Subgroups — C₅₃×#9753 #27217265

Order 13568.

502 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3147	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1502	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1480	C₂	1	2	yes	maximal, complemented
6	C₂×#135×C₅₃	—	2	1	no	—
7	C₂×#135×C₅₃	—	2	1	no	—
8	C₅₃×#532	C₂²	1	2	yes	—
9	C₅₃×#495	—	2	1	no	—
10	C₅₃×#495	—	2	1	no	—
11	C₅₃×#494	C₄	1	1	yes	characteristic, complemented
12	C₅₃×#493	C₄	1	1	yes	characteristic, complemented
13	C₅₃×#492	C₄	1	2	yes	complemented
14	C₅₃×#431	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#97×C₅₃	—	2	1	no	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#336	—	2	1	no	—
20	C₅₃×#335	—	2	1	no	—
21	C₅₃×#334	—	2	1	no	—
22	C₅₃×#327	—	2	4	no	—
23	C₅₃×D₄⋊C₈	—	2	4	no	—
24	C₂×#41×C₅₃	—	2	2	no	—
25	#135×C₅₃	—	4	2	no	—
26	#135×C₅₃	—	4	2	no	—
27	#135×C₅₃	—	4	2	no	—
28	#135×C₅₃	—	4	2	no	—
29	C₂×Q₁₆×C₅₃	—	2	1	no	—
30	C₂×Q₁₆×C₅₃	—	2	1	no	—
31	C₂×SD₁₆×C₅₃	—	2	2	no	—
32	C₂×SD₁₆×C₅₃	—	2	2	no	—
33	C₂×D₈×C₅₃	—	2	1	no	—
34	C₂×D₈×C₅₃	—	2	1	no	—
35	C₂²×C₈×C₅₃	—	2	1	no	—
36	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
39	#127×C₅₃	C₂×C₄	1	2	yes	—
40	#124×C₅₃	—	2	2	no	—
41	#121×C₅₃	—	2	2	no	—
42	#121×C₅₃	—	2	2	no	—
43	C₄×D₄×C₅₃	—	2	2	no	—
44	C₄×D₄×C₅₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
46	C₂×C₄²×C₅₃	—	2	1	no	—
47	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₅₃	—	2	2	no	—
49	C₄⋊C₈×C₅₃	—	2	1	no	—
50	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
51	C₂²⋊C₈×C₅₃	—	2	2	no	—
52	#97×C₅₃	—	4	2	no	—
53	C₄×C₈×C₅₃	—	2	2	no	—
54	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
57	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
58	#95×C₅₃	D₄	1	2	yes	—
59	#41×C₅₃	—	4	2	no	—
60	#41×C₅₃	—	4	2	no	—
61	#41×C₅₃	—	4	2	no	—
62	#41×C₅₃	—	4	2	no	—
63	C₂×Q₈×C₅₃	—	2	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	2	no	—
68	C₂²×C₄×C₅₃	—	2	2	no	—
69	C₂²×C₄×C₅₃	—	4	2	no	—
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
73	Q₁₆×C₅₃	—	4	2	no	—
74	Q₁₆×C₅₃	—	4	2	no	—
75	SD₁₆×C₅₃	—	4	4	no	—
76	SD₁₆×C₅₃	—	4	4	no	—
77	D₈×C₅₃	—	4	2	no	—
78	D₈×C₅₃	—	4	2	no	—
79	C₂×C₈×C₅₃	—	2	1	no	—
80	C₂×C₈×C₅₃	—	2	1	no	—
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	—	4	1	no	—
83	C₂×C₈×C₅₃	—	4	1	no	—
84	C₂×C₈×C₅₃	—	4	2	no	—
85	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈×C₅₃	D₈	1	2	yes	—
90	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
91	C₄⋊C₄×C₅₃	—	2	2	no	—
92	C₄⋊C₄×C₅₃	—	2	2	no	—
93	C₄⋊C₄×C₅₃	—	4	2	no	—
94	C₂²⋊C₄×C₅₃	—	4	2	no	—
95	C₂²⋊C₄×C₅₃	—	4	2	no	—
96	C₄²×C₅₃	—	4	1	no	—
97	C₄²×C₅₃	—	4	1	no	—
98	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₅₃	#41	1	1	yes	characteristic
102	C₂³×C₅₃	—	4	2	no	—
103	C₂³×C₅₃	#121	1	1	yes	characteristic
104	Q₈×C₅₃	—	4	2	no	—
105	Q₈×C₅₃	—	4	2	no	—
106	D₄×C₅₃	—	4	2	no	—
107	D₄×C₅₃	—	4	2	no	—
108	D₄×C₅₃	—	4	4	no	—
109	D₄×C₅₃	—	8	2	no	—
110	D₄×C₅₃	—	8	2	no	—
111	D₄×C₅₃	—	8	2	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	4	1	no	—
114	C₂×C₄×C₅₃	—	4	2	no	—
115	C₂×C₄×C₅₃	—	4	2	no	—
116	C₂×C₄×C₅₃	—	4	2	no	—
117	C₂×C₄×C₅₃	—	4	2	no	—
118	C₂×C₄×C₅₃	—	8	2	no	—
119	C₂×C₄×C₅₃	—	8	2	no	—
