Subgroups — #27282692

Order 13568.

388 conjugacy classes of subgroups.

The subgroup lattice has 272 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	#20274568	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810374	C₂	1	2	yes	maximal
4	#20272550	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₅₃	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	1	yes	characteristic
10	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
11	C₂×#13133128	—	2	1	no	—
12	#16810378	—	2	1	no	—
13	#16810374	—	2	4	no	—
14	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
16	C₄³×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#336	C₄	1	1	yes	characteristic
18	C₅₃×#336	C₄	1	1	yes	characteristic
19	C₂²×#88322	—	2	1	no	—
20	C₂²×#88322	—	2	1	no	—
21	C₂²×#88322	—	2	2	no	—
22	C₄⋊#88322	—	2	1	no	—
23	C₄⋊#88322	—	2	1	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₄⋊#88322	—	2	2	no	—
26	C₄⋊#88322	—	4	2	no	—
27	C₄⋊#88322	—	4	2	no	—
28	C₄⋊#88322	D₄	1	2	yes	—
29	C₄⋊#88322	Q₈	1	2	yes	—
30	#13133128	—	4	2	no	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₅₃	—	2	4	no	—
41	C₄×C₈×C₅₃	—	2	2	no	—
42	C₄×C₈×C₅₃	—	2	2	no	—
43	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₅₃	—	2	2	no	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	—	2	2	no	—
55	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
56	C₄²×C₅₃	—	2	2	no	—
57	C₄²×C₅₃	—	2	2	no	—
58	C₄²×C₅₃	—	2	2	no	—
59	C₄²×C₅₃	—	2	1	no	—
60	C₄²×C₅₃	—	2	1	no	—
61	C₄²×C₅₃	—	4	1	no	—
62	C₄²×C₅₃	SD₁₆	1	2	yes	—
63	C₄²×C₅₃	SD₁₆	1	2	yes	—
64	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
66	C₄²×C₅₃	#41	1	1	yes	characteristic
67	C₄²×C₅₃	#41	1	1	yes	characteristic
68	C₂×#88322	—	2	2	no	—
69	C₂×#88322	—	2	2	no	—
70	C₂×#88322	—	4	1	no	—
71	C₂×#88322	—	4	1	no	—
72	C₂×#88322	—	4	1	no	—
73	C₂×#88322	—	4	1	no	—
74	C₂×#88322	—	4	2	no	—
75	C₂×#88322	—	4	2	no	—
76	C₂×#88322	—	4	2	no	—
77	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	—	2	1	no	—
79	C₂×C₄×C₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	2	1	no	—
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	#106	1	2	yes	—
90	C₂×C₄×C₅₃	#106	1	2	yes	—
91	C₂×C₄×C₅₃	#108	1	2	yes	—
92	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
99	C₂×C₄×C₅₃	#131	1	2	yes	—
100	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
104	C₈×C₅₃	—	2	4	no	—
105	C₈×C₅₃	—	4	2	no	—
106	#88322	—	4	2	no	—
107	#88322	—	4	2	no	—
108	#88322	—	8	2	no	—
109	#9605	—	53	1	no	maximal, Sylow
110	C₂²×C₅₃	#389	1	1	yes	characteristic
111	C₂²×C₅₃	#423	1	1	yes	characteristic
112	C₂²×C₅₃	#424	1	1	yes	characteristic
113	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
114	C₂²×C₅₃	#428	1	1	yes	characteristic
115	C₂²×C₅₃	C₂×#108	1	1	yes	characteristic
116	C₂²×C₅₃	#430	1	1	yes	characteristic
117	C₄×C₅₃	—	2	2	no	—
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	—	2	2	no	—
120	C₄×C₅₃	—	2	1	no	—
121	C₄×C₅₃	—	2	1	no	—
122	C₄×C₅₃	—	4	1	no	—
123	C₄×C₅₃	#435	1	2	yes	—
124	C₄×C₅₃	#446	1	2	yes	—
125	C₄×C₅₃	#460	1	1	yes	characteristic
126	C₄×C₅₃	#461	1	1	yes	characteristic
127	C₄×C₅₃	#475	1	1	yes	characteristic
128	C₄×C₅₃	#476	1	1	yes	characteristic
129	#1842	—	53	1	no	—
130	#1767	—	53	1	no	—
131	C₄×C₄⋊C₈	D₅₃	1	1	yes	characteristic
132	C₂×#334	—	53	2	no	—
133	#1277	—	53	2	no	—
134	C₂×C₅₃	#1566	1	1	yes	characteristic
135	C₂×C₅₃	#1569	1	1	yes	characteristic
136	C₂×C₅₃	#1570	1	1	yes	characteristic
137	C₂×C₅₃	#1573	1	1	yes	characteristic
138	C₂×C₅₃	#1940	1	1	yes	characteristic
139	C₂×C₅₃	#1948	1	1	yes	characteristic
140	C₂×C₅₃	#1950	1	1	yes	characteristic
141	C₂×C₄⋊C₈	—	53	1	no	—
142	C₂×C₄⋊C₈	—	53	1	no	—
143	C₂×C₄⋊C₈	—	53	2	no	—
144	C₂×C₄⋊C₈	—	106	1	no	—
145	C₂×C₄⋊C₈	—	106	1	no	—
146	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
147	C₂×#97	—	106	1	no	—
148	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
149	C₄³	C₂×D₅₃	1	1	yes	characteristic
150	#336	—	106	1	no	—
