Subgroups — #27356129

Order 13568.

644 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285294	C₂	1	1	yes	maximal, characteristic
2	#20283781	C₂	1	2	yes	maximal, complemented
3	#20273628	C₂	1	2	yes	maximal, complemented
4	#20273543	C₂	1	4	yes	maximal, complemented
5	#20273538	C₂	1	2	yes	maximal, complemented
6	#20273131	C₂	1	2	yes	maximal, complemented
7	#20273004	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
9	#16811173	C₂²	1	2	yes	—
10	#16811028	C₄	1	1	yes	characteristic
11	#16811000	C₄	1	1	yes	characteristic
12	#16810970	C₄	1	2	yes	—
13	#16810859	C₄	1	2	yes	—
14	#16810859	C₄	1	2	yes	—
15	C₄×D₄₂₄	C₂²	1	2	yes	—
16	C₄×D₄₂₄	C₂²	1	2	yes	—
17	#16810702	—	2	2	no	—
18	#16810700	C₂²	1	4	yes	—
19	C₈×D₂₁₂	—	2	2	no	—
20	#16810688	C₂²	1	4	yes	—
21	#16810676	C₂²	1	1	yes	characteristic
22	#16810588	—	2	4	no	—
23	C₄×#11719917	C₂²	1	1	yes	characteristic
24	#16810350	C₂²	1	4	yes	—
25	#16810350	C₂²	1	4	yes	—
26	#16810345	C₂²	1	2	yes	—
27	#16810344	C₂²	1	2	yes	—
28	C₅₃×#431	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#335	C₂²	1	2	yes	—
31	C₅₃×#334	C₂²	1	1	yes	characteristic
32	C₅₃×#334	C₂²	1	1	yes	characteristic
33	C₂×#11719945	—	2	2	no	—
34	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
35	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
36	C₂×#11719917	—	2	1	no	—
37	C₂×#11719912	—	2	4	no	—
38	C₂×C₈×D₅₃	—	2	4	no	—
39	#13133217	C₂×C₄	1	2	yes	—
40	#13133214	C₂×C₄	1	2	yes	—
41	#13133205	—	2	2	no	—
42	#13133191	—	2	2	no	—
43	#13133191	C₂×C₄	1	4	yes	—
44	C₄×D₂₁₂	—	2	2	no	—
45	C₄×D₂₁₂	C₂³	1	4	yes	—
46	#13133151	C₂×C₄	1	2	yes	—
47	#13133151	C₂×C₄	1	2	yes	—
48	C₄⋊#88322	D₄	1	2	yes	—
49	#13133139	C₂×C₄	1	4	yes	—
50	#13133139	C₂×C₄	1	4	yes	—
51	C₂²⋊#88322	D₄	1	2	yes	—
52	#13133129	—	2	4	no	—
53	#13133129	—	2	4	no	—
54	#13133128	—	2	1	no	—
55	C₄×#88322	C₂³	1	2	yes	—
56	C₄×#88322	C₂³	1	2	yes	—
57	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
61	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
62	C₂²⋊C₈×C₅₃	—	2	2	no	—
63	C₄×C₈×C₅₃	—	2	2	no	—
64	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
71	C₂×C₂²⋊D₅₃	—	2	4	no	—
72	#11719945	—	8	2	no	—
73	#11719945	—	8	2	no	—
74	C₂×D₂₁₂	—	2	2	no	—
75	C₂×D₂₁₂	—	2	2	no	—
76	C₂×D₂₁₂	C₂²×C₄	1	4	yes	—
77	C₂×C₄×D₅₃	—	2	4	no	—
78	C₂×Dic₁₀₆	—	2	2	no	—
79	C₂×C₈×C₅₃	—	2	2	no	—
80	C₂×C₈×C₅₃	—	2	2	no	—
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	—	4	1	no	—
83	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₅₃	#41	1	1	yes	characteristic
90	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
91	#11719921	—	2	4	no	—
92	#11719921	—	2	4	no	—
93	C₄⋊Dic₅₃	—	2	2	no	—
94	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
95	#11719917	—	4	2	no	—
96	C₂×#88322	—	2	2	no	—
97	C₂×#88322	C₂×D₄	1	4	yes	—
98	D₄₂₄	—	2	4	no	—
99	D₄₂₄	—	4	2	no	—
100	#11719912	—	8	4	no	—
101	C₈×D₅₃	—	8	4	no	—
102	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
103	C₂²×D₅₃	—	2	4	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	2	1	no	—
