Subgroups — #27289793

Order 13568.

626 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810578	C₂	1	1	yes	maximal, characteristic
2	#20274453	C₂	1	2	yes	maximal
3	C₂×#16810388	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810381	C₂	1	2	yes	maximal
5	C₂×C₅₃×#341	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	2	yes	—
7	C₂²×#11719917	C₂²	1	1	yes	characteristic
8	C₂×#13133176	C₂²	1	1	yes	characteristic
9	#16810578	—	2	2	no	—
10	#16810578	—	2	2	no	—
11	C₂×#13133127	C₄	1	2	yes	—
12	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
13	#16810391	—	2	4	no	—
14	#16810388	—	2	2	no	—
15	#16810381	C₄	1	8	yes	complemented
16	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
17	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#341	—	2	2	no	—
19	C₂×#11719917	—	2	1	no	—
20	C₂×#11719917	—	2	1	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	—	2	2	no	—
27	C₂×#11719917	C₂×C₄	1	4	yes	—
28	C₂×#11719917	C₂×C₄	1	4	yes	—
29	C₂²×#88322	—	2	2	no	—
30	C₂²×#88322	C₂×C₄	1	2	yes	—
31	#13133176	—	2	1	no	—
32	#13133176	—	2	1	no	—
33	#13133176	—	2	1	no	—
34	#13133176	—	2	1	no	—
35	C₂×C₄⋊Dic₅₃	D₄	1	2	yes	—
36	C₂×C₂²⋊Dic₅₃	Q₈	1	2	yes	—
37	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
38	#13133127	—	2	4	no	—
39	C₈×Dic₅₃	—	2	4	no	—
40	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
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	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₅₃	—	2	2	no	—
48	C₂²×C₈×C₅₃	—	2	2	no	—
49	C₂²×C₈×C₅₃	—	2	2	no	—
50	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	D₄	1	2	yes	—
53	C₂²×C₈×C₅₃	D₄	1	2	yes	—
54	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	#41	1	2	yes	—
63	C₂²×C₄×C₅₃	#41	1	2	yes	—
64	C₂²×C₄×C₅₃	#41	1	2	yes	—
65	C₂²×Dic₅₃	C₄⋊C₄	1	4	yes	—
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	1	no	—
71	C₂×C₈×C₅₃	—	2	1	no	—
72	C₂×C₈×C₅₃	—	2	1	no	—
73	C₂×C₈×C₅₃	—	2	1	no	—
74	C₂×C₈×C₅₃	—	2	4	no	—
75	C₂×C₈×C₅₃	—	2	4	no	—
76	C₂×C₈×C₅₃	—	2	4	no	—
77	C₂×C₈×C₅₃	—	4	2	no	—
78	C₂×C₈×C₅₃	—	4	2	no	—
79	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
81	C₂²⋊Dic₅₃	—	2	2	no	—
82	C₂²⋊Dic₅₃	—	2	2	no	—
83	C₄⋊Dic₅₃	—	2	2	no	—
84	C₄⋊Dic₅₃	—	2	2	no	—
85	C₄×Dic₅₃	—	2	2	no	—
86	C₄×Dic₅₃	—	2	2	no	—
87	#11719917	—	2	4	no	—
88	#11719917	—	2	4	no	—
89	#11719917	—	2	4	no	—
90	#11719917	—	2	4	no	—
91	#11719917	—	4	2	no	—
92	#11719917	—	4	2	no	—
93	C₂×#88322	—	2	2	no	—
94	C₂×#88322	—	2	2	no	—
95	C₂×#88322	—	2	4	no	—
96	C₂×#88322	—	4	2	no	—
97	C₂×#88322	—	4	2	no	—
98	C₂×#88322	C₄²	1	4	yes	—
99	C₂³×C₅₃	—	4	2	no	—
100	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
102	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂³×C₅₃	#125	1	2	yes	—
105	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
108	C₂×C₄×C₅₃	#117	1	2	yes	—
109	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
111	C₂×C₄×C₅₃	#117	1	2	yes	—
