Subgroups — #27541798

Order 13568.

484 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287728	C₂	1	1	yes	maximal, characteristic
2	#20276949	C₂	1	2	yes	maximal, complemented
3	#20276940	C₂	1	1	yes	maximal, characteristic
4	#20276869	C₂	1	2	yes	maximal, complemented
5	#20276854	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276777	C₂	1	2	yes	maximal, complemented
7	#20276755	C₂	1	1	yes	maximal, characteristic
8	Q₈×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13133163	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#13133160	C₂	1	2	yes	maximal, complemented
11	C₅₃×#3383	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811358	—	2	1	no	—
13	C₂×#13133163	C₂²	1	1	yes	characteristic
14	C₂×#13133160	C₂²	1	2	yes	—
15	#16811007	C₂²	1	2	yes	—
16	#16811000	C₂²	1	1	yes	characteristic
17	#16810839	—	2	2	no	—
18	#16810824	—	2	1	no	—
19	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
20	C₄×D₄₂₄	C₂²	1	2	yes	—
21	#16810450	C₂²	1	2	yes	—
22	#16810448	C₂²	1	1	yes	characteristic
23	C₄⋊#11719908	C₂²	1	1	yes	characteristic
24	C₄⋊#11719908	C₂²	1	2	yes	—
25	#16810427	C₂²	1	1	yes	characteristic
26	#16810426	C₂²	1	2	yes	—
27	#16810426	C₂²	1	1	yes	characteristic
28	#16810423	C₂²	1	2	yes	—
29	#16810423	C₂²	1	2	yes	—
30	#16810423	C₂²	1	2	yes	—
31	C₄×#11719908	C₂²	1	2	yes	—
32	C₄×#11719908	C₂²	1	1	yes	characteristic
33	C₅₃×#557	—	2	1	no	—
34	C₅₃×#500	C₂²	1	1	yes	characteristic
35	C₅₃×#499	C₂²	1	2	yes	—
36	C₅₃×#479	—	2	2	no	—
37	C₅₃×#475	—	2	1	no	—
38	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
41	C₂×Dic₂₁₂	C₂³	1	1	yes	characteristic
42	C₂×#11719913	—	2	2	no	—
43	C₂×D₄₂₄	—	2	1	no	—
44	C₂×D₄₂₄	C₂³	1	2	yes	—
45	#13133214	—	2	1	no	—
46	#13133206	—	4	1	no	—
47	#13133205	—	2	2	no	—
48	#13133202	—	4	2	no	—
49	C₄×Dic₁₀₆	—	2	1	no	—
50	C₄×D₂₁₂	—	2	2	no	—
51	#13133179	—	4	1	no	—
52	#13133163	—	2	2	no	—
53	#13133160	—	2	4	no	—
54	C₂×#11719908	C₂³	1	2	yes	—
55	C₂×#11719908	C₂³	1	1	yes	characteristic
56	C₂×#11719908	C₂³	1	1	yes	characteristic
57	#13133151	C₂³	1	1	yes	characteristic
58	#13133142	—	2	1	no	—
59	#13133139	—	2	2	no	—
60	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₅₃	—	2	1	no	—
62	C₄⋊Q₈×C₅₃	—	4	1	no	—
63	#125×C₅₃	—	2	2	no	—
64	#125×C₅₃	—	4	2	no	—
65	C₄×Q₈×C₅₃	—	2	1	no	—
66	C₄×Q₈×C₅₃	—	2	2	no	—
67	C₄×Q₈×C₅₃	—	4	1	no	—
68	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
69	#107×C₅₃	—	2	1	no	—
70	#107×C₅₃	C₂³	1	1	yes	characteristic
71	#107×C₅₃	C₂³	1	2	yes	—
72	#106×C₅₃	—	2	2	no	—
