Subgroups — C₅₃×#18091 #27225603

Order 13568.

644 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3169	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2979	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1594	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1581	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1580	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1484	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1451	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#532	C₂²	1	2	yes	—
10	C₅₃×#500	C₄	1	1	yes	characteristic
11	C₅₃×#493	C₄	1	1	yes	characteristic
12	C₅₃×#486	C₄	1	2	yes	—
13	C₅₃×#466	C₄	1	2	yes	—
14	C₅₃×#466	C₄	1	2	yes	—
15	C₄×D₈×C₅₃	C₂²	1	2	yes	—
16	C₄×D₈×C₅₃	C₂²	1	2	yes	—
17	C₅₃×#436	—	2	2	no	—
18	C₅₃×#436	C₂²	1	4	yes	—
19	C₈×D₄×C₅₃	—	2	2	no	—
20	C₈×D₄×C₅₃	C₂²	1	4	yes	—
21	C₅₃×#431	C₂²	1	2	yes	—
22	C₅₃×#431	C₂²	1	1	yes	characteristic
23	C₅₃×#408	—	2	4	no	—
24	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#335	C₂²	1	2	yes	—
27	C₅₃×#334	C₂²	1	1	yes	characteristic
28	C₅₃×#334	C₂²	1	1	yes	characteristic
29	C₅₃×D₄⋊C₈	C₂²	1	4	yes	—
30	C₅₃×D₄⋊C₈	C₂²	1	4	yes	—
31	C₅₃×#322	C₂²	1	2	yes	—
32	C₅₃×#322	C₂²	1	2	yes	—
33	C₂×#41×C₅₃	—	2	2	no	—
34	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×M₁₆×C₅₃	—	2	4	no	—
37	C₂×M₁₆×C₅₃	—	2	1	no	—
38	C₂²×C₈×C₅₃	—	2	4	no	—
39	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
42	#127×C₅₃	C₂×C₄	1	2	yes	—
43	#124×C₅₃	—	2	2	no	—
44	#121×C₅₃	—	2	2	no	—
45	#121×C₅₃	C₂×C₄	1	4	yes	—
46	C₄×D₄×C₅₃	—	2	2	no	—
47	C₄×D₄×C₅₃	C₂³	1	4	yes	—
48	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	#107×C₅₃	C₂×C₄	1	2	yes	—
50	#107×C₅₃	C₂×C₄	1	2	yes	—
51	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
52	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
53	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
54	#102×C₅₃	C₂×C₄	1	4	yes	—
55	#102×C₅₃	C₂×C₄	1	4	yes	—
56	C₂²⋊C₈×C₅₃	—	2	4	no	—
57	C₂²⋊C₈×C₅₃	—	2	4	no	—
58	C₂²⋊C₈×C₅₃	—	2	2	no	—
59	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
60	#97×C₅₃	—	2	1	no	—
61	C₄×C₈×C₅₃	—	2	2	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	C₂³	1	2	yes	—
65	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₅₃	C₂³	1	2	yes	—
68	#41×C₅₃	—	8	2	no	—
69	#41×C₅₃	—	8	2	no	—
70	C₂×Q₈×C₅₃	—	2	2	no	—
71	C₂×D₄×C₅₃	—	2	4	no	—
72	C₂×D₄×C₅₃	—	2	2	no	—
73	C₂×D₄×C₅₃	—	2	2	no	—
74	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₅₃	—	2	4	no	—
76	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
79	D₈×C₅₃	—	2	4	no	—
80	D₈×C₅₃	—	4	2	no	—
81	M₁₆×C₅₃	—	4	2	no	—
82	M₁₆×C₅₃	—	8	4	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	—	4	1	no	—
88	C₂×C₈×C₅₃	—	8	4	no	—
89	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
94	C₄⋊C₄×C₅₃	—	2	2	no	—
95	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
96	C₂²⋊C₄×C₅₃	—	2	4	no	—
97	C₂²⋊C₄×C₅₃	—	2	4	no	—
98	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₅₃	#41	1	1	yes	characteristic
101	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
102	C₂³×C₅₃	—	2	4	no	—
103	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
104	Q₈×C₅₃	—	8	2	no	—
105	D₄×C₅₃	—	2	4	no	—
106	D₄×C₅₃	—	2	4	no	—
107	D₄×C₅₃	—	4	4	no	—
108	D₄×C₅₃	—	8	2	no	—
