Subgroups — #27299380

Order 13568.

428 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283517	C₂	1	1	yes	maximal, characteristic
2	#20276450	C₂	1	1	yes	maximal, characteristic
3	#20276442	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810401	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272829	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272816	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1773	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133239	C₂²	1	1	yes	characteristic
9	C₄×#11719945	—	2	1	no	—
10	C₂×#13133157	C₂²	1	1	yes	characteristic
11	C₂×#13133157	C₂²	1	1	yes	characteristic
12	#16810736	C₄	1	2	yes	complemented
13	#16810731	C₄	1	2	yes	complemented
14	#16810725	—	2	2	no	—
15	#16810661	C₂²	1	1	yes	characteristic
16	#16810620	—	2	1	no	—
17	#16810432	C₄	1	4	yes	complemented
18	#16810429	C₄	1	4	yes	complemented
19	#16810401	—	2	2	no	—
20	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#427	C₂²	1	1	yes	characteristic
22	C₅₃×#405	—	2	1	no	—
23	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#339	—	2	1	no	—
25	C₂×#11719945	—	2	1	no	—
26	#13133239	—	2	1	no	—
27	#13133239	—	2	1	no	—
28	#13133239	C₂×C₄	1	2	yes	—
29	#13133239	C₂×C₄	1	2	yes	—
30	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
31	C₂×#11719913	—	2	1	no	—
32	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₁₀₆	—	2	2	no	—
34	C₄×D₂₁₂	—	2	2	no	—
35	C₄×C₂²⋊D₅₃	—	4	2	no	—
36	#13133176	—	2	1	no	—
37	#13133174	—	4	1	no	—
38	C₄²×D₅₃	—	4	1	no	—
39	#13133157	—	2	1	no	—
40	#13133157	—	2	1	no	—
41	#13133157	—	2	1	no	—
42	#13133157	C₂×C₄	1	2	yes	complemented
43	C₂×#11719908	—	2	1	no	—
44	C₂×#11719908	C₂×C₄	1	2	yes	complemented
45	#13133151	—	2	1	no	—
46	#13133149	C₂×C₄	1	2	yes	—
47	#13133142	—	2	2	no	—
48	#13133139	—	2	2	no	—
49	C₂×M₁₆×C₅₃	—	2	1	no	—
50	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	#117×C₅₃	—	2	1	no	—
53	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
54	#108×C₅₃	—	2	1	no	—
55	#108×C₅₃	C₂×C₄	1	2	yes	complemented
56	#107×C₅₃	—	2	1	no	—
57	#106×C₅₃	C₂×C₄	1	2	yes	complemented
58	#97×C₅₃	—	4	1	no	—
59	#97×C₅₃	C₂×C₄	1	2	yes	—
60	#97×C₅₃	D₄	1	2	yes	—
61	C₄×C₈×C₅₃	—	4	1	no	—
62	C₂²×C₄×C₅₃	—	2	1	no	—
63	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₂²⋊D₅₃	—	4	1	no	—
65	#11719945	—	2	2	no	—
66	#11719945	—	2	2	no	—
67	#11719945	—	4	1	no	—
68	#11719945	—	4	1	no	—
69	C₂×D₂₁₂	—	2	1	no	—
70	C₂×C₄×D₅₃	—	4	1	no	—
71	C₂×C₄×D₅₃	—	4	2	no	—
72	C₂×Dic₁₀₆	—	2	1	no	—
73	M₁₆×C₅₃	—	2	2	no	—
74	M₁₆×C₅₃	—	4	1	no	—
75	C₂×C₈×C₅₃	—	2	1	no	—
76	C₂×C₈×C₅₃	—	2	1	no	—
77	C₂×C₈×C₅₃	—	4	1	no	—
78	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
84	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
85	C₄⋊C₄×C₅₃	—	2	2	no	—
86	C₂²⋊C₄×C₅₃	—	4	1	no	—
87	C₄²×C₅₃	—	2	1	no	—
88	C₄²×C₅₃	—	4	1	no	—
89	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
90	C₂²⋊Dic₅₃	—	4	1	no	—
91	#11719921	—	4	2	no	—
92	C₄⋊Dic₅₃	—	2	2	no	—
93	#11719919	—	4	2	no	—
94	C₄×Dic₅₃	—	4	2	no	—
95	C₄×Dic₅₃	—	4	1	no	—
96	Dic₂₁₂	—	2	1	no	—
97	Dic₂₁₂	C₄²	1	2	yes	—
98	D₄₂₄	—	2	1	no	—
99	D₄₂₄	C₄²	1	2	yes	—
100	#11719913	—	2	2	no	—
101	#11719913	—	2	2	no	—
102	#11719908	—	2	2	no	—
103	#11719908	—	2	2	no	—
104	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂²×D₅₃	—	4	1	no	—
