Subgroups — #27284227

Order 13568.

450 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274867	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	#20272598	C₂	1	2	yes	maximal
4	#20272598	C₂	1	2	yes	maximal
5	C₅₃×#2859	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	—	2	1	no	—
7	C₂×C₄⋊#88322	—	2	1	no	—
8	C₂×C₄⋊#88322	C₂²	1	2	yes	—
9	C₂×C₄⋊#88322	C₂²	1	2	yes	—
10	C₂×C₄×#88322	—	2	1	no	—
11	#16810378	—	2	1	no	—
12	C₂×#125×C₅₃	—	2	1	no	—
13	C₅₃×#389	—	2	1	no	—
14	C₅₃×#389	—	2	1	no	—
15	C₅₃×#389	C₄	1	2	yes	—
16	C₅₃×#389	C₄	1	2	yes	—
17	C₅₃×#384	C₂²	1	1	yes	characteristic
18	C₅₃×#384	C₂²	1	1	yes	characteristic
19	C₅₃×#382	—	2	1	no	—
20	C₄³×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×#88322	—	2	2	no	—
22	C₂²×#88322	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	4	no	—
25	C₄⋊#88322	—	4	2	no	—
26	C₄⋊#88322	—	4	2	no	—
27	C₄⋊#88322	Q₈	1	4	yes	—
28	C₄×#88322	—	4	2	no	—
29	#125×C₅₃	—	4	2	no	—
30	#125×C₅₃	—	4	2	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
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	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
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	2	no	—
47	#95×C₅₃	—	2	2	no	—
48	C₂²×C₄×C₅₃	—	2	2	no	—
49	C₂²×C₄×C₅₃	—	2	2	no	—
50	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂²⋊C₄	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	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₄⋊C₄×C₅₃	—	4	2	no	—
58	C₄⋊C₄×C₅₃	—	4	2	no	—
59	C₄⋊C₄×C₅₃	—	4	4	no	—
60	C₄⋊C₄×C₅₃	—	4	4	no	—
61	C₄⋊C₄×C₅₃	—	4	1	no	—
62	C₄⋊C₄×C₅₃	—	4	1	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	1	no	—
67	C₄⋊C₄×C₅₃	—	4	1	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	D₈	1	2	yes	—
74	C₄²×C₅₃	SD₁₆	1	2	yes	—
75	C₄²×C₅₃	SD₁₆	1	2	yes	—
76	C₄²×C₅₃	Q₁₆	1	2	yes	—
77	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
78	C₄²×C₅₃	#41	1	2	yes	—
79	C₂×#88322	—	2	4	no	—
80	C₂×#88322	—	4	2	no	—
81	C₂×#88322	—	4	2	no	—
82	C₂×#88322	—	4	2	no	—
83	C₂×#88322	—	4	2	no	—
84	C₂×#88322	—	4	2	no	—
85	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	4	2	no	—
97	C₂×C₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	—	4	2	no	—
99	C₂×C₄×C₅₃	—	4	2	no	—
100	C₂×C₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	—	4	2	no	—
102	C₂×C₄×C₅₃	#102	1	2	yes	—
103	C₂×C₄×C₅₃	#102	1	2	yes	—
104	C₂×C₄×C₅₃	#103	1	2	yes	—
105	C₂×C₄×C₅₃	#103	1	2	yes	—
106	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
107	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#135	1	2	yes	—
115	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
117	#88322	—	4	4	no	—
118	#88322	—	8	2	no	—
119	#9900	—	53	1	no	maximal, Sylow
120	C₂²×C₅₃	#391	1	1	yes	characteristic
121	C₂²×C₅₃	#413	1	1	yes	characteristic
122	C₂²×C₅₃	#413	1	1	yes	characteristic
123	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
124	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
125	C₂²×C₅₃	#416	1	1	yes	characteristic
126	C₂²×C₅₃	#417	1	1	yes	characteristic
127	C₄×C₅₃	—	2	2	no	—
128	C₄×C₅₃	—	2	2	no	—
129	C₄×C₅₃	—	2	2	no	—
130	C₄×C₅₃	—	2	2	no	—
131	C₄×C₅₃	—	2	2	no	—
132	C₄×C₅₃	—	8	2	no	—
133	C₄×C₅₃	—	8	2	no	—
134	C₄×C₅₃	#478	1	2	yes	—
135	C₄×C₅₃	#479	1	2	yes	—
136	C₄×C₅₃	#486	1	1	yes	characteristic
137	C₄×C₅₃	#487	1	1	yes	characteristic
138	C₄×C₅₃	#498	1	1	yes	characteristic
139	C₄×C₅₃	#500	1	1	yes	characteristic
140	#2859	D₅₃	1	1	yes	characteristic
141	#1842	—	53	1	no	—
142	C₄×C₄⋊C₈	—	53	1	no	—
143	#1296	—	53	2	no	—
144	#1296	—	53	2	no	—
145	C₂×C₅₃	#1540	1	1	yes	characteristic
146	C₂×C₅₃	#1542	1	1	yes	characteristic
147	C₂×C₅₃	#1547	1	1	yes	characteristic
148	C₂×C₅₃	#1550	1	1	yes	characteristic
149	C₂×C₅₃	#1989	1	1	yes	characteristic
150	C₂×C₅₃	#1990	1	1	yes	characteristic
151	C₂×C₅₃	#1998	1	1	yes	characteristic
152	C₂×#125	—	2	1	no	—
153	C₂×C₄⋊C₈	—	53	2	no	—
154	C₂×C₄⋊C₈	—	53	2	no	—
155	C₂×C₄⋊C₈	—	106	1	no	—
156	C₂×C₄⋊C₈	—	106	1	no	—
157	C₂×C₄×C₈	—	106	1	no	—
158	#389	—	2	1	no	—
159	#389	—	2	1	no	—
160	#389	Dic₅₃	1	2	yes	—