120	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
124	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
125	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₈×C₅₃	—	2	1	no	—
131	C₈×C₅₃	—	2	1	no	—
132	C₈×C₅₃	—	4	2	no	—
133	C₈×C₅₃	—	8	2	no	—
134	C₈×C₅₃	#102	1	2	yes	—
135	C₈×C₅₃	#102	1	2	yes	—
136	#9753	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂²×C₅₃	—	2	1	no	—
138	C₂²×C₅₃	—	2	1	no	—
139	C₂²×C₅₃	—	8	2	no	—
140	C₂²×C₅₃	—	8	2	no	—
141	C₂²×C₅₃	—	8	2	no	—
142	C₂²×C₅₃	#390	1	1	yes	characteristic
143	C₂²×C₅₃	#459	1	1	yes	characteristic
144	C₂²×C₅₃	#460	1	1	yes	characteristic
145	C₄×C₅₃	—	2	1	no	—
146	C₄×C₅₃	—	2	1	no	—
147	C₄×C₅₃	—	2	1	no	—
148	C₄×C₅₃	—	8	2	no	—
149	C₄×C₅₃	—	8	2	no	—
150	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
151	C₄×C₅₃	#418	1	1	yes	characteristic
152	C₄×C₅₃	#444	1	2	yes	—
153	C₄×C₅₃	#459	1	1	yes	characteristic
154	C₄×C₅₃	#460	1	1	yes	characteristic
155	#3147	C₂×C₅₃	1	1	yes	characteristic
156	#1770	C₂×C₅₃	1	1	yes	characteristic, complemented
157	#1564	C₂×C₅₃	1	1	yes	characteristic, complemented
158	#1502	C₂×C₅₃	1	2	yes	complemented
159	#1480	C₂×C₅₃	1	2	yes	complemented
160	C₂×C₅₃	—	2	1	no	—
161	C₂×C₅₃	—	2	1	no	—
162	C₂×C₅₃	—	16	2	no	—
163	C₂×C₅₃	#1975	1	1	yes	characteristic
164	C₂×C₅₃	#1976	1	1	yes	characteristic
165	C₂×C₅₃	#1982	1	1	yes	characteristic
166	C₂×#135	—	2	1	no	—
167	C₂×#135	—	2	1	no	—
168	#532	C₂²×C₅₃	1	2	yes	—
169	#495	—	2	1	no	—
170	#495	—	2	1	no	—
171	#494	C₄×C₅₃	1	1	yes	characteristic, complemented
172	#493	C₄×C₅₃	1	1	yes	characteristic, complemented
173	#492	C₄×C₅₃	1	2	yes	complemented
174	#431	C₂²×C₅₃	1	2	yes	—
175	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
176	C₂×#97	—	2	1	no	—
177	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
178	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
179	#336	—	2	1	no	—
180	#335	—	2	1	no	—
181	#334	—	2	1	no	—
182	#327	—	2	4	no	—
183	D₄⋊C₈	—	2	4	no	—
184	C₅₃	#9753	1	1	yes	characteristic, direct product factor, Sylow
185	C₂×#41	—	2	2	no	—
186	#135	—	4	2	no	—
187	#135	—	4	2	no	—
188	#135	—	4	2	no	—
189	#135	—	4	2	no	—
190	C₂×Q₁₆	—	2	1	no	—
191	C₂×Q₁₆	—	2	1	no	—
192	C₂×SD₁₆	—	2	2	no	—
193	C₂×SD₁₆	—	2	2	no	—
194	C₂×D₈	—	2	1	no	—
195	C₂×D₈	—	2	1	no	—
196	C₂²×C₈	—	2	1	no	—
197	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
198	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
199	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
200	#127	C₂×C₄×C₅₃	1	2	yes	—
201	#124	—	2	2	no	—
202	#121	—	2	2	no	—
203	#121	—	2	2	no	—
204	C₄×D₄	—	2	2	no	—
205	C₄×D₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
209	C₄⋊C₈	—	2	2	no	—
210	C₄⋊C₈	—	2	1	no	—
211	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
212	C₂²⋊C₈	—	2	2	no	—
213	#97	—	4	2	no	—
214	C₄×C₈	—	2	2	no	—
215	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
216	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
217	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
218	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
219	#95	D₄×C₅₃	1	2	yes	—
220	#41	—	4	2	no	—
221	#41	—	4	2	no	—
222	#41	—	4	2	no	—
223	#41	—	4	2	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
