Subgroups — #27283264

Order 13568.

498 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275635	C₂	1	2	yes	maximal, complemented
2	C₄×#13133149	C₂	1	1	yes	maximal, characteristic, complemented
3	#20273210	C₂	1	2	yes	maximal, complemented
4	#20273004	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#3169	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	1	yes	characteristic
7	C₂×#13133149	—	2	1	no	—
8	C₂×#13133144	—	2	4	no	—
9	C₄×#11719917	C₂²	1	1	yes	characteristic
10	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
11	#16810386	—	2	1	no	—
12	#16810357	—	2	4	no	—
13	#16810345	—	2	1	no	—
14	#16810344	—	2	1	no	—
15	C₅₃×#532	C₂²	1	2	yes	—
16	C₅₃×#500	C₄	1	1	yes	characteristic, complemented
17	C₅₃×#493	C₄	1	1	yes	characteristic, complemented
18	C₅₃×#486	C₄	1	2	yes	complemented
19	C₅₃×#466	—	2	1	no	—
20	C₅₃×#466	—	2	1	no	—
21	C₄×D₈×C₅₃	—	2	1	no	—
22	C₄×D₈×C₅₃	—	2	1	no	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#11719917	D₄	1	2	yes	—
25	C₂×C₄⋊Dic₅₃	—	2	2	no	—
26	C₂×C₄×Dic₅₃	—	2	2	no	—
27	#13133149	—	2	4	no	—
28	C₄⋊#88322	—	2	1	no	—
29	#13133144	—	4	8	no	—
30	C₂²⋊#88322	D₄	1	2	yes	—
31	#13133128	—	2	1	no	—
32	C₄×#88322	—	2	1	no	—
33	C₄×#88322	—	2	1	no	—
34	#13133124	—	2	2	no	—
35	C₂×#41×C₅₃	—	2	2	no	—
36	C₂×D₈×C₅₃	—	2	1	no	—
37	C₂×D₈×C₅₃	—	2	1	no	—
38	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
41	#127×C₅₃	C₂×C₄	1	2	yes	—
42	#124×C₅₃	—	2	2	no	—
43	#121×C₅₃	—	2	2	no	—
44	#121×C₅₃	—	2	2	no	—
45	C₄×D₄×C₅₃	—	2	2	no	—
46	C₄×D₄×C₅₃	—	2	2	no	—
47	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
48	#107×C₅₃	—	2	1	no	—
49	#107×C₅₃	—	2	1	no	—
50	#102×C₅₃	—	2	2	no	—
51	#102×C₅₃	—	2	2	no	—
52	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
55	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
56	#41×C₅₃	—	4	4	no	—
57	#41×C₅₃	—	4	4	no	—
58	C₂×Q₈×C₅₃	—	2	2	no	—
59	C₂×D₄×C₅₃	—	2	2	no	—
60	C₂×D₄×C₅₃	—	2	2	no	—
61	C₂×D₄×C₅₃	—	2	2	no	—
62	C₂×D₄×C₅₃	—	4	2	no	—
63	C₂²×C₄×C₅₃	—	4	2	no	—
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×Dic₅₃	—	4	2	no	—
68	D₈×C₅₃	—	4	2	no	—
69	D₈×C₅₃	—	4	2	no	—
70	C₂×C₈×C₅₃	—	2	2	no	—
71	C₂×C₈×C₅₃	—	4	1	no	—
72	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₄⋊C₄×C₅₃	—	2	2	no	—
77	C₄⋊C₄×C₅₃	—	2	2	no	—
78	C₂²⋊C₄×C₅₃	—	4	2	no	—
79	C₂²⋊C₄×C₅₃	—	4	2	no	—
80	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₅₃	#41	1	1	yes	characteristic
83	C₄²×C₅₃	#41	1	1	yes	characteristic
84	C₄⋊Dic₅₃	—	2	4	no	—
85	C₄⋊Dic₅₃	—	4	2	no	—
86	C₄×Dic₅₃	—	4	4	no	—
87	#11719917	—	2	4	no	—
88	C₂×#88322	—	2	2	no	—
89	C₂×#88322	—	2	2	no	—
90	C₂³×C₅₃	—	4	2	no	—
91	C₂³×C₅₃	#127	1	1	yes	characteristic
92	Q₈×C₅₃	—	4	4	no	—
93	D₄×C₅₃	—	4	2	no	—
94	D₄×C₅₃	—	4	2	no	—
95	D₄×C₅₃	—	4	4	no	—
96	D₄×C₅₃	—	8	2	no	—
97	D₄×C₅₃	—	8	2	no	—
98	D₄×C₅₃	—	8	2	no	—
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	2	1	no	—
101	C₂×C₄×C₅₃	—	4	1	no	—
102	C₂×C₄×C₅₃	—	4	2	no	—
103	C₂×C₄×C₅₃	—	8	4	no	—
