Subgroups — D₅₃×#1376 #27298943

Order 13568.

670 conjugacy classes of subgroups.

The subgroup lattice has 310 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	#20272797	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272796	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1376	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×C₄×#88322	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
17	C₄×C₈×D₅₃	—	2	4	no	—
18	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄×D₅₃	—	2	1	no	—
21	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
23	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×D₅₃	—	2	2	no	—
25	C₂×C₈×D₅₃	—	2	2	no	—
26	C₂×C₈×D₅₃	C₈	1	2	yes	—
27	C₂×C₈×D₅₃	C₈	1	2	yes	—
28	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
29	C₂×C₈×D₅₃	D₄	1	4	yes	—
30	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
31	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
32	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
33	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
34	C₂×C₄×Dic₅₃	—	2	1	no	—
35	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄²×D₅₃	—	2	2	no	—
37	C₄²×D₅₃	C₈	1	4	yes	—
38	#13133157	—	2	2	no	—
39	#13133157	C₂×C₄	1	4	yes	complemented
40	#13133156	—	2	8	no	—
41	C₂×#11719908	—	2	2	no	—
42	C₁₆×D₅₃	—	2	8	no	—
43	C₄×#88322	—	2	2	no	—
44	C₈×Dic₅₃	—	2	2	no	—
45	C₈×Dic₅₃	—	2	2	no	—
46	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
49	M₃₂×C₅₃	—	2	2	no	—
50	M₃₂×C₅₃	C₂×C₄	1	4	yes	complemented
51	C₂×C₁₆×C₅₃	—	2	2	no	—
52	C₄×C₈×C₅₃	—	2	2	no	—
53	C₂³×D₅₃	M₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	—	2	1	no	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×Dic₅₃	—	2	1	no	—
57	C₂²×Dic₅₃	M₁₆	1	1	yes	characteristic
58	C₂×C₄×D₅₃	—	2	8	no	—
59	C₂×C₄×D₅₃	—	2	4	no	—
60	C₂×C₄×D₅₃	C₄²	1	2	yes	—
61	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
62	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
65	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
66	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
67	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
68	C₂×C₈×C₅₃	—	2	1	no	—
69	C₂×C₈×C₅₃	—	2	1	no	—
70	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
71	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
75	C₄²×C₅₃	—	2	1	no	—
76	C₄²×C₅₃	C₂×C₈	1	2	yes	—
77	C₄×Dic₅₃	—	2	1	no	—
78	C₄×Dic₅₃	—	2	4	no	—
79	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
80	C₂×#88322	—	2	1	no	—
81	C₂×#88322	—	2	1	no	—
82	C₂×#88322	C₄²	1	2	yes	—
83	C₂×#88322	C₂²⋊C₄	1	2	yes	—
84	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
85	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
86	C₂×#88322	C₂×C₈	1	2	yes	—
87	C₈×D₅₃	—	2	4	no	—
88	C₈×D₅₃	—	2	4	no	—
89	C₈×D₅₃	C₂²⋊C₄	1	4	yes	—
90	C₈×D₅₃	C₂²⋊C₄	1	4	yes	—
91	C₈×D₅₃	C₄⋊C₄	1	4	yes	—
92	C₈×D₅₃	C₄⋊C₄	1	4	yes	—
93	C₁₆×C₅₃	—	2	4	no	—
94	#11719908	—	2	4	no	—
95	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
96	C₂²×D₅₃	—	2	4	no	—
97	C₂²×D₅₃	#97	1	2	yes	—
98	C₂²×D₅₃	C₂²⋊C₈	1	2	yes	—
99	C₂²×D₅₃	C₄⋊C₈	1	1	yes	characteristic
100	C₂²×D₅₃	C₄⋊C₈	1	1	yes	characteristic
101	C₂×C₄×C₅₃	—	2	4	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
109	C₂×Dic₅₃	—	2	4	no	—
110	C₂×Dic₅₃	—	2	2	no	—
111	C₂×Dic₅₃	#97	1	2	yes	—
