Subgroups — #27277163

Order 13568.

526 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×D₂₁₂	C₂	1	1	yes	maximal, characteristic
2	#20274712	C₂	1	2	yes	maximal
3	C₂×#16810376	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272598	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1834	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₅₃	—	2	1	no	—
7	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
8	C₈×D₂₁₂	—	2	2	no	—
9	C₈×D₂₁₂	C₄	1	4	yes	complemented
10	C₂×C₄⋊#88322	C₂²	1	2	yes	—
11	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
12	C₂×#13133137	—	2	2	no	—
13	C₂×#13133129	—	2	1	no	—
14	#16810385	—	2	2	no	—
15	#16810376	—	2	2	no	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#384	C₂²	1	2	yes	—
18	C₅₃×#340	—	2	1	no	—
19	C₅₃×#340	—	2	1	no	—
20	C₂²×D₂₁₂	D₄	1	1	yes	characteristic
21	C₂²×C₄×D₅₃	—	2	1	no	—
22	C₂²×#88322	—	2	1	no	—
23	C₂²×#88322	—	2	2	no	—
24	C₂×C₈×D₅₃	—	2	4	no	—
25	C₂×C₈×D₅₃	—	4	4	no	—
26	C₄×D₂₁₂	—	2	1	no	—
27	C₄×D₂₁₂	—	2	1	no	—
28	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
29	C₄×D₂₁₂	Q₈	1	2	yes	—
30	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
31	C₂×#11719921	—	2	1	no	—
32	C₄⋊#88322	—	2	1	no	—
33	C₄⋊#88322	—	2	2	no	—
34	C₄⋊#88322	—	2	2	no	—
35	C₄⋊#88322	C₂×C₄	1	2	yes	—
36	#13133137	—	4	4	no	—
37	#13133129	—	2	4	no	—
38	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
42	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
43	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₅₃	D₄	1	2	yes	—
46	C₂³×D₅₃	—	2	1	no	—
47	C₂²×C₄×C₅₃	—	4	2	no	—
48	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
51	C₂²×Dic₅₃	—	2	1	no	—
52	C₂×D₂₁₂	—	2	1	no	—
53	C₂×D₂₁₂	—	2	1	no	—
54	C₂×D₂₁₂	—	2	1	no	—
55	C₂×D₂₁₂	C₄⋊C₄	1	2	yes	—
56	C₂×D₂₁₂	D₈	1	2	yes	—
57	C₂×D₂₁₂	Q₁₆	1	2	yes	—
58	C₂×C₄×D₅₃	—	2	2	no	—
59	C₂×C₄×D₅₃	—	2	2	no	—
60	C₂×C₄×D₅₃	—	4	2	no	—
61	C₂×C₄×D₅₃	—	4	2	no	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	—	2	2	no	—
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
66	C₄⋊C₄×C₅₃	—	4	2	no	—
67	C₄⋊C₄×C₅₃	—	4	2	no	—
68	C₄⋊C₄×C₅₃	—	4	2	no	—
69	C₄⋊C₄×C₅₃	—	4	2	no	—
70	C₄⋊C₄×C₅₃	—	8	2	no	—
71	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
74	C₄²×C₅₃	#41	1	1	yes	characteristic
75	#11719921	—	2	2	no	—
76	#11719921	—	2	2	no	—
77	C₄⋊Dic₅₃	—	2	1	no	—
78	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
79	C₂×#88322	—	2	2	no	—
80	C₂×#88322	—	4	2	no	—
81	C₂×#88322	—	4	2	no	—
82	C₂×#88322	—	4	2	no	—
83	C₂×#88322	—	4	2	no	—
84	C₈×D₅₃	—	4	8	no	—
85	C₂³×C₅₃	#121	1	1	yes	characteristic
86	C₂²×D₅₃	—	2	2	no	—
87	C₂²×D₅₃	—	2	2	no	—
88	C₂²×D₅₃	—	2	2	no	—
89	C₂²×D₅₃	—	4	2	no	—
90	C₂²×D₅₃	—	4	2	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	8	2	no	—
93	C₂×C₄×C₅₃	—	8	2	no	—
94	C₂×C₄×C₅₃	—	8	2	no	—
95	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#135	1	2	yes	—
105	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
106	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
107	C₂×Dic₅₃	—	2	2	no	—
