Subgroups — #27541696

Order 13568.

514 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287137	C₂	1	1	yes	maximal, characteristic, complemented
2	#20287062	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276908	C₂	1	2	yes	maximal, complemented
4	#20276902	C₂	1	2	yes	maximal, complemented
5	#20276859	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276849	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276819	C₂	1	2	yes	maximal, complemented
8	C₄×D₈₄₈	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276548	C₂	1	1	yes	maximal, characteristic, complemented
10	#20276502	C₂	1	2	yes	maximal, complemented
11	C₅₃×#2254	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811339	—	2	1	no	—
13	#16811276	—	2	1	no	—
14	C₂×D₈₄₈	C₂²	1	1	yes	characteristic
15	C₂×#13133156	C₂²	1	2	yes	—
16	#16810950	—	2	2	no	—
17	#16810938	—	2	2	no	—
18	#16810910	—	2	1	no	—
19	#16810900	—	2	1	no	—
20	#16810860	C₂²	1	2	yes	—
21	#16810860	C₂²	1	2	yes	—
22	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
23	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
24	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
25	#16810660	C₂²	1	2	yes	—
26	#107×D₅₃	C₂²	1	2	yes	—
27	#16810450	C₂²	1	2	yes	—
28	#16810449	C₂²	1	1	yes	characteristic
29	#16810448	C₂²	1	2	yes	—
30	C₄⋊#11719908	C₂²	1	1	yes	characteristic
31	#16810425	C₂²	1	1	yes	characteristic
32	#16810425	C₂²	1	1	yes	characteristic
33	#16810423	C₂²	1	2	yes	—
34	#16810423	C₂²	1	2	yes	—
35	#16810402	C₂²	1	2	yes	—
36	C₅₃×#500	C₂²	1	1	yes	characteristic
37	C₅₃×#500	C₂²	1	1	yes	characteristic
38	C₅₃×#367	C₂²	1	2	yes	—
39	C₅₃×#367	C₂²	1	2	yes	—
40	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×#11719949	—	4	1	no	—
42	C₂×D₄₂₄	C₂³	1	1	yes	characteristic
43	C₂×D₄₂₄	C₂³	1	1	yes	characteristic
44	C₂×C₈×D₅₃	C₂³	1	2	yes	—
45	#13133206	—	4	2	no	—
46	#13133202	—	4	2	no	—
47	#13133193	—	4	1	no	—
48	#13133192	—	2	2	no	—
49	#13133192	—	2	2	no	—
50	#13133192	—	4	1	no	—
51	C₄×D₂₁₂	—	2	1	no	—
52	C₄×D₂₁₂	—	2	1	no	—
53	C₄×D₂₁₂	D₄	1	1	yes	characteristic
54	#13133175	—	2	2	no	—
55	#13133175	—	4	1	no	—
56	C₄⋊C₄×D₅₃	—	2	2	no	—
57	D₈₄₈	—	2	2	no	—
58	#13133156	—	2	4	no	—
59	C₂×#11719908	C₂³	1	2	yes	—
60	#13133151	C₂³	1	1	yes	characteristic
61	#13133151	C₂³	1	1	yes	characteristic
62	#13133150	—	2	2	no	—
63	#13133148	—	2	2	no	—
64	C₄⋊#88322	D₄	1	1	yes	characteristic
65	#13133139	—	2	1	no	—
66	#13133139	—	2	1	no	—
67	#13133137	—	2	2	no	—
68	#13133137	—	2	2	no	—
69	#13133129	D₄	1	2	yes	—
70	C₄⋊Q₈×C₅₃	—	2	1	no	—
71	C₄⋊Q₈×C₅₃	—	2	1	no	—
72	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
