Subgroups — #27541741

Order 13568.

506 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287719	C₂	1	1	yes	maximal, characteristic, complemented
2	#20287567	C₂	1	1	yes	maximal, characteristic
3	#20276941	C₂	1	2	yes	maximal, complemented
4	#20276855	C₂	1	2	yes	maximal, complemented
5	#20276854	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276759	C₂	1	2	yes	maximal
7	#20276757	C₂	1	1	yes	maximal, characteristic
8	C₁₆×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13133162	C₂	1	1	yes	maximal, characteristic
10	#20276564	C₂	1	2	yes	maximal
11	C₅₃×#2241	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811348	—	2	1	no	—
13	#16811288	—	2	1	no	—
14	C₂×#13133162	C₂²	1	1	yes	characteristic
15	C₂×#13133160	C₂²	1	2	yes	—
16	#16811027	C₂²	1	2	yes	—
17	#16811001	C₂²	1	2	yes	—
18	#16810902	—	2	2	no	—
19	#16810900	—	2	1	no	—
20	#16810831	—	2	2	no	—
21	#16810830	—	2	1	no	—
22	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
23	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
24	Q₁₆×Dic₅₃	C₂²	1	2	yes	—
25	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
26	#16810705	C₂²	1	2	yes	—
27	#16810451	C₂²	1	1	yes	characteristic
28	#16810450	C₂²	1	2	yes	—
29	C₄⋊#11719908	C₂²	1	1	yes	characteristic
30	#16810441	C₂²	1	2	yes	—
31	#16810428	C₂²	1	1	yes	characteristic
32	#16810427	C₂²	1	2	yes	—
33	#16810426	C₂²	1	2	yes	—
34	#16810425	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₅₃	C₂²	1	2	yes	—
36	C₅₃×#500	C₂²	1	1	yes	characteristic
37	C₅₃×#494	C₂²	1	1	yes	characteristic
38	C₅₃×#358	C₂²	1	2	yes	—
39	C₅₃×#358	C₂²	1	2	yes	—
40	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×Q₈×D₅₃	—	4	1	no	—
42	C₂×Dic₂₁₂	C₂³	1	1	yes	characteristic
43	C₂×#11719925	—	2	2	no	—
44	C₂×D₄₂₄	C₂³	1	1	yes	characteristic
45	#13133216	—	2	2	no	—
46	#13133215	—	2	2	no	—
47	#13133208	—	4	1	no	—
48	#13133207	—	4	2	no	—
49	#13133195	—	4	2	no	—
50	#13133193	—	4	1	no	—
51	Q₈×Dic₅₃	—	2	2	no	—
52	C₄×Dic₁₀₆	—	2	1	no	—
53	C₄×Dic₁₀₆	D₄	1	1	yes	characteristic
54	#13133182	—	4	1	no	—
55	C₄×D₂₁₂	—	2	1	no	—
56	#13133179	—	2	2	no	—
57	#13133162	—	2	2	no	—
58	#13133160	—	2	4	no	—
59	C₂×#11719908	C₂³	1	2	yes	—
60	#13133151	C₂³	1	1	yes	characteristic
61	#13133151	C₂³	1	1	yes	characteristic
62	#13133150	—	2	2	no	—
63	C₄⋊#88322	D₄	1	1	yes	characteristic
64	#13133146	D₄	1	2	yes	—
65	#13133142	—	2	1	no	—
66	#13133141	—	2	2	no	—
67	#13133139	—	2	1	no	—
68	#13133137	—	2	2	no	—
69	C₈×Dic₅₃	C₂³	1	2	yes	—
70	C₂×Q₁₆×C₅₃	—	2	1	no	—