233	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
234	Q₁₆	—	4	2	no	—
235	Q₁₆	—	4	2	no	—
236	SD₁₆	—	4	4	no	—
237	SD₁₆	—	4	4	no	—
238	D₈	—	4	2	no	—
239	D₈	—	4	2	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
247	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
248	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
249	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
250	C₂×C₈	D₈×C₅₃	1	2	yes	—
251	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₄²	—	4	1	no	—
258	C₄²	—	4	1	no	—
259	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
260	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
261	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
262	C₄²	#41×C₅₃	1	1	yes	characteristic
263	C₂³	—	4	2	no	—
264	C₂³	#121×C₅₃	1	1	yes	characteristic
265	Q₈	—	4	2	no	—
266	Q₈	—	4	2	no	—
267	D₄	—	4	2	no	—
268	D₄	—	4	2	no	—
269	D₄	—	4	4	no	—
270	D₄	—	8	2	no	—
271	D₄	—	8	2	no	—
272	D₄	—	8	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	8	2	no	—
280	C₂×C₄	—	8	2	no	—
281	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
282	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
283	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
284	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
285	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
286	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
287	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
288	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
289	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
290	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	4	2	no	—
294	C₈	—	8	2	no	—
295	C₈	#102×C₅₃	1	2	yes	—
296	C₈	#102×C₅₃	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	8	2	no	—
302	C₂²	C₅₃×#390	1	1	yes	characteristic
303	C₂²	C₅₃×#459	1	1	yes	characteristic
304	C₂²	C₅₃×#460	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	8	2	no	—
309	C₄	—	8	2	no	—
310	C₄	C₂×#102×C₅₃	1	1	yes	characteristic
311	C₄	C₅₃×#418	1	1	yes	characteristic
312	C₄	C₅₃×#444	1	2	yes	—
313	C₄	C₅₃×#459	1	1	yes	characteristic
314	C₄	C₅₃×#460	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	16	2	no	—
318	C₂	C₅₃×#1975	1	1	yes	characteristic
319	C₂	C₅₃×#1976	1	1	yes	characteristic
320	C₂	C₅₃×#1982	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#121
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#390
prime radical	C₅₃×#3147	C₂
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×#1770	C₂	semidirect
3. C₅₃×#1564	C₂	semidirect
4. C₅₃×#1502	C₂	semidirect
5. C₅₃×#1480	C₂	semidirect
11. C₅₃×#494	C₄	semidirect
12. C₅₃×#493	C₄	semidirect
13. C₅₃×#492	C₄	semidirect
136. #9753	C₅₃	direct (N × H)
156. #1770	C₂×C₅₃	semidirect
157. #1564	C₂×C₅₃	semidirect
158. #1502	C₂×C₅₃	semidirect
159. #1480	C₂×C₅₃	semidirect
171. #494	C₄×C₅₃	semidirect
172. #493	C₄×C₅₃	semidirect
173. #492	C₄×C₅₃	semidirect
184. C₅₃	#9753	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×#3147	2
C₅₃×#1770	2
C₅₃×#1564	2
C₅₃×#1502	2
C₅₃×#1480	2
#9753	53

Subgroups — C53×#9753 #27217265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9753 #27217265