Subgroups — #27277192

Order 13568.

434 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810753	C₂	1	1	yes	maximal, characteristic
2	#20274875	C₂	1	1	yes	maximal, characteristic
3	#20274721	C₂	1	2	yes	maximal
4	#20272599	C₂	1	2	yes	maximal
5	C₂×C₅₃×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
7	#16810753	—	2	2	no	—
8	#16810753	C₄	1	4	yes	complemented
9	C₂×#13133151	—	2	1	no	—
10	C₂×#13133151	—	2	1	no	—
11	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
12	C₂×#13133146	—	2	1	no	—
13	C₂×#13133142	—	2	2	no	—
14	C₂×#13133127	—	2	1	no	—
15	#16810493	C₂²	1	2	yes	—
16	#16810386	—	2	2	no	—
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#334	—	2	2	no	—
20	C₂²×Dic₁₀₆	D₄	1	1	yes	characteristic
21	C₂²×#88322	—	2	1	no	—
22	C₄×Dic₁₀₆	—	2	1	no	—
23	C₄×Dic₁₀₆	—	2	1	no	—
24	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
25	C₄×Dic₁₀₆	Q₈	1	2	yes	—
26	C₂×C₄⋊Dic₅₃	—	2	2	no	—
27	C₂×C₄⋊Dic₅₃	—	2	2	no	—
28	C₂×C₄⋊Dic₅₃	—	2	2	no	—
29	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
30	C₂×#11719919	—	2	1	no	—
31	C₂×C₄×Dic₅₃	—	2	1	no	—
32	#13133151	—	4	2	no	—
33	#13133151	—	4	2	no	—
34	C₄⋊#88322	—	2	1	no	—
35	C₄⋊#88322	C₂×C₄	1	2	yes	—
36	#13133146	—	2	4	no	—
37	#13133142	—	4	4	no	—
38	#13133127	—	2	4	no	—
39	C₂²×C₈×C₅₃	—	2	2	no	—
40	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₅₃	—	2	2	no	—
44	C₄⋊C₈×C₅₃	—	2	2	no	—
45	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₅₃	D₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
50	C₂²×Dic₅₃	—	2	1	no	—
51	C₂²×Dic₅₃	—	2	1	no	—
52	C₂²×Dic₅₃	—	4	2	no	—
53	C₂×Dic₁₀₆	—	2	2	no	—
54	C₂×Dic₁₀₆	—	2	2	no	—
55	C₂×Dic₁₀₆	—	2	1	no	—
56	C₂×Dic₁₀₆	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₅₃	—	2	1	no	—
58	C₂×C₈×C₅₃	—	2	1	no	—
59	C₂×C₈×C₅₃	—	4	2	no	—
60	C₂×C₈×C₅₃	—	4	2	no	—
61	C₂×C₈×C₅₃	—	4	2	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	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
69	C₄²×C₅₃	#41	1	1	yes	characteristic
70	C₄⋊Dic₅₃	—	2	1	no	—
71	C₄⋊Dic₅₃	—	4	2	no	—
72	C₄⋊Dic₅₃	—	4	2	no	—
73	C₄⋊Dic₅₃	—	4	2	no	—
74	C₄⋊Dic₅₃	—	4	2	no	—
75	C₄⋊Dic₅₃	—	8	2	no	—
76	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
77	#11719919	—	2	2	no	—
78	#11719919	—	2	2	no	—
79	C₄×Dic₅₃	—	2	2	no	—
80	C₄×Dic₅₃	—	2	2	no	—
81	C₂×#88322	—	2	2	no	—
82	C₂×#88322	—	4	1	no	—
83	C₂×#88322	—	4	1	no	—
84	C₂³×C₅₃	#121	1	1	yes	characteristic
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
95	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
96	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#137	1	2	yes	—
98	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
99	C₂×Dic₅₃	—	2	2	no	—
100	C₂×Dic₅₃	—	2	2	no	—
101	C₂×Dic₅₃	—	4	1	no	—
102	C₂×Dic₅₃	—	4	1	no	—
103	C₂×Dic₅₃	—	4	1	no	—
104	C₂×Dic₅₃	—	4	1	no	—
105	C₂×Dic₅₃	—	8	2	no	—
106	C₂×Dic₅₃	—	8	2	no	—
107	C₂×Dic₅₃	—	8	2	no	—
108	Dic₁₀₆	—	2	4	no	—
109	Dic₁₀₆	—	4	2	no	—
110	C₈×C₅₃	—	2	2	no	—
