Subgroups — #27283339

Order 13568.

654 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275643	C₂	1	2	yes	maximal, complemented
2	C₈×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16810376	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16810367	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#493	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
8	C₂×#13133138	—	2	4	no	—
9	C₂×C₈×Dic₅₃	—	2	1	no	—
10	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
11	#16810377	—	2	1	no	—
12	#16810376	—	2	2	no	—
13	#16810367	—	2	4	no	—
14	C₂²×D₈×C₅₃	—	2	1	no	—
15	C₂²×D₈×C₅₃	—	2	1	no	—
16	C₂×#127×C₅₃	C₂²	1	2	yes	—
17	C₅₃×#493	—	2	2	no	—
18	C₅₃×#493	C₄	1	4	yes	complemented
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×#88322	—	2	1	no	—
21	C₂²×#88322	—	2	2	no	—
22	C₂×C₄⋊Dic₅₃	D₄	1	2	yes	—
23	C₂×C₄×Dic₅₃	—	2	1	no	—
24	C₄⋊#88322	—	2	1	no	—
25	C₄⋊#88322	—	2	2	no	—
26	C₄⋊#88322	—	2	2	no	—
27	C₄⋊#88322	D₄	1	2	yes	—
28	#13133138	—	4	8	no	—
29	C₈×Dic₅₃	—	4	2	no	—
30	#13133124	—	2	1	no	—
31	C₂²×D₄×C₅₃	—	2	2	no	—
32	C₂²×D₄×C₅₃	—	2	2	no	—
33	C₂²×D₄×C₅₃	—	2	2	no	—
34	C₂×D₈×C₅₃	—	2	4	no	—
35	C₂×D₈×C₅₃	—	2	4	no	—
36	C₂×D₈×C₅₃	—	4	2	no	—
37	C₂×D₈×C₅₃	—	4	2	no	—
38	C₂×D₈×C₅₃	—	4	2	no	—
39	C₂×D₈×C₅₃	—	4	2	no	—
40	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
42	#127×C₅₃	—	2	2	no	—
43	#127×C₅₃	—	2	2	no	—
44	#127×C₅₃	—	2	2	no	—
45	#127×C₅₃	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
47	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₅₃	D₄	1	2	yes	—
49	C₂⁴×C₅₃	—	4	2	no	—
50	C₂×D₄×C₅₃	—	2	4	no	—
51	C₂×D₄×C₅₃	—	2	4	no	—
52	C₂×D₄×C₅₃	—	2	4	no	—
53	C₂×D₄×C₅₃	—	4	2	no	—
54	C₂×D₄×C₅₃	—	4	2	no	—
55	C₂×D₄×C₅₃	—	4	2	no	—
56	C₂×D₄×C₅₃	—	4	2	no	—
57	C₂×D₄×C₅₃	—	4	4	no	—
58	C₂×D₄×C₅₃	—	4	2	no	—
59	C₂×D₄×C₅₃	—	4	4	no	—
60	C₂×D₄×C₅₃	—	4	2	no	—
61	C₂×D₄×C₅₃	—	4	2	no	—
62	C₂×D₄×C₅₃	—	8	4	no	—
63	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
66	C₂²×Dic₅₃	—	2	2	no	—
67	D₈×C₅₃	—	4	8	no	—
68	D₈×C₅₃	—	4	8	no	—
69	C₂×C₈×C₅₃	—	2	1	no	—
70	C₂×C₈×C₅₃	—	2	1	no	—
71	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₅₃	D₈	1	2	yes	—
74	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
75	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₅₃	#41	1	1	yes	characteristic
79	C₄⋊Dic₅₃	—	2	2	no	—
80	C₄⋊Dic₅₃	D₈	1	4	yes	complemented
81	C₄×Dic₅₃	—	4	1	no	—
82	C₄×Dic₅₃	—	4	1	no	—
83	C₂×#88322	—	2	2	no	—
84	C₂×#88322	—	4	1	no	—
85	C₂×#88322	—	4	1	no	—
86	C₂×#88322	—	4	2	no	—
87	C₂×#88322	—	4	4	no	—
88	C₂³×C₅₃	—	4	4	no	—
89	C₂³×C₅₃	—	8	2	no	—
90	C₂³×C₅₃	—	8	2	no	—
91	C₂³×C₅₃	—	8	4	no	—
92	C₂³×C₅₃	—	8	4	no	—
93	C₂³×C₅₃	#121	1	1	yes	characteristic
94	D₄×C₅₃	—	4	4	no	—
95	D₄×C₅₃	—	4	4	no	—
96	D₄×C₅₃	—	4	8	no	—
97	D₄×C₅₃	—	8	4	no	—
98	D₄×C₅₃	—	8	4	no	—
99	D₄×C₅₃	—	8	2	no	—
100	D₄×C₅₃	—	8	2	no	—
101	D₄×C₅₃	—	8	2	no	—
102	D₄×C₅₃	—	8	2	no	—
103	C₂×C₄×C₅₃	—	2	2	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	#124	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
111	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#135	1	2	yes	—
114	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×Dic₅₃	—	2	4	no	—
