Subgroups — D₅₃×#2248 #27539482

Order 13568.

918 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×SD₃₂×D₅₃	C₂	1	4	yes	maximal, complemented
2	#494×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#493×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20276972	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276971	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276970	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276969	C₂	1	4	yes	maximal, complemented
8	C₄×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#2248	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₁₆×D₅₃	—	2	1	no	—
11	C₂×Q₁₆×D₅₃	C₂²	1	2	yes	—
12	C₂×D₈×D₅₃	—	2	1	no	—
13	C₂×D₈×D₅₃	C₂²	1	2	yes	—
14	C₄⋊Q₈×D₅₃	—	2	1	no	—
15	#127×D₅₃	—	2	1	no	—
16	C₂×#13133162	C₂²	1	4	yes	complemented
17	C₂×#13133161	C₂²	1	4	yes	complemented
18	C₂×#13133160	C₂²	1	4	yes	complemented
19	SD₃₂×D₅₃	—	2	16	no	—
20	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
21	#16811003	C₂²	1	1	yes	characteristic, complemented
22	#16811002	—	2	1	no	—
23	#16811001	—	2	1	no	—
24	#16811001	C₂²	1	2	yes	complemented
25	#16811000	C₂²	1	1	yes	characteristic
26	#16810999	—	2	1	no	—
27	#16810998	—	2	1	no	—
28	#16810998	C₂²	1	2	yes	—
29	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
30	C₄×#11719908	C₂²	1	1	yes	characteristic, complemented
31	C₁₆×Dic₅₃	C₂²	1	2	yes	complemented
32	C₂×SD₃₂×C₅₃	C₂²	1	4	yes	complemented
33	C₅₃×#494	C₂²	1	1	yes	characteristic, complemented
34	C₅₃×#493	C₂²	1	1	yes	characteristic
35	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
36	C₂×Q₈×D₅₃	—	2	2	no	—
37	C₂×D₄×D₅₃	—	2	2	no	—
38	C₂×D₄×D₅₃	—	4	2	no	—
39	C₂×Dic₂₁₂	—	2	1	no	—
40	C₂×Dic₂₁₂	C₂³	1	2	yes	—
41	C₂×#11719925	—	2	2	no	—
42	C₂×#11719925	—	2	2	no	—
43	Q₁₆×D₅₃	—	2	8	no	—
44	Q₁₆×D₅₃	—	4	4	no	—
45	C₂×D₄₂₄	—	2	1	no	—
46	C₂×D₄₂₄	C₂³	1	2	yes	—
47	C₂×#11719922	—	2	2	no	—
48	C₂×#11719922	—	2	2	no	—
49	D₈×D₅₃	—	2	8	no	—
50	D₈×D₅₃	—	4	4	no	—
51	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
52	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
53	#13133217	—	2	1	no	—
54	#13133216	—	4	2	no	—
55	#13133215	—	2	2	no	—
56	#13133214	—	2	1	no	—
57	#13133213	—	2	2	no	—
58	#13133213	—	4	2	no	—
59	C₄⋊C₄×D₅₃	—	4	2	no	—
60	C₄²×D₅₃	D₄	1	1	yes	characteristic
61	#13133162	—	2	8	no	—
62	#13133161	—	2	8	no	—
63	#13133160	—	2	8	no	—
