Subgroups — #27537401

Order 13568.

682 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811056	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133240	C₂	1	2	yes	maximal, complemented
3	#20276592	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276590	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276586	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276582	C₂	1	2	yes	maximal, complemented
7	#20276577	C₂	1	2	yes	maximal, complemented
8	#20276498	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276421	C₂	1	2	yes	maximal, complemented
10	#346×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2176	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133240	C₂²	1	2	yes	—
13	C₄×#11719949	—	2	2	no	—
14	#16811056	C₄	1	8	yes	complemented
15	C₂×#13133160	C₂²	1	2	yes	—
16	C₂×D₈₄₈	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
19	Q₁₆×Dic₅₃	C₂²	1	2	yes	—
20	C₄×#11719924	—	2	2	no	—
21	#16810708	—	2	2	no	—
22	C₄×D₄₂₄	C₂²	1	2	yes	—
23	#16810705	C₂²	1	2	yes	—
24	#16810660	C₂²	1	2	yes	—
25	#16810618	—	2	2	no	—
26	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
27	#16810451	C₂²	1	1	yes	characteristic
28	#16810448	C₂²	1	2	yes	—
29	#16810427	C₂²	1	2	yes	—
30	#16810425	C₂²	1	2	yes	—
31	#16810423	C₂²	1	2	yes	—
32	#16810399	C₂²	1	1	yes	characteristic
33	#16810398	C₂²	1	1	yes	characteristic
34	#16810398	C₂²	1	1	yes	characteristic
35	C₂×Q₃₂×C₅₃	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
37	C₅₃×#367	C₂²	1	1	yes	characteristic
38	C₅₃×#358	C₂²	1	2	yes	—
39	C₅₃×#346	C₂²	1	1	yes	characteristic
40	C₂×#11719949	—	2	2	no	—
41	#13133240	—	2	4	no	—
42	#13133240	C₂×C₄	1	8	yes	complemented
43	C₂×#11719924	—	2	2	no	—
44	C₂×D₄₂₄	C₂³	1	2	yes	—
45	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
46	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
47	Q₈×Dic₅₃	—	2	2	no	—
48	C₄×D₂₁₂	—	2	2	no	—
49	C₄×D₂₁₂	—	4	2	no	—
50	#13133179	—	2	2	no	—
51	#13133179	—	4	2	no	—
52	#13133175	—	2	2	no	—
53	#13133175	—	4	2	no	—
54	C₄²×D₅₃	—	4	2	no	—
55	C₄²×D₅₃	D₄	1	1	yes	characteristic
56	#13133160	C₂×C₄	1	8	yes	complemented
57	D₈₄₈	C₂×C₄	1	4	yes	complemented
58	C₂×#11719908	C₂³	1	1	yes	characteristic
59	C₂×#11719908	C₂³	1	1	yes	characteristic
60	C₁₆×D₅₃	—	2	2	no	—
61	C₁₆×D₅₃	C₂×C₄	1	4	yes	complemented
62	#13133151	C₂³	1	2	yes	—
63	#13133148	—	2	2	no	—
64	#13133141	—	2	2	no	—
65	#13133139	—	2	2	no	—
66	#13133137	—	2	2	no	—
67	C₄×#88322	D₄	1	1	yes	characteristic
68	C₈×Dic₅₃	D₄	1	1	yes	characteristic
69	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
70	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
71	C₄×Q₈×C₅₃	—	2	2	no	—
