Subgroups — #27277181

Order 13568.

434 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274872	C₂	1	1	yes	maximal, characteristic
2	#20274720	C₂	1	2	yes	maximal
3	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
4	#20272599	C₂	1	2	yes	maximal
5	C₂×C₅₃×#446	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133149	—	2	1	no	—
7	C₂×#13133149	—	2	1	no	—
8	C₂×C₄⋊#88322	C₂²	1	2	yes	—
9	C₂×#13133141	—	2	2	no	—
10	#16810493	C₂²	1	2	yes	—
11	#16810385	—	2	2	no	—
12	#16810376	—	2	2	no	—
13	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#446	—	2	2	no	—
15	C₅₃×#446	C₄	1	4	yes	complemented
16	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×#97×C₅₃	—	2	1	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×#88322	—	2	2	no	—
21	C₂×C₄⋊Dic₅₃	—	2	2	no	—
22	C₂×C₄⋊Dic₅₃	—	2	2	no	—
23	C₂×C₄⋊Dic₅₃	—	2	2	no	—
24	#13133149	—	4	2	no	—
25	#13133149	—	4	2	no	—
26	C₄⋊#88322	—	2	2	no	—
27	C₄⋊#88322	—	2	2	no	—
28	#13133141	—	4	4	no	—
29	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	—	2	1	no	—
31	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₄×Q₈×C₅₃	—	2	1	no	—
34	C₄×Q₈×C₅₃	—	2	1	no	—
35	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
36	C₄×Q₈×C₅₃	Q₈	1	2	yes	—
37	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
38	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	—	2	1	no	—
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₅₃	—	2	4	no	—
42	C₄⋊C₈×C₅₃	—	2	1	no	—
43	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
44	#97×C₅₃	—	2	4	no	—
45	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₅₃	D₄	1	2	yes	—
47	C₂×Q₈×C₅₃	—	2	1	no	—
48	C₂×Q₈×C₅₃	—	2	2	no	—
49	C₂×Q₈×C₅₃	—	2	2	no	—
50	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅₃	—	2	1	no	—
52	C₂²×C₄×C₅₃	—	2	1	no	—
53	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
56	C₂²×Dic₅₃	—	4	2	no	—
57	C₂×C₈×C₅₃	—	2	1	no	—
58	C₂×C₈×C₅₃	—	2	1	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	—	4	1	no	—
61	C₂×C₈×C₅₃	—	4	1	no	—
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
65	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
66	C₄⋊C₄×C₅₃	—	2	2	no	—
67	C₄⋊C₄×C₅₃	—	2	1	no	—
68	C₄⋊C₄×C₅₃	—	2	2	no	—
69	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
70	C₄²×C₅₃	—	2	2	no	—
71	C₄²×C₅₃	—	2	2	no	—
72	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
75	C₄²×C₅₃	#41	1	1	yes	characteristic
76	C₄⋊Dic₅₃	—	4	2	no	—
77	C₄⋊Dic₅₃	—	4	2	no	—
78	C₄⋊Dic₅₃	—	4	2	no	—
79	C₄⋊Dic₅₃	—	4	2	no	—
80	C₄⋊Dic₅₃	—	8	2	no	—
81	C₂×#88322	—	4	2	no	—
82	C₂×#88322	—	4	2	no	—
83	C₂×#88322	—	4	2	no	—
84	C₂³×C₅₃	#121	1	1	yes	characteristic
85	Q₈×C₅₃	—	2	4	no	—
86	Q₈×C₅₃	—	4	1	no	—
87	Q₈×C₅₃	—	4	1	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	4	2	no	—
92	C₂×C₄×C₅₃	—	4	2	no	—
93	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
101	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
103	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#137	1	2	yes	—
105	C₂×Dic₅₃	—	8	2	no	—
106	C₂×Dic₅₃	—	8	2	no	—
107	C₂×Dic₅₃	—	8	2	no	—
108	C₈×C₅₃	—	2	2	no	—
109	C₈×C₅₃	—	4	2	no	—
110	C₈×C₅₃	#103	1	4	yes	complemented
111	#88322	—	8	2	no	—
112	#8651	—	53	1	no	maximal, Sylow
113	C₂²×C₅₃	#380	1	1	yes	characteristic
114	C₂²×C₅₃	#460	1	1	yes	characteristic
115	C₂²×C₅₃	#462	1	1	yes	characteristic
116	C₂²×C₅₃	#468	1	2	yes	—
117	C₂²×C₅₃	#471	1	2	yes	—
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	—	4	2	no	—
121	C₄×C₅₃	—	4	2	no	—
122	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
123	C₄×C₅₃	#417	1	1	yes	characteristic
124	C₄×C₅₃	#428	1	2	yes	—
125	C₄×C₅₃	#469	1	1	yes	characteristic
126	C₄×C₅₃	#470	1	1	yes	characteristic
127	C₄×C₅₃	#475	1	1	yes	characteristic
128	C₄×C₅₃	#477	1	1	yes	characteristic
129	Dic₅₃	—	16	2	no	—
130	C₂×#446	D₅₃	1	1	yes	characteristic
131	#1844	—	53	1	no	—
132	#1803	—	53	2	no	—
133	C₂×#335	—	53	1	no	—
134	#1296	—	53	2	no	—
135	C₂×C₅₃	#1686	1	2	yes	—
136	C₂×C₅₃	#1692	1	2	yes	—
137	C₂×C₅₃	#1803	1	1	yes	characteristic
138	C₂×C₅₃	#1809	1	1	yes	characteristic
139	C₂×C₅₃	#1815	1	1	yes	characteristic
140	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
141	#446	—	2	2	no	—
142	#446	Dic₅₃	1	4	yes	complemented
143	C₂×#106	—	106	1	no	—
144	C₂×#106	—	106	1	no	—
145	C₂×C₄⋊C₈	—	2	1	no	—
146	C₂×C₄⋊C₈	—	53	2	no	—
147	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
