Subgroups — D₅₃⋊#1579 #28316370

Order 13568.

718 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₅₃⋊C₈	C₂	1	2	yes	maximal
2	#20290310	C₂	1	2	yes	maximal, complemented
3	#20290309	C₂	1	4	yes	maximal, complemented
4	#20290308	C₂	1	2	yes	maximal, complemented
5	D₅₃⋊#334	C₂	1	1	yes	maximal, characteristic
6	D₅₃⋊(Q₈⋊C₈)	C₂	1	2	yes	maximal
7	#20289909	C₂	1	1	yes	maximal, characteristic
8	C₄×Q₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
9	#16811672	—	2	4	no	—
10	#16811672	C₂²	1	4	yes	—
11	Q₈×#88324	C₂²	1	4	yes	—
12	D₅₃⋊(C₄⋊C₈)	—	2	2	no	—
13	D₅₃⋊(C₄⋊C₈)	C₂²	1	2	yes	—
14	C₄×D₅₃⋊C₈	—	2	2	no	—
15	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
17	#16811485	C₂²	1	2	yes	—
18	#16811484	C₂²	1	2	yes	—
19	#16811473	C₂²	1	4	yes	—
20	Q₈⋊#88324	C₂²	1	4	yes	—
21	#16811464	C₂²	1	2	yes	—
22	#16811464	C₂²	1	2	yes	—
23	C₂×Q₁₆×D₅₃	C₄	1	1	yes	characteristic
24	C₄×Q₈×D₅₃	C₂²	1	2	yes	—
25	C₄×Dic₂₁₂	C₄	1	1	yes	characteristic
26	Q₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
27	#16810717	C₂²	1	1	yes	characteristic
28	C₄×#11719925	C₄	1	2	yes	—
29	#16810715	C₂²	1	2	yes	—
30	#107×D₅₃	C₄	1	1	yes	characteristic
31	#103×D₅₃	C₄	1	2	yes	—
32	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
33	C₄×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
34	C₂×D₅₃⋊C₈	—	2	2	no	—
35	C₂×D₅₃⋊C₈	—	2	1	no	—
36	C₂×D₅₃⋊C₈	D₄	1	2	yes	—
37	#13133284	—	2	2	no	—
38	#13133284	—	2	4	no	—
39	C₄⋊#88324	—	2	4	no	—
40	C₄⋊#88324	C₂³	1	4	yes	—
41	C₄×#88324	—	2	1	no	—
42	C₄×#88324	—	2	4	no	—
43	C₄×#88324	—	2	4	no	—
44	C₄×#88324	D₄	1	2	yes	—
45	C₄×#88324	C₂³	1	2	yes	—
46	C₄×#88324	C₂³	1	2	yes	—
47	C₂×Q₈×D₅₃	C₂×C₄	1	2	yes	—
48	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
49	C₂×#11719925	C₂×C₄	1	2	yes	—
50	Q₁₆×D₅₃	C₈	1	8	yes	complemented
51	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
53	Q₈×Dic₅₃	C₂³	1	2	yes	—
54	C₄×Dic₁₀₆	—	2	2	no	—
55	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
56	#13133182	—	2	2	no	—
57	#13133182	C₂³	1	2	yes	—
58	C₄⋊C₄×D₅₃	—	2	2	no	—
59	C₄⋊C₄×D₅₃	C₂×C₄	1	2	yes	—
60	C₄²×D₅₃	—	2	2	no	—
61	C₄²×D₅₃	C₂³	1	1	yes	characteristic
62	#13133151	C₂×C₄	1	1	yes	characteristic
63	#13133150	C₂×C₄	1	2	yes	—
64	#13133142	C₂×C₄	1	2	yes	—
65	#13133141	C₂×C₄	1	2	yes	—
66	#13133140	C₂×C₄	1	2	yes	—
67	C₄×#88322	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
69	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
70	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
72	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
73	#103×C₅₃	C₂×C₄	1	2	yes	—
74	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
