Subgroups — D₅₃×#2176 #27537385

Order 13568.

670 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₁₆×D₅₃	C₂	1	2	yes	maximal
3	#20276587	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276586	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276585	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276584	C₂	1	2	yes	maximal, complemented
7	#20276583	C₂	1	2	yes	maximal, complemented
8	#367×D₅₃	C₂	1	1	yes	maximal, characteristic
9	#358×D₅₃	C₂	1	2	yes	maximal
10	#346×D₅₃	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#2176	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₅₃	C₂²	1	2	yes	—
13	C₄×Q₈×D₅₃	—	2	2	no	—
14	C₂×Dic₄₂₄	C₂²	1	1	yes	characteristic
15	C₂×#13133163	C₂²	1	2	yes	—
16	Q₃₂×D₅₃	C₄	1	8	yes	complemented
17	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
19	C₄×Dic₂₁₂	C₂²	1	2	yes	—
20	Q₁₆×Dic₅₃	C₂²	1	2	yes	—
21	#16810717	C₂²	1	2	yes	—
22	C₄×#11719925	—	2	2	no	—
23	#16810715	—	2	2	no	—
24	#107×D₅₃	C₂²	1	2	yes	—
25	#103×D₅₃	—	2	2	no	—
26	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
27	#16810451	C₂²	1	1	yes	characteristic
28	#16810450	C₂²	1	2	yes	—
29	#16810428	C₂²	1	2	yes	—
30	#16810427	C₂²	1	2	yes	—
31	#16810426	C₂²	1	2	yes	—
32	#16810399	C₂²	1	1	yes	characteristic
33	#16810398	C₂²	1	1	yes	characteristic
34	#16810398	C₂²	1	1	yes	characteristic
35	C₂×Q₃₂×C₅₃	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
37	C₅₃×#367	C₂²	1	1	yes	characteristic
38	C₅₃×#358	C₂²	1	2	yes	—
39	C₅₃×#346	C₂²	1	1	yes	characteristic
40	C₂×Q₈×D₅₃	—	2	2	no	—
41	C₂×Dic₂₁₂	C₂³	1	2	yes	—
42	C₂×#11719925	—	2	2	no	—
43	Q₁₆×D₅₃	—	2	4	no	—
44	Q₁₆×D₅₃	C₂×C₄	1	8	yes	—
45	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
46	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
47	Q₈×Dic₅₃	—	2	2	no	—
48	C₄×Dic₁₀₆	—	2	2	no	—
49	C₄×Dic₁₀₆	—	4	2	no	—
50	#13133182	—	2	2	no	—
51	#13133182	—	4	2	no	—
52	C₄⋊C₄×D₅₃	—	2	2	no	—
53	C₄⋊C₄×D₅₃	—	4	2	no	—
54	C₄²×D₅₃	—	4	2	no	—
55	C₄²×D₅₃	D₄	1	1	yes	characteristic
56	Dic₄₂₄	C₂×C₄	1	4	yes	complemented
57	#13133163	C₂×C₄	1	8	yes	complemented
58	C₂×#11719908	C₂³	1	1	yes	characteristic
59	C₂×#11719908	C₂³	1	1	yes	characteristic
60	C₁₆×D₅₃	—	2	2	no	—
61	C₁₆×D₅₃	C₂×C₄	1	4	yes	—
62	#13133151	C₂³	1	2	yes	—
63	#13133150	—	2	2	no	—
64	#13133142	—	2	2	no	—
65	#13133141	—	2	2	no	—
66	#13133140	—	2	2	no	—
67	C₄×#88322	D₄	1	1	yes	characteristic
68	C₈×Dic₅₃	D₄	1	1	yes	characteristic
69	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
70	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
71	C₄×Q₈×C₅₃	—	2	2	no	—
