Subgroups — D₅₃×#2177 #27537391

Order 13568.

798 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₄×D₈×D₅₃	C₂	1	2	yes	maximal, complemented
3	#20276592	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276591	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276590	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276589	C₂	1	2	yes	maximal, complemented
7	#20276588	C₂	1	2	yes	maximal, complemented
8	#367×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#357×D₅₃	C₂	1	2	yes	maximal, complemented
10	#346×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2177	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×D₅₃	C₂²	1	2	yes	—
13	C₄×D₄×D₅₃	—	2	2	no	—
14	C₂×D₈₄₈	C₂²	1	1	yes	characteristic
15	C₂×#13133158	C₂²	1	2	yes	—
16	D₁₆×D₅₃	C₄	1	8	yes	complemented
17	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
19	C₄×D₄₂₄	C₂²	1	2	yes	complemented
20	D₈×Dic₅₃	C₂²	1	2	yes	complemented
21	#16810705	C₂²	1	2	yes	complemented
22	C₄×#11719922	—	2	2	no	—
23	#16810703	—	2	2	no	—
24	#107×D₅₃	C₂²	1	2	yes	—
25	#102×D₅₃	—	2	2	no	—
26	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
27	#16810451	C₂²	1	1	yes	characteristic, complemented
28	#16810450	C₂²	1	2	yes	complemented
29	#16810425	C₂²	1	2	yes	complemented
30	#16810424	C₂²	1	2	yes	complemented
31	#16810423	C₂²	1	2	yes	complemented
32	#16810399	C₂²	1	1	yes	characteristic, complemented
33	#16810398	C₂²	1	1	yes	characteristic, complemented
34	#16810398	C₂²	1	1	yes	characteristic, complemented
35	C₂×D₁₆×C₅₃	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₅₃	C₂²	1	2	yes	complemented
37	C₅₃×#367	C₂²	1	1	yes	characteristic, complemented
38	C₅₃×#357	C₂²	1	2	yes	complemented
39	C₅₃×#346	C₂²	1	1	yes	characteristic, complemented
40	C₂×D₄×D₅₃	—	2	2	no	—
41	C₂²×C₄×D₅₃	—	4	2	no	—
42	C₂×D₄₂₄	C₂³	1	2	yes	—
43	C₂×#11719922	—	2	2	no	—
44	D₈×D₅₃	—	2	4	no	—
45	D₈×D₅₃	C₂×C₄	1	8	yes	complemented
46	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
47	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
48	C₄×D₂₁₂	—	2	2	no	—
49	D₄×Dic₅₃	—	2	2	no	—
50	#13133179	—	2	2	no	—
51	C₄×C₂²⋊D₅₃	—	4	2	no	—
52	#13133177	—	4	2	no	—
53	C₄⋊C₄×D₅₃	—	2	2	no	—
54	C₂²⋊C₄×D₅₃	—	4	2	no	—
55	C₄²×D₅₃	D₄	1	1	yes	characteristic
56	D₈₄₈	C₂×C₄	1	4	yes	complemented
57	#13133158	C₂×C₄	1	8	yes	complemented
58	C₂×#11719908	C₂³	1	1	yes	characteristic
59	C₂×#11719908	C₂³	1	1	yes	characteristic
60	C₁₆×D₅₃	—	2	2	no	—