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	—	4	2	no	—
108	C₂×C₄×C₅₃	—	8	1	no	—
109	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
115	C₂²⋊D₅₃	—	4	2	no	—
116	C₂²⋊D₅₃	—	4	2	no	—
117	C₂×Dic₅₃	—	4	1	no	—
118	C₂×Dic₅₃	—	4	2	no	—
119	D₂₁₂	—	2	2	no	—
120	D₂₁₂	—	4	1	no	—
121	C₄×D₅₃	—	4	2	no	—
122	C₄×D₅₃	—	4	2	no	—
123	C₄×D₅₃	—	8	4	no	—
124	Dic₁₀₆	—	2	2	no	—
125	Dic₁₀₆	—	4	1	no	—
126	C₈×C₅₃	—	2	2	no	—
127	C₈×C₅₃	—	4	2	no	—
128	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
129	C₈×C₅₃	#117	1	1	yes	characteristic
130	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
131	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
132	#12649	—	53	1	no	maximal, Sylow
133	C₂²×C₅₃	—	4	1	no	—
134	C₂²×C₅₃	#418	1	1	yes	characteristic
135	C₂²×C₅₃	#440	1	1	yes	characteristic
136	C₂²×C₅₃	#442	1	1	yes	characteristic
137	C₂×D₅₃	—	4	2	no	—
138	C₂×D₅₃	—	8	2	no	—
139	C₄×C₅₃	—	4	2	no	—
140	C₄×C₅₃	—	8	2	no	—
141	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
142	C₄×C₅₃	#417	1	1	yes	characteristic
143	C₄×C₅₃	#440	1	1	yes	characteristic
144	C₄×C₅₃	#442	1	1	yes	characteristic
145	Dic₅₃	—	4	2	no	—
146	Dic₅₃	—	8	2	no	—
147	#2943	—	53	1	no	—
148	#2145	—	53	1	no	—
149	#2143	—	53	1	no	—
150	C₂×#347	—	53	1	no	—
151	#1773	D₅₃	1	1	yes	characteristic, complemented
152	#1389	—	53	1	no	—
153	#1384	—	53	1	no	—
154	C₂×C₅₃	—	2	1	no	—
155	C₂×C₅₃	—	2	1	no	—
156	C₂×C₅₃	—	2	1	no	—
157	C₂×C₅₃	#1762	1	1	yes	characteristic
158	D₅₃	—	8	2	no	—
159	C₂×#135	—	53	1	no	—
160	C₄×#41	—	106	1	no	—
161	C₂×M₃₂	—	53	1	no	—
162	C₂×M₃₂	—	53	1	no	—
163	#442	—	53	2	no	—
164	#441	—	53	2	no	—
165	#440	—	106	2	no	—
166	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
167	#427	—	53	1	no	—
168	#427	C₂×D₅₃	1	1	yes	characteristic
169	#416	—	106	1	no	—
170	#405	—	2	1	no	—
171	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
172	#360	—	53	4	no	—
173	#359	—	53	4	no	—
174	#347	—	106	2	no	—
175	#339	—	2	1	no	—
176	C₅₃	#12649	1	1	yes	characteristic, complemented, Sylow
177	C₂×#41	—	106	1	no	—
178	#135	—	53	2	no	—
179	#135	—	53	2	no	—
180	#135	—	106	1	no	—
181	#135	—	106	1	no	—
182	C₂×Q₁₆	—	53	1	no	—
183	C₂×SD₁₆	—	106	1	no	—
184	C₂×D₈	—	53	1	no	—
185	C₂×M₁₆	—	2	1	no	—
186	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
187	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
188	C₄×Q₈	—	106	2	no	—
189	C₄×D₄	—	106	2	no	—
190	C₄×D₄	—	212	2	no	—
191	#117	—	2	1	no	—
192	#117	—	106	1	no	—
193	#117	—	212	1	no	—
194	C₂×C₄²	—	212	1	no	—
195	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
196	M₃₂	—	53	2	no	—
197	M₃₂	—	106	1	no	—
198	M₃₂	—	106	1	no	—
199	M₃₂	—	106	1	no	—
200	C₂×C₁₆	—	53	2	no	—
201	C₂×C₁₆	—	106	1	no	—
202	#108	—	2	1	no	—
203	#108	C₄×D₅₃	1	2	yes	complemented
204	#107	—	2	1	no	—
205	#107	—	106	1	no	—
206	#106	—	53	2	no	—
207	#106	C₄×D₅₃	1	2	yes	complemented
208	#103	—	106	2	no	—
209	#102	—	106	2	no	—
210	#97	—	4	1	no	—
211	#97	C₄×D₅₃	1	2	yes	—
212	#97	D₂₁₂	1	2	yes	—
213	C₄×C₈	—	4	1	no	—
214	#41	—	106	2	no	—
215	#41	—	106	2	no	—
216	#41	—	212	1	no	—
217	#41	—	212	1	no	—
218	C₂×Q₈	—	106	1	no	—
219	C₂×D₄	—	106	1	no	—
220	C₂×D₄	—	212	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	212	2	no	—
223	C₂²×C₄	—	212	1	no	—
