Subgroups — #27292365

Order 13568.

634 conjugacy classes of subgroups.

The subgroup lattice has 304 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	1	yes	maximal, characteristic
2	C₂×#16810391	C₂	1	2	yes	maximal
3	C₂×#16810388	C₂	1	1	yes	maximal, characteristic
4	#20274436	C₂	1	2	yes	maximal
5	C₅₃×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	—
8	C₂×#13133151	—	2	4	no	—
9	C₂×#13133151	—	2	2	no	—
10	#16810391	—	2	4	no	—
11	#16810388	C₄	1	4	yes	complemented
12	#16810385	—	2	4	no	—
13	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#336	—	2	2	no	—
17	C₂³×Dic₅₃	—	2	2	no	—
18	C₂×#11719917	—	2	2	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	C₂×C₄	1	4	yes	—
24	C₂²×#88322	—	2	2	no	—
25	C₂×C₄⋊Dic₅₃	—	2	2	no	—
26	C₂×C₄⋊Dic₅₃	—	2	2	no	—
27	C₂×C₄⋊Dic₅₃	—	2	2	no	—
28	C₂×C₄⋊Dic₅₃	—	2	2	no	—
29	C₂×C₄⋊Dic₅₃	—	2	2	no	—
30	C₂×C₄⋊Dic₅₃	D₄	1	4	yes	—
31	#13133151	—	2	8	no	—
32	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
33	C₂×M₁₆×C₅₃	—	2	2	no	—
34	C₂×M₁₆×C₅₃	—	2	1	no	—
35	C₂×M₁₆×C₅₃	—	2	2	no	—
36	C₂×M₁₆×C₅₃	Q₈	1	2	yes	—
37	C₂²×C₈×C₅₃	—	2	2	no	—
38	C₂²×C₈×C₅₃	—	2	1	no	—
39	C₂²×C₈×C₅₃	—	2	2	no	—
40	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₅₃	D₄	1	2	yes	—
42	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
43	#117×C₅₃	—	2	2	no	—
44	#117×C₅₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
46	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
47	C₂×C₄²×C₅₃	—	2	1	no	—
48	C₂⁴×C₅₃	#41	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	—	2	2	no	—
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	2	yes	—
53	C₂²×C₄×C₅₃	C₂×D₄	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×C₄×C₅₃	#41	1	2	yes	—
59	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
60	C₂²×Dic₅₃	—	2	4	no	—
61	C₂²×Dic₅₃	—	4	2	no	—
62	C₂²×Dic₅₃	—	4	2	no	—
63	C₂²×Dic₅₃	—	8	4	no	—
64	M₁₆×C₅₃	—	2	4	no	—
65	M₁₆×C₅₃	—	4	2	no	—
66	C₂×C₈×C₅₃	—	2	4	no	—
67	C₂×C₈×C₅₃	—	2	2	no	—
68	C₂×C₈×C₅₃	—	2	4	no	—
69	C₂×C₈×C₅₃	—	2	2	no	—
70	C₂×C₈×C₅₃	—	2	2	no	—
71	C₂×C₈×C₅₃	—	4	1	no	—
72	C₂×C₈×C₅₃	—	4	1	no	—
73	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
74	C₄⋊C₄×C₅₃	—	2	4	no	—
75	C₂²⋊C₄×C₅₃	—	4	2	no	—
76	C₄²×C₅₃	—	4	1	no	—
77	C₄²×C₅₃	—	4	1	no	—
78	C₄⋊Dic₅₃	—	2	4	no	—
79	C₄⋊Dic₅₃	—	2	4	no	—
80	C₄⋊Dic₅₃	—	4	2	no	—
81	C₄⋊Dic₅₃	—	4	2	no	—
82	#11719917	—	2	4	no	—
83	#11719917	—	2	4	no	—
84	#11719917	—	4	2	no	—
85	#11719917	—	4	2	no	—
86	C₂×#88322	—	2	4	no	—
87	C₂×#88322	—	4	2	no	—
88	C₂×#88322	—	4	2	no	—
89	C₂³×C₅₃	—	2	2	no	—
90	C₂³×C₅₃	—	2	2	no	—
91	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
92	C₂³×C₅₃	#121	1	1	yes	characteristic
93	C₂³×C₅₃	#123	1	2	yes	—
94	C₂³×C₅₃	#125	1	1	yes	characteristic
95	C₂³×C₅₃	#126	1	2	yes	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	—	4	2	no	—
99	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#117	1	2	yes	—
101	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
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	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#123	1	2	yes	—
