Subgroups — #27541731

Order 13568.

482 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	#20287139	C₂	1	1	yes	maximal, characteristic
2	#20287107	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276988	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276926	C₂	1	2	yes	maximal, complemented
5	#20276894	C₂	1	2	yes	maximal, complemented
6	#20276868	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276827	C₂	1	2	yes	maximal, complemented
8	C₄×#13133161	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276546	C₂	1	1	yes	maximal, characteristic, complemented
10	#20276502	C₂	1	2	yes	maximal, complemented
11	C₅₃×#2226	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811338	—	2	1	no	—
13	#16811334	—	2	1	no	—
14	C₂×#13133161	C₂²	1	1	yes	characteristic
15	C₂×#13133156	C₂²	1	2	yes	—
16	#16811028	C₂²	1	1	yes	characteristic
17	#16810960	—	2	2	no	—
18	#16810945	—	2	2	no	—
19	#16810915	—	2	1	no	—
20	#16810913	—	2	1	no	—
21	#16810865	C₂²	1	2	yes	—
22	#16810856	C₂²	1	2	yes	—
23	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
24	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
25	#16810660	C₂²	1	2	yes	—
26	#107×D₅₃	C₂²	1	2	yes	—
27	#16810450	C₂²	1	1	yes	characteristic
28	#16810449	C₂²	1	2	yes	—
29	#16810448	C₂²	1	2	yes	—
30	#16810448	C₂²	1	2	yes	—
31	#16810428	C₂²	1	2	yes	—
32	#16810426	C₂²	1	1	yes	characteristic
33	#16810424	C₂²	1	1	yes	characteristic
34	#16810402	C₂²	1	2	yes	—
35	C₄×#11719908	C₂²	1	1	yes	characteristic
36	C₅₃×#500	C₂²	1	1	yes	characteristic
37	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#367	C₂²	1	2	yes	—
39	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
40	C₅₃×#357	C₂²	1	2	yes	—
41	C₂×Dic₂₁₂	C₂³	1	1	yes	characteristic
42	C₂×C₈×D₅₃	C₂³	1	2	yes	—
43	#13133217	—	2	1	no	—
44	#13133208	—	4	1	no	—
45	#13133203	—	4	2	no	—
46	#13133197	—	2	2	no	—
47	#13133193	—	4	2	no	—
48	#13133191	—	4	1	no	—
49	#13133188	—	2	2	no	—
50	C₄×Dic₁₀₆	—	2	1	no	—
51	C₄×D₂₁₂	D₄	1	1	yes	characteristic
52	#13133179	—	4	1	no	—
53	#13133176	—	4	1	no	—
54	#13133175	—	2	2	no	—
55	C₄⋊C₄×D₅₃	—	2	2	no	—
56	#13133161	—	2	2	no	—
57	#13133156	—	2	4	no	—
58	C₂×#11719908	C₂³	1	1	yes	characteristic
59	C₂×#11719908	C₂³	1	1	yes	characteristic
60	#13133151	—	2	1	no	—
61	#13133151	C₂³	1	2	yes	—
62	#13133151	C₂³	1	1	yes	characteristic
63	#13133150	—	2	2	no	—
64	#13133148	—	2	2	no	—
65	C₄⋊#88322	D₄	1	1	yes	characteristic
66	#13133142	—	2	1	no	—
67	#13133140	—	2	2	no	—
68	#13133138	—	2	2	no	—
69	#13133129	D₄	1	2	yes	—
70	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