64	C₂×#11719908	C₂³	1	2	yes	—
65	C₁₆×D₅₃	D₄	1	8	yes	complemented
66	C₄×#88322	D₄	1	1	yes	characteristic
67	C₈×Dic₅₃	D₄	1	1	yes	characteristic
68	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
69	C₂×Q₁₆×C₅₃	—	2	1	no	—
70	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
71	C₂×D₈×C₅₃	—	2	1	no	—
72	C₂×D₈×C₅₃	C₂³	1	2	yes	—
73	C₄⋊Q₈×C₅₃	—	2	1	no	—
74	#127×C₅₃	—	2	1	no	—
75	SD₃₂×C₅₃	—	2	8	no	—
76	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
77	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₂³×D₅₃	—	4	2	no	—
79	C₂×Q₈×C₅₃	—	2	2	no	—
80	C₂×D₄×C₅₃	—	2	2	no	—
81	C₂×D₄×C₅₃	—	4	2	no	—
82	C₂×C₂²⋊D₅₃	—	4	4	no	—
83	C₂×C₂²⋊D₅₃	—	4	2	no	—
84	Q₈×D₅₃	—	4	4	no	—
85	Q₈×D₅₃	—	4	4	no	—
86	D₄×D₅₃	—	4	4	no	—
87	D₄×D₅₃	—	4	4	no	—
88	D₄×D₅₃	—	8	8	no	—
89	C₂×D₂₁₂	—	2	2	no	—
90	C₂×D₂₁₂	—	4	2	no	—
91	C₂×C₄×D₅₃	—	4	2	no	—
92	C₂×C₄×D₅₃	D₈	1	2	yes	—
93	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂×Dic₁₀₆	—	2	2	no	—
95	C₂×Dic₁₀₆	—	4	2	no	—
96	Q₁₆×C₅₃	—	2	4	no	—
97	Q₁₆×C₅₃	—	4	2	no	—
98	D₈×C₅₃	—	2	4	no	—
99	D₈×C₅₃	—	4	2	no	—
100	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
102	C₄⋊C₄×C₅₃	—	4	2	no	—
103	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
104	#11719925	—	4	4	no	—
105	#11719925	—	4	4	no	—
106	#11719922	—	4	4	no	—
107	#11719922	—	4	4	no	—
108	C₄⋊Dic₅₃	—	4	2	no	—
109	#11719919	—	4	4	no	—
110	C₄×Dic₅₃	D₈	1	2	yes	—
111	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
112	C₂×#88322	C₂×D₄	1	1	yes	characteristic
113	C₂×#88322	C₂×D₄	1	1	yes	characteristic
114	Dic₂₁₂	—	2	4	no	—
115	Dic₂₁₂	—	4	2	no	—
116	D₄₂₄	—	2	4	no	—
117	D₄₂₄	—	4	2	no	—
118	C₈×D₅₃	D₈	1	4	yes	—
119	C₈×D₅₃	C₂×D₄	1	4	yes	—
120	C₁₆×C₅₃	C₂×D₄	1	4	yes	complemented
121	#11719908	C₂×D₄	1	4	yes	complemented
122	C₂³×C₅₃	—	4	2	no	—
123	C₂²×D₅₃	—	4	2	no	—
124	C₂²×D₅₃	—	8	4	no	—
125	C₂²×D₅₃	—	8	8	no	—
126	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
127	Q₈×C₅₃	—	4	2	no	—
128	Q₈×C₅₃	—	4	2	no	—
129	D₄×C₅₃	—	4	2	no	—
130	D₄×C₅₃	—	4	2	no	—
131	D₄×C₅₃	—	8	4	no	—
132	C₂×C₄×C₅₃	—	4	2	no	—
133	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
135	C₂²⋊D₅₃	—	8	4	no	—
136	C₂²⋊D₅₃	—	8	8	no	—
137	C₂×Dic₅₃	—	4	2	no	—
138	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
139	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
140	D₂₁₂	—	4	2	no	—
141	D₂₁₂	—	4	2	no	—
142	D₂₁₂	—	8	4	no	—