72	Q₃₂×C₅₃	C₂×C₄	1	4	yes	complemented
73	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
75	#107×C₅₃	C₂³	1	2	yes	—
76	#103×C₅₃	—	2	2	no	—
77	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₂×Q₈×C₅₃	—	2	2	no	—
79	Q₈×D₅₃	—	2	8	no	—
80	Q₈×D₅₃	—	4	4	no	—
81	C₂×C₄×D₅₃	—	4	2	no	—
82	C₂×C₄×D₅₃	—	4	2	no	—
83	C₂×C₄×D₅₃	D₈	1	2	yes	—
84	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂×Dic₁₀₆	—	2	2	no	—
86	C₂×Dic₁₀₆	—	4	2	no	—
87	Q₁₆×C₅₃	—	2	2	no	—
88	Q₁₆×C₅₃	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₅₃	—	2	2	no	—
92	C₄⋊C₄×C₅₃	—	4	2	no	—
93	C₄²×C₅₃	—	4	2	no	—
94	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
95	#11719925	—	2	8	no	—
96	C₄⋊Dic₅₃	—	2	2	no	—
97	C₄⋊Dic₅₃	—	4	2	no	—
98	#11719919	—	4	2	no	—
99	#11719919	—	4	2	no	—
100	#11719919	—	4	2	no	—
101	C₄×Dic₅₃	—	4	2	no	—
102	C₄×Dic₅₃	—	4	2	no	—
103	C₄×Dic₅₃	—	4	2	no	—
104	C₄×Dic₅₃	D₈	1	2	yes	—
105	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂×#88322	C₂×D₄	1	1	yes	characteristic
107	C₂×#88322	C₂×D₄	1	1	yes	characteristic
108	Dic₂₁₂	—	2	2	no	—
109	Dic₂₁₂	C₂²×C₄	1	4	yes	—
110	C₈×D₅₃	—	2	2	no	—
111	C₈×D₅₃	C₂²×C₄	1	2	yes	—
112	C₈×D₅₃	#41	1	2	yes	—
113	C₁₆×C₅₃	—	2	1	no	—
114	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
115	#11719908	—	2	1	no	—
116	#11719908	C₂²×C₄	1	2	yes	—
117	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
118	Q₈×C₅₃	—	2	4	no	—
119	Q₈×C₅₃	—	4	2	no	—
120	C₂×C₄×C₅₃	—	4	2	no	—
121	C₂×C₄×C₅₃	—	4	2	no	—
122	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×Dic₅₃	—	4	2	no	—
125	C₂×Dic₅₃	—	4	2	no	—
126	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
127	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
128	C₄×D₅₃	—	2	4	no	—
129	C₄×D₅₃	—	4	8	no	—
130	C₄×D₅₃	—	8	4	no	—
131	C₄×D₅₃	C₄×D₄	1	2	yes	—
132	C₄×D₅₃	#135	1	2	yes	—
133	Dic₁₀₆	—	2	4	no	—
134	Dic₁₀₆	—	4	2	no	—
135	Dic₁₀₆	—	4	8	no	—
136	C₈×C₅₃	—	2	1	no	—
137	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
138	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
139	#88322	—	2	1	no	—
140	#88322	C₄×D₄	1	2	yes	—
141	C₂×#2176	—	53	1	no	maximal, Sylow
142	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
143	C₂×D₅₃	C₄×D₈	1	2	yes	—
144	C₂×D₅₃	#510	1	4	yes	—
145	C₄×C₅₃	—	2	2	no	—
146	C₄×C₅₃	—	4	4	no	—
147	C₄×C₅₃	—	8	2	no	—
148	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
149	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
150	Dic₅₃	—	2	2	no	—
151	Dic₅₃	—	4	4	no	—
152	Dic₅₃	—	8	2	no	—
153	Dic₅₃	C₄×D₈	1	2	yes	—
154	C₂²×Q₃₂	—	53	1	no	—
155	C₂×C₄×Q₁₆	—	53	2	no	—
156	#2176	—	53	1	no	—
