Subgroups — #27539394

Order 13568.

598 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285226	C₂	1	1	yes	maximal, characteristic, complemented
2	#20285144	C₂	1	1	yes	maximal, characteristic
3	#20276948	C₂	1	4	yes	maximal, complemented
4	#20276942	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276940	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276939	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276421	C₂	1	4	yes	maximal, complemented
8	C₄×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#2241	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13133240	—	2	1	no	—
11	C₂×#13133240	C₂²	1	2	yes	—
12	#16811178	—	2	1	no	—
13	#16811174	—	2	1	no	—
14	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
15	#16811028	C₂²	1	1	yes	characteristic
16	#16811024	—	2	1	no	—
17	#16811024	C₂²	1	2	yes	—
18	#16811019	—	2	1	no	—
19	#16811008	—	2	1	no	—
20	#16811008	C₂²	1	2	yes	—
21	#16811000	C₂²	1	1	yes	characteristic
22	#16810997	—	2	1	no	—
23	#16810660	—	2	1	no	—
24	#16810660	C₂²	1	2	yes	—
25	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
26	#16810427	C₂²	1	4	yes	—
27	#16810425	C₂²	1	4	yes	—
28	#16810423	C₂²	1	4	yes	—
29	C₄×#11719908	C₂²	1	1	yes	characteristic
30	C₁₆×Dic₅₃	C₂²	1	2	yes	—
31	C₅₃×#500	C₂²	1	1	yes	characteristic
32	C₅₃×#494	C₂²	1	1	yes	characteristic
33	C₅₃×#358	C₂²	1	4	yes	—
34	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×#11719949	—	2	2	no	—
36	#13133240	—	2	8	no	—
37	#13133240	—	4	4	no	—
38	C₂×#11719924	—	2	2	no	—
39	C₂×#11719924	—	2	2	no	—
40	C₂×D₄₂₄	—	2	1	no	—
41	C₂×D₄₂₄	C₂³	1	2	yes	—
42	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
43	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
44	#13133217	—	2	1	no	—
45	#13133214	—	2	1	no	—
46	#13133208	—	2	2	no	—
47	#13133206	—	2	2	no	—
48	#13133197	—	4	1	no	—
49	#13133193	—	4	1	no	—
50	#13133192	—	4	1	no	—
51	#13133192	—	4	1	no	—
52	Q₈×Dic₅₃	—	4	1	no	—
53	#13133182	—	4	1	no	—
54	#13133179	—	4	1	no	—
55	#13133179	—	4	1	no	—
56	#13133175	—	2	2	no	—
57	C₄²×D₅₃	D₄	1	1	yes	characteristic
58	C₂×#11719908	C₂³	1	2	yes	—
59	C₁₆×D₅₃	—	2	4	no	—
60	#13133151	—	2	1	no	—
61	#13133151	C₂³	1	2	yes	—
62	#13133148	—	2	2	no	—
63	#13133148	—	2	2	no	—
64	C₄×#88322	D₄	1	1	yes	characteristic
65	C₈×Dic₅₃	D₄	1	1	yes	characteristic
66	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
67	C₂×Q₁₆×C₅₃	—	2	1	no	—
68	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₅₃	—	2	1	no	—
70	C₄⋊Q₈×C₅₃	—	2	1	no	—
71	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
72	#107×C₅₃	—	2	1	no	—
73	#107×C₅₃	C₂³	1	2	yes	—