61	C₁₆×D₅₃	C₂×C₄	1	4	yes	complemented
62	#13133151	C₂³	1	2	yes	—
63	#13133150	—	2	2	no	—
64	#13133139	—	2	2	no	—
65	#13133138	—	2	2	no	—
66	#13133137	—	2	2	no	—
67	C₄×#88322	D₄	1	1	yes	characteristic
68	C₈×Dic₅₃	D₄	1	1	yes	characteristic
69	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
70	C₂×D₈×C₅₃	C₂³	1	2	yes	—
71	C₄×D₄×C₅₃	—	2	2	no	—
72	D₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
73	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
75	#107×C₅₃	C₂³	1	2	yes	—
76	#102×C₅₃	—	2	2	no	—
77	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₂³×D₅₃	—	4	2	no	—
79	C₂×D₄×C₅₃	—	2	2	no	—
80	C₂²×C₄×C₅₃	—	4	2	no	—
81	C₂×C₂²⋊D₅₃	—	4	2	no	—
82	C₂²×Dic₅₃	—	4	2	no	—
83	D₄×D₅₃	—	2	8	no	—
84	D₄×D₅₃	—	4	4	no	—
85	C₂×D₂₁₂	—	2	2	no	—
86	C₂×C₄×D₅₃	—	4	2	no	—
87	C₂×C₄×D₅₃	—	4	2	no	—
88	C₂×C₄×D₅₃	—	4	2	no	—
89	C₂×C₄×D₅₃	—	8	8	no	—
90	C₂×C₄×D₅₃	D₈	1	2	yes	—
91	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
92	D₈×C₅₃	—	2	2	no	—
93	D₈×C₅₃	C₂²×C₄	1	4	yes	complemented
94	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₅₃	—	2	2	no	—
97	C₂²⋊C₄×C₅₃	—	4	2	no	—
98	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
99	C₂²⋊Dic₅₃	—	4	2	no	—
100	#11719922	—	2	8	no	—
101	#11719921	—	4	2	no	—
102	#11719921	—	4	2	no	—
103	C₄⋊Dic₅₃	—	2	2	no	—
104	#11719919	—	4	2	no	—
105	C₄×Dic₅₃	D₈	1	2	yes	—
106	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
107	C₂×#88322	C₂×D₄	1	1	yes	characteristic
108	C₂×#88322	C₂×D₄	1	1	yes	characteristic
109	D₄₂₄	—	2	2	no	—
110	D₄₂₄	C₂²×C₄	1	4	yes	complemented
111	C₈×D₅₃	—	2	2	no	—
112	C₈×D₅₃	C₂²×C₄	1	2	yes	—
113	C₈×D₅₃	#41	1	2	yes	—
114	C₁₆×C₅₃	—	2	1	no	—
115	C₁₆×C₅₃	C₂²×C₄	1	2	yes	complemented
116	#11719908	—	2	1	no	—
117	#11719908	C₂²×C₄	1	2	yes	complemented
118	C₂³×C₅₃	—	4	2	no	—
119	C₂²×D₅₃	—	4	2	no	—
120	C₂²×D₅₃	—	4	8	no	—
121	C₂²×D₅₃	—	8	4	no	—
122	C₂²×D₅₃	—	8	4	no	—
123	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
124	D₄×C₅₃	—	2	4	no	—
125	D₄×C₅₃	—	4	2	no	—
126	C₂×C₄×C₅₃	—	4	2	no	—
127	C₂×C₄×C₅₃	—	8	4	no	—
128	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
130	C₂²⋊D₅₃	—	4	8	no	—
131	C₂×Dic₅₃	—	4	2	no	—
132	C₂×Dic₅₃	—	8	4	no	—
133	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
134	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
135	D₂₁₂	—	2	4	no	—
136	D₂₁₂	—	4	2	no	—
137	C₄×D₅₃	—	2	4	no	—
138	C₄×D₅₃	—	8	4	no	—
139	C₄×D₅₃	—	8	4	no	—
140	C₄×D₅₃	—	8	4	no	—
141	C₄×D₅₃	C₄×D₄	1	2	yes	—