148	C₂×#103	—	106	2	no	—
149	C₂×#97	—	2	1	no	—
150	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
151	#384	—	53	2	no	—
152	#340	—	106	2	no	—
153	#335	—	106	2	no	—
154	C₅₃	#8651	1	1	yes	characteristic, complemented, Sylow
155	C₂²×Q₈	D₂₁₂	1	1	yes	characteristic
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	—	106	2	no	—
158	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
159	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
160	C₄×Q₈	—	2	1	no	—
161	C₄×Q₈	—	2	1	no	—
162	C₄×Q₈	Dic₁₀₆	1	2	yes	—
163	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
164	C₂×C₄⋊C₄	—	2	1	no	—
165	C₂×C₄⋊C₄	—	106	2	no	—
166	C₂×C₄⋊C₄	—	106	2	no	—
167	C₂×C₄⋊C₄	—	106	2	no	—
168	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
169	C₂×C₄²	—	2	1	no	—
170	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
171	#106	—	212	2	no	—
172	#106	—	212	2	no	—
173	C₄⋊C₈	—	2	4	no	—
174	C₄⋊C₈	—	2	1	no	—
175	C₄⋊C₈	—	106	2	no	—
176	C₄⋊C₈	—	106	2	no	—
177	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
178	#103	—	212	4	no	—
179	#97	—	2	4	no	—
180	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
181	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
182	C₂×Q₈	—	2	1	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	—	2	2	no	—
185	C₂×Q₈	C₄⋊Dic₅₃	1	2	yes	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	212	2	no	—
189	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
190	C₂²×C₄	#11719945	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	212	2	no	—
198	C₂×C₈	—	212	2	no	—
199	C₂×C₈	—	212	2	no	—
200	C₂×C₈	#11719924	1	2	yes	—
201	C₂×C₈	#11719925	1	2	yes	—
202	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
203	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	424	2	no	—
208	C₄⋊C₄	—	212	2	no	—
209	C₄⋊C₄	—	212	2	no	—
210	C₄⋊C₄	—	212	2	no	—
211	C₄⋊C₄	—	212	2	no	—
212	C₄⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
213	C₄²	—	2	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
216	C₄²	#11719945	1	1	yes	characteristic
217	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
218	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
219	C₂³	#13133191	1	1	yes	characteristic
220	Q₈	—	2	4	no	—
221	Q₈	—	4	1	no	—
222	Q₈	—	4	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	424	2	no	—
229	C₂×C₄	—	424	2	no	—
230	C₂×C₄	—	424	2	no	—
231	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
232	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
233	C₂×C₄	#13133176	1	1	yes	characteristic
234	C₂×C₄	#13133191	1	1	yes	characteristic
235	C₂×C₄	#13133196	1	1	yes	characteristic
236	C₂×C₄	#13133196	1	1	yes	characteristic
237	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
238	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
239	C₂×C₄	#13133242	1	1	yes	characteristic
240	C₂×C₄	#13133245	1	2	yes	—
241	C₂×C₄	#13133247	1	1	yes	characteristic
242	C₂×C₄	#13133250	1	2	yes	—
243	C₈	—	2	2	no	—
244	C₈	—	4	2	no	—
245	C₈	—	424	2	no	—
246	C₈	#13133141	1	4	yes	complemented
247	C₂²	#16810483	1	1	yes	characteristic
248	C₂²	#16810823	1	1	yes	characteristic
249	C₂²	#16810833	1	1	yes	characteristic
250	C₂²	#16810869	1	2	yes	—
251	C₂²	#16810886	1	2	yes	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	4	2	no	—
255	C₄	—	4	2	no	—
256	C₄	—	848	2	no	—
257	C₄	C₂×#13133141	1	1	yes	characteristic
258	C₄	#16810624	1	1	yes	characteristic
259	C₄	#16810665	1	2	yes	—
260	C₄	#16810876	1	1	yes	characteristic
261	C₄	#16810881	1	1	yes	characteristic
262	C₄	#16810908	1	1	yes	characteristic
263	C₄	#16810918	1	1	yes	characteristic
264	C₂	#20274187	1	2	yes	—
265	C₂	#20274221	1	2	yes	—
266	C₂	#20274720	1	1	yes	characteristic
267	C₂	#20274748	1	1	yes	characteristic
268	C₂	#20274774	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133191
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#446	C₂
socle, prime radical	C₂³×C₅₃	#121
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#446	D₅₃
hypocenter	C₅₃	#8651
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
15. C₅₃×#446	C₄	semidirect
110. C₈×C₅₃	#103	semidirect
142. #446	Dic₅₃	semidirect
154. C₅₃	#8651	semidirect
246. C₈	#13133141	semidirect

Maximal subgroups

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

subgroup	index
#20274872	2
#20274720	2
C₂×#16810376	2
#20272599	2
C₂×C₅₃×#446	2
#8651	53