71	C₄⋊Q₈×C₅₃	—	2	1	no	—
72	C₄×D₄×C₅₃	—	2	1	no	—
73	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
74	#107×C₅₃	—	2	1	no	—
75	#107×C₅₃	C₂³	1	2	yes	—
76	#107×C₅₃	C₂³	1	1	yes	characteristic
77	#102×C₅₃	—	2	1	no	—
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₅₃	—	4	1	no	—
80	C₂×D₄×C₅₃	—	2	1	no	—
81	C₂²×C₄×C₅₃	—	4	1	no	—
82	C₂×C₂²⋊D₅₃	—	4	2	no	—
83	C₂×D₂₁₂	D₈	1	1	yes	characteristic
84	C₂×C₄×D₅₃	—	4	2	no	—
85	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
86	C₂×Dic₁₀₆	—	2	1	no	—
87	C₂×Dic₁₀₆	—	4	1	no	—
88	D₈×C₅₃	—	2	1	no	—
89	D₈×C₅₃	—	2	1	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	1	no	—
93	C₄⋊C₄×C₅₃	—	2	2	no	—
94	C₄⋊C₄×C₅₃	—	4	1	no	—
95	C₂²⋊C₄×C₅₃	—	4	1	no	—
96	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂²⋊Dic₅₃	—	4	2	no	—
98	#11719921	—	4	2	no	—
99	#11719921	—	4	2	no	—
100	#11719921	D₈	1	2	yes	—
101	C₄⋊Dic₅₃	—	2	2	no	—
102	C₄⋊Dic₅₃	—	2	1	no	—
103	C₄⋊Dic₅₃	—	4	1	no	—
104	C₄⋊Dic₅₃	—	4	2	no	—
105	C₄⋊Dic₅₃	D₈	1	1	yes	characteristic
106	#11719919	—	4	1	no	—
107	#11719919	—	4	2	no	—
108	C₄×Dic₅₃	—	4	1	no	—
109	C₄×Dic₅₃	—	4	2	no	—
110	C₂×#88322	C₂×D₄	1	2	yes	—
111	Dic₂₁₂	—	2	1	no	—
112	Dic₂₁₂	—	2	1	no	—
113	C₈×D₅₃	—	2	2	no	—
114	C₈×D₅₃	—	2	2	no	—
115	C₁₆×C₅₃	—	2	2	no	—
116	#11719908	—	2	1	no	—
117	#11719908	#41	1	2	yes	—
118	C₂³×C₅₃	—	4	1	no	—
119	C₂²×D₅₃	C₂×D₈	1	2	yes	—
120	Q₈×C₅₃	—	16	1	no	—
121	D₄×C₅₃	—	4	1	no	—
122	D₄×C₅₃	—	4	1	no	—
123	C₂×C₄×C₅₃	—	4	2	no	—
124	C₂×C₄×C₅₃	—	4	1	no	—
125	C₂×C₄×C₅₃	—	8	2	no	—
126	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
129	C₂²⋊D₅₃	—	16	2	no	—
130	C₂×Dic₅₃	—	4	1	no	—
131	C₂×Dic₅₃	—	4	2	no	—
132	C₂×Dic₅₃	—	4	1	no	—
133	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
134	D₂₁₂	—	2	2	no	—
135	C₄×D₅₃	—	2	2	no	—
136	C₄×D₅₃	—	2	2	no	—
137	C₄×D₅₃	—	16	2	no	—
138	Dic₁₀₆	—	4	1	no	—
139	Dic₁₀₆	—	4	1	no	—
140	Dic₁₀₆	—	16	1	no	—
141	C₈×C₅₃	—	2	1	no	—
142	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
143	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
144	#88322	—	2	2	no	—
145	#33924	—	53	1	no	maximal, Sylow
146	C₂²×C₅₃	—	8	1	no	—
147	C₂²×C₅₃	—	8	1	no	—
148	C₂²×C₅₃	—	8	1	no	—
149	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
150	C₂×D₅₃	—	2	2	no	—
151	C₂×D₅₃	—	2	2	no	—
152	C₂×D₅₃	—	2	2	no	—
153	C₄×C₅₃	—	2	1	no	—
154	C₄×C₅₃	—	8	1	no	—
155	C₄×C₅₃	—	8	2	no	—
156	C₄×C₅₃	#510	1	2	yes	—
157	C₄×C₅₃	#547	1	1	yes	characteristic
158	C₄×C₅₃	#578	1	1	yes	characteristic
159	Dic₅₃	—	2	2	no	—
160	Dic₅₃	—	8	1	no	—
161	Dic₅₃	—	8	2	no	—
162	Dic₅₃	—	8	1	no	—
163	#3341	—	53	1	no	—
164	#3338	—	53	1	no	—
165	#2252	—	53	1	no	—
