Subgroups — C₅₃×#9843 #27217355

Order 13568.

516 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3167	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1994	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1962	C₂	1	1	yes	maximal, characteristic
4	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1566	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1493	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1492	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#533	C₂²	1	1	yes	characteristic
9	C₅₃×#532	C₂²	1	1	yes	characteristic
10	C₅₃×#498	C₄	1	1	yes	characteristic, complemented
11	C₅₃×#492	C₄	1	1	yes	characteristic, complemented
12	C₅₃×#487	C₄	1	1	yes	characteristic, complemented
13	C₅₃×#486	C₄	1	1	yes	characteristic, complemented
14	C₅₃×#465	—	2	2	no	—
15	C₄×SD₁₆×C₅₃	—	2	2	no	—
16	C₅₃×#423	C₂²	1	1	yes	characteristic
17	C₅₃×#423	C₂²	1	1	yes	characteristic
18	C₂×#104×C₅₃	—	2	2	no	—
19	C₂×C₄×C₈×C₅₃	—	2	2	no	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₄³×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#339	—	2	2	no	—
24	C₅₃×#334	—	2	2	no	—
25	C₅₃×Q₈⋊C₈	—	2	2	no	—
26	C₅₃×D₄⋊C₈	—	2	2	no	—
27	C₂×#41×C₅₃	—	2	1	no	—
28	C₂×SD₁₆×C₅₃	—	2	2	no	—
29	C₂×M₁₆×C₅₃	—	2	1	no	—
30	C₂×M₁₆×C₅₃	—	2	1	no	—
31	C₂²×C₈×C₅₃	—	2	1	no	—
32	C₂²×C₈×C₅₃	D₄	1	2	yes	—
33	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	#127×C₅₃	C₂×C₄	1	1	yes	characteristic
37	#125×C₅₃	—	2	1	no	—
38	#124×C₅₃	—	2	1	no	—
39	C₂²⋊Q₈×C₅₃	—	2	1	no	—
40	C₂²⋊Q₈×C₅₃	—	2	1	no	—
41	#121×C₅₃	—	2	1	no	—
42	#121×C₅₃	—	2	1	no	—
43	C₄×Q₈×C₅₃	—	2	1	no	—
44	C₄×Q₈×C₅₃	—	2	1	no	—
45	C₄×D₄×C₅₃	—	2	1	no	—
46	C₄×D₄×C₅₃	—	2	1	no	—
47	#117×C₅₃	—	2	1	no	—
48	C₂×C₄²×C₅₃	—	2	1	no	—
49	C₂×C₄²×C₅₃	—	2	1	no	—
50	C₂×C₄²×C₅₃	D₄	1	2	yes	—
51	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
52	#106×C₅₃	—	2	2	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	#104×C₅₃	—	4	4	no	—
58	#103×C₅₃	—	2	2	no	—
59	#102×C₅₃	—	2	2	no	—
60	C₄×C₈×C₅₃	—	2	8	no	—
61	C₄×C₈×C₅₃	—	2	1	no	—
62	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₅₃	D₄	1	2	yes	—
65	C₄×C₈×C₅₃	D₄	1	2	yes	—
66	#41×C₅₃	—	4	2	no	—
67	#41×C₅₃	—	4	2	no	—
68	C₂×Q₈×C₅₃	—	2	1	no	—
69	C₂×Q₈×C₅₃	—	2	1	no	—
70	C₂×Q₈×C₅₃	—	2	1	no	—
71	C₂×D₄×C₅₃	—	2	1	no	—
72	C₂×D₄×C₅₃	—	2	1	no	—
73	C₂×D₄×C₅₃	—	2	1	no	—
74	C₂×D₄×C₅₃	—	4	1	no	—
75	C₂²×C₄×C₅₃	—	2	2	no	—
76	C₂²×C₄×C₅₃	—	4	1	no	—
77	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
80	SD₁₆×C₅₃	—	4	4	no	—
81	M₁₆×C₅₃	—	4	2	no	—
82	M₁₆×C₅₃	—	4	2	no	—
83	C₂×C₈×C₅₃	—	2	4	no	—
84	C₂×C₈×C₅₃	—	2	2	no	—
85	C₂×C₈×C₅₃	—	2	4	no	—
86	C₂×C₈×C₅₃	—	4	1	no	—
87	C₂×C₈×C₅₃	—	4	1	no	—
88	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
89	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
90	C₄⋊C₄×C₅₃	—	2	1	no	—
91	C₄⋊C₄×C₅₃	—	2	1	no	—
92	C₄⋊C₄×C₅₃	—	2	1	no	—
93	C₄⋊C₄×C₅₃	—	2	1	no	—
94	C₄⋊C₄×C₅₃	—	2	1	no	—
95	C₄⋊C₄×C₅₃	—	2	1	no	—
96	C₄⋊C₄×C₅₃	—	4	1	no	—
97	C₄⋊C₄×C₅₃	—	4	1	no	—
98	C₂²⋊C₄×C₅₃	—	4	1	no	—
99	C₂²⋊C₄×C₅₃	—	4	1	no	—
100	C₂²⋊C₄×C₅₃	—	4	1	no	—
101	C₄²×C₅₃	—	2	4	no	—
102	C₄²×C₅₃	—	2	4	no	—
103	C₄²×C₅₃	—	2	2	no	—
104	C₄²×C₅₃	—	4	1	no	—
105	C₄²×C₅₃	SD₁₆	1	4	yes	complemented
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	1	no	—
111	C₂³×C₅₃	#121	1	1	yes	characteristic
112	Q₈×C₅₃	—	4	2	no	—
113	Q₈×C₅₃	—	4	2	no	—
114	Q₈×C₅₃	—	4	2	no	—
115	D₄×C₅₃	—	4	2	no	—
116	D₄×C₅₃	—	4	2	no	—
117	D₄×C₅₃	—	4	2	no	—
118	D₄×C₅₃	—	8	2	no	—
119	C₂×C₄×C₅₃	—	2	4	no	—
120	C₂×C₄×C₅₃	—	2	4	no	—