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₂×Q₈×C₅₃	—	2	2	no	—
76	C₂×Q₈×C₅₃	—	4	1	no	—
77	#11719949	—	4	4	no	—
78	#11719949	—	4	4	no	—
79	C₂×D₂₁₂	—	2	2	no	—
80	C₂×D₂₁₂	—	4	2	no	—
81	C₂×D₂₁₂	—	4	1	no	—
82	C₂×D₂₁₂	—	4	1	no	—
83	C₂×C₄×D₅₃	—	4	2	no	—
84	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
85	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
86	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
87	C₂×Dic₁₀₆	—	4	1	no	—
88	Q₁₆×C₅₃	—	2	4	no	—
89	Q₁₆×C₅₃	—	4	2	no	—
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	1	no	—
94	C₄⋊C₄×C₅₃	—	4	1	no	—
95	C₄⋊C₄×C₅₃	—	4	1	no	—
96	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
97	#11719924	—	4	4	no	—
98	#11719924	—	4	4	no	—
99	#11719921	—	4	2	no	—
100	#11719921	—	4	2	no	—
101	#11719921	—	4	2	no	—
102	C₄⋊Dic₅₃	—	2	2	no	—
103	C₄⋊Dic₅₃	—	4	1	no	—
104	C₄⋊Dic₅₃	—	4	2	no	—
105	#11719919	—	4	2	no	—
106	C₄×Dic₅₃	—	4	2	no	—
107	C₄×Dic₅₃	D₈	1	1	yes	characteristic
108	C₄×Dic₅₃	D₈	1	1	yes	characteristic
109	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂×#88322	C₂×D₄	1	1	yes	characteristic
111	C₂×#88322	C₂×D₄	1	1	yes	characteristic
112	D₄₂₄	—	2	4	no	—
113	D₄₂₄	—	4	2	no	—
114	C₈×D₅₃	SD₁₆	1	4	yes	—
115	C₈×D₅₃	#41	1	4	yes	—
116	C₁₆×C₅₃	—	2	2	no	—
117	#11719908	—	2	2	no	—
118	C₂²×D₅₃	—	4	2	no	—
119	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
120	Q₈×C₅₃	—	4	2	no	—
121	Q₈×C₅₃	—	4	2	no	—
122	Q₈×C₅₃	—	8	2	no	—
123	C₂×C₄×C₅₃	—	4	2	no	—
124	C₂×C₄×C₅₃	—	4	2	no	—
125	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×Dic₅₃	—	4	2	no	—
129	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
132	D₂₁₂	—	4	2	no	—
133	D₂₁₂	—	4	2	no	—
134	D₂₁₂	—	8	4	no	—
135	D₂₁₂	—	8	2	no	—
136	D₂₁₂	—	8	2	no	—
137	C₄×D₅₃	—	2	2	no	—
138	C₄×D₅₃	—	2	2	no	—
139	C₄×D₅₃	—	8	4	no	—
140	C₄×D₅₃	#124	1	2	yes	—
141	C₄×D₅₃	C₂×SD₁₆	1	2	yes	—
142	Dic₁₀₆	—	8	2	no	—
143	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
144	C₈×C₅₃	C₂×#41	1	2	yes	—
145	#88322	#124	1	2	yes	—
146	#88322	C₂×SD₁₆	1	2	yes	—
147	#33697	—	53	1	no	maximal, Sylow
148	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
149	C₂×D₅₃	—	2	1	no	—
150	C₂×D₅₃	—	2	1	no	—
151	C₂×D₅₃	—	8	2	no	—
152	C₂×D₅₃	—	8	2	no	—
153	C₂×D₅₃	—	8	2	no	—
154	C₂×D₅₃	#486	1	2	yes	—
155	C₄×C₅₃	—	8	2	no	—
156	C₄×C₅₃	—	8	2	no	—
157	C₄×C₅₃	#508	1	2	yes	—
158	C₄×C₅₃	#508	1	2	yes	—
159	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
160	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
161	Dic₅₃	—	2	1	no	—
162	Dic₅₃	—	2	1	no	—
163	Dic₅₃	—	8	2	no	—
164	Dic₅₃	#486	1	2	yes	—
165	#3160	—	53	1	no	—
166	#3147	—	53	1	no	—