166	#2237	—	53	2	no	—
167	#2232	—	53	2	no	—
168	#2227	—	53	1	no	—
169	#2226	D₅₃	1	1	yes	characteristic, complemented
170	#2220	—	53	2	no	—
171	C₄×SD₃₂	—	53	1	no	—
172	#2169	—	53	1	no	—
173	#2159	—	53	2	no	—
174	C₂×C₅₃	—	16	1	no	—
175	C₂×C₅₃	#3334	1	1	yes	characteristic
176	C₂×C₅₃	C₂×#510	1	1	yes	characteristic
177	C₂×C₅₃	#3415	1	1	yes	characteristic
178	D₅₃	—	4	2	no	—
179	#555	—	106	1	no	—
180	#554	—	106	1	no	—
181	C₂×SD₃₂	—	53	1	no	—
182	C₂×M₃₂	—	53	2	no	—
183	#500	—	53	1	no	—
184	#500	C₂×D₅₃	1	1	yes	characteristic
185	#484	—	106	2	no	—
186	#482	—	106	2	no	—
187	#477	—	106	1	no	—
188	#476	—	106	1	no	—
189	#467	—	53	2	no	—
190	#466	—	53	2	no	—
191	C₄×Q₁₆	—	53	1	no	—
192	C₄×D₈	C₂×D₅₃	1	1	yes	characteristic
193	C₈×D₄	—	53	1	no	—
194	#427	—	53	2	no	—
195	C₂×#107	—	53	2	no	—
196	#367	—	53	1	no	—
197	#367	C₂×D₅₃	1	2	yes	—
198	#366	—	53	2	no	—
199	#366	—	53	2	no	—
200	#366	—	53	2	no	—
201	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
202	#358	—	53	2	no	—
203	#358	—	53	1	no	—
204	#357	—	53	1	no	—
205	#357	C₂×D₅₃	1	2	yes	—
206	C₂²⋊C₁₆	—	53	2	no	—
207	C₄×C₁₆	—	53	1	no	—
208	C₅₃	#33924	1	1	yes	characteristic, complemented, Sylow
209	C₂×Q₁₆	—	53	1	no	—
210	C₂×D₈	C₂²×D₅₃	1	1	yes	characteristic
211	C₂²×C₈	—	53	2	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	106	1	no	—
214	#125	—	212	1	no	—
215	#124	—	212	2	no	—
216	C₂²⋊Q₈	—	106	2	no	—
217	C₂²⋊Q₈	—	212	2	no	—
218	#121	—	106	2	no	—
219	#121	—	212	1	no	—
220	C₄×Q₈	—	106	1	no	—
221	C₄×D₄	—	2	1	no	—
222	C₄×D₄	—	53	1	no	—
223	C₄×D₄	—	212	1	no	—
224	#117	—	106	2	no	—
225	#117	—	212	1	no	—
226	C₂×C₄⋊C₄	—	106	2	no	—
227	SD₃₂	—	106	2	no	—
228	M₃₂	—	106	4	no	—
229	C₂×C₁₆	—	53	1	no	—
230	C₂×C₁₆	—	53	1	no	—
231	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
232	#107	—	2	1	no	—
233	#107	—	53	1	no	—
234	#107	—	53	2	no	—
235	#107	—	106	2	no	—
236	#107	—	106	1	no	—
237	#107	C₂²×D₅₃	1	2	yes	—
238	#107	C₂²×D₅₃	1	1	yes	characteristic
239	#106	—	106	2	no	—
240	C₄⋊C₈	—	53	1	no	—
241	#103	—	106	2	no	—
242	#103	—	106	1	no	—
243	#102	—	2	1	no	—
244	#102	—	106	2	no	—
245	C₂²⋊C₈	—	53	2	no	—
246	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	106	1	no	—
249	C₂×Q₈	—	212	1	no	—
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	53	1	no	—
252	C₂×D₄	—	212	2	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	53	2	no	—
255	C₂²×C₄	—	212	2	no	—
256	Q₁₆	—	106	1	no	—
257	Q₁₆	—	106	1	no	—
258	D₈	—	2	1	no	—
259	D₈	—	2	1	no	—
260	C₂×C₈	—	53	2	no	—
261	C₂×C₈	—	106	2	no	—
262	C₂×C₈	—	106	2	no	—
263	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
