Subgroups — #27289992

Order 13568.

982 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133151	C₂	1	2	yes	maximal
2	C₂×#16810386	C₂	1	2	yes	maximal
3	#20272945	C₂	1	2	yes	maximal
4	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₅₃	C₂²	1	4	yes	—
6	C₂×#13133151	—	2	4	no	—
7	C₂×#13133151	—	2	4	no	—
8	C₂×#13133151	C₄	1	8	yes	complemented
9	C₂×#13133124	—	2	4	no	—
10	#16810386	—	2	4	no	—
11	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	—	2	4	no	—
15	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	complemented
16	C₂³×Dic₅₃	—	2	4	no	—
17	C₂×C₄⋊Dic₅₃	—	2	4	no	—
18	C₂×C₄⋊Dic₅₃	—	2	4	no	—
19	C₂×C₄⋊Dic₅₃	—	2	4	no	—
20	C₂×C₄⋊Dic₅₃	—	2	4	no	—
21	C₂×C₄⋊Dic₅₃	—	2	4	no	—
22	C₂×C₄⋊Dic₅₃	C₂×C₄	1	8	yes	—
23	#13133151	—	2	16	no	—
24	#13133124	—	4	8	no	—
25	C₂³×C₄×C₅₃	D₄	1	2	yes	—
26	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	—	2	2	no	—
28	C₂²×C₈×C₅₃	—	2	2	no	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₅₃	D₄	1	2	yes	—
33	C₂²×C₈×C₅₃	D₄	1	2	yes	—
34	C₂²×C₈×C₅₃	D₄	1	4	yes	—
35	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
36	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
37	C₂×C₄²×C₅₃	—	2	2	no	—
38	C₂×C₄²×C₅₃	—	2	2	no	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
41	C₄×C₈×C₅₃	—	2	8	no	—
42	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	—	2	4	no	—
44	C₂²×C₄×C₅₃	—	2	4	no	—
45	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₅₃	D₈	1	4	yes	—
47	C₂²×C₄×C₅₃	Q₁₆	1	4	yes	—
48	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	#41	1	2	yes	—
57	C₂²×C₄×C₅₃	#41	1	2	yes	—
58	C₂²×Dic₅₃	—	2	8	no	—
59	C₂²×Dic₅₃	—	4	4	no	—
60	C₂²×Dic₅₃	—	4	4	no	—
61	C₂²×Dic₅₃	—	4	8	no	—
62	C₂²×Dic₅₃	—	4	8	no	—
63	C₂×C₈×C₅₃	—	2	2	no	—
64	C₂×C₈×C₅₃	—	2	2	no	—
65	C₂×C₈×C₅₃	—	2	4	no	—
66	C₂×C₈×C₅₃	—	2	4	no	—
67	C₂×C₈×C₅₃	C₄²	1	4	yes	—
68	C₂×C₈×C₅₃	C₂²⋊C₄	1	8	yes	—
69	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
70	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
71	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
72	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
73	C₄²×C₅₃	—	2	2	no	—
74	C₄²×C₅₃	—	2	2	no	—
75	C₄²×C₅₃	—	2	4	no	—
76	C₄⋊Dic₅₃	—	2	8	no	—
77	C₄⋊Dic₅₃	—	2	8	no	—
78	C₄⋊Dic₅₃	—	4	4	no	—
79	C₄⋊Dic₅₃	—	4	4	no	—
80	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂³×C₅₃	C₄×D₄	1	2	yes	—
82	C₂³×C₅₃	#121	1	2	yes	—
83	C₂³×C₅₃	#127	1	1	yes	characteristic
84	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
85	C₂³×C₅₃	C₂×D₈	1	2	yes	—
86	C₂³×C₅₃	C₂×Q₁₆	1	2	yes	—
87	C₂³×C₅₃	#135	1	4	yes	—
88	C₂×C₄×C₅₃	—	2	4	no	—
89	C₂×C₄×C₅₃	—	2	4	no	—
90	C₂×C₄×C₅₃	—	2	8	no	—
91	C₂×C₄×C₅₃	—	2	8	no	—
92	C₂×C₄×C₅₃	#107	1	8	yes	complemented
93	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
95	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
103	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
104	C₂×C₄×C₅₃	#121	1	2	yes	—
105	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
106	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₅₃	#124	1	2	yes	—
108	C₂×C₄×C₅₃	#125	1	2	yes	—
109	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
111	C₂×Dic₅₃	—	4	8	no	—
112	C₂×Dic₅₃	—	4	16	no	—
113	C₂×Dic₅₃	—	8	8	no	—
114	C₂×Dic₅₃	—	8	4	no	—
115	C₂×Dic₅₃	—	8	4	no	—
