Subgroups — #27539626

Order 13568.

602 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285291	C₂	1	2	yes	maximal, complemented
2	#20276989	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276963	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276937	C₂	1	2	yes	maximal, complemented
5	#20276802	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276800	C₂	1	2	yes	maximal, complemented
7	#20276798	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₈₄₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13133158	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811170	—	2	2	no	—
13	C₂×D₈₄₈	C₂²	1	1	yes	characteristic
14	C₂×#13133158	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	#16810998	C₂²	1	2	yes	—
19	#16810968	—	2	2	no	—
20	#16810860	C₂²	1	2	yes	—
21	#16810858	—	2	2	no	—
22	#16810856	C₂²	1	2	yes	—
23	C₄×D₄₂₄	C₂²	1	2	yes	—
24	C₄×#11719922	—	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	#16810425	C₂²	1	2	yes	—
29	#16810424	C₂²	1	2	yes	—
30	#16810423	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×D₁₆×C₅₃	C₂²	1	1	yes	characteristic
35	C₄×D₈×C₅₃	C₂²	1	2	yes	—
36	C₅₃×#366	C₂²	1	1	yes	characteristic
37	C₅₃×#357	C₂²	1	2	yes	—
38	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
39	C₂×#11719945	—	4	2	no	—
40	C₂×D₄₂₄	C₂³	1	2	yes	—
41	C₂×#11719922	—	2	2	no	—
42	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
43	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
44	#13133216	—	2	2	no	—
45	#13133213	—	2	2	no	—
46	#13133203	—	4	2	no	—
47	#13133192	—	2	2	no	—
48	#13133190	—	4	2	no	—
49	#13133188	—	2	2	no	—
50	C₄×D₂₁₂	—	2	2	no	—
51	C₄×C₂²⋊D₅₃	—	4	2	no	—
52	C₄²×D₅₃	D₄	1	1	yes	characteristic
53	D₈₄₈	—	2	2	no	—
54	#13133158	—	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	#13133139	—	2	2	no	—
62	#13133138	—	2	2	no	—
63	#13133137	—	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×D₈×C₅₃	C₂³	1	2	yes	—
68	C₄×D₄×C₅₃	—	2	2	no	—
69	D₁₆×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	#102×C₅₃	—	2	2	no	—
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₂×D₄×C₅₃	—	2	2	no	—
76	C₂²×C₄×C₅₃	—	4	2	no	—
77	C₂×C₂²⋊D₅₃	—	4	2	no	—
78	C₂×C₂²⋊D₅₃	—	4	2	no	—
79	C₂×C₂²⋊D₅₃	—	4	2	no	—
80	#11719945	—	8	8	no	—
81	C₂×D₂₁₂	—	2	2	no	—
82	C₂×D₂₁₂	—	4	2	no	—
83	C₂×C₄×D₅₃	—	4	2	no	—
84	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
85	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
86	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
87	C₂×Dic₁₀₆	—	4	2	no	—
88	D₈×C₅₃	—	2	2	no	—
89	D₈×C₅₃	—	2	2	no	—
90	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₅₃	—	2	2	no	—
