Subgroups — C₅₃×#13019 #27220531

Order 13568.

496 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	C₂²×SD₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2948	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#361	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#2141	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1979	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1382	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
9	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#518	—	2	1	no	—
11	C₂×SD₃₂×C₅₃	—	2	2	no	—
12	C₂×SD₃₂×C₅₃	—	2	2	no	—
13	C₂×SD₃₂×C₅₃	C₄	1	4	yes	complemented
14	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#442	—	2	2	no	—
17	C₄×D₈×C₅₃	—	2	2	no	—
18	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#427	C₂²	1	1	yes	characteristic
20	C₅₃×#419	—	2	1	no	—
21	C₅₃×#418	—	2	1	no	—
22	C₂×#101×C₅₃	—	2	1	no	—
23	C₅₃×#405	C₂²	1	1	yes	characteristic
24	C₅₃×#361	—	2	2	no	—
25	C₅₃×#358	—	2	2	no	—
26	C₂²×Q₈×C₅₃	—	2	1	no	—
27	C₂²×D₄×C₅₃	—	2	1	no	—
28	C₂×Q₁₆×C₅₃	—	2	1	no	—
29	C₂×Q₁₆×C₅₃	—	2	1	no	—
30	C₂×Q₁₆×C₅₃	—	2	1	no	—
31	C₂×Q₁₆×C₅₃	—	2	1	no	—
32	C₂×Q₁₆×C₅₃	—	2	1	no	—
33	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
34	C₂×D₈×C₅₃	—	2	1	no	—
35	C₂×D₈×C₅₃	—	2	1	no	—
36	C₂×D₈×C₅₃	—	2	1	no	—
37	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×D₈×C₅₃	D₄	1	2	yes	—
39	C₂×D₈×C₅₃	D₄	1	2	yes	—
40	C₂×M₁₆×C₅₃	—	2	1	no	—
41	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₄×D₄×C₅₃	—	2	4	no	—
44	C₄×D₄×C₅₃	—	4	2	no	—
45	#117×C₅₃	—	2	1	no	—
46	#117×C₅₃	D₄	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₅₃	—	4	2	no	—
48	SD₃₂×C₅₃	—	2	8	no	—
49	M₃₂×C₅₃	—	2	1	no	—
50	M₃₂×C₅₃	—	2	1	no	—
51	C₂×C₁₆×C₅₃	—	2	1	no	—
52	C₂×C₁₆×C₅₃	—	2	1	no	—
53	C₂×C₁₆×C₅₃	—	2	1	no	—
54	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
55	#108×C₅₃	—	2	1	no	—
56	#108×C₅₃	—	2	1	no	—
57	#107×C₅₃	D₄	1	2	yes	—
58	#106×C₅₃	—	2	1	no	—
59	#103×C₅₃	—	4	2	no	—
60	#102×C₅₃	—	2	4	no	—
61	#101×C₅₃	—	4	2	no	—
62	#97×C₅₃	—	2	1	no	—
63	C₄×C₈×C₅₃	—	2	1	no	—
64	C₂⁴×C₅₃	—	4	1	no	—
65	C₂×Q₈×C₅₃	—	2	2	no	—
66	C₂×Q₈×C₅₃	—	4	1	no	—
67	C₂×Q₈×C₅₃	—	4	1	no	—
68	C₂×Q₈×C₅₃	—	4	1	no	—
69	C₂×Q₈×C₅₃	—	4	1	no	—
70	C₂×Q₈×C₅₃	—	4	1	no	—
71	C₂×D₄×C₅₃	—	2	2	no	—
72	C₂×D₄×C₅₃	—	2	2	no	—
73	C₂×D₄×C₅₃	—	2	2	no	—
74	C₂×D₄×C₅₃	—	4	1	no	—
75	C₂×D₄×C₅₃	—	4	1	no	—
76	C₂×D₄×C₅₃	—	4	1	no	—
77	C₂²×C₄×C₅₃	—	4	4	no	—
78	C₂²×C₄×C₅₃	—	4	1	no	—
79	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
80	Q₁₆×C₅₃	—	2	2	no	—
81	Q₁₆×C₅₃	—	2	2	no	—
82	Q₁₆×C₅₃	—	4	1	no	—
83	Q₁₆×C₅₃	—	4	1	no	—
84	D₈×C₅₃	—	2	2	no	—
85	D₈×C₅₃	—	2	2	no	—
86	D₈×C₅₃	—	2	2	no	—
87	D₈×C₅₃	C₂²⋊C₄	1	4	yes	—
88	M₁₆×C₅₃	—	2	1	no	—
89	M₁₆×C₅₃	—	2	1	no	—
90	M₁₆×C₅₃	—	4	1	no	—
91	M₁₆×C₅₃	—	4	1	no	—
92	C₂×C₈×C₅₃	—	2	1	no	—
93	C₂×C₈×C₅₃	—	4	1	no	—
94	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
99	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
100	C₄⋊C₄×C₅₃	—	2	2	no	—
101	C₄⋊C₄×C₅₃	D₈	1	2	yes	—
102	C₂²⋊C₄×C₅₃	—	2	1	no	—
103	C₂²⋊C₄×C₅₃	—	4	4	no	—
104	C₂²⋊C₄×C₅₃	—	4	1	no	—
105	C₄²×C₅₃	—	2	1	no	—
106	C₄²×C₅₃	—	4	1	no	—
107	C₁₆×C₅₃	—	2	2	no	—
108	C₁₆×C₅₃	—	4	1	no	—
109	C₂³×C₅₃	—	4	2	no	—
110	C₂³×C₅₃	—	4	2	no	—
111	C₂³×C₅₃	—	4	2	no	—
112	C₂³×C₅₃	—	16	2	no	—
113	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
114	Q₈×C₅₃	—	4	2	no	—
115	Q₈×C₅₃	—	4	2	no	—
116	Q₈×C₅₃	—	8	1	no	—
117	Q₈×C₅₃	—	8	1	no	—
118	D₄×C₅₃	—	2	4	no	—
119	D₄×C₅₃	—	4	2	no	—
120	D₄×C₅₃	—	4	2	no	—
