Subgroups — #27541688

Order 13568.

502 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287147	C₂	1	1	yes	maximal, characteristic, complemented
2	#20287064	C₂	1	1	yes	maximal, characteristic
3	#20276990	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276916	C₂	1	2	yes	maximal, complemented
5	#20276904	C₂	1	2	yes	maximal, complemented
6	#20276857	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276852	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276825	C₂	1	2	yes	maximal, complemented
9	#20276548	C₂	1	1	yes	maximal, characteristic, complemented
10	#20276502	C₂	1	2	yes	maximal, complemented
11	C₄×SD₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811346	—	2	1	no	—
13	#16811278	—	2	1	no	—
14	C₂×#13133156	C₂²	1	2	yes	—
15	#16811028	C₂²	1	1	yes	characteristic
16	#16811028	C₂²	1	1	yes	characteristic
17	#16810963	—	2	2	no	—
18	#16810935	—	2	2	no	—
19	#16810908	—	2	1	no	—
20	#16810903	—	2	1	no	—
21	#16810861	C₂²	1	2	yes	—
22	#16810856	C₂²	1	2	yes	—
23	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
24	#16810660	C₂²	1	2	yes	—
25	#107×D₅₃	C₂²	1	2	yes	—
26	#16810450	C₂²	1	2	yes	—
27	#16810449	C₂²	1	2	yes	—
28	#16810449	C₂²	1	2	yes	—
29	#16810448	C₂²	1	2	yes	—
30	C₄⋊#11719908	C₂²	1	1	yes	characteristic
31	#16810427	C₂²	1	2	yes	—
32	#16810424	C₂²	1	2	yes	—
33	#16810402	C₂²	1	2	yes	—
34	C₂×SD₃₂×C₅₃	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#367	C₂²	1	1	yes	characteristic
38	C₅₃×#358	C₂²	1	1	yes	characteristic
39	C₅₃×#357	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×#11719945	—	4	1	no	—
42	C₂×C₈×D₅₃	C₂³	1	2	yes	—
43	#13133217	—	2	1	no	—
44	#13133217	—	2	1	no	—
45	#13133212	—	4	2	no	—
46	#13133205	—	4	1	no	—
47	#13133199	—	4	2	no	—
48	#13133196	—	4	1	no	—
49	#13133193	—	2	2	no	—
50	#13133188	—	2	2	no	—
51	C₄×Dic₁₀₆	—	4	1	no	—
52	C₄×D₂₁₂	D₄	1	1	yes	characteristic
53	#13133175	—	2	2	no	—
54	C₄⋊C₄×D₅₃	—	2	2	no	—
55	#13133156	—	2	4	no	—
56	C₂×#11719908	C₂³	1	2	yes	—
57	#13133151	—	2	1	no	—
58	#13133151	—	2	1	no	—
59	#13133151	C₂³	1	2	yes	—
60	#13133151	C₂³	1	2	yes	—
61	#13133150	—	2	2	no	—
62	#13133148	—	2	2	no	—
63	C₄⋊#88322	D₄	1	1	yes	characteristic
64	#13133141	—	2	2	no	—
65	#13133138	—	2	2	no	—
66	#13133129	D₄	1	2	yes	—
67	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₄×Q₈×C₅₃	—	2	1	no	—
70	C₄×D₄×C₅₃	—	2	1	no	—
71	SD₃₂×C₅₃	—	2	2	no	—
72	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
74	#107×C₅₃	C₂³	1	1	yes	characteristic
75	#107×C₅₃	C₂³	1	1	yes	characteristic
76	#103×C₅₃	—	2	1	no	—
77	#102×C₅₃	—	2	1	no	—
