Subgroups — #27272439

Order 13568.

370 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274858	C₂	1	1	yes	maximal, characteristic
2	#20274461	C₂	1	1	yes	maximal, characteristic
3	#20272538	C₂	1	2	yes	maximal
4	#20272538	C₂	1	2	yes	maximal
5	C₅₃×#2864	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×#88322	—	2	1	no	—
7	#16810378	—	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	C₅₃×#389	C₄	1	1	yes	characteristic
13	C₅₃×#389	C₄	1	1	yes	characteristic
14	C₅₃×#389	C₂²	1	1	yes	characteristic
15	C₅₃×#389	C₂²	1	1	yes	characteristic
16	C₅₃×#387	—	2	1	no	—
17	C₅₃×#387	—	2	1	no	—
18	C₅₃×#387	—	2	1	no	—
19	C₅₃×#387	—	2	1	no	—
20	C₅₃×#382	C₄	1	1	yes	characteristic
21	C₅₃×#382	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	—	2	4	no	—
26	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
27	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	#95×C₅₃	—	2	1	no	—
42	#95×C₅₃	—	2	1	no	—
43	#95×C₅₃	—	2	1	no	—
44	#95×C₅₃	—	2	1	no	—
45	#95×C₅₃	—	2	1	no	—
46	#95×C₅₃	—	2	1	no	—
47	#95×C₅₃	—	2	1	no	—
48	#95×C₅₃	—	2	1	no	—
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	—	2	1	no	—
51	C₂²×C₄×C₅₃	—	2	1	no	—
52	C₂²×C₄×C₅₃	—	2	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₄⋊C₄×C₅₃	—	2	2	no	—
61	C₄⋊C₄×C₅₃	—	2	2	no	—
62	C₄⋊C₄×C₅₃	—	4	2	no	—
63	C₄⋊C₄×C₅₃	—	4	1	no	—
64	C₄⋊C₄×C₅₃	—	4	1	no	—
65	C₄⋊C₄×C₅₃	—	4	2	no	—
66	C₄⋊C₄×C₅₃	—	4	2	no	—
67	C₄⋊C₄×C₅₃	—	4	2	no	—
68	C₄²×C₅₃	—	2	2	no	—
69	C₄²×C₅₃	—	2	2	no	—
70	C₄²×C₅₃	—	2	2	no	—
71	C₄²×C₅₃	—	2	2	no	—
72	C₄²×C₅₃	—	2	2	no	—
73	C₄²×C₅₃	—	4	1	no	—
74	C₄²×C₅₃	—	4	1	no	—
75	C₂×#88322	—	2	4	no	—
76	C₂×#88322	—	4	2	no	—
77	C₂×#88322	—	4	2	no	—
78	C₂×#88322	—	4	2	no	—
79	C₂×#88322	—	4	2	no	—
80	C₂×#88322	—	4	2	no	—
81	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	1	no	—
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	—	4	1	no	—
99	C₂×C₄×C₅₃	—	4	1	no	—
100	C₂×C₄×C₅₃	—	4	1	no	—
101	C₂×C₄×C₅₃	—	4	1	no	—
102	C₂×C₄×C₅₃	—	4	1	no	—
103	C₂×C₄×C₅₃	—	4	1	no	—
104	C₂×C₄×C₅₃	—	4	1	no	—
105	C₂×C₄×C₅₃	—	4	1	no	—
106	C₂×C₄×C₅₃	—	4	1	no	—
107	C₂×C₄×C₅₃	—	4	1	no	—
108	C₂×C₄×C₅₃	—	4	1	no	—
109	C₂×C₄×C₅₃	—	4	1	no	—
110	C₂×C₄×C₅₃	#101	1	2	yes	—
111	C₂×C₄×C₅₃	#104	1	2	yes	—
112	C₂×C₄×C₅₃	#131	1	2	yes	—
113	C₂×C₄×C₅₃	#135	1	2	yes	—
114	C₂×C₄×C₅₃	#135	1	2	yes	—
115	#88322	—	4	4	no	—
116	#88322	—	8	2	no	—
117	#7657	—	53	1	no	maximal, Sylow
118	C₂²×C₅₃	#408	1	1	yes	characteristic
119	C₂²×C₅₃	#410	1	1	yes	characteristic
120	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
121	C₂²×C₅₃	#416	1	1	yes	characteristic
122	C₂²×C₅₃	#416	1	1	yes	characteristic
123	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
124	C₂²×C₅₃	#421	1	1	yes	characteristic
125	C₄×C₅₃	—	2	2	no	—
126	C₄×C₅₃	—	2	2	no	—
127	C₄×C₅₃	—	2	2	no	—
128	C₄×C₅₃	—	2	2	no	—
129	C₄×C₅₃	—	2	2	no	—
130	C₄×C₅₃	—	4	1	no	—
131	C₄×C₅₃	—	4	1	no	—
132	C₄×C₅₃	—	8	1	no	—
133	C₄×C₅₃	—	8	1	no	—
134	C₄×C₅₃	—	8	1	no	—
135	C₄×C₅₃	—	8	1	no	—
136	#2864	D₅₃	1	1	yes	characteristic
137	#1839	—	53	1	no	—
138	#1744	—	53	1	no	—
139	#1271	—	53	2	no	—
140	#1271	—	53	2	no	—
141	C₂×C₅₃	#1470	1	1	yes	characteristic
142	C₂×C₅₃	#1482	1	1	yes	characteristic
143	C₂×C₅₃	#1494	1	1	yes	characteristic
144	C₂×C₅₃	#1500	1	1	yes	characteristic
145	C₂×C₅₃	#1521	1	1	yes	characteristic
146	C₂×C₅₃	#1531	1	1	yes	characteristic
147	C₂×C₅₃	#1544	1	1	yes	characteristic
148	C₂×C₄×C₈	—	106	1	no	—
149	#389	Dic₅₃	1	1	yes	characteristic
