Subgroups — D₅₃×#1377 #27298949

Order 13568.

630 conjugacy classes of subgroups.

The subgroup lattice has 316 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×#97×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#20272799	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272798	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1377	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₅₃	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₅₃	C₄	1	1	yes	characteristic
9	C₂×#13133157	C₂²	1	2	yes	—
10	C₂×#13133156	—	2	2	no	—
11	M₃₂×D₅₃	—	2	4	no	—
12	M₃₂×D₅₃	C₄	1	8	yes	complemented
13	C₂×C₁₆×D₅₃	—	2	2	no	—
14	C₂×#13133128	C₄	1	1	yes	characteristic
15	C₂×#13133127	C₄	1	1	yes	characteristic
16	C₂×#13133127	C₂²	1	1	yes	characteristic
17	#97×D₅₃	—	2	2	no	—
18	#97×D₅₃	—	2	2	no	—
19	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
20	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄×D₅₃	C₈	1	2	yes	—
22	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
24	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×D₅₃	—	2	1	no	—
26	C₂×C₈×D₅₃	—	2	1	no	—
27	C₂×C₈×D₅₃	—	2	2	no	—
28	C₂×C₈×D₅₃	C₈	1	4	yes	—
29	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
30	C₂×C₈×D₅₃	D₄	1	2	yes	—
31	C₂×C₈×D₅₃	D₄	1	2	yes	—
32	C₂×C₈×D₅₃	D₄	1	2	yes	—
33	C₂×C₈×D₅₃	Q₈	1	2	yes	—
34	C₂×C₄×Dic₅₃	C₈	1	2	yes	—
35	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄²×D₅₃	—	2	1	no	—
37	C₄²×D₅₃	—	2	1	no	—
38	C₄²×D₅₃	—	2	1	no	—
39	C₄²×D₅₃	—	2	1	no	—
40	#13133157	—	2	2	no	—
41	#13133157	C₂×C₄	1	4	yes	complemented
42	#13133156	—	2	8	no	—
43	C₂×#11719908	—	2	2	no	—
44	C₁₆×D₅₃	—	2	8	no	—
45	#13133128	—	2	2	no	—
46	#13133127	—	2	2	no	—
47	#13133127	—	2	2	no	—
48	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
51	M₃₂×C₅₃	—	2	2	no	—
52	M₃₂×C₅₃	C₂×C₄	1	4	yes	complemented
53	C₂×C₁₆×C₅₃	—	2	2	no	—
54	#97×C₅₃	—	2	2	no	—
55	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
59	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
60	C₂×C₄×D₅₃	—	2	2	no	—
61	C₂×C₄×D₅₃	—	2	2	no	—
62	C₂×C₄×D₅₃	—	4	4	no	—
63	C₂×C₄×D₅₃	C₄²	1	2	yes	—
64	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
67	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
68	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
69	C₂×C₄×D₅₃	M₁₆	1	1	yes	characteristic
70	C₂×C₄×D₅₃	M₁₆	1	1	yes	characteristic
71	C₂×C₈×C₅₃	—	2	1	no	—
72	C₂×C₈×C₅₃	—	2	1	no	—
73	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
78	C₄²×C₅₃	—	2	1	no	—
79	C₄²×C₅₃	—	2	1	no	—
80	C₄×Dic₅₃	—	2	1	no	—
81	C₄×Dic₅₃	—	2	1	no	—
82	C₄×Dic₅₃	—	2	2	no	—
83	C₄×Dic₅₃	—	2	2	no	—
84	C₂×#88322	—	2	1	no	—
85	C₂×#88322	—	2	1	no	—
86	C₂×#88322	C₄²	1	2	yes	—
87	C₂×#88322	C₂²⋊C₄	1	2	yes	—
88	C₂×#88322	C₄⋊C₄	1	2	yes	—
89	C₂×#88322	C₂×C₈	1	2	yes	—
90	C₈×D₅₃	—	2	4	no	—
91	C₈×D₅₃	—	2	4	no	—
92	C₈×D₅₃	C₂²⋊C₄	1	4	yes	—
93	C₈×D₅₃	C₂²⋊C₄	1	4	yes	—
94	C₈×D₅₃	C₄⋊C₄	1	4	yes	—
95	C₈×D₅₃	C₄⋊C₄	1	4	yes	—
96	C₁₆×C₅₃	—	2	4	no	—
97	#11719908	—	2	4	no	—
98	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
99	C₂²×D₅₃	—	2	4	no	—
100	C₂²×D₅₃	C₄×C₈	1	2	yes	—
101	C₂²×D₅₃	C₂²⋊C₈	1	1	yes	characteristic
102	C₂²×D₅₃	C₂²⋊C₈	1	1	yes	characteristic
103	C₂²×D₅₃	C₄⋊C₈	1	2	yes	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	4	2	no	—
106	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
112	C₂×Dic₅₃	—	2	2	no	—
113	C₂×Dic₅₃	—	4	2	no	—
