Subgroups — C₅₃×#8673 #27216185

Order 13568.

386 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2965	C₂	1	1	yes	maximal, characteristic
2	C₄×#108×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1489	C₂	1	2	yes	maximal
5	C₅₃×#1299	C₂	1	2	yes	maximal, complemented
6	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#446	C₄	1	1	yes	characteristic
8	C₅₃×#446	C₄	1	1	yes	characteristic
9	C₅₃×#436	C₄	1	1	yes	characteristic
10	C₅₃×#436	C₄	1	1	yes	characteristic
11	C₅₃×#435	—	2	1	no	—
12	C₅₃×#435	—	2	1	no	—
13	C₅₃×#431	—	2	1	no	—
14	C₅₃×#431	C₂²	1	1	yes	characteristic
15	C₂×#108×C₅₃	—	2	1	no	—
16	C₅₃×#423	C₂²	1	2	yes	—
17	C₄×M₁₆×C₅₃	—	2	1	no	—
18	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#341	—	2	1	no	—
21	C₅₃×#335	—	2	1	no	—
22	C₅₃×#335	—	2	1	no	—
23	C₅₃×Q₈⋊C₈	—	2	2	no	—
24	C₅₃×D₄⋊C₈	—	2	2	no	—
25	C₂×#41×C₅₃	Q₈	1	1	yes	characteristic
26	C₂×M₁₆×C₅₃	—	2	1	no	—
27	C₂×M₁₆×C₅₃	—	2	1	no	—
28	C₂×M₁₆×C₅₃	—	2	2	no	—
29	C₂×M₁₆×C₅₃	—	2	2	no	—
30	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
34	C₄×D₄×C₅₃	—	2	1	no	—
35	C₄×D₄×C₅₃	—	2	1	no	—
36	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
38	#117×C₅₃	—	2	1	no	—
39	#117×C₅₃	Q₈	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	—	2	1	no	—
41	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	#108×C₅₃	—	2	4	no	—
43	C₄⋊C₈×C₅₃	—	2	1	no	—
44	C₄⋊C₈×C₅₃	—	2	1	no	—
45	C₄⋊C₈×C₅₃	—	2	2	no	—
46	C₄⋊C₈×C₅₃	—	2	2	no	—
47	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂²⋊C₈×C₅₃	—	2	1	no	—
50	C₂²⋊C₈×C₅₃	—	2	1	no	—
51	C₂²⋊C₈×C₅₃	—	2	1	no	—
52	#97×C₅₃	—	2	1	no	—
53	#97×C₅₃	—	2	1	no	—
54	#97×C₅₃	—	2	2	no	—
55	C₄×C₈×C₅₃	—	2	1	no	—
56	C₄×C₈×C₅₃	—	2	2	no	—
57	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
60	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
61	#41×C₅₃	—	4	2	no	—
62	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂×D₄×C₅₃	—	2	1	no	—
64	C₂×D₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	—	2	1	no	—
66	C₂²×C₄×C₅₃	—	2	1	no	—
67	C₂²×C₄×C₅₃	—	2	1	no	—
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
71	M₁₆×C₅₃	—	2	4	no	—
72	M₁₆×C₅₃	—	4	2	no	—
73	M₁₆×C₅₃	—	4	2	no	—
74	M₁₆×C₅₃	—	4	2	no	—
75	M₁₆×C₅₃	—	8	2	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	2	1	no	—
78	C₂×C₈×C₅₃	—	2	1	no	—
79	C₂×C₈×C₅₃	—	4	1	no	—
80	C₂×C₈×C₅₃	—	4	2	no	—
81	C₂×C₈×C₅₃	—	4	2	no	—
82	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₄⋊C₄×C₅₃	—	2	1	no	—
87	C₄⋊C₄×C₅₃	—	2	1	no	—
88	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
89	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
90	C₂²⋊C₄×C₅₃	—	2	1	no	—
91	C₂²⋊C₄×C₅₃	—	2	1	no	—
92	C₂²⋊C₄×C₅₃	—	2	1	no	—
93	C₄²×C₅₃	—	2	1	no	—
94	C₄²×C₅₃	—	2	1	no	—
95	C₄²×C₅₃	—	2	1	no	—
96	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₅₃	#41	1	1	yes	characteristic
100	C₂³×C₅₃	—	2	1	no	—
101	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
102	Q₈×C₅₃	—	2	2	no	—
103	D₄×C₅₃	—	2	2	no	—
104	D₄×C₅₃	—	4	2	no	—
105	C₂×C₄×C₅₃	—	2	1	no	—
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	2	no	—
108	C₂×C₄×C₅₃	—	2	1	no	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	—	2	1	no	—