121	D₄×C₅₃	—	4	2	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	4	2	no	—
124	C₂×C₄×C₅₃	—	4	2	no	—
125	C₂×C₄×C₅₃	—	8	1	no	—
126	C₂×C₄×C₅₃	—	8	1	no	—
127	C₂×C₄×C₅₃	—	16	4	no	—
128	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
130	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
133	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₈×C₅₃	—	4	1	no	—
135	C₈×C₅₃	—	8	1	no	—
136	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
138	C₈×C₅₃	#121	1	1	yes	characteristic
139	C₈×C₅₃	#123	1	1	yes	characteristic
140	#13019	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂²×C₅₃	—	2	2	no	—
142	C₂²×C₅₃	—	4	4	no	—
143	C₂²×C₅₃	—	8	4	no	—
144	C₂²×C₅₃	—	16	2	no	—
145	C₂²×C₅₃	—	16	2	no	—
146	C₂²×C₅₃	#442	1	1	yes	characteristic
147	C₂²×C₅₃	#447	1	1	yes	characteristic
148	C₂²×C₅₃	#463	1	1	yes	characteristic
149	C₄×C₅₃	—	4	2	no	—
150	C₄×C₅₃	—	8	2	no	—
151	C₄×C₅₃	—	8	2	no	—
152	C₄×C₅₃	#386	1	1	yes	characteristic
153	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
154	C₄×C₅₃	#449	1	1	yes	characteristic
155	C₄×C₅₃	#459	1	1	yes	characteristic
156	C₂²×SD₃₂	C₂×C₅₃	1	1	yes	characteristic
157	#2948	C₂×C₅₃	1	1	yes	characteristic
158	C₂×#361	C₂×C₅₃	1	1	yes	characteristic
159	#2141	C₂×C₅₃	1	1	yes	characteristic, complemented
160	#1979	C₂×C₅₃	1	1	yes	characteristic, complemented
161	#1382	C₂×C₅₃	1	1	yes	characteristic, complemented
162	#1379	C₂×C₅₃	1	1	yes	characteristic, complemented
163	C₂×C₅₃	—	2	1	no	—
164	C₂×C₅₃	—	2	1	no	—
165	C₂×C₅₃	—	8	4	no	—
166	C₂×C₅₃	#1879	1	1	yes	characteristic
167	C₂×C₅₃	#2194	1	2	yes	—
168	C₂²×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
169	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
170	#518	—	2	1	no	—
171	C₂×SD₃₂	—	2	2	no	—
172	C₂×SD₃₂	—	2	2	no	—
173	C₂×SD₃₂	C₄×C₅₃	1	4	yes	complemented
174	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
175	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
176	#442	—	2	2	no	—
177	C₄×D₈	—	2	2	no	—
178	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
179	#427	C₂²×C₅₃	1	1	yes	characteristic
180	#419	—	2	1	no	—
181	#418	—	2	1	no	—
182	C₂×#101	—	2	1	no	—
183	#405	C₂²×C₅₃	1	1	yes	characteristic
184	#361	—	2	2	no	—
185	#358	—	2	2	no	—
186	C₅₃	#13019	1	1	yes	characteristic, direct product factor, Sylow
187	C₂²×Q₈	—	2	1	no	—
188	C₂²×D₄	—	2	1	no	—
189	C₂×Q₁₆	—	2	1	no	—
190	C₂×Q₁₆	—	2	1	no	—
191	C₂×Q₁₆	—	2	1	no	—
192	C₂×Q₁₆	—	2	1	no	—
193	C₂×Q₁₆	—	2	1	no	—
194	C₂×Q₁₆	C₂×C₄×C₅₃	1	2	yes	complemented
195	C₂×D₈	—	2	1	no	—
196	C₂×D₈	—	2	1	no	—
197	C₂×D₈	—	2	1	no	—
198	C₂×D₈	C₂×C₄×C₅₃	1	2	yes	—
199	C₂×D₈	D₄×C₅₃	1	2	yes	—
200	C₂×D₈	D₄×C₅₃	1	2	yes	—
201	C₂×M₁₆	—	2	1	no	—
202	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
203	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
204	C₄×D₄	—	2	4	no	—
205	C₄×D₄	—	4	2	no	—
206	#117	—	2	1	no	—
207	#117	D₄×C₅₃	1	1	yes	characteristic
208	C₂×C₂²⋊C₄	—	4	2	no	—
209	SD₃₂	—	2	8	no	—
210	M₃₂	—	2	1	no	—
211	M₃₂	—	2	1	no	—
212	C₂×C₁₆	—	2	1	no	—
213	C₂×C₁₆	—	2	1	no	—
214	C₂×C₁₆	—	2	1	no	—
215	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	complemented
216	#108	—	2	1	no	—
217	#108	—	2	1	no	—
218	#107	D₄×C₅₃	1	2	yes	—
219	#106	—	2	1	no	—
220	#103	—	4	2	no	—
221	#102	—	2	4	no	—
222	#101	—	4	2	no	—
223	#97	—	2	1	no	—
224	C₄×C₈	—	2	1	no	—
225	C₂⁴	—	4	1	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	4	1	no	—
229	C₂×Q₈	—	4	1	no	—
230	C₂×Q₈	—	4	1	no	—