109	D₄×C₅₃	—	8	4	no	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	—	2	1	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	2	2	no	—
114	C₂×C₄×C₅₃	—	2	4	no	—
115	C₂×C₄×C₅₃	—	4	4	no	—
116	C₂×C₄×C₅₃	—	4	4	no	—
117	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
123	C₈×C₅₃	—	2	1	no	—
124	C₈×C₅₃	—	2	1	no	—
125	C₈×C₅₃	—	2	1	no	—
126	C₈×C₅₃	—	2	4	no	—
127	C₈×C₅₃	—	4	2	no	—
128	C₈×C₅₃	—	4	2	no	—
129	C₈×C₅₃	—	4	2	no	—
130	C₈×C₅₃	#131	1	2	yes	—
131	#18091	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂²×C₅₃	—	2	1	no	—
133	C₂²×C₅₃	—	2	1	no	—
134	C₂²×C₅₃	—	2	1	no	—
135	C₂²×C₅₃	—	4	4	no	—
136	C₂²×C₅₃	—	4	4	no	—
137	C₂²×C₅₃	—	4	4	no	—
138	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
139	C₄×C₅₃	—	2	1	no	—
140	C₄×C₅₃	—	2	1	no	—
141	C₄×C₅₃	—	2	1	no	—
142	C₄×C₅₃	—	4	4	no	—
143	C₄×C₅₃	#444	1	2	yes	—
144	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
145	C₄×C₅₃	#568	1	1	yes	characteristic
146	C₄×C₅₃	#576	1	2	yes	—
147	#3169	C₂×C₅₃	1	1	yes	characteristic
148	#2979	C₂×C₅₃	1	2	yes	complemented
149	#1594	C₂×C₅₃	1	2	yes	complemented
150	#1581	C₂×C₅₃	1	4	yes	complemented
151	#1580	C₂×C₅₃	1	2	yes	complemented
152	#1564	C₂×C₅₃	1	1	yes	characteristic, complemented
153	#1484	C₂×C₅₃	1	2	yes	complemented
154	#1451	C₂×C₅₃	1	1	yes	characteristic, complemented
155	C₂×C₅₃	—	4	1	no	—
156	C₂×C₅₃	—	8	4	no	—
157	C₂×C₅₃	#2935	1	1	yes	characteristic
158	C₂×C₅₃	#2948	1	1	yes	characteristic
159	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
160	#532	C₂²×C₅₃	1	2	yes	—
161	#500	C₄×C₅₃	1	1	yes	characteristic
162	#493	C₄×C₅₃	1	1	yes	characteristic
163	#486	C₄×C₅₃	1	2	yes	—
164	#466	C₄×C₅₃	1	2	yes	—
165	#466	C₄×C₅₃	1	2	yes	—
166	C₄×D₈	C₂²×C₅₃	1	2	yes	—
167	C₄×D₈	C₂²×C₅₃	1	2	yes	—
168	#436	—	2	2	no	—
169	#436	C₂²×C₅₃	1	4	yes	—
170	C₈×D₄	—	2	2	no	—
171	C₈×D₄	C₂²×C₅₃	1	4	yes	—
172	#431	C₂²×C₅₃	1	2	yes	—
173	#431	C₂²×C₅₃	1	1	yes	characteristic
174	#408	—	2	4	no	—
175	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
176	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
177	#335	C₂²×C₅₃	1	2	yes	—
178	#334	C₂²×C₅₃	1	1	yes	characteristic
179	#334	C₂²×C₅₃	1	1	yes	characteristic
180	D₄⋊C₈	C₂²×C₅₃	1	4	yes	—
181	D₄⋊C₈	C₂²×C₅₃	1	4	yes	—
182	#322	C₂²×C₅₃	1	2	yes	—
183	#322	C₂²×C₅₃	1	2	yes	—
184	C₅₃	#18091	1	1	yes	characteristic, direct product factor, Sylow
185	C₂×#41	—	2	2	no	—
186	C₂×D₈	C₂×C₄×C₅₃	1	2	yes	—
187	C₂×D₈	C₂×C₄×C₅₃	1	2	yes	—
188	C₂×M₁₆	—	2	1	no	—
189	C₂×M₁₆	—	2	4	no	—
190	C₂²×C₈	—	2	4	no	—
191	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
192	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
193	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
194	#127	C₂×C₄×C₅₃	1	2	yes	—
195	#124	—	2	2	no	—
196	#121	—	2	2	no	—
197	#121	C₂×C₄×C₅₃	1	4	yes	—
198	C₄×D₄	—	2	2	no	—
199	C₄×D₄	C₂³×C₅₃	1	4	yes	—
200	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
201	#107	C₂×C₄×C₅₃	1	2	yes	—
202	#107	C₂×C₄×C₅₃	1	2	yes	—
203	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
204	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
205	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
206	#102	C₂×C₄×C₅₃	1	4	yes	—
207	#102	C₂×C₄×C₅₃	1	4	yes	—
208	C₂²⋊C₈	—	2	2	no	—
209	C₂²⋊C₈	—	2	4	no	—
