Subgroups — #27299366

Order 13568.

486 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810736	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810435	C₂	1	1	yes	maximal, characteristic
3	#20276430	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810401	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272820	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272815	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1773	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄₂₄	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₂₁₂	—	2	1	no	—
10	C₂×#13133157	C₂²	1	1	yes	characteristic
11	C₂×#13133157	C₂²	1	1	yes	characteristic
12	#16810736	—	2	2	no	—
13	#16810736	C₄	1	4	yes	complemented
14	C₂×#13133149	C₂²	1	1	yes	characteristic
15	C₂×#13133139	—	2	1	no	—
16	#16810435	C₄	1	4	yes	complemented
17	#16810423	C₄	1	4	yes	complemented
18	#16810401	—	2	2	no	—
19	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#427	C₂²	1	1	yes	characteristic
21	C₅₃×#405	—	2	1	no	—
22	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#339	—	2	1	no	—
24	C₂²×D₂₁₂	—	2	1	no	—
25	C₂²×C₄×D₅₃	—	4	1	no	—
26	C₂×D₄₂₄	—	2	1	no	—
27	C₂×D₄₂₄	—	2	1	no	—
28	C₂×D₄₂₄	—	2	1	no	—
29	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
30	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
31	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
32	C₄×D₂₁₂	—	2	4	no	—
33	C₄×D₂₁₂	—	4	2	no	—
34	C₂×C₄⋊Dic₅₃	—	2	1	no	—
35	C₂×#11719921	—	4	1	no	—
36	#13133157	—	2	1	no	—
37	#13133157	—	2	1	no	—
38	#13133157	—	2	1	no	—
39	#13133157	C₂×C₄	1	2	yes	complemented
40	C₂×#11719908	—	2	1	no	—
41	C₂×#11719908	C₂×C₄	1	2	yes	complemented
42	#13133149	—	2	1	no	—
43	#13133149	C₂×C₄	1	2	yes	—
44	#13133139	—	2	4	no	—
45	C₂×M₁₆×C₅₃	—	2	1	no	—
46	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	#117×C₅₃	—	2	1	no	—
49	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
50	#108×C₅₃	—	2	1	no	—
51	#108×C₅₃	C₂×C₄	1	2	yes	complemented
52	#107×C₅₃	C₂×C₄	1	2	yes	complemented
53	#106×C₅₃	—	2	1	no	—
54	#97×C₅₃	—	4	1	no	—
55	#97×C₅₃	C₂×C₄	1	2	yes	—
56	#97×C₅₃	D₄	1	2	yes	—
57	C₄×C₈×C₅₃	—	4	1	no	—
58	C₂³×D₅₃	—	4	1	no	—
59	C₂²×C₄×C₅₃	—	2	1	no	—
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×Dic₅₃	—	4	1	no	—
62	C₂×D₂₁₂	—	2	2	no	—
63	C₂×D₂₁₂	—	2	2	no	—
64	C₂×D₂₁₂	—	2	2	no	—
65	C₂×D₂₁₂	—	4	1	no	—
66	C₂×D₂₁₂	—	4	1	no	—
67	C₂×D₂₁₂	—	4	1	no	—
68	C₂×C₄×D₅₃	—	4	4	no	—
69	C₂×C₄×D₅₃	—	8	4	no	—
70	M₁₆×C₅₃	—	2	2	no	—
71	M₁₆×C₅₃	—	4	1	no	—
72	C₂×C₈×C₅₃	—	2	1	no	—
73	C₂×C₈×C₅₃	—	2	1	no	—
74	C₂×C₈×C₅₃	—	4	1	no	—
75	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
81	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
82	C₄⋊C₄×C₅₃	—	2	2	no	—
83	C₂²⋊C₄×C₅₃	—	4	1	no	—
84	C₄²×C₅₃	—	2	1	no	—
85	C₄²×C₅₃	—	4	1	no	—
86	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
87	#11719921	—	4	4	no	—
88	C₄⋊Dic₅₃	—	2	2	no	—
89	C₄⋊Dic₅₃	—	4	1	no	—
90	D₄₂₄	—	2	2	no	—
91	D₄₂₄	—	2	2	no	—
92	D₄₂₄	—	2	2	no	—
93	D₄₂₄	C₄²	1	4	yes	—
94	#11719908	—	2	2	no	—
95	#11719908	—	2	2	no	—
96	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂²×D₅₃	—	4	2	no	—
98	C₂²×D₅₃	—	4	2	no	—
