Subgroups — #27366734

Order 13568.

586 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285229	C₂	1	1	yes	maximal, characteristic
2	#20284755	C₂	1	2	yes	maximal
3	C₂×#16810753	C₂	1	1	yes	maximal, characteristic
4	#20274244	C₂	1	2	yes	maximal, complemented
5	#20274234	C₂	1	2	yes	maximal, complemented
6	#20274222	C₂	1	4	yes	maximal, complemented
7	#20273370	C₂	1	2	yes	maximal
8	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic
9	#16811180	C₂²	1	2	yes	—
10	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
11	#16811028	C₂²	1	1	yes	characteristic
12	#16811028	C₂²	1	1	yes	characteristic
13	#16811025	—	2	1	no	—
14	#16811025	—	2	1	no	—
15	#16811021	C₂²	1	2	yes	—
16	#16810959	—	2	2	no	—
17	#16810919	C₂²	1	4	yes	—
18	#16810834	C₂²	1	4	yes	—
19	#16810781	—	2	2	no	—
20	#16810753	—	2	1	no	—
21	#16810753	—	2	1	no	—
22	#16810753	C₂²	1	2	yes	—
23	#16810753	C₂²	1	2	yes	—
24	C₄×Dic₂₁₂	—	2	4	no	—
25	#16810661	—	2	1	no	—
26	#16810661	—	2	1	no	—
27	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
28	C₂×#13133146	—	2	1	no	—
29	C₂×#13133127	—	2	1	no	—
30	#16810371	C₂²	1	4	yes	—
31	#16810370	C₂²	1	4	yes	—
32	C₅₃×#431	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₅₃×#335	C₂²	1	2	yes	—
35	C₅₃×#334	C₂²	1	2	yes	—
36	C₂²×Dic₁₀₆	D₄	1	1	yes	characteristic
37	C₂×Dic₂₁₂	—	2	4	no	—
38	C₂²×#88322	—	2	1	no	—
39	#13133217	C₂³	1	2	yes	—
40	#13133217	C₂³	1	2	yes	—
41	#13133209	—	2	2	no	—
42	#13133196	—	2	2	no	—
43	#13133196	—	2	2	no	—
44	C₄×Dic₁₀₆	—	2	1	no	—
45	C₄×Dic₁₀₆	—	2	1	no	—
46	C₄×Dic₁₀₆	—	2	2	no	—
47	C₄×Dic₁₀₆	—	2	2	no	—
48	C₄×Dic₁₀₆	D₄	1	2	yes	—
49	C₄×Dic₁₀₆	C₂³	1	2	yes	—
50	#13133176	—	2	2	no	—
51	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
52	C₂×#11719919	—	2	1	no	—
53	C₂×C₄×Dic₅₃	—	2	1	no	—
54	#13133151	—	2	1	no	—
55	#13133151	—	2	1	no	—
56	#13133151	—	2	1	no	—
57	#13133151	—	2	1	no	—
58	#13133151	—	2	4	no	—
59	#13133149	—	2	2	no	—
60	#13133149	—	2	2	no	—
61	C₄⋊#88322	—	2	1	no	—
62	C₄⋊#88322	C₂³	1	2	yes	—
63	#13133146	—	2	2	no	—
64	#13133146	—	2	2	no	—
65	#13133142	—	2	4	no	—
66	#13133142	—	2	4	no	—
67	#13133127	—	2	2	no	—
68	#13133127	—	2	2	no	—
69	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
70	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
71	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
72	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
73	C₂²⋊C₈×C₅₃	—	2	2	no	—
74	C₄×C₈×C₅₃	—	2	2	no	—
75	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
76	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
82	C₂²×Dic₅₃	—	2	1	no	—
83	C₂²×Dic₅₃	—	2	1	no	—
84	C₂×Dic₁₀₆	—	2	1	no	—
85	C₂×Dic₁₀₆	—	2	2	no	—
86	C₂×Dic₁₀₆	—	2	2	no	—
87	C₂×Dic₁₀₆	—	4	2	no	—
88	C₂×Dic₁₀₆	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₅₃	—	2	2	no	—
90	C₂×C₈×C₅₃	—	2	4	no	—
91	C₂×C₈×C₅₃	Q₁₆	1	4	yes	complemented
92	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₅₃	#41	1	1	yes	characteristic
99	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
100	C₂²⋊Dic₅₃	—	4	2	no	—
101	C₄⋊Dic₅₃	—	2	1	no	—
102	C₄⋊Dic₅₃	—	2	2	no	—
103	C₄⋊Dic₅₃	—	2	2	no	—
104	C₄⋊Dic₅₃	—	2	2	no	—
105	C₄⋊Dic₅₃	—	2	2	no	—
106	C₄⋊Dic₅₃	C₂×D₄	1	2	yes	—
107	#11719919	—	2	2	no	—
108	#11719919	—	2	2	no	—
109	#11719919	—	4	2	no	—
110	#11719919	—	4	2	no	—
111	C₄×Dic₅₃	—	2	2	no	—
112	C₄×Dic₅₃	—	2	2	no	—
113	C₄×Dic₅₃	—	4	2	no	—
114	C₂×#88322	—	2	2	no	—