73	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
74	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
75	#103×C₅₃	—	2	2	no	—
76	#103×C₅₃	—	2	1	no	—
77	C₄×C₈×C₅₃	C₂³	1	2	yes	—
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₅₃	—	2	1	no	—
80	C₂×Q₈×C₅₃	—	4	1	no	—
81	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
82	C₂×D₂₁₂	—	2	2	no	—
83	C₂×D₂₁₂	—	4	1	no	—
84	C₂×D₂₁₂	—	4	1	no	—
85	C₂×C₄×D₅₃	—	4	2	no	—
86	C₂×Dic₁₀₆	—	2	1	no	—
87	Q₁₆×C₅₃	—	2	1	no	—
88	Q₁₆×C₅₃	#41	1	2	yes	—
89	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₅₃	—	2	1	no	—
93	C₄⋊C₄×C₅₃	—	2	2	no	—
94	C₄⋊C₄×C₅₃	—	4	1	no	—
95	C₄⋊C₄×C₅₃	—	4	2	no	—
96	C₄⋊C₄×C₅₃	—	4	2	no	—
97	C₄⋊C₄×C₅₃	—	4	2	no	—
98	C₄⋊C₄×C₅₃	—	4	1	no	—
99	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
100	C₄⋊C₄×C₅₃	D₈	1	2	yes	—
101	C₄²×C₅₃	—	4	2	no	—
102	C₄²×C₅₃	—	4	1	no	—
103	C₄²×C₅₃	C₂×D₄	1	2	yes	—
104	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
105	#11719921	—	4	2	no	—
106	#11719921	—	4	2	no	—
107	#11719921	—	4	2	no	—
108	C₄⋊Dic₅₃	—	2	1	no	—
109	#11719919	—	4	1	no	—
110	C₄×Dic₅₃	—	4	1	no	—
111	Dic₂₁₂	—	2	1	no	—
112	Dic₂₁₂	—	2	1	no	—
113	D₄₂₄	—	2	2	no	—
114	D₄₂₄	—	2	2	no	—
115	D₄₂₄	—	8	1	no	—
116	#11719913	—	8	2	no	—
117	#11719908	—	2	2	no	—
118	#11719908	—	2	1	no	—
119	#11719908	—	2	1	no	—
120	C₂²×D₅₃	—	4	2	no	—
121	Q₈×C₅₃	—	2	2	no	—
122	Q₈×C₅₃	—	2	2	no	—
123	Q₈×C₅₃	—	4	1	no	—
124	Q₈×C₅₃	—	8	2	no	—
125	C₂×C₄×C₅₃	—	4	1	no	—
126	C₂×C₄×C₅₃	—	4	2	no	—
127	C₂×C₄×C₅₃	—	4	1	no	—
128	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×Dic₅₃	—	4	1	no	—
134	C₂×Dic₅₃	—	4	1	no	—
135	D₂₁₂	—	4	2	no	—
136	D₂₁₂	—	4	2	no	—
137	D₂₁₂	—	8	2	no	—
138	D₂₁₂	—	16	1	no	—
139	C₄×D₅₃	—	16	2	no	—
140	Dic₁₀₆	—	4	1	no	—
141	Dic₁₀₆	—	4	1	no	—
142	C₈×C₅₃	—	2	2	no	—
143	C₈×C₅₃	—	2	1	no	—
144	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
145	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
146	#33932	—	53	1	no	maximal, Sylow
147	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
148	C₂×D₅₃	—	8	2	no	—
149	C₂×D₅₃	—	8	2	no	—
150	C₂×D₅₃	—	8	2	no	—
151	C₄×C₅₃	—	2	2	no	—
152	C₄×C₅₃	—	2	2	no	—
153	C₄×C₅₃	—	2	1	no	—
154	C₄×C₅₃	—	4	2	no	—
155	C₄×C₅₃	—	8	2	no	—
156	C₄×C₅₃	—	8	1	no	—
157	C₄×C₅₃	#508	1	2	yes	—
158	C₄×C₅₃	#548	1	1	yes	characteristic
159	C₄×C₅₃	#577	1	1	yes	characteristic
160	Dic₅₃	—	8	1	no	—
161	Dic₅₃	—	8	1	no	—
162	#3396	—	53	1	no	—
163	#3383	D₅₃	1	1	yes	characteristic, complemented
164	#2242	—	53	2	no	—
165	#2240	—	53	1	no	—
166	#2228	—	53	2	no	—
167	#2225	—	53	1	no	—
168	#2211	—	53	2	no	—
