Subgroups — #27539680

Order 13568.

508 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285347	C₂	1	2	yes	maximal, complemented
2	#20276989	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276973	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276950	C₂	1	2	yes	maximal, complemented
5	#20276826	C₂	1	2	yes	maximal, complemented
6	#20276806	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276576	C₂	1	2	yes	maximal, complemented
8	#20276573	C₂	1	1	yes	maximal, characteristic, complemented
9	D₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#20276311	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2215	C₂	1	1	yes	maximal, characteristic
12	#16811240	—	2	2	no	—
13	C₂×Dic₄₂₄	C₂²	1	1	yes	characteristic
14	C₂×#13133161	C₂²	1	2	yes	—
15	C₂×#13133157	C₂²	1	1	yes	characteristic
16	#16811027	C₂²	1	2	yes	—
17	#16811006	C₂²	1	2	yes	—
18	#16810981	—	2	2	no	—
19	#16810867	—	2	2	no	—
20	#16810864	C₂²	1	2	yes	—
21	#16810720	—	2	2	no	—
22	#16810717	C₂²	1	2	yes	—
23	D₈×Dic₅₃	C₂²	1	2	yes	—
24	#16810578	C₂²	1	1	yes	characteristic
25	#16810449	C₂²	1	1	yes	characteristic
26	#16810448	C₂²	1	2	yes	—
27	#16810428	C₂²	1	2	yes	—
28	#16810426	C₂²	1	2	yes	—
29	#16810424	C₂²	1	2	yes	—
30	#16810398	C₂²	1	1	yes	characteristic
31	#16810398	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	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
36	C₅₃×#466	C₂²	1	2	yes	—
37	C₅₃×#366	C₂²	1	1	yes	characteristic
38	C₅₃×#357	C₂²	1	2	yes	—
39	C₂×Dic₂₁₂	C₂³	1	2	yes	—
40	C₂×#11719923	—	2	2	no	—
41	C₂×#11719917	D₄	1	1	yes	characteristic
42	#13133216	—	2	2	no	—
43	#13133210	—	4	2	no	—
44	#13133204	—	2	2	no	—
45	#13133204	—	4	2	no	—
46	#13133198	—	4	2	no	—
47	#13133196	—	2	2	no	—
48	#13133194	—	4	2	no	—
49	#13133182	—	2	2	no	—
50	D₄×Dic₅₃	—	2	2	no	—
51	#13133176	D₄	1	1	yes	characteristic
52	Dic₄₂₄	—	2	2	no	—
53	#13133161	—	2	4	no	—
54	#13133157	—	2	2	no	—
55	C₂×#11719908	C₂³	1	1	yes	characteristic
56	C₂×#11719908	C₂³	1	1	yes	characteristic
57	#13133151	C₂³	1	2	yes	—
58	#13133150	—	2	2	no	—
59	#13133142	—	2	2	no	—
60	#13133140	—	2	2	no	—
61	#13133138	—	2	2	no	—
62	#13133127	D₄	1	1	yes	characteristic
63	#13133127	D₄	1	1	yes	characteristic
64	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
65	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
66	C₂×D₈×C₅₃	C₂³	1	2	yes	—
67	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	#121×C₅₃	—	2	2	no	—
69	D₁₆×C₅₃	—	2	2	no	—
70	C₂×C₁₆×C₅₃	—	2	2	no	—
71	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
72	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
73	#107×C₅₃	C₂³	1	2	yes	—
74	#102×C₅₃	—	2	2	no	—
75	C₂×D₄×C₅₃	—	2	2	no	—
76	C₂×D₄×C₅₃	—	4	2	no	—
77	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×Dic₅₃	—	4	2	no	—
79	C₂×Dic₁₀₆	—	2	2	no	—
80	C₂×Dic₁₀₆	—	4	2	no	—
81	D₈×C₅₃	—	2	2	no	—
82	D₈×C₅₃	—	2	2	no	—
83	C₂×C₈×C₅₃	—	2	1	no	—
84	C₂×C₈×C₅₃	—	2	1	no	—
85	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
87	C₄⋊C₄×C₅₃	—	2	2	no	—
88	C₂²⋊C₄×C₅₃	—	4	2	no	—
89	C₂²⋊Dic₅₃	—	4	2	no	—
90	C₂²⋊Dic₅₃	—	4	2	no	—
91	C₂²⋊Dic₅₃	—	4	2	no	—
92	C₂²⋊Dic₅₃	—	4	2	no	—
93	C₂²⋊Dic₅₃	D₈	1	1	yes	characteristic
94	C₂²⋊Dic₅₃	D₈	1	1	yes	characteristic
95	#11719923	—	4	4	no	—
96	C₄⋊Dic₅₃	—	2	2	no	—
97	C₄⋊Dic₅₃	D₈	1	1	yes	characteristic
98	C₄⋊Dic₅₃	D₈	1	1	yes	characteristic
99	#11719919	—	4	2	no	—
