Subgroups — #27292973

Order 13568.

614 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810672	C₂	1	1	yes	maximal, characteristic
2	#20274442	C₂	1	2	yes	maximal
3	#20274408	C₂	1	2	yes	maximal
4	#20274408	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×#338	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	2	yes	—
7	C₂²×#11719917	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
9	#16810672	—	2	2	no	—
10	#16810672	C₄	1	4	yes	complemented
11	C₂×#13133153	—	2	2	no	—
12	#16810388	—	2	4	no	—
13	#16810377	—	2	4	no	—
14	#16810377	—	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	C₂³×Dic₅₃	Q₈	1	2	yes	—
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	1	no	—
25	C₂×#11719917	—	2	2	no	—
26	C₂×#11719917	—	2	2	no	—
27	C₂×#11719917	C₂×C₄	1	4	yes	—
28	C₂×#11719917	Q₈	1	2	yes	—
29	C₂²×#88322	—	2	1	no	—
30	C₂²×#88322	—	2	2	no	—
31	C₂×C₄×Dic₅₃	—	2	1	no	—
32	C₂×C₄×Dic₅₃	—	2	2	no	—
33	C₂×C₄×Dic₅₃	—	2	2	no	—
34	C₂×C₄×Dic₅₃	—	2	1	no	—
35	#13133153	—	2	8	no	—
36	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₅₃	—	2	2	no	—
38	C₂×M₁₆×C₅₃	—	2	2	no	—
39	C₂×M₁₆×C₅₃	—	2	2	no	—
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	C₂×C₄	1	2	yes	—
46	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
47	C₂²×C₈×C₅₃	D₄	1	4	yes	—
48	C₂²×C₈×C₅₃	D₄	1	2	yes	—
49	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
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×D₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	#41	1	2	yes	—
56	C₂²×C₄×C₅₃	#41	1	2	yes	—
57	C₂²×C₄×C₅₃	#41	1	2	yes	—
58	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
60	C₂²×Dic₅₃	—	2	4	no	—
61	C₂²×Dic₅₃	—	2	2	no	—
62	C₂²×Dic₅₃	—	4	4	no	—
63	M₁₆×C₅₃	—	2	4	no	—
64	M₁₆×C₅₃	—	2	4	no	—
65	M₁₆×C₅₃	—	4	2	no	—
66	M₁₆×C₅₃	—	4	2	no	—
67	M₁₆×C₅₃	—	4	2	no	—
68	M₁₆×C₅₃	—	4	2	no	—
69	C₂×C₈×C₅₃	—	2	2	no	—
70	C₂×C₈×C₅₃	—	2	2	no	—
71	C₂×C₈×C₅₃	—	2	4	no	—
72	C₂×C₈×C₅₃	—	2	4	no	—
73	C₂×C₈×C₅₃	—	2	4	no	—
74	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
75	C₄×Dic₅₃	—	4	1	no	—
76	C₄×Dic₅₃	—	4	2	no	—
77	C₄×Dic₅₃	—	4	1	no	—
78	#11719917	—	2	4	no	—
79	#11719917	—	2	4	no	—
80	#11719917	—	2	4	no	—
81	#11719917	—	4	2	no	—
82	#11719917	—	4	2	no	—
83	#11719917	—	4	2	no	—
84	C₂×#88322	—	2	4	no	—
85	C₂×#88322	—	2	2	no	—
86	C₂×#88322	—	4	2	no	—
87	C₂×#88322	—	4	1	no	—
88	C₂×#88322	—	4	2	no	—
89	C₂×#88322	—	4	1	no	—
90	C₂³×C₅₃	—	4	2	no	—
91	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
92	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
93	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂³×C₅₃	#125	1	2	yes	—
95	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#117	1	2	yes	—
98	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#121	1	2	yes	—
101	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