167	#2242	—	53	4	no	—
168	#2241	—	53	1	no	—
169	#2241	D₅₃	1	1	yes	characteristic, complemented
170	#2240	—	53	1	no	—
171	#2240	—	53	1	no	—
172	#2138	—	53	4	no	—
173	C₂×C₄×C₁₆	—	53	1	no	—
174	C₂×C₅₃	C₂×#486	1	1	yes	characteristic
175	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
176	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
177	D₅₃	—	2	2	no	—
178	D₅₃	—	16	2	no	—
179	C₂×#135	—	53	2	no	—
180	C₂×#135	—	106	1	no	—
181	#533	—	106	1	no	—
182	#532	—	106	1	no	—
183	C₂²×C₁₆	—	53	2	no	—
184	#500	—	53	1	no	—
185	#500	C₂×D₅₃	1	1	yes	characteristic
186	#499	—	53	2	no	—
187	#499	—	106	1	no	—
188	#498	—	106	1	no	—
189	#495	—	53	2	no	—
190	#495	—	106	1	no	—
191	#494	C₂×D₅₃	1	1	yes	characteristic
192	#493	—	53	1	no	—
193	#492	—	106	1	no	—
194	#427	—	53	2	no	—
195	#427	—	106	1	no	—
196	C₂×C₄×C₈	—	53	1	no	—
197	#358	—	53	4	no	—
198	#358	C₂×D₅₃	1	4	yes	—
199	#357	—	53	4	no	—
200	#357	—	53	4	no	—
201	C₄×C₁₆	—	53	2	no	—
202	C₄×C₁₆	—	53	1	no	—
203	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
204	C₅₃	#33697	1	1	yes	characteristic, complemented, Sylow
205	C₂×#41	—	106	2	no	—
206	#135	—	106	8	no	—
207	#135	—	212	4	no	—
208	C₂×Q₁₆	—	2	1	no	—
209	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
210	C₂×SD₁₆	—	106	2	no	—
211	C₂×SD₁₆	—	106	2	no	—
212	C₂×D₈	—	53	2	no	—
213	C₂×D₈	—	106	1	no	—
214	C₂²×C₈	—	53	1	no	—
215	C₂²×C₈	—	53	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	—	2	1	no	—
218	C₄⋊Q₈	—	106	1	no	—
219	#127	—	106	1	no	—
220	#125	—	106	2	no	—
221	#124	—	106	2	no	—
222	C₂²⋊Q₈	—	212	1	no	—
223	C₂²⋊Q₈	—	212	1	no	—
224	#121	—	212	1	no	—
225	#121	—	212	1	no	—
226	C₄×Q₈	—	212	1	no	—
227	C₄×Q₈	—	212	1	no	—
228	C₄×D₄	—	212	1	no	—
229	C₄×D₄	—	212	1	no	—
230	#117	—	106	2	no	—
231	C₂×C₄²	—	53	1	no	—
232	C₂×C₁₆	—	53	2	no	—
233	C₂×C₁₆	—	106	4	no	—
234	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
235	#107	—	2	1	no	—
236	#107	—	53	2	no	—
237	#107	—	106	1	no	—
238	#107	C₂²×D₅₃	1	2	yes	—
239	#106	—	106	2	no	—
240	#106	—	106	2	no	—
241	C₄×C₈	—	53	1	no	—
242	C₄×C₈	—	53	1	no	—
243	C₄×C₈	—	53	1	no	—
244	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
245	#41	—	212	4	no	—
246	#41	—	212	4	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	212	1	no	—
250	C₂×D₄	—	106	2	no	—
251	C₂×D₄	—	212	2	no	—
252	C₂×D₄	—	212	1	no	—
253	C₂×D₄	—	212	1	no	—
254	C₂²×C₄	—	53	1	no	—
255	C₂²×C₄	—	53	1	no	—
256	C₂²×C₄	—	53	1	no	—
257	C₂²×C₄	—	212	2	no	—
258	Q₁₆	—	2	4	no	—
259	Q₁₆	—	4	2	no	—
260	SD₁₆	—	212	4	no	—
261	SD₁₆	—	212	4	no	—
262	D₈	—	106	4	no	—
263	D₈	—	212	2	no	—
264	C₂×C₈	—	53	1	no	—