75	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
76	Q₈×D₅₃	—	2	4	no	—
77	Q₈×D₅₃	C₂×C₈	1	8	yes	—
78	C₂×C₄×D₅₃	—	2	2	no	—
79	C₂×C₄×D₅₃	—	2	2	no	—
80	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
83	C₂×Dic₁₀₆	—	2	2	no	—
84	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
85	C₂×#88324	—	2	4	no	—
86	C₂×#88324	—	2	2	no	—
87	C₂×#88324	C₂×D₄	1	4	yes	—
88	D₅₃⋊C₈	—	2	4	no	—
89	D₅₃⋊C₈	—	4	2	no	—
90	D₅₃⋊C₈	—	4	4	no	—
91	Q₁₆×C₅₃	C₂×C₈	1	4	yes	complemented
92	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₄⋊C₄×C₅₃	—	2	2	no	—
95	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
96	C₄²×C₅₃	—	2	2	no	—
97	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	#11719925	C₂×C₈	1	8	yes	complemented
99	C₄⋊Dic₅₃	—	2	2	no	—
100	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
101	#11719919	—	2	2	no	—
102	#11719919	—	2	2	no	—
103	#11719919	—	2	2	no	—
104	C₄×Dic₅₃	—	2	2	no	—
105	C₄×Dic₅₃	—	2	2	no	—
106	C₄×Dic₅₃	—	2	2	no	—
107	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
108	C₄×Dic₅₃	#41	1	1	yes	characteristic
109	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
110	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
111	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
112	Dic₂₁₂	C₂×C₈	1	4	yes	complemented
113	C₈×D₅₃	—	2	2	no	—
114	C₈×D₅₃	C₂×C₈	1	4	yes	—
115	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
116	Q₈×C₅₃	—	2	2	no	—
117	Q₈×C₅₃	C₂²×C₈	1	4	yes	—
118	C₂×C₄×C₅₃	—	2	2	no	—
119	C₂×C₄×C₅₃	—	2	2	no	—
120	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×Dic₅₃	—	2	2	no	—
124	C₂×Dic₅₃	—	2	2	no	—
125	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
126	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×Dic₅₃	C₂×#41	1	1	yes	characteristic
128	C₄×D₅₃	—	2	8	no	—
129	C₄×D₅₃	—	2	2	no	—
130	C₄×D₅₃	—	4	4	no	—
131	C₄×D₅₃	C₂²×C₈	1	2	yes	—
132	C₄×D₅₃	#131	1	2	yes	—
133	C₄×D₅₃	#137	1	4	yes	—
134	Dic₁₀₆	—	2	2	no	—
135	Dic₁₀₆	—	2	8	no	—
136	Dic₁₀₆	C₂²×C₈	1	4	yes	—
137	#88324	—	2	4	no	—
138	#88324	—	4	2	no	—
139	#88324	—	4	4	no	—
140	C₈×C₅₃	—	2	1	no	—
141	C₈×C₅₃	C₂²×C₈	1	2	yes	—
142	#88322	—	2	1	no	—
143	#88322	C₂²×C₈	1	2	yes	—
144	C₂×#1579	—	53	1	no	maximal, Sylow
145	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
146	C₂×D₅₃	C₈×D₄	1	2	yes	—
147	C₂×D₅₃	#441	1	2	yes	—
148	C₂×D₅₃	#444	1	2	yes	—
149	C₄×C₅₃	—	2	4	no	—
150	C₄×C₅₃	—	2	1	no	—
151	C₄×C₅₃	—	4	2	no	—
152	C₄×C₅₃	#441	1	2	yes	—
153	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
154	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
155	Dic₅₃	—	2	4	no	—
156	Dic₅₃	—	2	1	no	—
157	Dic₅₃	—	4	2	no	—