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₅₃	—	2	1	no	—
80	C₂×D₄×C₅₃	—	2	1	no	—
81	C₂²×C₄×C₅₃	—	4	1	no	—
82	C₂×C₂²⋊D₅₃	—	4	2	no	—
83	#11719945	—	8	4	no	—
84	C₂×D₂₁₂	D₈	1	1	yes	characteristic
85	C₂×C₄×D₅₃	—	4	2	no	—
86	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
87	C₂×Dic₁₀₆	—	4	1	no	—
88	C₂×Dic₁₀₆	—	4	1	no	—
89	Q₁₆×C₅₃	—	2	1	no	—
90	Q₁₆×C₅₃	—	2	1	no	—
91	D₈×C₅₃	—	2	1	no	—
92	D₈×C₅₃	—	2	1	no	—
93	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₅₃	—	2	1	no	—
96	C₄⋊C₄×C₅₃	—	2	1	no	—
97	C₄⋊C₄×C₅₃	—	4	1	no	—
98	C₂²⋊C₄×C₅₃	—	4	1	no	—
99	C₄²×C₅₃	—	4	1	no	—
100	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₂²⋊Dic₅₃	—	4	2	no	—
102	#11719921	—	4	2	no	—
103	#11719921	—	4	2	no	—
104	#11719921	D₈	1	2	yes	—
105	C₄⋊Dic₅₃	—	2	2	no	—
106	C₄⋊Dic₅₃	—	2	2	no	—
107	C₄⋊Dic₅₃	—	4	1	no	—
108	C₄⋊Dic₅₃	—	4	1	no	—
109	C₄⋊Dic₅₃	D₈	1	1	yes	characteristic
110	#11719919	—	4	2	no	—
111	#11719919	—	4	2	no	—
112	C₄×Dic₅₃	—	4	2	no	—
113	C₂×#88322	C₂×D₄	1	2	yes	—
114	C₈×D₅₃	—	2	2	no	—
115	C₈×D₅₃	—	2	2	no	—
116	C₁₆×C₅₃	—	2	1	no	—
117	C₁₆×C₅₃	#41	1	2	yes	complemented
118	#11719908	—	2	2	no	—
119	C₂³×C₅₃	—	4	1	no	—
120	C₂²×D₅₃	C₂×D₈	1	2	yes	—
121	Q₈×C₅₃	—	4	1	no	—
122	Q₈×C₅₃	—	4	1	no	—
123	D₄×C₅₃	—	4	1	no	—
124	D₄×C₅₃	—	4	1	no	—
125	C₂×C₄×C₅₃	—	4	1	no	—
126	C₂×C₄×C₅₃	—	4	1	no	—
127	C₂×C₄×C₅₃	—	4	1	no	—
128	C₂×C₄×C₅₃	—	8	2	no	—
129	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
132	C₂²⋊D₅₃	—	8	4	no	—
133	C₂×Dic₅₃	—	4	2	no	—
134	C₂×Dic₅₃	—	4	2	no	—
135	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
136	D₂₁₂	—	2	2	no	—
137	C₄×D₅₃	—	2	2	no	—
138	C₄×D₅₃	—	2	2	no	—
139	C₄×D₅₃	—	8	4	no	—
140	Dic₁₀₆	—	8	2	no	—
141	Dic₁₀₆	—	8	2	no	—
142	C₈×C₅₃	—	2	1	no	—
143	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
144	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
145	#88322	—	2	2	no	—
146	#33920	—	53	1	no	maximal, Sylow
147	C₂²×C₅₃	—	8	1	no	—
148	C₂²×C₅₃	—	8	1	no	—
149	C₂²×C₅₃	—	8	1	no	—
150	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
151	C₂×D₅₃	—	2	2	no	—
152	C₂×D₅₃	—	2	2	no	—
153	C₂×D₅₃	—	2	2	no	—
154	C₄×C₅₃	—	2	1	no	—
155	C₄×C₅₃	—	8	1	no	—
156	C₄×C₅₃	—	8	1	no	—
157	C₄×C₅₃	—	8	1	no	—
158	C₄×C₅₃	#509	1	2	yes	—
159	C₄×C₅₃	#547	1	1	yes	characteristic
160	C₄×C₅₃	#578	1	1	yes	characteristic
161	Dic₅₃	—	2	2	no	—
162	Dic₅₃	—	8	2	no	—
163	Dic₅₃	—	8	2	no	—
164	#3342	—	53	1	no	—
165	#3334	—	53	1	no	—
166	#2252	—	53	1	no	—
167	#2236	—	53	2	no	—
168	#2234	—	53	2	no	—
169	#2225	—	53	1	no	—
170	#2224	—	53	1	no	—
171	#2220	—	53	2	no	—
172	C₄×SD₃₂	D₅₃	1	1	yes	characteristic, complemented
