Subgroups — #27272285

Order 13568.

410 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274832	C₂	1	1	yes	maximal, characteristic
2	#20274464	C₂	1	1	yes	maximal, characteristic
3	#20272538	C₂	1	2	yes	maximal
4	#20272538	C₂	1	2	yes	maximal
5	C₄×#125×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	—	2	1	no	—
7	C₂×#13133128	—	2	1	no	—
8	#16810378	—	2	1	no	—
9	#16810378	—	2	1	no	—
10	#16810378	C₂²	1	2	yes	—
11	#16810378	C₂²	1	2	yes	—
12	C₂×#125×C₅₃	C₄	1	1	yes	characteristic
13	C₅₃×#389	C₂²	1	1	yes	characteristic
14	C₅₃×#389	C₂²	1	1	yes	characteristic
15	C₅₃×#387	—	2	1	no	—
16	C₅₃×#387	—	2	1	no	—
17	C₅₃×#382	C₄	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
19	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
20	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
22	C₄³×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×#88322	—	2	2	no	—
24	C₂²×#88322	—	2	2	no	—
25	C₄⋊#88322	—	4	2	no	—
26	#13133128	—	2	4	no	—
27	#125×C₅₃	—	2	2	no	—
28	#125×C₅₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	—	2	1	no	—
38	C₂×C₄²×C₅₃	—	2	1	no	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	#95×C₅₃	—	2	1	no	—
47	#95×C₅₃	—	2	1	no	—
48	#95×C₅₃	—	2	1	no	—
49	#95×C₅₃	—	2	1	no	—
50	#95×C₅₃	—	2	1	no	—
51	#95×C₅₃	—	2	1	no	—
52	C₂²×C₄×C₅₃	—	2	1	no	—
53	C₂²×C₄×C₅₃	—	2	1	no	—
54	C₂²×C₄×C₅₃	—	2	1	no	—
55	C₂²×C₄×C₅₃	—	2	1	no	—
56	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₄⋊C₄×C₅₃	—	2	4	no	—
64	C₄⋊C₄×C₅₃	—	2	4	no	—
65	C₄⋊C₄×C₅₃	—	2	1	no	—
66	C₄⋊C₄×C₅₃	—	2	1	no	—
67	C₄⋊C₄×C₅₃	—	4	1	no	—
68	C₄⋊C₄×C₅₃	—	4	1	no	—
69	C₄⋊C₄×C₅₃	—	4	2	no	—
70	C₄⋊C₄×C₅₃	—	4	1	no	—
71	C₄⋊C₄×C₅₃	—	4	1	no	—
72	C₄⋊C₄×C₅₃	M₁₆	1	2	yes	—
73	C₄⋊C₄×C₅₃	M₁₆	1	2	yes	—
74	C₄²×C₅₃	—	2	2	no	—
75	C₄²×C₅₃	—	2	2	no	—
76	C₄²×C₅₃	—	2	1	no	—
77	C₄²×C₅₃	—	2	1	no	—
78	C₄²×C₅₃	—	2	2	no	—
79	C₄²×C₅₃	—	2	1	no	—
80	C₄²×C₅₃	—	2	1	no	—
81	C₄²×C₅₃	—	4	1	no	—
82	C₄²×C₅₃	—	4	1	no	—
83	C₄²×C₅₃	—	4	2	no	—
84	C₄²×C₅₃	—	4	2	no	—
85	C₂×#88322	—	2	4	no	—
86	C₂×#88322	—	4	2	no	—
87	C₂×#88322	—	4	2	no	—
88	C₂×#88322	—	4	2	no	—
89	C₂×#88322	—	4	2	no	—
90	C₂×#88322	—	4	2	no	—
91	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	1	no	—
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	—	2	1	no	—
98	C₂×C₄×C₅₃	—	2	1	no	—
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	2	1	no	—
101	C₂×C₄×C₅₃	—	2	1	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	2	1	no	—
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	1	no	—
108	C₂×C₄×C₅₃	—	2	1	no	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	4	1	no	—
111	C₂×C₄×C₅₃	—	4	1	no	—
112	C₂×C₄×C₅₃	—	4	1	no	—
113	C₂×C₄×C₅₃	—	4	1	no	—
114	C₂×C₄×C₅₃	—	4	1	no	—
115	C₂×C₄×C₅₃	—	4	1	no	—
116	C₂×C₄×C₅₃	—	4	1	no	—
117	C₂×C₄×C₅₃	—	4	1	no	—
118	C₂×C₄×C₅₃	—	4	1	no	—
119	C₂×C₄×C₅₃	—	4	1	no	—
120	C₂×C₄×C₅₃	#101	1	2	yes	—
121	C₂×C₄×C₅₃	#104	1	2	yes	—
122	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
124	C₂×C₄×C₅₃	#135	1	2	yes	—
125	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
127	#88322	—	4	4	no	—
128	#88322	—	8	2	no	—
129	#7619	—	53	1	no	maximal, Sylow
130	C₂²×C₅₃	#407	1	1	yes	characteristic
131	C₂²×C₅₃	#410	1	1	yes	characteristic
132	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
133	C₂²×C₅₃	#416	1	1	yes	characteristic
134	C₂²×C₅₃	#417	1	1	yes	characteristic
135	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
136	C₂²×C₅₃	#421	1	1	yes	characteristic
137	C₄×C₅₃	—	2	2	no	—
138	C₄×C₅₃	—	2	1	no	—
139	C₄×C₅₃	—	2	1	no	—
140	C₄×C₅₃	—	2	2	no	—
141	C₄×C₅₃	—	2	1	no	—
142	C₄×C₅₃	—	2	1	no	—
143	C₄×C₅₃	—	2	2	no	—
144	C₄×C₅₃	—	4	1	no	—
145	C₄×C₅₃	—	4	2	no	—
146	C₄×C₅₃	—	4	1	no	—
147	C₄×C₅₃	—	4	2	no	—
148	C₄×C₅₃	—	8	1	no	—
149	C₄×C₅₃	—	8	1	no	—
150	C₄×#125	D₅₃	1	1	yes	characteristic
151	#1831	—	53	1	no	—
152	#1745	—	53	1	no	—
153	#1271	—	53	2	no	—
154	#1271	—	53	2	no	—
155	C₂×C₅₃	#1467	1	1	yes	characteristic
156	C₂×C₅₃	#1488	1	1	yes	characteristic
157	C₂×C₅₃	#1489	1	1	yes	characteristic
158	C₂×C₅₃	#1500	1	1	yes	characteristic