112	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
117	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
118	C₂×C₄×C₅₃	#123	1	2	yes	—
119	C₂×C₄×C₅₃	#123	1	2	yes	—
120	C₂×C₄×C₅₃	#124	1	2	yes	—
121	C₂×C₄×C₅₃	#126	1	2	yes	—
122	C₂×C₄×C₅₃	#126	1	2	yes	—
123	C₂×Dic₅₃	—	2	4	no	—
124	C₂×Dic₅₃	—	4	4	no	—
125	C₈×C₅₃	—	2	2	no	—
126	C₈×C₅₃	—	2	2	no	—
127	C₈×C₅₃	—	4	4	no	—
128	#88322	—	2	4	no	—
129	#88322	—	4	4	no	—
130	#10815	—	53	1	no	maximal, Sylow
131	C₂²×C₅₃	—	2	2	no	—
132	C₂²×C₅₃	—	2	2	no	—
133	C₂²×C₅₃	—	2	2	no	—
134	C₂²×C₅₃	—	4	2	no	—
135	C₂²×C₅₃	—	4	2	no	—
136	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₅₃	#380	1	1	yes	characteristic
138	C₂²×C₅₃	#382	1	1	yes	characteristic
139	C₂²×C₅₃	#387	1	2	yes	—
140	C₂²×C₅₃	#387	1	2	yes	—
141	C₄×C₅₃	#376	1	1	yes	characteristic
142	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
143	C₄×C₅₃	#381	1	1	yes	characteristic
144	C₄×C₅₃	#383	1	1	yes	characteristic
145	C₄×C₅₃	#388	1	2	yes	—
146	C₄×C₅₃	#388	1	2	yes	—
147	Dic₅₃	—	4	4	no	—
148	C₂×#405	—	53	1	no	—
149	#1742	—	53	2	no	—
150	C₂×#341	—	53	1	no	—
151	C₂×#341	D₅₃	1	1	yes	characteristic
152	C₂×#338	—	53	2	no	—
153	C₂×C₅₃	—	2	1	no	—
154	C₂×C₅₃	—	2	1	no	—
155	C₂×C₅₃	—	2	2	no	—
156	C₂×C₅₃	—	2	2	no	—
157	C₂×C₅₃	#1434	1	1	yes	characteristic
158	C₂×C₅₃	#1813	1	2	yes	—
159	C₂²×M₁₆	—	53	1	no	—
160	C₂²×M₁₆	—	53	2	no	—
161	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
162	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
163	C₂×#117	—	53	1	no	—
164	#405	—	106	2	no	—
165	#405	—	106	2	no	—
166	C₂×#97	—	53	2	no	—
167	C₂×C₄×C₈	—	53	2	no	—
168	#343	—	106	4	no	—
169	#341	—	2	2	no	—
170	#341	—	106	2	no	—
171	#338	—	53	8	no	—
172	C₅₃	#10815	1	1	yes	characteristic, complemented, Sylow
173	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	53	4	no	—
180	C₂×M₁₆	—	53	4	no	—
181	C₂×M₁₆	—	106	1	no	—
182	C₂×M₁₆	—	106	2	no	—
183	C₂×M₁₆	—	106	1	no	—
184	C₂×M₁₆	—	106	2	no	—
185	C₂×M₁₆	—	106	2	no	—
186	C₂×M₁₆	—	106	2	no	—
187	C₂×M₁₆	—	106	2	no	—
188	C₂×M₁₆	—	106	2	no	—
189	C₂×M₁₆	C₄×D₅₃	1	4	yes	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	—	53	2	no	—
194	C₂²×C₈	—	106	2	no	—
195	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
196	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
197	C₂²×C₈	D₂₁₂	1	2	yes	—
198	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
199	#117	—	106	1	no	—
200	#117	—	106	1	no	—
201	#117	—	106	1	no	—
202	#117	—	106	1	no	—
203	C₂×C₄⋊C₄	—	53	2	no	—
204	C₂×C₂²⋊C₄	—	53	2	no	—
205	C₂×C₄²	—	53	2	no	—
206	#97	—	106	4	no	—
207	C₄×C₈	—	106	4	no	—
208	C₂⁴	Q₈×D₅₃	1	1	yes	characteristic
209	C₂²×C₄	—	53	4	no	—
210	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
212	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
213	C₂²×C₄	#11719945	1	2	yes	—
214	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
215	C₂²×C₄	#11719947	1	2	yes	—
216	C₂²×C₄	#11719947	1	1	yes	characteristic