104	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
108	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
110	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
111	C₂×Dic₅₃	—	4	4	no	—
112	C₂×Dic₅₃	—	8	2	no	—
113	C₂×Dic₅₃	—	8	2	no	—
114	C₈×C₅₃	—	2	1	no	—
115	C₈×C₅₃	—	2	1	no	—
116	C₈×C₅₃	—	2	1	no	—
117	C₈×C₅₃	#104	1	2	yes	complemented
118	#88322	—	4	2	no	—
119	#88322	—	4	2	no	—
120	#9721	—	53	1	no	maximal, Sylow
121	C₂²×C₅₃	—	2	1	no	—
122	C₂²×C₅₃	—	2	1	no	—
123	C₂²×C₅₃	—	8	2	no	—
124	C₂²×C₅₃	—	8	2	no	—
125	C₂²×C₅₃	—	8	2	no	—
126	C₂²×C₅₃	#390	1	1	yes	characteristic
127	C₂²×C₅₃	#496	1	2	yes	—
128	C₄×C₅₃	—	2	1	no	—
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	—	2	1	no	—
131	C₄×C₅₃	—	8	2	no	—
132	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
133	C₄×C₅₃	#421	1	1	yes	characteristic
134	C₄×C₅₃	#442	1	2	yes	—
135	C₄×C₅₃	#469	1	2	yes	—
136	Dic₅₃	—	8	4	no	—
137	#3169	D₅₃	1	1	yes	characteristic
138	#1980	—	53	2	no	—
139	C₄×#106	—	53	1	no	—
140	#1502	—	53	2	no	—
141	#1451	—	53	1	no	—
142	C₂×C₅₃	—	2	2	no	—
143	C₂×C₅₃	—	16	2	no	—
144	C₂×C₅₃	#1972	1	1	yes	characteristic
145	C₂×C₅₃	#1976	1	1	yes	characteristic
146	C₂×C₅₃	#1980	1	1	yes	characteristic
147	#532	C₂×D₅₃	1	2	yes	—
148	#500	Dic₅₃	1	1	yes	characteristic, complemented
149	#493	Dic₅₃	1	1	yes	characteristic, complemented
150	#486	Dic₅₃	1	2	yes	complemented
151	#466	—	2	1	no	—
152	#466	—	2	1	no	—
153	C₄×D₈	—	2	1	no	—
154	C₄×D₈	—	2	1	no	—
155	#431	—	53	1	no	—
156	C₂×#106	—	106	1	no	—
157	C₂×#104	—	106	4	no	—
158	C₄×M₁₆	—	53	1	no	—
159	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
160	C₄×C₄⋊C₄	—	53	2	no	—
161	#340	—	106	1	no	—
162	#327	—	106	4	no	—
163	#322	—	106	1	no	—
164	#322	—	106	1	no	—
165	C₅₃	#9721	1	1	yes	characteristic, complemented, Sylow
166	C₂×#41	—	2	2	no	—
167	C₂×D₈	—	2	1	no	—
168	C₂×D₈	—	2	1	no	—
169	C₂×M₁₆	—	53	2	no	—
170	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
171	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
172	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
173	#127	C₂×Dic₅₃	1	2	yes	—
174	#124	—	2	2	no	—
175	#121	—	2	2	no	—
176	#121	—	2	2	no	—
177	C₄×D₄	—	2	2	no	—
178	C₄×D₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	106	2	no	—
180	C₂×C₄²	—	106	2	no	—
181	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
182	#107	—	2	1	no	—
183	#107	—	2	1	no	—
184	#106	—	106	4	no	—
185	C₄⋊C₈	—	106	1	no	—
186	#104	—	212	8	no	—
187	#102	—	2	2	no	—
188	#102	—	2	2	no	—
189	C₂²⋊C₈	—	53	2	no	—
190	#97	—	106	1	no	—
191	C₄×C₈	—	106	1	no	—
192	C₄×C₈	—	106	1	no	—
193	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
194	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
195	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
196	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
197	#95	—	106	2	no	—
198	#41	—	4	4	no	—
199	#41	—	4	4	no	—
200	C₂×Q₈	—	2	2	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	—	212	2	no	—
