Subgroups — C₅₃×#8650 #27216162

Order 13568.

510 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#436	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1844	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1802	C₂	1	2	yes	maximal
4	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1296	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₅₃	—	2	1	no	—
7	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#436	—	2	2	no	—
9	C₅₃×#436	C₄	1	4	yes	complemented
10	C₂×#106×C₅₃	—	2	1	no	—
11	C₂×#106×C₅₃	—	2	1	no	—
12	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#102×C₅₃	—	2	2	no	—
15	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#384	C₂²	1	2	yes	—
18	C₅₃×#340	—	2	2	no	—
19	C₅₃×#335	—	2	2	no	—
20	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₅₃	—	2	1	no	—
22	C₂×M₁₆×C₅₃	—	2	4	no	—
23	C₂×M₁₆×C₅₃	—	4	2	no	—
24	C₂×M₁₆×C₅₃	—	4	2	no	—
25	C₂²×C₈×C₅₃	—	2	1	no	—
26	C₂²×C₈×C₅₃	—	2	2	no	—
27	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₄×D₄×C₅₃	—	2	1	no	—
30	C₄×D₄×C₅₃	—	2	1	no	—
31	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
32	C₄×D₄×C₅₃	Q₈	1	2	yes	—
33	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	#106×C₅₃	—	4	2	no	—
40	#106×C₅₃	—	4	2	no	—
41	C₄⋊C₈×C₅₃	—	2	1	no	—
42	C₄⋊C₈×C₅₃	—	2	2	no	—
43	C₄⋊C₈×C₅₃	—	2	2	no	—
44	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
45	#102×C₅₃	—	4	4	no	—
46	C₂²⋊C₈×C₅₃	—	2	4	no	—
47	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₅₃	D₄	1	2	yes	—
49	C₂⁴×C₅₃	—	2	1	no	—
50	C₂×D₄×C₅₃	—	2	1	no	—
51	C₂×D₄×C₅₃	—	2	2	no	—
52	C₂×D₄×C₅₃	—	2	2	no	—
53	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	—	2	1	no	—
55	C₂²×C₄×C₅₃	—	2	2	no	—
56	C₂²×C₄×C₅₃	—	2	2	no	—
57	C₂²×C₄×C₅₃	—	4	1	no	—
58	C₂²×C₄×C₅₃	—	4	1	no	—
59	C₂²×C₄×C₅₃	—	4	1	no	—
60	C₂²×C₄×C₅₃	—	4	1	no	—
61	C₂²×C₄×C₅₃	—	4	2	no	—
62	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
65	M₁₆×C₅₃	—	4	8	no	—
66	C₂×C₈×C₅₃	—	2	1	no	—
67	C₂×C₈×C₅₃	—	2	1	no	—
68	C₂×C₈×C₅₃	—	2	2	no	—
69	C₂×C₈×C₅₃	—	4	1	no	—
70	C₂×C₈×C₅₃	—	4	1	no	—
71	C₂×C₈×C₅₃	—	4	2	no	—
72	C₂×C₈×C₅₃	—	4	2	no	—
73	C₂×C₈×C₅₃	—	4	2	no	—
74	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₅₃	D₈	1	2	yes	—
77	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
78	C₄⋊C₄×C₅₃	—	2	1	no	—
79	C₄⋊C₄×C₅₃	—	4	2	no	—
80	C₄⋊C₄×C₅₃	—	4	2	no	—
81	C₄⋊C₄×C₅₃	—	4	2	no	—
82	C₄⋊C₄×C₅₃	—	4	2	no	—
83	C₄⋊C₄×C₅₃	—	8	2	no	—
84	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
85	C₂²⋊C₄×C₅₃	—	2	2	no	—
86	C₂²⋊C₄×C₅₃	—	2	2	no	—
87	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
90	C₄²×C₅₃	#41	1	1	yes	characteristic
91	C₂³×C₅₃	—	2	2	no	—
92	C₂³×C₅₃	—	4	1	no	—
93	C₂³×C₅₃	—	4	1	no	—
94	C₂³×C₅₃	—	4	2	no	—
95	C₂³×C₅₃	—	4	2	no	—
96	C₂³×C₅₃	#121	1	1	yes	characteristic
97	D₄×C₅₃	—	2	4	no	—
98	D₄×C₅₃	—	4	1	no	—
99	D₄×C₅₃	—	4	1	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	4	1	no	—
103	C₂×C₄×C₅₃	—	4	1	no	—
104	C₂×C₄×C₅₃	—	4	2	no	—
105	C₂×C₄×C₅₃	—	4	2	no	—
106	C₂×C₄×C₅₃	—	4	2	no	—
107	C₂×C₄×C₅₃	—	4	2	no	—
108	C₂×C₄×C₅₃	—	8	2	no	—
109	C₂×C₄×C₅₃	—	8	2	no	—
110	C₂×C₄×C₅₃	—	8	2	no	—
