Subgroups — C₅₃×#8196 #27215708

Order 13568.

500 conjugacy classes of subgroups.

The subgroup lattice has 320 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×#104×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₄×#97×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1476	C₂	1	1	yes	maximal, characteristic
5	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1278	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1276	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#446	C₄	1	2	yes	complemented
10	C₅₃×#436	C₄	1	2	yes	complemented
11	C₅₃×#435	—	2	2	no	—
12	C₅₃×#431	—	2	1	no	—
13	C₅₃×#431	C₂²	1	1	yes	characteristic
14	C₅₃×#431	C₂²	1	1	yes	characteristic
15	C₂×#104×C₅₃	C₄	1	2	yes	—
16	C₄×M₁₆×C₅₃	—	2	1	no	—
17	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×#97×C₅₃	—	2	2	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₄³×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#339	C₄	1	2	yes	—
23	C₅₃×Q₈⋊C₈	C₄	1	2	yes	complemented
24	C₅₃×D₄⋊C₈	C₄	1	2	yes	complemented
25	C₅₃×#322	—	2	1	no	—
26	C₈²×C₅₃	—	2	1	no	—
27	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×M₁₆×C₅₃	—	2	2	no	—
29	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₅₃	—	2	1	no	—
31	C₂²×C₈×C₅₃	D₄	1	2	yes	—
32	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄×Q₈×C₅₃	C₂×C₄	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	C₂×C₄	1	1	yes	characteristic
38	#117×C₅₃	—	2	1	no	—
39	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	—	2	1	no	—
41	C₂×C₄²×C₅₃	—	2	1	no	—
42	C₂×C₄²×C₅₃	—	2	1	no	—
43	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₅₃	—	2	2	no	—
46	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
47	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
48	#104×C₅₃	—	2	4	no	—
49	C₂²⋊C₈×C₅₃	—	2	2	no	—
50	C₂²⋊C₈×C₅₃	—	2	1	no	—
51	C₂²⋊C₈×C₅₃	—	2	1	no	—
52	#97×C₅₃	—	2	8	no	—
53	#97×C₅₃	—	2	2	no	—
54	#97×C₅₃	—	2	1	no	—
55	C₄×C₈×C₅₃	—	2	1	no	—
56	C₄×C₈×C₅₃	—	2	1	no	—
57	C₄×C₈×C₅₃	—	2	1	no	—
58	C₄×C₈×C₅₃	—	2	1	no	—
59	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
62	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
63	#41×C₅₃	—	2	2	no	—
64	#41×C₅₃	—	2	2	no	—
65	C₂×Q₈×C₅₃	C₄²	1	1	yes	characteristic
66	C₂×D₄×C₅₃	—	2	1	no	—
67	C₂×D₄×C₅₃	C₄²	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	—	2	2	no	—
69	C₂²×C₄×C₅₃	—	2	1	no	—
70	C₂²×C₄×C₅₃	—	2	1	no	—
71	C₂²×C₄×C₅₃	—	2	1	no	—
72	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
75	M₁₆×C₅₃	—	2	2	no	—
76	M₁₆×C₅₃	—	4	4	no	—
77	M₁₆×C₅₃	M₁₆	1	4	yes	complemented
78	C₂×C₈×C₅₃	—	2	4	no	—
79	C₂×C₈×C₅₃	—	2	2	no	—
80	C₂×C₈×C₅₃	—	2	2	no	—
81	C₂×C₈×C₅₃	—	2	4	no	—
82	C₂×C₈×C₅₃	—	2	2	no	—
83	C₂×C₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
85	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
86	C₄⋊C₄×C₅₃	—	2	1	no	—
87	C₄⋊C₄×C₅₃	—	2	1	no	—
88	C₄⋊C₄×C₅₃	C₄²	1	1	yes	characteristic
89	C₄⋊C₄×C₅₃	C₄²	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	4	no	—
94	C₄²×C₅₃	—	2	1	no	—
95	C₄²×C₅₃	—	2	1	no	—
96	C₄²×C₅₃	—	2	1	no	—
97	C₄²×C₅₃	—	2	2	no	—
98	C₄²×C₅₃	—	2	2	no	—
99	C₄²×C₅₃	—	4	2	no	—
100	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₅₃	#41	1	1	yes	characteristic
104	C₂³×C₅₃	—	2	1	no	—
105	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
106	Q₈×C₅₃	—	2	1	no	—
107	Q₈×C₅₃	—	2	1	no	—
108	D₄×C₅₃	—	2	1	no	—
109	D₄×C₅₃	—	2	1	no	—
110	D₄×C₅₃	—	4	2	no	—