102	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#123	1	2	yes	—
104	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
106	C₂×C₄×C₅₃	#123	1	2	yes	—
107	C₂×C₄×C₅₃	#123	1	2	yes	—
108	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#125	1	2	yes	—
111	C₂×C₄×C₅₃	#126	1	2	yes	—
112	C₂×C₄×C₅₃	#126	1	2	yes	—
113	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
114	C₂×Dic₅₃	—	2	8	no	—
115	C₂×Dic₅₃	—	2	4	no	—
116	C₂×Dic₅₃	—	4	4	no	—
117	C₂×Dic₅₃	—	4	2	no	—
118	C₂×Dic₅₃	—	4	2	no	—
119	C₈×C₅₃	—	2	4	no	—
120	C₈×C₅₃	—	4	4	no	—
121	#88322	—	4	4	no	—
122	#88322	—	4	2	no	—
123	#11375	—	53	1	no	maximal, Sylow
124	C₂²×C₅₃	—	2	4	no	—
125	C₂²×C₅₃	—	2	2	no	—
126	C₂²×C₅₃	—	4	1	no	—
127	C₂²×C₅₃	—	4	2	no	—
128	C₂²×C₅₃	—	4	1	no	—
129	C₂²×C₅₃	#387	1	2	yes	—
130	C₂²×C₅₃	#391	1	1	yes	characteristic
131	C₂²×C₅₃	#398	1	2	yes	—
132	C₂²×C₅₃	#399	1	1	yes	characteristic
133	C₂²×C₅₃	#400	1	1	yes	characteristic
134	C₄×C₅₃	#381	1	1	yes	characteristic
135	C₄×C₅₃	#387	1	2	yes	—
136	C₄×C₅₃	#390	1	1	yes	characteristic
137	C₄×C₅₃	#397	1	2	yes	—
138	C₄×C₅₃	#399	1	1	yes	characteristic
139	C₄×C₅₃	#400	1	1	yes	characteristic
140	Dic₅₃	—	4	4	no	—
141	C₂×#430	—	53	1	no	—
142	#1738	—	53	2	no	—
143	C₂×#338	D₅₃	1	1	yes	characteristic
144	#1728	—	53	2	no	—
145	#1728	—	53	1	no	—
146	C₂×C₅₃	—	2	2	no	—
147	C₂×C₅₃	—	2	1	no	—
148	C₂×C₅₃	—	2	2	no	—
149	C₂×C₅₃	—	2	1	no	—
150	C₂×C₅₃	#1445	1	1	yes	characteristic
151	C₂×C₅₃	#2091	1	2	yes	—
152	C₂²×M₁₆	—	53	2	no	—
153	C₂²×M₁₆	—	53	1	no	—
154	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
155	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
156	C₂²×C₄²	—	53	1	no	—
157	#430	—	53	4	no	—
158	#430	—	106	2	no	—
159	C₂×#108	—	106	2	no	—
160	#341	—	106	4	no	—
161	#338	—	2	2	no	—
162	#336	—	106	4	no	—
163	#336	—	106	2	no	—
164	C₅₃	#11375	1	1	yes	characteristic, complemented, Sylow
165	C₂³×C₄	—	53	2	no	—
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	—	2	2	no	—
175	C₂×M₁₆	—	53	4	no	—
176	C₂×M₁₆	—	53	2	no	—
177	C₂×M₁₆	—	106	2	no	—
178	C₂×M₁₆	—	106	1	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	—	106	2	no	—
185	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
186	C₂×M₁₆	C₂²⋊D₅₃	1	2	yes	—
187	C₂²×C₈	—	106	2	no	—
188	C₂²×C₈	—	106	1	no	—
189	C₂²×C₈	D₂₁₂	1	4	yes	—
190	C₂²×C₈	D₂₁₂	1	2	yes	—
191	C₂×C₄²	—	106	1	no	—
192	C₂×C₄²	—	106	2	no	—
193	C₂×C₄²	—	106	2	no	—
194	C₂×C₄²	—	106	1	no	—
195	#108	—	106	8	no	—
196	C₂⁴	Q₈×D₅₃	1	1	yes	characteristic
197	C₂²×C₄	—	106	4	no	—
198	C₂²×C₄	—	106	2	no	—
199	C₂²×C₄	—	212	4	no	—
200	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
203	C₂²×C₄	#11719945	1	2	yes	—
204	C₂²×C₄	#11719947	1	2	yes	—
205	C₂²×C₄	#11719947	1	2	yes	—
206	C₂²×C₄	#11719947	1	1	yes	characteristic
207	C₂²×C₄	#11719947	1	1	yes	characteristic
208	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