115	C₂×#88322	—	4	2	no	—
116	Dic₂₁₂	—	4	8	no	—
117	C₂³×C₅₃	#121	1	1	yes	characteristic
118	C₂×C₄×C₅₃	—	2	2	no	—
119	C₂×C₄×C₅₃	—	4	1	no	—
120	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
121	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
122	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
124	C₂×C₄×C₅₃	#137	1	2	yes	—
125	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
128	C₂×Dic₅₃	—	2	2	no	—
129	C₂×Dic₅₃	—	2	2	no	—
130	C₂×Dic₅₃	—	4	2	no	—
131	C₂×Dic₅₃	—	4	2	no	—
132	C₂×Dic₅₃	—	4	2	no	—
133	C₂×Dic₅₃	—	4	2	no	—
134	Dic₁₀₆	—	2	4	no	—
135	Dic₁₀₆	—	4	2	no	—
136	Dic₁₀₆	—	4	4	no	—
137	Dic₁₀₆	—	8	2	no	—
138	C₈×C₅₃	—	2	1	no	—
139	C₈×C₅₃	—	2	1	no	—
140	C₈×C₅₃	—	4	4	no	—
141	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
142	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
143	#88322	—	4	2	no	—
144	#19150	—	53	1	no	maximal, Sylow
145	C₂²×C₅₃	—	2	1	no	—
146	C₂²×C₅₃	—	2	1	no	—
147	C₂²×C₅₃	#462	1	2	yes	—
148	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
149	C₄×C₅₃	—	2	1	no	—
150	C₄×C₅₃	—	2	1	no	—
151	C₄×C₅₃	—	2	1	no	—
152	C₄×C₅₃	#462	1	2	yes	—
153	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
154	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
155	C₄×C₅₃	#575	1	2	yes	—
156	Dic₅₃	—	4	2	no	—
157	Dic₅₃	—	4	2	no	—
158	Dic₅₃	—	8	2	no	—
159	Dic₅₃	—	8	2	no	—
160	#3160	—	53	1	no	—
161	#3104	—	53	2	no	—
162	C₂×#446	—	53	1	no	—
163	#1696	—	53	2	no	—
164	#1694	—	53	2	no	—
165	#1692	—	53	4	no	—
166	#1564	D₅₃	1	1	yes	characteristic
167	#1542	—	53	2	no	—
168	C₂×C₅₃	—	2	2	no	—
169	C₂×C₅₃	C₂×#462	1	1	yes	characteristic
170	C₂×C₅₃	#3054	1	1	yes	characteristic
171	C₂×C₅₃	#3067	1	1	yes	characteristic
172	#533	—	53	2	no	—
173	C₂×C₄×Q₈	—	53	1	no	—
174	#500	—	53	1	no	—
175	#500	—	53	1	no	—
176	#499	—	106	1	no	—
177	#499	—	106	1	no	—
178	#498	—	53	2	no	—
179	#484	—	106	2	no	—
180	#477	—	53	4	no	—
181	#462	—	53	4	no	—
182	#451	—	106	2	no	—
183	#446	—	53	2	no	—
184	#446	—	53	2	no	—
185	#446	—	106	1	no	—
186	#446	—	106	1	no	—
187	C₄×Q₁₆	—	106	4	no	—
188	#431	C₂×D₅₃	1	2	yes	—
189	#427	—	106	1	no	—
190	#427	—	106	1	no	—
191	C₂×C₄⋊C₈	—	53	1	no	—
192	C₂×C₄⋊C₈	—	106	1	no	—
193	C₂×#97	—	106	1	no	—
194	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
195	#335	C₂×D₅₃	1	2	yes	—
196	#334	C₂×D₅₃	1	2	yes	—
197	#333	—	53	4	no	—
198	#332	—	53	4	no	—
199	C₅₃	#19150	1	1	yes	characteristic, complemented, Sylow
200	C₂²×Q₈	—	53	1	no	—
201	C₂×Q₁₆	—	106	4	no	—
202	C₂²×C₈	—	106	1	no	—
203	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
204	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
205	C₄⋊Q₈	—	53	2	no	—
206	C₄⋊Q₈	—	53	2	no	—
207	#125	—	106	2	no	—
208	C₂²⋊Q₈	—	106	2	no	—
209	C₂²⋊Q₈	—	106	2	no	—
210	C₄×Q₈	—	53	2	no	—
211	C₄×Q₈	—	53	2	no	—
212	C₄×Q₈	—	106	1	no	—
213	C₄×Q₈	—	106	1	no	—
214	C₄×Q₈	—	106	2	no	—
215	C₄×Q₈	—	106	2	no	—
216	#117	—	106	2	no	—
217	C₂×C₄⋊C₄	—	53	1	no	—
218	C₂×C₄⋊C₄	—	106	1	no	—
219	C₂×C₄²	—	106	1	no	—
220	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
221	#107	—	106	4	no	—
222	#107	—	106	1	no	—
223	#107	—	106	1	no	—
224	#107	—	106	1	no	—
225	#107	—	106	1	no	—
226	#106	—	106	2	no	—
227	#106	—	106	2	no	—
228	C₄⋊C₈	—	53	2	no	—
229	C₄⋊C₈	—	106	2	no	—
230	C₄⋊C₈	—	106	2	no	—
231	C₄⋊C₈	—	106	1	no	—
232	C₄⋊C₈	C₂²×D₅₃	1	4	yes	—
233	#103	—	106	4	no	—
234	#103	—	106	4	no	—