111	C₂×C₄×C₅₃	—	2	2	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	2	4	no	—
114	C₂×C₄×C₅₃	—	2	1	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	—	2	1	no	—
117	C₂×C₄×C₅₃	—	4	2	no	—
118	C₂×C₄×C₅₃	—	4	2	no	—
119	C₂×C₄×C₅₃	—	4	2	no	—
120	C₂×C₄×C₅₃	—	4	2	no	—
121	C₂×C₄×C₅₃	—	4	2	no	—
122	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
129	C₂×C₄×C₅₃	#131	1	2	yes	—
130	C₈×C₅₃	—	2	4	no	—
131	C₈×C₅₃	—	2	2	no	—
132	C₈×C₅₃	—	2	4	no	—
133	C₈×C₅₃	—	4	2	no	—
134	C₈×C₅₃	—	4	2	no	—
135	C₈×C₅₃	#104	1	4	yes	complemented
136	#8196	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂²×C₅₃	—	2	1	no	—
138	C₂²×C₅₃	—	2	1	no	—
139	C₂²×C₅₃	—	4	1	no	—
140	C₂²×C₅₃	—	4	1	no	—
141	C₂²×C₅₃	—	4	1	no	—
142	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
143	C₂²×C₅₃	#436	1	2	yes	—
144	C₄×C₅₃	—	2	1	no	—
145	C₄×C₅₃	—	2	1	no	—
146	C₄×C₅₃	—	4	4	no	—
147	C₄×C₅₃	—	4	1	no	—
148	C₄×C₅₃	—	4	1	no	—
149	C₄×C₅₃	—	4	1	no	—
150	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
151	C₄×C₅₃	#405	1	1	yes	characteristic
152	C₄×C₅₃	C₂×#104	1	2	yes	—
153	C₄×C₅₃	#435	1	2	yes	—
154	C₄×C₅₃	#444	1	2	yes	—
155	#2965	C₂×C₅₃	1	1	yes	characteristic
156	C₄×#104	C₂×C₅₃	1	1	yes	characteristic
157	C₄×#97	C₂×C₅₃	1	1	yes	characteristic, complemented
158	#1476	C₂×C₅₃	1	1	yes	characteristic
159	C₈×M₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
160	#1278	C₂×C₅₃	1	1	yes	characteristic, complemented
161	#1276	C₂×C₅₃	1	1	yes	characteristic, complemented
162	C₂×C₅₃	—	2	2	no	—
163	C₂×C₅₃	—	8	1	no	—
164	C₂×C₅₃	#1753	1	1	yes	characteristic
165	C₂×C₅₃	C₄×#104	1	1	yes	characteristic
166	C₂×C₅₃	#1765	1	1	yes	characteristic
167	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
168	#446	C₄×C₅₃	1	2	yes	complemented
169	#436	C₄×C₅₃	1	2	yes	complemented
170	#435	—	2	2	no	—
171	#431	—	2	1	no	—
172	#431	C₂²×C₅₃	1	1	yes	characteristic
173	#431	C₂²×C₅₃	1	1	yes	characteristic
174	C₂×#104	C₄×C₅₃	1	2	yes	—
175	C₄×M₁₆	—	2	1	no	—
176	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
177	C₂×#97	—	2	2	no	—
178	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
179	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
180	C₄³	C₂²×C₅₃	1	1	yes	characteristic
181	#339	C₄×C₅₃	1	2	yes	—
182	Q₈⋊C₈	C₄×C₅₃	1	2	yes	complemented
183	D₄⋊C₈	C₄×C₅₃	1	2	yes	complemented
184	#322	—	2	1	no	—
185	C₈²	—	2	1	no	—
186	C₅₃	#8196	1	1	yes	characteristic, direct product factor, Sylow
187	C₂×#41	C₂×C₄×C₅₃	1	1	yes	characteristic
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	D₄×C₅₃	1	2	yes	—
192	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
193	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
194	C₄×D₄	—	2	1	no	—
195	C₄×D₄	—	2	1	no	—
196	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
197	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
198	#117	—	2	1	no	—
199	#117	C₂×C₄×C₅₃	1	1	yes	characteristic
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	2	1	no	—
202	C₂×C₄²	—	2	1	no	—
203	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
204	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
205	C₄⋊C₈	—	2	2	no	—
206	C₄⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
207	C₄⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
208	#104	—	2	4	no	—
209	C₂²⋊C₈	—	2	1	no	—
210	C₂²⋊C₈	—	2	1	no	—
211	C₂²⋊C₈	—	2	2	no	—
212	#97	—	2	8	no	—
213	#97	—	2	2	no	—
214	#97	—	2	1	no	—
215	C₄×C₈	—	2	1	no	—
216	C₄×C₈	—	2	1	no	—
217	C₄×C₈	—	2	1	no	—