110	C₂×C₄×C₅₃	#124	1	2	yes	—
111	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#125	1	2	yes	—
114	C₂×C₄×C₅₃	#125	1	2	yes	—
115	C₂×C₄×C₅₃	#126	1	2	yes	—
116	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
117	C₂×Dic₅₃	—	4	4	no	—
118	C₂×Dic₅₃	—	8	4	no	—
119	C₂×Dic₅₃	—	8	4	no	—
120	C₂×Dic₅₃	—	8	4	no	—
121	C₈×C₅₃	—	2	4	no	—
122	C₈×C₅₃	—	4	2	no	—
123	#88322	—	4	4	no	—
124	#11270	—	53	1	no	maximal, Sylow
125	C₂²×C₅₃	—	2	4	no	—
126	C₂²×C₅₃	—	2	2	no	—
127	C₂²×C₅₃	—	2	2	no	—
128	C₂²×C₅₃	—	2	2	no	—
129	C₂²×C₅₃	—	2	2	no	—
130	C₂²×C₅₃	#387	1	2	yes	—
131	C₂²×C₅₃	#389	1	1	yes	characteristic
132	C₂²×C₅₃	#394	1	1	yes	characteristic
133	C₂²×C₅₃	#397	1	1	yes	characteristic
134	C₂²×C₅₃	#400	1	2	yes	—
135	C₂²×C₅₃	#472	1	2	yes	—
136	C₂²×C₅₃	#473	1	2	yes	—
137	C₄×C₅₃	—	8	2	no	—
138	C₄×C₅₃	#382	1	1	yes	characteristic
139	C₄×C₅₃	#385	1	1	yes	characteristic
140	C₄×C₅₃	#388	1	2	yes	—
141	C₄×C₅₃	#393	1	1	yes	characteristic
142	C₄×C₅₃	#398	1	2	yes	—
143	C₄×C₅₃	#400	1	1	yes	characteristic
144	Dic₅₃	—	8	4	no	—
145	C₂²×#107	—	53	1	no	—
146	C₂×#343	—	53	2	no	—
147	C₂×#341	—	53	1	no	—
148	#1736	—	53	2	no	—
149	#1729	D₅₃	1	1	yes	characteristic
150	C₂×C₅₃	—	2	2	no	—
151	C₂×C₅₃	—	2	2	no	—
152	C₂×C₅₃	#1442	1	1	yes	characteristic
153	C₂×C₅₃	#1952	1	2	yes	—
154	C₂×C₅₃	#2045	1	1	yes	characteristic
155	C₂×C₅₃	#2047	1	1	yes	characteristic
156	C₂×C₅₃	#2078	1	2	yes	—
157	C₂²×M₁₆	—	53	2	no	—
158	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
159	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
160	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
161	C₂²×C₄⋊C₄	—	53	2	no	—
162	C₂×#107	—	106	4	no	—
163	C₂×#107	—	106	2	no	—
164	#343	—	106	4	no	—
165	#341	—	53	4	no	—
166	#340	—	106	4	no	—
167	#336	—	2	2	no	—
168	C₅₃	#11270	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₄	—	106	2	no	—
170	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
171	C₂×M₁₆	—	2	1	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	53	4	no	—
175	C₂×M₁₆	—	106	2	no	—
176	C₂×M₁₆	—	106	2	no	—
177	C₂×M₁₆	—	106	2	no	—
178	C₂×M₁₆	—	106	2	no	—
179	C₂×M₁₆	—	106	2	no	—
180	C₂×M₁₆	Dic₁₀₆	1	2	yes	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	1	no	—
184	C₂²×C₈	—	106	2	no	—
185	C₂²×C₈	Dic₁₀₆	1	2	yes	—
186	C₂²×C₈	C₄×D₅₃	1	2	yes	—
187	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
188	#117	—	2	2	no	—
189	#117	—	2	2	no	—
190	C₂×C₄⋊C₄	—	53	4	no	—
191	C₂×C₄⋊C₄	—	106	2	no	—
192	C₂×C₄⋊C₄	—	106	2	no	—
193	C₂×C₄⋊C₄	—	106	2	no	—
194	C₂×C₄⋊C₄	—	106	2	no	—
195	C₂×C₄⋊C₄	—	106	2	no	—
196	C₂×C₄⋊C₄	D₂₁₂	1	2	yes	—
197	C₂×C₂²⋊C₄	—	2	1	no	—
198	C₂×C₄²	—	2	1	no	—
199	#107	—	106	8	no	—
200	C₂⁴	#11719945	1	1	yes	characteristic
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	106	4	no	—
203	C₂²×C₄	—	424	4	no	—
204	C₂²×C₄	—	212	2	no	—
205	C₂²×C₄	—	212	2	no	—
206	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
207	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