150	#389	Dic₅₃	1	1	yes	characteristic
151	#389	C₂×D₅₃	1	1	yes	characteristic
152	#389	C₂×D₅₃	1	1	yes	characteristic
153	#387	—	2	1	no	—
154	#387	—	2	1	no	—
155	#387	—	2	1	no	—
156	#387	—	2	1	no	—
157	#382	Dic₅₃	1	1	yes	characteristic
158	#382	Dic₅₃	1	1	yes	characteristic
159	C₄³	C₂×D₅₃	1	1	yes	characteristic
160	#336	—	53	2	no	—
161	#336	—	53	2	no	—
162	#336	—	106	1	no	—
163	#336	—	106	1	no	—
164	#336	—	106	1	no	—
165	C₅₃	#7657	1	1	yes	characteristic, complemented, Sylow
166	C₂²×C₈	—	106	2	no	—
167	C₂²×C₈	—	106	2	no	—
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	—	2	1	no	—
172	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
173	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
174	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
175	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
176	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
177	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
178	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
179	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
180	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
181	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
182	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
183	C₄×C₈	—	106	4	no	—
184	#95	—	2	1	no	—
185	#95	—	2	1	no	—
186	#95	—	2	1	no	—
187	#95	—	2	1	no	—
188	#95	—	2	1	no	—
189	#95	—	2	1	no	—
190	#95	—	2	1	no	—
191	#95	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
197	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
198	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
199	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
200	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
203	C₂×C₈	—	106	4	no	—
204	C₂×C₈	—	212	2	no	—
205	C₂×C₈	—	212	2	no	—
206	C₂×C₈	—	212	2	no	—
207	C₂×C₈	—	212	2	no	—
208	C₂×C₈	—	212	2	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	4	1	no	—
212	C₄⋊C₄	—	4	1	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	4	1	no	—
223	C₄²	—	4	1	no	—
224	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	#13133136	1	2	yes	—
254	C₂×C₄	#13133144	1	2	yes	—
255	C₂×C₄	#13133224	1	2	yes	—
256	C₂×C₄	#13133237	1	2	yes	—
257	C₂×C₄	#13133237	1	2	yes	—
258	C₈	—	424	2	no	—
259	C₈	—	212	4	no	—
260	C₂²	#16810590	1	1	yes	characteristic
261	C₂²	#16810597	1	1	yes	characteristic
262	C₂²	C₂×#13133136	1	1	yes	characteristic
263	C₂²	#16810619	1	1	yes	characteristic
264	C₂²	#16810619	1	1	yes	characteristic
265	C₂²	C₂×#13133144	1	1	yes	characteristic
266	C₂²	#16810640	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	8	1	no	—
275	C₄	—	8	1	no	—
276	C₄	—	8	1	no	—
277	C₄	—	8	1	no	—
278	C₂	#20273071	1	1	yes	characteristic
279	C₂	#20273123	1	1	yes	characteristic
280	C₂	#20273175	1	1	yes	characteristic
281	C₂	#20273201	1	1	yes	characteristic
282	C₂	#20273292	1	1	yes	characteristic
283	C₂	#20273330	1	1	yes	characteristic
284	C₂	#20273380	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2864	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2864	D₅₃
hypocenter	C₅₃	#7657
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2864

Complements

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

N	H ≅ G/N	type
165. C₅₃	#7657	semidirect

Maximal subgroups

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

subgroup	index
#20274858	2
#20274461	2
#20272538	2
C₅₃×#2864	2
#7657	53