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	—	2	2	no	—
108	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
114	C₂²⋊D₅₃	—	8	4	no	—
115	C₂×Dic₅₃	—	2	4	no	—
116	D₂₁₂	—	2	4	no	—
117	D₂₁₂	—	2	4	no	—
118	D₂₁₂	—	4	4	no	—
119	D₂₁₂	—	8	2	no	—
120	C₄×D₅₃	—	4	4	no	—
121	C₄×D₅₃	—	4	4	no	—
122	Dic₁₀₆	—	8	2	no	—
123	C₈×C₅₃	—	2	1	no	—
124	C₈×C₅₃	—	2	1	no	—
125	C₈×C₅₃	—	2	1	no	—
126	C₈×C₅₃	—	4	2	no	—
127	C₈×C₅₃	—	4	2	no	—
128	C₈×C₅₃	#131	1	2	yes	—
129	#88322	—	2	4	no	—
130	#88322	—	4	2	no	—
131	#18091	—	53	1	no	maximal, Sylow
132	C₂²×C₅₃	—	2	1	no	—
133	C₂²×C₅₃	—	2	1	no	—
134	C₂²×C₅₃	—	2	1	no	—
135	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂×D₅₃	—	4	4	no	—
137	C₂×D₅₃	—	4	4	no	—
138	C₂×D₅₃	—	4	4	no	—
139	C₄×C₅₃	—	2	1	no	—
140	C₄×C₅₃	—	2	1	no	—
141	C₄×C₅₃	—	2	1	no	—
142	C₄×C₅₃	#444	1	2	yes	—
143	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
144	C₄×C₅₃	#568	1	1	yes	characteristic
145	C₄×C₅₃	#576	1	2	yes	—
146	Dic₅₃	—	4	4	no	—
147	#3169	—	53	1	no	—
148	#2979	—	53	2	no	—
149	#1594	—	53	2	no	—
150	#1581	—	53	4	no	—
151	#1580	—	53	2	no	—
152	#1564	D₅₃	1	1	yes	characteristic, complemented
153	#1484	—	53	2	no	—
154	#1451	—	53	1	no	—
155	C₂×C₅₃	—	4	1	no	—
156	C₂×C₅₃	#2935	1	1	yes	characteristic
157	C₂×C₅₃	#2948	1	1	yes	characteristic
158	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
159	D₅₃	—	8	4	no	—
160	#532	—	53	2	no	—
161	#500	—	53	1	no	—
162	#493	—	53	1	no	—
163	#486	—	53	2	no	—
164	#466	—	53	2	no	—
165	#466	—	53	2	no	—
166	C₄×D₈	—	53	2	no	—
167	C₄×D₈	—	53	2	no	—
168	#436	—	53	4	no	—
169	#436	—	106	2	no	—
170	C₈×D₄	—	53	4	no	—
171	C₈×D₄	—	106	2	no	—
172	#431	—	53	1	no	—
173	#431	C₂×D₅₃	1	2	yes	—
174	#408	—	106	4	no	—
175	C₄×M₁₆	—	53	1	no	—
176	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
177	#335	C₂×D₅₃	1	2	yes	—
178	#334	C₂×D₅₃	1	1	yes	characteristic
179	#334	C₂×D₅₃	1	1	yes	characteristic
180	D₄⋊C₈	—	53	4	no	—
181	D₄⋊C₈	—	53	4	no	—
182	#322	—	53	2	no	—
183	#322	—	53	2	no	—
184	C₅₃	#18091	1	1	yes	characteristic, complemented, Sylow
185	C₂×#41	—	106	2	no	—
186	C₂×D₈	—	53	2	no	—
187	C₂×D₈	—	53	2	no	—
188	C₂×M₁₆	—	106	4	no	—
189	C₂×M₁₆	—	106	1	no	—
190	C₂²×C₈	—	106	4	no	—
191	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
192	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
193	C₄⋊Q₈	—	53	2	no	—
194	#127	—	53	2	no	—
195	#124	—	106	2	no	—
196	#121	—	53	4	no	—
197	#121	—	106	2	no	—
198	C₄×D₄	—	53	4	no	—
199	C₄×D₄	—	106	2	no	—
200	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
201	#107	—	53	2	no	—
202	#107	—	53	2	no	—
203	C₄⋊C₈	—	53	2	no	—
204	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
205	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
206	#102	—	53	4	no	—
207	#102	—	53	4	no	—
208	C₂²⋊C₈	—	2	2	no	—
209	C₂²⋊C₈	—	53	2	no	—
210	C₂²⋊C₈	—	106	4	no	—
211	C₂²⋊C₈	—	106	4	no	—
212	#97	—	106	1	no	—
213	C₄×C₈	—	2	2	no	—
214	C₄×C₈	—	53	2	no	—
215	C₄×C₈	—	53	2	no	—
216	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
217	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
