Subgroups — D₅₃×#2150 #27537068

Order 13568.

798 conjugacy classes of subgroups.

The subgroup lattice has 340 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	2	yes	maximal
2	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#20276471	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276470	C₂	1	2	yes	maximal, complemented
5	#20276469	C₂	1	4	yes	maximal, complemented
6	C₄⋊C₁₆×D₅₃	C₂	1	4	yes	maximal, complemented
7	C₅₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₅₃	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄²×D₅₃	C₄	1	1	yes	characteristic
11	C₂×#13133157	C₂²	1	2	yes	—
12	C₂×#13133156	C₂²	1	4	yes	—
13	M₃₂×D₅₃	—	2	8	no	—
14	C₂×C₁₆×D₅₃	C₂²	1	4	yes	—
15	C₂×C₄×#88322	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
18	C₄×C₈×D₅₃	C₄	1	2	yes	—
19	C₄×C₈×D₅₃	C₄	1	2	yes	—
20	C₄×C₈×D₅₃	C₂²	1	2	yes	—
21	C₄×C₈×D₅₃	C₂²	1	2	yes	—
22	C₄⋊#11719908	C₂²	1	4	yes	complemented
23	#16810441	C₂²	1	8	yes	complemented
24	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
27	C₂²×C₄×D₅₃	C₈	1	2	yes	—
28	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
30	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₅₃	—	2	4	no	—
32	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₅₃	D₄	1	4	yes	—
37	C₂×C₈×D₅₃	Q₈	1	4	yes	—
38	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
39	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
40	C₂×C₄×Dic₅₃	C₈	1	2	yes	—
41	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄²×D₅₃	C₈	1	1	yes	characteristic
43	C₄²×D₅₃	C₈	1	1	yes	characteristic
44	C₄²×D₅₃	C₈	1	1	yes	characteristic
45	C₄²×D₅₃	C₈	1	1	yes	characteristic
46	C₄²×D₅₃	C₂×C₄	1	2	yes	—
47	C₄²×D₅₃	C₂×C₄	1	2	yes	—
48	#13133157	—	2	4	no	—
49	#13133156	—	2	8	no	—
50	C₂×#11719908	C₂³	1	4	yes	—
51	C₁₆×D₅₃	—	2	8	no	—
52	C₄×#88322	C₂×C₄	1	2	yes	—
53	C₄×#88322	C₂×C₄	1	2	yes	—
54	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
55	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
56	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
57	C₈×Dic₅₃	C₂³	1	2	yes	—
58	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
61	M₃₂×C₅₃	—	2	4	no	—
62	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
63	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₅₃	C₂³	1	2	yes	—
65	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
69	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
70	C₂×C₄×D₅₃	—	2	8	no	—
71	C₂×C₄×D₅₃	C₄⋊C₄	1	4	yes	—
72	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
73	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
74	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
75	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
76	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
77	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
78	C₂×C₄×D₅₃	M₁₆	1	4	yes	—
79	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
86	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
87	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
88	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
89	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
90	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
91	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
92	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
93	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
94	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
95	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
96	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
97	C₂×#88322	—	2	2	no	—
98	C₂×#88322	C₄⋊C₄	1	4	yes	—
99	C₂×#88322	C₂²×C₄	1	2	yes	—
100	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
