Subgroups — #27349673

Order 13568.

914 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#431×D₅₃	C₂	1	2	yes	maximal
2	C₂×C₄×#11719912	C₂	1	1	yes	maximal, characteristic
3	#20272999	C₂	1	2	yes	maximal, complemented
4	#20272998	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272994	C₂	1	4	yes	maximal, complemented
6	C₈×#11719912	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11719912	C₄	1	2	yes	—
9	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
10	#16810676	C₂²	1	2	yes	—
11	#16810675	C₄	1	4	yes	—
12	#16810674	C₂²	1	4	yes	—
13	#16810673	C₄	1	4	yes	—
14	C₄⋊C₈×D₅₃	C₄	1	4	yes	—
15	C₂²⋊C₈×D₅₃	C₄	1	4	yes	—
16	C₄×#11719912	C₄	1	4	yes	—
17	C₄×#11719912	C₂²	1	2	yes	—
18	C₄×#11719912	C₂²	1	2	yes	—
19	C₂×#13133127	C₄	1	2	yes	—
20	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
21	C₄×C₈×D₅₃	C₂²	1	4	yes	—
22	#16810345	C₂²	1	4	yes	complemented
23	#16810344	C₂²	1	2	yes	complemented
24	C₈×#88322	C₂²	1	4	yes	complemented
25	C₈×#88322	C₂²	1	2	yes	complemented
26	C₅₃×#431	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#322	C₂²	1	2	yes	complemented
29	C₈²×C₅₃	C₂²	1	2	yes	complemented
30	C₂²×C₄×D₅₃	C₂×C₄	1	2	yes	—
31	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×#11719912	C₈	1	16	yes	complemented
33	C₂×#11719912	C₂×C₄	1	4	yes	—
34	C₂×#11719912	C₂×C₄	1	4	yes	—
35	C₂²×#88322	C₂×C₄	1	2	yes	—
36	C₂×C₈×D₅₃	C₂×C₄	1	8	yes	—
37	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
39	C₄²×D₅₃	C₂×C₄	1	2	yes	—
40	C₄²×D₅₃	C₂×C₄	1	2	yes	—
41	C₄²×D₅₃	C₂×C₄	1	2	yes	—
42	C₄²×D₅₃	C₂³	1	2	yes	—
43	C₄⋊#88322	C₂×C₄	1	4	yes	—
44	#13133146	C₂×C₄	1	8	yes	—
45	C₂²⋊#88322	C₂×C₄	1	4	yes	—
46	#13133129	C₂×C₄	1	8	yes	—
47	#13133127	C₂×C₄	1	4	yes	—
48	#13133127	C₂×C₄	1	4	yes	—
49	C₄×#88322	C₂×C₄	1	2	yes	—
50	C₄×#88322	C₂³	1	4	yes	—
51	C₄×#88322	C₂³	1	2	yes	—
52	C₈×Dic₅₃	C₂×C₄	1	8	yes	—
53	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
56	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
57	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₅₃	C₂³	1	4	yes	—
59	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₂³×D₅₃	C₄²	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×Dic₅₃	C₄²	1	2	yes	—
65	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
66	C₂×C₄×D₅₃	—	4	4	no	—
67	C₂×C₄×D₅₃	C₄²	1	2	yes	—
68	C₂×C₄×D₅₃	C₄²	1	2	yes	—
69	C₂×C₄×D₅₃	C₄²	1	2	yes	—
70	C₂×C₄×D₅₃	C₂×C₈	1	8	yes	—
71	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
72	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
73	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₅₃	C₂×C₈	1	8	yes	complemented
75	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
76	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
78	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
80	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
81	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
82	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
83	C₄×Dic₅₃	C₄²	1	2	yes	—
84	C₄×Dic₅₃	C₄²	1	2	yes	—
85	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
86	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
87	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
88	C₂×#88322	C₂×C₈	1	8	yes	complemented
89	C₂×#88322	C₂²×C₄	1	8	yes	—
90	C₂×#88322	C₂²×C₄	1	4	yes	—