117	C₂×Dic₅₃	—	4	2	no	—
118	C₂×Dic₅₃	—	4	2	no	—
119	C₈×C₅₃	—	2	2	no	—
120	C₈×C₅₃	#102	1	4	yes	complemented
121	#88322	—	4	2	no	—
122	#88322	—	8	2	no	—
123	#9734	—	53	1	no	maximal, Sylow
124	C₂²×C₅₃	—	8	4	no	—
125	C₂²×C₅₃	—	8	4	no	—
126	C₂²×C₅₃	—	8	4	no	—
127	C₂²×C₅₃	—	16	4	no	—
128	C₂²×C₅₃	—	16	4	no	—
129	C₂²×C₅₃	#390	1	1	yes	characteristic
130	C₂²×C₅₃	#459	1	1	yes	characteristic
131	C₂²×C₅₃	#460	1	1	yes	characteristic
132	C₂²×C₅₃	#466	1	2	yes	—
133	C₂²×C₅₃	#472	1	2	yes	—
134	C₄×C₅₃	—	2	1	no	—
135	C₄×C₅₃	—	2	1	no	—
136	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
137	C₄×C₅₃	#418	1	1	yes	characteristic
138	C₄×C₅₃	C₄×D₈	1	2	yes	—
139	C₄×C₅₃	#459	1	1	yes	characteristic
140	C₄×C₅₃	#460	1	1	yes	characteristic
141	C₄×C₅₃	#493	1	1	yes	characteristic
142	C₄×C₅₃	#495	1	1	yes	characteristic
143	Dic₅₃	—	4	4	no	—
144	C₂×#493	D₅₃	1	1	yes	characteristic
145	#1982	—	53	2	no	—
146	C₈×C₄⋊C₄	—	53	1	no	—
147	C₂×#335	—	53	1	no	—
148	C₂×#331	—	53	2	no	—
149	C₂×C₅₃	—	16	4	no	—
150	C₂×C₅₃	#1667	1	2	yes	—
151	C₂×C₅₃	#1668	1	2	yes	—
152	C₂×C₅₃	#1970	1	1	yes	characteristic
153	C₂×C₅₃	#1978	1	1	yes	characteristic
154	C₂×C₅₃	#1982	1	1	yes	characteristic
155	C₂²×D₈	—	2	1	no	—
156	C₂²×D₈	—	2	1	no	—
157	C₂×#127	C₂×D₅₃	1	2	yes	—
158	#493	—	2	2	no	—
159	#493	Dic₅₃	1	4	yes	complemented
160	C₂×C₄⋊C₈	—	53	1	no	—
161	C₂×C₄⋊C₈	—	53	2	no	—
162	C₂×#102	—	106	4	no	—
163	C₂×C₄×C₈	—	106	1	no	—
164	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
165	C₄×C₄⋊C₄	—	53	1	no	—
166	#336	—	106	1	no	—
167	#335	—	106	2	no	—
168	#331	—	106	4	no	—
169	C₅₃	#9734	1	1	yes	characteristic, complemented, Sylow
170	C₂²×D₄	—	2	2	no	—
171	C₂²×D₄	—	2	2	no	—
172	C₂²×D₄	—	2	2	no	—
173	C₂×D₈	—	2	4	no	—
174	C₂×D₈	—	2	4	no	—
175	C₂×D₈	—	4	2	no	—
176	C₂×D₈	—	4	2	no	—
177	C₂×D₈	—	4	2	no	—
178	C₂×D₈	—	4	2	no	—
179	C₂²×C₈	—	106	1	no	—
180	C₂²×C₈	—	106	2	no	—
181	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
182	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
183	#127	—	2	2	no	—
184	#127	—	2	2	no	—
185	#127	—	2	2	no	—
186	#127	C₂×Dic₅₃	1	4	yes	—
187	C₂×C₄⋊C₄	—	53	2	no	—
188	C₂×C₄²	—	106	1	no	—
189	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
190	C₄⋊C₈	—	53	2	no	—
191	C₄⋊C₈	—	106	2	no	—
192	C₄⋊C₈	—	106	1	no	—
193	C₄⋊C₈	—	106	2	no	—
194	#102	—	212	8	no	—
195	C₄×C₈	—	212	2	no	—
196	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
197	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
198	#95	—	106	1	no	—
199	C₂⁴	—	4	2	no	—
200	C₂×D₄	—	2	4	no	—
201	C₂×D₄	—	2	4	no	—
202	C₂×D₄	—	2	4	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂×D₄	—	4	2	no	—
206	C₂×D₄	—	4	4	no	—
207	C₂×D₄	—	4	2	no	—
208	C₂×D₄	—	4	2	no	—
209	C₂×D₄	—	4	4	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	8	4	no	—
213	C₂²×C₄	—	106	2	no	—
214	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
215	C₂²×C₄	#11719947	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
217	D₈	—	4	8	no	—
218	D₈	—	4	8	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	106	2	no	—
222	C₂×C₈	—	212	2	no	—
223	C₂×C₈	—	212	4	no	—
224	C₂×C₈	—	212	1	no	—
225	C₂×C₈	—	212	1	no	—
