Subgroups — C₅₃×#19660 #27227172

Order 13568.

914 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3226	C₂	1	2	yes	maximal, complemented
2	C₅₃×#3205	C₂	1	2	yes	maximal, complemented
3	C₅₃×#3147	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1722	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1721	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1706	C₂	1	4	yes	maximal, complemented
8	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₅₃	—	2	2	no	—
10	C₂²×SD₁₆×C₅₃	—	2	2	no	—
11	C₅₃×#532	C₂²	1	2	yes	—
12	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#500	C₂²	1	4	yes	complemented
15	C₅₃×#495	—	2	2	no	—
16	C₅₃×#494	C₂²	1	1	yes	characteristic, complemented
17	C₅₃×#493	C₂²	1	4	yes	—
18	C₅₃×#493	C₂²	1	1	yes	characteristic, complemented
19	C₅₃×#492	—	2	2	no	—
20	C₅₃×#492	C₂²	1	2	yes	—
21	C₅₃×#492	C₂²	1	2	yes	—
22	C₅₃×#492	C₂²	1	2	yes	complemented
23	C₅₃×#486	C₂²	1	4	yes	—
24	C₅₃×#486	C₂²	1	4	yes	complemented
25	C₅₃×#480	—	2	4	no	—
26	C₅₃×#474	—	2	4	no	—
27	C₅₃×#459	—	2	4	no	—
28	C₅₃×#447	—	2	4	no	—
29	C₅₃×#431	C₂²	1	2	yes	—
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	C₂×#41×C₅₃	—	2	2	no	—
34	C₂²×Q₈×C₅₃	—	2	1	no	—
35	C₂²×D₄×C₅₃	—	2	1	no	—
36	C₂²×D₄×C₅₃	—	2	2	no	—
37	#135×C₅₃	—	4	8	no	—
38	C₂×Q₁₆×C₅₃	—	2	2	no	—
39	C₂×SD₁₆×C₅₃	—	2	4	no	—
40	C₂×SD₁₆×C₅₃	—	2	4	no	—
41	C₂×SD₁₆×C₅₃	—	2	4	no	—
42	C₂×SD₁₆×C₅₃	—	4	8	no	—
43	C₂×D₈×C₅₃	—	2	2	no	—
44	C₂×D₈×C₅₃	—	2	8	no	—
45	C₂²×C₈×C₅₃	D₄	1	2	yes	—
46	C₄⋊Q₈×C₅₃	—	2	1	no	—
47	C₄⋊Q₈×C₅₃	—	2	2	no	—
48	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
49	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
50	#127×C₅₃	—	2	1	no	—
51	#127×C₅₃	—	2	2	no	—
52	#127×C₅₃	C₂³	1	2	yes	—
53	#127×C₅₃	C₂³	1	2	yes	—
54	#124×C₅₃	—	2	2	no	—
55	#121×C₅₃	—	2	2	no	—
56	#121×C₅₃	—	2	2	no	—
57	C₄×D₄×C₅₃	—	2	2	no	—
58	C₄×D₄×C₅₃	—	2	2	no	—
59	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
60	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
61	#107×C₅₃	—	2	8	no	—
62	C₄⋊C₈×C₅₃	D₄	1	4	yes	—
63	#102×C₅₃	—	2	8	no	—
64	#102×C₅₃	—	2	8	no	—
65	C₂²⋊C₈×C₅₃	D₄	1	4	yes	—
66	C₄×C₈×C₅₃	D₄	1	2	yes	—
67	C₄×C₈×C₅₃	C₂³	1	4	yes	—
68	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
70	C₂⁴×C₅₃	—	4	1	no	—
71	#41×C₅₃	—	4	8	no	—
72	C₂×Q₈×C₅₃	—	2	2	no	—
73	C₂×Q₈×C₅₃	—	2	2	no	—
74	C₂×Q₈×C₅₃	—	4	1	no	—
75	C₂×Q₈×C₅₃	—	4	4	no	—
76	C₂×D₄×C₅₃	—	2	2	no	—
77	C₂×D₄×C₅₃	—	2	4	no	—
78	C₂×D₄×C₅₃	—	2	2	no	—
79	C₂×D₄×C₅₃	—	2	2	no	—
80	C₂×D₄×C₅₃	—	2	2	no	—
81	C₂×D₄×C₅₃	—	4	2	no	—
82	C₂×D₄×C₅₃	—	4	2	no	—
83	C₂×D₄×C₅₃	—	4	1	no	—
84	C₂×D₄×C₅₃	—	4	4	no	—