71	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
72	C₄⋊Q₈×C₅₃	—	2	1	no	—
73	C₄⋊Q₈×C₅₃	—	2	1	no	—
74	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
75	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
76	#107×C₅₃	—	2	1	no	—
77	#107×C₅₃	C₂³	1	2	yes	—
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₅₃	—	2	2	no	—
80	C₂×Q₈×C₅₃	—	4	1	no	—
81	Q₈×D₅₃	—	8	4	no	—
82	C₂×D₂₁₂	—	2	1	no	—
83	C₂×C₄×D₅₃	—	4	1	no	—
84	C₂×C₄×D₅₃	—	4	2	no	—
85	C₂×Dic₁₀₆	—	2	1	no	—
86	C₂×Dic₁₀₆	—	4	2	no	—
87	C₂×Dic₁₀₆	D₈	1	1	yes	characteristic
88	Q₁₆×C₅₃	—	2	2	no	—
89	Q₁₆×C₅₃	—	2	2	no	—
90	Q₁₆×C₅₃	—	8	1	no	—
91	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₅₃	—	2	2	no	—
94	C₄⋊C₄×C₅₃	—	4	1	no	—
95	C₄⋊C₄×C₅₃	—	4	1	no	—
96	C₄⋊C₄×C₅₃	—	4	1	no	—
97	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
98	#11719925	—	8	2	no	—
99	#11719921	—	4	1	no	—
100	#11719921	—	4	2	no	—
101	C₄⋊Dic₅₃	—	2	1	no	—
102	C₄⋊Dic₅₃	—	2	1	no	—
103	C₄⋊Dic₅₃	—	4	2	no	—
104	C₄⋊Dic₅₃	D₈	1	1	yes	characteristic
105	#11719919	—	4	1	no	—
106	#11719919	—	4	2	no	—
107	#11719919	—	4	2	no	—
108	#11719919	—	4	2	no	—
109	#11719919	D₈	1	2	yes	—
110	C₄×Dic₅₃	—	4	1	no	—
111	C₄×Dic₅₃	—	4	2	no	—
112	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
113	C₂×#88322	C₂×D₄	1	2	yes	—
114	Dic₂₁₂	—	2	1	no	—
115	Dic₂₁₂	—	2	1	no	—
116	D₄₂₄	—	2	1	no	—
117	D₄₂₄	#41	1	2	yes	—
118	C₁₆×C₅₃	—	2	1	no	—
119	C₁₆×C₅₃	—	2	1	no	—
120	#11719908	—	2	2	no	—
121	C₂²×D₅₃	—	4	1	no	—
122	Q₈×C₅₃	—	4	2	no	—
123	Q₈×C₅₃	—	4	2	no	—
124	Q₈×C₅₃	—	8	2	no	—
125	C₂×C₄×C₅₃	—	4	2	no	—
126	C₂×C₄×C₅₃	—	4	2	no	—
127	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×Dic₅₃	—	4	1	no	—
131	C₂×Dic₅₃	—	4	1	no	—
132	C₂×Dic₅₃	—	4	1	no	—
133	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
134	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
135	D₂₁₂	—	2	2	no	—
136	D₂₁₂	—	4	1	no	—
137	C₄×D₅₃	—	8	2	no	—
138	C₄×D₅₃	—	8	4	no	—
139	Dic₁₀₆	—	4	1	no	—
140	Dic₁₀₆	—	4	1	no	—
141	Dic₁₀₆	—	8	4	no	—
142	Dic₁₀₆	SD₃₂	1	4	yes	complemented
143	C₈×C₅₃	—	2	1	no	—
144	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
145	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
146	#88322	—	2	2	no	—
147	#33926	—	53	1	no	maximal, Sylow
148	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
149	C₂×D₅₃	—	4	2	no	—
150	C₂×D₅₃	—	8	1	no	—
151	C₂×D₅₃	—	8	1	no	—
152	C₄×C₅₃	—	2	1	no	—
153	C₄×C₅₃	—	8	2	no	—
154	C₄×C₅₃	—	8	2	no	—
155	C₄×C₅₃	C₂×SD₃₂	1	2	yes	—
156	C₄×C₅₃	#548	1	1	yes	characteristic