143	C₄×D₅₃	—	8	4	no	—
144	C₄×D₅₃	SD₃₂	1	8	yes	complemented
145	C₄×D₅₃	#127	1	2	yes	—
146	C₄×D₅₃	C₂×D₈	1	2	yes	—
147	Dic₁₀₆	—	4	2	no	—
148	Dic₁₀₆	—	4	2	no	—
149	Dic₁₀₆	—	8	4	no	—
150	C₈×C₅₃	C₂×D₈	1	2	yes	—
151	C₈×C₅₃	C₂²×D₄	1	2	yes	—
152	#88322	#127	1	2	yes	—
153	#88322	C₂×D₈	1	2	yes	—
154	C₂×#2248	—	53	1	no	maximal, Sylow
155	C₂²×C₅₃	—	8	2	no	—
156	C₂²×C₅₃	—	8	4	no	—
157	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
158	C₂×D₅₃	—	8	2	no	—
159	C₂×D₅₃	—	8	4	no	—
160	C₂×D₅₃	—	16	8	no	—
161	C₂×D₅₃	#493	1	2	yes	—
162	C₂×D₅₃	C₂×SD₃₂	1	4	yes	—
163	C₄×C₅₃	—	8	2	no	—
164	C₄×C₅₃	C₂×SD₃₂	1	4	yes	complemented
165	C₄×C₅₃	C₂×#127	1	1	yes	characteristic
166	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
167	Dic₅₃	—	8	2	no	—
168	Dic₅₃	#493	1	2	yes	—
169	Dic₅₃	C₂×SD₃₂	1	4	yes	complemented
170	C₂²×SD₃₂	—	53	4	no	—
171	C₂×#494	—	53	1	no	—
172	C₂×#493	—	53	1	no	—
173	#2248	—	53	4	no	—
174	#2248	—	53	1	no	—
175	#2248	—	53	1	no	—
176	#2248	—	53	1	no	—
177	#2248	D₅₃	1	1	yes	characteristic, direct product factor
178	C₂×C₄×C₁₆	—	53	1	no	—
179	C₂×C₅₃	—	16	2	no	—
180	C₂×C₅₃	C₂×#493	1	1	yes	characteristic
181	C₂×C₅₃	C₂²×SD₃₂	1	2	yes	—
182	D₅₃	—	16	2	no	—
183	D₅₃	#2248	1	4	yes	direct product factor
184	C₂²×Q₁₆	—	53	2	no	—
185	C₂²×Q₁₆	—	106	1	no	—
186	C₂²×D₈	—	53	2	no	—
187	C₂²×D₈	—	106	1	no	—
188	C₂×C₄⋊Q₈	—	106	1	no	—
189	C₂×#127	—	106	1	no	—
190	C₂×SD₃₂	—	53	4	no	—
191	C₂×SD₃₂	—	53	4	no	—
192	C₂×SD₃₂	—	53	4	no	—
193	C₂×SD₃₂	—	106	16	no	—
194	C₂×SD₃₂	C₂×D₅₃	1	4	yes	complemented
195	C₂²×C₁₆	—	53	2	no	—
196	#494	—	53	2	no	—
197	#494	—	53	1	no	—
198	#494	—	106	1	no	—
199	#494	—	106	1	no	—
200	#494	C₂×D₅₃	1	1	yes	characteristic, complemented
201	#493	—	53	2	no	—
202	#493	—	53	1	no	—
203	#493	—	106	1	no	—
204	#493	—	106	1	no	—
205	#493	C₂×D₅₃	1	1	yes	characteristic
206	C₂×C₄×C₈	—	53	1	no	—
207	C₄×C₁₆	—	53	2	no	—
208	C₄×C₁₆	—	53	1	no	—
209	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
210	C₅₃	C₂×#2248	1	1	yes	characteristic, complemented, Sylow
211	C₂²×Q₈	—	106	2	no	—
212	C₂²×D₄	—	106	2	no	—
213	C₂²×D₄	—	212	2	no	—
214	C₂×Q₁₆	—	2	1	no	—
215	C₂×Q₁₆	—	53	2	no	—
216	C₂×Q₁₆	—	106	8	no	—
217	C₂×Q₁₆	—	106	2	no	—
218	C₂×Q₁₆	—	106	2	no	—
219	C₂×Q₁₆	—	106	1	no	—
220	C₂×Q₁₆	—	212	4	no	—
221	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
222	C₂×D₈	—	2	1	no	—
223	C₂×D₈	—	53	2	no	—
