Subgroups — #27539611

Order 13568.

514 conjugacy classes of subgroups.

The subgroup lattice has 308 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
2	#500×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#20276999	C₂	1	4	yes	maximal, complemented
4	#20276997	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276995	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276988	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276500	C₂	1	4	yes	maximal
8	C₄×#13133156	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#2252	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811178	—	2	1	no	—
11	C₄⋊Q₈×D₅₃	—	2	1	no	—
12	C₂×#13133156	C₂²	1	2	yes	—
13	#16811028	C₂²	1	1	yes	characteristic
14	#16811028	C₂²	1	1	yes	characteristic
15	#16811027	—	2	1	no	—
16	#16811027	C₂²	1	2	yes	—
17	#16811026	—	2	1	no	—
18	#16811024	—	2	1	no	—
19	#16811024	C₂²	1	2	yes	—
20	#16811019	—	2	1	no	—
21	#16810660	—	2	1	no	—
22	#16810660	C₂²	1	2	yes	—
23	#107×D₅₃	—	2	1	no	—
24	#107×D₅₃	C₂²	1	2	yes	—
25	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
26	#16810451	C₂²	1	4	yes	—
27	#16810450	C₂²	1	4	yes	—
28	#16810448	C₂²	1	4	yes	—
29	#16810398	C₂²	1	2	yes	—
30	C₄×#11719908	C₂²	1	1	yes	characteristic
31	C₅₃×#500	C₂²	1	1	yes	characteristic
32	C₅₃×#500	C₂²	1	1	yes	characteristic
33	C₅₃×#366	C₂²	1	4	yes	—
34	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×Q₈×D₅₃	—	4	1	no	—
36	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
37	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
38	#13133217	—	2	1	no	—
39	#13133217	—	2	1	no	—
40	#13133216	—	2	2	no	—
41	#13133216	—	4	1	no	—
42	#13133215	—	4	1	no	—
43	#13133208	—	2	2	no	—
44	#13133197	—	4	1	no	—
45	#13133193	—	4	1	no	—
46	Q₈×Dic₅₃	—	4	1	no	—
47	#13133182	—	4	1	no	—
48	#13133175	—	2	2	no	—
49	C₄⋊C₄×D₅₃	—	2	2	no	—
50	C₄⋊C₄×D₅₃	—	4	1	no	—
51	C₄²×D₅₃	D₄	1	1	yes	characteristic
52	#13133156	Q₈	1	8	yes	complemented
53	C₂×#11719908	C₂³	1	2	yes	—
54	#13133151	—	2	1	no	—
55	#13133151	—	2	1	no	—
56	#13133151	C₂³	1	2	yes	—
57	#13133151	C₂³	1	2	yes	—
58	#13133150	—	2	2	no	—
59	#13133150	—	2	2	no	—
60	#13133148	—	2	2	no	—
61	#13133148	—	2	2	no	—
62	C₄×#88322	D₄	1	1	yes	characteristic
63	C₈×Dic₅₃	D₄	1	1	yes	characteristic
64	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
65	C₄⋊Q₈×C₅₃	—	2	1	no	—
66	C₄⋊Q₈×C₅₃	—	2	1	no	—
67	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
68	#107×C₅₃	—	2	1	no	—
69	#107×C₅₃	—	2	1	no	—
70	#107×C₅₃	C₂³	1	2	yes	—
71	#107×C₅₃	C₂³	1	2	yes	—
72	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
73	C₂×Q₈×C₅₃	—	4	1	no	—
74	C₂×Q₈×C₅₃	—	4	1	no	—
75	Q₈×D₅₃	—	8	4	no	—
76	C₂×C₄×D₅₃	—	4	2	no	—
77	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
78	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
79	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
80	C₂×Dic₁₀₆	—	4	1	no	—
81	C₂×Dic₁₀₆	—	4	1	no	—
82	C₂×Dic₁₀₆	—	4	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	—	2	2	no	—
87	C₄⋊C₄×C₅₃	—	4	1	no	—
88	C₄⋊C₄×C₅₃	—	4	1	no	—
89	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
90	#11719921	—	4	2	no	—
91	C₄⋊Dic₅₃	—	2	2	no	—
92	C₄⋊Dic₅₃	—	2	2	no	—
93	C₄⋊Dic₅₃	—	4	1	no	—
