Subgroups — D₅₃×#2151 #27537072

Order 13568.

734 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#405×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20276475	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276474	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276473	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276472	C₂	1	4	yes	maximal, complemented
8	C₄⋊C₁₆×D₅₃	C₂	1	4	yes	maximal, complemented
9	C₅₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×M₁₆×D₅₃	C₄	1	1	yes	characteristic
11	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	#117×D₅₃	C₄	1	1	yes	characteristic
13	C₂×#13133157	C₂²	1	1	yes	characteristic
14	C₂×#13133156	C₂²	1	2	yes	—
15	M₃₂×D₅₃	—	2	4	no	—
16	C₂²×#11719908	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×D₅₃	—	2	4	no	—
18	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
19	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
20	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
21	#16810581	C₄	1	1	yes	characteristic
22	#16810580	C₄	1	2	yes	—
23	#16810579	C₄	1	1	yes	characteristic
24	#16810578	C₂²	1	1	yes	characteristic
25	#16810577	C₂²	1	2	yes	—
26	#97×D₅₃	C₄	1	2	yes	—
27	C₄×C₈×D₅₃	C₂²	1	2	yes	—
28	C₄⋊#11719908	C₂²	1	4	yes	complemented
29	#16810441	C₂²	1	4	yes	complemented
30	#16810441	C₂²	1	4	yes	complemented
31	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
32	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×#405	C₂²	1	1	yes	characteristic
34	C₅₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
35	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
37	C₂×#11719912	C₂×C₄	1	2	yes	—
38	M₁₆×D₅₃	—	2	4	no	—
39	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₅₃	D₄	1	4	yes	—
44	C₂×C₈×D₅₃	Q₈	1	4	yes	—
45	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
46	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
47	#13133176	C₂×C₄	1	1	yes	characteristic
48	#13133175	C₈	1	2	yes	—
49	#13133174	C₂×C₄	1	2	yes	—
50	#13133173	C₈	1	2	yes	—
51	C₄⋊C₄×D₅₃	C₈	1	2	yes	—
52	C₂²⋊C₄×D₅₃	C₈	1	2	yes	—
53	C₄²×D₅₃	C₂×C₄	1	2	yes	—
54	#13133157	—	2	2	no	—
55	#13133156	—	2	4	no	—
56	C₂×#11719908	—	2	2	no	—
57	C₂×#11719908	C₂³	1	2	yes	—
58	C₂×#11719908	C₂³	1	2	yes	—
59	C₁₆×D₅₃	—	2	4	no	—
60	C₁₆×D₅₃	—	2	4	no	—
61	C₁₆×D₅₃	—	2	4	no	—
62	#13133128	C₂×C₄	1	2	yes	—
63	#13133127	C₂×C₄	1	2	yes	—
64	#13133127	C₂×C₄	1	2	yes	—
65	C₄×#88322	C₂×C₄	1	2	yes	—
66	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
67	C₈×Dic₅₃	C₂³	1	2	yes	—
68	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
70	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
71	M₃₂×C₅₃	—	2	2	no	—
72	C₂×C₁₆×C₅₃	—	2	2	no	—
73	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
74	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
75	#97×C₅₃	C₂×C₄	1	2	yes	—
76	C₄×C₈×C₅₃	C₂³	1	2	yes	—
77	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
80	C₂×C₄×D₅₃	C₄⋊C₄	1	4	yes	—