72	Q₃₂×C₅₃	C₂×C₄	1	4	yes	complemented
73	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
75	#107×C₅₃	C₂³	1	2	yes	—
76	#103×C₅₃	—	2	2	no	—
77	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₂×Q₈×C₅₃	—	2	2	no	—
79	#11719949	—	2	8	no	—
80	#11719949	—	4	4	no	—
81	C₂×D₂₁₂	—	2	2	no	—
82	C₂×D₂₁₂	—	4	2	no	—
83	C₂×C₄×D₅₃	—	4	2	no	—
84	C₂×C₄×D₅₃	—	4	2	no	—
85	C₂×C₄×D₅₃	—	4	2	no	—
86	C₂×C₄×D₅₃	D₈	1	2	yes	—
87	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
88	Q₁₆×C₅₃	—	2	2	no	—
89	Q₁₆×C₅₃	C₂²×C₄	1	4	yes	—
90	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₅₃	—	2	2	no	—
93	C₄⋊C₄×C₅₃	—	4	2	no	—
94	C₄²×C₅₃	—	4	2	no	—
95	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
96	#11719924	—	2	8	no	—
97	#11719921	—	4	2	no	—
98	#11719921	—	4	2	no	—
99	#11719921	—	4	2	no	—
100	C₄⋊Dic₅₃	—	2	2	no	—
101	C₄⋊Dic₅₃	—	4	2	no	—
102	C₄×Dic₅₃	—	4	2	no	—
103	C₄×Dic₅₃	—	4	2	no	—
104	C₄×Dic₅₃	D₈	1	2	yes	—
105	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂×#88322	C₂×D₄	1	1	yes	characteristic
107	C₂×#88322	C₂×D₄	1	1	yes	characteristic
108	D₄₂₄	—	2	2	no	—
109	D₄₂₄	C₂²×C₄	1	4	yes	—
110	C₈×D₅₃	—	2	2	no	—
111	C₈×D₅₃	C₂²×C₄	1	2	yes	—
112	C₈×D₅₃	#41	1	2	yes	—
113	C₁₆×C₅₃	—	2	1	no	—
114	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
115	#11719908	—	2	1	no	—
116	#11719908	C₂²×C₄	1	2	yes	—
117	C₂²×D₅₃	—	4	2	no	—
118	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
119	Q₈×C₅₃	—	2	4	no	—
120	Q₈×C₅₃	—	4	2	no	—
121	C₂×C₄×C₅₃	—	4	2	no	—
122	C₂×C₄×C₅₃	—	4	2	no	—
123	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×Dic₅₃	—	4	2	no	—
126	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
127	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
128	D₂₁₂	—	2	4	no	—
129	D₂₁₂	—	4	8	no	—
130	D₂₁₂	—	4	2	no	—
131	C₄×D₅₃	—	4	2	no	—
132	C₄×D₅₃	—	4	8	no	—
133	C₄×D₅₃	—	8	4	no	—
134	C₄×D₅₃	—	8	4	no	—
135	C₄×D₅₃	C₄×D₄	1	2	yes	—
136	C₄×D₅₃	#135	1	2	yes	—
137	C₈×C₅₃	—	2	1	no	—
138	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
139	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
140	#88322	—	2	1	no	—
141	#88322	C₄×D₄	1	2	yes	—
142	#33451	—	53	1	no	maximal, Sylow
143	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
144	C₂×D₅₃	—	2	2	no	—
145	C₂×D₅₃	—	4	4	no	—
146	C₂×D₅₃	—	8	2	no	—
147	C₂×D₅₃	—	8	2	no	—
148	C₂×D₅₃	C₄×D₈	1	2	yes	—
149	C₄×C₅₃	—	2	2	no	—
150	C₄×C₅₃	—	4	4	no	—
151	C₄×C₅₃	—	8	2	no	—
152	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
153	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
154	Dic₅₃	—	2	2	no	—
155	Dic₅₃	—	8	2	no	—
156	Dic₅₃	C₄×D₈	1	2	yes	—
157	C₂×#508	—	53	1	no	—
158	C₄×#135	—	53	2	no	—