85	C₂×D₄×C₅₃	—	8	4	no	—
86	C₂²×C₄×C₅₃	—	4	1	no	—
87	C₂²×C₄×C₅₃	—	4	2	no	—
88	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
90	Q₁₆×C₅₃	—	4	4	no	—
91	SD₁₆×C₅₃	—	4	8	no	—
92	SD₁₆×C₅₃	—	4	8	no	—
93	SD₁₆×C₅₃	—	4	8	no	—
94	D₈×C₅₃	—	4	4	no	—
95	D₈×C₅₃	—	8	8	no	—
96	C₂×C₈×C₅₃	D₈	1	8	yes	complemented
97	C₂×C₈×C₅₃	C₂×D₄	1	8	yes	—
98	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
100	C₄⋊C₄×C₅₃	—	2	2	no	—
101	C₄⋊C₄×C₅₃	—	2	4	no	—
102	C₄⋊C₄×C₅₃	—	2	2	no	—
103	C₄⋊C₄×C₅₃	—	4	2	no	—
104	C₂²⋊C₄×C₅₃	—	4	2	no	—
105	C₂²⋊C₄×C₅₃	—	4	2	no	—
106	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₂×D₄	1	2	yes	—
108	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
109	C₂³×C₅₃	—	4	2	no	—
110	C₂³×C₅₃	—	4	2	no	—
111	C₂³×C₅₃	—	8	4	no	—
112	C₂³×C₅₃	—	8	2	no	—
113	C₂³×C₅₃	#127	1	1	yes	characteristic
114	Q₈×C₅₃	—	4	4	no	—
115	Q₈×C₅₃	—	4	4	no	—
116	Q₈×C₅₃	—	8	2	no	—
117	D₄×C₅₃	—	4	4	no	—
118	D₄×C₅₃	—	4	4	no	—
119	D₄×C₅₃	—	4	4	no	—
120	D₄×C₅₃	—	4	8	no	—
121	D₄×C₅₃	—	4	4	no	—
122	D₄×C₅₃	—	8	4	no	—
123	D₄×C₅₃	—	8	4	no	—
124	D₄×C₅₃	—	8	2	no	—
125	C₂×C₄×C₅₃	—	2	4	no	—
126	C₂×C₄×C₅₃	—	4	2	no	—
127	C₂×C₄×C₅₃	—	4	2	no	—
128	C₂×C₄×C₅₃	—	8	2	no	—
129	C₂×C₄×C₅₃	—	8	4	no	—
130	C₂×C₄×C₅₃	#127	1	2	yes	—
131	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
133	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
135	C₈×C₅₃	—	2	8	no	—
136	C₈×C₅₃	C₂×D₈	1	4	yes	—
137	C₈×C₅₃	C₂×D₈	1	4	yes	—
138	#19660	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂²×C₅₃	—	2	2	no	—
140	C₂²×C₅₃	—	8	2	no	—
141	C₂²×C₅₃	—	8	2	no	—
142	C₂²×C₅₃	—	8	4	no	—
143	C₂²×C₅₃	—	8	2	no	—
144	C₂²×C₅₃	—	8	2	no	—
145	C₂²×C₅₃	—	16	4	no	—
146	C₂²×C₅₃	#493	1	2	yes	—
147	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
148	C₄×C₅₃	—	2	2	no	—
149	C₄×C₅₃	—	8	2	no	—
150	C₄×C₅₃	—	8	2	no	—
151	C₄×C₅₃	#493	1	2	yes	—
152	C₄×C₅₃	C₂²×D₈	1	2	yes	—
153	C₄×C₅₃	#577	1	4	yes	—
154	#3226	C₂×C₅₃	1	2	yes	complemented
155	#3205	C₂×C₅₃	1	2	yes	complemented
156	#3147	C₂×C₅₃	1	1	yes	characteristic, complemented
157	C₂×#492	C₂×C₅₃	1	1	yes	characteristic, complemented
158	#1722	C₂×C₅₃	1	2	yes	complemented
159	#1721	C₂×C₅₃	1	2	yes	complemented
160	#1706	C₂×C₅₃	1	4	yes	complemented
161	#1450	C₂×C₅₃	1	1	yes	characteristic, complemented
162	C₂×C₅₃	—	2	2	no	—
163	C₂×C₅₃	—	16	2	no	—
164	C₂×C₅₃	—	16	2	no	—
165	C₂×C₅₃	C₂×#493	1	1	yes	characteristic
166	C₂×C₅₃	#3154	1	2	yes	—
167	C₂×#135	—	2	2	no	—
168	C₂²×SD₁₆	—	2	2	no	—
169	#532	C₂²×C₅₃	1	2	yes	—
170	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
171	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