157	C₄×C₅₃	#577	1	1	yes	characteristic
158	Dic₅₃	—	2	2	no	—
159	Dic₅₃	—	8	1	no	—
160	Dic₅₃	—	8	1	no	—
161	Dic₅₃	—	8	1	no	—
162	Dic₅₃	C₂×SD₃₂	1	4	yes	—
163	#3395	—	53	1	no	—
164	Q₈×D₈	—	53	1	no	—
165	#2241	—	53	2	no	—
166	#2241	D₅₃	1	1	yes	characteristic, complemented
167	#2225	—	53	1	no	—
168	#2225	—	53	2	no	—
169	#2208	—	53	1	no	—
170	#2208	—	53	2	no	—
171	Q₈×C₁₆	—	53	1	no	—
172	C₄×SD₃₂	—	53	2	no	—
173	C₄×SD₃₂	—	53	1	no	—
174	C₂×C₅₃	#3358	1	1	yes	characteristic
175	C₂×C₅₃	C₂²×SD₃₂	1	1	yes	characteristic
176	C₂×C₅₃	#3417	1	1	yes	characteristic
177	D₅₃	—	8	2	no	—
178	#557	—	106	1	no	—
179	D₄×Q₈	—	106	1	no	—
180	C₂×SD₃₂	—	53	2	no	—
181	C₂×SD₃₂	—	53	1	no	—
182	#500	—	53	2	no	—
183	#500	C₂×D₅₃	1	1	yes	characteristic
184	#494	—	53	2	no	—
185	#494	C₂×D₅₃	1	1	yes	characteristic
186	#474	—	106	2	no	—
187	#474	—	106	1	no	—
188	#462	—	106	1	no	—
189	#462	—	106	2	no	—
190	C₈×Q₈	—	53	1	no	—
191	C₄×Q₁₆	—	53	2	no	—
192	C₄×Q₁₆	—	53	1	no	—
193	C₄×D₈	—	53	2	no	—
194	C₄×D₈	—	53	1	no	—
195	#367	—	53	1	no	—
196	#367	—	53	2	no	—
197	C₄⋊C₁₆	—	53	2	no	—
198	C₄⋊C₁₆	—	53	1	no	—
199	#358	—	53	2	no	—
200	#358	—	53	1	no	—
201	#358	—	53	2	no	—
202	#358	C₂×D₅₃	1	2	yes	—
203	#358	C₂×D₅₃	1	2	yes	—
204	#357	—	53	1	no	—
205	C₄×C₁₆	—	53	2	no	—
206	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
207	C₅₃	#33926	1	1	yes	characteristic, complemented, Sylow
208	C₂²×Q₈	—	212	1	no	—
209	C₂×Q₁₆	—	2	1	no	—
210	C₂×Q₁₆	—	53	1	no	—
211	C₂×Q₁₆	—	106	2	no	—
212	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
213	C₂×D₈	—	53	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	106	2	no	—
217	C₄⋊Q₈	—	106	2	no	—
218	#125	—	212	2	no	—
219	#125	—	212	1	no	—
220	C₂²⋊Q₈	—	212	2	no	—
221	C₂²⋊Q₈	—	212	1	no	—
222	C₄×Q₈	—	53	1	no	—
223	C₄×Q₈	—	106	2	no	—
224	C₄×Q₈	—	106	1	no	—
225	C₄×Q₈	—	212	1	no	—
226	C₄×D₄	—	106	2	no	—
227	C₄×D₄	—	106	1	no	—
228	SD₃₂	—	106	2	no	—
229	SD₃₂	—	106	4	no	—
230	C₂×C₁₆	—	53	2	no	—
231	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
232	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
233	#107	—	2	1	no	—
234	#107	—	53	1	no	—
235	#107	—	53	1	no	—
236	#107	—	106	2	no	—
237	#107	C₂²×D₅₃	1	2	yes	—
238	C₄⋊C₈	—	53	2	no	—
239	C₄⋊C₈	—	53	1	no	—
240	#103	—	106	2	no	—
241	#103	—	106	1	no	—
242	#102	—	106	2	no	—
243	#102	—	106	1	no	—
244	C₄×C₈	—	53	2	no	—
245	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	53	1	no	—
249	C₂×Q₈	—	106	1	no	—
250	C₂×Q₈	—	424	4	no	—
251	C₂×Q₈	—	212	2	no	—
252	C₂×D₄	—	106	1	no	—