81	C₂×C₄×D₅₃	C₂×C₈	1	4	yes	—
82	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄×D₅₃	M₁₆	1	4	yes	—
85	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
87	M₁₆×C₅₃	—	2	2	no	—
88	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
92	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
94	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
95	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
96	C₂²⋊Dic₅₃	C₂×C₈	1	2	yes	—
97	#11719921	C₂×C₈	1	4	yes	—
98	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
99	#11719919	C₂×C₈	1	4	yes	—
100	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
101	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
102	#11719917	—	2	2	no	—
103	C₂×#88322	C₄⋊C₄	1	4	yes	—
104	C₂×#88322	C₂²×C₄	1	2	yes	—
105	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
106	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
107	#11719912	—	2	4	no	—
108	C₈×D₅₃	—	2	4	no	—
109	C₈×D₅₃	C₄⋊C₄	1	4	yes	—
110	C₈×D₅₃	C₄⋊C₄	1	4	yes	—
111	C₈×D₅₃	C₂×D₄	1	2	yes	—
112	C₈×D₅₃	C₂×D₄	1	2	yes	—
113	C₈×D₅₃	C₂×Q₈	1	2	yes	—
114	C₈×D₅₃	C₂×Q₈	1	2	yes	—
115	C₁₆×C₅₃	—	2	2	no	—
116	C₁₆×C₅₃	—	2	2	no	—
117	#11719908	—	2	2	no	—
118	#11719908	—	2	2	no	—
119	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
120	C₂²×D₅₃	—	2	4	no	—
121	C₂²×D₅₃	C₄⋊C₈	1	4	yes	—
122	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
123	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
126	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
128	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
129	C₂×Dic₅₃	C₄⋊C₈	1	4	yes	—
130	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
131	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
132	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
133	C₄×D₅₃	—	2	8	no	—
134	C₄×D₅₃	C₄⋊C₈	1	4	yes	—
135	C₄×D₅₃	C₄⋊C₈	1	4	yes	—
136	C₄×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
137	C₄×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
138	C₄×D₅₃	C₂×M₁₆	1	2	yes	—
139	C₄×D₅₃	C₂×M₁₆	1	2	yes	—
140	C₈×C₅₃	—	2	2	no	—
141	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
142	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
143	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
144	#88322	—	2	2	no	—
145	#88322	C₂×C₄⋊C₄	1	2	yes	—
146	#88322	C₂×C₄⋊C₄	1	2	yes	—
147	C₂×#2151	—	53	1	no	maximal, Sylow
148	C₂²×C₅₃	—	2	2	no	—
149	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
150	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
151	C₂×D₅₃	—	2	2	no	—
152	C₂×D₅₃	—	2	8	no	—
153	C₂×D₅₃	C₂×C₄⋊C₈	1	2	yes	—
154	C₂×D₅₃	C₂×C₄⋊C₈	1	2	yes	—
155	C₂×D₅₃	#504	1	4	yes	—
156	C₄×C₅₃	—	2	4	no	—
157	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
158	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
159	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
160	Dic₅₃	—	2	4	no	—
161	Dic₅₃	C₂×C₄⋊C₈	1	2	yes	—
162	Dic₅₃	C₂×C₄⋊C₈	1	2	yes	—
163	C₂²×M₃₂	—	53	1	no	—