74	#107×C₅₃	—	2	1	no	—
75	#107×C₅₃	—	2	1	no	—
76	#107×C₅₃	C₂³	1	2	yes	—
77	#107×C₅₃	C₂³	1	2	yes	—
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₅₃	—	4	1	no	—
80	C₂×Q₈×C₅₃	—	4	1	no	—
81	#11719949	—	8	4	no	—
82	C₂×D₂₁₂	—	2	1	no	—
83	C₂×D₂₁₂	—	2	1	no	—
84	C₂×D₂₁₂	—	4	2	no	—
85	C₂×D₂₁₂	—	4	2	no	—
86	C₂×D₂₁₂	D₈	1	1	yes	characteristic
87	C₂×C₄×D₅₃	—	4	2	no	—
88	C₂×C₄×D₅₃	—	4	1	no	—
89	C₂×C₄×D₅₃	—	4	1	no	—
90	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₅₃	—	2	2	no	—
94	C₄⋊C₄×C₅₃	—	2	2	no	—
95	C₄⋊C₄×C₅₃	—	4	1	no	—
96	C₄⋊C₄×C₅₃	—	4	1	no	—
97	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
98	#11719921	—	4	2	no	—
99	#11719921	—	4	1	no	—
100	#11719921	—	4	1	no	—
101	#11719921	—	4	2	no	—
102	#11719921	—	4	2	no	—
103	#11719921	D₈	1	2	yes	—
104	C₄⋊Dic₅₃	—	2	1	no	—
105	C₄⋊Dic₅₃	—	2	1	no	—
106	C₄⋊Dic₅₃	D₈	1	1	yes	characteristic
107	#11719919	—	4	2	no	—
108	C₄×Dic₅₃	—	4	2	no	—
109	C₂×#88322	C₂×D₄	1	2	yes	—
110	D₄₂₄	—	2	1	no	—
111	D₄₂₄	—	2	1	no	—
112	D₄₂₄	—	2	1	no	—
113	D₄₂₄	—	2	1	no	—
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	C₂²×D₅₃	—	4	1	no	—
120	C₂²×D₅₃	—	4	1	no	—
121	C₂²×D₅₃	C₂×D₈	1	2	yes	—
122	Q₈×C₅₃	—	8	2	no	—
123	Q₈×C₅₃	—	16	1	no	—
124	C₂×C₄×C₅₃	—	4	2	no	—
125	C₂×C₄×C₅₃	—	4	2	no	—
126	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×Dic₅₃	—	4	1	no	—
130	C₂×Dic₅₃	—	4	1	no	—
131	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
132	D₂₁₂	—	2	2	no	—
133	D₂₁₂	—	4	1	no	—
134	D₂₁₂	—	4	1	no	—
135	D₂₁₂	—	4	1	no	—
136	D₂₁₂	—	4	1	no	—
137	D₂₁₂	—	8	4	no	—
138	D₂₁₂	—	16	2	no	—
139	C₄×D₅₃	—	2	2	no	—
140	C₄×D₅₃	—	2	2	no	—
141	C₄×D₅₃	—	8	4	no	—
142	C₄×D₅₃	—	8	2	no	—
143	C₄×D₅₃	—	8	2	no	—
144	C₈×C₅₃	—	2	1	no	—
145	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
146	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
147	#88322	—	2	2	no	—
148	#33921	—	53	1	no	maximal, Sylow
149	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
150	C₂×D₅₃	—	2	2	no	—
151	C₂×D₅₃	—	2	2	no	—
152	C₂×D₅₃	—	2	2	no	—
153	C₂×D₅₃	—	8	1	no	—
154	C₂×D₅₃	—	8	1	no	—
155	C₂×D₅₃	—	8	1	no	—
156	C₂×D₅₃	—	8	1	no	—
157	C₂×D₅₃	—	8	1	no	—
158	C₂×D₅₃	—	8	1	no	—
159	C₄×C₅₃	—	2	1	no	—
160	C₄×C₅₃	—	8	2	no	—
161	C₄×C₅₃	—	8	2	no	—
162	C₄×C₅₃	#509	1	2	yes	—
163	C₄×C₅₃	#547	1	1	yes	characteristic
164	C₄×C₅₃	#578	1	1	yes	characteristic
165	Dic₅₃	—	2	2	no	—
166	Dic₅₃	—	8	1	no	—
167	Dic₅₃	—	8	1	no	—
168	#3341	—	53	1	no	—
169	#3334	—	53	1	no	—
170	#2254	D₅₃	1	1	yes	characteristic, complemented
171	#2234	—	53	2	no	—
172	#2233	—	53	2	no	—
173	#2226	—	53	1	no	—
174	#2224	—	53	1	no	—
175	#2218	—	53	2	no	—
176	C₄×D₁₆	—	53	1	no	—