101	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
102	C₈×D₅₃	—	2	4	no	—
103	C₈×D₅₃	—	2	4	no	—
104	C₈×D₅₃	C₄⋊C₄	1	4	yes	—
105	C₈×D₅₃	C₄⋊C₄	1	4	yes	—
106	C₈×D₅₃	C₂×D₄	1	2	yes	—
107	C₈×D₅₃	C₂×D₄	1	2	yes	—
108	C₈×D₅₃	C₂×Q₈	1	2	yes	—
109	C₈×D₅₃	C₂×Q₈	1	2	yes	—
110	C₁₆×C₅₃	—	2	4	no	—
111	#11719908	—	2	4	no	—
112	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
113	C₂²×D₅₃	—	2	4	no	—
114	C₂²×D₅₃	C₄⋊C₈	1	4	yes	—
115	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
116	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₅₃	—	2	4	no	—
118	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
120	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
122	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
123	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×Dic₅₃	—	2	4	no	—
125	C₂×Dic₅₃	C₄⋊C₈	1	4	yes	—
126	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
127	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
128	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
130	C₄×D₅₃	—	2	4	no	—
131	C₄×D₅₃	—	2	4	no	—
132	C₄×D₅₃	—	2	4	no	—
133	C₄×D₅₃	C₄⋊C₈	1	4	yes	—
134	C₄×D₅₃	C₄⋊C₈	1	4	yes	—
135	C₄×D₅₃	M₃₂	1	8	yes	complemented
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×#2150	—	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	C₂×M₃₂	1	4	yes	—
156	C₄×C₅₃	—	2	2	no	—
157	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
158	C₄×C₅₃	C₂×M₃₂	1	4	yes	complemented
159	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
160	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
161	Dic₅₃	—	2	2	no	—
162	Dic₅₃	C₂×C₄⋊C₈	1	2	yes	—
163	Dic₅₃	C₂×C₄⋊C₈	1	2	yes	—
164	Dic₅₃	C₂×M₃₂	1	4	yes	complemented
165	C₂²×M₃₂	—	53	2	no	—
166	C₂²×C₄×C₈	—	53	1	no	—
167	#2150	—	53	2	no	—
168	#2150	—	53	1	no	—
169	#2150	—	53	4	no	—
170	#2150	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	C₂²×M₃₂	1	2	yes	—
175	D₅₃	—	2	2	no	—
176	D₅₃	#2150	1	4	yes	direct product factor
177	C₂³×C₈	—	53	1	no	—
178	C₂³×C₈	—	53	1	no	—
179	C₂²×C₄²	—	53	1	no	—
180	C₂×M₃₂	—	53	2	no	—
181	C₂×M₃₂	—	53	4	no	—
182	C₂×M₃₂	—	106	8	no	—
183	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
184	C₂²×C₁₆	—	53	4	no	—
185	C₂×C₄×C₈	—	53	1	no	—
186	C₂×C₄×C₈	—	53	1	no	—
187	C₂×C₄×C₈	—	53	1	no	—
188	C₂×C₄×C₈	—	53	2	no	—
189	C₂×C₄×C₈	—	53	2	no	—
190	C₂×C₄×C₈	—	53	2	no	—
191	C₂×C₄×C₈	—	53	2	no	—
192	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
193	C₄⋊C₁₆	—	53	8	no	—
194	C₄⋊C₁₆	—	53	4	no	—
195	C₄⋊C₁₆	C₂×D₅₃	1	4	yes	complemented
196	C₅₃	C₂×#2150	1	1	yes	characteristic, complemented, Sylow
197	C₂³×C₄	—	53	1	no	—
198	C₂³×C₄	—	53	2	no	—
199	C₂²×C₈	—	53	1	no	—
200	C₂²×C₈	—	53	1	no	—
201	C₂²×C₈	—	53	1	no	—
202	C₂²×C₈	—	53	4	no	—
203	C₂²×C₈	—	53	4	no	—
204	C₂²×C₈	—	53	1	no	—
205	C₂²×C₈	—	53	1	no	—
206	C₂²×C₈	—	53	1	no	—
207	C₂²×C₈	—	53	2	no	—
208	C₂²×C₈	—	53	2	no	—
209	C₂²×C₈	—	106	4	no	—
210	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
211	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
212	C₂×C₄²	—	53	1	no	—
213	C₂×C₄²	—	53	2	no	—
214	C₂×C₄²	—	53	2	no	—
215	C₂×C₄²	—	53	2	no	—
216	C₂×C₄²	—	53	1	no	—
217	C₂×C₄²	—	53	1	no	—
218	C₂×C₄²	—	53	1	no	—
219	C₂×C₄²	—	53	1	no	—
220	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
221	M₃₂	—	2	4	no	—
222	M₃₂	—	106	8	no	—
223	M₃₂	—	106	4	no	—
224	C₂×C₁₆	—	53	4	no	—
225	C₂×C₁₆	—	106	8	no	—
226	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
227	C₄×C₈	—	53	2	no	—
228	C₄×C₈	—	53	2	no	—
229	C₄×C₈	—	53	2	no	—
230	C₄×C₈	—	53	2	no	—
231	C₄×C₈	—	53	2	no	—
232	C₄×C₈	—	53	2	no	—
233	C₄×C₈	C₄×D₅₃	1	2	yes	—
234	C₄×C₈	C₂²×D₅₃	1	2	yes	—
235	C₂⁴	—	53	1	no	—
236	C₂²×C₄	—	53	1	no	—
237	C₂²×C₄	—	53	4	no	—
238	C₂²×C₄	—	53	2	no	—
239	C₂²×C₄	—	53	1	no	—