159	#2177	—	53	1	no	—
160	#2177	—	53	1	no	—
161	#2176	—	53	1	no	—
162	#2176	D₅₃	1	1	yes	characteristic, complemented
163	#2175	—	53	2	no	—
164	#2175	—	53	2	no	—
165	#2158	—	53	1	no	—
166	#2138	—	53	2	no	—
167	C₂×#346	—	53	1	no	—
168	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
169	C₂×C₅₃	#3414	1	2	yes	—
170	D₅₃	—	2	2	no	—
171	D₅₃	—	8	4	no	—
172	C₂×#135	—	53	2	no	—
173	C₄×#41	—	106	2	no	—
174	#508	—	53	8	no	—
175	C₂×Q₃₂	C₂×D₅₃	1	1	yes	characteristic
176	C₂×SD₃₂	—	53	2	no	—
177	C₂×D₁₆	—	53	1	no	—
178	C₂²×C₁₆	—	53	1	no	—
179	C₂²×C₁₆	—	53	1	no	—
180	C₄×Q₁₆	—	53	2	no	—
181	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
182	C₄×SD₁₆	—	106	2	no	—
183	C₄×SD₁₆	—	106	2	no	—
184	C₄×D₈	—	53	2	no	—
185	C₄×D₈	—	53	2	no	—
186	#427	—	53	2	no	—
187	#416	—	106	2	no	—
188	C₂×C₄×C₈	—	53	1	no	—
189	#367	—	53	1	no	—
190	#367	C₂×D₅₃	1	1	yes	characteristic
191	#366	—	53	2	no	—
192	#358	—	53	2	no	—
193	#358	C₂×D₅₃	1	2	yes	—
194	#357	—	53	2	no	—
195	#357	—	53	2	no	—
196	#346	—	53	1	no	—
197	#346	—	53	1	no	—
198	#346	—	53	1	no	—
199	#346	C₂×D₅₃	1	1	yes	characteristic
200	C₅₃	#33451	1	1	yes	characteristic, complemented, Sylow
201	C₂×#41	—	106	2	no	—
202	#135	—	53	8	no	—
203	#135	—	106	4	no	—
204	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
205	C₂×SD₁₆	—	106	2	no	—
206	C₂×D₈	—	53	2	no	—
207	C₂²×C₈	—	53	1	no	—
208	C₂²×C₈	—	53	1	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	—	106	2	no	—
211	C₄×D₄	—	106	2	no	—
212	C₄×D₄	—	106	2	no	—
213	C₄×D₄	—	212	2	no	—
214	C₄×D₄	—	212	2	no	—
215	#117	—	106	2	no	—
216	#117	—	212	2	no	—
217	C₂×C₄²	—	53	1	no	—
218	C₂×C₄²	—	212	2	no	—
219	Q₃₂	C₄×D₅₃	1	4	yes	complemented
220	SD₃₂	—	53	8	no	—
221	D₁₆	—	53	4	no	—
222	C₂×C₁₆	—	53	1	no	—
223	C₂×C₁₆	—	53	4	no	—
224	C₂×C₁₆	—	53	1	no	—
225	C₂×C₁₆	—	106	2	no	—
226	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
227	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
228	#107	—	53	2	no	—
229	#107	C₂²×D₅₃	1	2	yes	—
230	#106	—	106	2	no	—
231	#103	—	2	2	no	—
232	#103	—	106	2	no	—
233	#102	—	106	2	no	—
234	#102	—	106	2	no	—
235	C₄×C₈	—	53	1	no	—
236	C₄×C₈	—	53	1	no	—
237	C₄×C₈	—	53	1	no	—
238	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
239	#41	—	106	8	no	—
240	#41	—	212	4	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×D₄	—	106	2	no	—
243	C₂×D₄	—	212	2	no	—
244	C₂²×C₄	—	53	1	no	—
245	C₂²×C₄	—	53	2	no	—
246	C₂²×C₄	—	212	2	no	—
247	C₂²×C₄	—	212	2	no	—
248	C₂²×C₄	—	212	2	no	—
249	Q₁₆	—	2	2	no	—
250	Q₁₆	C₂×C₄×D₅₃	1	4	yes	—
251	SD₁₆	—	106	8	no	—
252	D₈	—	53	4	no	—
253	D₈	—	106	2	no	—
254	C₂×C₈	—	53	1	no	—
255	C₂×C₈	—	53	2	no	—