99	C₂²×D₅₃	—	4	2	no	—
100	C₂²×D₅₃	—	8	2	no	—
101	C₂²×D₅₃	—	8	2	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	4	2	no	—
104	C₂×C₄×C₅₃	—	4	2	no	—
105	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
111	C₂×Dic₅₃	—	4	2	no	—
112	C₂×Dic₅₃	—	8	2	no	—
113	D₂₁₂	—	2	4	no	—
114	D₂₁₂	—	4	2	no	—
115	D₂₁₂	—	4	2	no	—
116	D₂₁₂	—	4	2	no	—
117	C₄×D₅₃	—	8	8	no	—
118	C₈×C₅₃	—	2	2	no	—
119	C₈×C₅₃	—	4	2	no	—
120	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
121	C₈×C₅₃	#117	1	1	yes	characteristic
122	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
123	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
124	#12647	—	53	1	no	maximal, Sylow
125	C₂²×C₅₃	—	2	2	no	—
126	C₂²×C₅₃	#418	1	1	yes	characteristic
127	C₂²×C₅₃	#440	1	1	yes	characteristic
128	C₂²×C₅₃	#442	1	1	yes	characteristic
129	C₂×D₅₃	—	4	4	no	—
130	C₂×D₅₃	—	8	2	no	—
131	C₂×D₅₃	—	8	2	no	—
132	C₂×D₅₃	—	8	4	no	—
133	C₂×D₅₃	—	8	4	no	—
134	C₄×C₅₃	—	4	2	no	—
135	C₄×C₅₃	—	8	2	no	—
136	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₅₃	#417	1	1	yes	characteristic
138	C₄×C₅₃	#440	1	1	yes	characteristic
139	C₄×C₅₃	#442	1	1	yes	characteristic
140	Dic₅₃	—	8	2	no	—
141	C₂×#442	—	53	1	no	—
142	C₂×#361	—	53	1	no	—
143	#2140	—	53	1	no	—
144	C₂×#347	—	53	1	no	—
145	#1773	D₅₃	1	1	yes	characteristic, complemented
146	#1386	—	53	1	no	—
147	#1383	—	53	1	no	—
148	C₂×C₅₃	—	2	1	no	—
149	C₂×C₅₃	—	2	1	no	—
150	C₂×C₅₃	#1762	1	1	yes	characteristic
151	C₂×C₅₃	AGL(1,ℤ/16)	1	2	yes	—
152	D₅₃	—	8	4	no	—
153	C₂²×D₈	—	53	1	no	—
154	C₂×C₄×D₄	—	106	1	no	—
155	C₂×M₃₂	—	53	1	no	—
156	C₂×M₃₂	—	53	1	no	—
157	#442	—	53	4	no	—
158	#442	—	106	2	no	—
159	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
160	#427	C₂×D₅₃	1	1	yes	characteristic
161	C₂×#106	—	53	1	no	—
162	C₂×#102	—	106	1	no	—
163	#405	—	2	1	no	—
164	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
165	#361	—	53	4	no	—
166	#357	—	53	4	no	—
167	#347	—	106	2	no	—
168	#339	—	2	1	no	—
169	C₅₃	#12647	1	1	yes	characteristic, complemented, Sylow
170	C₂²×D₄	—	106	1	no	—
171	C₂³×C₄	—	212	1	no	—
172	C₂×D₈	—	53	2	no	—
173	C₂×D₈	—	53	2	no	—
174	C₂×D₈	—	53	2	no	—
175	C₂×D₈	—	106	1	no	—
176	C₂×D₈	—	106	1	no	—
177	C₂×D₈	—	106	1	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
181	C₄×D₄	—	106	4	no	—
182	C₄×D₄	—	212	2	no	—
183	#117	—	2	1	no	—
184	C₂×C₄⋊C₄	—	106	1	no	—
185	C₂×C₂²⋊C₄	—	212	1	no	—
186	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
187	M₃₂	—	53	2	no	—
188	M₃₂	—	106	1	no	—
189	M₃₂	—	106	1	no	—
190	M₃₂	—	106	1	no	—
191	C₂×C₁₆	—	53	2	no	—
192	C₂×C₁₆	—	106	1	no	—
193	#108	—	2	1	no	—
194	#108	C₄×D₅₃	1	2	yes	complemented
195	#107	C₄×D₅₃	1	2	yes	complemented
196	#106	—	2	1	no	—
197	#106	—	53	2	no	—
198	#106	—	106	1	no	—
199	#102	—	106	4	no	—
200	#97	—	4	1	no	—
201	#97	C₄×D₅₃	1	2	yes	—
202	#97	D₂₁₂	1	2	yes	—
203	C₄×C₈	—	4	1	no	—
204	C₂⁴	—	212	1	no	—
205	C₂×D₄	—	106	2	no	—
206	C₂×D₄	—	106	2	no	—
207	C₂×D₄	—	106	2	no	—
208	C₂×D₄	—	212	1	no	—
209	C₂×D₄	—	212	1	no	—
210	C₂×D₄	—	212	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	424	4	no	—
213	C₂²×C₄	—	212	4	no	—
214	C₂²×C₄	—	212	1	no	—
215	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
