Subgroups — #27275027

Order 13568.

488 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274464	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133128	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810365	C₂	1	2	yes	maximal
4	#20272596	C₂	1	2	yes	maximal
5	C₄×#125×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133128	—	2	1	no	—
7	C₂×#13133128	C₂²	1	2	yes	—
8	C₂×#13133128	C₂²	1	2	yes	—
9	C₂×C₄×#88322	—	2	1	no	—
10	#16810378	—	2	1	no	—
11	#16810378	—	2	1	no	—
12	#16810365	C₄	1	8	yes	complemented
13	C₂×#125×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#389	C₄	1	1	yes	characteristic
15	C₅₃×#389	C₄	1	1	yes	characteristic
16	C₅₃×#387	—	2	1	no	—
17	C₅₃×#387	—	2	1	no	—
18	C₅₃×#382	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
20	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
21	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
23	C₄³×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²×#88322	—	2	2	no	—
25	C₂²×#88322	—	2	2	no	—
26	#13133128	—	2	4	no	—
27	#13133128	—	2	2	no	—
28	#13133128	—	2	2	no	—
29	#13133128	C₂×C₄	1	4	yes	—
30	#13133128	C₂×C₄	1	4	yes	—
31	C₄×#88322	—	2	4	no	—
32	#125×C₅₃	—	2	2	no	—
33	#125×C₅₃	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
38	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	—	2	1	no	—
43	C₂×C₄²×C₅₃	—	2	1	no	—
44	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
51	#95×C₅₃	—	2	1	no	—
52	#95×C₅₃	—	2	1	no	—
53	#95×C₅₃	—	2	1	no	—
54	#95×C₅₃	—	2	1	no	—
55	#95×C₅₃	—	2	1	no	—
56	#95×C₅₃	—	2	1	no	—
57	C₂²×C₄×C₅₃	—	2	1	no	—
58	C₂²×C₄×C₅₃	—	2	1	no	—
59	C₂²×C₄×C₅₃	—	2	1	no	—
60	C₂²×C₄×C₅₃	—	2	1	no	—
61	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₅₃	—	2	1	no	—
69	C₄⋊C₄×C₅₃	—	2	1	no	—
70	C₄⋊C₄×C₅₃	—	2	4	no	—
71	C₄⋊C₄×C₅₃	—	2	4	no	—
72	C₄⋊C₄×C₅₃	—	4	2	no	—
73	C₄⋊C₄×C₅₃	—	4	1	no	—
74	C₄⋊C₄×C₅₃	—	4	1	no	—
75	C₄⋊C₄×C₅₃	—	4	2	no	—
76	C₄⋊C₄×C₅₃	C₄²	1	2	yes	—
77	C₄⋊C₄×C₅₃	C₄²	1	2	yes	—
78	C₄²×C₅₃	—	2	2	no	—
79	C₄²×C₅₃	—	2	2	no	—
80	C₄²×C₅₃	—	2	2	no	—
81	C₄²×C₅₃	—	4	1	no	—
82	C₄²×C₅₃	—	4	1	no	—
83	C₄²×C₅₃	—	4	1	no	—
84	C₄²×C₅₃	—	4	2	no	—
85	C₄²×C₅₃	—	4	2	no	—
86	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
87	C₄²×C₅₃	#41	1	2	yes	—
88	C₂×#88322	—	2	4	no	—
89	C₂×#88322	—	2	4	no	—
90	C₂×#88322	—	2	4	no	—
91	C₂×#88322	—	2	4	no	—
92	C₂×#88322	—	4	2	no	—
93	C₂×#88322	—	4	2	no	—
94	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	1	no	—
103	C₂×C₄×C₅₃	—	2	1	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	2	no	—
108	C₂×C₄×C₅₃	—	2	2	no	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	—	2	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	—	4	1	no	—
121	C₂×C₄×C₅₃	—	4	1	no	—
122	C₂×C₄×C₅₃	#101	1	2	yes	—
123	C₂×C₄×C₅₃	#104	1	2	yes	—
124	C₂×C₄×C₅₃	#104	1	2	yes	—
125	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
126	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
129	#88322	—	2	8	no	—
130	#88322	—	4	4	no	—
131	#8227	—	53	1	no	maximal, Sylow
132	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
134	C₂²×C₅₃	#413	1	1	yes	characteristic
135	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
136	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
137	C₂²×C₅₃	#421	1	1	yes	characteristic
138	C₂²×C₅₃	#421	1	1	yes	characteristic
139	C₄×C₅₃	—	2	1	no	—
140	C₄×C₅₃	—	2	1	no	—
141	C₄×C₅₃	—	2	2	no	—
142	C₄×C₅₃	—	2	2	no	—
143	C₄×C₅₃	—	2	2	no	—
144	C₄×C₅₃	—	2	2	no	—
145	C₄×C₅₃	—	2	2	no	—
146	C₄×C₅₃	—	4	2	no	—
147	C₄×C₅₃	—	4	2	no	—
148	C₄×C₅₃	—	4	1	no	—
149	C₄×C₅₃	—	8	1	no	—
150	C₄×C₅₃	—	8	1	no	—
151	C₄×#125	D₅₃	1	1	yes	characteristic
152	#1745	—	53	1	no	—
153	C₄×#97	—	53	1	no	—
154	C₂×#330	—	53	2	no	—
155	#1295	—	53	2	no	—
156	C₂×C₅₃	#1528	1	1	yes	characteristic
157	C₂×C₅₃	#1529	1	1	yes	characteristic
158	C₂×C₅₃	#1531	1	1	yes	characteristic
159	C₂×C₅₃	#1532	1	1	yes	characteristic
160	C₂×C₅₃	C₄×#101	1	1	yes	characteristic
161	C₂×C₅₃	C₄×#104	1	1	yes	characteristic
162	C₂×C₅₃	#1759	1	1	yes	characteristic
163	C₂×#125	C₂×D₅₃	1	1	yes	characteristic
164	C₂×#97	—	53	2	no	—