111	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
121	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
122	C₂×C₄×C₅₃	#137	1	2	yes	—
123	C₈×C₅₃	—	2	2	no	—
124	C₈×C₅₃	—	4	2	no	—
125	C₈×C₅₃	—	8	2	no	—
126	C₈×C₅₃	#102	1	4	yes	complemented
127	#8650	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂²×C₅₃	—	4	2	no	—
129	C₂²×C₅₃	—	4	2	no	—
130	C₂²×C₅₃	—	4	2	no	—
131	C₂²×C₅₃	—	8	2	no	—
132	C₂²×C₅₃	—	8	2	no	—
133	C₂²×C₅₃	#380	1	1	yes	characteristic
134	C₂²×C₅₃	#459	1	1	yes	characteristic
135	C₂²×C₅₃	#461	1	1	yes	characteristic
136	C₂²×C₅₃	#468	1	2	yes	—
137	C₂²×C₅₃	#471	1	2	yes	—
138	C₄×C₅₃	—	2	1	no	—
139	C₄×C₅₃	—	2	1	no	—
140	C₄×C₅₃	—	4	2	no	—
141	C₄×C₅₃	—	16	2	no	—
142	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
143	C₄×C₅₃	#417	1	1	yes	characteristic
144	C₄×C₅₃	#428	1	2	yes	—
145	C₄×C₅₃	#469	1	1	yes	characteristic
146	C₄×C₅₃	#470	1	1	yes	characteristic
147	C₄×C₅₃	#474	1	1	yes	characteristic
148	C₄×C₅₃	#476	1	1	yes	characteristic
149	C₂×#436	C₂×C₅₃	1	1	yes	characteristic
150	#1844	C₂×C₅₃	1	1	yes	characteristic, complemented
151	#1802	C₂×C₅₃	1	2	yes	—
152	C₂×#335	C₂×C₅₃	1	1	yes	characteristic, complemented
153	#1296	C₂×C₅₃	1	2	yes	complemented
154	C₂×C₅₃	—	8	2	no	—
155	C₂×C₅₃	#1685	1	2	yes	—
156	C₂×C₅₃	#1691	1	2	yes	—
157	C₂×C₅₃	#1802	1	1	yes	characteristic
158	C₂×C₅₃	#1809	1	1	yes	characteristic
159	C₂×C₅₃	#1815	1	1	yes	characteristic
160	C₂²×M₁₆	—	2	1	no	—
161	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
162	#436	—	2	2	no	—
163	#436	C₄×C₅₃	1	4	yes	complemented
164	C₂×#106	—	2	1	no	—
165	C₂×#106	—	2	1	no	—
166	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
167	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
168	C₂×#102	—	2	2	no	—
169	C₂×C₂²⋊C₈	—	2	1	no	—
170	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
171	#384	C₂²×C₅₃	1	2	yes	—
172	#340	—	2	2	no	—
173	#335	—	2	2	no	—
174	C₅₃	#8650	1	1	yes	characteristic, direct product factor, Sylow
175	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
176	C₂³×C₄	—	2	1	no	—
177	C₂×M₁₆	—	2	4	no	—
178	C₂×M₁₆	—	4	2	no	—
179	C₂×M₁₆	—	4	2	no	—
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
183	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
184	C₄×D₄	—	2	1	no	—
185	C₄×D₄	—	2	1	no	—
186	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
187	C₄×D₄	Q₈×C₅₃	1	2	yes	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
192	C₂×C₂²⋊C₄	—	2	1	no	—
193	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
194	#106	—	4	2	no	—
195	#106	—	4	2	no	—
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	—	2	2	no	—
198	C₄⋊C₈	—	2	2	no	—
199	C₄⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
200	#102	—	4	4	no	—
201	C₂²⋊C₈	—	2	4	no	—
202	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
203	C₄×C₈	D₄×C₅₃	1	2	yes	—
204	C₂⁴	—	2	1	no	—
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	C₄⋊C₄×C₅₃	1	2	yes	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	2	no	—
217	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
219	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