169	#2207	—	53	1	no	—
170	Q₈×C₁₆	—	53	1	no	—
171	C₄×Q₃₂	—	53	1	no	—
172	C₄×SD₃₂	—	53	2	no	—
173	C₂×C₅₃	#3358	1	1	yes	characteristic
174	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
175	C₂×C₅₃	#3416	1	1	yes	characteristic
176	D₅₃	—	16	2	no	—
177	#559	—	106	1	no	—
178	#557	—	2	1	no	—
179	C₂×Q₃₂	—	53	1	no	—
180	C₂×SD₃₂	—	53	2	no	—
181	#500	C₂×D₅₃	1	1	yes	characteristic
182	#499	C₂×D₅₃	1	2	yes	—
183	#495	—	53	2	no	—
184	#493	—	53	1	no	—
185	#479	—	2	2	no	—
186	#475	—	2	1	no	—
187	#463	—	106	2	no	—
188	#460	—	106	1	no	—
189	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
190	C₄×Q₁₆	—	53	1	no	—
191	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
192	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
193	C₄×D₈	—	53	2	no	—
194	#367	—	53	2	no	—
195	#366	—	53	1	no	—
196	C₄⋊C₁₆	—	53	1	no	—
197	C₄⋊C₁₆	—	53	2	no	—
198	#358	—	53	1	no	—
199	#358	—	53	1	no	—
200	#358	—	53	2	no	—
201	#357	—	53	2	no	—
202	#357	—	53	2	no	—
203	#357	—	53	2	no	—
204	C₄×C₁₆	—	53	2	no	—
205	C₄×C₁₆	—	53	1	no	—
206	C₅₃	#33932	1	1	yes	characteristic, complemented, Sylow
207	C₂×Q₁₆	—	53	1	no	—
208	C₂×Q₁₆	C₂²×D₅₃	1	1	yes	characteristic
209	C₂×SD₁₆	—	106	2	no	—
210	C₂×D₈	—	53	2	no	—
211	C₂×D₈	—	106	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	4	1	no	—
214	#127	—	106	1	no	—
215	#125	—	2	2	no	—
216	#125	—	4	2	no	—
217	#124	—	106	2	no	—
218	#124	—	212	1	no	—
219	#123	—	212	2	no	—
220	C₄×Q₈	—	2	1	no	—
221	C₄×Q₈	—	2	2	no	—
222	C₄×Q₈	—	4	1	no	—
223	C₄×Q₈	—	106	1	no	—
224	C₄×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
225	C₄×D₄	—	106	2	no	—
226	C₄×D₄	—	212	1	no	—
227	Q₃₂	—	106	2	no	—
228	SD₃₂	—	106	4	no	—
229	C₂×C₁₆	—	53	1	no	—
230	C₂×C₁₆	—	53	1	no	—
231	C₂×C₁₆	—	53	2	no	—
232	#107	—	2	1	no	—
233	#107	—	53	1	no	—
234	#107	C₂²×D₅₃	1	1	yes	characteristic
235	#107	C₂²×D₅₃	1	2	yes	—
236	#106	—	2	2	no	—
237	C₄⋊C₈	C₂²⋊D₅₃	1	1	yes	characteristic
238	C₄⋊C₈	C₂²⋊D₅₃	1	2	yes	—
239	#103	—	2	2	no	—
240	#103	—	2	1	no	—
241	#103	—	106	1	no	—
242	#102	—	106	2	no	—
243	C₄×C₈	C₂²×D₅₃	1	2	yes	—
244	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	106	1	no	—
248	C₂×Q₈	#11719922	1	1	yes	characteristic
249	C₂×D₄	—	106	2	no	—
250	C₂×D₄	—	212	1	no	—
251	C₂×D₄	—	212	1	no	—
252	C₂²×C₄	—	212	2	no	—
253	Q₁₆	—	2	1	no	—
254	Q₁₆	—	106	1	no	—
255	Q₁₆	—	106	1	no	—
256	Q₁₆	#11719949	1	2	yes	—
257	SD₁₆	—	424	2	no	—
258	D₈	—	106	2	no	—
259	D₈	—	106	2	no	—
260	D₈	—	424	1	no	—
261	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
262	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