94	C₄⋊Dic₅₃	—	4	1	no	—
95	C₄⋊Dic₅₃	—	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	C₈×D₅₃	Q₁₆	1	4	yes	—
106	C₈×D₅₃	C₂×Q₈	1	4	yes	—
107	C₁₆×C₅₃	C₂×Q₈	1	4	yes	complemented
108	#11719908	C₂×Q₈	1	4	yes	complemented
109	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
110	Q₈×C₅₃	—	8	2	no	—
111	Q₈×C₅₃	—	8	2	no	—
112	C₂×C₄×C₅₃	—	4	2	no	—
113	C₂×C₄×C₅₃	—	4	2	no	—
114	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×Dic₅₃	—	4	2	no	—
118	C₂×Dic₅₃	—	4	2	no	—
119	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
120	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
121	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
122	C₄×D₅₃	—	2	2	no	—
123	C₄×D₅₃	—	2	2	no	—
124	C₄×D₅₃	—	8	4	no	—
125	C₄×D₅₃	C₄⋊Q₈	1	2	yes	—
126	C₄×D₅₃	C₂×Q₁₆	1	2	yes	—
127	Dic₁₀₆	—	8	4	no	—
128	Dic₁₀₆	—	8	2	no	—
129	Dic₁₀₆	—	8	2	no	—
130	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
131	C₈×C₅₃	C₂²×Q₈	1	2	yes	—
132	#88322	C₄⋊Q₈	1	2	yes	—
133	#88322	C₂×Q₁₆	1	2	yes	—
134	#33731	—	53	1	no	maximal, Sylow
135	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
136	C₂×D₅₃	—	2	1	no	—
137	C₂×D₅₃	—	2	1	no	—
138	C₂×D₅₃	#500	1	2	yes	—
139	C₄×C₅₃	—	8	2	no	—
140	C₄×C₅₃	—	8	2	no	—
141	C₄×C₅₃	#509	1	2	yes	—
142	C₄×C₅₃	#510	1	2	yes	—
143	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
144	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
145	Dic₅₃	—	2	1	no	—
146	Dic₅₃	—	2	1	no	—
147	Dic₅₃	—	8	2	no	—
148	Dic₅₃	—	8	2	no	—
149	Dic₅₃	#500	1	2	yes	—
150	#3160	—	53	1	no	—
151	C₂×#500	—	53	1	no	—
152	#2255	—	53	4	no	—
153	#2254	—	53	1	no	—
154	#2254	—	53	1	no	—
155	#2252	—	53	1	no	—
156	#2252	D₅₃	1	1	yes	characteristic, complemented
157	#2159	—	53	4	no	—
158	C₄×M₃₂	—	53	1	no	—
159	C₂×C₅₃	C₂×#500	1	1	yes	characteristic
160	C₂×C₅₃	C₂×#509	1	1	yes	characteristic
161	C₂×C₅₃	C₂×#510	1	1	yes	characteristic
162	D₅₃	—	2	2	no	—
163	#533	—	106	1	no	—
164	C₂×C₄⋊Q₈	—	106	1	no	—
165	C₂×M₃₂	—	53	2	no	—
166	#500	—	53	2	no	—
167	#500	—	53	1	no	—
168	#500	—	53	1	no	—
169	#500	—	106	1	no	—
170	#500	—	106	1	no	—
171	#500	C₂×D₅₃	1	1	yes	characteristic
172	#500	C₂×D₅₃	1	1	yes	characteristic
173	#499	—	53	2	no	—
174	#499	—	106	1	no	—
175	#498	—	106	1	no	—
176	#427	—	53	2	no	—
177	#427	—	106	1	no	—
178	C₂×#107	—	53	2	no	—
179	C₂×#107	—	106	1	no	—
180	C₂×C₄×C₈	—	53	1	no	—
181	#367	—	53	4	no	—
182	#367	—	53	4	no	—
183	#366	—	53	4	no	—
184	#366	C₂×D₅₃	1	4	yes	—
185	#346	—	53	2	no	—
186	C₄×C₁₆	—	53	1	no	—
187	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
188	C₅₃	#33731	1	1	yes	characteristic, complemented, Sylow
189	C₂²×Q₈	—	212	1	no	—
190	C₂²×C₈	—	53	1	no	—
191	C₂²×C₈	—	53	1	no	—
192	C₄⋊Q₈	—	2	1	no	—
193	C₄⋊Q₈	—	2	1	no	—
194	C₄⋊Q₈	—	106	2	no	—
195	C₄⋊Q₈	—	106	1	no	—
196	C₄⋊Q₈	—	106	1	no	—
197	C₄⋊Q₈	—	212	1	no	—
198	C₄⋊Q₈	—	212	1	no	—
199	#125	—	106	2	no	—
200	C₂²⋊Q₈	—	212	1	no	—
201	C₂²⋊Q₈	—	212	1	no	—
202	C₄×Q₈	—	212	1	no	—
203	C₄×Q₈	—	212	1	no	—
204	#117	—	106	2	no	—
205	C₂×C₄⋊C₄	—	106	2	no	—
206	C₂×C₄⋊C₄	—	212	1	no	—
207	C₂×C₄²	—	53	1	no	—
208	M₃₂	—	53	8	no	—
209	C₂×C₁₆	—	53	2	no	—
210	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
211	#107	—	2	1	no	—
212	#107	—	2	1	no	—
