Subgroups — D₅₃×#1848 #27384545

Order 13568.

1046 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#108×D₅₃	C₂	1	4	yes	maximal
2	#423×D₅₃	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20274887	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274886	C₂	1	2	yes	maximal, complemented
6	#20274885	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₅₃	C₂²	1	4	yes	—
9	C₂²×C₈×D₅₃	C₂²	1	2	yes	—
10	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
11	C₂×#13133153	C₂²	1	4	yes	—
12	C₂×#13133152	C₂²	1	8	yes	—
13	#108×D₅₃	—	2	16	no	—
14	#16810647	C₂²	1	2	yes	—
15	#16810646	C₂²	1	4	yes	—
16	#16810645	—	2	4	no	—
17	C₄⋊C₈×D₅₃	—	2	4	no	—
18	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
20	C₄×C₈×D₅₃	C₄	1	2	yes	—
21	C₄×C₈×D₅₃	C₄	1	2	yes	—
22	C₄×C₈×D₅₃	C₄	1	2	yes	—
23	C₄×C₈×D₅₃	C₄	1	2	yes	—
24	C₂×#108×C₅₃	C₂²	1	4	yes	—
25	C₅₃×#423	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₂²×C₄×D₅₃	Q₈	1	2	yes	—
28	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
29	C₂×#11719917	C₂³	1	4	yes	—
30	C₂×#11719912	—	2	4	no	—
31	M₁₆×D₅₃	—	2	16	no	—
32	C₂²×#88322	C₂³	1	2	yes	—
33	C₂×C₈×D₅₃	—	2	4	no	—
34	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
35	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
36	C₂×C₈×D₅₃	D₄	1	8	yes	—
37	C₂×C₈×D₅₃	Q₈	1	8	yes	—
38	C₂×C₄×Dic₅₃	Q₈	1	2	yes	—
39	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
40	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
41	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
44	C₄²×D₅₃	D₄	1	2	yes	—
45	C₄²×D₅₃	D₄	1	2	yes	—
46	#13133153	—	2	8	no	—
47	#13133152	—	2	16	no	—
48	C₄⋊#88322	—	2	4	no	—
49	#13133146	—	2	8	no	—
50	C₄×#88322	C₂×C₄	1	2	yes	—
51	C₄×#88322	C₂×C₄	1	2	yes	—
52	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
53	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
54	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
55	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
56	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
57	#108×C₅₃	—	2	8	no	—
58	C₄⋊C₈×C₅₃	—	2	4	no	—
59	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
61	C₂³×D₅₃	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
64	C₂²×Dic₅₃	C₂×Q₈	1	2	yes	—
65	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₄×D₅₃	—	2	8	no	—
67	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
68	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
69	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
70	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
71	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×D₅₃	C₂×D₄	1	4	yes	—
74	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×D₅₃	C₂×Q₈	1	4	yes	—
77	M₁₆×C₅₃	—	2	8	no	—
78	C₂×C₈×C₅₃	—	2	4	no	—
79	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
80	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
81	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
82	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₅₃	C₂×D₄	1	2	yes	—
85	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
86	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
87	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
88	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
89	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
92	#11719917	—	2	8	no	—
93	C₂×#88322	—	2	4	no	—
94	C₂×#88322	C₂²×C₄	1	4	yes	—