159	C₂×C₅₃	#1512	1	1	yes	characteristic
160	C₂×C₅₃	#1528	1	1	yes	characteristic
161	C₂×C₅₃	#1552	1	1	yes	characteristic
162	C₂×#125	Dic₅₃	1	1	yes	characteristic
163	C₂×C₄⋊C₈	—	106	1	no	—
164	C₂×#97	—	106	1	no	—
165	#389	C₂×D₅₃	1	1	yes	characteristic
166	#389	C₂×D₅₃	1	1	yes	characteristic
167	#387	—	2	1	no	—
168	#387	—	2	1	no	—
169	#382	Dic₅₃	1	1	yes	characteristic
170	C₄×C₄⋊C₄	—	2	1	no	—
171	C₄×C₄⋊C₄	—	2	1	no	—
172	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
173	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
174	C₄³	C₂×D₅₃	1	1	yes	characteristic
175	#336	—	53	2	no	—
176	#336	—	53	2	no	—
177	#336	—	106	1	no	—
178	#336	—	106	1	no	—
179	C₅₃	#7619	1	1	yes	characteristic, complemented, Sylow
180	C₂²×C₈	—	106	2	no	—
181	C₂²×C₈	—	106	2	no	—
182	#125	—	2	2	no	—
183	#125	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
189	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
190	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
191	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	—	2	1	no	—
194	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
195	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
196	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
197	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
198	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
199	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
200	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
201	C₄⋊C₈	—	212	2	no	—
202	#97	—	106	4	no	—
203	#95	—	2	1	no	—
204	#95	—	2	1	no	—
205	#95	—	2	1	no	—
206	#95	—	2	1	no	—
207	#95	—	2	1	no	—
208	#95	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
214	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
215	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
216	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
217	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
220	C₂×C₈	—	106	4	no	—
221	C₂×C₈	—	212	2	no	—
222	C₂×C₈	—	212	2	no	—
223	C₂×C₈	—	212	2	no	—
224	C₂×C₈	—	212	2	no	—
225	C₂×C₈	—	212	2	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	4	1	no	—
231	C₄⋊C₄	—	4	1	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	1	no	—
234	C₄⋊C₄	—	4	1	no	—
235	C₄⋊C₄	#11719917	1	2	yes	—
236	C₄⋊C₄	#11719917	1	2	yes	—
237	C₄²	—	2	2	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	1	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	4	1	no	—
246	C₄²	—	4	2	no	—
247	C₄²	—	4	2	no	—
248	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	#13133136	1	2	yes	—
278	C₂×C₄	#13133144	1	2	yes	—
279	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
280	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
281	C₂×C₄	#13133237	1	2	yes	—
282	C₂×C₄	#13133242	1	1	yes	characteristic
283	C₂×C₄	#13133247	1	1	yes	characteristic
284	C₈	—	424	2	no	—
285	C₈	—	212	4	no	—
286	C₂²	#16810587	1	1	yes	characteristic
287	C₂²	#16810597	1	1	yes	characteristic
288	C₂²	C₂×#13133136	1	1	yes	characteristic
289	C₂²	#16810619	1	1	yes	characteristic
290	C₂²	#16810624	1	1	yes	characteristic
291	C₂²	C₂×#13133144	1	1	yes	characteristic
292	C₂²	#16810640	1	1	yes	characteristic
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	2	no	—
300	C₄	—	4	1	no	—
301	C₄	—	4	2	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	2	no	—
304	C₄	—	8	1	no	—
305	C₄	—	8	1	no	—
306	C₂	#20273059	1	1	yes	characteristic
307	C₂	#20273149	1	1	yes	characteristic
308	C₂	#20273154	1	1	yes	characteristic
309	C₂	#20273201	1	1	yes	characteristic
310	C₂	#20273253	1	1	yes	characteristic
311	C₂	#20273317	1	1	yes	characteristic
312	C₂	#20273415	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×#125×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#125	D₅₃
hypocenter	C₅₃	#7619
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
179. C₅₃	#7619	semidirect

Maximal subgroups

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

subgroup	index
#20274832	2
#20274464	2
#20272538	2
C₄×#125×C₅₃	2
#7619	53