Subgroups — C₅₃×#19651 #27227163

Order 13568.

806 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3229	C₂	1	2	yes	maximal, complemented
2	C₅₃×#3208	C₂	1	2	yes	maximal
3	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×#492	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1714	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1708	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1705	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×SD₁₆×C₅₃	—	2	2	no	—
10	C₅₃×#533	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#500	C₂²	1	2	yes	complemented
14	C₅₃×#499	—	2	2	no	—
15	C₅₃×#498	C₂²	1	4	yes	complemented
16	C₅₃×#498	C₂²	1	1	yes	characteristic, complemented
17	C₅₃×#498	C₂²	1	1	yes	characteristic, complemented
18	C₅₃×#492	—	2	2	no	—
19	C₅₃×#492	C₂²	1	4	yes	—
20	C₅₃×#492	C₂²	1	2	yes	—
21	C₅₃×#492	C₂²	1	2	yes	—
22	C₅₃×#487	C₂²	1	4	yes	complemented
23	C₅₃×#486	C₂²	1	4	yes	—
24	C₅₃×#483	—	2	4	no	—
25	C₅₃×#475	—	2	4	no	—
26	C₅₃×#460	—	2	4	no	—
27	C₅₃×#450	—	2	4	no	—
28	C₅₃×#431	C₂²	1	2	yes	—
29	C₅₃×#427	—	2	2	no	—
30	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#322	C₂²	1	2	yes	complemented
32	C₈²×C₅₃	C₂²	1	2	yes	complemented
33	C₂²×Q₈×C₅₃	—	2	1	no	—
34	C₂²×D₄×C₅₃	—	2	1	no	—
35	C₂²×D₄×C₅₃	—	2	2	no	—
36	C₂×SD₁₆×C₅₃	—	2	4	no	—
37	C₂×SD₁₆×C₅₃	—	2	4	no	—
38	C₂×SD₁₆×C₅₃	—	2	8	no	—
39	C₂×SD₁₆×C₅₃	—	4	8	no	—
40	C₂²×C₈×C₅₃	D₄	1	2	yes	—
41	C₄⋊Q₈×C₅₃	—	2	1	no	—
42	C₄⋊Q₈×C₅₃	—	2	2	no	—
43	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
44	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
45	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
46	#127×C₅₃	—	2	1	no	—
47	#127×C₅₃	—	2	2	no	—
48	#127×C₅₃	C₂³	1	2	yes	—
49	#125×C₅₃	—	2	2	no	—
50	C₂²⋊Q₈×C₅₃	—	2	2	no	—
51	C₂²⋊Q₈×C₅₃	—	2	2	no	—
52	C₄×Q₈×C₅₃	—	2	2	no	—
53	C₄×Q₈×C₅₃	—	2	2	no	—
54	#117×C₅₃	—	2	2	no	—
55	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
56	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
57	#107×C₅₃	—	2	4	no	—
58	#106×C₅₃	—	2	2	no	—
59	#106×C₅₃	—	2	2	no	—
60	#106×C₅₃	—	2	8	no	—
61	C₄⋊C₈×C₅₃	D₄	1	4	yes	—
62	#103×C₅₃	—	2	8	no	—
63	#102×C₅₃	—	2	8	no	—
64	C₂²⋊C₈×C₅₃	D₄	1	4	yes	—
65	C₄×C₈×C₅₃	D₄	1	2	yes	—
66	C₄×C₈×C₅₃	C₂³	1	4	yes	—
67	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₂⁴×C₅₃	—	4	1	no	—
70	C₂×Q₈×C₅₃	—	2	2	no	—
71	C₂×Q₈×C₅₃	—	2	2	no	—
72	C₂×Q₈×C₅₃	—	2	2	no	—
73	C₂×Q₈×C₅₃	—	4	1	no	—
74	C₂×Q₈×C₅₃	—	4	4	no	—
