Subgroups — D₅₃×#1385 #27299002

Order 13568.

554 conjugacy classes of subgroups.

The subgroup lattice has 298 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	#20272819	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272818	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1385	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×M₁₆×D₅₃	—	2	2	no	—
7	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
8	C₂×#13133157	C₂²	1	1	yes	characteristic
9	C₂×#13133156	C₄	1	2	yes	—
10	M₃₂×D₅₃	—	2	2	no	—
11	M₃₂×D₅₃	—	2	2	no	—
12	C₂×C₁₆×D₅₃	C₄	1	2	yes	—
13	C₂×#13133153	C₂²	1	2	yes	—
14	C₂×#13133152	—	2	2	no	—
15	#108×D₅₃	—	2	2	no	—
16	#108×D₅₃	—	2	2	no	—
17	#108×D₅₃	C₄	1	4	yes	—
18	#108×D₅₃	C₄	1	4	yes	—
19	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#108×C₅₃	C₂²	1	2	yes	—
21	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
22	C₂×#11719917	—	2	2	no	—
23	C₂×#11719912	—	4	2	no	—
24	M₁₆×D₅₃	—	2	4	no	—
25	M₁₆×D₅₃	—	2	4	no	—
26	M₁₆×D₅₃	—	4	2	no	—
27	M₁₆×D₅₃	—	4	2	no	—
28	C₂²×#88322	D₄	1	1	yes	characteristic
29	C₂×C₈×D₅₃	—	4	2	no	—
30	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₅₃	D₄	1	2	yes	—
35	C₂×C₈×D₅₃	D₄	1	2	yes	—
36	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
37	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
38	#13133157	—	2	2	no	—
39	#13133156	—	2	4	no	—
40	C₂×#11719908	C₂×C₄	1	2	yes	—
41	C₁₆×D₅₃	—	2	2	no	—
42	C₁₆×D₅₃	—	2	2	no	—
43	#13133153	—	2	2	no	—
44	#13133153	C₂×C₄	1	4	yes	—
45	#13133152	—	2	4	no	—
46	#13133152	—	2	4	no	—
47	C₂×M₁₆×C₅₃	—	2	2	no	—
48	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	M₃₂×C₅₃	—	2	2	no	—
50	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
51	#108×C₅₃	—	2	2	no	—
52	#108×C₅₃	C₂×C₄	1	4	yes	—
53	C₂³×D₅₃	Q₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₅₃	Q₁₆	1	1	yes	characteristic
56	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
57	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
61	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
62	C₂×C₄×D₅₃	SD₁₆	1	2	yes	—
63	C₂×C₄×D₅₃	SD₁₆	1	2	yes	—
64	M₁₆×C₅₃	—	2	4	no	—
65	M₁₆×C₅₃	—	4	2	no	—
66	C₂×C₈×C₅₃	—	4	2	no	—
67	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
71	#11719917	—	2	4	no	—
72	#11719917	—	4	2	no	—
73	C₂×#88322	—	4	2	no	—
74	C₂×#88322	C₂²⋊C₄	1	2	yes	—
75	C₂×#88322	C₂²⋊C₄	1	2	yes	—
76	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
77	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
78	#11719912	—	4	4	no	—
79	#11719912	—	4	4	no	—
80	C₈×D₅₃	—	4	4	no	—
81	C₈×D₅₃	—	4	4	no	—
82	C₈×D₅₃	C₄²	1	1	yes	characteristic
83	C₈×D₅₃	C₄²	1	1	yes	characteristic
84	C₈×D₅₃	C₄²	1	2	yes	—
85	C₈×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₈×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₈×D₅₃	C₂²⋊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	C₄⋊C₄	1	2	yes	—
92	C₁₆×C₅₃	—	2	2	no	—
93	#11719908	—	2	2	no	—
94	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
95	C₂²×D₅₃	—	4	2	no	—
96	C₂²×D₅₃	#103	1	2	yes	—
97	C₂²×D₅₃	#103	1	2	yes	—
98	C₂²×D₅₃	#107	1	1	yes	characteristic
99	C₂²×D₅₃	#107	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
