Subgroups — #27277152

Order 13568.

454 conjugacy classes of subgroups.

The subgroup lattice has 296 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	#20274875	C₂	1	1	yes	maximal, characteristic
3	#20274721	C₂	1	2	yes	maximal
4	#20272599	C₂	1	2	yes	maximal
5	C₂×C₅₃×#335	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×#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×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×#335	—	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	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	1	no	—
55	C₂×Dic₁₀₆	—	2	1	no	—
56	C₂×Dic₁₀₆	C₄⋊C₄	1	2	yes	—
57	C₂×Dic₁₀₆	D₈	1	2	yes	—
58	C₂×Dic₁₀₆	Q₁₆	1	2	yes	—
59	C₂×C₈×C₅₃	—	2	1	no	—
60	C₂×C₈×C₅₃	—	2	1	no	—
61	C₂×C₈×C₅₃	—	4	2	no	—
62	C₂×C₈×C₅₃	—	4	2	no	—
63	C₂×C₈×C₅₃	—	4	2	no	—
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
67	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
68	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
71	C₄²×C₅₃	#41	1	1	yes	characteristic
72	C₄⋊Dic₅₃	—	2	1	no	—
73	C₄⋊Dic₅₃	—	4	2	no	—
74	C₄⋊Dic₅₃	—	4	2	no	—
75	C₄⋊Dic₅₃	—	4	2	no	—
76	C₄⋊Dic₅₃	—	4	2	no	—
77	C₄⋊Dic₅₃	—	8	2	no	—
78	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
79	#11719919	—	2	2	no	—
80	#11719919	—	2	2	no	—
81	C₄×Dic₅₃	—	2	2	no	—
82	C₄×Dic₅₃	—	2	2	no	—
83	C₂×#88322	—	2	2	no	—
84	C₂×#88322	—	4	1	no	—
85	C₂×#88322	—	4	1	no	—
86	C₂³×C₅₃	#121	1	1	yes	characteristic
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
101	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
103	C₂×Dic₅₃	—	2	2	no	—
104	C₂×Dic₅₃	—	2	2	no	—
105	C₂×Dic₅₃	—	2	2	no	—
106	C₂×Dic₅₃	—	2	2	no	—
107	C₂×Dic₅₃	—	4	1	no	—
108	C₂×Dic₅₃	—	4	1	no	—
109	C₂×Dic₅₃	—	8	2	no	—
110	C₂×Dic₅₃	—	8	2	no	—
111	C₂×Dic₅₃	—	8	2	no	—
112	Dic₁₀₆	—	2	2	no	—
113	Dic₁₀₆	—	2	2	no	—
114	Dic₁₀₆	—	2	2	no	—
115	Dic₁₀₆	#107	1	4	yes	complemented
116	C₈×C₅₃	—	2	2	no	—
117	C₈×C₅₃	—	8	2	no	—
118	C₈×C₅₃	#103	1	4	yes	complemented
119	#88322	—	4	2	no	—
120	#8645	—	53	1	no	maximal, Sylow
121	C₂²×C₅₃	#380	1	1	yes	characteristic
122	C₂²×C₅₃	#460	1	1	yes	characteristic
123	C₂²×C₅₃	#462	1	1	yes	characteristic
124	C₂²×C₅₃	#466	1	1	yes	characteristic
125	C₂²×C₅₃	#467	1	1	yes	characteristic
126	C₂²×C₅₃	#472	1	1	yes	characteristic
127	C₂²×C₅₃	#473	1	1	yes	characteristic
128	C₄×C₅₃	—	2	1	no	—
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
131	C₄×C₅₃	#417	1	1	yes	characteristic
132	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
133	C₄×C₅₃	#427	1	1	yes	characteristic
134	C₄×C₅₃	#466	1	1	yes	characteristic
135	C₄×C₅₃	#467	1	1	yes	characteristic
136	C₄×C₅₃	#475	1	1	yes	characteristic
137	C₄×C₅₃	#477	1	1	yes	characteristic
138	Dic₅₃	—	2	4	no	—