142	C₄×D₅₃	#135	1	2	yes	—
143	C₈×C₅₃	—	2	1	no	—
144	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
145	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
146	#88322	—	2	1	no	—
147	#88322	C₄×D₄	1	2	yes	—
148	C₂×#2177	—	53	1	no	maximal, Sylow
149	C₂²×C₅₃	—	4	4	no	—
150	C₂²×C₅₃	—	8	2	no	—
151	C₂²×C₅₃	—	8	2	no	—
152	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
153	C₂×D₅₃	—	4	4	no	—
154	C₂×D₅₃	—	8	2	no	—
155	C₂×D₅₃	—	8	2	no	—
156	C₂×D₅₃	—	8	16	no	—
157	C₂×D₅₃	C₄×D₈	1	2	yes	—
158	C₂×D₅₃	#509	1	4	yes	—
159	C₄×C₅₃	—	2	2	no	—
160	C₄×C₅₃	—	8	2	no	—
161	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
162	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
163	Dic₅₃	—	2	2	no	—
164	Dic₅₃	—	8	2	no	—
165	Dic₅₃	C₄×D₈	1	2	yes	—
166	C₂²×D₁₆	—	53	1	no	—
167	C₂×C₄×D₈	—	53	2	no	—
168	#2177	—	53	1	no	—
169	#2177	—	53	2	no	—
170	#2177	—	53	1	no	—
171	#2177	—	53	2	no	—
172	#2177	—	53	1	no	—
173	#2177	D₅₃	1	1	yes	characteristic, direct product factor
174	C₂×#367	—	53	1	no	—
175	C₂×#357	—	53	2	no	—
176	C₂×#346	—	53	1	no	—
177	C₂×C₅₃	—	8	4	no	—
178	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
179	C₂×C₅₃	C₂×#509	1	2	yes	—
180	D₅₃	—	8	4	no	—
181	D₅₃	#2177	1	4	yes	direct product factor
182	C₂²×D₈	—	53	2	no	—
183	C₂×C₄×D₄	—	106	2	no	—
184	C₂×D₁₆	—	53	2	no	—
185	C₂×D₁₆	—	53	8	no	—
186	C₂×D₁₆	—	53	1	no	—
187	C₂×D₁₆	C₂×D₅₃	1	1	yes	characteristic
188	C₂²×C₁₆	—	53	1	no	—
189	C₂²×C₁₆	—	53	1	no	—
190	C₄×D₈	—	53	2	no	—
191	C₄×D₈	—	53	2	no	—
192	C₄×D₈	—	53	2	no	—
193	C₄×D₈	—	106	2	no	—
194	C₄×D₈	—	106	2	no	—
195	C₄×D₈	C₂×D₅₃	1	2	yes	complemented
196	C₂×#107	—	53	2	no	—
197	C₂×#102	—	106	2	no	—
198	C₂×C₄×C₈	—	53	1	no	—
199	#367	—	53	1	no	—
200	#367	—	53	2	no	—
201	#367	C₂×D₅₃	1	1	yes	characteristic, complemented
202	#357	—	53	2	no	—
203	#357	—	53	2	no	—
204	#357	—	53	2	no	—
205	#357	C₂×D₅₃	1	2	yes	complemented
206	#346	—	53	1	no	—
207	#346	—	53	1	no	—
208	#346	—	53	1	no	—
209	#346	C₂×D₅₃	1	1	yes	characteristic, complemented
210	C₅₃	C₂×#2177	1	1	yes	characteristic, complemented, Sylow
211	C₂²×D₄	—	106	2	no	—
212	C₂³×C₄	—	212	2	no	—
213	C₂×D₈	—	53	8	no	—
214	C₂×D₈	—	53	2	no	—
215	C₂×D₈	—	106	4	no	—
216	C₂×D₈	—	106	2	no	—
217	C₂×D₈	C₂²×D₅₃	1	2	yes	—
218	C₂²×C₈	—	53	1	no	—
219	C₂²×C₈	—	53	1	no	—
220	C₄×D₄	—	2	2	no	—
221	C₄×D₄	—	106	2	no	—
222	C₄×D₄	—	106	2	no	—
223	C₄×D₄	—	106	2	no	—
224	C₄×D₄	—	212	2	no	—
225	C₄×D₄	—	212	2	no	—
226	C₂×C₄⋊C₄	—	106	2	no	—
