Subgroups — #27277301

Order 13568.

366 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274879	C₂	1	1	yes	maximal, characteristic
2	#20273463	C₂	1	1	yes	maximal, characteristic
3	#20273279	C₂	1	2	yes	maximal
4	#20272575	C₂	1	2	yes	maximal
5	C₅₃×#2960	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	C₂×#13133151	—	2	1	no	—
8	C₂×#13133149	—	2	1	no	—
9	#16810489	C₂²	1	2	yes	—
10	#16810376	—	2	1	no	—
11	#16810374	—	2	1	no	—
12	#16810360	—	2	4	no	—
13	C₅₃×#533	C₂²	1	1	yes	characteristic
14	C₅₃×#446	—	2	1	no	—
15	C₅₃×#446	C₄	1	2	yes	complemented
16	C₈×Q₈×C₅₃	—	2	1	no	—
17	C₈×Q₈×C₅₃	C₄	1	2	yes	complemented
18	C₅₃×#424	—	2	1	no	—
19	C₅₃×#423	C₂²	1	1	yes	characteristic
20	C₅₃×#405	—	2	1	no	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₅₃	—	2	2	no	—
23	#13133151	—	4	2	no	—
24	#13133149	—	4	2	no	—
25	C₄⋊#88322	—	2	2	no	—
26	C₂²⋊#88322	—	2	2	no	—
27	C₄×#88322	—	2	2	no	—
28	#13133124	—	2	2	no	—
29	#13133124	—	2	2	no	—
30	C₂×M₁₆×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	Q₈	1	1	yes	characteristic
34	C₄⋊Q₈×C₅₃	Q₈	1	1	yes	characteristic
35	#125×C₅₃	—	2	1	no	—
36	C₂²⋊Q₈×C₅₃	—	2	1	no	—
37	C₂²⋊Q₈×C₅₃	D₄	1	2	yes	—
38	C₄×Q₈×C₅₃	—	2	1	no	—
39	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
40	#117×C₅₃	—	2	1	no	—
41	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₅₃	—	2	2	no	—
43	C₄⋊C₈×C₅₃	—	2	2	no	—
44	C₄⋊C₈×C₅₃	—	2	1	no	—
45	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
46	#97×C₅₃	—	2	2	no	—
47	C₄×C₈×C₅₃	—	2	2	no	—
48	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
52	C₂×Q₈×C₅₃	—	2	1	no	—
53	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
57	C₂²×Dic₅₃	—	4	2	no	—
58	M₁₆×C₅₃	—	4	1	no	—
59	M₁₆×C₅₃	—	4	1	no	—
60	C₂×C₈×C₅₃	—	2	1	no	—
61	C₂×C₈×C₅₃	—	2	1	no	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
68	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
69	C₄⋊C₄×C₅₃	—	2	2	no	—
70	C₄⋊C₄×C₅₃	—	2	2	no	—
71	C₄⋊C₄×C₅₃	—	2	1	no	—
72	C₄⋊C₄×C₅₃	—	2	1	no	—
73	C₄⋊C₄×C₅₃	—	2	1	no	—
74	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
75	C₂²⋊C₄×C₅₃	—	2	2	no	—
76	C₄²×C₅₃	—	2	2	no	—
77	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
80	C₄²×C₅₃	#41	1	1	yes	characteristic
81	C₄⋊Dic₅₃	—	4	2	no	—
82	C₄⋊Dic₅₃	—	4	2	no	—
83	C₂×#88322	—	4	2	no	—
84	C₂³×C₅₃	#121	1	1	yes	characteristic
85	Q₈×C₅₃	—	4	2	no	—
86	Q₈×C₅₃	—	4	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	4	1	no	—
90	C₂×C₄×C₅₃	—	4	1	no	—
91	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#117	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	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
99	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
101	C₂×Dic₅₃	—	8	2	no	—
102	C₂×Dic₅₃	—	8	2	no	—
103	C₂×Dic₅₃	—	8	2	no	—
104	C₈×C₅₃	—	2	1	no	—
105	C₈×C₅₃	—	2	1	no	—
106	C₈×C₅₃	—	4	2	no	—
107	C₈×C₅₃	#103	1	2	yes	—
108	C₈×C₅₃	#103	1	2	yes	—
109	#88322	—	8	2	no	—
110	#8675	—	53	1	no	maximal, Sylow
111	C₂²×C₅₃	—	2	1	no	—
112	C₂²×C₅₃	—	2	1	no	—
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	C₄×C₅₃	—	2	1	no	—
117	C₄×C₅₃	—	2	1	no	—
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	#475	1	1	yes	characteristic
125	C₄×C₅₃	#477	1	1	yes	characteristic
126	Dic₅₃	—	16	2	no	—
127	#2960	D₅₃	1	1	yes	characteristic
128	#1846	—	53	1	no	—
129	#1564	—	53	1	no	—
130	#1518	—	53	2	no	—
131	#1286	—	53	2	no	—
132	C₂×C₅₃	—	2	1	no	—
133	C₂×C₅₃	—	2	1	no	—
134	C₂×C₅₃	#1803	1	1	yes	characteristic
135	C₂×C₅₃	#1812	1	1	yes	characteristic
136	C₂×C₅₃	#1814	1	1	yes	characteristic
137	#533	C₂×D₅₃	1	1	yes	characteristic
138	#446	—	2	1	no	—
139	#446	Dic₅₃	1	2	yes	complemented
140	C₈×Q₈	—	2	1	no	—
141	C₈×Q₈	Dic₅₃	1	2	yes	complemented
142	#431	—	53	2	no	—
143	C₂×#107	—	106	1	no	—
144	C₂×#106	—	106	1	no	—
145	#424	—	2	1	no	—
146	#423	C₂×D₅₃	1	1	yes	characteristic
147	#405	—	2	1	no	—
148	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
149	#382	—	53	2	no	—
150	#335	—	106	1	no	—
151	#334	—	106	1	no	—
152	#328	—	106	4	no	—
153	C₅₃	#8675	1	1	yes	characteristic, complemented, Sylow