157	#2176	—	53	2	no	—
158	#2176	—	53	1	no	—
159	#2176	—	53	2	no	—
160	#2176	—	53	1	no	—
161	#2176	D₅₃	1	1	yes	characteristic, direct product factor
162	C₂×#367	—	53	1	no	—
163	C₂×#358	—	53	2	no	—
164	C₂×#346	—	53	1	no	—
165	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
166	C₂×C₅₃	C₂×#510	1	2	yes	—
167	D₅₃	#2176	1	4	yes	direct product factor
168	C₂²×Q₁₆	—	53	2	no	—
169	C₂×C₄×Q₈	—	106	2	no	—
170	C₂×Q₃₂	—	53	2	no	—
171	C₂×Q₃₂	—	53	8	no	—
172	C₂×Q₃₂	—	53	1	no	—
173	C₂×Q₃₂	C₂×D₅₃	1	1	yes	characteristic
174	C₂²×C₁₆	—	53	1	no	—
175	C₂²×C₁₆	—	53	1	no	—
176	C₄×Q₁₆	—	53	2	no	—
177	C₄×Q₁₆	—	53	2	no	—
178	C₄×Q₁₆	—	53	2	no	—
179	C₄×Q₁₆	—	106	2	no	—
180	C₄×Q₁₆	—	106	2	no	—
181	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
182	C₂×#107	—	53	2	no	—
183	C₂×#103	—	106	2	no	—
184	C₂×C₄×C₈	—	53	1	no	—
185	#367	—	53	1	no	—
186	#367	—	53	2	no	—
187	#367	C₂×D₅₃	1	1	yes	characteristic
188	#358	—	53	2	no	—
189	#358	—	53	2	no	—
190	#358	—	53	2	no	—
191	#358	C₂×D₅₃	1	2	yes	—
192	#346	—	53	1	no	—
193	#346	—	53	1	no	—
194	#346	—	53	1	no	—
195	#346	C₂×D₅₃	1	1	yes	characteristic
196	C₅₃	C₂×#2176	1	1	yes	characteristic, complemented, Sylow
197	C₂²×Q₈	—	106	2	no	—
198	C₂×Q₁₆	—	53	8	no	—
199	C₂×Q₁₆	—	53	2	no	—
200	C₂×Q₁₆	—	106	4	no	—
201	C₂×Q₁₆	—	106	2	no	—
202	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
203	C₂²×C₈	—	53	1	no	—
204	C₂²×C₈	—	53	1	no	—
205	C₄×Q₈	—	2	2	no	—
206	C₄×Q₈	—	106	2	no	—
207	C₄×Q₈	—	106	2	no	—
208	C₄×Q₈	—	106	2	no	—
209	C₄×Q₈	—	212	2	no	—
210	C₄×Q₈	—	212	2	no	—
211	C₂×C₄⋊C₄	—	106	2	no	—
212	C₂×C₄⋊C₄	—	212	2	no	—
213	C₂×C₄²	—	53	1	no	—
214	C₂×C₄²	—	212	2	no	—
215	Q₃₂	—	53	8	no	—
216	Q₃₂	—	53	4	no	—
217	Q₃₂	C₄×D₅₃	1	4	yes	complemented
218	C₂×C₁₆	—	53	1	no	—
219	C₂×C₁₆	—	53	4	no	—
220	C₂×C₁₆	—	53	1	no	—
221	C₂×C₁₆	—	106	2	no	—
222	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
223	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
224	#107	—	53	2	no	—
225	#107	—	106	2	no	—
226	#107	C₂²×D₅₃	1	2	yes	—
227	#103	—	2	2	no	—
228	#103	—	106	2	no	—
229	#103	—	106	2	no	—
230	#103	—	106	2	no	—
231	C₄×C₈	—	53	1	no	—
232	C₄×C₈	—	53	1	no	—
233	C₄×C₈	—	53	1	no	—
234	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	106	8	no	—
237	C₂×Q₈	—	106	2	no	—
238	C₂×Q₈	—	212	2	no	—
239	C₂×Q₈	—	212	4	no	—
240	C₂²×C₄	—	53	1	no	—
241	C₂²×C₄	—	53	2	no	—
242	C₂²×C₄	—	212	2	no	—
243	C₂²×C₄	—	212	2	no	—
244	Q₁₆	—	2	2	no	—
245	Q₁₆	—	53	4	no	—
246	Q₁₆	—	106	8	no	—
247	Q₁₆	—	106	2	no	—
248	Q₁₆	C₂×C₄×D₅₃	1	4	yes	—
249	C₂×C₈	—	53	1	no	—
250	C₂×C₈	—	53	2	no	—