114	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
115	C₂×Dic₅₃	C₂²⋊C₈	1	1	yes	characteristic
116	C₂×Dic₅₃	C₂²⋊C₈	1	1	yes	characteristic
117	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
118	C₄×D₅₃	—	4	4	no	—
119	C₄×D₅₃	—	4	4	no	—
120	C₄×D₅₃	#95	1	2	yes	—
121	C₄×D₅₃	#95	1	2	yes	—
122	C₄×D₅₃	#97	1	2	yes	—
123	C₄×D₅₃	#97	1	2	yes	—
124	C₄×D₅₃	C₂²⋊C₈	1	2	yes	—
125	C₄×D₅₃	C₂²⋊C₈	1	2	yes	—
126	C₄×D₅₃	C₄⋊C₈	1	2	yes	—
127	C₄×D₅₃	C₄⋊C₈	1	2	yes	—
128	C₈×C₅₃	—	2	2	no	—
129	C₈×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
130	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
131	#88322	—	2	2	no	—
132	#88322	#95	1	4	yes	—
133	C₂×#1377	—	53	1	no	maximal, Sylow
134	C₂²×C₅₃	—	2	2	no	—
135	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
136	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
137	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
138	C₂×D₅₃	—	2	2	no	—
139	C₂×D₅₃	—	2	4	no	—
140	C₂×D₅₃	—	2	4	no	—
141	C₂×D₅₃	#336	1	2	yes	—
142	C₂×D₅₃	#336	1	2	yes	—
143	C₂×D₅₃	#349	1	4	yes	—
144	C₄×C₅₃	—	4	2	no	—
145	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
146	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
147	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
148	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
149	Dic₅₃	—	4	2	no	—
150	Dic₅₃	#336	1	2	yes	—
151	Dic₅₃	#336	1	2	yes	—
152	C₂²×M₃₂	—	53	2	no	—
153	C₂²×#97	—	53	1	no	—
154	#1377	—	53	2	no	—
155	#1377	—	53	1	no	—
156	#1377	D₅₃	1	1	yes	characteristic, direct product factor
157	C₂×C₅₃	—	2	1	no	—
158	C₂×C₅₃	—	2	1	no	—
159	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
160	C₂×C₅₃	C₂×#349	1	2	yes	—
161	D₅₃	—	2	2	no	—
162	D₅₃	#1377	1	4	yes	direct product factor
163	C₂³×C₈	—	53	1	no	—
164	C₂³×C₈	—	53	1	no	—
165	C₂²×C₄²	—	53	1	no	—
166	C₂×M₃₂	—	53	2	no	—
167	C₂×M₃₂	—	53	8	no	—
168	C₂×M₃₂	—	106	2	no	—
169	C₂×M₃₂	—	106	4	no	—
170	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
171	C₂²×C₁₆	—	106	2	no	—
172	C₂×#97	—	53	1	no	—
173	C₂×#97	—	53	1	no	—
174	C₂×#97	—	53	1	no	—
175	C₂×#97	—	106	2	no	—
176	C₂×#97	—	106	2	no	—
177	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
178	C₅₃	C₂×#1377	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₄	—	53	1	no	—
180	C₂³×C₄	—	53	2	no	—
181	C₂²×C₈	—	53	1	no	—
182	C₂²×C₈	—	53	4	no	—
183	C₂²×C₈	—	53	2	no	—
184	C₂²×C₈	—	53	2	no	—
185	C₂²×C₈	—	53	2	no	—
186	C₂²×C₈	—	53	2	no	—
187	C₂²×C₈	—	53	1	no	—
188	C₂²×C₈	—	53	4	no	—
189	C₂²×C₈	—	106	1	no	—
190	C₂²×C₈	—	106	1	no	—
191	C₂²×C₈	—	106	2	no	—
192	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
193	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
194	C₂×C₄²	—	53	1	no	—
195	C₂×C₄²	—	53	2	no	—
196	C₂×C₄²	—	106	1	no	—
197	C₂×C₄²	—	106	1	no	—
198	C₂×C₄²	—	106	1	no	—
199	C₂×C₄²	—	106	1	no	—
200	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
201	M₃₂	—	2	2	no	—
202	M₃₂	—	53	4	no	—
203	M₃₂	—	106	8	no	—
204	M₃₂	—	106	2	no	—
205	M₃₂	C₄×D₅₃	1	4	yes	complemented
206	C₂×C₁₆	—	2	2	no	—
207	C₂×C₁₆	—	106	8	no	—
208	C₂×C₁₆	—	106	2	no	—
209	#97	—	2	2	no	—
210	#97	—	106	2	no	—
211	#97	—	106	2	no	—
212	#97	—	106	2	no	—
213	C₂⁴	—	53	1	no	—
214	C₂²×C₄	—	53	1	no	—
215	C₂²×C₄	—	53	2	no	—
216	C₂²×C₄	—	53	1	no	—
217	C₂²×C₄	—	53	1	no	—
218	C₂²×C₄	—	53	2	no	—
219	C₂²×C₄	—	53	2	no	—
220	C₂²×C₄	—	53	1	no	—
221	C₂²×C₄	—	53	1	no	—