177	#2169	—	53	1	no	—
178	#2159	—	53	2	no	—
179	C₂×C₅₃	#3334	1	1	yes	characteristic
180	C₂×C₅₃	C₂×#509	1	1	yes	characteristic
181	C₂×C₅₃	#3416	1	1	yes	characteristic
182	D₅₃	—	4	2	no	—
183	D₅₃	—	16	1	no	—
184	D₅₃	—	16	1	no	—
185	#555	—	106	1	no	—
186	#547	—	106	1	no	—
187	C₂×D₁₆	—	53	1	no	—
188	C₂×M₃₂	—	53	2	no	—
189	#500	C₂×D₅₃	1	1	yes	characteristic
190	#500	C₂×D₅₃	1	1	yes	characteristic
191	#482	—	106	2	no	—
192	#480	—	106	2	no	—
193	#476	—	106	1	no	—
194	#474	—	106	1	no	—
195	#466	—	53	2	no	—
196	#466	—	53	2	no	—
197	C₄×D₈	—	53	1	no	—
198	C₄×D₈	—	53	1	no	—
199	C₈×D₄	—	53	1	no	—
200	#427	—	53	2	no	—
201	C₂×#107	—	53	2	no	—
202	#367	—	53	2	no	—
203	#367	C₂×D₅₃	1	2	yes	—
204	#367	C₂×D₅₃	1	2	yes	—
205	#366	—	53	1	no	—
206	#366	—	53	2	no	—
207	C₄⋊C₁₆	—	53	1	no	—
208	#357	—	53	1	no	—
209	#357	—	53	1	no	—
210	#357	—	53	2	no	—
211	#357	—	53	2	no	—
212	C₂²⋊C₁₆	—	53	2	no	—
213	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
214	C₅₃	#33921	1	1	yes	characteristic, complemented, Sylow
215	C₂×#41	—	212	1	no	—
216	C₂×D₈	—	53	1	no	—
217	C₂×D₈	—	53	1	no	—
218	C₂²×C₈	—	53	2	no	—
219	C₄⋊Q₈	—	2	1	no	—
220	C₄⋊Q₈	—	2	1	no	—
221	#124	—	212	2	no	—
222	#123	—	212	2	no	—
223	C₂²⋊Q₈	—	212	1	no	—
224	#121	—	106	2	no	—
225	#121	—	106	2	no	—
226	#121	—	212	1	no	—
227	C₄×D₄	—	53	1	no	—
228	C₄×D₄	—	106	1	no	—
229	C₄×D₄	—	106	1	no	—
230	#117	—	106	2	no	—
231	#117	—	212	1	no	—
232	C₂×C₄⋊C₄	—	106	2	no	—
233	D₁₆	—	106	2	no	—
234	M₃₂	—	106	4	no	—
235	C₂×C₁₆	—	53	2	no	—
236	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
237	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
238	#107	—	2	1	no	—
239	#107	—	2	1	no	—
240	#107	—	53	1	no	—
241	#107	—	53	1	no	—
242	#107	—	106	2	no	—
243	#107	C₂²×D₅₃	1	2	yes	—
244	#107	C₂²×D₅₃	1	2	yes	—
245	#106	—	106	2	no	—
246	C₄⋊C₈	—	53	1	no	—
247	#102	—	106	2	no	—
248	#102	—	106	2	no	—
249	#102	—	106	1	no	—
250	#102	—	106	1	no	—
251	C₂²⋊C₈	—	53	2	no	—
252	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
253	#41	—	424	4	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×Q₈	—	4	1	no	—
256	C₂×D₄	—	53	1	no	—
257	C₂×D₄	—	106	1	no	—
258	C₂×D₄	—	106	1	no	—
259	C₂×D₄	—	212	2	no	—
260	C₂×D₄	—	212	2	no	—
261	C₂²×C₄	—	53	2	no	—
262	C₂²×C₄	—	212	2	no	—
263	C₂²×C₄	—	212	1	no	—
264	C₂²×C₄	—	212	1	no	—
265	D₈	—	106	1	no	—
266	D₈	—	106	1	no	—
267	D₈	—	106	1	no	—
268	D₈	—	106	1	no	—
269	C₂×C₈	—	53	2	no	—
270	C₂×C₈	—	106	2	no	—
271	C₂×C₈	—	106	2	no	—
272	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
273	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	53	1	no	—
279	C₄⋊C₄	—	106	1	no	—