139	Dic₅₃	—	4	2	no	—
140	Dic₅₃	—	16	2	no	—
141	C₂×C₈×Q₈	—	53	1	no	—
142	#1845	—	53	1	no	—
143	#1803	—	53	2	no	—
144	C₂×#335	D₅₃	1	1	yes	characteristic
145	#1296	—	53	2	no	—
146	C₂×C₅₃	#1668	1	1	yes	characteristic
147	C₂×C₅₃	#1670	1	1	yes	characteristic
148	C₂×C₅₃	#1674	1	1	yes	characteristic
149	C₂×C₅₃	#1676	1	1	yes	characteristic
150	C₂×C₅₃	#1803	1	1	yes	characteristic
151	C₂×C₅₃	#1808	1	1	yes	characteristic
152	C₂×C₅₃	#1816	1	1	yes	characteristic
153	C₂×C₄×Q₈	—	53	1	no	—
154	C₈×Q₈	—	53	4	no	—
155	C₈×Q₈	—	106	2	no	—
156	C₂×#107	—	106	1	no	—
157	C₂×#107	—	106	1	no	—
158	C₂×C₄⋊C₈	—	53	1	no	—
159	C₂×C₄⋊C₈	—	106	1	no	—
160	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
161	C₂×#103	—	106	2	no	—
162	C₂×C₄×C₈	—	106	1	no	—
163	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
164	#384	—	53	2	no	—
165	#340	—	106	2	no	—
166	#335	—	2	2	no	—
167	C₅₃	#8645	1	1	yes	characteristic, complemented, Sylow
168	C₂²×Q₈	—	53	1	no	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	106	1	no	—
171	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
172	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
173	C₄×Q₈	—	53	2	no	—
174	C₄×Q₈	—	53	2	no	—
175	C₄×Q₈	—	106	1	no	—
176	C₄×Q₈	—	106	1	no	—
177	C₂×C₄⋊C₄	—	53	1	no	—
178	C₂×C₄⋊C₄	—	106	1	no	—
179	C₂×C₄⋊C₄	—	106	2	no	—
180	C₂×C₄⋊C₄	—	106	2	no	—
181	C₂×C₄⋊C₄	—	106	2	no	—
182	C₂×C₄²	—	106	1	no	—
183	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
184	#107	—	212	2	no	—
185	#107	—	212	2	no	—
186	C₄⋊C₈	—	2	2	no	—
187	C₄⋊C₈	—	2	2	no	—
188	C₄⋊C₈	—	53	2	no	—
189	C₄⋊C₈	—	106	4	no	—
190	C₄⋊C₈	—	106	1	no	—
191	#103	—	212	4	no	—
192	C₄×C₈	—	106	4	no	—
193	C₄×C₈	C₄×D₅₃	1	2	yes	—
194	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
195	C₂×Q₈	—	53	2	no	—
196	C₂×Q₈	—	53	2	no	—
197	C₂×Q₈	—	53	2	no	—
198	C₂×Q₈	—	106	1	no	—
199	C₂×Q₈	—	106	1	no	—
200	C₂×Q₈	—	106	1	no	—
201	C₂²×C₄	—	106	1	no	—
202	C₂²×C₄	—	106	1	no	—
203	C₂²×C₄	—	212	2	no	—
204	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
205	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
206	C₂²×C₄	#11719949	1	1	yes	characteristic
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	106	2	no	—
213	C₂×C₈	—	212	1	no	—
214	C₂×C₈	—	212	1	no	—
215	C₂×C₈	#11719913	1	2	yes	—
216	C₂×C₈	Dic₂₁₂	1	2	yes	—
217	C₂×C₈	#11719921	1	2	yes	—
218	C₂×C₈	#11719921	1	2	yes	—
219	C₄⋊C₄	—	53	2	no	—
220	C₄⋊C₄	—	106	2	no	—
221	C₄⋊C₄	—	106	1	no	—
222	C₄⋊C₄	—	106	2	no	—
223	C₄⋊C₄	—	424	2	no	—
224	C₄⋊C₄	—	212	2	no	—
225	C₄⋊C₄	—	212	2	no	—
226	C₄⋊C₄	—	212	2	no	—
227	C₄⋊C₄	—	212	2	no	—
228	C₄²	—	106	2	no	—
229	C₄²	—	106	2	no	—
230	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
231	C₄²	#11719947	1	1	yes	characteristic