75	C₂×D₄×C₅₃	—	2	4	no	—
76	C₂×D₄×C₅₃	—	2	2	no	—
77	C₂×D₄×C₅₃	—	4	2	no	—
78	C₂×D₄×C₅₃	—	4	1	no	—
79	C₂×D₄×C₅₃	—	4	4	no	—
80	C₂×D₄×C₅₃	—	8	4	no	—
81	C₂²×C₄×C₅₃	—	4	1	no	—
82	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
84	SD₁₆×C₅₃	—	4	8	no	—
85	SD₁₆×C₅₃	—	4	8	no	—
86	SD₁₆×C₅₃	—	8	8	no	—
87	C₂×C₈×C₅₃	SD₁₆	1	8	yes	complemented
88	C₂×C₈×C₅₃	C₂×D₄	1	8	yes	—
89	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
91	C₄⋊C₄×C₅₃	—	2	2	no	—
92	C₄⋊C₄×C₅₃	—	2	4	no	—
93	C₄⋊C₄×C₅₃	—	2	2	no	—
94	C₄⋊C₄×C₅₃	—	2	2	no	—
95	C₄⋊C₄×C₅₃	—	2	2	no	—
96	C₄⋊C₄×C₅₃	—	4	2	no	—
97	C₄⋊C₄×C₅₃	—	4	2	no	—
98	C₄⋊C₄×C₅₃	—	4	2	no	—
99	C₂²⋊C₄×C₅₃	—	4	2	no	—
100	C₄²×C₅₃	—	4	2	no	—
101	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₂×D₄	1	2	yes	—
103	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
104	C₂³×C₅₃	—	4	2	no	—
105	C₂³×C₅₃	—	8	4	no	—
106	C₂³×C₅₃	—	8	2	no	—
107	C₂³×C₅₃	#127	1	1	yes	characteristic
108	Q₈×C₅₃	—	4	4	no	—
109	Q₈×C₅₃	—	4	4	no	—
110	Q₈×C₅₃	—	4	4	no	—
111	Q₈×C₅₃	—	8	2	no	—
112	D₄×C₅₃	—	4	4	no	—
113	D₄×C₅₃	—	4	8	no	—
114	D₄×C₅₃	—	8	4	no	—
115	D₄×C₅₃	—	8	2	no	—
116	C₂×C₄×C₅₃	—	2	4	no	—
117	C₂×C₄×C₅₃	—	4	2	no	—
118	C₂×C₄×C₅₃	—	4	2	no	—
119	C₂×C₄×C₅₃	—	4	2	no	—
120	C₂×C₄×C₅₃	—	8	2	no	—
121	C₂×C₄×C₅₃	#127	1	2	yes	—
122	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
124	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
126	C₈×C₅₃	—	2	8	no	—
127	C₈×C₅₃	C₂×SD₁₆	1	4	yes	—
128	C₈×C₅₃	C₂×SD₁₆	1	4	yes	—
129	#19651	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂²×C₅₃	—	2	2	no	—
131	C₂²×C₅₃	—	8	2	no	—
132	C₂²×C₅₃	—	8	4	no	—
133	C₂²×C₅₃	—	16	4	no	—
134	C₂²×C₅₃	#492	1	2	yes	—
135	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
136	C₄×C₅₃	—	2	2	no	—
137	C₄×C₅₃	—	8	2	no	—
138	C₄×C₅₃	—	8	2	no	—
139	C₄×C₅₃	—	8	2	no	—
140	C₄×C₅₃	#492	1	2	yes	—
141	C₄×C₅₃	C₂²×SD₁₆	1	2	yes	—
142	C₄×C₅₃	#577	1	4	yes	—
143	#3229	C₂×C₅₃	1	2	yes	complemented
144	#3208	C₂×C₅₃	1	2	yes	—
145	#3160	C₂×C₅₃	1	1	yes	characteristic, complemented
146	C₂×#492	C₂×C₅₃	1	1	yes	characteristic
147	#1714	C₂×C₅₃	1	4	yes	complemented
148	#1708	C₂×C₅₃	1	2	yes	complemented
149	#1705	C₂×C₅₃	1	2	yes	complemented
150	#1450	C₂×C₅₃	1	1	yes	characteristic, complemented
151	C₂×C₅₃	—	2	2	no	—
152	C₂×C₅₃	—	16	2	no	—
153	C₂×C₅₃	C₂×#492	1	1	yes	characteristic
154	C₂×C₅₃	#3154	1	2	yes	—
155	C₂²×SD₁₆	—	2	2	no	—
156	#533	C₂²×C₅₃	1	2	yes	—
157	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