235	C₂²⋊C₈	—	2	2	no	—
236	#97	—	106	2	no	—
237	#97	—	106	2	no	—
238	C₄×C₈	—	2	2	no	—
239	C₄×C₈	D₂₁₂	1	1	yes	characteristic
240	C₄×C₈	D₂₁₂	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
243	C₂×Q₈	—	53	2	no	—
244	C₂×Q₈	—	106	1	no	—
245	C₂×Q₈	—	106	2	no	—
246	C₂×Q₈	—	106	2	no	—
247	C₂×Q₈	—	212	2	no	—
248	C₂²×C₄	—	106	1	no	—
249	C₂²×C₄	—	106	1	no	—
250	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
251	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
252	C₂²×C₄	#11719949	1	1	yes	characteristic
253	Q₁₆	—	212	8	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	106	2	no	—
257	C₂×C₈	—	212	2	no	—
258	C₂×C₈	Dic₂₁₂	1	4	yes	complemented
259	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
260	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
261	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
262	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
263	C₄⋊C₄	—	53	2	no	—
264	C₄⋊C₄	—	106	1	no	—
265	C₄⋊C₄	—	106	2	no	—
266	C₄⋊C₄	—	106	2	no	—
267	C₄⋊C₄	—	106	2	no	—
268	C₄⋊C₄	—	106	2	no	—
269	C₄⋊C₄	—	106	2	no	—
270	C₄⋊C₄	—	106	2	no	—
271	C₄⋊C₄	—	212	2	no	—
272	C₄⋊C₄	—	212	2	no	—
273	C₂²⋊C₄	—	212	2	no	—
274	C₄²	—	106	2	no	—
275	C₄²	—	106	2	no	—
276	C₄²	—	212	2	no	—
277	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
278	C₄²	D₄×D₅₃	1	1	yes	characteristic
279	C₄²	#11719949	1	1	yes	characteristic
280	C₄²	C₂³×D₅₃	1	1	yes	characteristic
281	C₂³	#13133192	1	1	yes	characteristic
282	Q₈	—	106	4	no	—
283	Q₈	—	424	2	no	—
284	Q₈	—	212	2	no	—
285	Q₈	—	212	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	106	2	no	—
289	C₂×C₄	—	106	2	no	—
290	C₂×C₄	—	212	2	no	—
291	C₂×C₄	—	212	2	no	—
292	C₂×C₄	—	212	2	no	—
293	C₂×C₄	—	212	2	no	—
294	C₂×C₄	#13133192	1	1	yes	characteristic
295	C₂×C₄	#13133192	1	1	yes	characteristic
296	C₂×C₄	#13133192	1	1	yes	characteristic
297	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
298	C₂×C₄	#13133250	1	2	yes	—
299	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
300	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
301	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	4	4	no	—
305	C₈	—	212	2	no	—
306	C₈	C₂×Dic₂₁₂	1	2	yes	—
307	C₈	C₂×Dic₂₁₂	1	2	yes	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	#16810834	1	2	yes	—
311	C₂²	C₂×#13133192	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	424	2	no	—
316	C₄	—	424	2	no	—
317	C₄	—	212	2	no	—
318	C₄	—	212	2	no	—
319	C₄	#16810834	1	2	yes	—
320	C₄	C₂²×Dic₂₁₂	1	1	yes	characteristic
321	C₄	C₂×#13133250	1	1	yes	characteristic
322	C₄	#16811420	1	2	yes	—
323	C₂	—	2	2	no	—
324	C₂	C₂×#16810834	1	1	yes	characteristic
325	C₂	#20284325	1	1	yes	characteristic
326	C₂	#20284436	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133192
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1564	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂³×C₅₃	#121
hypercenter	#1564	D₅₃
hypocenter	C₅₃	#19150
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
4. #20274244	C₂	semidirect
5. #20274234	C₂	semidirect
6. #20274222	C₂	semidirect
91. C₂×C₈×C₅₃	Q₁₆	semidirect
199. C₅₃	#19150	semidirect
258. C₂×C₈	Dic₂₁₂	semidirect

Maximal subgroups

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

subgroup	index
#20285229	2
#20284755	2
C₂×#16810753	2
#20274244	2
#20274234	2
#20274222	2
#20273370	2
C₅₃×#1564	2
#19150	53