164	C₂³×C₁₆	—	53	1	no	—
165	C₂×#405	—	53	1	no	—
166	#2151	—	53	1	no	—
167	#2151	—	53	1	no	—
168	#2151	—	53	1	no	—
169	#2151	—	53	4	no	—
170	#2151	D₅₃	1	1	yes	characteristic, direct product factor
171	C₂×C₄⋊C₁₆	—	53	4	no	—
172	C₂×C₅₃	—	2	2	no	—
173	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
174	C₂×C₅₃	C₂×#504	1	2	yes	—
175	D₅₃	—	2	2	no	—
176	D₅₃	#2151	1	4	yes	direct product factor
177	C₂²×M₁₆	—	53	1	no	—
178	C₂³×C₈	—	53	1	no	—
179	C₂×#117	—	53	1	no	—
180	C₂×M₃₂	—	53	1	no	—
181	C₂×M₃₂	—	53	2	no	—
182	C₂×M₃₂	—	106	4	no	—
183	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
184	C₂²×C₁₆	—	53	1	no	—
185	C₂²×C₁₆	—	53	2	no	—
186	C₂²×C₁₆	—	53	2	no	—
187	C₂²×C₁₆	—	53	2	no	—
188	C₂²×C₁₆	—	106	4	no	—
189	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
190	#405	—	53	1	no	—
191	#405	—	53	1	no	—
192	#405	—	53	1	no	—
193	#405	—	53	2	no	—
194	#405	—	53	2	no	—
195	#405	C₂×D₅₃	1	1	yes	characteristic
196	C₂×#97	—	53	2	no	—
197	C₂×C₄×C₈	—	53	2	no	—
198	C₄⋊C₁₆	—	53	4	no	—
199	C₄⋊C₁₆	—	53	4	no	—
200	C₄⋊C₁₆	—	53	4	no	—
201	C₄⋊C₁₆	C₂×D₅₃	1	4	yes	complemented
202	C₅₃	C₂×#2151	1	1	yes	characteristic, complemented, Sylow
203	C₂³×C₄	—	53	1	no	—
204	C₂×M₁₆	—	53	1	no	—
205	C₂×M₁₆	—	53	2	no	—
206	C₂×M₁₆	—	106	4	no	—
207	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
208	C₂²×C₈	—	53	1	no	—
209	C₂²×C₈	—	53	1	no	—
210	C₂²×C₈	—	53	1	no	—
211	C₂²×C₈	—	53	4	no	—
212	C₂²×C₈	—	53	4	no	—
213	C₂²×C₈	—	53	1	no	—
214	C₂²×C₈	—	53	1	no	—
215	C₂²×C₈	—	53	2	no	—
216	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
217	#117	—	53	1	no	—
218	#117	—	53	2	no	—
219	#117	—	53	2	no	—
220	#117	—	53	2	no	—
221	#117	C₄×D₅₃	1	1	yes	characteristic
222	C₂×C₄⋊C₄	—	53	2	no	—
223	C₂×C₂²⋊C₄	—	53	2	no	—
224	C₂×C₄²	—	53	2	no	—
225	M₃₂	—	2	2	no	—
226	M₃₂	—	106	4	no	—
227	M₃₂	—	106	2	no	—
228	C₂×C₁₆	—	2	2	no	—
229	C₂×C₁₆	—	53	2	no	—
230	C₂×C₁₆	—	53	2	no	—
231	C₂×C₁₆	—	106	4	no	—
232	C₂×C₁₆	—	106	4	no	—
233	C₂×C₁₆	—	106	4	no	—
234	C₂×C₁₆	—	106	2	no	—
235	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
236	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
237	#97	—	53	2	no	—
238	#97	—	53	2	no	—
239	#97	—	53	2	no	—
240	#97	C₄×D₅₃	1	2	yes	—
241	C₄×C₈	—	53	2	no	—
242	C₄×C₈	—	53	2	no	—
243	C₄×C₈	—	53	2	no	—
244	C₄×C₈	C₂²×D₅₃	1	2	yes	—
245	C₂⁴	—	53	1	no	—
246	C₂²×C₄	—	53	1	no	—
247	C₂²×C₄	—	53	4	no	—
248	C₂²×C₄	—	53	1	no	—
249	C₂²×C₄	—	53	1	no	—
250	C₂²×C₄	—	53	4	no	—
251	C₂²×C₄	—	53	1	no	—
252	C₂²×C₄	—	53	1	no	—
253	C₂²×C₄	—	53	4	no	—
254	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
255	M₁₆	—	2	2	no	—
256	M₁₆	—	106	4	no	—
257	M₁₆	—	106	2	no	—
258	C₂×C₈	—	53	4	no	—
259	C₂×C₈	—	53	1	no	—
260	C₂×C₈	—	53	1	no	—
261	C₂×C₈	—	53	4	no	—
262	C₂×C₈	—	53	4	no	—
263	C₂×C₈	—	53	2	no	—
264	C₂×C₈	—	53	2	no	—
265	C₂×C₈	—	53	2	no	—
266	C₂×C₈	—	53	2	no	—
267	C₂×C₈	—	53	2	no	—
268	C₂×C₈	—	106	4	no	—
269	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