226	C₂×C₈	#11719922	1	2	yes	—
227	C₂×C₈	#11719923	1	2	yes	—
228	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
229	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
230	C₄⋊C₄	—	53	4	no	—
231	C₄⋊C₄	—	106	2	no	—
232	C₄²	—	212	1	no	—
233	C₄²	—	212	1	no	—
234	C₄²	D₄×D₅₃	1	1	yes	characteristic
235	C₄²	#11719947	1	1	yes	characteristic
236	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
237	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
238	C₂³	—	4	4	no	—
239	C₂³	—	8	2	no	—
240	C₂³	—	8	2	no	—
241	C₂³	—	8	4	no	—
242	C₂³	—	8	4	no	—
243	C₂³	#13133188	1	1	yes	characteristic
244	D₄	—	4	4	no	—
245	D₄	—	4	4	no	—
246	D₄	—	4	8	no	—
247	D₄	—	8	4	no	—
248	D₄	—	8	4	no	—
249	D₄	—	8	2	no	—
250	D₄	—	8	2	no	—
251	D₄	—	8	2	no	—
252	D₄	—	8	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	106	4	no	—
255	C₂×C₄	—	212	2	no	—
256	C₂×C₄	—	212	2	no	—
257	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
258	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
259	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
260	C₂×C₄	#13133188	1	1	yes	characteristic
261	C₂×C₄	#13133204	1	1	yes	characteristic
262	C₂×C₄	#13133213	1	1	yes	characteristic
263	C₂×C₄	D₈×D₅₃	1	2	yes	—
264	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
265	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
266	C₂×C₄	#13133236	1	2	yes	—
267	C₂×C₄	#13133243	1	1	yes	characteristic
268	C₂×C₄	#13133243	1	1	yes	characteristic
269	C₈	—	2	2	no	—
270	C₈	—	424	2	no	—
271	C₈	—	212	2	no	—
272	C₈	#13133138	1	4	yes	complemented
273	C₂²	—	8	4	no	—
274	C₂²	—	8	4	no	—
275	C₂²	—	8	4	no	—
276	C₂²	—	16	4	no	—
277	C₂²	—	16	4	no	—
278	C₂²	#16810526	1	1	yes	characteristic
279	C₂²	#16810815	1	1	yes	characteristic
280	C₂²	#16810820	1	1	yes	characteristic
281	C₂²	#16810856	1	2	yes	—
282	C₂²	#16810890	1	2	yes	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	212	4	no	—
286	C₄	C₂×#13133138	1	1	yes	characteristic
287	C₄	#16810628	1	1	yes	characteristic
288	C₄	D₈×Dic₅₃	1	2	yes	—
289	C₄	#16810815	1	1	yes	characteristic
290	C₄	#16810820	1	1	yes	characteristic
291	C₄	#16810998	1	1	yes	characteristic
292	C₄	#16811006	1	1	yes	characteristic
293	C₂	—	16	4	no	—
294	C₂	#20274067	1	2	yes	—
295	C₂	#20274072	1	2	yes	—
296	C₂	#20275582	1	1	yes	characteristic
297	C₂	#20275626	1	1	yes	characteristic
298	C₂	#20275643	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133188
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting, prime radical	C₂×C₅₃×#493	C₂
socle	C₂³×C₅₃	#121
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#493	D₅₃
hypocenter	C₅₃	#9734
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. #20275643	C₂	semidirect
2. C₈×C₄⋊Dic₅₃	C₂	semidirect
3. C₂×#16810376	C₂	semidirect
4. C₂×#16810367	C₂	semidirect
18. C₅₃×#493	C₄	semidirect
80. C₄⋊Dic₅₃	D₈	semidirect
120. C₈×C₅₃	#102	semidirect
159. #493	Dic₅₃	semidirect
169. C₅₃	#9734	semidirect
272. C₈	#13133138	semidirect

Maximal subgroups

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

subgroup	index
#20275643	2
C₈×C₄⋊Dic₅₃	2
C₂×#16810376	2
C₂×#16810367	2
C₂×C₅₃×#493	2
#9734	53