Subgroups — #27272140

Order 13568.

426 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	C₄×#13133209	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	#20272539	C₂	1	2	yes	maximal
4	#20272537	C₂	1	2	yes	maximal
5	C₅₃×#1839	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133209	C₄	1	1	yes	characteristic
7	C₂×C₄⋊#88322	—	2	1	no	—
8	C₂×C₄×#88322	—	2	1	no	—
9	#16810521	C₂²	1	1	yes	characteristic
10	#16810521	C₂²	1	1	yes	characteristic
11	#16810507	—	2	1	no	—
12	#16810507	—	2	1	no	—
13	#16810489	C₄	1	1	yes	characteristic
14	C₄×#11719919	—	2	1	no	—
15	C₄×#11719919	—	2	1	no	—
16	C₄×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
17	C₄×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
18	#16810378	C₂²	1	2	yes	—
19	C₄³×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#336	—	2	1	no	—
21	C₅₃×#336	—	2	1	no	—
22	C₅₃×#336	C₂²	1	2	yes	—
23	C₂²×#88322	—	2	2	no	—
24	#13133209	—	2	2	no	—
25	#13133209	C₈	1	4	yes	complemented
26	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×#11719919	—	2	1	no	—
31	C₂×#11719919	—	2	1	no	—
32	C₂×#11719919	—	2	1	no	—
33	C₂×#11719919	—	2	1	no	—
34	C₂×C₄×Dic₅₃	—	2	1	no	—
35	C₂×C₄×Dic₅₃	—	2	1	no	—
36	C₄⋊#88322	—	2	4	no	—
37	C₄×#88322	—	4	2	no	—
38	#13133124	—	2	1	no	—
39	#13133124	—	2	1	no	—
40	#13133124	—	2	1	no	—
41	#13133124	—	2	1	no	—
42	#13133124	—	2	1	no	—
43	#13133124	—	2	1	no	—
44	C₂²×C₈×C₅₃	—	2	2	no	—
45	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	C₂×C₄	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	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₅₃	—	2	1	no	—
60	C₂²×Dic₅₃	—	2	1	no	—
61	C₂²×Dic₅₃	—	2	1	no	—
62	C₂²×Dic₅₃	—	2	1	no	—
63	C₂×C₈×C₅₃	—	4	2	no	—
64	C₂×C₈×C₅₃	—	4	2	no	—
65	C₂×C₈×C₅₃	—	4	2	no	—
66	C₄²×C₅₃	—	2	2	no	—
67	C₄²×C₅₃	—	2	2	no	—
68	C₄²×C₅₃	—	2	2	no	—
69	C₄²×C₅₃	—	2	1	no	—
70	C₄²×C₅₃	—	2	1	no	—
71	C₄²×C₅₃	—	4	1	no	—
72	C₄²×C₅₃	—	4	1	no	—
73	C₄²×C₅₃	C₂×C₈	1	2	yes	—
74	C₄²×C₅₃	M₁₆	1	2	yes	—
75	C₄⋊Dic₅₃	—	2	1	no	—
76	C₄⋊Dic₅₃	—	2	1	no	—
77	C₄⋊Dic₅₃	—	4	1	no	—
78	C₄⋊Dic₅₃	—	4	1	no	—
79	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
80	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
81	#11719919	—	2	4	no	—
82	#11719919	—	2	4	no	—
83	#11719919	—	4	2	no	—
84	#11719919	—	4	1	no	—
85	#11719919	—	4	1	no	—
86	C₄×Dic₅₃	—	4	2	no	—
87	C₄×Dic₅₃	—	4	2	no	—
88	C₂×#88322	—	2	4	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	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	C₂²⋊C₈	1	2	yes	—
107	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
108	C₂×C₄×C₅₃	#101	1	2	yes	—
109	C₂×C₄×C₅₃	#104	1	2	yes	—