111	C₈×C₅₃	—	8	2	no	—
112	C₈×C₅₃	#103	1	4	yes	complemented
113	#88322	—	4	2	no	—
114	#8653	—	53	1	no	maximal, Sylow
115	C₂²×C₅₃	#380	1	1	yes	characteristic
116	C₂²×C₅₃	#460	1	1	yes	characteristic
117	C₂²×C₅₃	#462	1	1	yes	characteristic
118	C₂²×C₅₃	#469	1	1	yes	characteristic
119	C₂²×C₅₃	#470	1	1	yes	characteristic
120	C₂²×C₅₃	#472	1	1	yes	characteristic
121	C₂²×C₅₃	#473	1	1	yes	characteristic
122	C₄×C₅₃	—	2	1	no	—
123	C₄×C₅₃	—	2	1	no	—
124	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
125	C₄×C₅₃	#417	1	1	yes	characteristic
126	C₄×C₅₃	#428	1	2	yes	—
127	C₄×C₅₃	#468	1	2	yes	—
128	C₄×C₅₃	#475	1	1	yes	characteristic
129	C₄×C₅₃	#477	1	1	yes	characteristic
130	Dic₅₃	—	4	2	no	—
131	Dic₅₃	—	4	2	no	—
132	Dic₅₃	—	16	2	no	—
133	C₂×#446	—	53	1	no	—
134	#1845	—	53	1	no	—
135	#1803	—	53	2	no	—
136	C₂×#334	D₅₃	1	1	yes	characteristic
137	#1296	—	53	2	no	—
138	C₂×C₅₃	#1688	1	1	yes	characteristic
139	C₂×C₅₃	#1690	1	1	yes	characteristic
140	C₂×C₅₃	#1694	1	1	yes	characteristic
141	C₂×C₅₃	#1696	1	1	yes	characteristic
142	C₂×C₅₃	#1803	1	1	yes	characteristic
143	C₂×C₅₃	#1809	1	1	yes	characteristic
144	C₂×C₅₃	#1816	1	1	yes	characteristic
145	C₂×C₄×Q₈	—	53	1	no	—
146	#446	—	53	4	no	—
147	#446	—	106	2	no	—
148	C₂×#107	—	106	1	no	—
149	C₂×#107	—	106	1	no	—
150	C₂×C₄⋊C₈	—	53	1	no	—
151	C₂×C₄⋊C₈	—	106	1	no	—
152	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
153	C₂×#103	—	106	2	no	—
154	C₂×#97	—	106	1	no	—
155	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
156	#384	—	53	2	no	—
157	#340	—	106	2	no	—
158	#334	—	2	2	no	—
159	C₅₃	#8653	1	1	yes	characteristic, complemented, Sylow
160	C₂²×Q₈	—	53	1	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	106	1	no	—
163	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
164	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
165	C₄×Q₈	—	53	2	no	—
166	C₄×Q₈	—	53	2	no	—
167	C₄×Q₈	—	106	1	no	—
168	C₄×Q₈	—	106	1	no	—
169	C₂×C₄⋊C₄	—	53	1	no	—
170	C₂×C₄⋊C₄	—	106	1	no	—
171	C₂×C₄⋊C₄	—	106	2	no	—
172	C₂×C₄⋊C₄	—	106	2	no	—
173	C₂×C₄⋊C₄	—	106	2	no	—
174	C₂×C₄²	—	106	1	no	—
175	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
176	#107	—	212	2	no	—
177	#107	—	212	2	no	—
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	—	53	2	no	—
181	C₄⋊C₈	—	106	4	no	—
182	C₄⋊C₈	—	106	1	no	—
183	#103	—	212	4	no	—
184	#97	—	106	4	no	—
185	C₄×C₈	C₄×D₅₃	1	2	yes	—
186	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
187	C₂×Q₈	—	53	2	no	—
188	C₂×Q₈	—	106	1	no	—
189	C₂×Q₈	—	106	2	no	—
190	C₂×Q₈	—	106	2	no	—
191	C₂²×C₄	—	106	1	no	—
192	C₂²×C₄	—	106	1	no	—
193	C₂²×C₄	—	212	2	no	—
194	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
195	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
196	C₂²×C₄	#11719949	1	1	yes	characteristic
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	106	2	no	—
203	C₂×C₈	—	212	1	no	—
204	C₂×C₈	—	212	1	no	—
205	C₂×C₈	#11719913	1	2	yes	—
206	C₂×C₈	Dic₂₁₂	1	2	yes	—
