Subgroups — #27272623

Order 13568.

378 conjugacy classes of subgroups.

The subgroup lattice has 306 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	#20274568	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	C₂×C₄⋊#88322	—	2	1	no	—
8	C₂×#13133128	—	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	C₄⋊#88322	—	4	2	no	—
27	#13133128	—	4	2	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	1	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	C₂×C₄	1	1	yes	characteristic
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×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	C₂×C₄	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	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
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	#95×C₅₃	—	2	1	no	—
50	#95×C₅₃	—	2	1	no	—
51	C₂²×C₄×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	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	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₅₃	—	2	2	no	—
63	C₄⋊C₄×C₅₃	—	2	2	no	—
64	C₄⋊C₄×C₅₃	—	4	2	no	—
65	C₄⋊C₄×C₅₃	—	4	1	no	—
66	C₄⋊C₄×C₅₃	—	4	1	no	—
67	C₄⋊C₄×C₅₃	—	4	2	no	—
68	C₄⋊C₄×C₅₃	—	4	1	no	—
69	C₄⋊C₄×C₅₃	—	4	1	no	—
70	C₄⋊C₄×C₅₃	—	4	1	no	—
71	C₄⋊C₄×C₅₃	—	4	1	no	—
72	C₄²×C₅₃	—	2	2	no	—
73	C₄²×C₅₃	—	2	2	no	—
74	C₄²×C₅₃	—	2	2	no	—
75	C₄²×C₅₃	—	2	1	no	—
76	C₄²×C₅₃	—	2	1	no	—
77	C₄²×C₅₃	—	2	1	no	—
78	C₄²×C₅₃	—	2	1	no	—
79	C₄²×C₅₃	—	4	1	no	—
80	C₄²×C₅₃	—	4	1	no	—
81	C₂×#88322	—	2	4	no	—
82	C₂×#88322	—	4	2	no	—
83	C₂×#88322	—	4	2	no	—
84	C₂×#88322	—	4	2	no	—
85	C₂×#88322	—	4	2	no	—
86	C₂×#88322	—	4	2	no	—
87	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	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	—	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	—	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	#101	1	2	yes	—
117	C₂×C₄×C₅₃	#104	1	2	yes	—
118	C₂×C₄×C₅₃	#131	1	2	yes	—
119	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
122	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
123	#88322	—	4	4	no	—
124	#88322	—	8	2	no	—
125	#7697	—	53	1	no	maximal, Sylow
126	C₂²×C₅₃	#408	1	1	yes	characteristic
127	C₂²×C₅₃	#410	1	1	yes	characteristic
128	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
129	C₂²×C₅₃	#417	1	1	yes	characteristic
130	C₂²×C₅₃	#417	1	1	yes	characteristic
131	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
132	C₂²×C₅₃	#421	1	1	yes	characteristic
133	C₄×C₅₃	—	2	2	no	—
134	C₄×C₅₃	—	2	2	no	—
135	C₄×C₅₃	—	2	1	no	—
136	C₄×C₅₃	—	2	1	no	—
137	C₄×C₅₃	—	2	1	no	—
138	C₄×C₅₃	—	2	1	no	—
139	C₄×C₅₃	—	2	2	no	—
140	C₄×C₅₃	—	4	1	no	—
141	C₄×C₅₃	—	4	1	no	—
142	C₄×C₅₃	—	8	1	no	—
143	C₄×C₅₃	—	8	1	no	—
144	C₄×C₅₃	—	8	1	no	—
145	C₄×C₅₃	—	8	1	no	—
146	#2864	D₅₃	1	1	yes	characteristic
147	#1831	—	53	1	no	—
148	#1767	—	53	1	no	—
149	#1271	—	53	2	no	—
150	#1271	—	53	2	no	—
151	C₂×C₅₃	#1470	1	1	yes	characteristic
152	C₂×C₅₃	#1483	1	1	yes	characteristic
153	C₂×C₅₃	#1495	1	1	yes	characteristic
154	C₂×C₅₃	#1509	1	1	yes	characteristic
155	C₂×C₅₃	#1512	1	1	yes	characteristic
156	C₂×C₅₃	#1531	1	1	yes	characteristic
157	C₂×C₅₃	#1561	1	1	yes	characteristic
158	C₂×C₄⋊C₈	—	106	1	no	—
159	C₂×C₄⋊C₈	—	106	1	no	—
160	C₂×#97	—	106	1	no	—
161	#389	Dic₅₃	1	1	yes	characteristic
162	#389	Dic₅₃	1	1	yes	characteristic
163	#389	C₂×D₅₃	1	1	yes	characteristic
164	#389	C₂×D₅₃	1	1	yes	characteristic
165	#387	—	2	1	no	—
166	#387	—	2	1	no	—
167	#387	—	2	1	no	—
168	#387	—	2	1	no	—