154	C₂×M₁₆	—	2	1	no	—
155	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
156	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
157	C₄⋊Q₈	Dic₁₀₆	1	1	yes	characteristic
158	C₄⋊Q₈	Dic₁₀₆	1	1	yes	characteristic
159	#125	—	2	1	no	—
160	C₂²⋊Q₈	—	2	1	no	—
161	C₂²⋊Q₈	D₂₁₂	1	2	yes	—
162	C₄×Q₈	—	2	1	no	—
163	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
164	#117	—	2	1	no	—
165	C₂×C₄⋊C₄	—	106	2	no	—
166	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
167	#107	—	212	2	no	—
168	#106	—	212	2	no	—
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	1	no	—
172	C₄⋊C₈	—	106	2	no	—
173	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
174	C₂²⋊C₈	—	106	2	no	—
175	#97	—	2	2	no	—
176	C₄×C₈	—	2	2	no	—
177	C₄×C₈	—	106	2	no	—
178	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
179	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
180	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
182	#95	—	106	2	no	—
183	#95	—	106	2	no	—
184	C₂×Q₈	—	2	1	no	—
185	C₂×Q₈	C₄⋊Dic₅₃	1	2	yes	—
186	C₂²×C₄	—	212	2	no	—
187	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
188	C₂²×C₄	#11719945	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
190	M₁₆	—	4	1	no	—
191	M₁₆	—	4	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	212	2	no	—
196	C₂×C₈	#11719924	1	2	yes	—
197	C₂×C₈	#11719925	1	2	yes	—
198	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
199	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
200	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
201	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	212	2	no	—
208	C₄⋊C₄	—	212	2	no	—
209	C₄⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
210	C₂²⋊C₄	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
213	C₄²	#11719945	1	1	yes	characteristic
214	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
215	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
216	C₂³	#13133191	1	1	yes	characteristic
217	Q₈	—	4	2	no	—
218	Q₈	—	4	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	424	2	no	—
224	C₂×C₄	—	424	2	no	—
225	C₂×C₄	—	424	2	no	—
226	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
227	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
228	C₂×C₄	#13133176	1	1	yes	characteristic
229	C₂×C₄	#13133191	1	1	yes	characteristic
230	C₂×C₄	#13133196	1	1	yes	characteristic
231	C₂×C₄	#13133196	1	1	yes	characteristic
232	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
233	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
234	C₂×C₄	#13133242	1	1	yes	characteristic
235	C₂×C₄	#13133247	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	4	2	no	—
239	C₈	—	424	2	no	—
240	C₈	#13133141	1	2	yes	—
241	C₈	#13133141	1	2	yes	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	#16810483	1	1	yes	characteristic
245	C₂²	#16810823	1	1	yes	characteristic
246	C₂²	#16810833	1	1	yes	characteristic
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	4	2	no	—
252	C₄	—	4	2	no	—
253	C₄	—	848	2	no	—
254	C₄	C₂×#13133141	1	1	yes	characteristic
255	C₄	#16810624	1	1	yes	characteristic
256	C₄	#16810908	1	1	yes	characteristic
257	C₄	#16810918	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	#20274720	1	1	yes	characteristic
261	C₂	#20274761	1	1	yes	characteristic
262	C₂	#20274769	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810483
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2960	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂³×C₅₃	#121
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2960	D₅₃
hypocenter	C₅₃	#8675
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

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

N	H ≅ G/N	type
15. C₅₃×#446	C₄	semidirect
17. C₈×Q₈×C₅₃	C₄	semidirect
139. #446	Dic₅₃	semidirect
141. C₈×Q₈	Dic₅₃	semidirect
153. C₅₃	#8675	semidirect

Maximal subgroups

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

subgroup	index
#20274879	2
#20273463	2
#20273279	2
#20272575	2
C₅₃×#2960	2
#8675	53