264	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	53	1	no	—
269	C₄⋊C₄	—	106	2	no	—
270	C₄⋊C₄	—	106	1	no	—
271	C₄⋊C₄	—	212	2	no	—
272	C₄⋊C₄	—	212	2	no	—
273	C₄⋊C₄	—	212	1	no	—
274	C₄⋊C₄	—	212	1	no	—
275	C₂²⋊C₄	—	4	1	no	—
276	C₂²⋊C₄	—	53	2	no	—
277	C₂²⋊C₄	—	212	2	no	—
278	C₂²⋊C₄	—	212	2	no	—
279	C₂²⋊C₄	—	212	2	no	—
280	C₄²	—	212	2	no	—
281	C₄²	—	212	1	no	—
282	C₄²	D₄×D₅₃	1	1	yes	characteristic
283	C₁₆	—	2	2	no	—
284	C₁₆	—	53	2	no	—
285	C₁₆	—	106	1	no	—
286	C₂³	—	4	1	no	—
287	C₂³	—	53	2	no	—
288	Q₈	—	16	1	no	—
289	Q₈	—	848	1	no	—
290	Q₈	—	212	1	no	—
291	Q₈	—	212	1	no	—
292	D₄	—	4	1	no	—
293	D₄	—	4	1	no	—
294	D₄	—	848	2	no	—
295	D₄	—	106	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	—	53	2	no	—
300	C₂×C₄	—	848	2	no	—
301	C₂×C₄	—	106	2	no	—
302	C₂×C₄	—	106	2	no	—
303	C₂×C₄	—	212	2	no	—
304	C₂×C₄	—	212	1	no	—
305	C₂×C₄	—	212	1	no	—
306	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
307	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
308	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	106	2	no	—
311	C₈	#41×D₅₃	1	1	yes	characteristic
312	C₈	#13133265	1	1	yes	characteristic
313	C₂²	—	8	1	no	—
314	C₂²	—	8	1	no	—
315	C₂²	—	8	1	no	—
316	C₂²	—	106	2	no	—
317	C₂²	—	106	2	no	—
318	C₂²	—	106	2	no	—
319	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	8	1	no	—
322	C₄	—	8	2	no	—
323	C₄	—	106	2	no	—
324	C₄	—	424	1	no	—
325	C₄	—	424	1	no	—
326	C₄	—	424	2	no	—
327	C₄	#16811062	1	2	yes	—
328	C₄	#16811275	1	1	yes	characteristic
329	C₄	#16811439	1	1	yes	characteristic
330	C₂	—	16	1	no	—
331	C₂	—	212	2	no	—
332	C₂	#20287061	1	1	yes	characteristic
333	C₂	C₂×#16811062	1	1	yes	characteristic
334	C₂	#20287855	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×#2226	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#20287139	C₂
hypercenter	#2226	D₅₃
hypocenter	C₅₃	#33924
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2226

Complements

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

N	H ≅ G/N	type
2. #20287107	C₂	semidirect
3. #20276988	C₂	semidirect
4. #20276926	C₂	semidirect
5. #20276894	C₂	semidirect
6. #20276868	C₂	semidirect
7. #20276827	C₂	semidirect
8. C₄×#13133161	C₂	semidirect
9. #20276546	C₂	semidirect
10. #20276502	C₂	semidirect
11. C₅₃×#2226	C₂	semidirect
169. #2226	D₅₃	semidirect
208. C₅₃	#33924	semidirect

Maximal subgroups

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

subgroup	index
#20287139	2
#20287107	2
#20276988	2
#20276926	2
#20276894	2
#20276868	2
#20276827	2
C₄×#13133161	2
#20276546	2
#20276502	2
C₅₃×#2226	2
#33924	53