110	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#135	1	2	yes	—
113	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
115	C₂×Dic₅₃	—	2	2	no	—
116	C₂×Dic₅₃	—	2	2	no	—
117	C₂×Dic₅₃	—	4	1	no	—
118	C₂×Dic₅₃	—	4	1	no	—
119	C₂×Dic₅₃	—	4	1	no	—
120	C₂×Dic₅₃	—	4	1	no	—
121	C₂×Dic₅₃	—	4	1	no	—
122	C₂×Dic₅₃	—	4	1	no	—
123	C₂×Dic₅₃	—	4	1	no	—
124	C₂×Dic₅₃	—	4	1	no	—
125	C₂×Dic₅₃	—	4	1	no	—
126	C₂×Dic₅₃	—	4	1	no	—
127	C₈×C₅₃	—	8	2	no	—
128	#88322	—	4	4	no	—
129	#7585	—	53	1	no	maximal, Sylow
130	C₂²×C₅₃	C₂×C₂²⋊C₈	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	2	no	—
142	C₄×C₅₃	—	2	1	no	—
143	C₄×C₅₃	—	2	1	no	—
144	C₄×C₅₃	—	2	2	no	—
145	C₄×C₅₃	—	2	2	no	—
146	Dic₅₃	—	4	2	no	—
147	Dic₅₃	—	4	2	no	—
148	Dic₅₃	—	8	1	no	—
149	Dic₅₃	—	8	1	no	—
150	C₄×#125	—	53	1	no	—
151	#1839	D₅₃	1	1	yes	characteristic
152	C₄×C₄⋊C₈	—	53	1	no	—
153	#1271	—	53	2	no	—
154	#1271	—	53	2	no	—
155	C₂×C₅₃	#1461	1	1	yes	characteristic
156	C₂×C₅₃	#1476	1	1	yes	characteristic
157	C₂×C₅₃	#1477	1	1	yes	characteristic
158	C₂×C₅₃	#1509	1	1	yes	characteristic
159	C₂×C₅₃	#1521	1	1	yes	characteristic
160	C₂×C₅₃	#1528	1	1	yes	characteristic
161	C₂×C₅₃	#1552	1	1	yes	characteristic
162	C₂×#125	—	53	1	no	—
163	C₂×C₄⋊C₈	—	106	1	no	—
164	C₂×C₄×C₈	—	106	1	no	—
165	#389	—	53	1	no	—
166	#389	—	53	1	no	—
167	#387	—	106	1	no	—
168	#387	—	106	1	no	—
169	#382	—	53	1	no	—
170	C₄×C₄⋊C₄	—	53	1	no	—
171	C₄×C₄⋊C₄	—	53	1	no	—
172	C₄×C₄⋊C₄	—	106	1	no	—
173	C₄×C₄⋊C₄	—	106	1	no	—
174	C₄³	C₂×D₅₃	1	1	yes	characteristic
175	#336	—	2	1	no	—
176	#336	—	2	1	no	—
177	#336	—	53	2	no	—
178	#336	C₂×D₅₃	1	2	yes	—
179	C₅₃	#7585	1	1	yes	characteristic, complemented, Sylow
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	106	2	no	—
182	#125	—	53	4	no	—
183	#125	—	106	2	no	—
184	C₂×C₄⋊C₄	—	53	1	no	—
185	C₂×C₄⋊C₄	—	53	1	no	—
186	C₂×C₄⋊C₄	—	53	1	no	—
187	C₂×C₄⋊C₄	—	53	1	no	—
188	C₂×C₄⋊C₄	—	106	1	no	—
189	C₂×C₄⋊C₄	—	106	1	no	—
190	C₂×C₄⋊C₄	—	106	1	no	—
191	C₂×C₄⋊C₄	—	106	1	no	—
192	C₂×C₄²	—	106	1	no	—
193	C₂×C₄²	—	106	1	no	—
194	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
195	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
196	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
197	C₂×C₄²	D₂₁₂	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	—	106	4	no	—
202	C₄×C₈	—	212	2	no	—
203	#95	—	106	1	no	—
204	#95	—	106	1	no	—
205	#95	—	106	1	no	—
206	#95	—	106	1	no	—
207	#95	—	106	1	no	—
208	#95	—	106	1	no	—
209	C₂²×C₄	—	106	1	no	—
210	C₂²×C₄	—	106	1	no	—
211	C₂²×C₄	—	106	1	no	—
212	C₂²×C₄	—	106	1	no	—
213	C₂²×C₄	#11719921	1	1	yes	characteristic
214	C₂²×C₄	#11719921	1	1	yes	characteristic
215	C₂²×C₄	#11719921	1	1	yes	characteristic
