Subgroups — D₅₃×#2161 #27537145

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 308 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	1	yes	maximal, characteristic
2	C₂×#108×D₅₃	C₂	1	2	yes	maximal
3	#20276510	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276509	C₂	1	2	yes	maximal, complemented
5	#20276508	C₂	1	4	yes	maximal, complemented
6	#368×D₅₃	C₂	1	4	yes	maximal, complemented
7	C₅₃×#2161	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₅₃	—	2	2	no	—
9	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
10	C₂×#13133157	C₂²	1	1	yes	characteristic
11	C₂×#13133156	C₂²	1	2	yes	—
12	M₃₂×D₅₃	C₄	1	4	yes	—
13	M₃₂×D₅₃	C₄	1	4	yes	—
14	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
15	C₂×#13133153	C₂²	1	2	yes	—
16	C₂×#13133152	—	2	2	no	—
17	#108×D₅₃	—	2	2	no	—
18	#108×D₅₃	—	2	2	no	—
19	#108×D₅₃	C₂²	1	4	yes	—
20	#108×D₅₃	C₂²	1	4	yes	—
21	#16810453	C₂²	1	4	yes	complemented
22	#16810452	C₂²	1	8	yes	complemented
23	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#108×C₅₃	C₂²	1	2	yes	—
25	C₅₃×#368	C₂²	1	4	yes	complemented
26	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
27	C₂×#11719917	—	2	2	no	—
28	C₂×#11719912	—	4	2	no	—
29	M₁₆×D₅₃	—	2	4	no	—
30	M₁₆×D₅₃	—	2	4	no	—
31	M₁₆×D₅₃	—	4	2	no	—
32	M₁₆×D₅₃	—	4	2	no	—
33	C₂²×#88322	D₄	1	1	yes	characteristic
34	C₂×C₈×D₅₃	—	4	2	no	—
35	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
38	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
39	C₂×C₈×D₅₃	Q₈	1	2	yes	—
40	C₂×C₈×D₅₃	Q₈	1	2	yes	—
41	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
42	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
43	#13133157	C₂×C₄	1	4	yes	—
44	#13133156	C₂×C₄	1	8	yes	—
45	C₂×#11719908	C₂³	1	2	yes	—
46	C₁₆×D₅₃	C₂×C₄	1	4	yes	—
47	C₁₆×D₅₃	C₂×C₄	1	4	yes	—
48	#13133153	—	2	2	no	—
49	#13133153	C₂³	1	4	yes	—
50	#13133152	—	2	4	no	—
51	#13133152	—	2	4	no	—
52	C₂×M₁₆×C₅₃	—	2	2	no	—
53	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
55	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
56	#108×C₅₃	—	2	2	no	—
57	#108×C₅₃	C₂³	1	4	yes	—
58	C₂³×D₅₃	Q₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×Dic₅₃	Q₁₆	1	1	yes	characteristic
61	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
62	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
63	C₂×C₄×D₅₃	D₈	1	2	yes	—
64	C₂×C₄×D₅₃	D₈	1	2	yes	—
65	C₂×C₄×D₅₃	Q₁₆	1	1	yes	characteristic
66	C₂×C₄×D₅₃	Q₁₆	1	1	yes	characteristic
67	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
69	M₁₆×C₅₃	—	2	4	no	—
70	M₁₆×C₅₃	—	4	2	no	—
71	C₂×C₈×C₅₃	—	4	2	no	—
72	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
75	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
76	#11719917	—	2	4	no	—
77	#11719917	—	4	2	no	—
78	C₂×#88322	—	4	2	no	—
79	C₂×#88322	C₄⋊C₄	1	2	yes	—
80	C₂×#88322	C₄⋊C₄	1	2	yes	—
81	C₂×#88322	C₂×D₄	1	1	yes	characteristic
82	C₂×#88322	C₂×D₄	1	1	yes	characteristic
83	#11719912	—	4	4	no	—
84	#11719912	—	4	4	no	—
85	C₈×D₅₃	—	4	4	no	—
86	C₈×D₅₃	—	4	4	no	—
87	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
88	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
89	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
90	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
91	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₈×D₅₃	C₂²×C₄	1	2	yes	—
94	C₈×D₅₃	C₂×Q₈	1	1	yes	characteristic
95	C₈×D₅₃	C₂×Q₈	1	1	yes	characteristic
96	C₈×D₅₃	C₂×Q₈	1	2	yes	—
97	C₁₆×C₅₃	C₂²×C₄	1	4	yes	—
98	#11719908	C₂²×C₄	1	4	yes	—