222	C₂²×C₄	—	53	2	no	—
223	C₂²×C₄	—	53	2	no	—
224	C₂²×C₄	—	106	2	no	—
225	C₂²×C₄	—	106	2	no	—
226	C₂²×C₄	—	212	4	no	—
227	C₂²×C₄	C₈×D₅₃	1	2	yes	—
228	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	53	2	no	—
232	C₂×C₈	—	53	2	no	—
233	C₂×C₈	—	53	2	no	—
234	C₂×C₈	—	53	4	no	—
235	C₂×C₈	—	53	4	no	—
236	C₂×C₈	—	53	4	no	—
237	C₂×C₈	—	53	4	no	—
238	C₂×C₈	—	53	2	no	—
239	C₂×C₈	—	106	4	no	—
240	C₂×C₈	—	106	4	no	—
241	C₂×C₈	—	106	1	no	—
242	C₂×C₈	—	106	1	no	—
243	C₂×C₈	C₈×D₅₃	1	2	yes	—
244	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
245	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
246	C₂×C₈	D₄×D₅₃	1	2	yes	—
247	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
248	C₄²	—	2	1	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	106	2	no	—
251	C₄²	—	106	2	no	—
252	C₄²	—	106	1	no	—
253	C₄²	—	106	1	no	—
254	C₁₆	—	2	4	no	—
255	C₁₆	—	106	4	no	—
256	C₂³	—	53	1	no	—
257	C₂³	—	53	2	no	—
258	C₂³	—	53	1	no	—
259	C₂³	—	53	2	no	—
260	C₂³	—	106	4	no	—
261	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	53	2	no	—
265	C₂×C₄	—	53	1	no	—
266	C₂×C₄	—	53	1	no	—
267	C₂×C₄	—	53	2	no	—
268	C₂×C₄	—	53	2	no	—
269	C₂×C₄	—	53	2	no	—
270	C₂×C₄	—	53	2	no	—
271	C₂×C₄	—	53	2	no	—
272	C₂×C₄	—	53	2	no	—
273	C₂×C₄	—	53	2	no	—
274	C₂×C₄	—	53	2	no	—
275	C₂×C₄	—	53	2	no	—
276	C₂×C₄	—	106	2	no	—
277	C₂×C₄	—	212	4	no	—
278	C₂×C₄	—	212	4	no	—
279	C₂×C₄	—	212	2	no	—
280	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
281	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
282	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
283	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
284	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
285	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	—	53	4	no	—
288	C₈	—	106	2	no	—
289	C₈	C₂²⋊C₄×D₅₃	1	2	yes	—
290	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
291	C₂²	—	2	2	no	—
292	C₂²	—	53	4	no	—
293	C₂²	—	53	2	no	—
294	C₂²	—	53	2	no	—
295	C₂²	—	106	4	no	—
296	C₂²	—	106	4	no	—
297	C₂²	—	106	2	no	—
298	C₂²	C₄×C₈×D₅₃	1	1	yes	characteristic
299	C₂²	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
300	C₂²	C₄⋊C₈×D₅₃	1	1	yes	characteristic
301	C₄	—	4	2	no	—
302	C₄	—	53	2	no	—
303	C₄	—	53	2	no	—
304	C₄	—	212	2	no	—
305	C₄	#95×D₅₃	1	1	yes	characteristic
306	C₄	#97×D₅₃	1	1	yes	characteristic
307	C₄	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
308	C₄	C₄⋊C₈×D₅₃	1	1	yes	characteristic
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	—	53	4	no	—
312	C₂	—	106	2	no	—
313	C₂	#336×D₅₃	1	1	yes	characteristic
314	C₂	#349×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1377	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1377	D₅₃
hypocenter	C₅₃	C₂×#1377
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1377

Complements

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

N	H ≅ G/N	type
3. #20272799	C₂	semidirect
4. #20272798	C₂	semidirect
5. C₅₃×#1377	C₂	semidirect
12. M₃₂×D₅₃	C₄	semidirect
41. #13133157	C₂×C₄	semidirect
52. M₃₂×C₅₃	C₂×C₄	semidirect
156. #1377	D₅₃	direct (N × H)
162. D₅₃	#1377	direct (N × H)
178. C₅₃	C₂×#1377	semidirect
205. M₃₂	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×M₃₂×D₅₃	2
C₂×#97×D₅₃	2
#20272799	2
#20272798	2
C₅₃×#1377	2
C₂×#1377	53

Subgroups — D53×#1377 #27298949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#1377 #27298949