161	#389	Dic₅₃	1	2	yes	—
162	#384	C₂×D₅₃	1	1	yes	characteristic
163	#384	C₂×D₅₃	1	1	yes	characteristic
164	#382	—	2	1	no	—
165	C₄³	C₂×D₅₃	1	1	yes	characteristic
166	#336	—	106	1	no	—
167	C₅₃	#9900	1	1	yes	characteristic, complemented, Sylow
168	C₂²×C₈	—	106	2	no	—
169	C₂²×C₈	—	106	2	no	—
170	#125	—	4	2	no	—
171	#125	—	4	2	no	—
172	C₂×C₄⋊C₄	—	2	1	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
179	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
180	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
181	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
182	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
183	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
184	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
185	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
186	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
187	C₄⋊C₈	—	53	4	no	—
188	C₄⋊C₈	—	106	2	no	—
189	C₄⋊C₈	—	106	4	no	—
190	C₄⋊C₈	—	212	2	no	—
191	C₄⋊C₈	—	212	2	no	—
192	C₄×C₈	—	212	2	no	—
193	#95	—	2	2	no	—
194	#95	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
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	C₂²×Dic₅₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
204	C₂×C₈	—	106	4	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	C₂×C₈	—	212	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	4	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	—	4	1	no	—
215	C₄⋊C₄	—	4	1	no	—
216	C₄⋊C₄	—	4	1	no	—
217	C₄⋊C₄	—	4	1	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄⋊C₄	—	4	1	no	—
220	C₄⋊C₄	—	4	1	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	—	4	1	no	—
225	C₄²	—	4	1	no	—
226	C₄²	#11719922	1	2	yes	—
227	C₄²	#11719923	1	2	yes	—
228	C₄²	#11719924	1	2	yes	—
229	C₄²	#11719925	1	2	yes	—
230	C₄²	Q₈×D₅₃	1	2	yes	—
231	C₄²	#11719949	1	2	yes	—
232	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
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	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	#13133138	1	2	yes	—
250	C₂×C₄	#13133138	1	2	yes	—
251	C₂×C₄	#13133141	1	2	yes	—
252	C₂×C₄	#13133141	1	2	yes	—
253	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
254	C₂×C₄	#13133193	1	2	yes	—
255	C₂×C₄	#13133206	1	1	yes	characteristic
256	C₂×C₄	#13133215	1	1	yes	characteristic
257	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
258	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
259	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
260	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
261	C₂×C₄	#13133237	1	2	yes	—
262	C₂×C₄	#13133242	1	1	yes	characteristic
263	C₂×C₄	#13133247	1	1	yes	characteristic
264	C₈	—	424	2	no	—
265	C₈	—	212	4	no	—
266	C₂²	#16810530	1	1	yes	characteristic
267	C₂²	#16810607	1	1	yes	characteristic
268	C₂²	#16810607	1	1	yes	characteristic
269	C₂²	C₂×#13133138	1	1	yes	characteristic
270	C₂²	C₂×#13133141	1	1	yes	characteristic
271	C₂²	#16810619	1	1	yes	characteristic
272	C₂²	#16810624	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	8	2	no	—
279	C₄	—	8	2	no	—
280	C₄	#16810920	1	2	yes	—
281	C₄	#16810927	1	2	yes	—
282	C₄	#16810971	1	1	yes	characteristic
283	C₄	#16810975	1	1	yes	characteristic
284	C₄	#16811019	1	1	yes	characteristic
285	C₄	#16811026	1	1	yes	characteristic
286	C₂	#20273364	1	1	yes	characteristic
287	C₂	#20273372	1	1	yes	characteristic
288	C₂	#20273392	1	1	yes	characteristic
289	C₂	#20273405	1	1	yes	characteristic
290	C₂	#20275673	1	1	yes	characteristic
291	C₂	#20275677	1	1	yes	characteristic
292	C₂	#20275709	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×#2859	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2859	D₅₃
hypocenter	C₅₃	#9900
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

Complements

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

N	H ≅ G/N	type
167. C₅₃	#9900	semidirect

Maximal subgroups

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

subgroup	index
#20274867	2
C₄×C₄⋊#88322	2
#20272598	2
C₅₃×#2859	2
#9900	53