227	C₂×C₂²⋊C₄	—	212	2	no	—
228	C₂×C₄²	—	53	1	no	—
229	D₁₆	—	53	8	no	—
230	D₁₆	—	53	4	no	—
231	D₁₆	C₄×D₅₃	1	4	yes	complemented
232	C₂×C₁₆	—	53	1	no	—
233	C₂×C₁₆	—	53	4	no	—
234	C₂×C₁₆	—	53	1	no	—
235	C₂×C₁₆	—	106	2	no	—
236	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
237	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
238	#107	—	53	2	no	—
239	#107	—	106	2	no	—
240	#107	C₂²×D₅₃	1	2	yes	—
241	#102	—	2	2	no	—
242	#102	—	106	2	no	—
243	#102	—	106	2	no	—
244	#102	—	106	2	no	—
245	C₄×C₈	—	53	1	no	—
246	C₄×C₈	—	53	1	no	—
247	C₄×C₈	—	53	1	no	—
248	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
249	C₂⁴	—	212	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	106	8	no	—
252	C₂×D₄	—	106	2	no	—
253	C₂×D₄	—	212	4	no	—
254	C₂×D₄	—	212	2	no	—
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	—	53	1	no	—
257	C₂²×C₄	—	53	2	no	—
258	C₂²×C₄	—	424	8	no	—
259	C₂²×C₄	—	212	2	no	—
260	C₂²×C₄	—	212	2	no	—
261	C₂²×C₄	—	212	2	no	—
262	C₂²×C₄	—	212	2	no	—
263	D₈	—	2	2	no	—
264	D₈	—	53	4	no	—
265	D₈	—	106	8	no	—
266	D₈	—	106	2	no	—
267	D₈	C₂×C₄×D₅₃	1	4	yes	complemented
268	C₂×C₈	—	53	1	no	—
269	C₂×C₈	—	53	2	no	—
270	C₂×C₈	—	53	2	no	—
271	C₂×C₈	—	53	1	no	—
272	C₂×C₈	—	106	2	no	—
273	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
274	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	106	2	no	—
277	C₄⋊C₄	—	212	2	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₂²⋊C₄	—	212	2	no	—
280	C₂²⋊C₄	—	212	2	no	—
281	C₂²⋊C₄	—	212	2	no	—
282	C₄²	—	53	2	no	—
283	C₄²	—	53	1	no	—
284	C₄²	D₄×D₅₃	1	1	yes	characteristic
285	C₁₆	—	2	1	no	—
286	C₁₆	—	53	2	no	—
287	C₁₆	—	106	1	no	—
288	C₁₆	C₂×C₄×D₅₃	1	2	yes	complemented
289	C₂³	—	4	2	no	—
290	C₂³	—	53	1	no	—
291	C₂³	—	424	4	no	—
292	C₂³	—	424	4	no	—
293	C₂³	—	212	8	no	—
294	C₂³	—	212	2	no	—
295	D₄	—	2	4	no	—
296	D₄	—	4	2	no	—
297	D₄	—	106	4	no	—
298	D₄	—	212	8	no	—
299	D₄	—	212	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	8	4	no	—
302	C₂×C₄	—	53	1	no	—
303	C₂×C₄	—	53	2	no	—
304	C₂×C₄	—	53	2	no	—
305	C₂×C₄	—	53	2	no	—
306	C₂×C₄	—	106	4	no	—
307	C₂×C₄	—	424	4	no	—
308	C₂×C₄	—	424	4	no	—
309	C₂×C₄	—	424	4	no	—
310	C₂×C₄	—	424	4	no	—
311	C₂×C₄	—	212	2	no	—
312	C₂×C₄	D₈×D₅₃	1	2	yes	—
313	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	53	2	no	—
316	C₈	—	106	1	no	—
317	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
318	C₈	#41×D₅₃	1	1	yes	characteristic
319	C₂²	—	4	4	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	53	2	no	—