165	C₂×#97	—	53	2	no	—
166	C₂×#97	—	106	1	no	—
167	C₂×C₄×C₈	—	106	1	no	—
168	#389	Dic₅₃	1	1	yes	characteristic
169	#389	Dic₅₃	1	1	yes	characteristic
170	#387	—	2	1	no	—
171	#387	—	2	1	no	—
172	#382	C₂×D₅₃	1	1	yes	characteristic
173	C₄×C₄⋊C₄	—	2	1	no	—
174	C₄×C₄⋊C₄	—	2	1	no	—
175	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
176	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
177	C₄³	C₂×D₅₃	1	1	yes	characteristic
178	#336	—	106	1	no	—
179	#336	—	106	1	no	—
180	#330	—	53	8	no	—
181	C₅₃	#8227	1	1	yes	characteristic, complemented, Sylow
182	C₂²×C₈	—	106	2	no	—
183	C₂²×C₈	—	106	2	no	—
184	#125	—	2	2	no	—
185	#125	C₄×D₅₃	1	4	yes	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
191	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
192	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
193	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
194	C₂×C₄²	—	2	1	no	—
195	C₂×C₄²	—	2	1	no	—
196	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
197	C₂×C₄²	C₂×Dic₅₃	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	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
202	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
203	#97	—	53	4	no	—
204	#97	—	53	4	no	—
205	#97	—	106	4	no	—
206	#97	—	106	2	no	—
207	#97	—	106	2	no	—
208	C₄×C₈	—	106	4	no	—
209	#95	—	2	1	no	—
210	#95	—	2	1	no	—
211	#95	—	2	1	no	—
212	#95	—	2	1	no	—
213	#95	—	2	1	no	—
214	#95	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
220	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
221	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
222	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
223	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
226	C₂×C₈	—	106	4	no	—
227	C₂×C₈	—	106	4	no	—
228	C₂×C₈	—	106	4	no	—
229	C₂×C₈	—	106	4	no	—
230	C₂×C₈	—	212	2	no	—
231	C₂×C₈	—	212	2	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	C₄×Dic₅₃	1	2	yes	—
241	C₄⋊C₄	C₄×Dic₅₃	1	2	yes	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	4	1	no	—
246	C₄²	—	4	1	no	—
247	C₄²	—	4	1	no	—
248	C₄²	—	4	2	no	—
249	C₄²	—	4	2	no	—
250	C₄²	C₂×C₄×D₅₃	1	2	yes	—
251	C₄²	#11719945	1	2	yes	—
252	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
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	1	no	—
262	C₂×C₄	—	2	1	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	—	2	1	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	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	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	#13133136	1	2	yes	—
281	C₂×C₄	#13133144	1	2	yes	—
282	C₂×C₄	#13133144	1	2	yes	—
283	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
284	C₂×C₄	#13133176	1	1	yes	characteristic
285	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
286	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
287	C₈	—	106	8	no	—
288	C₈	—	212	4	no	—
289	C₂²	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
290	C₂²	C₂×#13133136	1	1	yes	characteristic
291	C₂²	#16810607	1	1	yes	characteristic
292	C₂²	C₂×#13133144	1	1	yes	characteristic
293	C₂²	C₂×#13133144	1	1	yes	characteristic
294	C₂²	#16810640	1	1	yes	characteristic
295	C₂²	#16810640	1	1	yes	characteristic
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	2	no	—
303	C₄	—	4	2	no	—
304	C₄	—	4	2	no	—
305	C₄	—	4	1	no	—
306	C₄	—	8	1	no	—
307	C₄	—	8	1	no	—
308	C₂	#20273317	1	1	yes	characteristic
309	C₂	#20273322	1	1	yes	characteristic
310	C₂	#20273330	1	1	yes	characteristic
311	C₂	#20273335	1	1	yes	characteristic
312	C₂	C₄×#13133136	1	1	yes	characteristic
313	C₂	C₄×#13133144	1	1	yes	characteristic
314	C₂	#20274525	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
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	#8227
LCS	G ▷ C₂×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
12. #16810365	C₄	semidirect
181. C₅₃	#8227	semidirect

Maximal subgroups

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

subgroup	index
#20274464	2
C₄×#13133128	2
C₂×#16810365	2
#20272596	2
C₄×#125×C₅₃	2
#8227	53