101	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
104	C₂×Dic₅₃	#103	1	2	yes	—
105	C₂×Dic₅₃	#103	1	2	yes	—
106	C₂×Dic₅₃	#107	1	1	yes	characteristic
107	C₂×Dic₅₃	#107	1	1	yes	characteristic
108	C₄×D₅₃	#95	1	1	yes	characteristic
109	C₄×D₅₃	#95	1	2	yes	—
110	C₄×D₅₃	#95	1	1	yes	characteristic
111	C₄×D₅₃	#102	1	2	yes	—
112	C₄×D₅₃	#102	1	2	yes	—
113	C₄×D₅₃	#102	1	2	yes	—
114	C₄×D₅₃	#102	1	2	yes	—
115	C₄×D₅₃	#106	1	2	yes	—
116	C₄×D₅₃	#106	1	1	yes	characteristic
117	C₄×D₅₃	#106	1	1	yes	characteristic
118	C₈×C₅₃	—	4	4	no	—
119	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
120	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
122	#88322	—	4	4	no	—
123	#88322	#95	1	1	yes	characteristic
124	#88322	#95	1	1	yes	characteristic
125	#88322	#95	1	2	yes	—
126	C₂×#1385	—	53	1	no	maximal, Sylow
127	C₂²×C₅₃	—	4	1	no	—
128	C₂²×C₅₃	C₂×#103	1	2	yes	—
129	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
130	C₂×D₅₃	—	2	4	no	—
131	C₂×D₅₃	—	2	2	no	—
132	C₂×D₅₃	—	4	2	no	—
133	C₂×D₅₃	—	4	1	no	—
134	C₂×D₅₃	#340	1	2	yes	—
135	C₂×D₅₃	#340	1	1	yes	characteristic
136	C₂×D₅₃	#340	1	1	yes	characteristic
137	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
138	C₄×C₅₃	C₂×#102	1	2	yes	—
139	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
140	Dic₅₃	#340	1	1	yes	characteristic
141	Dic₅₃	#340	1	2	yes	—
142	Dic₅₃	#340	1	1	yes	characteristic
143	C₂²×M₃₂	—	53	1	no	—
144	C₂²×#108	—	53	2	no	—
145	#1385	—	53	1	no	—
146	#1385	—	53	2	no	—
147	#1385	D₅₃	1	1	yes	characteristic, direct product factor
148	C₂×C₅₃	—	2	2	no	—
149	C₂×C₅₃	—	2	1	no	—
150	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
151	D₅₃	—	2	2	no	—
152	D₅₃	—	2	1	no	—
153	D₅₃	#1385	1	2	yes	direct product factor
154	C₂²×M₁₆	—	106	2	no	—
155	C₂³×C₈	—	53	1	no	—
156	C₂×M₃₂	—	53	1	no	—
157	C₂×M₃₂	—	53	2	no	—
158	C₂×M₃₂	—	106	2	no	—
159	C₂×M₃₂	—	106	2	no	—
160	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
161	C₂²×C₁₆	—	53	2	no	—
162	C₂×#108	—	53	2	no	—
163	C₂×#108	—	53	4	no	—
164	C₂×#108	—	53	4	no	—
165	C₂×#108	—	106	2	no	—
166	C₂×#108	—	106	2	no	—
167	C₂×#108	—	106	2	no	—
168	C₂×#108	C₂×D₅₃	1	2	yes	—
169	C₅₃	C₂×#1385	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₄	—	53	1	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	106	4	no	—
173	C₂×M₁₆	—	106	4	no	—
174	C₂×M₁₆	—	106	2	no	—
175	C₂×M₁₆	—	212	2	no	—
176	C₂×M₁₆	—	212	2	no	—
177	C₂×M₁₆	—	212	2	no	—
178	C₂²×C₈	—	53	1	no	—
179	C₂²×C₈	—	53	2	no	—
180	C₂²×C₈	—	53	2	no	—
181	C₂²×C₈	—	53	1	no	—
182	C₂²×C₈	—	53	1	no	—
183	C₂²×C₈	—	53	2	no	—
184	C₂²×C₈	—	53	2	no	—
185	C₂²×C₈	—	53	1	no	—
186	C₂²×C₈	—	53	1	no	—
187	C₂²×C₈	—	212	2	no	—
188	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
189	M₃₂	—	2	2	no	—
190	M₃₂	—	106	4	no	—
191	M₃₂	—	106	2	no	—
192	C₂×C₁₆	—	53	2	no	—
193	C₂×C₁₆	—	106	2	no	—
194	C₂×C₁₆	—	106	2	no	—
195	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
196	#108	—	2	2	no	—
197	#108	—	53	4	no	—
198	#108	—	106	4	no	—
199	#108	—	106	4	no	—
200	#108	—	106	2	no	—
201	#108	C₄×D₅₃	1	4	yes	—
202	C₂⁴	—	53	1	no	—
203	C₂²×C₄	—	53	1	no	—
204	C₂²×C₄	—	53	2	no	—