116	C₈×C₅₃	—	2	4	no	—
117	C₈×C₅₃	#95	1	8	yes	complemented
118	#10862	—	53	1	no	maximal, Sylow
119	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₅₃	#380	1	1	yes	characteristic
121	C₂²×C₅₃	#384	1	1	yes	characteristic
122	C₂²×C₅₃	#389	1	2	yes	—
123	C₂²×C₅₃	#390	1	2	yes	—
124	C₂²×C₅₃	C₂×#107	1	2	yes	—
125	C₂²×C₅₃	#427	1	2	yes	—
126	C₂²×C₅₃	C₄×D₈	1	4	yes	—
127	C₂²×C₅₃	C₄×Q₁₆	1	4	yes	—
128	C₂²×C₅₃	#466	1	4	yes	—
129	C₂²×C₅₃	#467	1	4	yes	—
130	C₂²×C₅₃	#493	1	1	yes	characteristic
131	C₂²×C₅₃	#494	1	1	yes	characteristic
132	C₂²×C₅₃	#495	1	2	yes	—
133	C₂²×C₅₃	#499	1	2	yes	—
134	C₂²×C₅₃	#500	1	2	yes	—
135	C₄×C₅₃	—	2	8	no	—
136	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
137	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
138	C₄×C₅₃	#380	1	1	yes	characteristic
139	C₄×C₅₃	#382	1	1	yes	characteristic
140	C₄×C₅₃	#389	1	2	yes	—
141	C₄×C₅₃	#391	1	2	yes	—
142	Dic₅₃	—	8	8	no	—
143	C₂²×#107	—	53	2	no	—
144	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
145	C₂×#340	—	53	2	no	—
146	C₄⋊#95	—	53	2	no	—
147	C₂×C₅₃	C₄⋊#95	1	1	yes	characteristic
148	C₂×C₅₃	C₄×#107	1	2	yes	—
149	C₂×C₅₃	#1808	1	2	yes	—
150	C₂×C₅₃	#1845	1	1	yes	characteristic
151	C₂×C₅₃	#1846	1	1	yes	characteristic
152	C₂×C₅₃	#1942	1	4	yes	—
153	C₂×C₅₃	#1969	1	2	yes	—
154	C₂×C₅₃	#1970	1	2	yes	—
155	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
156	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
157	C₂²×C₄⋊C₄	—	53	4	no	—
158	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
159	C₂×#107	—	53	8	no	—
160	C₂×#107	—	106	4	no	—
161	C₂×#107	—	106	4	no	—
162	C₂×C₄×C₈	—	2	4	no	—
163	C₂×C₄×C₈	Dic₅₃	1	4	yes	complemented
164	C₂×#95	—	106	4	no	—
165	#340	—	106	4	no	—
166	C₅₃	#10862	1	1	yes	characteristic, complemented, Sylow
167	C₂³×C₄	—	106	4	no	—
168	C₂³×C₄	D₂₁₂	1	2	yes	—
169	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	Dic₁₀₆	1	2	yes	—
173	C₂²×C₈	Dic₁₀₆	1	2	yes	—
174	C₂²×C₈	C₄×D₅₃	1	4	yes	—
175	C₂²×C₈	D₂₁₂	1	2	yes	—
176	C₂²×C₈	D₂₁₂	1	2	yes	—
177	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
178	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
179	C₂²×C₈	C₂²⋊D₅₃	1	4	yes	—
180	C₂×C₄⋊C₄	—	53	8	no	—
181	C₂×C₄⋊C₄	—	106	4	no	—
182	C₂×C₄⋊C₄	—	106	4	no	—
183	C₂×C₄⋊C₄	—	106	4	no	—
184	C₂×C₄⋊C₄	—	106	4	no	—
185	C₂×C₄⋊C₄	—	106	4	no	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	Dic₁₀₆	1	2	yes	—
189	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
190	#107	—	106	16	no	—
191	C₄×C₈	—	2	8	no	—
192	#95	—	212	8	no	—
193	C₂⁴	C₂×D₂₁₂	1	1	yes	characteristic
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	106	8	no	—
197	C₂²×C₄	—	212	4	no	—
198	C₂²×C₄	—	212	4	no	—
199	C₂²×C₄	—	212	8	no	—
200	C₂²×C₄	—	212	8	no	—
201	C₂²×C₄	D₄₂₄	1	4	yes	—
202	C₂²×C₄	Dic₂₁₂	1	4	yes	—
203	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
204	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
205	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
206	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
208	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
210	C₂²×C₄	#11719945	1	2	yes	—
211	C₂²×C₄	#11719945	1	2	yes	—
212	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
219	C₂×C₈	#11719919	1	8	yes	—