251	C₂×C₈	—	53	2	no	—
252	C₂×C₈	—	53	1	no	—
253	C₂×C₈	—	106	2	no	—
254	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
255	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	106	2	no	—
259	C₄⋊C₄	—	212	2	no	—
260	C₄⋊C₄	—	212	2	no	—
261	C₄⋊C₄	—	212	2	no	—
262	C₄⋊C₄	—	212	2	no	—
263	C₄²	—	4	2	no	—
264	C₄²	—	53	2	no	—
265	C₄²	—	53	1	no	—
266	C₄²	—	212	2	no	—
267	C₄²	—	212	2	no	—
268	C₄²	—	212	2	no	—
269	C₄²	D₄×D₅₃	1	1	yes	characteristic
270	C₁₆	—	2	1	no	—
271	C₁₆	—	53	2	no	—
272	C₁₆	—	106	1	no	—
273	C₁₆	C₂×C₄×D₅₃	1	2	yes	—
274	C₂³	—	53	1	no	—
275	Q₈	—	2	4	no	—
276	Q₈	—	4	2	no	—
277	Q₈	—	106	4	no	—
278	Q₈	—	212	8	no	—
279	Q₈	—	212	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	53	1	no	—
283	C₂×C₄	—	53	2	no	—
284	C₂×C₄	—	53	2	no	—
285	C₂×C₄	—	53	2	no	—
286	C₂×C₄	—	106	4	no	—
287	C₂×C₄	—	424	4	no	—
288	C₂×C₄	—	212	8	no	—
289	C₂×C₄	—	212	2	no	—
290	C₂×C₄	—	212	2	no	—
291	C₂×C₄	D₈×D₅₃	1	2	yes	—
292	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	53	2	no	—
295	C₈	—	106	1	no	—
296	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
297	C₈	#41×D₅₃	1	1	yes	characteristic
298	C₂²	—	53	2	no	—
299	C₂²	—	53	4	no	—
300	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
301	C₄	—	2	2	no	—
302	C₄	—	4	4	no	—
303	C₄	—	8	2	no	—
304	C₄	—	53	2	no	—
305	C₄	—	106	2	no	—
306	C₄	—	424	2	no	—
307	C₄	—	212	4	no	—
308	C₄	C₄×D₄×D₅₃	1	1	yes	characteristic
309	C₄	#135×D₅₃	1	1	yes	characteristic
310	C₂	—	53	4	no	—
311	C₂	C₄×D₈×D₅₃	1	1	yes	characteristic
312	C₂	#510×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2176	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂²×D₅₃	C₂×D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2176	D₅₃
hypocenter	C₅₃	C₂×#2176
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2176

Complements

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

N	H ≅ G/N	type
3. #20276587	C₂	semidirect
4. #20276586	C₂	semidirect
5. #20276585	C₂	semidirect
6. #20276584	C₂	semidirect
7. #20276583	C₂	semidirect
11. C₅₃×#2176	C₂	semidirect
16. Q₃₂×D₅₃	C₄	semidirect
56. Dic₄₂₄	C₂×C₄	semidirect
57. #13133163	C₂×C₄	semidirect
72. Q₃₂×C₅₃	C₂×C₄	semidirect
161. #2176	D₅₃	direct (N × H)
167. D₅₃	#2176	direct (N × H)
196. C₅₃	C₂×#2176	semidirect
217. Q₃₂	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×Q₃₂×D₅₃	2
C₄×Q₁₆×D₅₃	2
#20276587	2
#20276586	2
#20276585	2
#20276584	2
#20276583	2
#367×D₅₃	2
#358×D₅₃	2
#346×D₅₃	2
C₅₃×#2176	2
C₂×#2176	53

Subgroups — D53×#2176 #27537385

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#2176 #27537385