265	C₂×C₈	—	53	1	no	—
266	C₂×C₈	—	53	4	no	—
267	C₂×C₈	—	53	4	no	—
268	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
269	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	106	2	no	—
275	C₄⋊C₄	—	212	2	no	—
276	C₄⋊C₄	—	212	2	no	—
277	C₄⋊C₄	—	212	1	no	—
278	C₂²⋊C₄	—	212	2	no	—
279	C₂²⋊C₄	—	212	2	no	—
280	C₂²⋊C₄	—	212	2	no	—
281	C₄²	—	53	1	no	—
282	C₄²	—	53	1	no	—
283	C₄²	—	53	1	no	—
284	C₄²	—	212	2	no	—
285	C₄²	D₄×D₅₃	1	1	yes	characteristic
286	C₁₆	—	2	2	no	—
287	C₁₆	—	106	2	no	—
288	C₂³	—	53	1	no	—
289	C₂³	—	212	2	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	4	2	no	—
292	Q₈	—	8	2	no	—
293	Q₈	—	424	2	no	—
294	D₄	—	424	4	no	—
295	D₄	—	424	2	no	—
296	D₄	—	424	2	no	—
297	D₄	—	212	2	no	—
298	D₄	—	212	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	53	1	no	—
302	C₂×C₄	—	53	2	no	—
303	C₂×C₄	—	53	2	no	—
304	C₂×C₄	—	53	1	no	—
305	C₂×C₄	—	53	1	no	—
306	C₂×C₄	—	106	2	no	—
307	C₂×C₄	—	106	2	no	—
308	C₂×C₄	—	424	4	no	—
309	C₂×C₄	—	212	2	no	—
310	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
311	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
312	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
313	C₈	—	53	2	no	—
314	C₈	—	53	2	no	—
315	C₈	SD₁₆×D₅₃	1	2	yes	—
316	C₈	#41×D₅₃	1	2	yes	—
317	C₂²	—	53	2	no	—
318	C₂²	—	106	1	no	—
319	C₂²	—	106	1	no	—
320	C₂²	—	424	2	no	—
321	C₂²	—	424	2	no	—
322	C₂²	—	424	2	no	—
323	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	—	53	2	no	—
327	C₄	—	106	1	no	—
328	C₄	—	106	1	no	—
329	C₄	—	424	2	no	—
330	C₄	#16811054	1	2	yes	—
331	C₄	#16811056	1	2	yes	—
332	C₄	#124×D₅₃	1	1	yes	characteristic
333	C₄	C₂×SD₁₆×D₅₃	1	1	yes	characteristic
334	C₂	—	848	2	no	—
335	C₂	—	106	2	no	—
336	C₂	#486×D₅₃	1	1	yes	characteristic
337	C₂	C₂×#16811054	1	1	yes	characteristic
338	C₂	C₂×#16811056	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2241	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#20285144	C₂
hypercenter	#2241	D₅₃
hypocenter	C₅₃	#33697
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2241

Complements

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

N	H ≅ G/N	type
1. #20285226	C₂	semidirect
3. #20276948	C₂	semidirect
4. #20276942	C₂	semidirect
5. #20276940	C₂	semidirect
6. #20276939	C₂	semidirect
7. #20276421	C₂	semidirect
8. C₄×C₁₆×D₅₃	C₂	semidirect
9. C₅₃×#2241	C₂	semidirect
169. #2241	D₅₃	semidirect
204. C₅₃	#33697	semidirect

Maximal subgroups

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

subgroup	index
#20285226	2
#20285144	2
#20276948	2
#20276942	2
#20276940	2
#20276939	2
#20276421	2
C₄×C₁₆×D₅₃	2
C₅₃×#2241	2
#33697	53