93	C₂²⋊C₄×C₅₃	—	4	2	no	—
94	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂²⋊Dic₅₃	—	4	2	no	—
96	#11719922	—	4	4	no	—
97	#11719921	—	4	2	no	—
98	#11719921	—	4	2	no	—
99	C₄⋊Dic₅₃	—	2	2	no	—
100	#11719919	—	4	2	no	—
101	C₄×Dic₅₃	D₈	1	1	yes	characteristic
102	C₄×Dic₅₃	D₈	1	1	yes	characteristic
103	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
104	C₂×#88322	C₂×D₄	1	1	yes	characteristic
105	C₂×#88322	C₂×D₄	1	1	yes	characteristic
106	D₄₂₄	—	2	2	no	—
107	D₄₂₄	—	2	2	no	—
108	C₈×D₅₃	—	2	1	no	—
109	C₈×D₅₃	—	2	1	no	—
110	C₈×D₅₃	—	2	2	no	—
111	C₁₆×C₅₃	—	2	1	no	—
112	C₁₆×C₅₃	#41	1	2	yes	complemented
113	#11719908	—	2	1	no	—
114	#11719908	#41	1	2	yes	complemented
115	C₂³×C₅₃	—	4	2	no	—
116	C₂²×D₅₃	—	4	2	no	—
117	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
118	D₄×C₅₃	—	4	2	no	—
119	D₄×C₅₃	—	4	2	no	—
120	C₂×C₄×C₅₃	—	4	2	no	—
121	C₂×C₄×C₅₃	—	8	4	no	—
122	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
125	C₂²⋊D₅₃	—	8	4	no	—
126	C₂²⋊D₅₃	—	8	4	no	—
127	C₂²⋊D₅₃	—	8	4	no	—
128	C₂×Dic₅₃	—	4	2	no	—
129	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
132	D₂₁₂	—	4	2	no	—
133	D₂₁₂	—	4	2	no	—
134	D₂₁₂	—	8	4	no	—
135	C₄×D₅₃	—	2	1	no	—
136	C₄×D₅₃	—	2	1	no	—
137	C₄×D₅₃	—	2	2	no	—
138	C₄×D₅₃	—	8	4	no	—
139	C₄×D₅₃	D₁₆	1	4	yes	complemented
140	Dic₁₀₆	—	8	4	no	—
141	C₈×C₅₃	—	2	1	no	—
142	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
143	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
144	#88322	—	2	1	no	—
145	#88322	#135	1	2	yes	—
146	#33733	—	53	1	no	maximal, Sylow
147	C₂²×C₅₃	—	8	2	no	—
148	C₂²×C₅₃	—	8	2	no	—
149	C₂²×C₅₃	—	8	2	no	—
150	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
151	C₂×D₅₃	—	2	1	no	—
152	C₂×D₅₃	—	2	1	no	—
153	C₂×D₅₃	—	8	2	no	—
154	C₂×D₅₃	—	8	2	no	—
155	C₂×D₅₃	—	8	2	no	—
156	C₂×D₅₃	C₂×D₁₆	1	2	yes	—
157	C₄×C₅₃	—	2	1	no	—
158	C₄×C₅₃	—	8	2	no	—
159	C₄×C₅₃	C₂×D₁₆	1	2	yes	—
160	C₄×C₅₃	#532	1	1	yes	characteristic
161	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
162	Dic₅₃	—	2	1	no	—
163	Dic₅₃	—	8	2	no	—
164	Dic₅₃	C₂×D₁₆	1	2	yes	—
165	Dic₅₃	#508	1	2	yes	—
166	#3169	—	53	2	no	—
167	#2252	—	53	1	no	—
168	#2246	—	53	1	no	—
169	#2240	—	53	2	no	—
170	#2215	—	53	1	no	—
171	#2215	—	53	1	no	—
172	#2215	—	53	2	no	—
173	C₄×D₁₆	—	53	2	no	—
174	C₄×D₁₆	—	53	1	no	—
175	C₄×D₁₆	D₅₃	1	1	yes	characteristic, complemented
176	C₂×C₄×C₁₆	—	53	1	no	—
177	C₂×C₅₃	—	16	2	no	—
178	C₂×C₅₃	#3169	1	1	yes	characteristic
179	C₂×C₅₃	C₂²×D₁₆	1	1	yes	characteristic
180	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
181	D₅₃	—	2	2	no	—