99	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
100	C₂²×D₅₃	—	4	2	no	—
101	C₂²×D₅₃	#107	1	2	yes	—
102	C₂²×D₅₃	#107	1	2	yes	—
103	C₂²×D₅₃	C₂×Q₁₆	1	1	yes	characteristic
104	C₂²×D₅₃	C₂×Q₁₆	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
109	C₂×Dic₅₃	#107	1	2	yes	—
110	C₂×Dic₅₃	#107	1	2	yes	—
111	C₂×Dic₅₃	C₂×Q₁₆	1	1	yes	characteristic
112	C₂×Dic₅₃	C₂×Q₁₆	1	1	yes	characteristic
113	C₄×D₅₃	#107	1	2	yes	—
114	C₄×D₅₃	#107	1	2	yes	—
115	C₄×D₅₃	#107	1	2	yes	—
116	C₄×D₅₃	#107	1	2	yes	—
117	C₄×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₄×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
119	C₄×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₄×D₅₃	C₂×D₈	1	1	yes	characteristic
121	C₄×D₅₃	C₂×D₈	1	2	yes	—
122	C₄×D₅₃	C₂×D₈	1	1	yes	characteristic
123	C₈×C₅₃	—	4	4	no	—
124	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
125	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
126	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
127	#88322	—	4	4	no	—
128	#88322	C₂×C₄⋊C₄	1	1	yes	characteristic
129	#88322	C₂×C₄⋊C₄	1	1	yes	characteristic
130	#88322	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×#2161	—	53	1	no	maximal, Sylow
132	C₂²×C₅₃	—	4	1	no	—
133	C₂²×C₅₃	C₂×#107	1	2	yes	—
134	C₂²×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
135	C₂×D₅₃	—	2	4	no	—
136	C₂×D₅₃	—	2	2	no	—
137	C₂×D₅₃	—	4	2	no	—
138	C₂×D₅₃	—	4	1	no	—
139	C₂×D₅₃	C₂×#107	1	2	yes	—
140	C₂×D₅₃	C₂×#107	1	1	yes	characteristic
141	C₂×D₅₃	C₂×#107	1	1	yes	characteristic
142	C₄×C₅₃	C₂×#107	1	2	yes	—
143	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
145	Dic₅₃	C₂×#107	1	1	yes	characteristic
146	Dic₅₃	C₂×#107	1	2	yes	—
147	Dic₅₃	C₂×#107	1	1	yes	characteristic
148	C₂²×M₃₂	—	53	1	no	—
149	C₂²×#108	—	53	2	no	—
150	#2161	—	53	1	no	—
151	#2161	—	53	2	no	—
152	#2161	—	53	4	no	—
153	#2161	D₅₃	1	1	yes	characteristic, direct product factor
154	C₂×#368	—	53	4	no	—
155	C₂×C₅₃	—	2	1	no	—
156	C₂×C₅₃	—	2	2	no	—
157	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
158	D₅₃	—	2	2	no	—
159	D₅₃	—	2	1	no	—
160	D₅₃	#2161	1	2	yes	direct product factor
161	C₂²×M₁₆	—	106	2	no	—
162	C₂³×C₈	—	53	1	no	—
163	C₂×M₃₂	—	53	1	no	—
164	C₂×M₃₂	—	53	4	no	—
165	C₂×M₃₂	—	53	4	no	—
166	C₂×M₃₂	—	53	2	no	—
167	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
168	C₂²×C₁₆	—	53	2	no	—
169	C₂×#108	—	53	2	no	—
170	C₂×#108	—	53	4	no	—
171	C₂×#108	—	53	4	no	—
172	C₂×#108	—	106	2	no	—
173	C₂×#108	—	106	2	no	—
174	C₂×#108	—	106	2	no	—
175	C₂×#108	C₂×D₅₃	1	2	yes	—
176	#368	—	53	4	no	—
177	#368	—	53	8	no	—
178	#368	C₂×D₅₃	1	4	yes	complemented
179	C₅₃	C₂×#2161	1	1	yes	characteristic, complemented, Sylow
180	C₂³×C₄	—	53	1	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	106	4	no	—
183	C₂×M₁₆	—	106	4	no	—
184	C₂×M₁₆	—	106	2	no	—
185	C₂×M₁₆	—	212	2	no	—
186	C₂×M₁₆	—	212	2	no	—
187	C₂×M₁₆	—	212	2	no	—
188	C₂²×C₈	—	53	1	no	—
189	C₂²×C₈	—	53	2	no	—
190	C₂²×C₈	—	53	2	no	—
191	C₂²×C₈	—	53	1	no	—
192	C₂²×C₈	—	53	1	no	—
193	C₂²×C₈	—	53	2	no	—
194	C₂²×C₈	—	53	2	no	—
195	C₂²×C₈	—	53	1	no	—
196	C₂²×C₈	—	53	1	no	—
197	C₂²×C₈	—	212	2	no	—
198	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
199	M₃₂	—	53	4	no	—
200	M₃₂	—	53	8	no	—
201	M₃₂	C₄×D₅₃	1	4	yes	—
202	C₂×C₁₆	—	53	2	no	—
203	C₂×C₁₆	—	53	4	no	—
204	C₂×C₁₆	—	53	4	no	—
205	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
206	#108	—	2	2	no	—
207	#108	—	53	4	no	—
208	#108	—	106	4	no	—
209	#108	—	106	4	no	—