220	M₁₆	—	4	8	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
230	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
231	C₂×C₈	D₈×C₅₃	1	2	yes	—
232	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	—	8	2	no	—
239	C₄⋊C₄	C₄⋊C₄×C₅₃	1	2	yes	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
243	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
244	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
245	C₄²	#41×C₅₃	1	1	yes	characteristic
246	C₂³	—	2	2	no	—
247	C₂³	—	4	1	no	—
248	C₂³	—	4	1	no	—
249	C₂³	—	4	2	no	—
250	C₂³	—	4	2	no	—
251	C₂³	#121×C₅₃	1	1	yes	characteristic
252	D₄	—	2	4	no	—
253	D₄	—	4	1	no	—
254	D₄	—	4	1	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	8	2	no	—
264	C₂×C₄	—	8	2	no	—
265	C₂×C₄	—	8	2	no	—
266	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
267	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
268	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
269	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
270	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
271	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
272	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
273	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
274	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
275	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
276	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
277	C₂×C₄	#137×C₅₃	1	2	yes	—
278	C₈	—	2	2	no	—
279	C₈	—	4	2	no	—
280	C₈	—	8	2	no	—
281	C₈	#102×C₅₃	1	4	yes	complemented
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	8	2	no	—
286	C₂²	—	8	2	no	—
287	C₂²	C₅₃×#380	1	1	yes	characteristic
288	C₂²	C₅₃×#459	1	1	yes	characteristic
289	C₂²	C₅₃×#461	1	1	yes	characteristic
290	C₂²	C₅₃×#468	1	2	yes	—
291	C₂²	C₅₃×#471	1	2	yes	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	4	2	no	—
295	C₄	—	16	2	no	—
296	C₄	C₂×#102×C₅₃	1	1	yes	characteristic
297	C₄	C₅₃×#417	1	1	yes	characteristic
298	C₄	C₅₃×#428	1	2	yes	—
299	C₄	C₅₃×#469	1	1	yes	characteristic
300	C₄	C₅₃×#470	1	1	yes	characteristic
301	C₄	C₅₃×#474	1	1	yes	characteristic
302	C₄	C₅₃×#476	1	1	yes	characteristic
303	C₂	—	8	2	no	—
304	C₂	C₅₃×#1685	1	2	yes	—
305	C₂	C₅₃×#1691	1	2	yes	—
306	C₂	C₅₃×#1802	1	1	yes	characteristic
307	C₂	C₅₃×#1809	1	1	yes	characteristic
308	C₂	C₅₃×#1815	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	#121
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂²×D₄×C₅₃	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#1844	C₂	semidirect
4. C₂×C₅₃×#335	C₂	semidirect
5. C₅₃×#1296	C₂	semidirect
9. C₅₃×#436	C₄	semidirect
126. C₈×C₅₃	#102	semidirect
127. #8650	C₅₃	direct (N × H)
150. #1844	C₂×C₅₃	semidirect
152. C₂×#335	C₂×C₅₃	semidirect
153. #1296	C₂×C₅₃	semidirect
163. #436	C₄×C₅₃	semidirect
174. C₅₃	#8650	direct (N × H)
281. C₈	#102×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#436	2
C₅₃×#1844	2
C₅₃×#1802	2
C₂×C₅₃×#335	2
C₅₃×#1296	2
#8650	53

Subgroups — C53×#8650 #27216162

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8650 #27216162