no	—
145	#9830	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂²×C₅₃	—	8	2	no	—
147	C₂²×C₅₃	—	8	2	no	—
148	C₂²×C₅₃	—	8	2	no	—
149	C₂²×C₅₃	—	16	1	no	—
150	C₂²×C₅₃	—	16	1	no	—
151	C₂²×C₅₃	—	16	1	no	—
152	C₂²×C₅₃	—	16	1	no	—
153	C₂²×C₅₃	#390	1	1	yes	characteristic
154	C₂²×C₅₃	#493	1	1	yes	characteristic
155	C₂²×C₅₃	#495	1	1	yes	characteristic
156	C₂²×C₅₃	#496	1	1	yes	characteristic
157	C₂²×C₅₃	#496	1	1	yes	characteristic
158	C₂²×C₅₃	#497	1	1	yes	characteristic
159	C₂²×C₅₃	#497	1	1	yes	characteristic
160	C₄×C₅₃	—	8	2	no	—
161	C₄×C₅₃	—	16	1	no	—
162	C₄×C₅₃	—	16	1	no	—
163	C₄×C₅₃	#417	1	2	yes	—
164	C₄×C₅₃	C₄×D₈	1	2	yes	—
165	C₄×C₅₃	#440	1	2	yes	—
166	C₄×C₅₃	#466	1	2	yes	—
167	C₄×C₅₃	#468	1	2	yes	—
168	C₄×C₅₃	#489	1	2	yes	—
169	C₂×#486	C₂×C₅₃	1	1	yes	characteristic
170	#1990	C₂×C₅₃	1	1	yes	characteristic, complemented
171	#1957	C₂×C₅₃	1	1	yes	characteristic
172	#1957	C₂×C₅₃	1	1	yes	characteristic
173	#1956	C₂×C₅₃	1	1	yes	characteristic, complemented
174	#1845	C₂×C₅₃	1	1	yes	characteristic, complemented
175	C₂×#322	C₂×C₅₃	1	1	yes	characteristic, complemented
176	C₂×C₅₃	—	16	2	no	—
177	C₂×C₅₃	#1719	1	1	yes	characteristic
178	C₂×C₅₃	#1720	1	1	yes	characteristic
179	C₂×C₅₃	#1721	1	1	yes	characteristic
180	C₂×C₅₃	#1722	1	1	yes	characteristic
181	C₂×C₅₃	#1959	1	1	yes	characteristic
182	C₂×C₅₃	#1970	1	1	yes	characteristic
183	C₂×C₅₃	#1973	1	1	yes	characteristic
184	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
185	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
186	#486	—	2	2	no	—
187	#486	C₄×C₅₃	1	4	yes	complemented
188	C₂×#107	—	2	2	no	—
189	C₂×#103	—	2	2	no	—
190	C₂×#102	—	2	2	no	—
191	C₂×#102	—	2	2	no	—
192	C₂×#102	—	2	2	no	—
193	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
194	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
195	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
196	#384	C₂²×C₅₃	1	1	yes	characteristic
197	#384	C₂²×C₅₃	1	1	yes	characteristic
198	#340	—	2	2	no	—
199	#322	—	2	2	no	—
200	C₅₃	#9830	1	1	yes	characteristic, direct product factor, Sylow
201	C₂²×Q₈	—	2	1	no	—
202	C₂²×D₄	—	2	1	no	—
203	C₂²×D₄	—	2	1	no	—
204	C₂²×D₄	—	2	1	no	—
205	C₂²×C₈	D₄×C₅₃	1	2	yes	—
206	C₂²×C₈	D₄×C₅₃	1	2	yes	—
207	C₂²×C₈	D₄×C₅₃	1	2	yes	—
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
212	#127	—	2	1	no	—
213	#127	—	2	1	no	—
214	#127	—	2	1	no	—
215	#127	C₂×C₄×C₅₃	1	2	yes	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
225	#107	—	4	4	no	—
226	#103	—	4	4	no	—
227	#102	—	2	8	no	—
228	#102	—	4	4	no	—
229	#102	—	4	4	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	2	1	no	—
232	C₄×C₈	—	2	1	no	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
235	C₄×C₈	D₄×C₅₃	1	2	yes	—
236	C₂⁴	—	4	1	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	4	1	no	—
239	C₂×Q₈	—	4	2	no	—
240	C₂×Q₈	—	4	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	—	4	2	no	—