263	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	106	1	no	—
272	C₄⋊C₄	—	212	1	no	—
273	C₄⋊C₄	#11719922	1	1	yes	characteristic
274	C₄⋊C₄	#11719922	1	2	yes	—
275	C₂²⋊C₄	—	212	2	no	—
276	C₂²⋊C₄	—	212	2	no	—
277	C₂²⋊C₄	—	212	2	no	—
278	C₄²	—	4	2	no	—
279	C₄²	—	4	1	no	—
280	C₄²	—	212	1	no	—
281	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
282	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
283	C₁₆	—	106	2	no	—
284	C₁₆	—	106	1	no	—
285	C₁₆	—	106	1	no	—
286	C₂³	—	212	2	no	—
287	Q₈	—	2	2	no	—
288	Q₈	—	2	2	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	8	2	no	—
291	Q₈	—	212	1	no	—
292	Q₈	—	212	1	no	—
293	D₄	—	848	1	no	—
294	D₄	—	424	2	no	—
295	D₄	—	212	2	no	—
296	D₄	—	212	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	848	2	no	—
301	C₂×C₄	—	212	1	no	—
302	C₂×C₄	—	212	1	no	—
303	C₂×C₄	C₂×#11719922	1	2	yes	—
304	C₂×C₄	C₂×#11719922	1	2	yes	—
305	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
306	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
307	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	2	no	—
310	C₈	C₂×#11719949	1	1	yes	characteristic
311	C₈	#13133265	1	1	yes	characteristic
312	C₂²	—	424	2	no	—
313	C₂²	—	424	2	no	—
314	C₂²	—	424	2	no	—
315	C₂²	C₂²×#11719922	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	1	no	—
319	C₄	—	4	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	1	no	—
322	C₄	—	424	1	no	—
323	C₄	—	424	1	no	—
324	C₄	#16811053	1	2	yes	—
325	C₄	#16811284	1	1	yes	characteristic
326	C₄	#16811431	1	1	yes	characteristic
327	C₂	—	848	2	no	—
328	C₂	#20287350	1	1	yes	characteristic
329	C₂	C₂×#16811053	1	1	yes	characteristic
330	C₂	#20287859	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#3383	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#16811000	C₂²
hypercenter	#3383	D₅₃
hypocenter	C₅₃	#33932
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3383

Complements

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

N	H ≅ G/N	type
2. #20276949	C₂	semidirect
4. #20276869	C₂	semidirect
5. #20276854	C₂	semidirect
6. #20276777	C₂	semidirect
8. Q₈×#11719908	C₂	semidirect
9. C₄×#13133163	C₂	semidirect
10. C₄×#13133160	C₂	semidirect
11. C₅₃×#3383	C₂	semidirect
163. #3383	D₅₃	semidirect
206. C₅₃	#33932	semidirect

Maximal subgroups

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

subgroup	index
#20287728	2
#20276949	2
#20276940	2
#20276869	2
#20276854	2
#20276777	2
#20276755	2
Q₈×#11719908	2
C₄×#13133163	2
C₄×#13133160	2
C₅₃×#3383	2
#33932	53