232	C₄²	Q₈×D₅₃	1	1	yes	characteristic
233	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
234	C₂³	#13133192	1	1	yes	characteristic
235	Q₈	—	53	4	no	—
236	Q₈	—	106	2	no	—
237	Q₈	—	106	2	no	—
238	Q₈	—	106	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	106	2	no	—
242	C₂×C₄	—	106	2	no	—
243	C₂×C₄	—	106	2	no	—
244	C₂×C₄	—	106	2	no	—
245	C₂×C₄	—	424	2	no	—
246	C₂×C₄	—	424	2	no	—
247	C₂×C₄	—	424	2	no	—
248	C₂×C₄	—	212	1	no	—
249	C₂×C₄	—	212	1	no	—
250	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
251	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
252	C₂×C₄	#13133175	1	1	yes	characteristic
253	C₂×C₄	#13133188	1	1	yes	characteristic
254	C₂×C₄	#13133193	1	1	yes	characteristic
255	C₂×C₄	#13133197	1	1	yes	characteristic
256	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
257	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
258	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
259	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
260	C₂×C₄	#13133236	1	1	yes	characteristic
261	C₂×C₄	#13133240	1	1	yes	characteristic
262	C₂×C₄	#13133245	1	1	yes	characteristic
263	C₂×C₄	#13133250	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	8	2	no	—
266	C₈	—	212	2	no	—
267	C₈	#13133142	1	4	yes	complemented
268	C₂²	#16810480	1	1	yes	characteristic
269	C₂²	#16810824	1	1	yes	characteristic
270	C₂²	#16810834	1	1	yes	characteristic
271	C₂²	#16810860	1	1	yes	characteristic
272	C₂²	#16810865	1	1	yes	characteristic
273	C₂²	#16810894	1	1	yes	characteristic
274	C₂²	#16810899	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	848	2	no	—
278	C₄	—	106	4	no	—
279	C₄	—	212	2	no	—
280	C₄	C₂×#13133142	1	1	yes	characteristic
281	C₄	#16810625	1	1	yes	characteristic
282	C₄	#107×D₅₃	1	1	yes	characteristic
283	C₄	#16810660	1	1	yes	characteristic
284	C₄	#16810856	1	1	yes	characteristic
285	C₄	#16810861	1	1	yes	characteristic
286	C₄	#16810909	1	1	yes	characteristic
287	C₄	#16810919	1	1	yes	characteristic
288	C₂	#20274076	1	1	yes	characteristic
289	C₂	#20274086	1	1	yes	characteristic
290	C₂	#20274110	1	1	yes	characteristic
291	C₂	#20274120	1	1	yes	characteristic
292	C₂	#20274721	1	1	yes	characteristic
293	C₂	#20274741	1	1	yes	characteristic
294	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×#335	C₂
socle, prime radical	C₂³×C₅₃	#121
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#335	D₅₃
hypocenter	C₅₃	#8645
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
8. C₈×Dic₁₀₆	C₄	semidirect
115. Dic₁₀₆	#107	semidirect
118. C₈×C₅₃	#103	semidirect
167. C₅₃	#8645	semidirect
267. 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
#20274875	2
#20274721	2
#20272599	2
C₂×C₅₃×#335	2
#8645	53