108	C₂×Dic₅₃	—	4	2	no	—
109	D₂₁₂	—	2	2	no	—
110	D₂₁₂	—	2	2	no	—
111	D₂₁₂	—	2	2	no	—
112	D₂₁₂	#107	1	4	yes	—
113	C₄×D₅₃	—	4	2	no	—
114	C₄×D₅₃	—	4	2	no	—
115	C₄×D₅₃	—	4	4	no	—
116	C₈×C₅₃	—	2	2	no	—
117	C₈×C₅₃	—	2	2	no	—
118	#88322	—	4	2	no	—
119	#88322	—	8	2	no	—
120	#8648	—	53	1	no	maximal, Sylow
121	C₂²×C₅₃	#380	1	1	yes	characteristic
122	C₂²×C₅₃	#463	1	2	yes	—
123	C₂²×C₅₃	#466	1	1	yes	characteristic
124	C₂²×C₅₃	#467	1	1	yes	characteristic
125	C₂²×C₅₃	#471	1	2	yes	—
126	C₂×D₅₃	—	2	4	no	—
127	C₂×D₅₃	—	4	2	no	—
128	C₂×D₅₃	—	4	2	no	—
129	C₂×D₅₃	—	4	4	no	—
130	C₂×D₅₃	—	4	4	no	—
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	—	16	2	no	—
134	C₄×C₅₃	#416	1	1	yes	characteristic
135	C₄×C₅₃	#418	1	1	yes	characteristic
136	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
137	C₄×C₅₃	#427	1	1	yes	characteristic
138	C₄×C₅₃	#466	1	1	yes	characteristic
139	C₄×C₅₃	#467	1	1	yes	characteristic
140	C₄×C₅₃	#478	1	2	yes	—
141	Dic₅₃	—	4	2	no	—
142	C₂×C₈×D₄	—	53	1	no	—
143	#1834	D₅₃	1	1	yes	characteristic, complemented
144	#1802	—	53	2	no	—
145	C₂×#335	—	53	1	no	—
146	#1296	—	53	2	no	—
147	C₂×C₅₃	#1681	1	2	yes	—
148	C₂×C₅₃	#1683	1	2	yes	—
149	C₂×C₅₃	#1804	1	1	yes	characteristic
150	C₂×C₅₃	#1808	1	1	yes	characteristic
151	C₂×C₅₃	#1815	1	1	yes	characteristic
152	D₅₃	—	4	4	no	—
153	C₂³×C₈	—	106	1	no	—
154	C₂×C₄×D₄	—	53	1	no	—
155	C₈×D₄	—	53	4	no	—
156	C₈×D₄	—	106	2	no	—
157	C₂×C₄⋊C₈	—	53	1	no	—
158	C₂×C₄⋊C₈	—	53	2	no	—
159	C₂×#102	—	106	2	no	—
160	C₂×C₂²⋊C₈	—	106	1	no	—
161	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
162	#384	C₂×D₅₃	1	2	yes	—
163	#340	—	2	1	no	—
164	#340	—	2	1	no	—
165	#340	—	106	2	no	—
166	#335	—	106	2	no	—
167	C₅₃	#8648	1	1	yes	characteristic, complemented, Sylow
168	C₂²×D₄	—	53	1	no	—
169	C₂³×C₄	—	106	1	no	—
170	C₂²×C₈	—	106	1	no	—
171	C₂²×C₈	—	106	4	no	—
172	C₂²×C₈	—	106	2	no	—
173	C₂²×C₈	—	212	4	no	—
174	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
175	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
176	C₄×D₄	—	53	2	no	—
177	C₄×D₄	—	53	2	no	—
178	C₄×D₄	—	106	1	no	—
179	C₄×D₄	—	106	1	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	53	1	no	—
184	C₂×C₂²⋊C₄	—	106	1	no	—
185	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
186	C₄⋊C₈	—	53	2	no	—
187	C₄⋊C₈	—	106	1	no	—
188	C₄⋊C₈	—	106	2	no	—
189	C₄⋊C₈	—	106	2	no	—
190	#102	—	212	4	no	—
191	C₂²⋊C₈	—	106	4	no	—
192	C₄×C₈	C₄×D₅₃	1	2	yes	—
193	C₄×C₈	D₂₁₂	1	2	yes	—
194	C₂⁴	—	106	1	no	—
195	C₂×D₄	—	53	2	no	—
196	C₂×D₄	—	53	2	no	—
197	C₂×D₄	—	53	2	no	—
198	C₂×D₄	—	106	1	no	—
199	C₂×D₄	—	106	1	no	—
200	C₂×D₄	—	106	1	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	—	106	1	no	—
203	C₂²×C₄	—	106	2	no	—
204	C₂²×C₄	—	106	2	no	—
205	C₂²×C₄	—	212	2	no	—
206	C₂²×C₄	—	212	2	no	—
207	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
208	C₂²×C₄	#11719947	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	106	2	no	—
213	C₂×C₈	—	212	8	no	—
214	C₂×C₈	—	212	2	no	—