95	C₂×#88322	C₂×D₄	1	4	yes	—
96	C₂×#88322	C₂×Q₈	1	4	yes	—
97	#11719912	—	4	8	no	—
98	C₈×D₅₃	—	4	8	no	—
99	C₈×D₅₃	C₄⋊C₄	1	16	yes	—
100	C₈×D₅₃	C₄⋊C₄	1	16	yes	—
101	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
102	C₂²×D₅₃	—	2	4	no	—
103	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂²×D₅₃	C₄⋊Q₈	1	4	yes	—
106	C₂×C₄×C₅₃	—	2	4	no	—
107	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
111	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
113	C₂×Dic₅₃	—	2	4	no	—
114	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×Dic₅₃	C₄⋊Q₈	1	4	yes	—
119	C₄×D₅₃	—	2	4	no	—
120	C₄×D₅₃	—	2	4	no	—
121	C₄×D₅₃	—	2	4	no	—
122	C₄×D₅₃	#108	1	8	yes	complemented
123	C₄×D₅₃	C₂×C₄⋊C₄	1	4	yes	—
124	C₄×D₅₃	C₂×C₄⋊C₄	1	4	yes	—
125	C₄×D₅₃	#127	1	2	yes	—
126	C₄×D₅₃	#127	1	2	yes	—
127	C₄×D₅₃	C₄⋊Q₈	1	2	yes	—
128	C₄×D₅₃	C₄⋊Q₈	1	2	yes	—
129	C₈×C₅₃	—	4	4	no	—
130	C₈×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
131	#88322	—	4	4	no	—
132	#88322	C₂×C₄⋊C₄	1	8	yes	—
133	C₂×#1848	—	53	1	no	maximal, Sylow
134	C₂²×C₅₃	—	2	2	no	—
135	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
136	C₂²×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
137	C₂×D₅₃	—	2	2	no	—
138	C₂×D₅₃	—	2	8	no	—
139	C₂×D₅₃	#384	1	2	yes	—
140	C₂×D₅₃	#384	1	2	yes	—
141	C₂×D₅₃	C₂×#108	1	4	yes	—
142	C₄×C₅₃	—	2	2	no	—
143	C₄×C₅₃	C₂×#108	1	4	yes	complemented
144	C₄×C₅₃	C₂²×C₄⋊C₄	1	2	yes	—
145	C₄×C₅₃	C₂×#127	1	1	yes	characteristic
146	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
147	Dic₅₃	—	2	2	no	—
148	Dic₅₃	#384	1	2	yes	—
149	Dic₅₃	#384	1	2	yes	—
150	Dic₅₃	C₂×#108	1	4	yes	complemented
151	C₂²×#108	—	53	4	no	—
152	C₂×#423	—	53	2	no	—
153	C₂²×C₄×C₈	—	53	1	no	—
154	#1848	—	53	4	no	—
155	#1848	—	53	2	no	—
156	#1848	—	53	1	no	—
157	#1848	D₅₃	1	1	yes	characteristic, direct product factor
158	C₂×C₅₃	—	2	2	no	—
159	C₂×C₅₃	C₂×#384	1	1	yes	characteristic
160	C₂×C₅₃	C₂²×#108	1	2	yes	—
161	D₅₃	—	2	2	no	—
162	D₅₃	#1848	1	4	yes	direct product factor
163	C₂²×M₁₆	—	53	4	no	—
164	C₂³×C₈	—	53	2	no	—
165	C₂²×C₄²	—	53	1	no	—
166	C₂×#108	—	53	8	no	—
167	C₂×#108	—	53	4	no	—
168	C₂×#108	—	106	16	no	—
169	C₂×#108	C₂×D₅₃	1	4	yes	—
170	#423	—	53	4	no	—
171	#423	—	53	2	no	—
172	#423	—	106	4	no	—
173	#423	C₂×D₅₃	1	2	yes	—
174	C₂×C₄⋊C₈	—	106	4	no	—
175	C₂×C₄×C₈	—	53	2	no	—
176	C₂×C₄×C₈	—	53	1	no	—
177	C₂×C₄×C₈	—	53	2	no	—
178	C₂×C₄×C₈	—	53	2	no	—
179	C₂×C₄×C₈	—	53	2	no	—
180	C₂×C₄×C₈	—	53	2	no	—
181	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
182	C₅₃	C₂×#1848	1	1	yes	characteristic, complemented, Sylow
183	C₂³×C₄	—	53	1	no	—
184	C₂³×C₄	—	53	2	no	—
185	C₂×M₁₆	—	53	4	no	—
186	C₂×M₁₆	—	106	4	no	—
187	C₂×M₁₆	—	106	16	no	—
188	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
189	C₂²×C₈	—	53	2	no	—
190	C₂²×C₈	—	53	4	no	—
191	C₂²×C₈	—	53	4	no	—
192	C₂²×C₈	—	53	8	no	—
193	C₂²×C₈	—	53	8	no	—
194	C₂²×C₈	—	106	4	no	—
195	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
196	C₂×C₄²	—	53	2	no	—
197	C₂×C₄²	—	53	1	no	—
198	C₂×C₄²	—	53	1	no	—
199	C₂×C₄²	—	53	1	no	—
200	C₂×C₄²	—	53	1	no	—
201	C₂×C₄²	—	53	1	no	—
202	C₂×C₄²	—	53	2	no	—
203	C₂×C₄²	—	53	2	no	—
204	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
205	#108	—	2	8	no	—
206	#108	—	106	16	no	—
207	#108	—	106	8	no	—
208	C₄⋊C₈	—	2	4	no	—
209	C₄⋊C₈	—	106	8	no	—
210	C₄⋊C₈	—	106	4	no	—
211	C₄×C₈	—	53	4	no	—
212	C₄×C₈	—	53	4	no	—
213	C₄×C₈	—	53	2	no	—