256	C₂×C₈	—	53	2	no	—
257	C₂×C₈	—	53	1	no	—
258	C₂×C₈	—	106	2	no	—
259	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
260	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	106	2	no	—
264	C₄⋊C₄	—	212	2	no	—
265	C₂²⋊C₄	—	212	2	no	—
266	C₂²⋊C₄	—	212	2	no	—
267	C₂²⋊C₄	—	212	2	no	—
268	C₄²	—	4	2	no	—
269	C₄²	—	53	2	no	—
270	C₄²	—	53	1	no	—
271	C₄²	—	212	2	no	—
272	C₄²	—	212	2	no	—
273	C₄²	D₄×D₅₃	1	1	yes	characteristic
274	C₁₆	—	2	1	no	—
275	C₁₆	—	53	2	no	—
276	C₁₆	—	106	1	no	—
277	C₁₆	C₂×C₄×D₅₃	1	2	yes	—
278	C₂³	—	53	1	no	—
279	C₂³	—	212	2	no	—
280	Q₈	—	2	4	no	—
281	Q₈	—	4	2	no	—
282	D₄	—	106	4	no	—
283	D₄	—	212	8	no	—
284	D₄	—	212	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	53	1	no	—
288	C₂×C₄	—	53	2	no	—
289	C₂×C₄	—	53	2	no	—
290	C₂×C₄	—	53	2	no	—
291	C₂×C₄	—	424	4	no	—
292	C₂×C₄	—	424	4	no	—
293	C₂×C₄	—	212	2	no	—
294	C₂×C₄	—	212	8	no	—
295	C₂×C₄	—	212	2	no	—
296	C₂×C₄	D₈×D₅₃	1	2	yes	—
297	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	53	2	no	—
300	C₈	—	106	1	no	—
301	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
302	C₈	#41×D₅₃	1	1	yes	characteristic
303	C₂²	—	53	2	no	—
304	C₂²	—	106	2	no	—
305	C₂²	—	424	2	no	—
306	C₂²	—	424	2	no	—
307	C₂²	—	212	4	no	—
308	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	4	4	no	—
311	C₄	—	8	2	no	—
312	C₄	—	53	2	no	—
313	C₄	—	106	2	no	—
314	C₄	—	424	2	no	—
315	C₄	C₄×D₄×D₅₃	1	1	yes	characteristic
316	C₄	#135×D₅₃	1	1	yes	characteristic
317	C₂	—	106	2	no	—
318	C₂	—	424	4	no	—
319	C₂	C₄×D₈×D₅₃	1	1	yes	characteristic
320	C₂	#20287850	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2176	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×#16811056	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2176	D₅₃
hypocenter	C₅₃	#33451
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2176

Complements

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

N	H ≅ G/N	type
2. C₄×#13133240	C₂	semidirect
3. #20276592	C₂	semidirect
4. #20276590	C₂	semidirect
5. #20276586	C₂	semidirect
6. #20276582	C₂	semidirect
7. #20276577	C₂	semidirect
8. #20276498	C₂	semidirect
9. #20276421	C₂	semidirect
10. #346×D₅₃	C₂	semidirect
11. C₅₃×#2176	C₂	semidirect
14. #16811056	C₄	semidirect
42. #13133240	C₂×C₄	semidirect
56. #13133160	C₂×C₄	semidirect
57. D₈₄₈	C₂×C₄	semidirect
61. C₁₆×D₅₃	C₂×C₄	semidirect
72. Q₃₂×C₅₃	C₂×C₄	semidirect
162. #2176	D₅₃	semidirect
200. C₅₃	#33451	semidirect
219. Q₃₂	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×#16811056	2
C₄×#13133240	2
#20276592	2
#20276590	2
#20276586	2
#20276582	2
#20276577	2
#20276498	2
#20276421	2
#346×D₅₃	2
C₅₃×#2176	2
#33451	53