231	C₂×Q₈	—	4	1	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	4	1	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂²×C₄	—	4	4	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
241	Q₁₆	—	2	2	no	—
242	Q₁₆	—	2	2	no	—
243	Q₁₆	—	4	1	no	—
244	Q₁₆	—	4	1	no	—
245	D₈	—	2	2	no	—
246	D₈	—	2	2	no	—
247	D₈	—	2	2	no	—
248	D₈	C₂²⋊C₄×C₅₃	1	4	yes	—
249	M₁₆	—	2	1	no	—
250	M₁₆	—	2	1	no	—
251	M₁₆	—	4	1	no	—
252	M₁₆	—	4	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
256	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
257	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
258	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
259	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
260	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	D₈×C₅₃	1	2	yes	—
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	4	4	no	—
265	C₂²⋊C₄	—	4	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	4	1	no	—
268	C₁₆	—	2	2	no	—
269	C₁₆	—	4	1	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	—	4	2	no	—
273	C₂³	—	16	2	no	—
274	C₂³	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
275	Q₈	—	4	2	no	—
276	Q₈	—	4	2	no	—
277	Q₈	—	8	1	no	—
278	Q₈	—	8	1	no	—
279	D₄	—	2	4	no	—
280	D₄	—	4	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	8	1	no	—
287	C₂×C₄	—	8	1	no	—
288	C₂×C₄	—	16	4	no	—
289	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
290	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
291	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
292	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
293	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
294	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
295	C₈	—	4	1	no	—
296	C₈	—	8	1	no	—
297	C₈	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
298	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
299	C₈	#121×C₅₃	1	1	yes	characteristic
300	C₈	#123×C₅₃	1	1	yes	characteristic
301	C₂²	—	2	2	no	—
302	C₂²	—	4	4	no	—
303	C₂²	—	8	4	no	—
304	C₂²	—	16	2	no	—
305	C₂²	—	16	2	no	—
306	C₂²	C₅₃×#442	1	1	yes	characteristic
307	C₂²	C₅₃×#447	1	1	yes	characteristic
308	C₂²	C₅₃×#463	1	1	yes	characteristic
309	C₄	—	4	2	no	—
310	C₄	—	8	2	no	—
311	C₄	—	8	2	no	—
312	C₄	C₅₃×#386	1	1	yes	characteristic
313	C₄	C₄×D₈×C₅₃	1	1	yes	characteristic
314	C₄	C₅₃×#449	1	1	yes	characteristic
315	C₄	C₅₃×#459	1	1	yes	characteristic
316	C₂	—	2	1	no	—
317	C₂	—	2	1	no	—
318	C₂	—	8	4	no	—
319	C₂	C₅₃×#1879	1	1	yes	characteristic
320	C₂	C₅₃×#2194	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	#447
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₂²×D₈×C₅₃	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₅₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#2141	C₂	semidirect
5. C₅₃×#1979	C₂	semidirect
6. C₅₃×#1382	C₂	semidirect
7. C₅₃×#1379	C₂	semidirect
13. C₂×SD₃₂×C₅₃	C₄	semidirect
33. C₂×Q₁₆×C₅₃	C₂×C₄	semidirect
54. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
140. #13019	C₅₃	direct (N × H)
159. #2141	C₂×C₅₃	semidirect
160. #1979	C₂×C₅₃	semidirect
161. #1382	C₂×C₅₃	semidirect
162. #1379	C₂×C₅₃	semidirect
173. C₂×SD₃₂	C₄×C₅₃	semidirect
186. C₅₃	#13019	direct (N × H)
194. C₂×Q₁₆	C₂×C₄×C₅₃	semidirect
215. C₂×C₁₆	C₂×C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×SD₃₂×C₅₃	2
C₅₃×#2948	2
C₂×C₅₃×#361	2
C₅₃×#2141	2
C₅₃×#1979	2
C₅₃×#1382	2
C₅₃×#1379	2
#13019	53

Subgroups — C53×#13019 #27220531

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13019 #27220531