Subgroups — #27537106

Order 13568.

636 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287817	C₂	1	1	yes	maximal, characteristic
2	#20283323	C₂	1	1	yes	maximal, characteristic
3	#20283306	C₂	1	1	yes	maximal, characteristic
4	#20276489	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276488	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276486	C₂	1	4	yes	maximal, complemented
7	C₂×#16810445	C₂	1	1	yes	maximal, characteristic, complemented
8	#365×D₅₃	C₂	1	4	yes	maximal, complemented
9	C₂×C₅₃×#365	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811389	—	2	1	no	—
11	C₂×#13133223	C₂²	1	1	yes	characteristic
12	#16811104	—	2	1	no	—
13	#16811041	C₄	1	4	yes	complemented
14	#16811039	C₄	1	4	yes	complemented
15	C₂×#13133157	C₂²	1	1	yes	characteristic
16	M₃₂×D₅₃	C₂²	1	4	yes	—
17	#16810671	—	2	1	no	—
18	#16810670	—	2	2	no	—
19	C₂×#13133153	C₂²	1	1	yes	characteristic
20	#108×D₅₃	C₂²	1	4	yes	—
21	#16810663	—	2	1	no	—
22	#16810662	—	2	2	no	—
23	#16810661	C₂²	1	1	yes	characteristic
24	#16810658	C₂²	1	2	yes	—
25	#106×D₅₃	C₂²	1	2	yes	—
26	#16810447	C₂²	1	2	yes	—
27	#16810447	C₂²	1	2	yes	—
28	#16810446	C₂²	1	2	yes	—
29	#16810446	C₂²	1	2	yes	—
30	#16810445	C₂²	1	4	yes	—
31	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
34	C₅₃×#365	C₂²	1	4	yes	—
35	C₂×#11719945	D₄	1	1	yes	characteristic
36	#13133225	—	4	2	no	—
37	#13133223	—	4	2	no	—
38	#13133223	C₂×C₄	1	2	yes	—
39	#13133223	C₂×C₄	1	2	yes	—
40	#13133223	Q₈	1	2	yes	—
41	#13133223	Q₈	1	2	yes	—
42	C₂×#11719917	—	2	1	no	—
43	C₂×#11719917	D₄	1	1	yes	characteristic
44	C₂×#11719912	D₄	1	2	yes	—
45	M₁₆×D₅₃	—	2	4	no	—
46	C₂×C₈×D₅₃	C₂³	1	2	yes	—
47	#13133182	—	4	1	no	—
48	#13133179	—	4	1	no	—
49	C₄×C₂²⋊D₅₃	—	4	2	no	—
50	#13133176	—	2	1	no	—
51	#13133175	—	2	2	no	—
52	C₄⋊C₄×D₅₃	—	2	2	no	—
53	#13133157	C₂×C₄	1	2	yes	complemented
54	#13133157	C₂³	1	2	yes	—
55	#13133156	C₂×C₄	1	8	yes	complemented
56	C₂×#11719908	C₂×C₄	1	2	yes	complemented
57	C₁₆×D₅₃	C₂×C₄	1	8	yes	complemented
58	#13133153	—	2	1	no	—
59	#13133153	C₂³	1	2	yes	—
60	#13133152	—	2	4	no	—
61	#13133151	—	2	1	no	—
62	#13133150	—	2	2	no	—
63	#13133149	C₂³	1	1	yes	characteristic
64	#13133149	C₂³	1	1	yes	characteristic
65	#13133148	—	2	2	no	—
66	C₂×M₁₆×C₅₃	—	2	1	no	—
67	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
69	M₃₂×C₅₃	C₂×C₄	1	2	yes	complemented
70	M₃₂×C₅₃	C₂³	1	2	yes	—
71	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
72	#108×C₅₃	—	2	1	no	—
73	#108×C₅₃	C₂³	1	2	yes	—
74	#106×C₅₃	—	2	1	no	—
75	#106×C₅₃	C₂³	1	2	yes	—
76	C₂²×C₄×C₅₃	—	4	1	no	—
77	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₂²⋊D₅₃	SD₁₆	1	2	yes	—
79	#11719945	C₄⋊C₄	1	2	yes	—
80	#11719945	C₄⋊C₄	1	2	yes	—
81	#11719945	SD₁₆	1	2	yes	—