173	#2169	—	53	1	no	—
174	#2159	—	53	2	no	—
175	C₂×C₅₃	—	16	1	no	—
176	C₂×C₅₃	#3334	1	1	yes	characteristic
177	C₂×C₅₃	C₂×#509	1	1	yes	characteristic
178	C₂×C₅₃	#3417	1	1	yes	characteristic
179	D₅₃	—	4	2	no	—
180	#556	—	106	1	no	—
181	#547	—	106	1	no	—
182	C₂×SD₃₂	C₂×D₅₃	1	1	yes	characteristic
183	C₂×M₃₂	—	53	2	no	—
184	#500	—	53	1	no	—
185	#500	—	53	1	no	—
186	#485	—	106	2	no	—
187	#480	—	106	2	no	—
188	#475	—	106	1	no	—
189	#474	—	106	1	no	—
190	#467	—	53	2	no	—
191	#466	—	53	2	no	—
192	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
193	C₄×D₈	C₂×D₅₃	1	1	yes	characteristic
194	C₈×D₄	—	53	1	no	—
195	#427	—	53	2	no	—
196	C₂×#107	—	53	2	no	—
197	#367	—	53	2	no	—
198	#367	C₂×D₅₃	1	1	yes	characteristic
199	#366	—	53	2	no	—
200	#366	—	53	2	no	—
201	#366	—	53	2	no	—
202	C₄⋊C₁₆	—	53	1	no	—
203	#358	—	53	2	no	—
204	#358	C₂×D₅₃	1	1	yes	characteristic
205	#357	—	53	2	no	—
206	#357	C₂×D₅₃	1	1	yes	characteristic
207	C₂²⋊C₁₆	—	53	2	no	—
208	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
209	C₅₃	#33920	1	1	yes	characteristic, complemented, Sylow
210	C₂×#41	—	212	1	no	—
211	C₂×Q₁₆	C₂²×D₅₃	1	1	yes	characteristic
212	C₂×D₈	C₂²×D₅₃	1	1	yes	characteristic
213	C₂²×C₈	—	53	2	no	—
214	C₄⋊Q₈	—	106	1	no	—
215	C₄⋊Q₈	—	106	1	no	—
216	#126	—	212	2	no	—
217	#124	—	212	1	no	—
218	#123	—	212	2	no	—
219	C₂²⋊Q₈	—	106	2	no	—
220	C₂²⋊Q₈	—	212	1	no	—
221	#121	—	106	2	no	—
222	C₄×Q₈	—	2	1	no	—
223	C₄×Q₈	—	212	1	no	—
224	C₄×D₄	—	2	1	no	—
225	C₄×D₄	—	53	1	no	—
226	#117	—	106	2	no	—
227	C₂×C₄⋊C₄	—	106	2	no	—
228	SD₃₂	—	2	2	no	—
229	M₃₂	—	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	—	53	2	no	—
234	#107	—	53	2	no	—
235	#107	—	106	2	no	—
236	#107	—	106	1	no	—
237	#107	—	106	1	no	—
238	#107	C₂²×D₅₃	1	1	yes	characteristic
239	#107	C₂²×D₅₃	1	1	yes	characteristic
240	#106	—	106	2	no	—
241	C₄⋊C₈	—	53	1	no	—
242	#103	—	2	1	no	—
243	#103	—	106	2	no	—
244	#102	—	2	1	no	—
245	#102	—	106	2	no	—
246	C₂²⋊C₈	—	53	2	no	—
247	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
248	#41	—	424	4	no	—
249	C₂×Q₈	—	2	1	no	—
250	C₂×Q₈	—	212	1	no	—
251	C₂×Q₈	—	212	1	no	—
252	C₂×D₄	—	2	1	no	—
253	C₂×D₄	—	53	1	no	—
254	C₂×D₄	—	212	2	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	—	53	2	no	—
257	C₂²×C₄	—	212	2	no	—
258	Q₁₆	—	2	1	no	—
259	Q₁₆	—	2	1	no	—
260	D₈	—	2	1	no	—
261	D₈	—	2	1	no	—
262	C₂×C₈	—	53	2	no	—
263	C₂×C₈	—	106	2	no	—
264	C₂×C₈	—	106	2	no	—
265	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
266	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	53	1	no	—
