Subgroups — C₅₃×#33683 #27241195

Order 13568.

556 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3177	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#2948	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2237	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2228	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2208	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2187	C₂	1	2	yes	maximal
7	C₅₃×#2175	C₂	1	2	yes	maximal, complemented
8	C₄×SD₃₂×C₅₃	C₂	1	2	yes	maximal, complemented
9	C₅₃×#2164	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#541	—	2	1	no	—
12	C₅₃×#518	—	2	1	no	—
13	C₂×SD₃₂×C₅₃	C₂²	1	2	yes	—
14	C₂×SD₃₂×C₅₃	C₂²	1	2	yes	—
15	C₅₃×#499	C₂²	1	1	yes	characteristic, complemented
16	C₅₃×#494	C₂²	1	1	yes	characteristic, complemented
17	C₅₃×#491	—	2	1	no	—
18	C₅₃×#470	—	2	1	no	—
19	C₅₃×#467	C₂²	1	2	yes	—
20	C₅₃×#442	—	2	1	no	—
21	C₅₃×#442	—	2	1	no	—
22	C₅₃×#441	—	2	1	no	—
23	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	complemented
24	C₄×D₈×C₅₃	C₂²	1	2	yes	—
25	C₄×D₈×C₅₃	C₂²	1	2	yes	—
26	C₅₃×#427	C₂²	1	1	yes	characteristic
27	C₅₃×#418	—	2	1	no	—
28	C₅₃×#405	C₂²	1	1	yes	characteristic
29	C₅₃×#367	C₂²	1	2	yes	complemented
30	C₅₃×#366	C₂²	1	2	yes	complemented
31	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	complemented
32	C₅₃×#358	C₂²	1	2	yes	complemented
33	C₅₃×#358	C₂²	1	2	yes	complemented
34	C₅₃×#357	C₂²	1	2	yes	—
35	C₅₃×#357	C₂²	1	2	yes	—
36	C₅₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
37	C₅₃×#346	C₂²	1	1	yes	characteristic, complemented
38	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
39	C₂²×D₄×C₅₃	—	2	1	no	—
40	C₂×Q₁₆×C₅₃	—	2	1	no	—
41	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
42	C₂×D₈×C₅₃	—	2	4	no	—
43	C₂×D₈×C₅₃	—	2	1	no	—
44	C₂×D₈×C₅₃	C₂³	1	2	yes	—
45	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
47	C₄⋊Q₈×C₅₃	—	2	1	no	—
48	#126×C₅₃	—	4	2	no	—
49	#125×C₅₃	—	2	1	no	—
50	#125×C₅₃	—	4	2	no	—
51	C₂²⋊Q₈×C₅₃	—	2	2	no	—
52	C₄×Q₈×C₅₃	—	2	2	no	—
53	C₄×D₄×C₅₃	—	2	2	no	—
54	C₄×D₄×C₅₃	—	2	2	no	—
55	C₄×D₄×C₅₃	—	4	2	no	—
56	#117×C₅₃	—	2	1	no	—
57	#117×C₅₃	D₄	1	1	yes	characteristic
58	C₂×C₂²⋊C₄×C₅₃	—	4	2	no	—
59	SD₃₂×C₅₃	—	2	4	no	—
60	SD₃₂×C₅₃	—	2	4	no	—
61	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
62	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
63	#107×C₅₃	C₂³	1	2	yes	—
64	#107×C₅₃	C₂³	1	2	yes	—
65	#106×C₅₃	—	2	1	no	—
66	#106×C₅₃	—	2	1	no	—
67	#103×C₅₃	—	2	2	no	—
68	#103×C₅₃	—	2	2	no	—
69	#102×C₅₃	—	2	2	no	—
70	#102×C₅₃	—	2	2	no	—
71	#97×C₅₃	D₄	1	1	yes	characteristic
72	#97×C₅₃	D₄	1	1	yes	characteristic