158	Dic₅₃	C₈×D₄	1	2	yes	—
159	Dic₅₃	C₂×#137	1	2	yes	—
160	C₂×C₈×Q₈	—	53	2	no	—
161	C₂×C₄×Q₁₆	—	53	1	no	—
162	#1579	—	53	2	no	—
163	#1579	—	53	4	no	—
164	#1579	—	53	2	no	—
165	C₂×#334	—	53	1	no	—
166	C₂×Q₈⋊C₈	—	53	2	no	—
167	C₂×#322	—	53	1	no	—
168	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
169	C₂×C₅₃	C₂×#441	1	1	yes	characteristic
170	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
171	D₅₃	#1579	1	4	yes	complemented
172	C₂²×Q₁₆	—	53	1	no	—
173	C₂×C₄×Q₈	—	53	2	no	—
174	C₈×Q₈	—	53	4	no	—
175	C₈×Q₈	—	53	4	no	—
176	C₈×Q₈	—	106	4	no	—
177	C₄×Q₁₆	—	53	1	no	—
178	C₄×Q₁₆	—	53	2	no	—
179	C₄×Q₁₆	—	53	1	no	—
180	C₄×Q₁₆	—	53	2	no	—
181	C₄×Q₁₆	—	53	1	no	—
182	C₄×Q₁₆	#6157	1	1	yes	characteristic
183	C₂×#107	—	53	1	no	—
184	C₂×C₄⋊C₈	—	53	2	no	—
185	C₂×C₄⋊C₈	—	106	2	no	—
186	C₂×#103	—	53	2	no	—
187	C₂×C₄×C₈	—	53	1	no	—
188	C₂×C₄×C₈	—	53	1	no	—
189	C₂×C₄×C₈	—	53	1	no	—
190	C₂×C₄×C₈	—	106	2	no	—
191	#334	—	53	2	no	—
192	#334	—	53	2	no	—
193	Q₈⋊C₈	—	53	4	no	—
194	Q₈⋊C₈	—	53	4	no	—
195	#322	—	53	2	no	—
196	#322	—	53	2	no	—
197	C₅₃	C₂×#1579	1	1	yes	characteristic, complemented, Sylow
198	C₂²×Q₈	—	53	2	no	—
199	C₂×Q₁₆	—	53	2	no	—
200	C₂×Q₁₆	—	53	8	no	—
201	C₂×Q₁₆	—	53	1	no	—
202	C₂×Q₁₆	C₂×#6157	1	1	yes	characteristic
203	C₂²×C₈	—	53	1	no	—
204	C₂²×C₈	—	53	2	no	—
205	C₂²×C₈	—	53	1	no	—
206	C₂²×C₈	—	106	1	no	—
207	C₂²×C₈	—	106	2	no	—
208	C₄×Q₈	—	53	2	no	—
209	C₄×Q₈	—	53	2	no	—
210	C₄×Q₈	—	53	2	no	—
211	C₄×Q₈	—	106	2	no	—
212	C₄×Q₈	—	106	2	no	—
213	C₄×Q₈	C₂×#6157	1	2	yes	—
214	C₂×C₄⋊C₄	—	53	2	no	—
215	C₂×C₄⋊C₄	—	106	2	no	—
216	C₂×C₄²	—	53	1	no	—
217	C₂×C₄²	—	106	2	no	—
218	#107	—	53	1	no	—
219	#107	—	53	2	no	—
220	#107	C₂×#6157	1	1	yes	characteristic
221	C₄⋊C₈	—	53	4	no	—
222	C₄⋊C₈	—	106	2	no	—
223	C₄⋊C₈	—	106	4	no	—
224	C₄⋊C₈	—	106	4	no	—
225	#103	—	53	2	no	—
226	#103	—	53	2	no	—
227	#103	—	53	2	no	—
228	#103	C₂×#6157	1	2	yes	—
229	C₄×C₈	—	53	2	no	—
230	C₄×C₈	—	53	2	no	—
231	C₄×C₈	—	53	1	no	—
232	C₄×C₈	—	53	1	no	—
233	C₄×C₈	—	53	2	no	—
234	C₄×C₈	—	53	1	no	—
235	C₄×C₈	—	106	1	no	—
236	C₄×C₈	—	106	4	no	—
237	C₄×C₈	—	106	4	no	—
238	C₄×C₈	C₂×#6157	1	1	yes	characteristic
239	C₂×Q₈	—	53	8	no	—
240	C₂×Q₈	—	53	2	no	—
241	C₂×Q₈	—	106	2	no	—
242	C₂×Q₈	—	106	4	no	—
243	C₂×Q₈	C₂²×#6157	1	2	yes	—
244	C₂²×C₄	—	53	1	no	—
245	C₂²×C₄	—	53	1	no	—
246	C₂²×C₄	—	53	1	no	—
247	C₂²×C₄	—	106	2	no	—
248	C₂²×C₄	—	106	2	no	—
249	Q₁₆	—	53	8	no	—
250	Q₁₆	—	53	4	no	—
251	Q₁₆	D₅₃⋊C₈	1	4	yes	complemented
252	C₂×C₈	—	53	1	no	—
253	C₂×C₈	—	53	4	no	—
254	C₂×C₈	—	53	4	no	—
255	C₂×C₈	—	53	1	no	—
256	C₂×C₈	—	106	2	no	—
257	C₂×C₈	—	106	4	no	—