205	C₂²×C₄	—	53	2	no	—
206	C₂²×C₄	—	53	1	no	—
207	C₂²×C₄	—	53	1	no	—
208	C₂²×C₄	—	53	1	no	—
209	C₂²×C₄	—	53	1	no	—
210	C₂²×C₄	—	53	2	no	—
211	C₂²×C₄	—	53	2	no	—
212	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
213	M₁₆	—	2	4	no	—
214	M₁₆	—	4	2	no	—
215	M₁₆	—	106	4	no	—
216	M₁₆	—	212	4	no	—
217	M₁₆	—	212	4	no	—
218	M₁₆	—	212	2	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	53	2	no	—
221	C₂×C₈	—	53	2	no	—
222	C₂×C₈	—	53	1	no	—
223	C₂×C₈	—	53	1	no	—
224	C₂×C₈	—	53	1	no	—
225	C₂×C₈	—	53	1	no	—
226	C₂×C₈	—	53	2	no	—
227	C₂×C₈	—	53	1	no	—
228	C₂×C₈	—	53	1	no	—
229	C₂×C₈	—	53	2	no	—
230	C₂×C₈	—	53	2	no	—
231	C₂×C₈	—	53	2	no	—
232	C₂×C₈	—	53	2	no	—
233	C₂×C₈	—	53	2	no	—
234	C₂×C₈	—	212	4	no	—
235	C₂×C₈	—	212	4	no	—
236	C₂×C₈	—	212	2	no	—
237	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
239	C₂×C₈	D₄×D₅₃	1	2	yes	—
240	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
241	C₁₆	—	2	2	no	—
242	C₁₆	—	106	2	no	—
243	C₂³	—	53	2	no	—
244	C₂³	—	53	1	no	—
245	C₂³	—	53	2	no	—
246	C₂³	—	53	1	no	—
247	C₂³	—	212	2	no	—
248	C₂³	Q₁₆×D₅₃	1	1	yes	characteristic
249	C₂×C₄	—	53	2	no	—
250	C₂×C₄	—	53	2	no	—
251	C₂×C₄	—	53	1	no	—
252	C₂×C₄	—	53	1	no	—
253	C₂×C₄	—	53	2	no	—
254	C₂×C₄	—	53	2	no	—
255	C₂×C₄	—	53	2	no	—
256	C₂×C₄	—	53	2	no	—
257	C₂×C₄	—	53	1	no	—
258	C₂×C₄	—	53	2	no	—
259	C₂×C₄	—	53	1	no	—
260	C₂×C₄	—	53	1	no	—
261	C₂×C₄	—	53	1	no	—
262	C₂×C₄	—	53	2	no	—
263	C₂×C₄	C₂²⋊C₄×D₅₃	1	2	yes	—
264	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
265	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
266	C₂×C₄	SD₁₆×D₅₃	1	2	yes	—
267	C₈	—	4	4	no	—
268	C₈	—	53	1	no	—
269	C₈	—	53	1	no	—
270	C₈	—	53	2	no	—
271	C₈	—	212	4	no	—
272	C₈	C₄²×D₅₃	1	1	yes	characteristic
273	C₈	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
274	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
275	C₂²	—	4	1	no	—
276	C₂²	—	53	1	no	—
277	C₂²	—	53	2	no	—
278	C₂²	—	53	1	no	—
279	C₂²	—	106	4	no	—
280	C₂²	—	106	2	no	—
281	C₂²	—	212	2	no	—
282	C₂²	—	212	1	no	—
283	C₂²	#103×D₅₃	1	2	yes	—
284	C₂²	#107×D₅₃	1	1	yes	characteristic
285	C₄	—	53	1	no	—
286	C₄	—	53	2	no	—
287	C₄	—	53	1	no	—
288	C₄	#95×D₅₃	1	1	yes	characteristic
289	C₄	#102×D₅₃	1	2	yes	—
290	C₄	#106×D₅₃	1	1	yes	characteristic
291	C₂	—	2	2	no	—
292	C₂	—	2	1	no	—
293	C₂	—	53	2	no	—
294	C₂	—	106	2	no	—
295	C₂	—	106	1	no	—
296	C₂	#340×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#106×D₅₃
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1385	C₂
socle	C₂×C₅₃	C₂×#340
prime radical	C₂³×D₅₃	Q₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1385	D₅₃
hypocenter	C₅₃	C₂×#1385
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

Complements

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

N	H ≅ G/N	type
3. #20272819	C₂	semidirect
4. #20272818	C₂	semidirect
5. C₅₃×#1385	C₂	semidirect
147. #1385	D₅₃	direct (N × H)
153. D₅₃	#1385	direct (N × H)
169. C₅₃	C₂×#1385	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
#20272819	2
#20272818	2
C₅₃×#1385	2
C₂×#1385	53

Subgroups — D53×#1385 #27299002

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#1385 #27299002