Subgroups — #27539633

Order 13568.

514 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285298	C₂	1	2	yes	maximal
2	#20276989	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276966	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276941	C₂	1	2	yes	maximal, complemented
5	#20276807	C₂	1	1	yes	maximal, characteristic
6	#20276805	C₂	1	2	yes	maximal
7	#20276803	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13133163	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
11	C₄×Q₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811177	—	2	2	no	—
13	C₂×Dic₄₂₄	C₂²	1	1	yes	characteristic
14	C₂×#13133163	C₂²	1	2	yes	—
15	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
17	#16811027	C₂²	1	2	yes	—
18	#16811001	C₂²	1	2	yes	—
19	#16810972	—	2	2	no	—
20	#16810865	C₂²	1	2	yes	—
21	#16810863	—	2	2	no	—
22	#16810861	C₂²	1	2	yes	—
23	C₄×Dic₂₁₂	C₂²	1	2	yes	—
24	C₄×#11719925	—	2	2	no	—
25	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
26	#16810449	C₂²	1	1	yes	characteristic
27	#16810448	C₂²	1	2	yes	—
28	#16810428	C₂²	1	2	yes	—
29	#16810427	C₂²	1	2	yes	—
30	#16810426	C₂²	1	2	yes	—
31	C₄×#11719908	C₂²	1	1	yes	characteristic
32	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
34	C₂×Q₃₂×C₅₃	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
36	C₅₃×#366	C₂²	1	1	yes	characteristic
37	C₅₃×#358	C₂²	1	2	yes	—
38	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
39	C₂×Dic₂₁₂	C₂³	1	2	yes	—
40	C₂×#11719925	—	2	2	no	—
41	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
42	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
43	#13133216	—	2	2	no	—
44	#13133215	—	2	2	no	—
45	#13133207	—	4	2	no	—
46	#13133197	—	2	2	no	—
47	#13133195	—	4	2	no	—
48	#13133193	—	2	2	no	—
49	C₄×Dic₁₀₆	—	2	2	no	—
50	C₄×Dic₁₀₆	—	4	2	no	—
51	#13133174	—	4	2	no	—
52	C₄²×D₅₃	D₄	1	1	yes	characteristic
53	Dic₄₂₄	—	2	2	no	—
54	#13133163	—	2	4	no	—
55	C₂×#11719908	C₂³	1	1	yes	characteristic
56	C₂×#11719908	C₂³	1	1	yes	characteristic
57	C₁₆×D₅₃	—	2	2	no	—
58	C₁₆×D₅₃	—	2	2	no	—
59	#13133151	C₂³	1	2	yes	—
60	#13133150	—	2	2	no	—
61	#13133142	—	2	2	no	—
62	#13133141	—	2	2	no	—
63	#13133140	—	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	C₂³	1	2	yes	—
68	C₄×Q₈×C₅₃	—	2	2	no	—
69	Q₃₂×C₅₃	—	2	2	no	—
70	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
71	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
72	#107×C₅₃	C₂³	1	2	yes	—
73	#103×C₅₃	—	2	2	no	—
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₂×Q₈×C₅₃	—	2	2	no	—
76	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
77	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
78	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
79	C₂×Dic₁₀₆	—	2	2	no	—
80	C₂×Dic₁₀₆	—	4	2	no	—
81	Q₁₆×C₅₃	—	2	2	no	—
82	Q₁₆×C₅₃	—	2	2	no	—
83	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
85	C₄⋊C₄×C₅₃	—	2	2	no	—
86	C₄⋊C₄×C₅₃	—	4	2	no	—
87	C₄²×C₅₃	—	4	2	no	—
88	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
89	#11719925	—	4	4	no	—
90	#11719921	—	4	2	no	—
91	#11719921	—	4	2	no	—
92	C₄⋊Dic₅₃	—	2	2	no	—
93	C₄⋊Dic₅₃	—	4	2	no	—
94	#11719919	—	4	2	no	—
95	#11719919	—	4	2	no	—
96	#11719919	—	4	2	no	—
97	#11719919	—	4	2	no	—
98	#11719919	—	4	2	no	—
99	C₄×Dic₅₃	—	4	2	no	—
100	C₄×Dic₅₃	D₈	1	1	yes	characteristic
