Subgroups — C₅₃×#9746 #27217258

Order 13568.

526 conjugacy classes of subgroups.

The subgroup lattice has 358 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×#1982	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1981	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₃×#331	C₂	1	1	yes	maximal, characteristic
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×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#103×C₅₃	—	2	2	no	—
16	C₂×#102×C₅₃	—	2	1	no	—
17	C₂×#102×C₅₃	—	2	1	no	—
18	C₂×#102×C₅₃	—	2	2	no	—
19	C₂×#97×C₅₃	—	2	1	no	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#336	—	2	1	no	—
23	C₅₃×#334	—	2	2	no	—
24	C₅₃×#332	—	2	2	no	—
25	C₅₃×#331	—	2	2	no	—
26	C₂²×Q₈×C₅₃	—	2	1	no	—
27	C₂²×D₄×C₅₃	—	2	1	no	—
28	C₂²×D₄×C₅₃	—	2	1	no	—
29	C₂²×D₄×C₅₃	—	2	1	no	—
30	C₂²×C₈×C₅₃	—	2	1	no	—
31	C₂²×C₈×C₅₃	—	2	1	no	—
32	C₂²×C₈×C₅₃	—	2	1	no	—
33	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
35	C₄⋊Q₈×C₅₃	—	2	1	no	—
36	C₄⋊Q₈×C₅₃	—	2	1	no	—
37	C₄⋊Q₈×C₅₃	—	2	1	no	—
38	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
39	#127×C₅₃	—	2	1	no	—
40	#127×C₅₃	—	2	1	no	—
41	#127×C₅₃	—	2	1	no	—
42	#127×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
44	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
45	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
46	C₂×C₄²×C₅₃	—	2	1	no	—
47	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₅₃	—	2	1	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	D₄	1	2	yes	—
54	#103×C₅₃	—	4	4	no	—
55	#102×C₅₃	—	2	4	no	—
56	#102×C₅₃	—	2	4	no	—
57	#102×C₅₃	—	4	4	no	—
58	#97×C₅₃	—	4	2	no	—
59	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₅₃	D₄	1	2	yes	—
61	#95×C₅₃	—	2	1	no	—
62	C₂⁴×C₅₃	—	4	1	no	—
63	C₂×Q₈×C₅₃	—	2	2	no	—
64	C₂×Q₈×C₅₃	—	4	1	no	—
65	C₂×Q₈×C₅₃	—	4	2	no	—
66	C₂×Q₈×C₅₃	—	4	2	no	—
67	C₂×D₄×C₅₃	—	2	2	no	—
68	C₂×D₄×C₅₃	—	2	2	no	—
69	C₂×D₄×C₅₃	—	2	2	no	—
70	C₂×D₄×C₅₃	—	4	1	no	—
71	C₂×D₄×C₅₃	—	4	1	no	—
72	C₂×D₄×C₅₃	—	4	1	no	—
73	C₂×D₄×C₅₃	—	4	1	no	—
74	C₂×D₄×C₅₃	—	4	1	no	—
75	C₂×D₄×C₅₃	—	4	2	no	—
76	C₂×D₄×C₅₃	—	4	1	no	—
77	C₂×D₄×C₅₃	—	4	2	no	—
78	C₂×D₄×C₅₃	—	4	1	no	—
79	C₂×D₄×C₅₃	—	8	2	no	—
80	C₂²×C₄×C₅₃	—	2	2	no	—
81	C₂²×C₄×C₅₃	—	4	1	no	—
82	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
85	C₂×C₈×C₅₃	—	2	1	no	—
86	C₂×C₈×C₅₃	—	2	1	no	—
87	C₂×C₈×C₅₃	—	2	1	no	—
88	C₂×C₈×C₅₃	—	2	1	no	—
89	C₂×C₈×C₅₃	—	2	2	no	—
90	C₂×C₈×C₅₃	—	4	1	no	—
91	C₂×C₈×C₅₃	—	4	1	no	—
92	C₂×C₈×C₅₃	—	4	1	no	—
93	C₂×C₈×C₅₃	—	4	1	no	—
94	C₂×C₈×C₅₃	—	4	2	no	—
95	C₂×C₈×C₅₃	—	4	2	no	—
96	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
97	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
98	C₄⋊C₄×C₅₃	—	2	2	no	—
99	C₄⋊C₄×C₅₃	—	2	2	no	—
100	C₄⋊C₄×C₅₃	—	2	2	no	—
101	C₄⋊C₄×C₅₃	—	2	2	no	—
102	C₄⋊C₄×C₅₃	—	4	1	no	—
103	C₄⋊C₄×C₅₃	—	4	1	no	—
104	C₄²×C₅₃	—	4	1	no	—
105	C₄²×C₅₃	—	4	1	no	—
106	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₅₃	#41	1	1	yes	characteristic
110	C₂³×C₅₃	—	4	2	no	—
111	C₂³×C₅₃	—	8	1	no	—
112	C₂³×C₅₃	—	8	1	no	—
113	C₂³×C₅₃	—	8	2	no	—
114	C₂³×C₅₃	—	8	2	no	—
115	C₂³×C₅₃	#121	1	1	yes	characteristic
116	Q₈×C₅₃	—	4	4	no	—
117	Q₈×C₅₃	—	8	1	no	—
118	Q₈×C₅₃	—	8	1	no	—
119	D₄×C₅₃	—	4	2	no	—
120	D₄×C₅₃	—	4	2	no	—
121	D₄×C₅₃	—	4	4	no	—
122	D₄×C₅₃	—	8	2	no	—
123	D₄×C₅₃	—	8	2	no	—
124	D₄×C₅₃	—	8	1	no	—
125	D₄×C₅₃	—	8	1	no	—
126	D₄×C₅₃	—	8	1	no	—
127	D₄×C₅₃	—	8	1	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	2	no	—
132	C₂×C₄×C₅₃	—	4	2	no	—
133	C₂×C₄×C₅₃	—	8	2	no	—
134	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
138	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