121	C₂×C₄×C₅₃	—	2	4	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	4	1	no	—
125	C₂×C₄×C₅₃	—	4	1	no	—
126	C₂×C₄×C₅₃	—	4	1	no	—
127	C₂×C₄×C₅₃	—	8	1	no	—
128	C₂×C₄×C₅₃	—	8	1	no	—
129	C₂×C₄×C₅₃	C₂×C₂²⋊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	#121	1	1	yes	characteristic
133	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
134	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
136	C₂×C₄×C₅₃	#135	1	2	yes	—
137	C₈×C₅₃	—	2	4	no	—
138	C₈×C₅₃	—	2	2	no	—
139	C₈×C₅₃	—	8	1	no	—
140	C₈×C₅₃	—	8	1	no	—
141	C₈×C₅₃	#104	1	4	yes	complemented
142	#9843	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂²×C₅₃	—	2	1	no	—
144	C₂²×C₅₃	—	2	1	no	—
145	C₂²×C₅₃	—	8	1	no	—
146	C₂²×C₅₃	—	8	1	no	—
147	C₂²×C₅₃	—	8	1	no	—
148	C₂²×C₅₃	#390	1	1	yes	characteristic
149	C₂²×C₅₃	#465	1	2	yes	—
150	C₄×C₅₃	—	2	8	no	—
151	C₄×C₅₃	—	2	1	no	—
152	C₄×C₅₃	—	2	1	no	—
153	C₄×C₅₃	—	8	1	no	—
154	C₄×C₅₃	—	8	1	no	—
155	C₄×C₅₃	—	8	1	no	—
156	C₄×C₅₃	C₂×#104	1	2	yes	—
157	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
158	C₄×C₅₃	#443	1	2	yes	—
159	C₄×C₅₃	#492	1	1	yes	characteristic
160	C₄×C₅₃	#495	1	1	yes	characteristic
161	#3167	C₂×C₅₃	1	1	yes	characteristic
162	#1994	C₂×C₅₃	1	1	yes	characteristic, complemented
163	#1962	C₂×C₅₃	1	1	yes	characteristic
164	C₄²×C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
165	#1566	C₂×C₅₃	1	1	yes	characteristic, complemented
166	#1493	C₂×C₅₃	1	1	yes	characteristic, complemented
167	#1492	C₂×C₅₃	1	1	yes	characteristic
168	C₂×C₅₃	—	2	2	no	—
169	C₂×C₅₃	—	16	1	no	—
170	C₂×C₅₃	#1962	1	1	yes	characteristic
171	C₂×C₅₃	#1968	1	1	yes	characteristic
172	C₂×C₅₃	#1974	1	1	yes	characteristic
173	#533	C₂²×C₅₃	1	1	yes	characteristic
174	#532	C₂²×C₅₃	1	1	yes	characteristic
175	#498	C₄×C₅₃	1	1	yes	characteristic, complemented
176	#492	C₄×C₅₃	1	1	yes	characteristic, complemented
177	#487	C₄×C₅₃	1	1	yes	characteristic, complemented
178	#486	C₄×C₅₃	1	1	yes	characteristic, complemented
179	#465	—	2	2	no	—
180	C₄×SD₁₆	—	2	2	no	—
181	#423	C₂²×C₅₃	1	1	yes	characteristic
182	#423	C₂²×C₅₃	1	1	yes	characteristic
183	C₂×#104	—	2	2	no	—
184	C₂×C₄×C₈	—	2	2	no	—
185	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
186	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
187	C₄³	C₂²×C₅₃	1	1	yes	characteristic
188	#339	—	2	2	no	—
189	#334	—	2	2	no	—
190	Q₈⋊C₈	—	2	2	no	—
191	D₄⋊C₈	—	2	2	no	—
192	C₅₃	#9843	1	1	yes	characteristic, direct product factor, Sylow
193	C₂×#41	—	2	1	no	—
194	C₂×SD₁₆	—	2	2	no	—
195	C₂×M₁₆	—	2	1	no	—
196	C₂×M₁₆	—	2	1	no	—
197	C₂²×C₈	—	2	1	no	—
198	C₂²×C₈	D₄×C₅₃	1	2	yes	—
199	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
200	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
201	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
202	#127	C₂×C₄×C₅₃	1	1	yes	characteristic
203	#125	—	2	1	no	—
204	#124	—	2	1	no	—
205	C₂²⋊Q₈	—	2	1	no	—
206	C₂²⋊Q₈	—	2	1	no	—
207	#121	—	2	1	no	—
208	#121	—	2	1	no	—
209	C₄×Q₈	—	2	1	no	—
210	C₄×Q₈	—	2	1	no	—
211	C₄×D₄	—	2	1	no	—
212	C₄×D₄	—	2	1	no	—
213	#117	—	2	1	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	—	2	1	no	—
216	C₂×C₄²	D₄×C₅₃	1	2	yes	—
217	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
218	#106	—	2	2	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	2	1	no	—
221	C₄⋊C₈	—	2	1	no	—
222	C₄⋊C₈	—	2	1	no	—
223	#104	—	4	4	no	—
224	#103	—	2	2	no	—
225	#102	—	2	2	no	—
226	C₄×C₈	—	2	8	no	—
227	C₄×C₈	—	2	1	no	—