100	#11719919	—	4	2	no	—
101	#11719919	—	4	2	no	—
102	C₄×Dic₅₃	—	4	2	no	—
103	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
104	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
105	#11719917	—	2	2	no	—
106	C₂×#88322	C₂×D₄	1	1	yes	characteristic
107	C₂×#88322	C₂×D₄	1	1	yes	characteristic
108	Dic₂₁₂	—	2	2	no	—
109	Dic₂₁₂	—	2	2	no	—
110	C₁₆×C₅₃	—	2	1	no	—
111	C₁₆×C₅₃	#41	1	2	yes	—
112	#11719908	—	2	1	no	—
113	#11719908	#41	1	2	yes	—
114	C₂³×C₅₃	—	4	2	no	—
115	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
116	D₄×C₅₃	—	4	2	no	—
117	D₄×C₅₃	—	4	2	no	—
118	D₄×C₅₃	—	16	2	no	—
119	C₂×C₄×C₅₃	—	2	1	no	—
120	C₂×C₄×C₅₃	—	2	1	no	—
121	C₂×C₄×C₅₃	—	4	2	no	—
122	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×Dic₅₃	—	4	2	no	—
125	C₂×Dic₅₃	—	4	2	no	—
126	C₂×Dic₅₃	—	16	2	no	—
127	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
128	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
129	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
131	Dic₁₀₆	—	4	2	no	—
132	Dic₁₀₆	—	4	2	no	—
133	Dic₁₀₆	—	16	2	no	—
134	C₈×C₅₃	—	2	1	no	—
135	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
136	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
137	#88322	—	2	1	no	—
138	#88322	#135	1	2	yes	—
139	#33739	—	53	1	no	maximal, Sylow
140	C₂²×C₅₃	—	2	1	no	—
141	C₂²×C₅₃	—	2	1	no	—
142	C₂²×C₅₃	—	2	1	no	—
143	C₂²×C₅₃	—	8	2	no	—
144	C₂²×C₅₃	—	8	2	no	—
145	C₂²×C₅₃	—	8	2	no	—
146	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
147	C₄×C₅₃	—	2	1	no	—
148	C₄×C₅₃	—	8	2	no	—
149	C₄×C₅₃	#532	1	1	yes	characteristic
150	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
151	Dic₅₃	—	2	1	no	—
152	Dic₅₃	—	2	1	no	—
153	Dic₅₃	—	2	1	no	—
154	Dic₅₃	—	2	1	no	—
155	Dic₅₃	—	8	2	no	—
156	Dic₅₃	—	8	2	no	—
157	#3175	—	53	2	no	—
158	#2252	—	53	1	no	—
159	#2249	—	53	1	no	—
160	#2243	—	53	2	no	—
161	#2220	—	53	2	no	—
162	#2216	—	53	1	no	—
163	#2215	D₅₃	1	1	yes	characteristic
164	#2175	—	53	2	no	—
165	C₄×Q₃₂	—	53	1	no	—
166	C₄×D₁₆	—	53	1	no	—
167	#2108	—	53	1	no	—
168	C₂×C₅₃	—	4	1	no	—
169	C₂×C₅₃	—	16	2	no	—
170	C₂×C₅₃	#3169	1	1	yes	characteristic
171	C₂×C₅₃	#3415	1	1	yes	characteristic
172	C₂×C₅₃	#3417	1	1	yes	characteristic
173	#542	—	106	2	no	—
174	C₂×Q₃₂	—	53	1	no	—
175	C₂×SD₃₂	—	53	2	no	—
176	C₂×D₁₆	C₂×D₅₃	1	1	yes	characteristic
177	C₂×M₃₂	—	53	1	no	—
178	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
179	#500	—	53	2	no	—
180	#495	—	53	2	no	—
181	#489	—	106	2	no	—
182	#468	—	106	2	no	—
183	#467	—	53	2	no	—
184	#466	C₂×D₅₃	1	2	yes	—
185	#440	—	106	2	no	—
186	C₄×Q₁₆	—	53	2	no	—
187	C₄×D₈	—	53	2	no	—
188	#405	—	53	1	no	—
189	#366	—	53	1	no	—
190	#366	—	53	2	no	—
191	#366	C₂×D₅₃	1	1	yes	characteristic
192	#358	—	53	2	no	—
193	#358	—	53	2	no	—
194	#357	—	53	2	no	—
195	#357	C₂×D₅₃	1	2	yes	—
196	#346	—	53	1	no	—
197	#346	—	53	1	no	—
198	C₄×C₁₆	—	53	1	no	—
199	C₄×C₁₆	—	53	1	no	—
200	C₅₃	#33739	1	1	yes	characteristic, complemented, Sylow
201	C₂×Q₁₆	—	53	2	no	—
202	C₂×SD₁₆	—	106	2	no	—
203	C₂×D₈	C₂²×D₅₃	1	2	yes	—
204	C₂×M₁₆	—	53	1	no	—
205	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
206	C₄⋊Q₈	—	106	2	no	—
207	#126	—	212	2	no	—
208	#124	—	106	2	no	—
209	#124	—	212	2	no	—
210	#123	—	212	2	no	—
211	C₂²⋊Q₈	—	106	2	no	—
212	C₂²⋊Q₈	—	212	2	no	—
213	#121	—	2	2	no	—
214	C₄×Q₈	—	106	2	no	—
215	C₄×D₄	—	106	2	no	—