271	C₄⋊C₄	—	106	2	no	—
272	C₄⋊C₄	—	106	2	no	—
273	C₄⋊C₄	—	212	2	no	—
274	C₄⋊C₄	—	212	2	no	—
275	C₄⋊C₄	—	212	1	no	—
276	C₄⋊C₄	—	212	1	no	—
277	C₂²⋊C₄	—	4	1	no	—
278	C₂²⋊C₄	—	53	2	no	—
279	C₂²⋊C₄	—	212	2	no	—
280	C₂²⋊C₄	—	212	2	no	—
281	C₂²⋊C₄	—	212	2	no	—
282	C₄²	—	4	1	no	—
283	C₄²	—	212	2	no	—
284	C₄²	D₄×D₅₃	1	1	yes	characteristic
285	C₁₆	—	2	1	no	—
286	C₁₆	—	106	2	no	—
287	C₁₆	#11719945	1	2	yes	complemented
288	C₂³	—	4	1	no	—
289	C₂³	—	53	2	no	—
290	Q₈	—	4	1	no	—
291	Q₈	—	4	1	no	—
292	Q₈	—	424	2	no	—
293	Q₈	—	424	2	no	—
294	D₄	—	4	1	no	—
295	D₄	—	4	1	no	—
296	D₄	—	106	2	no	—
297	D₄	—	424	4	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	—	53	2	no	—
303	C₂×C₄	—	106	2	no	—
304	C₂×C₄	—	106	2	no	—
305	C₂×C₄	—	424	4	no	—
306	C₂×C₄	—	212	2	no	—
307	C₂×C₄	—	212	2	no	—
308	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
309	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
310	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	106	2	no	—
313	C₈	C₂×#11719945	1	1	yes	characteristic
314	C₈	#13133264	1	1	yes	characteristic
315	C₂²	—	8	1	no	—
316	C₂²	—	8	1	no	—
317	C₂²	—	8	1	no	—
318	C₂²	—	106	2	no	—
319	C₂²	—	106	2	no	—
320	C₂²	—	106	2	no	—
321	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	8	1	no	—
324	C₄	—	8	1	no	—
325	C₄	—	8	1	no	—
326	C₄	—	106	2	no	—
327	C₄	—	424	2	no	—
328	C₄	—	424	2	no	—
329	C₄	#16811060	1	2	yes	—
330	C₄	#16811278	1	1	yes	characteristic
331	C₄	#16811437	1	1	yes	characteristic
332	C₂	—	16	1	no	—
333	C₂	—	212	2	no	—
334	C₂	#20287064	1	1	yes	characteristic
335	C₂	C₂×#16811060	1	1	yes	characteristic
336	C₂	#20287865	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₄×SD₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#20287064	C₂
hypercenter	C₄×SD₃₂	D₅₃
hypocenter	C₅₃	#33920
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×SD₃₂

Complements

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

N	H ≅ G/N	type
1. #20287147	C₂	semidirect
3. #20276990	C₂	semidirect
4. #20276916	C₂	semidirect
5. #20276904	C₂	semidirect
6. #20276857	C₂	semidirect
7. #20276852	C₂	semidirect
8. #20276825	C₂	semidirect
9. #20276548	C₂	semidirect
10. #20276502	C₂	semidirect
11. C₄×SD₃₂×C₅₃	C₂	semidirect
117. C₁₆×C₅₃	#41	semidirect
172. C₄×SD₃₂	D₅₃	semidirect
209. C₅₃	#33920	semidirect
287. C₁₆	#11719945	semidirect

Maximal subgroups

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

subgroup	index
#20287147	2
#20287064	2
#20276990	2
#20276916	2
#20276904	2
#20276857	2
#20276852	2
#20276825	2
#20276548	2
#20276502	2
C₄×SD₃₂×C₅₃	2
#33920	53