215	C₂×C₈	—	212	2	no	—
216	C₂×C₈	—	212	2	no	—
217	C₂×C₈	—	212	2	no	—
218	C₂×C₈	#11719921	1	2	yes	—
219	C₂×C₈	#11719921	1	2	yes	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	—	8	2	no	—
225	C₄⋊C₄	—	53	2	no	—
226	C₄⋊C₄	—	106	1	no	—
227	C₂²⋊C₄	—	106	2	no	—
228	C₂²⋊C₄	—	106	2	no	—
229	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
230	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
231	C₄²	Q₈×D₅₃	1	1	yes	characteristic
232	C₄²	#11719949	1	1	yes	characteristic
233	C₂³	—	106	2	no	—
234	C₂³	—	106	2	no	—
235	C₂³	—	106	2	no	—
236	C₂³	—	212	2	no	—
237	C₂³	—	212	2	no	—
238	C₂³	#13133188	1	1	yes	characteristic
239	D₄	—	53	4	no	—
240	D₄	—	106	2	no	—
241	D₄	—	106	2	no	—
242	D₄	—	106	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	8	2	no	—
245	C₂×C₄	—	8	2	no	—
246	C₂×C₄	—	8	2	no	—
247	C₂×C₄	—	106	2	no	—
248	C₂×C₄	—	212	2	no	—
249	C₂×C₄	—	212	2	no	—
250	C₂×C₄	—	212	2	no	—
251	C₂×C₄	—	212	4	no	—
252	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
253	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
254	C₂×C₄	#13133175	1	1	yes	characteristic
255	C₂×C₄	#13133192	1	1	yes	characteristic
256	C₂×C₄	#13133193	1	1	yes	characteristic
257	C₂×C₄	#13133197	1	1	yes	characteristic
258	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
259	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
260	C₂×C₄	#13133236	1	1	yes	characteristic
261	C₂×C₄	#13133237	1	2	yes	—
262	C₂×C₄	#13133240	1	1	yes	characteristic
263	C₂×C₄	#13133242	1	2	yes	—
264	C₈	—	2	2	no	—
265	C₈	—	2	2	no	—
266	C₈	—	424	2	no	—
267	C₈	—	212	2	no	—
268	C₂²	—	106	4	no	—
269	C₂²	—	212	2	no	—
270	C₂²	—	212	2	no	—
271	C₂²	—	212	4	no	—
272	C₂²	—	212	4	no	—
273	C₂²	#16810480	1	1	yes	characteristic
274	C₂²	#16810835	1	2	yes	—
275	C₂²	#16810856	1	1	yes	characteristic
276	C₂²	#16810861	1	1	yes	characteristic
277	C₂²	#16810883	1	2	yes	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	16	2	no	—
281	C₄	—	212	2	no	—
282	C₄	#16810617	1	1	yes	characteristic
283	C₄	#16810627	1	1	yes	characteristic
284	C₄	#107×D₅₃	1	1	yes	characteristic
285	C₄	#16810660	1	1	yes	characteristic
286	C₄	#16810860	1	1	yes	characteristic
287	C₄	#16810865	1	1	yes	characteristic
288	C₄	#16810920	1	2	yes	—
289	C₂	—	212	4	no	—
290	C₂	#20274149	1	2	yes	—
291	C₂	#20274163	1	2	yes	—
292	C₂	#20274722	1	1	yes	characteristic
293	C₂	#20274741	1	1	yes	characteristic
294	C₂	#20274775	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133188
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1834	C₂
socle	C₂³×C₅₃	#121
prime radical	C₂²×D₂₁₂	D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1834	D₅₃
hypocenter	C₅₃	#8648
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
3. C₂×#16810376	C₂	semidirect
4. #20272598	C₂	semidirect
5. C₅₃×#1834	C₂	semidirect
9. C₈×D₂₁₂	C₄	semidirect
143. #1834	D₅₃	semidirect
167. C₅₃	#8648	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₂₁₂	2
#20274712	2
C₂×#16810376	2
#20272598	2
C₅₃×#1834	2
#8648	53