213	#107	—	53	2	no	—
214	#107	—	53	2	no	—
215	#107	—	106	2	no	—
216	#107	—	106	2	no	—
217	#107	—	106	1	no	—
218	#107	—	106	1	no	—
219	#107	C₂²×D₅₃	1	2	yes	—
220	#107	C₂²×D₅₃	1	2	yes	—
221	#106	—	106	2	no	—
222	#106	—	106	2	no	—
223	C₄×C₈	—	53	1	no	—
224	C₄×C₈	—	53	1	no	—
225	C₄×C₈	—	53	1	no	—
226	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	4	1	no	—
229	C₂×Q₈	—	424	4	no	—
230	C₂×Q₈	—	212	2	no	—
231	C₂×Q₈	—	212	1	no	—
232	C₂×Q₈	—	212	1	no	—
233	C₂²×C₄	—	53	1	no	—
234	C₂²×C₄	—	53	1	no	—
235	C₂²×C₄	—	53	1	no	—
236	C₂²×C₄	—	212	2	no	—
237	C₂×C₈	—	53	1	no	—
238	C₂×C₈	—	53	1	no	—
239	C₂×C₈	—	53	4	no	—
240	C₂×C₈	—	53	4	no	—
241	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
242	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	4	1	no	—
246	C₄⋊C₄	—	4	1	no	—
247	C₄⋊C₄	—	106	2	no	—
248	C₄⋊C₄	—	106	2	no	—
249	C₄⋊C₄	—	212	2	no	—
250	C₄⋊C₄	—	212	2	no	—
251	C₄⋊C₄	—	212	2	no	—
252	C₄⋊C₄	—	212	2	no	—
253	C₄⋊C₄	—	212	1	no	—
254	C₄⋊C₄	—	212	1	no	—
255	C₂²⋊C₄	—	212	2	no	—
256	C₄²	—	53	1	no	—
257	C₄²	—	53	1	no	—
258	C₄²	—	53	1	no	—
259	C₄²	—	212	2	no	—
260	C₄²	D₄×D₅₃	1	1	yes	characteristic
261	C₁₆	—	53	4	no	—
262	C₁₆	Q₈×D₅₃	1	4	yes	complemented
263	C₂³	—	53	1	no	—
264	Q₈	—	8	2	no	—
265	Q₈	—	8	2	no	—
266	Q₈	—	424	4	no	—
267	Q₈	—	424	2	no	—
268	Q₈	—	424	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	53	1	no	—
272	C₂×C₄	—	53	2	no	—
273	C₂×C₄	—	53	2	no	—
274	C₂×C₄	—	53	1	no	—
275	C₂×C₄	—	53	1	no	—
276	C₂×C₄	—	106	2	no	—
277	C₂×C₄	—	106	2	no	—
278	C₂×C₄	—	424	4	no	—
279	C₂×C₄	—	212	2	no	—
280	C₂×C₄	—	212	2	no	—
281	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
282	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
283	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
284	C₈	—	53	2	no	—
285	C₈	—	53	2	no	—
286	C₈	Q₁₆×D₅₃	1	2	yes	—
287	C₈	C₂×Q₈×D₅₃	1	2	yes	—
288	C₂²	—	53	2	no	—
289	C₂²	—	106	1	no	—
290	C₂²	—	106	1	no	—
291	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
292	C₄	—	8	2	no	—
293	C₄	—	8	2	no	—
294	C₄	—	53	2	no	—
295	C₄	—	106	1	no	—
296	C₄	—	106	1	no	—
297	C₄	—	424	2	no	—
298	C₄	—	424	2	no	—
299	C₄	#16811057	1	2	yes	—
300	C₄	#16811065	1	2	yes	—
301	C₄	C₄⋊Q₈×D₅₃	1	1	yes	characteristic
302	C₄	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
303	C₂	—	106	2	no	—
304	C₂	#500×D₅₃	1	1	yes	characteristic
305	C₂	C₂×#16811057	1	1	yes	characteristic
306	C₂	C₂×#16811065	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×#2252	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂²×D₅₃	C₂×D₈
hypercenter	#2252	D₅₃
hypocenter	C₅₃	#33731
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

Complements

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

N	H ≅ G/N	type
3. #20276999	C₂	semidirect
4. #20276997	C₂	semidirect
5. #20276995	C₂	semidirect
6. #20276988	C₂	semidirect
9. C₅₃×#2252	C₂	semidirect
52. #13133156	Q₈	semidirect
107. C₁₆×C₅₃	C₂×Q₈	semidirect
108. #11719908	C₂×Q₈	semidirect
156. #2252	D₅₃	semidirect
188. C₅₃	#33731	semidirect
262. C₁₆	Q₈×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#20285226	2
#500×D₅₃	2
#20276999	2
#20276997	2
#20276995	2
#20276988	2
#20276500	2
C₄×#13133156	2
C₅₃×#2252	2
#33731	53