Subgroups — #27277142

Order 13568.

454 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	C₂×C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	#20274872	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	C₈×Dic₁₀₆	—	2	2	no	—
8	C₈×Dic₁₀₆	C₄	1	4	yes	complemented
9	C₂×#13133149	—	2	1	no	—
10	C₂×#13133149	—	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×C₈×Dic₅₃	—	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	#13133149	—	4	2	no	—
33	#13133149	—	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	C₈×Dic₅₃	—	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	1	no	—
54	C₂×Dic₁₀₆	—	2	2	no	—
55	C₂×Dic₁₀₆	C₄⋊C₄	1	2	yes	—
56	C₂×Dic₁₀₆	SD₁₆	1	4	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	2	no	—
86	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
93	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#135	1	2	yes	—
96	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
98	C₂×Dic₅₃	—	2	2	no	—
99	C₂×Dic₅₃	—	2	4	no	—
100	C₂×Dic₅₃	—	2	2	no	—
101	C₂×Dic₅₃	—	4	2	no	—
102	C₂×Dic₅₃	—	8	2	no	—
103	C₂×Dic₅₃	—	8	2	no	—
104	C₂×Dic₅₃	—	8	2	no	—
105	Dic₁₀₆	—	2	2	no	—
106	Dic₁₀₆	—	2	4	no	—
107	Dic₁₀₆	#106	1	4	yes	complemented
108	C₈×C₅₃	—	2	2	no	—
109	C₈×C₅₃	—	8	2	no	—
110	C₈×C₅₃	#103	1	4	yes	complemented
111	#88322	—	4	2	no	—
112	#8643	—	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	#465	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	C₂×#103	1	1	yes	characteristic
121	C₄×C₅₃	#417	1	1	yes	characteristic
122	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
123	C₄×C₅₃	#427	1	1	yes	characteristic
124	C₄×C₅₃	#465	1	2	yes	—
125	C₄×C₅₃	#475	1	1	yes	characteristic
126	C₄×C₅₃	#477	1	1	yes	characteristic
127	Dic₅₃	—	2	4	no	—
128	Dic₅₃	—	4	2	no	—
129	Dic₅₃	—	16	2	no	—
130	C₂×C₈×Q₈	—	53	1	no	—
131	#1844	—	53	1	no	—
132	#1803	—	53	2	no	—
133	C₂×#334	D₅₃	1	1	yes	characteristic
134	#1296	—	53	2	no	—
135	C₂×C₅₃	#1666	1	2	yes	—
136	C₂×C₅₃	#1672	1	2	yes	—
137	C₂×C₅₃	#1803	1	1	yes	characteristic
138	C₂×C₅₃	#1807	1	1	yes	characteristic
139	C₂×C₅₃	#1815	1	1	yes	characteristic
140	C₂×C₄×Q₈	—	53	1	no	—
141	C₈×Q₈	—	53	4	no	—
142	C₈×Q₈	—	106	2	no	—
143	C₂×#106	—	106	1	no	—
144	C₂×#106	—	106	1	no	—
145	C₂×C₄⋊C₈	—	53	1	no	—
146	C₂×C₄⋊C₈	—	106	1	no	—
147	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
148	C₂×#103	—	106	2	no	—
149	C₂×C₄×C₈	—	106	1	no	—
150	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
151	#384	—	53	2	no	—
152	#340	—	106	2	no	—
153	#334	—	2	2	no	—
154	C₅₃	#8643	1	1	yes	characteristic, complemented, Sylow
155	C₂²×Q₈	—	53	1	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	106	1	no	—
158	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
159	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
160	C₄×Q₈	—	53	2	no	—
161	C₄×Q₈	—	53	2	no	—
162	C₄×Q₈	—	106	1	no	—
163	C₄×Q₈	—	106	1	no	—
164	C₂×C₄⋊C₄	—	53	1	no	—
165	C₂×C₄⋊C₄	—	106	1	no	—
166	C₂×C₄⋊C₄	—	106	2	no	—
167	C₂×C₄⋊C₄	—	106	2	no	—
168	C₂×C₄⋊C₄	—	106	2	no	—
169	C₂×C₄²	—	106	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	2	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	53	2	no	—
176	C₄⋊C₈	—	106	4	no	—
177	C₄⋊C₈	—	106	1	no	—
178	#103	—	212	4	no	—