73	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₂⁴×C₅₃	—	4	1	no	—
76	C₂×Q₈×C₅₃	—	2	2	no	—
77	C₂×D₄×C₅₃	—	2	4	no	—
78	C₂×D₄×C₅₃	—	2	2	no	—
79	C₂×D₄×C₅₃	—	4	2	no	—
80	C₂×D₄×C₅₃	—	4	1	no	—
81	C₂²×C₄×C₅₃	—	4	2	no	—
82	C₂²×C₄×C₅₃	—	4	2	no	—
83	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
84	Q₁₆×C₅₃	—	2	2	no	—
85	Q₁₆×C₅₃	—	2	2	no	—
86	Q₁₆×C₅₃	—	4	2	no	—
87	D₈×C₅₃	—	2	4	no	—
88	D₈×C₅₃	—	2	2	no	—
89	D₈×C₅₃	#41	1	4	yes	—
90	M₁₆×C₅₃	—	4	1	no	—
91	C₂×C₈×C₅₃	—	2	2	no	—
92	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₅₃	—	2	2	no	—
97	C₄⋊C₄×C₅₃	—	2	2	no	—
98	C₄⋊C₄×C₅₃	—	4	2	no	—
99	C₄⋊C₄×C₅₃	—	4	2	no	—
100	C₄⋊C₄×C₅₃	—	4	2	no	—
101	C₄⋊C₄×C₅₃	—	4	2	no	—
102	C₄⋊C₄×C₅₃	D₈	1	2	yes	—
103	C₂²⋊C₄×C₅₃	—	4	1	no	—
104	C₂²⋊C₄×C₅₃	—	4	2	no	—
105	C₂²⋊C₄×C₅₃	—	4	2	no	—
106	C₂²⋊C₄×C₅₃	—	4	2	no	—
107	C₂²⋊C₄×C₅₃	D₈	1	2	yes	—
108	C₄²×C₅₃	—	4	1	no	—
109	C₄²×C₅₃	—	4	2	no	—
110	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₁₆×C₅₃	—	2	2	no	—
113	C₁₆×C₅₃	—	2	2	no	—
114	C₂³×C₅₃	—	4	4	no	—
115	C₂³×C₅₃	—	4	2	no	—
116	C₂³×C₅₃	—	16	2	no	—
117	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
118	Q₈×C₅₃	—	4	2	no	—
119	Q₈×C₅₃	—	4	2	no	—
120	D₄×C₅₃	—	2	4	no	—
121	D₄×C₅₃	—	4	4	no	—
122	D₄×C₅₃	—	4	2	no	—
123	C₂×C₄×C₅₃	—	2	1	no	—
124	C₂×C₄×C₅₃	—	2	1	no	—
125	C₂×C₄×C₅₃	—	4	2	no	—
126	C₂×C₄×C₅₃	—	4	2	no	—
127	C₂×C₄×C₅₃	—	4	2	no	—
128	C₂×C₄×C₅₃	—	16	2	no	—
129	C₂×C₄×C₅₃	—	16	2	no	—
130	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
134	C₈×C₅₃	—	2	1	no	—
135	C₈×C₅₃	—	2	1	no	—
136	C₈×C₅₃	—	2	1	no	—
137	C₈×C₅₃	C₂×#41	1	2	yes	—
138	#33683	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂²×C₅₃	—	2	1	no	—
140	C₂²×C₅₃	—	2	2	no	—
141	C₂²×C₅₃	—	4	4	no	—
142	C₂²×C₅₃	—	8	2	no	—
143	C₂²×C₅₃	—	8	2	no	—
144	C₂²×C₅₃	—	16	4	no	—
145	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
146	C₄×C₅₃	—	2	1	no	—
147	C₄×C₅₃	—	2	2	no	—
148	C₄×C₅₃	—	2	2	no	—
149	C₄×C₅₃	—	8	2	no	—
150	C₄×C₅₃	—	8	2	no	—
151	C₄×C₅₃	—	8	2	no	—
152	C₄×C₅₃	#525	1	1	yes	characteristic
153	C₄×C₅₃	#575	1	1	yes	characteristic
154	#3177	C₂×C₅₃	1	1	yes	characteristic, complemented
155	#2948	C₂×C₅₃	1	1	yes	characteristic
156	#2237	C₂×C₅₃	1	2	yes	complemented
157	#2228	C₂×C₅₃	1	2	yes	complemented
158	#2208	C₂×C₅₃	1	2	yes	complemented
159	#2187	C₂×C₅₃	1	2	yes	—
160	#2175	C₂×C₅₃	1	2	yes	complemented
161	C₄×SD₃₂	C₂×C₅₃	1	2	yes	complemented
162	#2164	C₂×C₅₃	1	1	yes	characteristic, complemented
163	C₂×C₅₃	—	4	1	no	—
164	C₂×C₅₃	—	8	4	no	—
165	C₂×C₅₃	#3100	1	1	yes	characteristic
166	C₂×C₅₃	#3416	1	2	yes	—
167	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
168	#541	—	2	1	no	—
169	#518	—	2	1	no	—
170	C₂×SD₃₂	C₂²×C₅₃	1	2	yes	—