158	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
159	#500	C₂²×C₅₃	1	2	yes	complemented
160	#499	—	2	2	no	—
161	#498	C₂²×C₅₃	1	1	yes	characteristic, complemented
162	#498	C₂²×C₅₃	1	1	yes	characteristic, complemented
163	#498	C₂²×C₅₃	1	4	yes	complemented
164	#492	—	2	2	no	—
165	#492	C₂²×C₅₃	1	2	yes	—
166	#492	C₂²×C₅₃	1	2	yes	—
167	#492	C₂²×C₅₃	1	4	yes	—
168	#487	C₂²×C₅₃	1	4	yes	complemented
169	#486	C₂²×C₅₃	1	4	yes	—
170	#483	—	2	4	no	—
171	#475	—	2	4	no	—
172	#460	—	2	4	no	—
173	#450	—	2	4	no	—
174	#431	C₂²×C₅₃	1	2	yes	—
175	#427	—	2	2	no	—
176	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
177	#322	C₂²×C₅₃	1	2	yes	complemented
178	C₈²	C₂²×C₅₃	1	2	yes	complemented
179	C₅₃	#19651	1	1	yes	characteristic, direct product factor, Sylow
180	C₂²×Q₈	—	2	1	no	—
181	C₂²×D₄	—	2	1	no	—
182	C₂²×D₄	—	2	2	no	—
183	C₂×SD₁₆	—	2	4	no	—
184	C₂×SD₁₆	—	2	4	no	—
185	C₂×SD₁₆	—	2	8	no	—
186	C₂×SD₁₆	—	4	8	no	—
187	C₂²×C₈	D₄×C₅₃	1	2	yes	—
188	C₄⋊Q₈	—	2	2	no	—
189	C₄⋊Q₈	—	2	1	no	—
190	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
191	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
192	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
193	#127	—	2	2	no	—
194	#127	—	2	1	no	—
195	#127	C₂³×C₅₃	1	2	yes	—
196	#125	—	2	2	no	—
197	C₂²⋊Q₈	—	2	2	no	—
198	C₂²⋊Q₈	—	2	2	no	—
199	C₄×Q₈	—	2	2	no	—
200	C₄×Q₈	—	2	2	no	—
201	#117	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
204	#107	—	2	4	no	—
205	#106	—	2	2	no	—
206	#106	—	2	2	no	—
207	#106	—	2	8	no	—
208	C₄⋊C₈	D₄×C₅₃	1	4	yes	—
209	#103	—	2	8	no	—
210	#102	—	2	8	no	—
211	C₂²⋊C₈	D₄×C₅₃	1	4	yes	—
212	C₄×C₈	D₄×C₅₃	1	2	yes	—
213	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
214	C₄×C₈	C₂³×C₅₃	1	4	yes	—
215	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
216	C₂⁴	—	4	1	no	—
217	C₂×Q₈	—	2	2	no	—
218	C₂×Q₈	—	2	2	no	—
219	C₂×Q₈	—	2	2	no	—
220	C₂×Q₈	—	4	1	no	—
221	C₂×Q₈	—	4	4	no	—
222	C₂×D₄	—	2	4	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	—	4	2	no	—
225	C₂×D₄	—	4	1	no	—
226	C₂×D₄	—	4	4	no	—
227	C₂×D₄	—	8	4	no	—
228	C₂²×C₄	—	4	1	no	—
229	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
230	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
231	SD₁₆	—	4	8	no	—
232	SD₁₆	—	4	8	no	—
233	SD₁₆	—	8	8	no	—
234	C₂×C₈	SD₁₆×C₅₃	1	8	yes	complemented
235	C₂×C₈	C₂×D₄×C₅₃	1	8	yes	—
236	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
237	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₄²	—	4	2	no	—