220	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
221	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
222	C₂×C₈	#11719921	1	8	yes	—
223	C₂×C₈	C₂²⋊Dic₅₃	1	4	yes	—
224	C₄⋊C₄	—	106	8	no	—
225	C₄⋊C₄	—	106	8	no	—
226	C₄⋊C₄	—	212	4	no	—
227	C₄⋊C₄	—	212	4	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	4	no	—
231	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
232	C₂³	C₄×D₂₁₂	1	2	yes	—
233	C₂³	#13133191	1	2	yes	—
234	C₂³	#13133214	1	1	yes	characteristic
235	C₂³	#13133217	1	1	yes	characteristic
236	C₂³	C₂×D₄₂₄	1	2	yes	—
237	C₂³	C₂×Dic₂₁₂	1	2	yes	—
238	C₂³	#13133239	1	4	yes	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	424	8	no	—
244	C₂×C₄	—	424	4	no	—
245	C₂×C₄	—	424	4	no	—
246	C₂×C₄	—	212	8	no	—
247	C₂×C₄	—	212	16	no	—
248	C₂×C₄	#13133151	1	8	yes	complemented
249	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
250	C₂×C₄	C₂×#11719921	1	2	yes	—
251	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
252	C₂×C₄	C₂×#11719919	1	2	yes	—
253	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
254	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
255	C₂×C₄	#13133176	1	1	yes	characteristic
256	C₂×C₄	#13133176	1	1	yes	characteristic
257	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
258	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
259	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
260	C₂×C₄	#13133191	1	2	yes	—
261	C₂×C₄	#13133196	1	2	yes	—
262	C₂×C₄	#13133196	1	2	yes	—
263	C₂×C₄	#13133205	1	2	yes	—
264	C₂×C₄	#13133209	1	2	yes	—
265	C₂×C₄	#13133217	1	1	yes	characteristic
266	C₂×C₄	#13133217	1	1	yes	characteristic
267	C₈	—	2	4	no	—
268	C₈	#13133124	1	8	yes	complemented
269	C₂²	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
270	C₂²	#16810483	1	1	yes	characteristic
271	C₂²	#16810493	1	1	yes	characteristic
272	C₂²	#16810521	1	2	yes	—
273	C₂²	#16810525	1	2	yes	—
274	C₂²	C₂×#13133151	1	2	yes	—
275	C₂²	#16810661	1	2	yes	—
276	C₂²	C₄×D₄₂₄	1	4	yes	—
277	C₂²	C₄×Dic₂₁₂	1	4	yes	—
278	C₂²	#16810859	1	4	yes	—
279	C₂²	#16810864	1	4	yes	—
280	C₂²	#16811000	1	1	yes	characteristic
281	C₂²	#16811003	1	1	yes	characteristic
282	C₂²	#16811007	1	2	yes	—
283	C₂²	#16811025	1	2	yes	—
284	C₂²	#16811028	1	2	yes	—
285	C₄	—	2	8	no	—
286	C₄	—	424	8	no	—
287	C₄	C₂×#13133124	1	1	yes	characteristic
288	C₄	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
289	C₄	#16810483	1	1	yes	characteristic
290	C₄	#16810489	1	1	yes	characteristic
291	C₄	#16810521	1	2	yes	—
292	C₄	#16810529	1	2	yes	—
293	C₂	#20272945	1	1	yes	characteristic
294	C₂	C₄×#13133151	1	2	yes	—
295	C₂	#20274744	1	2	yes	—
296	C₂	#20274875	1	1	yes	characteristic
297	C₂	#20274879	1	1	yes	characteristic
298	C₂	#20275448	1	4	yes	—
299	C₂	#20275576	1	2	yes	—
300	C₂	#20275581	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	#10862
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
8. C₂×#13133151	C₄	semidirect
15. C₂×C₄×C₈×C₅₃	C₄	semidirect
92. C₂×C₄×C₅₃	#107	semidirect
117. C₈×C₅₃	#95	semidirect
163. C₂×C₄×C₈	Dic₅₃	semidirect
166. C₅₃	#10862	semidirect
248. C₂×C₄	#13133151	semidirect
268. C₈	#13133124	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133151	2
C₂×#16810386	2
#20272945	2
C₂²×C₄×C₈×C₅₃	2
#10862	53