139	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
140	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
141	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
142	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
144	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
145	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
146	C₈×C₅₃	—	2	2	no	—
147	C₈×C₅₃	—	2	2	no	—
148	C₈×C₅₃	—	4	2	no	—
149	C₈×C₅₃	—	8	1	no	—
150	C₈×C₅₃	—	8	1	no	—
151	#9746	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂²×C₅₃	—	8	2	no	—
153	C₂²×C₅₃	—	8	2	no	—
154	C₂²×C₅₃	—	8	2	no	—
155	C₂²×C₅₃	—	16	2	no	—
156	C₂²×C₅₃	—	16	2	no	—
157	C₂²×C₅₃	#390	1	1	yes	characteristic
158	C₂²×C₅₃	#463	1	1	yes	characteristic
159	C₂²×C₅₃	#464	1	1	yes	characteristic
160	C₂²×C₅₃	#469	1	2	yes	—
161	C₂²×C₅₃	#471	1	2	yes	—
162	C₄×C₅₃	—	2	1	no	—
163	C₄×C₅₃	—	2	1	no	—
164	C₄×C₅₃	—	8	2	no	—
165	C₄×C₅₃	—	8	2	no	—
166	C₄×C₅₃	#416	1	1	yes	characteristic
167	C₄×C₅₃	#417	1	1	yes	characteristic
168	C₄×C₅₃	#442	1	2	yes	—
169	C₄×C₅₃	#463	1	1	yes	characteristic
170	C₄×C₅₃	#464	1	1	yes	characteristic
171	C₄×C₅₃	#496	1	1	yes	characteristic
172	C₄×C₅₃	#496	1	1	yes	characteristic
173	C₂×#486	C₂×C₅₃	1	1	yes	characteristic
174	#1982	C₂×C₅₃	1	1	yes	characteristic
175	#1981	C₂×C₅₃	1	1	yes	characteristic, complemented
176	#1770	C₂×C₅₃	1	1	yes	characteristic, complemented
177	C₂×#334	C₂×C₅₃	1	1	yes	characteristic, complemented
178	C₂×#332	C₂×C₅₃	1	1	yes	characteristic, complemented
179	C₂×#331	C₂×C₅₃	1	1	yes	characteristic
180	C₂×C₅₃	—	16	2	no	—
181	C₂×C₅₃	#1701	1	2	yes	—
182	C₂×C₅₃	#1702	1	2	yes	—
183	C₂×C₅₃	#1972	1	1	yes	characteristic
184	C₂×C₅₃	#1977	1	1	yes	characteristic
185	C₂×C₅₃	#1983	1	1	yes	characteristic
186	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
187	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
188	#486	—	2	2	no	—
189	#486	C₄×C₅₃	1	4	yes	complemented
190	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
191	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
192	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
193	C₂×#103	—	2	2	no	—
194	C₂×#102	—	2	1	no	—
195	C₂×#102	—	2	1	no	—
196	C₂×#102	—	2	2	no	—
197	C₂×#97	—	2	1	no	—
198	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
199	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
200	#336	—	2	1	no	—
201	#334	—	2	2	no	—
202	#332	—	2	2	no	—
203	#331	—	2	2	no	—
204	C₅₃	#9746	1	1	yes	characteristic, direct product factor, Sylow
205	C₂²×Q₈	—	2	1	no	—
206	C₂²×D₄	—	2	1	no	—
207	C₂²×D₄	—	2	1	no	—
208	C₂²×D₄	—	2	1	no	—
209	C₂²×C₈	—	2	1	no	—
210	C₂²×C₈	—	2	1	no	—
211	C₂²×C₈	—	2	1	no	—
212	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
213	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
218	#127	—	2	1	no	—
219	#127	—	2	1	no	—
220	#127	—	2	1	no	—
221	#127	C₂×C₄×C₅₃	1	2	yes	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
225	C₂×C₄²	—	2	1	no	—
226	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
227	C₄⋊C₈	—	2	1	no	—
228	C₄⋊C₈	—	2	1	no	—
229	C₄⋊C₈	—	2	1	no	—
230	C₄⋊C₈	—	2	1	no	—
231	C₄⋊C₈	—	2	1	no	—
232	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
233	#103	—	4	4	no	—
234	#102	—	2	4	no	—
235	#102	—	2	4	no	—
236	#102	—	4	4	no	—
237	#97	—	4	2	no	—
238	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
239	C₄×C₈	D₄×C₅₃	1	2	yes	—
240	#95	—	2	1	no	—
241	C₂⁴	—	4	1	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	—	4	2	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	2	2	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	4	1	no	—