207	C₂×C₈	#11719921	1	2	yes	—
208	C₂×C₈	#11719921	1	2	yes	—
209	C₄⋊C₄	—	53	2	no	—
210	C₄⋊C₄	—	106	2	no	—
211	C₄⋊C₄	—	106	1	no	—
212	C₄⋊C₄	—	106	2	no	—
213	C₄⋊C₄	—	424	2	no	—
214	C₄⋊C₄	—	212	2	no	—
215	C₄⋊C₄	—	212	2	no	—
216	C₄⋊C₄	—	212	2	no	—
217	C₄⋊C₄	—	212	2	no	—
218	C₄²	—	106	2	no	—
219	C₄²	—	106	2	no	—
220	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
221	C₄²	#11719947	1	1	yes	characteristic
222	C₄²	Q₈×D₅₃	1	1	yes	characteristic
223	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
224	C₂³	#13133192	1	1	yes	characteristic
225	Q₈	—	106	4	no	—
226	Q₈	—	212	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	106	2	no	—
230	C₂×C₄	—	106	2	no	—
231	C₂×C₄	—	424	2	no	—
232	C₂×C₄	—	424	2	no	—
233	C₂×C₄	—	424	2	no	—
234	C₂×C₄	—	212	1	no	—
235	C₂×C₄	—	212	1	no	—
236	C₂×C₄	—	212	1	no	—
237	C₂×C₄	—	212	1	no	—
238	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
239	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
240	C₂×C₄	#13133175	1	1	yes	characteristic
241	C₂×C₄	#13133188	1	1	yes	characteristic
242	C₂×C₄	#13133193	1	1	yes	characteristic
243	C₂×C₄	#13133197	1	1	yes	characteristic
244	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
245	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
246	C₂×C₄	#13133244	1	2	yes	—
247	C₂×C₄	#13133245	1	1	yes	characteristic
248	C₂×C₄	#13133246	1	2	yes	—
249	C₂×C₄	#13133250	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	8	2	no	—
252	C₈	—	212	2	no	—
253	C₈	#13133142	1	4	yes	complemented
254	C₂²	#16810480	1	1	yes	characteristic
255	C₂²	#16810824	1	1	yes	characteristic
256	C₂²	#16810834	1	1	yes	characteristic
257	C₂²	#16810877	1	1	yes	characteristic
258	C₂²	#16810882	1	1	yes	characteristic
259	C₂²	#16810894	1	1	yes	characteristic
260	C₂²	#16810899	1	1	yes	characteristic
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	848	2	no	—
264	C₄	—	212	2	no	—
265	C₄	—	212	2	no	—
266	C₄	C₂×#13133142	1	1	yes	characteristic
267	C₄	#16810625	1	1	yes	characteristic
268	C₄	#16810662	1	2	yes	—
269	C₄	#16810866	1	2	yes	—
270	C₄	#16810909	1	1	yes	characteristic
271	C₄	#16810919	1	1	yes	characteristic
272	C₂	#20274200	1	1	yes	characteristic
273	C₂	#20274210	1	1	yes	characteristic
274	C₂	#20274234	1	1	yes	characteristic
275	C₂	#20274244	1	1	yes	characteristic
276	C₂	#20274721	1	1	yes	characteristic
277	C₂	#20274745	1	1	yes	characteristic
278	C₂	#20274776	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133192
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#334	C₂
socle, prime radical	C₂³×C₅₃	#121
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#334	D₅₃
hypocenter	C₅₃	#8653
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
8. #16810753	C₄	semidirect
112. C₈×C₅₃	#103	semidirect
159. C₅₃	#8653	semidirect
253. C₈	#13133142	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810753	2
#20274875	2
#20274721	2
#20272599	2
C₂×C₅₃×#334	2
#8653	53