91	#11719912	C₂×C₈	1	16	yes	complemented
92	#11719912	C₂×C₈	1	16	yes	complemented
93	C₈×D₅₃	—	4	8	no	—
94	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
95	C₂²×D₅₃	—	4	2	no	—
96	C₂²×D₅₃	C₄×C₈	1	4	yes	—
97	C₂²×D₅₃	C₂×C₄²	1	2	yes	—
98	C₂×C₄×C₅₃	—	2	4	no	—
99	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
100	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
102	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
104	C₂×Dic₅₃	—	4	2	no	—
105	C₂×Dic₅₃	C₄×C₈	1	4	yes	—
106	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
107	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
108	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
109	C₄×D₅₃	—	2	4	no	—
110	C₄×D₅₃	—	2	4	no	—
111	C₄×D₅₃	—	4	2	no	—
112	C₄×D₅₃	—	4	2	no	—
113	C₄×D₅₃	C₄×C₈	1	8	yes	—
114	C₄×D₅₃	C₂²×C₈	1	8	yes	—
115	C₈×C₅₃	—	2	8	no	—
116	C₈×C₅₃	C₂²×C₈	1	4	yes	—
117	C₈×C₅₃	C₂²×C₈	1	4	yes	—
118	#88322	—	2	8	no	—
119	#88322	C₂²×C₈	1	8	yes	—
120	#17314	—	53	1	no	maximal, Sylow
121	C₂²×C₅₃	—	2	2	no	—
122	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
123	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
124	C₂×D₅₃	—	2	2	no	—
125	C₂×D₅₃	—	2	2	no	—
126	C₂×D₅₃	—	4	4	no	—
127	C₂×D₅₃	C₂×C₄×C₈	1	4	yes	—
128	C₄×C₅₃	—	2	2	no	—
129	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
130	C₄×C₅₃	C₂³×C₈	1	2	yes	—
131	C₄×C₅₃	#568	1	4	yes	—
132	Dic₅₃	—	2	2	no	—
133	Dic₅₃	—	2	2	no	—
134	Dic₅₃	C₂×C₄×C₈	1	4	yes	—
135	C₂×#431	—	53	2	no	—
136	C₂×C₄×M₁₆	—	53	1	no	—
137	#1450	—	53	2	no	—
138	#1450	—	53	1	no	—
139	#1450	D₅₃	1	1	yes	characteristic, complemented
140	C₈×M₁₆	—	53	4	no	—
141	C₈×M₁₆	—	53	4	no	—
142	C₂×C₅₃	—	2	2	no	—
143	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
144	C₂×C₅₃	#2873	1	2	yes	—
145	D₅₃	—	2	4	no	—
146	C₂²×M₁₆	—	53	2	no	—
147	C₂²×C₄²	—	53	1	no	—
148	#431	—	53	2	no	—
149	#431	—	53	4	no	—
150	#431	—	53	4	no	—
151	#431	—	53	4	no	—
152	#431	C₂×D₅₃	1	2	yes	—
153	C₂×C₄⋊C₈	—	53	4	no	—
154	C₂×C₂²⋊C₈	—	53	4	no	—
155	C₄×M₁₆	—	53	4	no	—
156	C₄×M₁₆	—	53	2	no	—
157	C₄×M₁₆	—	53	2	no	—
158	C₂×#97	—	53	2	no	—
159	C₂×C₄×C₈	—	53	1	no	—
160	C₂×C₄×C₈	—	53	4	no	—
161	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
162	#322	—	53	2	no	—
163	#322	—	53	4	no	—
164	#322	C₂×D₅₃	1	2	yes	complemented
165	C₈²	—	53	4	no	—
166	C₈²	—	53	2	no	—
167	C₈²	C₂×D₅₃	1	2	yes	complemented
168	C₅₃	#17314	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₄	—	53	1	no	—
170	C₂³×C₄	—	53	2	no	—
171	C₂×M₁₆	—	53	4	no	—
172	C₂×M₁₆	—	53	4	no	—
173	C₂×M₁₆	—	53	16	no	—
174	C₂²×C₈	—	53	2	no	—
175	C₂²×C₈	—	53	8	no	—
176	C₂²×C₈	C₄×D₅₃	1	2	yes	—
177	C₂×C₄²	—	53	1	no	—
178	C₂×C₄²	—	53	2	no	—
179	C₂×C₄²	—	53	2	no	—
180	C₂×C₄²	—	53	2	no	—
181	C₂×C₄²	—	53	2	no	—
182	C₂×C₄²	—	53	2	no	—
183	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
184	C₄⋊C₈	—	53	8	no	—
185	C₄⋊C₈	—	53	4	no	—
186	C₄⋊C₈	C₄×D₅₃	1	4	yes	—
187	C₂²⋊C₈	—	53	4	no	—
188	C₂²⋊C₈	—	53	8	no	—
189	C₂²⋊C₈	C₄×D₅₃	1	4	yes	—
190	#97	—	53	4	no	—
191	#97	—	53	4	no	—
192	C₄×C₈	—	53	8	no	—
193	C₄×C₈	—	53	2	no	—
194	C₄×C₈	—	53	2	no	—
195	C₄×C₈	—	53	4	no	—
196	C₄×C₈	C₄×D₅₃	1	2	yes	—
197	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
198	C₄×C₈	C₂²×D₅₃	1	4	yes	—
199	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
200	C₂⁴	—	53	1	no	—
201	C₂²×C₄	—	53	1	no	—