112	C₂×Dic₅₃	C₂²⋊C₈	1	2	yes	—
113	C₂×Dic₅₃	C₄⋊C₈	1	1	yes	characteristic
114	C₂×Dic₅₃	C₄⋊C₈	1	1	yes	characteristic
115	C₄×D₅₃	—	2	16	no	—
116	C₄×D₅₃	#95	1	2	yes	—
117	C₄×D₅₃	#95	1	2	yes	—
118	C₄×D₅₃	C₄×C₈	1	2	yes	—
119	C₄×D₅₃	C₄×C₈	1	2	yes	—
120	C₄×D₅₃	C₂²⋊C₈	1	2	yes	—
121	C₄×D₅₃	C₂²⋊C₈	1	2	yes	—
122	C₄×D₅₃	C₄⋊C₈	1	2	yes	—
123	C₄×D₅₃	C₄⋊C₈	1	2	yes	—
124	C₈×C₅₃	—	2	2	no	—
125	C₈×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
126	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
127	#88322	—	2	2	no	—
128	#88322	#95	1	2	yes	—
129	#88322	#95	1	2	yes	—
130	C₂×#1376	—	53	1	no	maximal, Sylow
131	C₂²×C₅₃	—	2	2	no	—
132	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
133	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
134	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
135	C₂×D₅₃	—	2	2	no	—
136	C₂×D₅₃	—	2	4	no	—
137	C₂×D₅₃	—	2	4	no	—
138	C₂×D₅₃	#336	1	2	yes	—
139	C₂×D₅₃	#336	1	2	yes	—
140	C₂×D₅₃	#364	1	4	yes	—
141	C₄×C₅₃	—	2	4	no	—
142	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
143	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
144	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
146	Dic₅₃	—	2	4	no	—
147	Dic₅₃	#336	1	2	yes	—
148	Dic₅₃	#336	1	2	yes	—
149	C₂²×M₃₂	—	53	2	no	—
150	C₂²×C₄×C₈	—	53	1	no	—
151	#1376	—	53	2	no	—
152	#1376	—	53	1	no	—
153	#1376	D₅₃	1	1	yes	characteristic, direct product factor
154	C₂×C₅₃	—	2	1	no	—
155	C₂×C₅₃	—	2	1	no	—
156	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
157	C₂×C₅₃	C₂×#364	1	2	yes	—
158	D₅₃	—	2	2	no	—
159	D₅₃	#1376	1	4	yes	direct product factor
160	C₂³×C₈	—	53	1	no	—
161	C₂³×C₈	—	53	1	no	—
162	C₂²×C₄²	—	53	1	no	—
163	C₂×M₃₂	—	53	2	no	—
164	C₂×M₃₂	—	53	8	no	—
165	C₂×M₃₂	—	106	2	no	—
166	C₂×M₃₂	—	106	4	no	—
167	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
168	C₂²×C₁₆	—	106	2	no	—
169	C₂×C₄×C₈	—	53	1	no	—
170	C₂×C₄×C₈	—	53	1	no	—
171	C₂×C₄×C₈	—	53	1	no	—
172	C₂×C₄×C₈	—	106	4	no	—
173	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
174	C₅₃	C₂×#1376	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₄	—	53	1	no	—
176	C₂³×C₄	—	106	1	no	—
177	C₂²×C₈	—	53	1	no	—
178	C₂²×C₈	—	53	4	no	—
179	C₂²×C₈	—	53	4	no	—
180	C₂²×C₈	—	53	1	no	—
181	C₂²×C₈	—	53	1	no	—
182	C₂²×C₈	—	53	1	no	—
183	C₂²×C₈	—	53	1	no	—
184	C₂²×C₈	—	53	1	no	—
185	C₂²×C₈	—	53	2	no	—
186	C₂²×C₈	—	53	2	no	—
187	C₂²×C₈	—	106	2	no	—
188	C₂²×C₈	—	106	2	no	—
189	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
190	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
191	C₂×C₄²	—	53	1	no	—
192	C₂×C₄²	—	53	4	no	—
193	C₂×C₄²	—	106	2	no	—
194	C₂×C₄²	—	106	1	no	—
195	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
196	M₃₂	—	2	2	no	—
197	M₃₂	—	53	4	no	—
198	M₃₂	—	106	8	no	—
199	M₃₂	—	106	2	no	—
200	M₃₂	C₄×D₅₃	1	4	yes	complemented
201	C₂×C₁₆	—	2	2	no	—
202	C₂×C₁₆	—	106	8	no	—
203	C₂×C₁₆	—	106	2	no	—
204	C₄×C₈	—	2	2	no	—
205	C₄×C₈	—	106	2	no	—
206	C₄×C₈	—	106	2	no	—
207	C₄×C₈	—	106	2	no	—
208	C₂⁴	—	53	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	53	1	no	—
211	C₂²×C₄	—	53	2	no	—
212	C₂²×C₄	—	53	2	no	—
213	C₂²×C₄	—	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	—	106	4	no	—
220	C₂²×C₄	—	106	8	no	—
221	C₂²×C₄	—	106	1	no	—
222	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	53	2	no	—