182	D₅₃	—	16	2	no	—
183	#532	—	106	2	no	—
184	C₂×D₁₆	—	53	2	no	—
185	C₂×D₁₆	—	53	1	no	—
186	C₂×D₁₆	C₂×D₅₃	1	1	yes	characteristic
187	C₂²×C₁₆	—	53	1	no	—
188	C₂²×C₁₆	—	53	1	no	—
189	#500	—	53	2	no	—
190	#493	—	53	2	no	—
191	#486	—	106	2	no	—
192	#466	—	53	2	no	—
193	#466	—	53	2	no	—
194	#466	—	106	2	no	—
195	C₄×D₈	—	53	2	no	—
196	C₄×D₈	—	106	2	no	—
197	C₄×D₈	C₂×D₅₃	1	2	yes	—
198	C₂×C₄×C₈	—	53	1	no	—
199	#366	—	53	1	no	—
200	#366	—	53	2	no	—
201	#366	C₂×D₅₃	1	1	yes	characteristic
202	#357	—	53	2	no	—
203	#357	—	53	2	no	—
204	#357	—	53	2	no	—
205	#357	C₂×D₅₃	1	2	yes	—
206	C₄×C₁₆	—	53	1	no	—
207	C₄×C₁₆	—	53	1	no	—
208	C₄×C₁₆	—	53	1	no	—
209	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
210	C₅₃	#33733	1	1	yes	characteristic, complemented, Sylow
211	C₂×#41	—	212	2	no	—
212	C₂×D₈	—	53	2	no	—
213	C₂×D₈	—	106	2	no	—
214	C₂×D₈	C₂²×D₅₃	1	2	yes	—
215	C₂²×C₈	—	53	1	no	—
216	C₂²×C₈	—	53	1	no	—
217	C₄⋊Q₈	—	106	2	no	—
218	#127	—	106	2	no	—
219	#124	—	212	2	no	—
220	#121	—	106	2	no	—
221	#121	—	106	2	no	—
222	#121	—	212	2	no	—
223	C₄×D₄	—	2	2	no	—
224	C₄×D₄	—	106	2	no	—
225	C₄×D₄	—	212	2	no	—
226	C₂×C₄²	—	53	1	no	—
227	D₁₆	—	2	2	no	—
228	D₁₆	—	106	4	no	—
229	D₁₆	—	106	2	no	—
230	C₂×C₁₆	—	53	1	no	—
231	C₂×C₁₆	—	53	1	no	—
232	C₂×C₁₆	—	106	2	no	—
233	C₂×C₁₆	—	106	2	no	—
234	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
235	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
236	#107	—	53	2	no	—
237	#107	—	106	2	no	—
238	#107	C₂²×D₅₃	1	2	yes	—
239	#102	—	2	2	no	—
240	#102	—	106	2	no	—
241	#102	—	106	2	no	—
242	#102	—	106	2	no	—
243	C₄×C₈	—	53	1	no	—
244	C₄×C₈	—	53	1	no	—
245	C₄×C₈	—	53	1	no	—
246	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
247	#41	—	424	8	no	—
248	C₂×Q₈	—	212	2	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	106	2	no	—
251	C₂×D₄	—	212	2	no	—
252	C₂×D₄	—	212	2	no	—
253	C₂×D₄	—	212	2	no	—
254	C₂×D₄	—	212	2	no	—
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	—	53	1	no	—
257	C₂²×C₄	—	53	1	no	—
258	C₂²×C₄	—	53	1	no	—
259	C₂²×C₄	—	212	2	no	—
260	D₈	—	2	2	no	—
261	D₈	—	2	2	no	—
262	D₈	—	106	2	no	—
263	D₈	—	106	2	no	—
264	D₈	—	212	4	no	—
265	C₂×C₈	—	53	1	no	—
266	C₂×C₈	—	53	1	no	—
267	C₂×C₈	—	106	2	no	—
268	C₂×C₈	—	106	1	no	—
269	C₂×C₈	—	106	1	no	—
270	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
271	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	106	2	no	—
274	C₄⋊C₄	—	212	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	212	2	no	—
277	C₂²⋊C₄	—	212	2	no	—
278	C₂²⋊C₄	—	212	2	no	—
279	C₄²	—	53	1	no	—
280	C₄²	—	53	1	no	—
281	C₄²	—	53	1	no	—