202	C₂²×C₄	—	53	2	no	—
203	C₂²×C₄	—	53	8	no	—
204	C₂²×C₄	—	53	2	no	—
205	C₂²×C₄	—	53	2	no	—
206	C₂²×C₄	—	53	2	no	—
207	C₂²×C₄	—	53	2	no	—
208	C₂²×C₄	—	53	2	no	—
209	C₂²×C₄	—	53	2	no	—
210	C₂²×C₄	—	212	4	no	—
211	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
212	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
213	M₁₆	—	53	16	no	—
214	M₁₆	—	53	16	no	—
215	C₂×C₈	—	53	8	no	—
216	C₂×C₈	—	53	4	no	—
217	C₂×C₈	—	53	8	no	—
218	C₂×C₈	—	212	8	no	—
219	C₂×C₈	C₈×D₅₃	1	8	yes	complemented
220	C₂×C₈	C₂×C₄×D₅₃	1	8	yes	—
221	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
222	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
223	C₄²	—	53	2	no	—
224	C₄²	—	53	2	no	—
225	C₄²	—	53	1	no	—
226	C₄²	—	53	1	no	—
227	C₄²	—	53	2	no	—
228	C₄²	—	53	2	no	—
229	C₄²	—	53	2	no	—
230	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
231	C₄²	C₂×C₄×D₅₃	1	2	yes	—
232	C₄²	C₂³×D₅₃	1	1	yes	characteristic
233	C₂³	—	53	2	no	—
234	C₂³	—	53	4	no	—
235	C₂³	—	212	2	no	—
236	C₂³	C₄²×D₅₃	1	1	yes	characteristic
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	53	2	no	—
239	C₂×C₄	—	53	4	no	—
240	C₂×C₄	—	53	8	no	—
241	C₂×C₄	—	53	8	no	—
242	C₂×C₄	—	53	2	no	—
243	C₂×C₄	—	53	2	no	—
244	C₂×C₄	—	106	4	no	—
245	C₂×C₄	—	106	4	no	—
246	C₂×C₄	—	212	2	no	—
247	C₂×C₄	—	212	2	no	—
248	C₂×C₄	—	212	2	no	—
249	C₂×C₄	C₄²×D₅₃	1	2	yes	—
250	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
251	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
252	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
253	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
254	C₈	—	2	8	no	—
255	C₈	—	53	8	no	—
256	C₈	—	106	8	no	—
257	C₈	C₂×C₈×D₅₃	1	4	yes	—
258	C₈	C₂×C₈×D₅₃	1	4	yes	—
259	C₂²	—	2	2	no	—
260	C₂²	—	53	4	no	—
261	C₂²	—	106	2	no	—
262	C₂²	—	106	2	no	—
263	C₂²	—	212	4	no	—
264	C₂²	C₄×C₈×D₅₃	1	2	yes	—
265	C₂²	C₂×C₄²×D₅₃	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	53	4	no	—
268	C₄	—	106	2	no	—
269	C₄	—	106	2	no	—
270	C₄	C₄×C₈×D₅₃	1	2	yes	—
271	C₄	C₂²×C₈×D₅₃	1	2	yes	—
272	C₄	#16811388	1	4	yes	—
273	C₂	—	2	2	no	—
274	C₂	—	106	4	no	—
275	C₂	C₂×C₄×C₈×D₅₃	1	1	yes	characteristic
276	C₂	#20283131	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₅₃
commutator	C₂×C₅₃	C₂²×C₄×C₈
Fitting	C₅₃×#1450	C₂
socle	C₂²×C₅₃	C₂²×C₄²
prime radical	C₂³×D₅₃	C₄²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1450	D₅₃
hypocenter	C₅₃	#17314
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ #1450

Complements

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

N	H ≅ G/N	type
3. #20272999	C₂	semidirect
4. #20272998	C₂	semidirect
5. #20272994	C₂	semidirect
6. C₈×#11719912	C₂	semidirect
7. C₅₃×#1450	C₂	semidirect
22. #16810345	C₂²	semidirect
23. #16810344	C₂²	semidirect
24. C₈×#88322	C₂²	semidirect
25. C₈×#88322	C₂²	semidirect
28. C₅₃×#322	C₂²	semidirect
29. C₈²×C₅₃	C₂²	semidirect
32. C₂×#11719912	C₈	semidirect
74. C₂×C₈×C₅₃	C₂×C₈	semidirect
88. C₂×#88322	C₂×C₈	semidirect
91. #11719912	C₂×C₈	semidirect
92. #11719912	C₂×C₈	semidirect
139. #1450	D₅₃	semidirect
164. #322	C₂×D₅₃	semidirect
167. C₈²	C₂×D₅₃	semidirect
168. C₅₃	#17314	semidirect
219. C₂×C₈	C₈×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#431×D₅₃	2
C₂×C₄×#11719912	2
#20272999	2
#20272998	2
#20272994	2
C₈×#11719912	2
C₅₃×#1450	2
#17314	53