172	#500	C₂²×C₅₃	1	4	yes	complemented
173	#495	—	2	2	no	—
174	#494	C₂²×C₅₃	1	1	yes	characteristic, complemented
175	#493	C₂²×C₅₃	1	4	yes	—
176	#493	C₂²×C₅₃	1	1	yes	characteristic, complemented
177	#492	—	2	2	no	—
178	#492	C₂²×C₅₃	1	2	yes	—
179	#492	C₂²×C₅₃	1	2	yes	—
180	#492	C₂²×C₅₃	1	2	yes	complemented
181	#486	C₂²×C₅₃	1	4	yes	—
182	#486	C₂²×C₅₃	1	4	yes	complemented
183	#480	—	2	4	no	—
184	#474	—	2	4	no	—
185	#459	—	2	4	no	—
186	#447	—	2	4	no	—
187	#431	C₂²×C₅₃	1	2	yes	—
188	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
189	#322	C₂²×C₅₃	1	2	yes	complemented
190	C₈²	C₂²×C₅₃	1	2	yes	complemented
191	C₅₃	#19660	1	1	yes	characteristic, direct product factor, Sylow
192	C₂×#41	—	2	2	no	—
193	C₂²×Q₈	—	2	1	no	—
194	C₂²×D₄	—	2	1	no	—
195	C₂²×D₄	—	2	2	no	—
196	#135	—	4	8	no	—
197	C₂×Q₁₆	—	2	2	no	—
198	C₂×SD₁₆	—	2	4	no	—
199	C₂×SD₁₆	—	2	4	no	—
200	C₂×SD₁₆	—	2	4	no	—
201	C₂×SD₁₆	—	4	8	no	—
202	C₂×D₈	—	2	2	no	—
203	C₂×D₈	—	2	8	no	—
204	C₂²×C₈	D₄×C₅₃	1	2	yes	—
205	C₄⋊Q₈	—	2	2	no	—
206	C₄⋊Q₈	—	2	1	no	—
207	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
208	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
209	#127	—	2	2	no	—
210	#127	—	2	1	no	—
211	#127	C₂³×C₅₃	1	2	yes	—
212	#127	C₂³×C₅₃	1	2	yes	—
213	#124	—	2	2	no	—
214	#121	—	2	2	no	—
215	#121	—	2	2	no	—
216	C₄×D₄	—	2	2	no	—
217	C₄×D₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
220	#107	—	2	8	no	—
221	C₄⋊C₈	D₄×C₅₃	1	4	yes	—
222	#102	—	2	8	no	—
223	#102	—	2	8	no	—
224	C₂²⋊C₈	D₄×C₅₃	1	4	yes	—
225	C₄×C₈	D₄×C₅₃	1	2	yes	—
226	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
227	C₄×C₈	C₂³×C₅₃	1	4	yes	—
228	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
229	C₂⁴	—	4	1	no	—
230	#41	—	4	8	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	4	1	no	—
234	C₂×Q₈	—	4	4	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	2	4	no	—
237	C₂×D₄	—	2	2	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	4	2	no	—
241	C₂×D₄	—	4	2	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	4	4	no	—
244	C₂×D₄	—	8	4	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
248	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
249	Q₁₆	—	4	4	no	—
250	SD₁₆	—	4	8	no	—
251	SD₁₆	—	4	8	no	—
252	SD₁₆	—	4	8	no	—
253	D₈	—	4	4	no	—
254	D₈	—	8	8	no	—
255	C₂×C₈	D₈×C₅₃	1	8	yes	complemented
256	C₂×C₈	C₂×D₄×C₅₃	1	8	yes	—
257	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
258	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
266	C₄²	C₂×D₄×C₅₃	1	2	yes	—
267	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