169	#382	Dic₅₃	1	1	yes	characteristic
170	#382	Dic₅₃	1	1	yes	characteristic
171	C₄³	C₂×D₅₃	1	1	yes	characteristic
172	#336	—	53	2	no	—
173	#336	—	53	2	no	—
174	#336	—	106	1	no	—
175	C₅₃	#7697	1	1	yes	characteristic, complemented, Sylow
176	C₂²×C₈	—	106	2	no	—
177	C₂²×C₈	—	106	2	no	—
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
183	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
184	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
185	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
186	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
187	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
188	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
189	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
190	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
191	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
192	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
193	C₄⋊C₈	—	106	4	no	—
194	C₄⋊C₈	—	212	2	no	—
195	#97	—	212	2	no	—
196	#95	—	2	1	no	—
197	#95	—	2	1	no	—
198	#95	—	2	1	no	—
199	#95	—	2	1	no	—
200	#95	—	2	1	no	—
201	#95	—	2	1	no	—
202	#95	—	2	1	no	—
203	#95	—	2	1	no	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
209	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
210	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
211	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
212	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
215	C₂×C₈	—	106	4	no	—
216	C₂×C₈	—	212	2	no	—
217	C₂×C₈	—	212	2	no	—
218	C₂×C₈	—	212	2	no	—
219	C₂×C₈	—	212	2	no	—
220	C₂×C₈	—	212	2	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	4	1	no	—
224	C₄⋊C₄	—	4	1	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	—	4	1	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₄⋊C₄	—	4	1	no	—
230	C₄⋊C₄	—	4	1	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	1	no	—
235	C₄²	—	2	1	no	—
236	C₄²	—	2	1	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	4	1	no	—
239	C₄²	—	4	1	no	—
240	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
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	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	#13133136	1	2	yes	—
270	C₂×C₄	#13133144	1	2	yes	—
271	C₂×C₄	#13133224	1	2	yes	—
272	C₂×C₄	#13133242	1	1	yes	characteristic
273	C₂×C₄	#13133242	1	1	yes	characteristic
274	C₂×C₄	#13133247	1	1	yes	characteristic
275	C₂×C₄	#13133247	1	1	yes	characteristic
276	C₈	—	424	2	no	—
277	C₈	—	212	4	no	—
278	C₂²	#16810590	1	1	yes	characteristic
279	C₂²	#16810597	1	1	yes	characteristic
280	C₂²	C₂×#13133136	1	1	yes	characteristic
281	C₂²	#16810624	1	1	yes	characteristic
282	C₂²	#16810624	1	1	yes	characteristic
283	C₂²	C₂×#13133144	1	1	yes	characteristic
284	C₂²	#16810640	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	4	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	8	1	no	—
295	C₄	—	8	1	no	—
296	C₄	—	8	1	no	—
297	C₄	—	8	1	no	—
298	C₂	#20273071	1	1	yes	characteristic
299	C₂	#20273128	1	1	yes	characteristic
300	C₂	#20273180	1	1	yes	characteristic
301	C₂	#20273240	1	1	yes	characteristic
302	C₂	#20273253	1	1	yes	characteristic
303	C₂	#20273330	1	1	yes	characteristic
304	C₂	#20273453	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	#7697
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
175. C₅₃	#7697	semidirect

Maximal subgroups

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

subgroup	index
#20274832	2
#20274568	2
#20272538	2
C₅₃×#2864	2
#7697	53