248	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
249	C₄²	C₂×D₄×C₅₃	1	2	yes	—
250	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
251	C₂³	—	4	2	no	—
252	C₂³	—	8	4	no	—
253	C₂³	—	8	2	no	—
254	C₂³	#127×C₅₃	1	1	yes	characteristic
255	Q₈	—	4	4	no	—
256	Q₈	—	4	4	no	—
257	Q₈	—	4	4	no	—
258	Q₈	—	8	2	no	—
259	D₄	—	4	4	no	—
260	D₄	—	4	8	no	—
261	D₄	—	8	4	no	—
262	D₄	—	8	2	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	8	2	no	—
268	C₂×C₄	#127×C₅₃	1	2	yes	—
269	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
270	C₂×C₄	C₂×SD₁₆×C₅₃	1	4	yes	—
271	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
272	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
273	C₈	—	2	8	no	—
274	C₈	C₂×SD₁₆×C₅₃	1	4	yes	—
275	C₈	C₂×SD₁₆×C₅₃	1	4	yes	—
276	C₂²	—	2	2	no	—
277	C₂²	—	8	2	no	—
278	C₂²	—	8	4	no	—
279	C₂²	—	16	4	no	—
280	C₂²	C₅₃×#492	1	2	yes	—
281	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	8	2	no	—
284	C₄	—	8	2	no	—
285	C₄	—	8	2	no	—
286	C₄	C₅₃×#492	1	2	yes	—
287	C₄	C₂²×SD₁₆×C₅₃	1	2	yes	—
288	C₄	C₅₃×#577	1	4	yes	—
289	C₂	—	2	2	no	—
290	C₂	—	16	2	no	—
291	C₂	C₂×C₅₃×#492	1	1	yes	characteristic
292	C₂	C₅₃×#3154	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₂×#127×C₅₃	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3229	C₂	semidirect
3. C₅₃×#3160	C₂	semidirect
5. C₅₃×#1714	C₂	semidirect
6. C₅₃×#1708	C₂	semidirect
7. C₅₃×#1705	C₂	semidirect
8. C₅₃×#1450	C₂	semidirect
13. C₅₃×#500	C₂²	semidirect
15. C₅₃×#498	C₂²	semidirect
16. C₅₃×#498	C₂²	semidirect
17. C₅₃×#498	C₂²	semidirect
22. C₅₃×#487	C₂²	semidirect
31. C₅₃×#322	C₂²	semidirect
32. C₈²×C₅₃	C₂²	semidirect
87. C₂×C₈×C₅₃	SD₁₆	semidirect
129. #19651	C₅₃	direct (N × H)
143. #3229	C₂×C₅₃	semidirect
145. #3160	C₂×C₅₃	semidirect
147. #1714	C₂×C₅₃	semidirect
148. #1708	C₂×C₅₃	semidirect
149. #1705	C₂×C₅₃	semidirect
150. #1450	C₂×C₅₃	semidirect
159. #500	C₂²×C₅₃	semidirect
161. #498	C₂²×C₅₃	semidirect
162. #498	C₂²×C₅₃	semidirect
163. #498	C₂²×C₅₃	semidirect
168. #487	C₂²×C₅₃	semidirect
177. #322	C₂²×C₅₃	semidirect
178. C₈²	C₂²×C₅₃	semidirect
179. C₅₃	#19651	direct (N × H)
234. C₂×C₈	SD₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3229	2
C₅₃×#3208	2
C₅₃×#3160	2
C₂×C₅₃×#492	2
C₅₃×#1714	2
C₅₃×#1708	2
C₅₃×#1705	2
C₅₃×#1450	2
#19651	53

Subgroups — C53×#19651 #27227163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19651 #27227163