240	C₂²×C₄	—	53	1	no	—
241	C₂²×C₄	—	53	4	no	—
242	C₂²×C₄	—	53	1	no	—
243	C₂²×C₄	—	53	1	no	—
244	C₂²×C₄	—	53	2	no	—
245	C₂²×C₄	—	53	2	no	—
246	C₂²×C₄	—	53	2	no	—
247	C₂²×C₄	—	53	2	no	—
248	C₂²×C₄	—	106	8	no	—
249	C₂²×C₄	C₈×D₅₃	1	2	yes	—
250	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	53	4	no	—
253	C₂×C₈	—	53	1	no	—
254	C₂×C₈	—	53	1	no	—
255	C₂×C₈	—	53	4	no	—
256	C₂×C₈	—	53	4	no	—
257	C₂×C₈	—	53	2	no	—
258	C₂×C₈	—	53	2	no	—
259	C₂×C₈	—	53	2	no	—
260	C₂×C₈	—	53	2	no	—
261	C₂×C₈	—	53	2	no	—
262	C₂×C₈	—	106	4	no	—
263	C₂×C₈	—	106	4	no	—
264	C₂×C₈	—	106	2	no	—
265	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
266	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
267	C₂×C₈	D₄×D₅₃	1	2	yes	—
268	C₂×C₈	Q₈×D₅₃	1	2	yes	—
269	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
270	C₄²	—	53	2	no	—
271	C₄²	—	53	2	no	—
272	C₄²	—	53	1	no	—
273	C₄²	—	53	2	no	—
274	C₄²	—	53	2	no	—
275	C₄²	—	53	1	no	—
276	C₄²	—	53	2	no	—
277	C₄²	C₈×D₅₃	1	1	yes	characteristic
278	C₄²	C₈×D₅₃	1	1	yes	characteristic
279	C₄²	C₂×C₄×D₅₃	1	2	yes	—
280	C₁₆	—	2	4	no	—
281	C₁₆	—	106	4	no	—
282	C₂³	—	53	1	no	—
283	C₂³	—	53	1	no	—
284	C₂³	—	53	4	no	—
285	C₂³	—	106	4	no	—
286	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	53	2	no	—
289	C₂×C₄	—	53	1	no	—
290	C₂×C₄	—	53	1	no	—
291	C₂×C₄	—	53	4	no	—
292	C₂×C₄	—	53	8	no	—
293	C₂×C₄	—	53	2	no	—
294	C₂×C₄	—	53	2	no	—
295	C₂×C₄	—	53	2	no	—
296	C₂×C₄	—	53	2	no	—
297	C₂×C₄	—	53	4	no	—
298	C₂×C₄	—	53	4	no	—
299	C₂×C₄	—	53	2	no	—
300	C₂×C₄	—	106	4	no	—
301	C₂×C₄	—	106	4	no	—
302	C₂×C₄	—	106	4	no	—
303	C₂×C₄	—	106	4	no	—
304	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
305	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
306	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
307	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
308	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
309	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	—	53	2	no	—
312	C₈	—	53	2	no	—
313	C₈	—	106	2	no	—
314	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
315	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
316	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
317	C₂²	—	2	2	no	—
318	C₂²	—	53	4	no	—
319	C₂²	—	53	2	no	—
320	C₂²	—	53	2	no	—
321	C₂²	—	106	8	no	—
322	C₂²	—	106	2	no	—
323	C₂²	C₄⋊C₈×D₅₃	1	2	yes	—
324	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
325	C₄	—	2	2	no	—
326	C₄	—	53	4	no	—
327	C₄	—	53	2	no	—
328	C₄	—	53	2	no	—
329	C₄	—	106	2	no	—
330	C₄	C₄⋊C₈×D₅₃	1	2	yes	—
331	C₄	M₃₂×D₅₃	1	4	yes	complemented
332	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
333	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	53	4	no	—
336	C₂	—	106	2	no	—
337	C₂	C₂×C₄⋊C₈×D₅₃	1	1	yes	characteristic
338	C₂	C₂×M₃₂×D₅₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2150	C₂
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2150	D₅₃
hypocenter	C₅₃	C₂×#2150
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
3. #20276471	C₂	semidirect
4. #20276470	C₂	semidirect
5. #20276469	C₂	semidirect
6. C₄⋊C₁₆×D₅₃	C₂	semidirect
7. C₅₃×#2150	C₂	semidirect
22. C₄⋊#11719908	C₂²	semidirect
23. #16810441	C₂²	semidirect
26. C₅₃×C₄⋊C₁₆	C₂²	semidirect
135. C₄×D₅₃	M₃₂	semidirect
158. C₄×C₅₃	C₂×M₃₂	semidirect
164. Dic₅₃	C₂×M₃₂	semidirect
170. #2150	D₅₃	direct (N × H)
176. D₅₃	#2150	direct (N × H)
195. C₄⋊C₁₆	C₂×D₅₃	semidirect
196. C₅₃	C₂×#2150	semidirect
331. C₄	M₃₂×D₅₃	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
C₂×C₄×C₈×D₅₃	2
#20276471	2
#20276470	2
#20276469	2
C₄⋊C₁₆×D₅₃	2
C₅₃×#2150	2
C₂×#2150	53

Subgroups — D53×#2150 #27537068

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#2150 #27537068