210	C₂²×C₄	#11719945	1	1	yes	characteristic
211	C₂²×C₄	D₄×D₅₃	1	2	yes	—
212	C₂²×C₄	#11719947	1	2	yes	—
213	C₂²×C₄	#11719947	1	2	yes	—
214	C₂²×C₄	#11719949	1	2	yes	—
215	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
216	M₁₆	—	2	4	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	106	4	no	—
219	M₁₆	—	106	4	no	—
220	M₁₆	—	212	2	no	—
221	M₁₆	—	212	2	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	106	4	no	—
230	C₂×C₈	—	212	2	no	—
231	C₂×C₈	—	212	2	no	—
232	C₂×C₈	#11719919	1	4	yes	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	106	4	no	—
235	C₄⋊C₄	—	106	4	no	—
236	C₄⋊C₄	—	212	2	no	—
237	C₄⋊C₄	—	212	2	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₄²	—	4	1	no	—
240	C₄²	—	4	1	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	C₄×Dic₁₀₆	1	1	yes	characteristic
244	C₂³	#13133191	1	1	yes	characteristic
245	C₂³	#13133202	1	2	yes	—
246	C₂³	#13133209	1	1	yes	characteristic
247	C₂³	#13133210	1	2	yes	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	424	4	no	—
252	C₂×C₄	—	424	4	no	—
253	C₂×C₄	—	424	4	no	—
254	C₂×C₄	—	212	4	no	—
255	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
256	C₂×C₄	#13133175	1	2	yes	—
257	C₂×C₄	#13133177	1	2	yes	—
258	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
259	C₂×C₄	#13133185	1	1	yes	characteristic
260	C₂×C₄	#13133192	1	2	yes	—
261	C₂×C₄	#13133194	1	2	yes	—
262	C₂×C₄	#13133194	1	2	yes	—
263	C₂×C₄	#13133196	1	1	yes	characteristic
264	C₂×C₄	#13133198	1	1	yes	characteristic
265	C₂×C₄	#13133201	1	2	yes	—
266	C₂×C₄	#13133203	1	2	yes	—
267	C₂×C₄	#13133204	1	1	yes	characteristic
268	C₂×C₄	#13133206	1	1	yes	characteristic
269	C₂×C₄	#13133208	1	2	yes	—
270	C₂×C₄	#13133208	1	2	yes	—
271	C₂×C₄	#13133212	1	2	yes	—
272	C₂×C₄	#13133217	1	1	yes	characteristic
273	C₈	—	2	4	no	—
274	C₈	—	4	2	no	—
275	C₈	—	212	4	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	4	no	—
281	C₂²	#16810510	1	2	yes	—
282	C₂²	#16810521	1	1	yes	characteristic
283	C₂²	#16810540	1	1	yes	characteristic
284	C₂²	#16810548	1	1	yes	characteristic
285	C₂²	#16810565	1	2	yes	—
286	C₂²	#16810893	1	2	yes	—
287	C₂²	#16810898	1	2	yes	—
288	C₄	—	8	2	no	—
289	C₄	—	424	4	no	—
290	C₄	#16810488	1	1	yes	characteristic
291	C₄	#16810495	1	1	yes	characteristic
292	C₄	#16810515	1	2	yes	—
293	C₄	#16810535	1	1	yes	characteristic
294	C₄	#16810555	1	2	yes	—
295	C₄	#16810562	1	1	yes	characteristic
296	C₂	—	2	2	no	—
297	C₂	—	2	2	no	—
298	C₂	#20272971	1	1	yes	characteristic
299	C₂	#20275500	1	2	yes	—
300	C₂	#20275971	1	1	yes	characteristic
301	C₂	#20275983	1	1	yes	characteristic
302	C₂	#20276155	1	2	yes	—

	subgroup	G/H
center	C₂³	#13133191
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1729	C₂
socle	C₂³×C₅₃	#121
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1729	D₅₃
hypocenter	C₅₃	#11270
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
11. #16810388	C₄	semidirect
168. C₅₃	#11270	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×#16810391	2
C₂×#16810388	2
#20274436	2
C₅₃×#1729	2
#11270	53