151	#336	Dic₅₃	1	1	yes	characteristic
152	#336	Dic₅₃	1	1	yes	characteristic
153	#334	—	106	4	no	—
154	C₅₃	#9605	1	1	yes	characteristic, complemented, Sylow
155	C₂²×C₈	—	106	1	no	—
156	C₂²×C₈	—	106	1	no	—
157	C₂²×C₈	—	106	2	no	—
158	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
159	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
160	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
161	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
162	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
163	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
164	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
165	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
166	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
167	C₄⋊C₈	—	2	4	no	—
168	C₄⋊C₈	—	53	2	no	—
169	C₄⋊C₈	—	53	2	no	—
170	C₄⋊C₈	—	106	2	no	—
171	C₄⋊C₈	—	106	1	no	—
172	C₄⋊C₈	—	106	1	no	—
173	C₄⋊C₈	—	106	2	no	—
174	C₄⋊C₈	—	212	2	no	—
175	C₄⋊C₈	—	212	2	no	—
176	#97	—	212	2	no	—
177	C₄×C₈	—	2	2	no	—
178	C₄×C₈	—	2	2	no	—
179	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
180	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
181	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
182	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
183	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
185	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	106	2	no	—
192	C₂×C₈	—	106	2	no	—
193	C₂×C₈	—	212	2	no	—
194	C₂×C₈	—	212	2	no	—
195	C₂×C₈	—	212	2	no	—
196	C₂×C₈	—	212	1	no	—
197	C₂×C₈	—	212	1	no	—
198	C₂×C₈	—	212	1	no	—
199	C₂×C₈	—	212	1	no	—
200	C₂×C₈	#11719917	1	4	yes	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	4	1	no	—
207	C₄²	#11719913	1	2	yes	—
208	C₄²	#11719913	1	2	yes	—
209	C₄²	D₄×D₅₃	1	1	yes	characteristic
210	C₄²	#11719947	1	1	yes	characteristic
211	C₄²	Q₈×D₅₃	1	1	yes	characteristic
212	C₄²	#11719949	1	1	yes	characteristic
213	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	#13133149	1	2	yes	—
226	C₂×C₄	#13133149	1	2	yes	—
227	C₂×C₄	#13133153	1	2	yes	—
228	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
229	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
230	C₂×C₄	#13133192	1	1	yes	characteristic
231	C₂×C₄	#13133197	1	1	yes	characteristic
232	C₂×C₄	#13133208	1	1	yes	characteristic
233	C₂×C₄	#13133216	1	1	yes	characteristic
234	C₂×C₄	C₂×#11719917	1	2	yes	—
235	C₂×C₄	#13133224	1	2	yes	—
236	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
237	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
238	C₂×C₄	#13133245	1	1	yes	characteristic
239	C₂×C₄	#13133250	1	1	yes	characteristic
240	C₈	—	2	4	no	—
241	C₈	—	4	2	no	—
242	C₈	—	424	2	no	—
243	C₈	—	212	2	no	—
244	C₈	—	212	2	no	—
245	C₂²	#16810522	1	1	yes	characteristic
246	C₂²	#16810647	1	1	yes	characteristic
247	C₂²	#16810650	1	1	yes	characteristic
248	C₂²	C₂×#13133149	1	1	yes	characteristic
249	C₂²	#16810665	1	1	yes	characteristic
250	C₂²	C₂×#13133153	1	1	yes	characteristic
251	C₂²	#16810672	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	4	1	no	—
258	C₄	#16810695	1	2	yes	—
259	C₄	#16810754	1	2	yes	—
260	C₄	#16810824	1	1	yes	characteristic
261	C₄	#16810829	1	1	yes	characteristic
262	C₄	#16810909	1	1	yes	characteristic
263	C₄	#16810914	1	1	yes	characteristic
264	C₂	#20273471	1	1	yes	characteristic
265	C₂	#20273482	1	1	yes	characteristic
266	C₂	#20273486	1	1	yes	characteristic
267	C₂	#20273500	1	1	yes	characteristic
268	C₂	#20275439	1	1	yes	characteristic
269	C₂	#20275479	1	1	yes	characteristic
270	C₂	#20275491	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×C₄⋊C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#9605
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
154. C₅₃	#9605	semidirect

Maximal subgroups

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

subgroup	index
#20274867	2
#20274568	2
C₂×#16810374	2
#20272550	2
C₄×C₄⋊C₈×C₅₃	2
#9605	53