323	C₂²	—	53	4	no	—
324	C₂²	—	424	16	no	—
325	C₂²	—	424	2	no	—
326	C₂²	—	424	2	no	—
327	C₂²	—	212	4	no	—
328	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
329	C₄	—	2	2	no	—
330	C₄	—	8	2	no	—
331	C₄	—	53	2	no	—
332	C₄	—	106	2	no	—
333	C₄	—	424	2	no	—
334	C₄	C₄×D₄×D₅₃	1	1	yes	characteristic
335	C₄	#135×D₅₃	1	1	yes	characteristic
336	C₂	—	8	4	no	—
337	C₂	—	53	4	no	—
338	C₂	—	424	4	no	—
339	C₂	C₄×D₈×D₅₃	1	1	yes	characteristic
340	C₂	#509×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2177	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×D₁₆×D₅₃	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2177	D₅₃
hypocenter	C₅₃	C₂×#2177
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2177

Complements

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

N	H ≅ G/N	type
2. C₄×D₈×D₅₃	C₂	semidirect
3. #20276592	C₂	semidirect
4. #20276591	C₂	semidirect
5. #20276590	C₂	semidirect
6. #20276589	C₂	semidirect
7. #20276588	C₂	semidirect
8. #367×D₅₃	C₂	semidirect
9. #357×D₅₃	C₂	semidirect
10. #346×D₅₃	C₂	semidirect
11. C₅₃×#2177	C₂	semidirect
16. D₁₆×D₅₃	C₄	semidirect
19. C₄×D₄₂₄	C₂²	semidirect
20. D₈×Dic₅₃	C₂²	semidirect
21. #16810705	C₂²	semidirect
27. #16810451	C₂²	semidirect
28. #16810450	C₂²	semidirect
29. #16810425	C₂²	semidirect
30. #16810424	C₂²	semidirect
31. #16810423	C₂²	semidirect
32. #16810399	C₂²	semidirect
33. #16810398	C₂²	semidirect
34. #16810398	C₂²	semidirect
36. C₄×D₈×C₅₃	C₂²	semidirect
37. C₅₃×#367	C₂²	semidirect
38. C₅₃×#357	C₂²	semidirect
39. C₅₃×#346	C₂²	semidirect
45. D₈×D₅₃	C₂×C₄	semidirect
56. D₈₄₈	C₂×C₄	semidirect
57. #13133158	C₂×C₄	semidirect
61. C₁₆×D₅₃	C₂×C₄	semidirect
72. D₁₆×C₅₃	C₂×C₄	semidirect
93. D₈×C₅₃	C₂²×C₄	semidirect
110. D₄₂₄	C₂²×C₄	semidirect
115. C₁₆×C₅₃	C₂²×C₄	semidirect
117. #11719908	C₂²×C₄	semidirect
173. #2177	D₅₃	direct (N × H)
181. D₅₃	#2177	direct (N × H)
195. C₄×D₈	C₂×D₅₃	semidirect
201. #367	C₂×D₅₃	semidirect
205. #357	C₂×D₅₃	semidirect
209. #346	C₂×D₅₃	semidirect
210. C₅₃	C₂×#2177	semidirect
231. D₁₆	C₄×D₅₃	semidirect
267. D₈	C₂×C₄×D₅₃	semidirect
288. C₁₆	C₂×C₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₆×D₅₃	2
C₄×D₈×D₅₃	2
#20276592	2
#20276591	2
#20276590	2
#20276589	2
#20276588	2
#367×D₅₃	2
#357×D₅₃	2
#346×D₅₃	2
C₅₃×#2177	2
C₂×#2177	53

Subgroups — D53×#2177 #27537391

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#2177 #27537391