218	C₄×C₈	—	2	1	no	—
219	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
220	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
221	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
222	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
223	#41	—	2	2	no	—
224	#41	—	2	2	no	—
225	C₂×Q₈	C₄²×C₅₃	1	1	yes	characteristic
226	C₂×D₄	—	2	1	no	—
227	C₂×D₄	C₄²×C₅₃	1	1	yes	characteristic
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
234	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
235	M₁₆	—	2	2	no	—
236	M₁₆	—	4	4	no	—
237	M₁₆	M₁₆×C₅₃	1	4	yes	complemented
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
245	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	C₄²×C₅₃	1	1	yes	characteristic
249	C₄⋊C₄	C₄²×C₅₃	1	1	yes	characteristic
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	2	1	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	4	2	no	—
260	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
261	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
262	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
263	C₄²	#41×C₅₃	1	1	yes	characteristic
264	C₂³	—	2	1	no	—
265	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
266	Q₈	—	2	1	no	—
267	Q₈	—	2	1	no	—
268	D₄	—	2	1	no	—
269	D₄	—	2	1	no	—
270	D₄	—	4	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
283	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
284	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
285	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
286	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
287	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
288	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
289	C₂×C₄	#131×C₅₃	1	2	yes	—
290	C₈	—	2	4	no	—
291	C₈	—	2	2	no	—
292	C₈	—	2	4	no	—
293	C₈	—	4	2	no	—
294	C₈	—	4	2	no	—
295	C₈	#104×C₅₃	1	4	yes	complemented
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	1	no	—
301	C₂²	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
302	C₂²	C₅₃×#436	1	2	yes	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	4	4	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	C₄×M₁₆×C₅₃	1	1	yes	characteristic
310	C₄	C₅₃×#405	1	1	yes	characteristic
311	C₄	C₂×#104×C₅₃	1	2	yes	—
312	C₄	C₅₃×#435	1	2	yes	—
313	C₄	C₅₃×#444	1	2	yes	—
314	C₂	—	2	2	no	—
315	C₂	—	8	1	no	—
316	C₂	C₅₃×#1753	1	1	yes	characteristic
317	C₂	C₄×#104×C₅₃	1	1	yes	characteristic
318	C₂	C₅₃×#1765	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₅₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×#97×C₅₃	C₂	semidirect
5. C₈×M₁₆×C₅₃	C₂	semidirect
6. C₅₃×#1278	C₂	semidirect
7. C₅₃×#1276	C₂	semidirect
9. C₅₃×#446	C₄	semidirect
10. C₅₃×#436	C₄	semidirect
23. C₅₃×Q₈⋊C₈	C₄	semidirect
24. C₅₃×D₄⋊C₈	C₄	semidirect
77. M₁₆×C₅₃	M₁₆	semidirect
135. C₈×C₅₃	#104	semidirect
136. #8196	C₅₃	direct (N × H)
157. C₄×#97	C₂×C₅₃	semidirect
159. C₈×M₁₆	C₂×C₅₃	semidirect
160. #1278	C₂×C₅₃	semidirect
161. #1276	C₂×C₅₃	semidirect
168. #446	C₄×C₅₃	semidirect
169. #436	C₄×C₅₃	semidirect
182. Q₈⋊C₈	C₄×C₅₃	semidirect
183. D₄⋊C₈	C₄×C₅₃	semidirect
186. C₅₃	#8196	direct (N × H)
237. M₁₆	M₁₆×C₅₃	semidirect
295. C₈	#104×C₅₃	semidirect

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×#104×C₅₃	2
C₄×#97×C₅₃	2
C₅₃×#1476	2
C₈×M₁₆×C₅₃	2
C₅₃×#1278	2
C₅₃×#1276	2
#8196	53

Subgroups — C53×#8196 #27215708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8196 #27215708