226	C₂×C₈	—	53	2	no	—
227	C₂×C₈	—	53	1	no	—
228	C₂×C₈	—	53	1	no	—
229	C₂×C₈	—	53	4	no	—
230	C₂×C₈	—	53	4	no	—
231	C₂×C₈	—	53	4	no	—
232	C₂×C₈	—	53	4	no	—
233	C₂×C₈	—	53	2	no	—
234	C₂×C₈	—	106	4	no	—
235	C₂×C₈	—	106	4	no	—
236	C₂×C₈	—	106	1	no	—
237	C₂×C₈	—	106	1	no	—
238	C₂×C₈	C₈×D₅₃	1	2	yes	—
239	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
240	C₂×C₈	D₄×D₅₃	1	2	yes	—
241	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
242	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
243	C₄²	—	2	1	no	—
244	C₄²	—	53	2	no	—
245	C₄²	—	106	4	no	—
246	C₄²	—	106	1	no	—
247	C₄²	C₈×D₅₃	1	2	yes	—
248	C₁₆	—	2	4	no	—
249	C₁₆	—	106	4	no	—
250	C₂³	—	53	1	no	—
251	C₂³	—	53	2	no	—
252	C₂³	—	53	2	no	—
253	C₂³	—	53	1	no	—
254	C₂³	—	106	4	no	—
255	C₂³	M₁₆×D₅₃	1	1	yes	characteristic
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	53	2	no	—
259	C₂×C₄	—	53	2	no	—
260	C₂×C₄	—	53	1	no	—
261	C₂×C₄	—	53	1	no	—
262	C₂×C₄	—	53	2	no	—
263	C₂×C₄	—	53	2	no	—
264	C₂×C₄	—	53	2	no	—
265	C₂×C₄	—	53	2	no	—
266	C₂×C₄	—	53	2	no	—
267	C₂×C₄	—	53	2	no	—
268	C₂×C₄	—	53	2	no	—
269	C₂×C₄	—	53	2	no	—
270	C₂×C₄	—	106	16	no	—
271	C₂×C₄	—	106	4	no	—
272	C₂×C₄	—	106	2	no	—
273	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
274	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
275	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
276	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
277	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
278	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
279	C₈	—	2	2	no	—
280	C₈	—	53	2	no	—
281	C₈	—	53	2	no	—
282	C₈	—	106	2	no	—
283	C₈	C₂²⋊C₄×D₅₃	1	2	yes	—
284	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
285	C₂²	—	2	2	no	—
286	C₂²	—	53	2	no	—
287	C₂²	—	53	4	no	—
288	C₂²	—	53	2	no	—
289	C₂²	—	106	4	no	—
290	C₂²	—	106	4	no	—
291	C₂²	—	106	2	no	—
292	C₂²	#97×D₅₃	1	1	yes	characteristic
293	C₂²	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
294	C₂²	C₄⋊C₈×D₅₃	1	1	yes	characteristic
295	C₄	—	2	4	no	—
296	C₄	—	53	2	no	—
297	C₄	—	53	2	no	—
298	C₄	—	106	4	no	—
299	C₄	#95×D₅₃	1	1	yes	characteristic
300	C₄	C₄×C₈×D₅₃	1	1	yes	characteristic
301	C₄	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
302	C₄	C₄⋊C₈×D₅₃	1	1	yes	characteristic
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	53	4	no	—
306	C₂	—	106	2	no	—
307	C₂	#336×D₅₃	1	1	yes	characteristic
308	C₂	#364×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1376	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×D₅₃	M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1376	D₅₃
hypocenter	C₅₃	C₂×#1376
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

Complements

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

N	H ≅ G/N	type
3. #20272797	C₂	semidirect
4. #20272796	C₂	semidirect
5. C₅₃×#1376	C₂	semidirect
12. M₃₂×D₅₃	C₄	semidirect
39. #13133157	C₂×C₄	semidirect
50. M₃₂×C₅₃	C₂×C₄	semidirect
153. #1376	D₅₃	direct (N × H)
159. D₅₃	#1376	direct (N × H)
174. C₅₃	C₂×#1376	semidirect
200. 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×C₄×C₈×D₅₃	2
#20272797	2
#20272796	2
C₅₃×#1376	2
C₂×#1376	53

Subgroups — D53×#1376 #27298943

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#1376 #27298943