258	C₂×C₈	—	106	2	no	—
259	C₂×C₈	—	106	4	no	—
260	C₂×C₈	—	212	4	no	—
261	C₂×C₈	—	212	2	no	—
262	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
263	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	53	2	no	—
266	C₄⋊C₄	—	106	2	no	—
267	C₄⋊C₄	—	106	2	no	—
268	C₄⋊C₄	—	106	2	no	—
269	C₄⋊C₄	—	106	2	no	—
270	C₄⋊C₄	C₂²×#6157	1	2	yes	—
271	C₄²	—	2	2	no	—
272	C₄²	—	53	1	no	—
273	C₄²	—	53	1	no	—
274	C₄²	—	53	1	no	—
275	C₄²	—	106	2	no	—
276	C₄²	—	106	2	no	—
277	C₄²	—	106	2	no	—
278	C₄²	C₂²×#6157	1	1	yes	characteristic
279	C₂³	—	53	1	no	—
280	Q₈	—	2	2	no	—
281	Q₈	—	53	4	no	—
282	Q₈	—	106	8	no	—
283	Q₈	—	106	2	no	—
284	Q₈	C₂×D₅₃⋊C₈	1	4	yes	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	53	1	no	—
288	C₂×C₄	—	53	1	no	—
289	C₂×C₄	—	53	4	no	—
290	C₂×C₄	—	53	2	no	—
291	C₂×C₄	—	53	2	no	—
292	C₂×C₄	—	53	1	no	—
293	C₂×C₄	—	106	2	no	—
294	C₂×C₄	—	106	8	no	—
295	C₂×C₄	—	106	2	no	—
296	C₂×C₄	—	106	2	no	—
297	C₂×C₄	—	212	4	no	—
298	C₂×C₄	D₄×#6157	1	1	yes	characteristic
299	C₂×C₄	D₄×#6157	1	1	yes	characteristic
300	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	53	2	no	—
303	C₈	—	106	4	no	—
304	C₈	—	106	1	no	—
305	C₈	—	212	4	no	—
306	C₈	—	212	2	no	—
307	C₈	C₂×D₅₃⋊C₈	1	2	yes	—
308	C₂²	—	53	2	no	—
309	C₂²	—	53	2	no	—
310	C₂²	—	53	2	no	—
311	C₂²	C₂×D₄×#6157	1	1	yes	characteristic
312	C₄	—	2	4	no	—
313	C₄	—	2	1	no	—
314	C₄	—	4	2	no	—
315	C₄	—	53	2	no	—
316	C₄	—	53	2	no	—
317	C₄	—	106	4	no	—
318	C₄	—	106	1	no	—
319	C₄	—	212	2	no	—
320	C₄	#16811663	1	2	yes	—
321	C₄	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
322	C₄	D₅₃⋊#131	1	1	yes	characteristic
323	C₂	—	53	4	no	—
324	C₂	D₄×D₅₃⋊C₈	1	1	yes	characteristic
325	C₂	C₂×#16811663	1	1	yes	characteristic
326	C₂	D₅₃⋊#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6157
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₄×Q₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂²×D₅₃	C₄×D₄
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	C₄×Q₁₆	#6157
hypocenter	C₅₃	C₂×#1579
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #20290310	C₂	semidirect
3. #20290309	C₂	semidirect
4. #20290308	C₂	semidirect
50. Q₁₆×D₅₃	C₈	semidirect
91. Q₁₆×C₅₃	C₂×C₈	semidirect
98. #11719925	C₂×C₈	semidirect
112. Dic₂₁₂	C₂×C₈	semidirect
171. D₅₃	#1579	semidirect
197. C₅₃	C₂×#1579	semidirect
251. Q₁₆	D₅₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₅₃⋊C₈	2
#20290310	2
#20290309	2
#20290308	2
D₅₃⋊#334	2
D₅₃⋊(Q₈⋊C₈)	2
#20289909	2
C₄×Q₁₆×D₅₃	2
C₂×#1579	53

Subgroups — D53⋊#1579 #28316370

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃⋊#1579 #28316370