216	C₂²×C₄	#11719921	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	106	4	no	—
224	C₂×C₈	—	212	2	no	—
225	C₂×C₈	—	212	2	no	—
226	C₄⋊C₄	—	53	2	no	—
227	C₄⋊C₄	—	53	2	no	—
228	C₄⋊C₄	—	106	4	no	—
229	C₄⋊C₄	—	106	4	no	—
230	C₄⋊C₄	—	106	1	no	—
231	C₄⋊C₄	—	106	1	no	—
232	C₄⋊C₄	—	212	1	no	—
233	C₄⋊C₄	—	212	1	no	—
234	C₄⋊C₄	—	212	2	no	—
235	C₄⋊C₄	—	212	1	no	—
236	C₄⋊C₄	—	212	1	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	2	1	no	—
242	C₄²	—	4	1	no	—
243	C₄²	—	4	1	no	—
244	C₄²	—	212	2	no	—
245	C₄²	—	212	2	no	—
246	C₄²	C₈×D₅₃	1	2	yes	—
247	C₄²	#11719912	1	2	yes	—
248	C₂³	C₂×#11719921	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	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	106	2	no	—
264	C₂×C₄	—	106	2	no	—
265	C₂×C₄	—	212	1	no	—
266	C₂×C₄	—	212	1	no	—
267	C₂×C₄	—	212	1	no	—
268	C₂×C₄	—	212	1	no	—
269	C₂×C₄	—	212	1	no	—
270	C₂×C₄	—	212	1	no	—
271	C₂×C₄	—	212	1	no	—
272	C₂×C₄	—	212	1	no	—
273	C₂×C₄	—	212	1	no	—
274	C₂×C₄	—	212	1	no	—
275	C₂×C₄	#13133129	1	2	yes	—
276	C₂×C₄	#13133129	1	2	yes	—
277	C₂×C₄	#13133135	1	2	yes	—
278	C₂×C₄	#13133143	1	2	yes	—
279	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
280	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
281	C₂×C₄	#13133237	1	2	yes	—
282	C₂×C₄	#13133245	1	1	yes	characteristic
283	C₂×C₄	#13133250	1	1	yes	characteristic
284	C₈	—	8	2	no	—
285	C₈	—	212	4	no	—
286	C₂²	C₂×#13133129	1	1	yes	characteristic
287	C₂²	#16810596	1	1	yes	characteristic
288	C₂²	C₂×#13133135	1	1	yes	characteristic
289	C₂²	#16810617	1	1	yes	characteristic
290	C₂²	#16810625	1	1	yes	characteristic
291	C₂²	C₂×#13133143	1	1	yes	characteristic
292	C₂²	#16810641	1	1	yes	characteristic
293	C₄	—	2	2	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	424	1	no	—
303	C₄	—	424	1	no	—
304	C₄	—	212	2	no	—
305	C₄	—	212	2	no	—
306	C₂	#20273038	1	1	yes	characteristic
307	C₂	#20273095	1	1	yes	characteristic
308	C₂	#20273103	1	1	yes	characteristic
309	C₂	#20273241	1	1	yes	characteristic
310	C₂	#20273290	1	1	yes	characteristic
311	C₂	#20273318	1	1	yes	characteristic
312	C₂	#20273416	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11719921
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1839	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1839	D₅₃
hypocenter	C₅₃	#7585
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1839

Complements

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

N	H ≅ G/N	type
25. #13133209	C₈	semidirect
179. C₅₃	#7585	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133209	2
C₄×C₄⋊#88322	2
#20272539	2
#20272537	2
C₅₃×#1839	2
#7585	53