82	#11719945	SD₁₆	1	2	yes	—
83	C₂×D₂₁₂	SD₁₆	1	1	yes	characteristic
84	C₂×C₄×D₅₃	—	4	2	no	—
85	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
86	C₂×Dic₁₀₆	SD₁₆	1	1	yes	characteristic
87	M₁₆×C₅₃	—	2	2	no	—
88	M₁₆×C₅₃	—	4	1	no	—
89	C₂×C₈×C₅₃	—	4	1	no	—
90	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₅₃	—	2	2	no	—
97	C₄⋊C₄×C₅₃	—	4	1	no	—
98	C₂²⋊Dic₅₃	—	4	1	no	—
99	#11719921	—	4	2	no	—
100	C₄⋊Dic₅₃	—	2	1	no	—
101	C₄⋊Dic₅₃	—	2	1	no	—
102	#11719919	—	4	2	no	—
103	C₄×Dic₅₃	—	4	1	no	—
104	C₄×Dic₅₃	—	4	2	no	—
105	#11719917	—	2	2	no	—
106	#11719917	—	4	1	no	—
107	#11719917	C₄⋊C₄	1	2	yes	—
108	#11719917	C₄⋊C₄	1	2	yes	—
109	C₂×#88322	—	4	1	no	—
110	C₂×#88322	C₂×D₄	1	2	yes	—
111	#11719912	—	4	4	no	—
112	#11719912	C₄⋊C₄	1	4	yes	—
113	#11719912	C₄⋊C₄	1	4	yes	—
114	C₈×D₅₃	—	4	4	no	—
115	C₈×D₅₃	C₂²×C₄	1	4	yes	—
116	C₈×D₅₃	C₂×Q₈	1	4	yes	—
117	C₁₆×C₅₃	C₂²×C₄	1	4	yes	—
118	#11719908	C₂²×C₄	1	4	yes	—
119	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
120	C₂²×D₅₃	C₂×SD₁₆	1	2	yes	—
121	C₂×C₄×C₅₃	—	4	2	no	—
122	C₂×C₄×C₅₃	—	16	1	no	—
123	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
129	C₂²⋊D₅₃	#106	1	4	yes	—
130	C₂²⋊D₅₃	#106	1	4	yes	—
131	C₂×Dic₅₃	—	4	1	no	—
132	C₂×Dic₅₃	—	4	1	no	—
133	C₂×Dic₅₃	C₂×SD₁₆	1	2	yes	—
134	D₂₁₂	#106	1	2	yes	—
135	D₂₁₂	#106	1	2	yes	—
136	C₄×D₅₃	—	16	2	no	—
137	C₄×D₅₃	C₂×C₄⋊C₄	1	4	yes	—
138	C₄×D₅₃	C₂×SD₁₆	1	4	yes	—
139	Dic₁₀₆	#106	1	2	yes	—
140	Dic₁₀₆	#106	1	2	yes	—
141	C₈×C₅₃	—	4	2	no	—
142	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
145	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
146	#88322	—	4	2	no	—
147	#88322	C₂×C₄⋊C₄	1	4	yes	—
148	#33404	—	53	1	no	maximal, Sylow
149	C₂²×C₅₃	—	4	1	no	—
150	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
151	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
152	C₂²×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
153	C₂×D₅₃	—	4	2	no	—
154	C₂×D₅₃	C₂×#106	1	4	yes	—
155	C₄×C₅₃	—	8	2	no	—
156	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
157	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
158	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
159	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
160	Dic₅₃	—	8	1	no	—
161	Dic₅₃	—	8	1	no	—
162	Dic₅₃	C₂×#106	1	4	yes	—
163	#3407	—	53	1	no	—
164	#2916	—	53	1	no	—
165	#2912	—	53	1	no	—
166	#2155	—	53	1	no	—
167	#2155	—	53	1	no	—
168	#2155	—	53	4	no	—
169	C₂×#365	—	53	1	no	—
170	C₂×#365	—	53	4	no	—
171	C₂×#365	D₅₃	1	1	yes	characteristic, complemented
172	C₂×C₅₃	—	2	1	no	—
173	C₂×C₅₃	—	2	1	no	—
174	C₂×C₅₃	—	2	1	no	—
175	C₂×C₅₃	C₂²×#106	1	1	yes	characteristic
176	D₅₃	—	2	4	no	—
177	#568	—	106	1	no	—
178	C₂×#131	—	53	1	no	—
179	#520	—	106	1	no	—
180	#504	—	53	4	no	—
181	#504	—	53	4	no	—
