Subgroups — #27289779

Order 13568.

612 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283133	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810395	C₂	1	2	yes	maximal
3	#20274441	C₂	1	2	yes	maximal
4	C₂×#16810381	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×#338	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	1	yes	characteristic
7	C₂²×#11719917	C₂²	1	2	yes	—
8	C₂×#13133176	C₂²	1	1	yes	characteristic
9	#16810581	—	2	2	no	—
10	#16810578	C₄	1	4	yes	—
11	M₁₆×Dic₅₃	—	2	2	no	—
12	#16810395	—	2	4	no	—
13	#16810387	C₄	1	8	yes	complemented
14	#16810381	—	2	2	no	—
15	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
16	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#338	—	2	2	no	—
18	C₂×#11719917	—	2	1	no	—
19	C₂×#11719917	—	2	2	no	—
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719917	—	2	2	no	—
22	C₂×#11719917	—	2	2	no	—
23	C₂×#11719917	—	2	2	no	—
24	C₂×#11719917	—	2	2	no	—
25	C₂×#11719917	—	2	2	no	—
26	C₂×#11719917	C₂×C₄	1	2	yes	—
27	C₂×#11719917	C₂×C₄	1	4	yes	—
28	C₂²×#88322	—	2	1	no	—
29	C₂²×#88322	C₂×C₄	1	4	yes	—
30	#13133176	—	2	1	no	—
31	#13133176	—	2	1	no	—
32	#13133176	C₂×C₄	1	2	yes	—
33	#13133176	D₄	1	2	yes	—
34	C₂×C₄⋊Dic₅₃	—	2	1	no	—
35	C₂×C₂²⋊Dic₅₃	Q₈	1	2	yes	—
36	C₂×C₄×Dic₅₃	—	2	1	no	—
37	#13133127	—	2	4	no	—
38	C₈×Dic₅₃	—	2	4	no	—
39	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
40	C₂×M₁₆×C₅₃	—	2	2	no	—
41	C₂×M₁₆×C₅₃	—	2	2	no	—
42	C₂×M₁₆×C₅₃	—	2	2	no	—
43	C₂×M₁₆×C₅₃	—	2	2	no	—
44	C₂×M₁₆×C₅₃	—	2	2	no	—
45	C₂×M₁₆×C₅₃	—	2	2	no	—
46	C₂×M₁₆×C₅₃	—	2	2	no	—
47	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
48	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
49	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
50	C₂²×C₈×C₅₃	—	2	2	no	—
51	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
52	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	#41	1	2	yes	—
61	C₂²×C₄×C₅₃	#41	1	2	yes	—
62	C₂²×C₄×C₅₃	#41	1	2	yes	—
63	C₂²×Dic₅₃	—	2	2	no	—
64	M₁₆×C₅₃	—	2	4	no	—
65	M₁₆×C₅₃	—	2	4	no	—
66	M₁₆×C₅₃	—	2	4	no	—
67	M₁₆×C₅₃	—	2	4	no	—
68	M₁₆×C₅₃	—	4	2	no	—
69	M₁₆×C₅₃	—	4	2	no	—
70	C₂×C₈×C₅₃	—	2	2	no	—
71	C₂×C₈×C₅₃	—	2	2	no	—
72	C₂×C₈×C₅₃	—	2	4	no	—
73	C₂×C₈×C₅₃	—	4	2	no	—
74	C₂×C₈×C₅₃	—	4	2	no	—
75	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
76	C₂²⋊Dic₅₃	—	2	2	no	—
77	C₂²⋊Dic₅₃	C₄⋊C₄	1	4	yes	—
78	C₄⋊Dic₅₃	—	2	2	no	—
79	C₄⋊Dic₅₃	—	2	2	no	—
80	C₄×Dic₅₃	—	2	2	no	—
81	C₄×Dic₅₃	—	2	2	no	—
82	#11719917	—	2	2	no	—
83	#11719917	—	2	4	no	—
84	#11719917	—	4	2	no	—
85	#11719917	—	4	2	no	—
86	#11719917	—	4	2	no	—
87	#11719917	—	4	2	no	—
88	#11719917	C₄²	1	4	yes	—
89	C₂×#88322	—	2	2	no	—
90	C₂×#88322	—	2	4	no	—
91	C₂×#88322	—	2	4	no	—
92	C₂×#88322	—	2	4	no	—
93	C₂×#88322	—	2	4	no	—
94	C₂³×C₅₃	—	8	1	no	—
95	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
97	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂³×C₅₃	#125	1	2	yes	—