171	C₂×SD₃₂	C₂²×C₅₃	1	2	yes	—
172	#499	C₂²×C₅₃	1	1	yes	characteristic, complemented
173	#494	C₂²×C₅₃	1	1	yes	characteristic, complemented
174	#491	—	2	1	no	—
175	#470	—	2	1	no	—
176	#467	C₂²×C₅₃	1	2	yes	—
177	#442	—	2	1	no	—
178	#442	—	2	1	no	—
179	#441	—	2	1	no	—
180	C₄×Q₁₆	C₂²×C₅₃	1	2	yes	complemented
181	C₄×D₈	C₂²×C₅₃	1	2	yes	—
182	C₄×D₈	C₂²×C₅₃	1	2	yes	—
183	#427	C₂²×C₅₃	1	1	yes	characteristic
184	#418	—	2	1	no	—
185	#405	C₂²×C₅₃	1	1	yes	characteristic
186	#367	C₂²×C₅₃	1	2	yes	complemented
187	#366	C₂²×C₅₃	1	2	yes	complemented
188	C₄⋊C₁₆	C₂²×C₅₃	1	2	yes	complemented
189	#358	C₂²×C₅₃	1	2	yes	complemented
190	#358	C₂²×C₅₃	1	2	yes	complemented
191	#357	C₂²×C₅₃	1	2	yes	—
192	#357	C₂²×C₅₃	1	2	yes	—
193	C₂²⋊C₁₆	C₂²×C₅₃	1	2	yes	—
194	#346	C₂²×C₅₃	1	1	yes	characteristic, complemented
195	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic, complemented
196	C₅₃	#33683	1	1	yes	characteristic, direct product factor, Sylow
197	C₂²×D₄	—	2	1	no	—
198	C₂×Q₁₆	—	2	1	no	—
199	C₂×Q₁₆	C₂³×C₅₃	1	2	yes	—
200	C₂×D₈	—	2	4	no	—
201	C₂×D₈	—	2	1	no	—
202	C₂×D₈	C₂³×C₅₃	1	2	yes	—
203	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
205	C₄⋊Q₈	—	2	1	no	—
206	#126	—	4	2	no	—
207	#125	—	2	1	no	—
208	#125	—	4	2	no	—
209	C₂²⋊Q₈	—	2	2	no	—
210	C₄×Q₈	—	2	2	no	—
211	C₄×D₄	—	2	2	no	—
212	C₄×D₄	—	2	2	no	—
213	C₄×D₄	—	4	2	no	—
214	#117	—	2	1	no	—
215	#117	D₄×C₅₃	1	1	yes	characteristic
216	C₂×C₂²⋊C₄	—	4	2	no	—
217	SD₃₂	—	2	4	no	—
218	SD₃₂	—	2	4	no	—
219	C₂×C₁₆	C₂³×C₅₃	1	2	yes	—
220	C₂×C₁₆	C₂³×C₅₃	1	2	yes	—
221	#107	C₂³×C₅₃	1	2	yes	—
222	#107	C₂³×C₅₃	1	2	yes	—
223	#106	—	2	1	no	—
224	#106	—	2	1	no	—
225	#103	—	2	2	no	—
226	#103	—	2	2	no	—
227	#102	—	2	2	no	—
228	#102	—	2	2	no	—
229	#97	D₄×C₅₃	1	1	yes	characteristic
230	#97	D₄×C₅₃	1	1	yes	characteristic
231	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
232	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
233	C₂⁴	—	4	1	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×D₄	—	2	4	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
242	Q₁₆	—	2	2	no	—
243	Q₁₆	—	2	2	no	—
244	Q₁₆	—	4	2	no	—
245	D₈	—	2	4	no	—
246	D₈	—	2	2	no	—
247	D₈	#41×C₅₃	1	4	yes	—
248	M₁₆	—	4	1	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
251	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
252	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
253	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	D₈×C₅₃	1	2	yes	—
261	C₂²⋊C₄	—	4	1	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	D₈×C₅₃	1	2	yes	—
266	C₄²	—	4	1	no	—
267	C₄²	—	4	2	no	—
268	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