224	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
225	Q₁₆	—	53	2	no	—
226	Q₁₆	—	106	1	no	—
227	SD₁₆	—	106	2	no	—
228	SD₁₆	—	106	2	no	—
229	D₈	—	53	2	no	—
230	D₈	—	106	1	no	—
231	M₁₆	—	2	2	no	—
232	M₁₆	—	4	1	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	#11719921	1	2	yes	—
237	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
239	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
240	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
241	C₂×C₈	#11719945	1	1	yes	characteristic
242	C₂×C₈	#11719945	1	1	yes	characteristic
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	106	2	no	—
245	C₄⋊C₄	—	212	2	no	—
246	C₂²⋊C₄	—	4	1	no	—
247	C₂²⋊C₄	—	212	2	no	—
248	C₂²⋊C₄	—	212	1	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	4	1	no	—
251	C₄²	—	212	2	no	—
252	C₄²	—	212	1	no	—
253	C₄²	#11719921	1	2	yes	—
254	C₁₆	—	106	2	no	—
255	C₁₆	—	106	2	no	—
256	C₂³	—	212	1	no	—
257	C₂³	#13133242	1	1	yes	characteristic
258	Q₈	—	106	2	no	—
259	Q₈	—	212	1	no	—
260	D₄	—	106	2	no	—
261	D₄	—	212	2	no	—
262	D₄	—	212	2	no	—
263	D₄	—	212	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	8	1	no	—
267	C₂×C₄	—	424	4	no	—
268	C₂×C₄	—	212	2	no	—
269	C₂×C₄	—	212	1	no	—
270	C₂×C₄	—	212	2	no	—
271	C₂×C₄	—	212	2	no	—
272	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
273	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
274	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
275	C₂×C₄	#13133242	1	1	yes	characteristic
276	C₂×C₄	#13133243	1	1	yes	characteristic
277	C₂×C₄	#13133248	1	1	yes	characteristic
278	C₈	—	2	2	no	—
279	C₈	—	4	2	no	—
280	C₈	C₄²×D₅₃	1	1	yes	characteristic
281	C₈	#13133174	1	1	yes	characteristic
282	C₈	C₄×D₂₁₂	1	1	yes	characteristic
283	C₈	C₄×D₂₁₂	1	1	yes	characteristic
284	C₂²	—	4	1	no	—
285	C₂²	—	424	2	no	—
286	C₂²	—	212	2	no	—
287	C₂²	#16810627	1	1	yes	characteristic
288	C₂²	#16810721	1	1	yes	characteristic
289	C₂²	#16810733	1	1	yes	characteristic
290	C₄	—	4	2	no	—
291	C₄	—	8	2	no	—
292	C₄	—	424	2	no	—
293	C₄	—	212	2	no	—
294	C₄	C₄×#11719921	1	1	yes	characteristic
295	C₄	#16810622	1	1	yes	characteristic
296	C₄	#16810721	1	1	yes	characteristic
297	C₄	#16810733	1	1	yes	characteristic
298	C₂	—	2	1	no	—
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	424	2	no	—
302	C₂	#20274543	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16810733
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1773	C₂
socle	C₂×C₅₃	#1762
prime radical	C₂×#13133239	C₂²
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1773	D₅₃
hypocenter	C₅₃	#12649
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ 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. #20272829	C₂	semidirect
6. #20272816	C₂	semidirect
7. C₅₃×#1773	C₂	semidirect
12. #16810736	C₄	semidirect
13. #16810731	C₄	semidirect
17. #16810432	C₄	semidirect
18. #16810429	C₄	semidirect
42. #13133157	C₂×C₄	semidirect
44. C₂×#11719908	C₂×C₄	semidirect
55. #108×C₅₃	C₂×C₄	semidirect
57. #106×C₅₃	C₂×C₄	semidirect
151. #1773	D₅₃	semidirect
176. C₅₃	#12649	semidirect
203. #108	C₄×D₅₃	semidirect
207. #106	C₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#20283517	2
#20276450	2
#20276442	2
C₂×#16810401	2
#20272829	2
#20272816	2
C₅₃×#1773	2
#12649	53