280	C₄⋊C₄	—	106	1	no	—
281	C₄⋊C₄	—	212	2	no	—
282	C₂²⋊C₄	—	53	2	no	—
283	C₂²⋊C₄	—	212	2	no	—
284	C₂²⋊C₄	—	212	2	no	—
285	C₂²⋊C₄	—	212	2	no	—
286	C₂²⋊C₄	—	212	1	no	—
287	C₂²⋊C₄	—	212	1	no	—
288	C₄²	—	212	2	no	—
289	C₄²	D₄×D₅₃	1	1	yes	characteristic
290	C₁₆	—	2	1	no	—
291	C₁₆	—	106	2	no	—
292	C₁₆	#11719949	1	2	yes	complemented
293	C₂³	—	53	2	no	—
294	C₂³	—	212	1	no	—
295	C₂³	—	212	1	no	—
296	Q₈	—	8	2	no	—
297	Q₈	—	16	1	no	—
298	D₄	—	848	2	no	—
299	D₄	—	106	2	no	—
300	D₄	—	424	4	no	—
301	D₄	—	212	1	no	—
302	D₄	—	212	1	no	—
303	D₄	—	212	1	no	—
304	D₄	—	212	1	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	53	2	no	—
308	C₂×C₄	—	106	2	no	—
309	C₂×C₄	—	106	2	no	—
310	C₂×C₄	—	424	4	no	—
311	C₂×C₄	—	424	2	no	—
312	C₂×C₄	—	424	2	no	—
313	C₂×C₄	—	212	1	no	—
314	C₂×C₄	—	212	1	no	—
315	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
316	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
317	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	106	2	no	—
320	C₈	C₂×#11719949	1	1	yes	characteristic
321	C₈	#13133265	1	1	yes	characteristic
322	C₂²	—	106	2	no	—
323	C₂²	—	106	2	no	—
324	C₂²	—	106	2	no	—
325	C₂²	—	424	1	no	—
326	C₂²	—	424	1	no	—
327	C₂²	—	424	1	no	—
328	C₂²	—	424	1	no	—
329	C₂²	—	424	1	no	—
330	C₂²	—	424	1	no	—
331	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	8	2	no	—
334	C₄	—	8	2	no	—
335	C₄	—	106	2	no	—
336	C₄	—	424	1	no	—
337	C₄	—	424	1	no	—
338	C₄	#16811060	1	2	yes	—
339	C₄	#16811276	1	1	yes	characteristic
340	C₄	#16811436	1	1	yes	characteristic
341	C₂	—	848	1	no	—
342	C₂	—	848	1	no	—
343	C₂	—	212	2	no	—
344	C₂	#20287062	1	1	yes	characteristic
345	C₂	C₂×#16811060	1	1	yes	characteristic
346	C₂	#20287857	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×#2254	C₂
socle	C₂²×C₅₃	C₂²×D₈
hypercenter	#2254	D₅₃
hypocenter	C₅₃	#33921
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

Complements

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

N	H ≅ G/N	type
1. #20287137	C₂	semidirect
2. #20287062	C₂	semidirect
3. #20276908	C₂	semidirect
4. #20276902	C₂	semidirect
5. #20276859	C₂	semidirect
6. #20276849	C₂	semidirect
7. #20276819	C₂	semidirect
8. C₄×D₈₄₈	C₂	semidirect
9. #20276548	C₂	semidirect
10. #20276502	C₂	semidirect
11. C₅₃×#2254	C₂	semidirect
117. C₁₆×C₅₃	#41	semidirect
170. #2254	D₅₃	semidirect
214. C₅₃	#33921	semidirect
292. C₁₆	#11719949	semidirect

Maximal subgroups

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

subgroup	index
#20287137	2
#20287062	2
#20276908	2
#20276902	2
#20276859	2
#20276849	2
#20276819	2
C₄×D₈₄₈	2
#20276548	2
#20276502	2
C₅₃×#2254	2
#33921	53