269	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
270	C₁₆	—	2	2	no	—
271	C₁₆	—	2	2	no	—
272	C₂³	—	4	4	no	—
273	C₂³	—	4	2	no	—
274	C₂³	—	16	2	no	—
275	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
276	Q₈	—	4	2	no	—
277	Q₈	—	4	2	no	—
278	D₄	—	2	4	no	—
279	D₄	—	4	4	no	—
280	D₄	—	4	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	16	2	no	—
287	C₂×C₄	—	16	2	no	—
288	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
289	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
290	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
291	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	C₂×#41×C₅₃	1	2	yes	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	4	4	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	16	4	no	—
302	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
303	C₄	—	2	1	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	8	2	no	—
307	C₄	—	8	2	no	—
308	C₄	—	8	2	no	—
309	C₄	C₅₃×#525	1	1	yes	characteristic
310	C₄	C₅₃×#575	1	1	yes	characteristic
311	C₂	—	4	1	no	—
312	C₂	—	8	4	no	—
313	C₂	C₅₃×#3100	1	1	yes	characteristic
314	C₂	C₅₃×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
prime radical	C₂²×D₈×C₅₃	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ #117×C₅₃ ◁ C₅₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3177	C₂	semidirect
3. C₅₃×#2237	C₂	semidirect
4. C₅₃×#2228	C₂	semidirect
5. C₅₃×#2208	C₂	semidirect
7. C₅₃×#2175	C₂	semidirect
8. C₄×SD₃₂×C₅₃	C₂	semidirect
9. C₅₃×#2164	C₂	semidirect
15. C₅₃×#499	C₂²	semidirect
16. C₅₃×#494	C₂²	semidirect
23. C₄×Q₁₆×C₅₃	C₂²	semidirect
29. C₅₃×#367	C₂²	semidirect
30. C₅₃×#366	C₂²	semidirect
31. C₅₃×C₄⋊C₁₆	C₂²	semidirect
32. C₅₃×#358	C₂²	semidirect
33. C₅₃×#358	C₂²	semidirect
37. C₅₃×#346	C₂²	semidirect
38. C₄×C₁₆×C₅₃	C₂²	semidirect
138. #33683	C₅₃	direct (N × H)
154. #3177	C₂×C₅₃	semidirect
156. #2237	C₂×C₅₃	semidirect
157. #2228	C₂×C₅₃	semidirect
158. #2208	C₂×C₅₃	semidirect
160. #2175	C₂×C₅₃	semidirect
161. C₄×SD₃₂	C₂×C₅₃	semidirect
162. #2164	C₂×C₅₃	semidirect
172. #499	C₂²×C₅₃	semidirect
173. #494	C₂²×C₅₃	semidirect
180. C₄×Q₁₆	C₂²×C₅₃	semidirect
186. #367	C₂²×C₅₃	semidirect
187. #366	C₂²×C₅₃	semidirect
188. C₄⋊C₁₆	C₂²×C₅₃	semidirect
189. #358	C₂²×C₅₃	semidirect
190. #358	C₂²×C₅₃	semidirect
194. #346	C₂²×C₅₃	semidirect
195. C₄×C₁₆	C₂²×C₅₃	semidirect
196. C₅₃	#33683	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×#3177	2
C₅₃×#2948	2
C₅₃×#2237	2
C₅₃×#2228	2
C₅₃×#2208	2
C₅₃×#2187	2
C₅₃×#2175	2
C₄×SD₃₂×C₅₃	2
C₅₃×#2164	2
#33683	53

Subgroups — C53×#33683 #27241195

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#33683 #27241195