268	C₂³	—	4	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	8	4	no	—
271	C₂³	—	8	2	no	—
272	C₂³	#127×C₅₃	1	1	yes	characteristic
273	Q₈	—	4	4	no	—
274	Q₈	—	4	4	no	—
275	Q₈	—	8	2	no	—
276	D₄	—	4	4	no	—
277	D₄	—	4	4	no	—
278	D₄	—	4	4	no	—
279	D₄	—	4	8	no	—
280	D₄	—	4	4	no	—
281	D₄	—	8	4	no	—
282	D₄	—	8	4	no	—
283	D₄	—	8	2	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	8	2	no	—
288	C₂×C₄	—	8	4	no	—
289	C₂×C₄	#127×C₅₃	1	2	yes	—
290	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
291	C₂×C₄	C₂×D₈×C₅₃	1	4	yes	—
292	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
293	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
294	C₈	—	2	8	no	—
295	C₈	C₂×D₈×C₅₃	1	4	yes	—
296	C₈	C₂×D₈×C₅₃	1	4	yes	—
297	C₂²	—	2	2	no	—
298	C₂²	—	8	2	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	8	4	no	—
301	C₂²	—	8	2	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	16	4	no	—
304	C₂²	C₅₃×#493	1	2	yes	—
305	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	8	2	no	—
308	C₄	—	8	2	no	—
309	C₄	C₅₃×#493	1	2	yes	—
310	C₄	C₂²×D₈×C₅₃	1	2	yes	—
311	C₄	C₅₃×#577	1	4	yes	—
312	C₂	—	2	2	no	—
313	C₂	—	16	2	no	—
314	C₂	—	16	2	no	—
315	C₂	C₂×C₅₃×#493	1	1	yes	characteristic
316	C₂	C₅₃×#3154	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
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×#3226	C₂	semidirect
2. C₅₃×#3205	C₂	semidirect
3. C₅₃×#3147	C₂	semidirect
4. C₂×C₅₃×#492	C₂	semidirect
5. C₅₃×#1722	C₂	semidirect
6. C₅₃×#1721	C₂	semidirect
7. C₅₃×#1706	C₂	semidirect
8. C₅₃×#1450	C₂	semidirect
14. C₅₃×#500	C₂²	semidirect
16. C₅₃×#494	C₂²	semidirect
18. C₅₃×#493	C₂²	semidirect
22. C₅₃×#492	C₂²	semidirect
24. C₅₃×#486	C₂²	semidirect
31. C₅₃×#322	C₂²	semidirect
32. C₈²×C₅₃	C₂²	semidirect
96. C₂×C₈×C₅₃	D₈	semidirect
138. #19660	C₅₃	direct (N × H)
154. #3226	C₂×C₅₃	semidirect
155. #3205	C₂×C₅₃	semidirect
156. #3147	C₂×C₅₃	semidirect
157. C₂×#492	C₂×C₅₃	semidirect
158. #1722	C₂×C₅₃	semidirect
159. #1721	C₂×C₅₃	semidirect
160. #1706	C₂×C₅₃	semidirect
161. #1450	C₂×C₅₃	semidirect
172. #500	C₂²×C₅₃	semidirect
174. #494	C₂²×C₅₃	semidirect
176. #493	C₂²×C₅₃	semidirect
180. #492	C₂²×C₅₃	semidirect
182. #486	C₂²×C₅₃	semidirect
189. #322	C₂²×C₅₃	semidirect
190. C₈²	C₂²×C₅₃	semidirect
191. C₅₃	#19660	direct (N × H)
255. C₂×C₈	D₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3226	2
C₅₃×#3205	2
C₅₃×#3147	2
C₂×C₅₃×#492	2
C₅₃×#1722	2
C₅₃×#1721	2
C₅₃×#1706	2
C₅₃×#1450	2
#19660	53

Subgroups — C53×#19660 #27227172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19660 #27227172