217	C₂²×C₄	#11719947	1	2	yes	—
218	C₂²×C₄	#11719949	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
220	M₁₆	—	2	4	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	4	2	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	—	106	4	no	—
225	M₁₆	—	106	4	no	—
226	M₁₆	—	106	4	no	—
227	M₁₆	—	106	4	no	—
228	M₁₆	—	212	2	no	—
229	M₁₆	—	212	2	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	53	4	no	—
240	C₂×C₈	—	106	2	no	—
241	C₂×C₈	—	106	2	no	—
242	C₂×C₈	—	106	4	no	—
243	C₂×C₈	—	212	2	no	—
244	C₂×C₈	—	212	2	no	—
245	C₂×C₈	#11719921	1	2	yes	—
246	C₂×C₈	#11719921	1	2	yes	—
247	C₄⋊C₄	—	106	2	no	—
248	C₄⋊C₄	—	106	2	no	—
249	C₂²⋊C₄	—	106	2	no	—
250	C₂²⋊C₄	—	106	2	no	—
251	C₄²	—	106	2	no	—
252	C₄²	—	106	2	no	—
253	C₂³	—	4	2	no	—
254	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
255	C₂³	#13133182	1	2	yes	—
256	C₂³	#13133193	1	1	yes	characteristic
257	C₂³	#13133197	1	1	yes	characteristic
258	C₂³	#13133207	1	2	yes	—
259	C₂×C₄	—	106	4	no	—
260	C₂×C₄	—	212	4	no	—
261	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
262	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
263	C₂×C₄	#13133173	1	2	yes	—
264	C₂×C₄	#13133173	1	1	yes	characteristic
265	C₂×C₄	#13133173	1	1	yes	characteristic
266	C₂×C₄	#13133174	1	1	yes	characteristic
267	C₂×C₄	#13133175	1	2	yes	—
268	C₂×C₄	#13133175	1	1	yes	characteristic
269	C₂×C₄	#13133177	1	2	yes	—
270	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
271	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
272	C₂×C₄	#13133188	1	1	yes	characteristic
273	C₂×C₄	#13133192	1	1	yes	characteristic
274	C₂×C₄	#13133198	1	2	yes	—
275	C₂×C₄	#13133201	1	2	yes	—
276	C₂×C₄	#13133203	1	2	yes	—
277	C₂×C₄	#13133210	1	2	yes	—
278	C₂×C₄	#13133212	1	2	yes	—
279	C₈	—	2	2	no	—
280	C₈	—	2	2	no	—
281	C₈	—	4	4	no	—
282	C₈	—	106	4	no	—
283	C₈	—	212	4	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	#16810474	1	1	yes	characteristic
290	C₂²	#16810480	1	1	yes	characteristic
291	C₂²	#16810487	1	1	yes	characteristic
292	C₂²	#16810504	1	2	yes	—
293	C₂²	#16810508	1	2	yes	—
294	C₄	—	212	4	no	—
295	C₄	#16810468	1	1	yes	characteristic
296	C₄	C₄×#11719921	1	1	yes	characteristic
297	C₄	#16810484	1	1	yes	characteristic
298	C₄	#16810490	1	1	yes	characteristic
299	C₄	#16810511	1	2	yes	—
300	C₄	#16810515	1	2	yes	—
301	C₂	—	2	2	no	—
302	C₂	—	2	2	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	#20272939	1	1	yes	characteristic
306	C₂	#20274762	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133175
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#341	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#341	D₅₃
hypocenter	C₅₃	#10815
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
15. #16810381	C₄	semidirect
172. C₅₃	#10815	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810578	2
#20274453	2
C₂×#16810388	2
C₂×#16810381	2
C₂×C₅₃×#341	2
#10815	53