101	C₄×Dic₅₃	D₈	1	1	yes	characteristic
102	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
103	C₂×#88322	C₂×D₄	1	1	yes	characteristic
104	C₂×#88322	C₂×D₄	1	1	yes	characteristic
105	Dic₂₁₂	—	2	2	no	—
106	Dic₂₁₂	—	2	2	no	—
107	C₈×D₅₃	—	2	1	no	—
108	C₈×D₅₃	—	2	1	no	—
109	C₈×D₅₃	—	2	2	no	—
110	C₁₆×C₅₃	—	2	1	no	—
111	C₁₆×C₅₃	#41	1	2	yes	—
112	#11719908	—	2	1	no	—
113	#11719908	#41	1	2	yes	—
114	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
115	Q₈×C₅₃	—	4	2	no	—
116	Q₈×C₅₃	—	4	2	no	—
117	C₂×C₄×C₅₃	—	4	2	no	—
118	C₂×C₄×C₅₃	—	4	2	no	—
119	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×Dic₅₃	—	4	2	no	—
123	C₂×Dic₅₃	—	4	2	no	—
124	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
125	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
126	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
127	C₄×D₅₃	—	2	1	no	—
128	C₄×D₅₃	—	2	1	no	—
129	C₄×D₅₃	—	2	2	no	—
130	C₄×D₅₃	Q₃₂	1	4	yes	complemented
131	Dic₁₀₆	—	4	2	no	—
132	Dic₁₀₆	—	4	2	no	—
133	Dic₁₀₆	—	8	4	no	—
134	C₈×C₅₃	—	2	1	no	—
135	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
136	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
137	#88322	—	2	1	no	—
138	#88322	#135	1	2	yes	—
139	#33734	—	53	1	no	maximal, Sylow
140	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
141	C₂×D₅₃	—	2	1	no	—
142	C₂×D₅₃	—	2	1	no	—
143	C₂×D₅₃	C₂×Q₃₂	1	2	yes	—
144	C₄×C₅₃	—	2	1	no	—
145	C₄×C₅₃	—	8	2	no	—
146	C₄×C₅₃	—	8	2	no	—
147	C₄×C₅₃	C₂×Q₃₂	1	2	yes	—
148	C₄×C₅₃	#532	1	1	yes	characteristic
149	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
150	Dic₅₃	—	2	1	no	—
151	Dic₅₃	—	8	2	no	—
152	Dic₅₃	—	8	2	no	—
153	Dic₅₃	C₂×Q₃₂	1	2	yes	—
154	Dic₅₃	#508	1	2	yes	—
155	#3170	—	53	2	no	—
156	#2252	—	53	1	no	—
157	#2247	—	53	1	no	—
158	#2241	—	53	2	no	—
159	#2216	—	53	1	no	—
160	#2216	—	53	1	no	—
161	#2216	—	53	2	no	—
162	C₄×Q₃₂	—	53	2	no	—
163	C₄×Q₃₂	—	53	1	no	—
164	C₄×Q₃₂	D₅₃	1	1	yes	characteristic, complemented
165	C₂×C₄×C₁₆	—	53	1	no	—
166	C₂×C₅₃	#3169	1	1	yes	characteristic
167	C₂×C₅₃	C₂²×Q₃₂	1	1	yes	characteristic
168	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
169	D₅₃	—	2	2	no	—
170	#533	—	106	2	no	—
171	C₂×Q₃₂	—	53	2	no	—
172	C₂×Q₃₂	—	53	1	no	—
173	C₂×Q₃₂	C₂×D₅₃	1	1	yes	characteristic
174	C₂²×C₁₆	—	53	1	no	—
175	C₂²×C₁₆	—	53	1	no	—
176	#500	—	53	2	no	—
177	#494	—	53	2	no	—
178	#487	—	106	2	no	—
179	#467	—	53	2	no	—
180	#467	—	53	2	no	—
181	#467	—	106	2	no	—
182	C₄×Q₁₆	—	53	2	no	—
183	C₄×Q₁₆	—	106	2	no	—
184	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
185	C₂×C₄×C₈	—	53	1	no	—
186	#366	—	53	1	no	—
187	#366	—	53	2	no	—
188	#366	C₂×D₅₃	1	1	yes	characteristic
189	#358	—	53	2	no	—
190	#358	—	53	2	no	—
191	#358	—	53	2	no	—
192	#358	C₂×D₅₃	1	2	yes	—
193	C₄×C₁₆	—	53	1	no	—
194	C₄×C₁₆	—	53	1	no	—
195	C₄×C₁₆	—	53	1	no	—
196	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
197	C₅₃	#33734	1	1	yes	characteristic, complemented, Sylow
198	C₂×Q₁₆	—	53	2	no	—
199	C₂×Q₁₆	—	106	2	no	—
200	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
201	C₂²×C₈	—	53	1	no	—
202	C₂²×C₈	—	53	1	no	—
203	C₄⋊Q₈	—	106	2	no	—
204	C₄⋊Q₈	—	106	2	no	—
205	#125	—	212	2	no	—
206	C₂²⋊Q₈	—	106	2	no	—
207	C₂²⋊Q₈	—	106	2	no	—
208	C₂²⋊Q₈	—	212	2	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	—	106	2	no	—
211	C₄×Q₈	—	212	2	no	—