112	C₂×C₄×C₅₃	—	2	1	no	—
113	C₂×C₄×C₅₃	—	4	2	no	—
114	C₂×C₄×C₅₃	—	4	1	no	—
115	C₂×C₄×C₅₃	—	4	1	no	—
116	C₂×C₄×C₅₃	—	4	1	no	—
117	C₂×C₄×C₅₃	—	4	1	no	—
118	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
121	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
122	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
124	C₈×C₅₃	—	2	1	no	—
125	C₈×C₅₃	—	2	1	no	—
126	C₈×C₅₃	—	2	1	no	—
127	C₈×C₅₃	—	4	4	no	—
128	C₈×C₅₃	—	4	1	no	—
129	C₈×C₅₃	—	4	1	no	—
130	C₈×C₅₃	—	8	2	no	—
131	C₈×C₅₃	#104	1	2	yes	—
132	#8673	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂²×C₅₃	—	2	1	no	—
134	C₂²×C₅₃	—	2	1	no	—
135	C₂²×C₅₃	—	2	1	no	—
136	C₂²×C₅₃	—	4	1	no	—
137	C₂²×C₅₃	—	4	1	no	—
138	C₂²×C₅₃	—	4	1	no	—
139	C₂²×C₅₃	#380	1	1	yes	characteristic
140	C₄×C₅₃	—	2	1	no	—
141	C₄×C₅₃	—	2	1	no	—
142	C₄×C₅₃	—	2	1	no	—
143	C₄×C₅₃	—	4	1	no	—
144	C₄×C₅₃	—	4	1	no	—
145	C₄×C₅₃	—	4	1	no	—
146	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
147	C₄×C₅₃	#421	1	1	yes	characteristic
148	C₄×C₅₃	#430	1	2	yes	—
149	C₄×C₅₃	#472	1	1	yes	characteristic
150	C₄×C₅₃	#473	1	1	yes	characteristic
151	#2965	C₂×C₅₃	1	1	yes	characteristic
152	C₄×#108	C₂×C₅₃	1	1	yes	characteristic, complemented
153	#1567	C₂×C₅₃	1	1	yes	characteristic, complemented
154	#1489	C₂×C₅₃	1	2	yes	—
155	#1299	C₂×C₅₃	1	2	yes	complemented
156	C₂×C₅₃	—	4	1	no	—
157	C₂×C₅₃	—	8	1	no	—
158	C₂×C₅₃	#1806	1	1	yes	characteristic
159	C₂×C₅₃	#1811	1	1	yes	characteristic
160	C₂×C₅₃	#1816	1	1	yes	characteristic
161	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
162	#446	C₄×C₅₃	1	1	yes	characteristic
163	#446	C₄×C₅₃	1	1	yes	characteristic
164	#436	C₄×C₅₃	1	1	yes	characteristic
165	#436	C₄×C₅₃	1	1	yes	characteristic
166	#435	—	2	1	no	—
167	#435	—	2	1	no	—
168	#431	—	2	1	no	—
169	#431	C₂²×C₅₃	1	1	yes	characteristic
170	C₂×#108	—	2	1	no	—
171	#423	C₂²×C₅₃	1	2	yes	—
172	C₄×M₁₆	—	2	1	no	—
173	C₄×M₁₆	C₂²×C₅₃	1	2	yes	—
174	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
175	#341	—	2	1	no	—
176	#335	—	2	1	no	—
177	#335	—	2	1	no	—
178	Q₈⋊C₈	—	2	2	no	—
179	D₄⋊C₈	—	2	2	no	—
180	C₅₃	#8673	1	1	yes	characteristic, direct product factor, Sylow
181	C₂×#41	Q₈×C₅₃	1	1	yes	characteristic
182	C₂×M₁₆	—	2	1	no	—
183	C₂×M₁₆	—	2	1	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
187	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
188	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
189	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
190	C₄×D₄	—	2	1	no	—
191	C₄×D₄	—	2	1	no	—
192	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
193	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
194	#117	—	2	1	no	—
195	#117	Q₈×C₅₃	1	1	yes	characteristic
196	C₂×C₄²	—	2	1	no	—
197	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
198	#108	—	2	4	no	—
199	C₄⋊C₈	—	2	1	no	—
200	C₄⋊C₈	—	2	1	no	—
201	C₄⋊C₈	—	2	2	no	—
202	C₄⋊C₈	—	2	2	no	—
203	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
204	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
205	C₂²⋊C₈	—	2	1	no	—
206	C₂²⋊C₈	—	2	1	no	—
207	C₂²⋊C₈	—	2	1	no	—
208	#97	—	2	1	no	—
209	#97	—	2	1	no	—