216	D₈	—	53	4	no	—
217	D₈	—	106	2	no	—
218	D₈	—	106	2	no	—
219	D₈	—	106	2	no	—
220	M₁₆	—	2	2	no	—
221	M₁₆	—	4	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	#11719921	1	2	yes	—
226	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
227	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
229	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
230	C₂×C₈	#11719945	1	1	yes	characteristic
231	C₂×C₈	#11719945	1	1	yes	characteristic
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	106	2	no	—
234	C₄⋊C₄	—	212	1	no	—
235	C₂²⋊C₄	—	4	1	no	—
236	C₂²⋊C₄	—	212	4	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	4	1	no	—
239	C₄²	#11719921	1	2	yes	—
240	C₁₆	—	106	2	no	—
241	C₁₆	—	106	2	no	—
242	C₂³	—	424	2	no	—
243	C₂³	—	424	2	no	—
244	C₂³	—	212	2	no	—
245	C₂³	—	212	2	no	—
246	C₂³	—	212	2	no	—
247	C₂³	#13133242	1	1	yes	characteristic
248	D₄	—	106	4	no	—
249	D₄	—	212	2	no	—
250	D₄	—	212	2	no	—
251	D₄	—	212	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	424	8	no	—
256	C₂×C₄	—	424	2	no	—
257	C₂×C₄	—	212	2	no	—
258	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
259	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
260	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
261	C₂×C₄	#13133242	1	1	yes	characteristic
262	C₂×C₄	#13133243	1	1	yes	characteristic
263	C₂×C₄	#13133248	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	4	2	no	—
266	C₈	C₄²×D₅₃	1	1	yes	characteristic
267	C₈	#13133174	1	1	yes	characteristic
268	C₈	C₄×D₂₁₂	1	1	yes	characteristic
269	C₈	C₄×D₂₁₂	1	1	yes	characteristic
270	C₂²	—	2	2	no	—
271	C₂²	—	424	2	no	—
272	C₂²	—	424	2	no	—
273	C₂²	—	424	4	no	—
274	C₂²	—	424	4	no	—
275	C₂²	—	212	4	no	—
276	C₂²	#16810627	1	1	yes	characteristic
277	C₂²	#16810721	1	1	yes	characteristic
278	C₂²	#16810733	1	1	yes	characteristic
279	C₄	—	4	2	no	—
280	C₄	—	8	2	no	—
281	C₄	—	424	2	no	—
282	C₄	C₄×#11719921	1	1	yes	characteristic
283	C₄	#16810622	1	1	yes	characteristic
284	C₄	#16810721	1	1	yes	characteristic
285	C₄	#16810733	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	—	424	4	no	—
289	C₂	#20274543	1	1	yes	characteristic
290	C₂	#20276613	1	2	yes	—

	subgroup	G/H
center	C₂²	#16810733
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1773	C₂
socle	C₂²×C₅₃	#442
prime radical	C₂²×D₄₂₄	C₂²
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1773	D₅₃
hypocenter	C₅₃	#12647
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1773

Complements

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

N	H ≅ G/N	type
4. C₂×#16810401	C₂	semidirect
5. #20272820	C₂	semidirect
6. #20272815	C₂	semidirect
7. C₅₃×#1773	C₂	semidirect
13. #16810736	C₄	semidirect
16. #16810435	C₄	semidirect
17. #16810423	C₄	semidirect
39. #13133157	C₂×C₄	semidirect
41. C₂×#11719908	C₂×C₄	semidirect
51. #108×C₅₃	C₂×C₄	semidirect
52. #107×C₅₃	C₂×C₄	semidirect
145. #1773	D₅₃	semidirect
169. C₅₃	#12647	semidirect
194. #108	C₄×D₅₃	semidirect
195. #107	C₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810736	2
C₂×#16810435	2
#20276430	2
C₂×#16810401	2
#20272820	2
#20272815	2
C₅₃×#1773	2
#12647	53