224	C₂×D₈	—	106	8	no	—
225	C₂×D₈	—	106	2	no	—
226	C₂×D₈	—	106	2	no	—
227	C₂×D₈	—	106	1	no	—
228	C₂×D₈	—	212	4	no	—
229	C₂×D₈	C₂²×D₅₃	1	2	yes	—
230	C₂²×C₈	—	53	1	no	—
231	C₂²×C₈	—	53	1	no	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	106	2	no	—
234	C₄⋊Q₈	—	106	1	no	—
235	C₄⋊Q₈	—	212	2	no	—
236	#127	—	2	1	no	—
237	#127	—	106	2	no	—
238	#127	—	106	1	no	—
239	#127	—	212	2	no	—
240	C₂×C₄⋊C₄	—	212	2	no	—
241	C₂×C₄²	—	53	1	no	—
242	SD₃₂	—	2	8	no	—
243	SD₃₂	—	106	8	no	—
244	SD₃₂	—	106	8	no	—
245	SD₃₂	—	106	8	no	—
246	C₂×C₁₆	—	53	2	no	—
247	C₂×C₁₆	—	53	8	no	—
248	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
249	C₄×C₈	—	53	1	no	—
250	C₄×C₈	—	53	1	no	—
251	C₄×C₈	—	53	1	no	—
252	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
253	C₂⁴	—	212	2	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	106	2	no	—
256	C₂×Q₈	—	212	2	no	—
257	C₂×Q₈	—	212	4	no	—
258	C₂×Q₈	—	212	4	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	4	2	no	—
261	C₂×D₄	—	106	2	no	—
262	C₂×D₄	—	424	8	no	—
263	C₂×D₄	—	212	4	no	—
264	C₂×D₄	—	212	4	no	—
265	C₂×D₄	—	212	4	no	—
266	C₂×D₄	—	212	2	no	—
267	C₂×D₄	—	212	2	no	—
268	C₂²×C₄	—	53	2	no	—
269	C₂²×C₄	—	53	1	no	—
270	C₂²×C₄	—	212	2	no	—
271	Q₁₆	—	2	4	no	—
272	Q₁₆	—	4	2	no	—
273	Q₁₆	—	106	4	no	—
274	Q₁₆	—	212	4	no	—
275	Q₁₆	—	212	4	no	—
276	Q₁₆	—	212	2	no	—
277	D₈	—	2	4	no	—
278	D₈	—	4	2	no	—
279	D₈	—	106	4	no	—
280	D₈	—	212	4	no	—
281	D₈	—	212	4	no	—
282	D₈	—	212	2	no	—
283	C₂×C₈	—	53	1	no	—
284	C₂×C₈	—	53	1	no	—
285	C₂×C₈	—	53	4	no	—
286	C₂×C₈	—	53	4	no	—
287	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
288	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
289	C₄⋊C₄	—	4	2	no	—
290	C₄⋊C₄	—	212	4	no	—
291	C₄⋊C₄	—	212	2	no	—
292	C₄²	—	53	2	no	—
293	C₄²	—	53	1	no	—
294	C₄²	D₄×D₅₃	1	1	yes	characteristic
295	C₁₆	—	53	4	no	—
296	C₁₆	D₄×D₅₃	1	4	yes	complemented
297	C₂³	—	4	2	no	—
298	C₂³	—	53	1	no	—
299	C₂³	—	424	8	no	—
300	C₂³	—	424	4	no	—
301	C₂³	—	212	2	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	424	4	no	—
305	Q₈	—	212	2	no	—
306	Q₈	—	212	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	4	2	no	—
309	D₄	—	8	4	no	—
310	D₄	—	424	8	no	—
311	D₄	—	424	4	no	—
312	D₄	—	424	4	no	—
313	D₄	—	212	2	no	—
314	D₄	—	212	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	53	2	no	—
317	C₂×C₄	—	53	1	no	—
318	C₂×C₄	—	53	8	no	—