179	C₄×C₈	—	106	4	no	—
180	C₄×C₈	C₄×D₅₃	1	2	yes	—
181	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
182	C₂×Q₈	—	53	2	no	—
183	C₂×Q₈	—	53	4	no	—
184	C₂×Q₈	—	106	1	no	—
185	C₂×Q₈	—	106	2	no	—
186	C₂²×C₄	—	106	1	no	—
187	C₂²×C₄	—	106	1	no	—
188	C₂²×C₄	—	212	2	no	—
189	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
190	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
191	C₂²×C₄	#11719949	1	1	yes	characteristic
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	106	2	no	—
198	C₂×C₈	—	212	1	no	—
199	C₂×C₈	—	212	1	no	—
200	C₂×C₈	#11719913	1	2	yes	—
201	C₂×C₈	Dic₂₁₂	1	2	yes	—
202	C₂×C₈	#11719921	1	2	yes	—
203	C₂×C₈	#11719921	1	2	yes	—
204	C₄⋊C₄	—	53	2	no	—
205	C₄⋊C₄	—	106	2	no	—
206	C₄⋊C₄	—	106	1	no	—
207	C₄⋊C₄	—	106	2	no	—
208	C₄⋊C₄	—	424	2	no	—
209	C₄⋊C₄	—	212	2	no	—
210	C₄⋊C₄	—	212	2	no	—
211	C₄⋊C₄	—	212	2	no	—
212	C₄⋊C₄	—	212	2	no	—
213	C₄²	—	106	2	no	—
214	C₄²	—	106	2	no	—
215	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
216	C₄²	#11719947	1	1	yes	characteristic
217	C₄²	Q₈×D₅₃	1	1	yes	characteristic
218	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
219	C₂³	#13133192	1	1	yes	characteristic
220	Q₈	—	53	4	no	—
221	Q₈	—	106	4	no	—
222	Q₈	—	106	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	106	4	no	—
225	C₂×C₄	—	106	2	no	—
226	C₂×C₄	—	106	2	no	—
227	C₂×C₄	—	424	2	no	—
228	C₂×C₄	—	424	2	no	—
229	C₂×C₄	—	424	2	no	—
230	C₂×C₄	—	212	2	no	—
231	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
232	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
233	C₂×C₄	#13133175	1	1	yes	characteristic
234	C₂×C₄	#13133188	1	1	yes	characteristic
235	C₂×C₄	#13133193	1	1	yes	characteristic
236	C₂×C₄	#13133197	1	1	yes	characteristic
237	C₂×C₄	SD₁₆×D₅₃	1	2	yes	—
238	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
239	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
240	C₂×C₄	#13133238	1	2	yes	—
241	C₂×C₄	#13133245	1	1	yes	characteristic
242	C₂×C₄	#13133250	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	8	2	no	—
245	C₈	—	212	2	no	—
246	C₈	#13133142	1	4	yes	complemented
247	C₂²	#16810480	1	1	yes	characteristic
248	C₂²	#16810824	1	1	yes	characteristic
249	C₂²	#16810834	1	1	yes	characteristic
250	C₂²	#16810853	1	2	yes	—
251	C₂²	#16810887	1	2	yes	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	848	2	no	—
255	C₄	—	106	4	no	—
256	C₄	—	212	2	no	—
257	C₄	C₂×#13133142	1	1	yes	characteristic
258	C₄	#16810625	1	1	yes	characteristic
259	C₄	#106×D₅₃	1	1	yes	characteristic
260	C₄	#16810658	1	1	yes	characteristic
261	C₄	#16810849	1	2	yes	—
262	C₄	#16810909	1	1	yes	characteristic
263	C₄	#16810919	1	1	yes	characteristic
264	C₂	#20274064	1	2	yes	—
265	C₂	#20274098	1	2	yes	—
266	C₂	#20274721	1	1	yes	characteristic
267	C₂	#20274737	1	1	yes	characteristic
268	C₂	#20274771	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	#8643
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. C₈×Dic₁₀₆	C₄	semidirect
107. Dic₁₀₆	#106	semidirect
110. C₈×C₅₃	#103	semidirect
154. C₅₃	#8643	semidirect
246. 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×C₈×Dic₁₀₆	2
#20274872	2
#20274721	2
#20272599	2
C₂×C₅₃×#334	2
#8643	53