253	C₂²×C₄	—	212	2	no	—
254	C₂²×C₄	—	212	1	no	—
255	Q₁₆	—	2	2	no	—
256	Q₁₆	—	2	2	no	—
257	Q₁₆	—	8	1	no	—
258	Q₁₆	—	106	1	no	—
259	Q₁₆	—	106	1	no	—
260	Q₁₆	—	424	2	no	—
261	D₈	—	53	2	no	—
262	D₈	—	106	1	no	—
263	C₂×C₈	—	53	2	no	—
264	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
265	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	53	2	no	—
271	C₄⋊C₄	—	53	1	no	—
272	C₄⋊C₄	—	106	1	no	—
273	C₄⋊C₄	—	106	1	no	—
274	C₄⋊C₄	—	212	2	no	—
275	C₄⋊C₄	—	212	2	no	—
276	C₄⋊C₄	—	212	2	no	—
277	C₄⋊C₄	—	212	2	no	—
278	C₄⋊C₄	—	212	1	no	—
279	C₂²⋊C₄	—	212	2	no	—
280	C₂²⋊C₄	—	212	1	no	—
281	C₄²	—	53	2	no	—
282	C₄²	—	212	2	no	—
283	C₄²	—	212	1	no	—
284	C₄²	D₄×D₅₃	1	1	yes	characteristic
285	C₁₆	—	2	1	no	—
286	C₁₆	—	2	1	no	—
287	C₁₆	—	106	2	no	—
288	C₂³	—	212	1	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	8	2	no	—
292	Q₈	—	53	4	no	—
293	Q₈	—	424	4	no	—
294	Q₈	—	212	1	no	—
295	Q₈	—	212	1	no	—
296	D₄	—	106	2	no	—
297	D₄	—	212	1	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	53	2	no	—
301	C₂×C₄	—	53	2	no	—
302	C₂×C₄	—	424	4	no	—
303	C₂×C₄	—	424	2	no	—
304	C₂×C₄	—	212	1	no	—
305	C₂×C₄	—	212	1	no	—
306	C₂×C₄	—	212	1	no	—
307	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
308	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	106	2	no	—
312	C₈	#41×D₅₃	1	1	yes	characteristic
313	C₈	#13133264	1	1	yes	characteristic
314	C₂²	—	424	1	no	—
315	C₂²	—	424	1	no	—
316	C₂²	—	212	2	no	—
317	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	53	4	no	—
322	C₄	—	106	2	no	—
323	C₄	—	424	1	no	—
324	C₄	—	424	1	no	—
325	C₄	—	424	1	no	—
326	C₄	SD₃₂×D₅₃	1	2	yes	—
327	C₄	#16811281	1	1	yes	characteristic
328	C₄	#16811429	1	1	yes	characteristic
329	C₂	—	424	2	no	—
330	C₂	#20287347	1	1	yes	characteristic
331	C₂	C₂×SD₃₂×D₅₃	1	1	yes	characteristic
332	C₂	#20287864	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	C₂×D₄₂₄	C₂³
hypercenter	#2241	D₅₃
hypocenter	C₅₃	#33926
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. #20287719	C₂	semidirect
3. #20276941	C₂	semidirect
4. #20276855	C₂	semidirect
5. #20276854	C₂	semidirect
8. C₁₆×Dic₁₀₆	C₂	semidirect
11. C₅₃×#2241	C₂	semidirect
142. Dic₁₀₆	SD₃₂	semidirect
166. #2241	D₅₃	semidirect
207. C₅₃	#33926	semidirect

Maximal subgroups

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

subgroup	index
#20287719	2
#20287567	2
#20276941	2
#20276855	2
#20276854	2
#20276759	2
#20276757	2
C₁₆×Dic₁₀₆	2
C₄×#13133162	2
#20276564	2
C₅₃×#2241	2
#33926	53