210	#108	—	106	2	no	—
211	#108	C₂²×D₅₃	1	4	yes	—
212	C₂⁴	—	53	1	no	—
213	C₂²×C₄	—	53	1	no	—
214	C₂²×C₄	—	53	2	no	—
215	C₂²×C₄	—	53	2	no	—
216	C₂²×C₄	—	53	2	no	—
217	C₂²×C₄	—	53	2	no	—
218	C₂²×C₄	—	53	1	no	—
219	C₂²×C₄	—	53	1	no	—
220	C₂²×C₄	—	53	1	no	—
221	C₂²×C₄	—	53	1	no	—
222	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
223	M₁₆	—	2	4	no	—
224	M₁₆	—	4	2	no	—
225	M₁₆	—	106	4	no	—
226	M₁₆	—	212	4	no	—
227	M₁₆	—	212	4	no	—
228	M₁₆	—	212	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	53	2	no	—
231	C₂×C₈	—	53	2	no	—
232	C₂×C₈	—	53	1	no	—
233	C₂×C₈	—	53	1	no	—
234	C₂×C₈	—	53	2	no	—
235	C₂×C₈	—	53	2	no	—
236	C₂×C₈	—	53	2	no	—
237	C₂×C₈	—	53	2	no	—
238	C₂×C₈	—	53	1	no	—
239	C₂×C₈	—	53	1	no	—
240	C₂×C₈	—	53	2	no	—
241	C₂×C₈	—	53	1	no	—
242	C₂×C₈	—	53	1	no	—
243	C₂×C₈	—	53	2	no	—
244	C₂×C₈	—	212	4	no	—
245	C₂×C₈	—	212	4	no	—
246	C₂×C₈	—	212	2	no	—
247	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
248	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
249	C₂×C₈	Q₈×D₅₃	1	2	yes	—
250	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
251	C₁₆	—	53	4	no	—
252	C₁₆	C₂×C₄×D₅₃	1	4	yes	—
253	C₂³	—	53	1	no	—
254	C₂³	—	53	2	no	—
255	C₂³	—	53	1	no	—
256	C₂³	—	53	2	no	—
257	C₂³	—	212	2	no	—
258	C₂³	Q₁₆×D₅₃	1	1	yes	characteristic
259	C₂×C₄	—	53	1	no	—
260	C₂×C₄	—	53	1	no	—
261	C₂×C₄	—	53	2	no	—
262	C₂×C₄	—	53	2	no	—
263	C₂×C₄	—	53	1	no	—
264	C₂×C₄	—	53	1	no	—
265	C₂×C₄	—	53	2	no	—
266	C₂×C₄	—	53	1	no	—
267	C₂×C₄	—	53	2	no	—
268	C₂×C₄	—	53	1	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	C₄⋊C₄×D₅₃	1	2	yes	—
274	C₂×C₄	D₈×D₅₃	1	2	yes	—
275	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
276	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
277	C₈	—	4	4	no	—
278	C₈	—	53	1	no	—
279	C₈	—	53	1	no	—
280	C₈	—	53	2	no	—
281	C₈	—	212	4	no	—
282	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
283	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
284	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
285	C₂²	—	4	1	no	—
286	C₂²	—	53	1	no	—
287	C₂²	—	53	2	no	—
288	C₂²	—	53	1	no	—
289	C₂²	—	106	2	no	—
290	C₂²	—	106	4	no	—
291	C₂²	—	212	1	no	—
292	C₂²	—	212	2	no	—
293	C₂²	#107×D₅₃	1	2	yes	—
294	C₂²	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
295	C₄	—	53	1	no	—
296	C₄	—	53	2	no	—
297	C₄	—	53	1	no	—
298	C₄	#107×D₅₃	1	2	yes	—
299	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
300	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	2	no	—
303	C₂	—	53	2	no	—
304	C₂	—	106	2	no	—
305	C₂	—	106	1	no	—
306	C₂	C₂×#107×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2161	C₂
socle	C₂×C₅₃	C₂²×#107
prime radical	C₂³×D₅₃	Q₁₆
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2161	D₅₃
hypocenter	C₅₃	C₂×#2161
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2161

Complements

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

N	H ≅ G/N	type
3. #20276510	C₂	semidirect
4. #20276509	C₂	semidirect
5. #20276508	C₂	semidirect
6. #368×D₅₃	C₂	semidirect
7. C₅₃×#2161	C₂	semidirect
21. #16810453	C₂²	semidirect
22. #16810452	C₂²	semidirect
25. C₅₃×#368	C₂²	semidirect
153. #2161	D₅₃	direct (N × H)
160. D₅₃	#2161	direct (N × H)
178. #368	C₂×D₅₃	semidirect
179. C₅₃	C₂×#2161	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×#108×D₅₃	2
#20276510	2
#20276509	2
#20276508	2
#368×D₅₃	2
C₅₃×#2161	2
C₂×#2161	53

Subgroups — D53×#2161 #27537145

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#2161 #27537145