216	#117	—	53	1	no	—
217	Q₃₂	—	106	2	no	—
218	SD₃₂	—	106	4	no	—
219	D₁₆	—	2	2	no	—
220	M₃₂	—	106	2	no	—
221	C₂×C₁₆	—	2	2	no	—
222	C₂×C₁₆	—	53	1	no	—
223	C₂×C₁₆	—	53	1	no	—
224	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
225	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
226	#107	—	53	2	no	—
227	#107	—	106	2	no	—
228	#107	C₂²×D₅₃	1	2	yes	—
229	#103	—	106	2	no	—
230	#103	—	106	2	no	—
231	#102	—	2	2	no	—
232	#102	—	106	2	no	—
233	#97	—	53	1	no	—
234	#97	—	53	1	no	—
235	C₄×C₈	—	53	1	no	—
236	C₄×C₈	—	53	1	no	—
237	C₂×Q₈	—	106	2	no	—
238	C₂×Q₈	—	212	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	4	2	no	—
241	C₂²×C₄	—	212	2	no	—
242	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
243	Q₁₆	—	106	2	no	—
244	Q₁₆	—	106	2	no	—
245	SD₁₆	—	212	4	no	—
246	D₈	—	2	2	no	—
247	D₈	—	2	2	no	—
248	M₁₆	—	106	2	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	53	1	no	—
252	C₂×C₈	—	53	1	no	—
253	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
254	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	53	1	no	—
257	C₄⋊C₄	—	53	1	no	—
258	C₄⋊C₄	—	106	2	no	—
259	C₄⋊C₄	—	212	2	no	—
260	C₄⋊C₄	—	212	2	no	—
261	C₄⋊C₄	—	212	2	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	53	1	no	—
264	C₂²⋊C₄	—	53	1	no	—
265	C₂²⋊C₄	—	212	2	no	—
266	C₂²⋊C₄	—	212	2	no	—
267	C₂²⋊C₄	—	212	2	no	—
268	C₂²⋊C₄	—	212	2	no	—
269	C₄²	—	53	1	no	—
270	C₄²	—	53	1	no	—
271	C₄²	—	212	2	no	—
272	C₁₆	—	2	1	no	—
273	C₁₆	—	53	2	no	—
274	C₁₆	—	106	1	no	—
275	C₁₆	#11719947	1	2	yes	—
276	C₂³	—	4	2	no	—
277	C₂³	D₈×D₅₃	1	1	yes	characteristic
278	Q₈	—	848	2	no	—
279	Q₈	—	212	2	no	—
280	Q₈	—	212	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	16	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	53	1	no	—
288	C₂×C₄	—	53	1	no	—
289	C₂×C₄	—	53	1	no	—
290	C₂×C₄	—	53	1	no	—
291	C₂×C₄	—	848	2	no	—
292	C₂×C₄	—	212	2	no	—
293	C₂×C₄	—	212	2	no	—
294	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
295	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	53	2	no	—
298	C₈	—	106	1	no	—
299	C₈	#41×D₅₃	1	1	yes	characteristic
300	C₈	C₂×#11719947	1	1	yes	characteristic
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	8	2	no	—
310	C₄	—	106	1	no	—
311	C₄	—	106	1	no	—
312	C₄	—	106	1	no	—
313	C₄	—	106	1	no	—
314	C₄	—	424	2	no	—
315	C₄	—	424	2	no	—
316	C₄	#16811168	1	1	yes	characteristic
317	C₄	#135×D₅₃	1	1	yes	characteristic
318	C₂	—	4	1	no	—
319	C₂	—	16	2	no	—
320	C₂	#20285289	1	1	yes	characteristic
321	C₂	#20287852	1	1	yes	characteristic
322	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, prime radical	C₅₃×#2215	C₂
socle	C₂²×C₅₃	C₂²×D₈
hypercenter	#2215	D₅₃
hypocenter	C₅₃	#33739
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2215

Complements

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

N	H ≅ G/N	type
1. #20285347	C₂	semidirect
2. #20276989	C₂	semidirect
3. #20276973	C₂	semidirect
4. #20276950	C₂	semidirect
5. #20276826	C₂	semidirect
6. #20276806	C₂	semidirect
7. #20276576	C₂	semidirect
8. #20276573	C₂	semidirect
9. D₁₆×Dic₅₃	C₂	semidirect
10. #20276311	C₂	semidirect
200. C₅₃	#33739	semidirect

Maximal subgroups

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

subgroup	index
#20285347	2
#20276989	2
#20276973	2
#20276950	2
#20276826	2
#20276806	2
#20276576	2
#20276573	2
D₁₆×Dic₅₃	2
#20276311	2
C₅₃×#2215	2
#33739	53