210	#97	—	2	2	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	—	2	2	no	—
213	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
214	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
215	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
216	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
217	#41	—	4	2	no	—
218	C₂×Q₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
219	C₂×D₄	—	2	1	no	—
220	C₂×D₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
225	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
226	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
227	M₁₆	—	2	4	no	—
228	M₁₆	—	4	2	no	—
229	M₁₆	—	4	2	no	—
230	M₁₆	—	4	2	no	—
231	M₁₆	—	8	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
239	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
240	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
241	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
245	C₄⋊C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
246	C₂²⋊C₄	—	2	1	no	—
247	C₂²⋊C₄	—	2	1	no	—
248	C₂²⋊C₄	—	2	1	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	2	1	no	—
251	C₄²	—	2	1	no	—
252	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
253	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
254	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
255	C₄²	#41×C₅₃	1	1	yes	characteristic
256	C₂³	—	2	1	no	—
257	C₂³	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
258	Q₈	—	2	2	no	—
259	D₄	—	2	2	no	—
260	D₄	—	4	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
275	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
276	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
277	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
278	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
279	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
280	C₈	—	2	1	no	—
281	C₈	—	2	1	no	—
282	C₈	—	2	1	no	—
283	C₈	—	4	4	no	—
284	C₈	—	4	1	no	—
285	C₈	—	4	1	no	—
286	C₈	—	8	2	no	—
287	C₈	#104×C₅₃	1	2	yes	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	1	no	—
294	C₂²	C₅₃×#380	1	1	yes	characteristic
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	4	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	C₂×#104×C₅₃	1	1	yes	characteristic
302	C₄	C₅₃×#421	1	1	yes	characteristic
303	C₄	C₅₃×#430	1	2	yes	—
304	C₄	C₅₃×#472	1	1	yes	characteristic
305	C₄	C₅₃×#473	1	1	yes	characteristic
306	C₂	—	4	1	no	—
307	C₂	—	8	1	no	—
308	C₂	C₅₃×#1806	1	1	yes	characteristic
309	C₂	C₅₃×#1811	1	1	yes	characteristic
310	C₂	C₅₃×#1816	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#121
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#380
prime radical	C₂×#41×C₅₃	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×#108×C₅₃	C₂	semidirect
3. C₅₃×#1567	C₂	semidirect
5. C₅₃×#1299	C₂	semidirect
132. #8673	C₅₃	direct (N × H)
152. C₄×#108	C₂×C₅₃	semidirect
153. #1567	C₂×C₅₃	semidirect
155. #1299	C₂×C₅₃	semidirect
180. C₅₃	#8673	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×#2965	2
C₄×#108×C₅₃	2
C₅₃×#1567	2
C₅₃×#1489	2
C₅₃×#1299	2
#8673	53

Subgroups — C53×#8673 #27216185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8673 #27216185