212	#117	—	212	2	no	—
213	C₂×C₄²	—	53	1	no	—
214	Q₃₂	—	2	2	no	—
215	Q₃₂	—	106	4	no	—
216	Q₃₂	—	106	2	no	—
217	C₂×C₁₆	—	53	1	no	—
218	C₂×C₁₆	—	53	1	no	—
219	C₂×C₁₆	—	106	2	no	—
220	C₂×C₁₆	—	106	2	no	—
221	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
222	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
223	#107	—	53	2	no	—
224	#107	—	106	2	no	—
225	#107	C₂²×D₅₃	1	2	yes	—
226	#103	—	2	2	no	—
227	#103	—	106	2	no	—
228	#103	—	106	2	no	—
229	#103	—	106	2	no	—
230	C₄×C₈	—	53	1	no	—
231	C₄×C₈	—	53	1	no	—
232	C₄×C₈	—	53	1	no	—
233	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	106	2	no	—
236	C₂×Q₈	—	212	2	no	—
237	C₂²×C₄	—	53	1	no	—
238	C₂²×C₄	—	53	1	no	—
239	C₂²×C₄	—	53	1	no	—
240	Q₁₆	—	2	2	no	—
241	Q₁₆	—	2	2	no	—
242	Q₁₆	—	106	2	no	—
243	Q₁₆	—	106	2	no	—
244	Q₁₆	—	212	4	no	—
245	C₂×C₈	—	53	1	no	—
246	C₂×C₈	—	53	1	no	—
247	C₂×C₈	—	106	2	no	—
248	C₂×C₈	—	106	1	no	—
249	C₂×C₈	—	106	1	no	—
250	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
251	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	106	2	no	—
255	C₄⋊C₄	—	212	2	no	—
256	C₄⋊C₄	—	212	2	no	—
257	C₄⋊C₄	—	212	2	no	—
258	C₄⋊C₄	—	212	2	no	—
259	C₄⋊C₄	—	212	2	no	—
260	C₄⋊C₄	—	212	2	no	—
261	C₂²⋊C₄	—	212	2	no	—
262	C₂²⋊C₄	—	212	2	no	—
263	C₄²	—	4	2	no	—
264	C₄²	—	53	1	no	—
265	C₄²	—	53	1	no	—
266	C₄²	—	53	1	no	—
267	C₄²	—	212	2	no	—
268	C₄²	D₄×D₅₃	1	1	yes	characteristic
269	C₁₆	—	2	1	no	—
270	C₁₆	—	53	2	no	—
271	C₁₆	—	106	1	no	—
272	C₁₆	#11719945	1	2	yes	—
273	C₂³	—	53	1	no	—
274	Q₈	—	4	2	no	—
275	Q₈	—	4	2	no	—
276	Q₈	—	424	4	no	—
277	Q₈	—	212	2	no	—
278	Q₈	—	212	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	53	1	no	—
282	C₂×C₄	—	53	1	no	—
283	C₂×C₄	—	53	4	no	—
284	C₂×C₄	—	53	1	no	—
285	C₂×C₄	—	106	2	no	—
286	C₂×C₄	—	106	1	no	—
287	C₂×C₄	—	106	1	no	—
288	C₂×C₄	—	212	2	no	—
289	C₂×C₄	—	212	2	no	—
290	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
291	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
292	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	53	2	no	—
295	C₈	—	106	1	no	—
296	C₈	#41×D₅₃	1	1	yes	characteristic
297	C₈	C₂×#11719945	1	1	yes	characteristic
298	C₂²	—	53	2	no	—
299	C₂²	—	106	1	no	—
300	C₂²	—	106	1	no	—
301	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	53	2	no	—
306	C₄	—	53	2	no	—
307	C₄	—	106	1	no	—
308	C₄	—	424	2	no	—
309	C₄	—	424	2	no	—
310	C₄	Q₃₂×D₅₃	1	2	yes	—
311	C₄	#16811170	1	1	yes	characteristic
312	C₄	#135×D₅₃	1	1	yes	characteristic
313	C₂	—	106	2	no	—
314	C₂	#20285291	1	1	yes	characteristic
315	C₂	C₂×Q₃₂×D₅₃	1	1	yes	characteristic
316	C₂	#508×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₄×Q₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂²×D₅₃	C₂×D₈
hypercenter	C₄×Q₃₂	D₅₃
hypocenter	C₅₃	#33734
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
2. #20276989	C₂	semidirect
3. #20276966	C₂	semidirect
4. #20276941	C₂	semidirect
8. C₄×Dic₄₂₄	C₂	semidirect
9. C₄×#13133163	C₂	semidirect
11. C₄×Q₃₂×C₅₃	C₂	semidirect
130. C₄×D₅₃	Q₃₂	semidirect
164. C₄×Q₃₂	D₅₃	semidirect
197. C₅₃	#33734	semidirect

Maximal subgroups

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

subgroup	index
#20285298	2
#20276989	2
#20276966	2
#20276941	2
#20276807	2
#20276805	2
#20276803	2
C₄×Dic₄₂₄	2
C₄×#13133163	2
C₄×C₁₆×D₅₃	2
C₄×Q₃₂×C₅₃	2
#33734	53