282	C₄²	D₄×D₅₃	1	1	yes	characteristic
283	C₁₆	—	2	1	no	—
284	C₁₆	—	53	2	no	—
285	C₁₆	—	106	1	no	—
286	C₁₆	#11719945	1	2	yes	complemented
287	C₂³	—	4	2	no	—
288	C₂³	—	53	1	no	—
289	C₂³	—	212	2	no	—
290	Q₈	—	424	4	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	424	4	no	—
294	D₄	—	424	4	no	—
295	D₄	—	424	4	no	—
296	D₄	—	424	4	no	—
297	D₄	—	212	2	no	—
298	D₄	—	212	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	8	4	no	—
301	C₂×C₄	—	53	1	no	—
302	C₂×C₄	—	53	1	no	—
303	C₂×C₄	—	53	4	no	—
304	C₂×C₄	—	53	1	no	—
305	C₂×C₄	—	106	2	no	—
306	C₂×C₄	—	106	1	no	—
307	C₂×C₄	—	106	1	no	—
308	C₂×C₄	—	424	4	no	—
309	C₂×C₄	—	212	2	no	—
310	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
311	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
312	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
313	C₈	—	2	1	no	—
314	C₈	—	53	2	no	—
315	C₈	—	106	1	no	—
316	C₈	#41×D₅₃	1	1	yes	characteristic
317	C₈	C₂×#11719945	1	1	yes	characteristic
318	C₂²	—	8	2	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	53	2	no	—
322	C₂²	—	106	1	no	—
323	C₂²	—	106	1	no	—
324	C₂²	—	424	2	no	—
325	C₂²	—	424	2	no	—
326	C₂²	—	424	2	no	—
327	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	8	2	no	—
330	C₄	—	53	2	no	—
331	C₄	—	53	2	no	—
332	C₄	—	106	1	no	—
333	C₄	—	424	2	no	—
334	C₄	D₁₆×D₅₃	1	2	yes	—
335	C₄	#16811170	1	1	yes	characteristic
336	C₄	#135×D₅₃	1	1	yes	characteristic
337	C₂	—	16	2	no	—
338	C₂	—	848	2	no	—
339	C₂	—	106	2	no	—
340	C₂	#20285291	1	1	yes	characteristic
341	C₂	C₂×D₁₆×D₅₃	1	1	yes	characteristic
342	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×D₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
hypercenter	C₄×D₁₆	D₅₃
hypocenter	C₅₃	#33733
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

Complements

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

N	H ≅ G/N	type
1. #20285291	C₂	semidirect
2. #20276989	C₂	semidirect
3. #20276963	C₂	semidirect
4. #20276937	C₂	semidirect
5. #20276802	C₂	semidirect
6. #20276800	C₂	semidirect
7. #20276798	C₂	semidirect
8. C₄×D₈₄₈	C₂	semidirect
9. C₄×#13133158	C₂	semidirect
10. C₄×C₁₆×D₅₃	C₂	semidirect
11. C₄×D₁₆×C₅₃	C₂	semidirect
112. C₁₆×C₅₃	#41	semidirect
114. #11719908	#41	semidirect
139. C₄×D₅₃	D₁₆	semidirect
175. C₄×D₁₆	D₅₃	semidirect
210. C₅₃	#33733	semidirect
286. C₁₆	#11719945	semidirect

Maximal subgroups

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

subgroup	index
#20285291	2
#20276989	2
#20276963	2
#20276937	2
#20276802	2
#20276800	2
#20276798	2
C₄×D₈₄₈	2
C₄×#13133158	2
C₄×C₁₆×D₅₃	2
C₄×D₁₆×C₅₃	2
#33733	53