182	C₂×M₃₂	—	53	1	no	—
183	C₂×M₃₂	—	53	4	no	—
184	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
185	#430	—	106	2	no	—
186	#430	—	106	1	no	—
187	C₂×#108	—	53	1	no	—
188	C₂×#108	—	53	4	no	—
189	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
190	#428	—	106	2	no	—
191	#428	—	106	1	no	—
192	#427	—	53	1	no	—
193	#427	—	53	2	no	—
194	C₂×#106	—	53	2	no	—
195	C₂×#106	C₂×D₅₃	1	1	yes	characteristic
196	#365	—	53	2	no	—
197	#365	—	53	2	no	—
198	#365	—	53	2	no	—
199	#365	—	53	2	no	—
200	#365	—	53	4	no	—
201	#365	C₂×D₅₃	1	4	yes	—
202	C₅₃	#33404	1	1	yes	characteristic, complemented, Sylow
203	C₂×#41	—	53	1	no	—
204	#131	—	53	2	no	—
205	#131	—	53	2	no	—
206	#131	—	53	2	no	—
207	#131	—	53	2	no	—
208	#131	—	212	2	no	—
209	#131	—	212	2	no	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂×M₁₆	—	53	1	no	—
212	C₂×M₁₆	—	53	2	no	—
213	C₂×M₁₆	—	106	4	no	—
214	C₂×M₁₆	—	106	1	no	—
215	C₂²×C₈	—	53	2	no	—
216	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
217	C₄×Q₈	—	212	1	no	—
218	C₄×D₄	—	212	1	no	—
219	C₄×D₄	—	212	2	no	—
220	#117	—	106	2	no	—
221	#117	—	106	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	106	2	no	—
224	M₃₂	—	53	2	no	—
225	M₃₂	—	53	2	no	—
226	M₃₂	—	53	8	no	—
227	M₃₂	C₄×D₅₃	1	2	yes	complemented
228	M₃₂	C₂²×D₅₃	1	2	yes	—
229	C₂×C₁₆	—	53	2	no	—
230	C₂×C₁₆	—	53	8	no	—
231	C₂×C₁₆	C₄×D₅₃	1	2	yes	complemented
232	#108	—	2	1	no	—
233	#108	—	53	2	no	—
234	#108	—	106	4	no	—
235	#108	—	106	1	no	—
236	#108	C₂²×D₅₃	1	2	yes	—
237	#107	—	106	2	no	—
238	#107	—	106	1	no	—
239	#106	—	2	1	no	—
240	#106	—	53	1	no	—
241	#106	—	53	1	no	—
242	#106	—	106	2	no	—
243	#106	C₂²×D₅₃	1	2	yes	—
244	#41	—	53	2	no	—
245	#41	—	53	2	no	—
246	#41	—	53	2	no	—
247	#41	—	53	2	no	—
248	C₂×Q₈	—	53	1	no	—
249	C₂×D₄	—	53	2	no	—
250	C₂×D₄	—	53	1	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	53	2	no	—
253	C₂²×C₄	—	212	2	no	—
254	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
255	M₁₆	—	2	2	no	—
256	M₁₆	—	4	1	no	—
257	M₁₆	—	53	2	no	—
258	M₁₆	—	53	2	no	—
259	M₁₆	—	53	4	no	—
260	M₁₆	—	53	4	no	—
261	M₁₆	—	106	2	no	—
262	M₁₆	—	212	4	no	—
263	M₁₆	—	212	1	no	—
264	C₂×C₈	—	4	1	no	—
265	C₂×C₈	—	53	2	no	—
266	C₂×C₈	—	53	4	no	—
267	C₂×C₈	—	53	4	no	—
268	C₂×C₈	—	212	4	no	—
269	C₂×C₈	—	212	1	no	—
270	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
271	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
272	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
273	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
274	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
275	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	106	1	no	—
279	C₄⋊C₄	—	106	1	no	—
280	C₄⋊C₄	—	212	2	no	—
281	C₂²⋊C₄	—	212	2	no	—
282	C₂²⋊C₄	—	212	1	no	—