319	C₂×C₄	—	53	2	no	—
320	C₂×C₄	—	53	2	no	—
321	C₂×C₄	—	424	4	no	—
322	C₂×C₄	—	212	2	no	—
323	C₂×C₄	D₈×D₅₃	1	2	yes	—
324	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
325	C₈	—	53	2	no	—
326	C₈	—	53	2	no	—
327	C₈	D₈×D₅₃	1	2	yes	—
328	C₈	C₂×D₄×D₅₃	1	2	yes	—
329	C₂²	—	8	4	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	53	4	no	—
332	C₂²	—	53	2	no	—
333	C₂²	—	848	8	no	—
334	C₂²	—	424	4	no	—
335	C₂²	—	424	2	no	—
336	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
337	C₄	—	8	2	no	—
338	C₄	—	53	4	no	—
339	C₄	—	53	2	no	—
340	C₄	—	424	2	no	—
341	C₄	SD₃₂×D₅₃	1	4	yes	complemented
342	C₄	#127×D₅₃	1	1	yes	characteristic
343	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
344	C₂	—	16	2	no	—
345	C₂	—	53	4	no	—
346	C₂	—	848	2	no	—
347	C₂	#493×D₅₃	1	1	yes	characteristic
348	C₂	C₂×SD₃₂×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2248	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#493×D₅₃	C₂
hypercenter	#2248	D₅₃
hypocenter	C₅₃	C₂×#2248
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2248

Complements

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

N	H ≅ G/N	type
1. C₂×SD₃₂×D₅₃	C₂	semidirect
2. #494×D₅₃	C₂	semidirect
4. #20276972	C₂	semidirect
5. #20276971	C₂	semidirect
6. #20276970	C₂	semidirect
7. #20276969	C₂	semidirect
8. C₄×C₁₆×D₅₃	C₂	semidirect
9. C₅₃×#2248	C₂	semidirect
16. C₂×#13133162	C₂²	semidirect
17. C₂×#13133161	C₂²	semidirect
18. C₂×#13133160	C₂²	semidirect
21. #16811003	C₂²	semidirect
24. #16811001	C₂²	semidirect
30. C₄×#11719908	C₂²	semidirect
31. C₁₆×Dic₅₃	C₂²	semidirect
32. C₂×SD₃₂×C₅₃	C₂²	semidirect
33. C₅₃×#494	C₂²	semidirect
35. C₄×C₁₆×C₅₃	C₂²	semidirect
65. C₁₆×D₅₃	D₄	semidirect
120. C₁₆×C₅₃	C₂×D₄	semidirect
121. #11719908	C₂×D₄	semidirect
144. C₄×D₅₃	SD₃₂	semidirect
164. C₄×C₅₃	C₂×SD₃₂	semidirect
169. Dic₅₃	C₂×SD₃₂	semidirect
177. #2248	D₅₃	direct (N × H)
183. D₅₃	#2248	direct (N × H)
194. C₂×SD₃₂	C₂×D₅₃	semidirect
200. #494	C₂×D₅₃	semidirect
209. C₄×C₁₆	C₂×D₅₃	semidirect
210. C₅₃	C₂×#2248	semidirect
296. C₁₆	D₄×D₅₃	semidirect
341. C₄	SD₃₂×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×SD₃₂×D₅₃	2
#494×D₅₃	2
#493×D₅₃	2
#20276972	2
#20276971	2
#20276970	2
#20276969	2
C₄×C₁₆×D₅₃	2
C₅₃×#2248	2
C₂×#2248	53

Subgroups — D53×#2248 #27539482

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#2248 #27539482