254	C₂×D₄	—	4	2	no	—
255	C₂×D₄	—	4	2	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂×D₄	—	8	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
263	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
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	2	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	4	1	no	—
271	C₂×C₈	—	4	1	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	—	4	2	no	—
274	C₂×C₈	—	4	2	no	—
275	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
276	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄²	—	4	1	no	—
284	C₄²	—	4	1	no	—
285	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
286	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
287	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
288	C₄²	#41×C₅₃	1	1	yes	characteristic
289	C₂³	—	4	2	no	—
290	C₂³	—	8	1	no	—
291	C₂³	—	8	1	no	—
292	C₂³	—	8	2	no	—
293	C₂³	—	8	2	no	—
294	C₂³	#121×C₅₃	1	1	yes	characteristic
295	Q₈	—	4	4	no	—
296	Q₈	—	8	1	no	—
297	Q₈	—	8	1	no	—
298	D₄	—	4	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	4	no	—
301	D₄	—	8	2	no	—
302	D₄	—	8	2	no	—
303	D₄	—	8	1	no	—
304	D₄	—	8	1	no	—
305	D₄	—	8	1	no	—
306	D₄	—	8	1	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	8	2	no	—
313	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
314	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
315	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
316	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
317	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
318	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
319	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
320	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
321	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
322	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
323	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
324	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
325	C₈	—	2	2	no	—
326	C₈	—	2	2	no	—
327	C₈	—	4	2	no	—
328	C₈	—	8	1	no	—
329	C₈	—	8	1	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	16	2	no	—
334	C₂²	—	16	2	no	—
335	C₂²	C₅₃×#390	1	1	yes	characteristic
336	C₂²	C₅₃×#463	1	1	yes	characteristic
337	C₂²	C₅₃×#464	1	1	yes	characteristic
338	C₂²	C₅₃×#469	1	2	yes	—
339	C₂²	C₅₃×#471	1	2	yes	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	8	2	no	—
343	C₄	—	8	2	no	—
344	C₄	C₅₃×#416	1	1	yes	characteristic
345	C₄	C₅₃×#417	1	1	yes	characteristic
346	C₄	C₅₃×#442	1	2	yes	—
347	C₄	C₅₃×#463	1	1	yes	characteristic
348	C₄	C₅₃×#464	1	1	yes	characteristic
349	C₄	C₅₃×#496	1	1	yes	characteristic
350	C₄	C₅₃×#496	1	1	yes	characteristic
351	C₂	—	16	2	no	—
352	C₂	C₅₃×#1701	1	2	yes	—
353	C₂	C₅₃×#1702	1	2	yes	—
354	C₂	C₅₃×#1972	1	1	yes	characteristic
355	C₂	C₅₃×#1977	1	1	yes	characteristic
356	C₂	C₅₃×#1983	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	#121
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
3. C₅₃×#1981	C₂	semidirect
4. C₅₃×#1770	C₂	semidirect
5. C₂×C₅₃×#334	C₂	semidirect
6. C₂×C₅₃×#332	C₂	semidirect
11. C₅₃×#486	C₄	semidirect
151. #9746	C₅₃	direct (N × H)
175. #1981	C₂×C₅₃	semidirect
176. #1770	C₂×C₅₃	semidirect
177. C₂×#334	C₂×C₅₃	semidirect
178. C₂×#332	C₂×C₅₃	semidirect
189. #486	C₄×C₅₃	semidirect
204. C₅₃	#9746	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×#1982	2
C₅₃×#1981	2
C₅₃×#1770	2
C₂×C₅₃×#334	2
C₂×C₅₃×#332	2
C₂×C₅₃×#331	2
#9746	53

Subgroups — C53×#9746 #27217258

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9746 #27217258