207	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
208	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
210	D₈	—	4	2	no	—
211	D₈	—	4	2	no	—
212	M₁₆	—	106	4	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	106	2	no	—
216	C₂×C₈	—	106	2	no	—
217	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
218	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
219	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
220	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	106	4	no	—
224	C₄⋊C₄	—	212	2	no	—
225	C₂²⋊C₄	—	4	2	no	—
226	C₂²⋊C₄	—	4	2	no	—
227	C₄²	—	212	4	no	—
228	C₄²	#11719947	1	1	yes	characteristic
229	C₄²	#11719947	1	1	yes	characteristic
230	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
231	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
232	C₂³	—	4	2	no	—
233	C₂³	#13133213	1	1	yes	characteristic
234	Q₈	—	4	4	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	4	4	no	—
238	D₄	—	8	2	no	—
239	D₄	—	8	2	no	—
240	D₄	—	8	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	8	4	no	—
246	C₂×C₄	—	424	2	no	—
247	C₂×C₄	—	424	2	no	—
248	C₂×C₄	—	212	4	no	—
249	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
250	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
251	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
252	C₂×C₄	#13133188	1	1	yes	characteristic
253	C₂×C₄	#13133188	1	1	yes	characteristic
254	C₂×C₄	#13133204	1	1	yes	characteristic
255	C₂×C₄	#13133241	1	4	yes	—
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	212	2	no	—
260	C₈	—	212	2	no	—
261	C₈	#13133144	1	2	yes	complemented
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	8	2	no	—
265	C₂²	—	8	2	no	—
266	C₂²	—	8	2	no	—
267	C₂²	#16810526	1	1	yes	characteristic
268	C₂²	#16811011	1	2	yes	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	8	2	no	—
273	C₄	—	424	4	no	—
274	C₄	C₂×#13133144	1	1	yes	characteristic
275	C₄	#16810640	1	1	yes	characteristic
276	C₄	#16810735	1	2	yes	—
277	C₄	#16810873	1	2	yes	—
278	C₂	—	2	2	no	—
279	C₂	—	16	2	no	—
280	C₂	#20275592	1	1	yes	characteristic
281	C₂	#20275614	1	1	yes	characteristic
282	C₂	#20275635	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133188
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting, prime radical	C₅₃×#3169	C₂
socle	C₂²×C₅₃	#390
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#3169	D₅₃
hypocenter	C₅₃	#9721
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #20275635	C₂	semidirect
2. C₄×#13133149	C₂	semidirect
3. #20273210	C₂	semidirect
4. #20273004	C₂	semidirect
16. C₅₃×#500	C₄	semidirect
17. C₅₃×#493	C₄	semidirect
18. C₅₃×#486	C₄	semidirect
117. C₈×C₅₃	#104	semidirect
148. #500	Dic₅₃	semidirect
149. #493	Dic₅₃	semidirect
150. #486	Dic₅₃	semidirect
165. C₅₃	#9721	semidirect
261. C₈	#13133144	semidirect

Maximal subgroups

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

subgroup	index
#20275635	2
C₄×#13133149	2
#20273210	2
#20273004	2
C₅₃×#3169	2
#9721	53