214	C₄×C₈	—	53	2	no	—
215	C₄×C₈	C₄×D₅₃	1	2	yes	—
216	C₄×C₈	C₄×D₅₃	1	2	yes	—
217	C₂⁴	—	53	1	no	—
218	C₂²×C₄	—	53	1	no	—
219	C₂²×C₄	—	53	2	no	—
220	C₂²×C₄	—	53	1	no	—
221	C₂²×C₄	—	53	1	no	—
222	C₂²×C₄	—	53	4	no	—
223	C₂²×C₄	—	53	1	no	—
224	C₂²×C₄	—	53	1	no	—
225	C₂²×C₄	—	53	4	no	—
226	C₂²×C₄	—	53	2	no	—
227	C₂²×C₄	—	53	2	no	—
228	C₂²×C₄	—	53	2	no	—
229	C₂²×C₄	—	53	2	no	—
230	C₂²×C₄	—	106	8	no	—
231	C₂²×C₄	Q₈×D₅₃	1	2	yes	—
232	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
233	M₁₆	—	2	8	no	—
234	M₁₆	—	106	8	no	—
235	M₁₆	—	212	8	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	53	4	no	—
238	C₂×C₈	—	53	4	no	—
239	C₂×C₈	—	53	4	no	—
240	C₂×C₈	—	53	16	no	—
241	C₂×C₈	—	53	16	no	—
242	C₂×C₈	—	106	4	no	—
243	C₂×C₈	—	212	8	no	—
244	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
245	C₂×C₈	D₄×D₅₃	1	4	yes	—
246	C₂×C₈	Q₈×D₅₃	1	4	yes	—
247	C₄²	—	53	2	no	—
248	C₄²	—	53	2	no	—
249	C₄²	—	53	2	no	—
250	C₄²	—	53	2	no	—
251	C₄²	—	53	1	no	—
252	C₄²	—	53	1	no	—
253	C₄²	—	53	2	no	—
254	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
255	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
256	C₄²	D₄×D₅₃	1	2	yes	—
257	C₂³	—	53	1	no	—
258	C₂³	—	53	1	no	—
259	C₂³	—	53	4	no	—
260	C₂³	—	106	4	no	—
261	C₂³	C₂×Q₈×D₅₃	1	1	yes	characteristic
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	53	1	no	—
264	C₂×C₄	—	53	1	no	—
265	C₂×C₄	—	53	2	no	—
266	C₂×C₄	—	53	4	no	—
267	C₂×C₄	—	53	4	no	—
268	C₂×C₄	—	53	4	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	8	no	—
274	C₂×C₄	—	53	2	no	—
275	C₂×C₄	—	106	4	no	—
276	C₂×C₄	—	106	4	no	—
277	C₂×C₄	—	106	4	no	—
278	C₂×C₄	—	106	4	no	—
279	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
280	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
281	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
282	C₂×C₄	C₂×D₄×D₅₃	1	2	yes	—
283	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
284	C₂×C₄	C₂×Q₈×D₅₃	1	2	yes	—
285	C₈	—	4	4	no	—
286	C₈	—	53	8	no	—
287	C₈	—	212	4	no	—
288	C₈	C₄⋊C₄×D₅₃	1	8	yes	—
289	C₂²	—	2	2	no	—
290	C₂²	—	53	2	no	—
291	C₂²	—	53	4	no	—
292	C₂²	—	53	2	no	—
293	C₂²	—	106	8	no	—
294	C₂²	—	106	2	no	—
295	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
296	C₂²	C₄⋊Q₈×D₅₃	1	2	yes	—
297	C₄	—	2	2	no	—
298	C₄	—	53	2	no	—
299	C₄	—	53	2	no	—
300	C₄	—	53	4	no	—
301	C₄	—	106	2	no	—
302	C₄	#108×D₅₃	1	4	yes	complemented
303	C₄	C₂×C₄⋊C₄×D₅₃	1	2	yes	—
304	C₄	#127×D₅₃	1	1	yes	characteristic
305	C₄	C₄⋊Q₈×D₅₃	1	1	yes	characteristic
306	C₂	—	2	2	no	—
307	C₂	—	53	4	no	—
308	C₂	—	106	2	no	—
309	C₂	#384×D₅₃	1	1	yes	characteristic
310	C₂	C₂×#108×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1848	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1848	D₅₃
hypocenter	C₅₃	C₂×#1848
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
4. #20274887	C₂	semidirect
5. #20274886	C₂	semidirect
6. #20274885	C₂	semidirect
7. C₅₃×#1848	C₂	semidirect
122. C₄×D₅₃	#108	semidirect
143. C₄×C₅₃	C₂×#108	semidirect
150. Dic₅₃	C₂×#108	semidirect
157. #1848	D₅₃	direct (N × H)
162. D₅₃	#1848	direct (N × H)
182. C₅₃	C₂×#1848	semidirect
302. C₄	#108×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#108×D₅₃	2
#423×D₅₃	2
C₂×C₄×C₈×D₅₃	2
#20274887	2
#20274886	2
#20274885	2
C₅₃×#1848	2
C₂×#1848	53

Subgroups — D53×#1848 #27384545

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#1848 #27384545