228	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
229	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
230	C₄×C₈	D₄×C₅₃	1	2	yes	—
231	C₄×C₈	D₄×C₅₃	1	2	yes	—
232	#41	—	4	2	no	—
233	#41	—	4	2	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×Q₈	—	2	1	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×D₄	—	2	1	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
245	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
246	SD₁₆	—	4	4	no	—
247	M₁₆	—	4	2	no	—
248	M₁₆	—	4	2	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
255	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₂²⋊C₄	—	4	1	no	—
265	C₂²⋊C₄	—	4	1	no	—
266	C₂²⋊C₄	—	4	1	no	—
267	C₄²	—	2	4	no	—
268	C₄²	—	2	4	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	4	1	no	—
271	C₄²	SD₁₆×C₅₃	1	4	yes	complemented
272	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
273	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
274	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
275	C₄²	#41×C₅₃	1	1	yes	characteristic
276	C₂³	—	4	1	no	—
277	C₂³	#121×C₅₃	1	1	yes	characteristic
278	Q₈	—	4	2	no	—
279	Q₈	—	4	2	no	—
280	Q₈	—	4	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	2	no	—
284	D₄	—	8	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	8	1	no	—
294	C₂×C₄	—	8	1	no	—
295	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
296	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
297	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
298	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
299	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
300	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
301	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
302	C₂×C₄	#135×C₅₃	1	2	yes	—
303	C₈	—	2	4	no	—
304	C₈	—	2	2	no	—
305	C₈	—	8	1	no	—
306	C₈	—	8	1	no	—
307	C₈	#104×C₅₃	1	4	yes	complemented
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	8	1	no	—
311	C₂²	—	8	1	no	—
312	C₂²	—	8	1	no	—
313	C₂²	C₅₃×#390	1	1	yes	characteristic
314	C₂²	C₅₃×#465	1	2	yes	—
315	C₄	—	2	8	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	8	1	no	—
319	C₄	—	8	1	no	—
320	C₄	—	8	1	no	—
321	C₄	C₂×#104×C₅₃	1	2	yes	—
322	C₄	C₄×SD₁₆×C₅₃	1	2	yes	—
323	C₄	C₅₃×#443	1	2	yes	—
324	C₄	C₅₃×#492	1	1	yes	characteristic
325	C₄	C₅₃×#495	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	16	1	no	—
328	C₂	C₅₃×#1962	1	1	yes	characteristic
329	C₂	C₅₃×#1968	1	1	yes	characteristic
330	C₂	C₅₃×#1974	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#127
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#390
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×#1994	C₂	semidirect
4. C₄²×C₈×C₅₃	C₂	semidirect
5. C₅₃×#1566	C₂	semidirect
6. C₅₃×#1493	C₂	semidirect
10. C₅₃×#498	C₄	semidirect
11. C₅₃×#492	C₄	semidirect
12. C₅₃×#487	C₄	semidirect
13. C₅₃×#486	C₄	semidirect
105. C₄²×C₅₃	SD₁₆	semidirect
141. C₈×C₅₃	#104	semidirect
142. #9843	C₅₃	direct (N × H)
162. #1994	C₂×C₅₃	semidirect
164. C₄²×C₈	C₂×C₅₃	semidirect
165. #1566	C₂×C₅₃	semidirect
166. #1493	C₂×C₅₃	semidirect
175. #498	C₄×C₅₃	semidirect
176. #492	C₄×C₅₃	semidirect
177. #487	C₄×C₅₃	semidirect
178. #486	C₄×C₅₃	semidirect
192. C₅₃	#9843	direct (N × H)
271. C₄²	SD₁₆×C₅₃	semidirect
307. C₈	#104×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3167	2
C₅₃×#1994	2
C₅₃×#1962	2
C₄²×C₈×C₅₃	2
C₅₃×#1566	2
C₅₃×#1493	2
C₅₃×#1492	2
#9843	53

Subgroups — C53×#9843 #27217355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9843 #27217355