251	C₂×D₄	—	4	2	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
267	C₂×C₈	D₈×C₅₃	1	4	yes	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	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	#41×C₅₃	1	1	yes	characteristic
282	C₂³	—	4	2	no	—
283	C₂³	—	8	1	no	—
284	C₂³	—	8	1	no	—
285	C₂³	—	8	1	no	—
286	C₂³	—	8	1	no	—
287	C₂³	—	8	1	no	—
288	C₂³	—	8	1	no	—
289	C₂³	#127×C₅₃	1	1	yes	characteristic
290	Q₈	—	4	4	no	—
291	Q₈	—	8	2	no	—
292	D₄	—	4	4	no	—
293	D₄	—	4	4	no	—
294	D₄	—	4	4	no	—
295	D₄	—	8	2	no	—
296	D₄	—	8	2	no	—
297	D₄	—	8	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	—	8	1	no	—
301	C₂×C₄	—	8	1	no	—
302	C₂×C₄	—	8	1	no	—
303	C₂×C₄	—	8	1	no	—
304	C₂×C₄	—	8	1	no	—
305	C₂×C₄	—	8	1	no	—
306	C₂×C₄	—	8	1	no	—
307	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
308	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
309	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
310	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
311	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
312	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
313	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
314	C₂×C₄	#135×C₅₃	1	2	yes	—
315	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
316	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
317	C₂×C₄	#137×C₅₃	1	2	yes	—
318	C₂×C₄	#137×C₅₃	1	2	yes	—
319	C₈	—	2	4	no	—
320	C₈	—	2	4	no	—
321	C₈	—	4	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	16	1	no	—
326	C₂²	—	16	1	no	—
327	C₂²	—	16	1	no	—
328	C₂²	—	16	1	no	—
329	C₂²	C₅₃×#390	1	1	yes	characteristic
330	C₂²	C₅₃×#493	1	1	yes	characteristic
331	C₂²	C₅₃×#495	1	1	yes	characteristic
332	C₂²	C₅₃×#496	1	1	yes	characteristic
333	C₂²	C₅₃×#496	1	1	yes	characteristic
334	C₂²	C₅₃×#497	1	1	yes	characteristic
335	C₂²	C₅₃×#497	1	1	yes	characteristic
336	C₄	—	8	2	no	—
337	C₄	—	16	1	no	—
338	C₄	—	16	1	no	—
339	C₄	C₅₃×#417	1	2	yes	—
340	C₄	C₄×D₈×C₅₃	1	2	yes	—
341	C₄	C₅₃×#440	1	2	yes	—
342	C₄	C₅₃×#466	1	2	yes	—
343	C₄	C₅₃×#468	1	2	yes	—
344	C₄	C₅₃×#489	1	2	yes	—
345	C₂	—	16	2	no	—
346	C₂	C₅₃×#1719	1	1	yes	characteristic
347	C₂	C₅₃×#1720	1	1	yes	characteristic
348	C₂	C₅₃×#1721	1	1	yes	characteristic
349	C₂	C₅₃×#1722	1	1	yes	characteristic
350	C₂	C₅₃×#1959	1	1	yes	characteristic
351	C₂	C₅₃×#1970	1	1	yes	characteristic
352	C₂	C₅₃×#1973	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₅₃×#1990	C₂	semidirect
5. C₅₃×#1956	C₂	semidirect
6. C₅₃×#1845	C₂	semidirect
7. C₂×C₅₃×#322	C₂	semidirect
11. C₅₃×#486	C₄	semidirect
145. #9830	C₅₃	direct (N × H)
170. #1990	C₂×C₅₃	semidirect
173. #1956	C₂×C₅₃	semidirect
174. #1845	C₂×C₅₃	semidirect
175. C₂×#322	C₂×C₅₃	semidirect
187. #486	C₄×C₅₃	semidirect
200. C₅₃	#9830	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#486	2
C₅₃×#1990	2
C₅₃×#1957	2
C₅₃×#1956	2
C₅₃×#1845	2
C₂×C₅₃×#322	2
#9830	53

Subgroups — C53×#9830 #27217342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9830 #27217342