100	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#117	1	2	yes	—
104	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
106	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#117	1	2	yes	—
108	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#123	1	2	yes	—
114	C₂×C₄×C₅₃	#123	1	2	yes	—
115	C₂×C₄×C₅₃	#124	1	2	yes	—
116	C₂×C₄×C₅₃	#126	1	2	yes	—
117	C₂×C₄×C₅₃	#126	1	2	yes	—
118	C₂×Dic₅₃	—	2	4	no	—
119	C₂×Dic₅₃	—	8	2	no	—
120	C₈×C₅₃	—	2	4	no	—
121	C₈×C₅₃	—	4	4	no	—
122	#88322	—	2	4	no	—
123	#88322	—	4	4	no	—
124	#10812	—	53	1	no	maximal, Sylow
125	C₂²×C₅₃	—	4	1	no	—
126	C₂²×C₅₃	—	4	1	no	—
127	C₂²×C₅₃	—	4	2	no	—
128	C₂²×C₅₃	—	4	1	no	—
129	C₂²×C₅₃	—	4	2	no	—
130	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
131	C₂²×C₅₃	#380	1	1	yes	characteristic
132	C₂²×C₅₃	#382	1	1	yes	characteristic
133	C₂²×C₅₃	#387	1	2	yes	—
134	C₂²×C₅₃	#387	1	2	yes	—
135	C₄×C₅₃	#376	1	1	yes	characteristic
136	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₅₃	#381	1	1	yes	characteristic
138	C₄×C₅₃	#383	1	1	yes	characteristic
139	C₄×C₅₃	#388	1	2	yes	—
140	C₄×C₅₃	#388	1	2	yes	—
141	Dic₅₃	—	4	4	no	—
142	#2873	—	53	1	no	—
143	C₂×#344	—	53	2	no	—
144	#1738	—	53	2	no	—
145	C₂×#338	—	53	1	no	—
146	C₂×#338	D₅₃	1	1	yes	characteristic
147	C₂×C₅₃	—	2	1	no	—
148	C₂×C₅₃	—	2	1	no	—
149	C₂×C₅₃	—	2	1	no	—
150	C₂×C₅₃	—	2	2	no	—
151	C₂×C₅₃	—	2	2	no	—
152	C₂×C₅₃	#1434	1	1	yes	characteristic
153	C₂²×M₁₆	—	53	2	no	—
154	C₂²×M₁₆	—	53	1	no	—
155	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
156	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
157	C₂×#117	—	53	1	no	—
158	#405	—	53	4	no	—
159	#405	—	106	2	no	—
160	C₄×M₁₆	—	106	2	no	—
161	#344	—	106	4	no	—
162	#341	—	53	8	no	—
163	#338	—	2	2	no	—
164	#338	—	106	2	no	—
165	C₅₃	#10812	1	1	yes	characteristic, complemented, Sylow
166	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	53	4	no	—
175	C₂×M₁₆	—	53	2	no	—
176	C₂×M₁₆	—	106	1	no	—
177	C₂×M₁₆	—	106	2	no	—
178	C₂×M₁₆	—	106	2	no	—
179	C₂×M₁₆	—	106	2	no	—
180	C₂×M₁₆	—	106	2	no	—
181	C₂×M₁₆	—	106	2	no	—
182	C₂×M₁₆	—	106	2	no	—
183	C₂×M₁₆	—	106	2	no	—
184	C₂×M₁₆	C₄×D₅₃	1	4	yes	—
185	C₂×M₁₆	D₂₁₂	1	2	yes	—
186	C₂×M₁₆	C₂²⋊D₅₃	1	2	yes	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	53	4	no	—
189	C₂²×C₈	—	106	1	no	—
190	C₂²×C₈	C₄×D₅₃	1	2	yes	—
191	#117	—	53	2	no	—
192	#117	—	53	2	no	—
193	#117	—	106	1	no	—
194	#117	—	106	1	no	—
195	C₂×C₄⋊C₄	—	106	1	no	—
196	C₂×C₂²⋊C₄	—	53	2	no	—
197	C₂×C₄²	—	106	1	no	—
198	#97	—	106	4	no	—
199	C₄×C₈	—	106	4	no	—
200	C₂⁴	Q₈×D₅₃	1	1	yes	characteristic
201	C₂²×C₄	—	106	2	no	—
202	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
204	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
205	C₂²×C₄	#11719945	1	2	yes	—
206	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
207	C₂²×C₄	#11719947	1	1	yes	characteristic
208	C₂²×C₄	#11719947	1	2	yes	—
209	C₂²×C₄	#11719947	1	2	yes	—
210	C₂²×C₄	#11719949	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