218	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
220	#41	—	424	2	no	—
221	#41	—	424	2	no	—
222	C₂×Q₈	—	106	2	no	—
223	C₂×D₄	—	53	4	no	—
224	C₂×D₄	—	106	4	no	—
225	C₂×D₄	—	106	2	no	—
226	C₂×D₄	—	106	2	no	—
227	C₂²×C₄	—	106	4	no	—
228	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
229	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
230	C₂²×C₄	#11719949	1	1	yes	characteristic
231	D₈	—	106	4	no	—
232	D₈	—	212	2	no	—
233	M₁₆	—	424	4	no	—
234	M₁₆	—	212	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	4	1	no	—
239	C₂×C₈	—	53	4	no	—
240	C₂×C₈	—	106	2	no	—
241	C₂×C₈	—	424	4	no	—
242	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
243	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
244	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
245	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
246	C₄⋊C₄	—	53	4	no	—
247	C₄⋊C₄	—	106	2	no	—
248	C₂²⋊C₄	—	106	4	no	—
249	C₂²⋊C₄	—	106	4	no	—
250	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
251	C₄²	D₄×D₅₃	1	1	yes	characteristic
252	C₄²	#11719949	1	1	yes	characteristic
253	C₄²	C₂³×D₅₃	1	1	yes	characteristic
254	C₂³	—	106	4	no	—
255	C₂³	#13133179	1	1	yes	characteristic
256	Q₈	—	424	2	no	—
257	D₄	—	106	4	no	—
258	D₄	—	106	4	no	—
259	D₄	—	424	2	no	—
260	D₄	—	424	4	no	—
261	D₄	—	212	4	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	106	4	no	—
267	C₂×C₄	—	212	4	no	—
268	C₂×C₄	—	212	4	no	—
269	C₂×C₄	#13133179	1	1	yes	characteristic
270	C₂×C₄	#13133179	1	1	yes	characteristic
271	C₂×C₄	#13133179	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
273	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
274	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	4	2	no	—
279	C₈	—	4	2	no	—
280	C₈	—	106	4	no	—
281	C₈	—	212	2	no	—
282	C₈	#13133223	1	2	yes	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	212	4	no	—
287	C₂²	—	212	4	no	—
288	C₂²	—	212	4	no	—
289	C₂²	C₂×#13133179	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	212	4	no	—
294	C₄	#16810744	1	2	yes	—
295	C₄	C₂×#13133223	1	1	yes	characteristic
296	C₄	#16811389	1	1	yes	characteristic
297	C₄	#16811425	1	2	yes	—
298	C₂	—	4	1	no	—
299	C₂	—	424	4	no	—
300	C₂	#20283458	1	1	yes	characteristic
301	C₂	#20283566	1	1	yes	characteristic
302	C₂	C₂×#16810744	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1564	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#20285294	C₂
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1564	D₅₃
hypocenter	C₅₃	#18091
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
2. #20283781	C₂	semidirect
3. #20273628	C₂	semidirect
4. #20273543	C₂	semidirect
5. #20273538	C₂	semidirect
6. #20273131	C₂	semidirect
7. #20273004	C₂	semidirect
8. C₅₃×#1564	C₂	semidirect
152. #1564	D₅₃	semidirect
184. C₅₃	#18091	semidirect

Maximal subgroups

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

subgroup	index
#20285294	2
#20283781	2
#20273628	2
#20273543	2
#20273538	2
#20273131	2
#20273004	2
C₅₃×#1564	2
#18091	53