270	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
271	C₂×C₈	D₄×D₅₃	1	2	yes	—
272	C₂×C₈	Q₈×D₅₃	1	2	yes	—
273	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
274	C₄⋊C₄	—	53	4	no	—
275	C₄⋊C₄	—	53	2	no	—
276	C₄⋊C₄	C₈×D₅₃	1	2	yes	—
277	C₂²⋊C₄	—	53	2	no	—
278	C₂²⋊C₄	—	53	4	no	—
279	C₂²⋊C₄	C₈×D₅₃	1	2	yes	—
280	C₄²	—	53	4	no	—
281	C₄²	—	53	2	no	—
282	C₄²	C₂×C₄×D₅₃	1	2	yes	—
283	C₁₆	—	2	2	no	—
284	C₁₆	—	2	2	no	—
285	C₁₆	—	106	2	no	—
286	C₁₆	—	106	2	no	—
287	C₂³	—	53	1	no	—
288	C₂³	—	53	1	no	—
289	C₂³	—	53	4	no	—
290	C₂³	—	106	4	no	—
291	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
292	C₂×C₄	—	53	1	no	—
293	C₂×C₄	—	53	1	no	—
294	C₂×C₄	—	53	4	no	—
295	C₂×C₄	—	53	2	no	—
296	C₂×C₄	—	53	2	no	—
297	C₂×C₄	—	53	2	no	—
298	C₂×C₄	—	53	2	no	—
299	C₂×C₄	—	53	4	no	—
300	C₂×C₄	—	53	4	no	—
301	C₂×C₄	—	53	4	no	—
302	C₂×C₄	—	106	8	no	—
303	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
304	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
305	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
306	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
307	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	53	2	no	—
310	C₈	—	53	2	no	—
311	C₈	—	106	2	no	—
312	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
313	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
314	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
315	C₂²	—	2	2	no	—
316	C₂²	—	53	4	no	—
317	C₂²	—	53	2	no	—
318	C₂²	—	53	2	no	—
319	C₂²	—	106	8	no	—
320	C₂²	—	106	2	no	—
321	C₂²	C₄⋊C₈×D₅₃	1	2	yes	—
322	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
323	C₄	—	2	4	no	—
324	C₄	—	53	2	no	—
325	C₄	—	53	2	no	—
326	C₄	—	106	4	no	—
327	C₄	C₄⋊C₈×D₅₃	1	2	yes	—
328	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
329	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
330	C₂	—	2	2	no	—
331	C₂	—	53	4	no	—
332	C₂	—	106	2	no	—
333	C₂	C₂×C₄⋊C₈×D₅₃	1	1	yes	characteristic
334	C₂	#504×D₅₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2151	C₂
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2151	D₅₃
hypocenter	C₅₃	C₂×#2151
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2151

Complements

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

N	H ≅ G/N	type
4. #20276475	C₂	semidirect
5. #20276474	C₂	semidirect
6. #20276473	C₂	semidirect
7. #20276472	C₂	semidirect
8. C₄⋊C₁₆×D₅₃	C₂	semidirect
9. C₅₃×#2151	C₂	semidirect
28. C₄⋊#11719908	C₂²	semidirect
29. #16810441	C₂²	semidirect
30. #16810441	C₂²	semidirect
34. C₅₃×C₄⋊C₁₆	C₂²	semidirect
170. #2151	D₅₃	direct (N × H)
176. D₅₃	#2151	direct (N × H)
201. C₄⋊C₁₆	C₂×D₅₃	semidirect
202. C₅₃	C₂×#2151	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂×D₅₃	2
C₂²×C₁₆×D₅₃	2
#405×D₅₃	2
#20276475	2
#20276474	2
#20276473	2
#20276472	2
C₄⋊C₁₆×D₅₃	2
C₅₃×#2151	2
C₂×#2151	53

Subgroups — D53×#2151 #27537072

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#2151 #27537072