283	C₄²	—	212	2	no	—
284	C₄²	—	212	1	no	—
285	C₁₆	—	53	4	no	—
286	C₁₆	C₂×C₄×D₅₃	1	4	yes	—
287	C₂³	—	53	2	no	—
288	C₂³	SD₁₆×D₅₃	1	1	yes	characteristic
289	Q₈	—	53	2	no	—
290	Q₈	—	53	2	no	—
291	D₄	—	53	4	no	—
292	D₄	—	53	4	no	—
293	D₄	—	53	2	no	—
294	D₄	—	53	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	16	1	no	—
297	C₂×C₄	—	53	2	no	—
298	C₂×C₄	—	53	4	no	—
299	C₂×C₄	—	53	4	no	—
300	C₂×C₄	—	848	2	no	—
301	C₂×C₄	—	212	1	no	—
302	C₂×C₄	—	212	1	no	—
303	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
304	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
305	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
306	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
307	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
308	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
309	C₈	—	4	2	no	—
310	C₈	—	53	4	no	—
311	C₈	—	212	2	no	—
312	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
313	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
314	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
315	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
316	C₂²	—	4	1	no	—
317	C₂²	—	53	4	no	—
318	C₂²	—	212	2	no	—
319	C₂²	#106×D₅₃	1	1	yes	characteristic
320	C₂²	#106×D₅₃	1	1	yes	characteristic
321	C₂²	C₂×SD₁₆×D₅₃	1	1	yes	characteristic
322	C₄	—	8	2	no	—
323	C₄	—	53	4	no	—
324	C₄	—	424	1	no	—
325	C₄	—	424	1	no	—
326	C₄	#106×D₅₃	1	1	yes	characteristic
327	C₄	#106×D₅₃	1	1	yes	characteristic
328	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
329	C₄	C₂×SD₁₆×D₅₃	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	106	4	no	—
334	C₂	C₂×#106×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#106×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#365	C₂
socle	C₂×C₅₃	C₂²×#106
prime radical	C₂×#11719945	D₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#365	D₅₃
hypocenter	C₅₃	#33404
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#365

Complements

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

N	H ≅ G/N	type
4. #20276489	C₂	semidirect
5. #20276488	C₂	semidirect
6. #20276486	C₂	semidirect
7. C₂×#16810445	C₂	semidirect
8. #365×D₅₃	C₂	semidirect
9. C₂×C₅₃×#365	C₂	semidirect
13. #16811041	C₄	semidirect
14. #16811039	C₄	semidirect
53. #13133157	C₂×C₄	semidirect
55. #13133156	C₂×C₄	semidirect
56. C₂×#11719908	C₂×C₄	semidirect
57. C₁₆×D₅₃	C₂×C₄	semidirect
69. M₃₂×C₅₃	C₂×C₄	semidirect
71. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
171. C₂×#365	D₅₃	semidirect
202. C₅₃	#33404	semidirect
227. M₃₂	C₄×D₅₃	semidirect
231. C₂×C₁₆	C₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#20287817	2
#20283323	2
#20283306	2
#20276489	2
#20276488	2
#20276486	2
C₂×#16810445	2
#365×D₅₃	2
C₂×C₅₃×#365	2
#33404	53