210	M₁₆	—	2	4	no	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	4	2	no	—
213	M₁₆	—	4	2	no	—
214	M₁₆	—	4	2	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	106	4	no	—
217	M₁₆	—	106	4	no	—
218	M₁₆	—	106	4	no	—
219	M₁₆	—	212	2	no	—
220	M₁₆	—	212	2	no	—
221	M₁₆	—	212	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	106	4	no	—
228	C₂×C₈	—	106	2	no	—
229	C₂×C₈	—	212	2	no	—
230	C₂×C₈	—	212	1	no	—
231	C₂×C₈	—	212	2	no	—
232	C₂×C₈	—	212	1	no	—
233	C₂×C₈	#11719921	1	4	yes	—
234	C₄²	—	212	2	no	—
235	C₄²	—	212	1	no	—
236	C₄²	—	212	1	no	—
237	C₂³	—	4	2	no	—
238	C₂³	#13133182	1	1	yes	characteristic
239	C₂³	#13133197	1	2	yes	—
240	C₂³	#13133197	1	1	yes	characteristic
241	C₂³	#13133207	1	2	yes	—
242	C₂³	#13133215	1	1	yes	characteristic
243	C₂×C₄	—	106	8	no	—
244	C₂×C₄	—	106	4	no	—
245	C₂×C₄	—	212	4	no	—
246	C₂×C₄	—	212	2	no	—
247	C₂×C₄	—	212	2	no	—
248	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
249	C₂×C₄	#13133173	1	2	yes	—
250	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
251	C₂×C₄	#13133182	1	1	yes	characteristic
252	C₂×C₄	#13133191	1	2	yes	—
253	C₂×C₄	#13133197	1	2	yes	—
254	C₂×C₄	#13133197	1	1	yes	characteristic
255	C₂×C₄	#13133198	1	2	yes	—
256	C₂×C₄	#13133201	1	2	yes	—
257	C₂×C₄	#13133201	1	2	yes	—
258	C₂×C₄	#13133201	1	1	yes	characteristic
259	C₂×C₄	#13133201	1	1	yes	characteristic
260	C₂×C₄	#13133204	1	1	yes	characteristic
261	C₂×C₄	#13133205	1	1	yes	characteristic
262	C₂×C₄	#13133207	1	2	yes	—
263	C₂×C₄	#13133210	1	2	yes	—
264	C₂×C₄	#13133210	1	2	yes	—
265	C₂×C₄	#13133214	1	1	yes	characteristic
266	C₈	—	2	4	no	—
267	C₈	—	4	4	no	—
268	C₈	—	212	4	no	—
269	C₈	—	212	2	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	1	no	—
275	C₂²	#16810508	1	2	yes	—
276	C₂²	#16810528	1	1	yes	characteristic
277	C₂²	#16810553	1	2	yes	—
278	C₂²	#16810560	1	1	yes	characteristic
279	C₂²	#16810563	1	1	yes	characteristic
280	C₄	—	212	4	no	—
281	C₄	#16810484	1	1	yes	characteristic
282	C₄	#16810508	1	2	yes	—
283	C₄	#16810525	1	1	yes	characteristic
284	C₄	#16810548	1	2	yes	—
285	C₄	#16810560	1	1	yes	characteristic
286	C₄	#16810563	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	2	1	no	—
289	C₂	—	2	2	no	—
290	C₂	—	2	1	no	—
291	C₂	#20272982	1	1	yes	characteristic
292	C₂	#20276232	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133214
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#338	C₂
socle	C₂²×C₅₃	#399
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#338	D₅₃
hypocenter	C₅₃	#11375
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#338

Complements

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

N	H ≅ G/N	type
10. #16810672	C₄	semidirect
164. C₅₃	#11375	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810672	2
#20274442	2
#20274408	2
C₂×C₅₃×#338	2
#11375	53