212	M₁₆	—	2	4	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	2	4	no	—
216	M₁₆	—	4	2	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	53	4	no	—
219	M₁₆	—	106	2	no	—
220	M₁₆	—	106	4	no	—
221	M₁₆	—	212	2	no	—
222	M₁₆	—	212	2	no	—
223	M₁₆	—	212	2	no	—
224	M₁₆	—	212	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	106	2	no	—
231	C₂×C₈	—	106	4	no	—
232	C₂×C₈	—	106	4	no	—
233	C₂×C₈	—	106	4	no	—
234	C₂×C₈	—	106	4	no	—
235	C₂×C₈	#11719921	1	4	yes	—
236	C₄⋊C₄	—	106	2	no	—
237	C₄⋊C₄	—	106	2	no	—
238	C₂²⋊C₄	—	53	4	no	—
239	C₂²⋊C₄	—	106	2	no	—
240	C₄²	—	106	2	no	—
241	C₄²	—	106	2	no	—
242	C₂³	—	8	1	no	—
243	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
244	C₂³	#13133182	1	2	yes	—
245	C₂³	#13133193	1	1	yes	characteristic
246	C₂³	#13133197	1	1	yes	characteristic
247	C₂³	#13133207	1	2	yes	—
248	C₂×C₄	—	106	4	no	—
249	C₂×C₄	—	424	2	no	—
250	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
251	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
252	C₂×C₄	#13133173	1	2	yes	—
253	C₂×C₄	#13133173	1	1	yes	characteristic
254	C₂×C₄	#13133173	1	1	yes	characteristic
255	C₂×C₄	#13133174	1	1	yes	characteristic
256	C₂×C₄	#13133175	1	2	yes	—
257	C₂×C₄	#13133175	1	1	yes	characteristic
258	C₂×C₄	#13133177	1	2	yes	—
259	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
260	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
261	C₂×C₄	#13133188	1	1	yes	characteristic
262	C₂×C₄	#13133192	1	1	yes	characteristic
263	C₂×C₄	#13133198	1	2	yes	—
264	C₂×C₄	#13133201	1	2	yes	—
265	C₂×C₄	#13133203	1	2	yes	—
266	C₂×C₄	#13133210	1	2	yes	—
267	C₂×C₄	#13133212	1	2	yes	—
268	C₈	—	2	4	no	—
269	C₈	—	4	4	no	—
270	C₈	—	106	4	no	—
271	C₈	—	212	4	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	4	2	no	—
277	C₂²	#16810474	1	1	yes	characteristic
278	C₂²	#16810480	1	1	yes	characteristic
279	C₂²	#16810487	1	1	yes	characteristic
280	C₂²	#16810504	1	2	yes	—
281	C₂²	#16810508	1	2	yes	—
282	C₄	—	212	4	no	—
283	C₄	#16810468	1	1	yes	characteristic
284	C₄	C₄×#11719921	1	1	yes	characteristic
285	C₄	#16810484	1	1	yes	characteristic
286	C₄	#16810490	1	1	yes	characteristic
287	C₄	#16810511	1	2	yes	—
288	C₄	#16810515	1	2	yes	—
289	C₂	—	2	1	no	—
290	C₂	—	2	2	no	—
291	C₂	—	2	2	no	—
292	C₂	—	2	1	no	—
293	C₂	—	2	1	no	—
294	C₂	#20272939	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×#11719921
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#338	C₂
socle	C₂×C₅₃	#1434
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#338	D₅₃
hypocenter	C₅₃	#10812
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ C₂×#338

Complements

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

N	H ≅ G/N	type
13. #16810387	C₄	semidirect
165. C₅₃	#10812	semidirect

Maximal subgroups

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

subgroup	index
#20283133	2
C₂×#16810395	2
#20274441	2
C₂×#16810381	2
C₂×C₅₃×#338	2
#10812	53