210	C₂²⋊C₈	—	2	4	no	—
211	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
212	#97	—	2	1	no	—
213	C₄×C₈	—	2	2	no	—
214	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
215	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
216	C₄×C₈	C₂³×C₅₃	1	2	yes	—
217	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
218	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
219	C₄×C₈	C₂³×C₅₃	1	2	yes	—
220	#41	—	8	2	no	—
221	#41	—	8	2	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
230	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
231	D₈	—	2	4	no	—
232	D₈	—	4	2	no	—
233	M₁₆	—	4	2	no	—
234	M₁₆	—	8	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	4	1	no	—
240	C₂×C₈	—	8	4	no	—
241	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
242	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
243	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
244	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
245	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
248	C₂²⋊C₄	—	2	4	no	—
249	C₂²⋊C₄	—	2	4	no	—
250	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
251	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
252	C₄²	#41×C₅₃	1	1	yes	characteristic
253	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
254	C₂³	—	2	4	no	—
255	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
256	Q₈	—	8	2	no	—
257	D₄	—	2	4	no	—
258	D₄	—	2	4	no	—
259	D₄	—	4	4	no	—
260	D₄	—	8	2	no	—
261	D₄	—	8	4	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
270	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
271	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
272	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
273	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
274	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	2	4	no	—
279	C₈	—	4	2	no	—
280	C₈	—	4	2	no	—
281	C₈	—	4	2	no	—
282	C₈	#131×C₅₃	1	2	yes	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	4	4	no	—
287	C₂²	—	4	4	no	—
288	C₂²	—	4	4	no	—
289	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	4	4	no	—
294	C₄	C₅₃×#444	1	2	yes	—
295	C₄	C₂×#131×C₅₃	1	1	yes	characteristic
296	C₄	C₅₃×#568	1	1	yes	characteristic
297	C₄	C₅₃×#576	1	2	yes	—
298	C₂	—	4	1	no	—
299	C₂	—	8	4	no	—
300	C₂	C₅₃×#2935	1	1	yes	characteristic
301	C₂	C₅₃×#2948	1	1	yes	characteristic
302	C₂	C₂×C₅₃×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₅₃×#3169	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₅₃×#431 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2979	C₂	semidirect
3. C₅₃×#1594	C₂	semidirect
4. C₅₃×#1581	C₂	semidirect
5. C₅₃×#1580	C₂	semidirect
6. C₅₃×#1564	C₂	semidirect
7. C₅₃×#1484	C₂	semidirect
8. C₅₃×#1451	C₂	semidirect
131. #18091	C₅₃	direct (N × H)
148. #2979	C₂×C₅₃	semidirect
149. #1594	C₂×C₅₃	semidirect
150. #1581	C₂×C₅₃	semidirect
151. #1580	C₂×C₅₃	semidirect
152. #1564	C₂×C₅₃	semidirect
153. #1484	C₂×C₅₃	semidirect
154. #1451	C₂×C₅₃	semidirect
184. C₅₃	#18091	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×#3169	2
C₅₃×#2979	2
C₅₃×#1594	2
C₅₃×#1581	2
C₅₃×#1580	2
C₅₃×#1564	2
C₅₃×#1484	2
C₅₃×#1451	2
#18091	53

Subgroups — C53×#18091 #27225603

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#18091 #27225603