Subgroups — #27541720

Order 13568.

470 conjugacy classes of subgroups.

The subgroup lattice has 326 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
2	#20287107	C₂	1	1	yes	maximal, characteristic
3	#20276995	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276925	C₂	1	2	yes	maximal, complemented
5	#20276911	C₂	1	2	yes	maximal, complemented
6	#20276858	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276822	C₂	1	2	yes	maximal, complemented
8	C₄×#13133163	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276546	C₂	1	1	yes	maximal, characteristic
10	#20276500	C₂	1	2	yes	maximal, complemented
11	C₅₃×#2227	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811346	—	2	1	no	—
13	#16811334	—	2	1	no	—
14	C₂×#13133163	C₂²	1	1	yes	characteristic
15	C₂×#13133156	C₂²	1	2	yes	—
16	#16811028	C₂²	1	1	yes	characteristic
17	#16810963	—	2	2	no	—
18	#16810960	—	2	2	no	—
19	#16810915	—	2	1	no	—
20	#16810908	—	2	1	no	—
21	#16810865	C₂²	1	2	yes	—
22	#16810861	C₂²	1	2	yes	—
23	C₄×Dic₂₁₂	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	#16810451	C₂²	1	2	yes	—
28	#16810450	C₂²	1	2	yes	—
29	#16810450	C₂²	1	2	yes	—
30	#16810448	C₂²	1	1	yes	characteristic
31	#16810428	C₂²	1	2	yes	—
32	#16810427	C₂²	1	1	yes	characteristic
33	#16810426	C₂²	1	1	yes	characteristic
34	#16810402	C₂²	1	2	yes	—
35	C₄×#11719908	C₂²	1	1	yes	characteristic
36	C₅₃×#500	C₂²	1	1	yes	characteristic
37	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#366	C₂²	1	2	yes	—
39	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
40	C₅₃×#358	C₂²	1	2	yes	—
41	C₂×Dic₂₁₂	C₂³	1	1	yes	characteristic
42	C₂×C₈×D₅₃	C₂³	1	2	yes	—
43	#13133217	—	2	1	no	—
44	#13133212	—	4	2	no	—
45	#13133208	—	4	1	no	—
46	#13133205	—	4	1	no	—
47	#13133197	—	2	2	no	—
48	#13133193	—	2	2	no	—
49	#13133193	—	4	2	no	—
50	C₄×Dic₁₀₆	—	2	1	no	—
51	C₄×Dic₁₀₆	—	4	1	no	—
52	C₄×D₂₁₂	D₄	1	1	yes	characteristic
53	#13133179	—	4	1	no	—
54	#13133175	—	2	2	no	—
55	C₄⋊C₄×D₅₃	—	2	2	no	—
56	#13133163	—	2	2	no	—
57	#13133156	—	2	4	no	—
58	C₂×#11719908	C₂³	1	1	yes	characteristic
59	C₂×#11719908	C₂³	1	1	yes	characteristic
60	#13133151	—	2	1	no	—
61	#13133151	C₂³	1	2	yes	—
62	#13133151	C₂³	1	1	yes	characteristic
63	#13133150	—	2	2	no	—
64	#13133148	—	2	2	no	—
65	C₄⋊#88322	D₄	1	1	yes	characteristic
66	#13133142	—	2	1	no	—
67	#13133141	—	2	2	no	—
68	#13133140	—	2	2	no	—
69	#13133129	D₄	1	2	yes	—
70	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
71	C₄⋊Q₈×C₅₃	—	2	1	no	—
72	C₄×Q₈×C₅₃	—	2	1	no	—
73	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
74	#107×C₅₃	—	2	1	no	—
75	#107×C₅₃	C₂³	1	2	yes	—
76	#107×C₅₃	C₂³	1	1	yes	characteristic
77	#103×C₅₃	—	2	1	no	—
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×D₂₁₂	D₈	1	1	yes	characteristic
82	C₂×C₄×D₅₃	—	4	2	no	—
83	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
84	C₂×Dic₁₀₆	—	2	1	no	—
85	C₂×Dic₁₀₆	—	4	1	no	—
86	Q₁₆×C₅₃	—	2	1	no	—
87	Q₁₆×C₅₃	—	2	1	no	—
88	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₅₃	—	2	1	no	—
91	C₄⋊C₄×C₅₃	—	2	2	no	—
92	C₄⋊C₄×C₅₃	—	4	1	no	—
93	C₄⋊C₄×C₅₃	—	4	1	no	—
94	C₄²×C₅₃	—	4	1	no	—
95	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
96	#11719921	—	4	2	no	—
97	#11719921	—	4	2	no	—
98	#11719921	—	4	2	no	—
99	#11719921	D₈	1	2	yes	—
100	C₄⋊Dic₅₃	—	2	2	no	—
101	C₄⋊Dic₅₃	—	2	1	no	—
102	C₄⋊Dic₅₃	—	4	1	no	—
103	C₄⋊Dic₅₃	—	4	2	no	—
104	C₄⋊Dic₅₃	D₈	1	1	yes	characteristic
105	#11719919	—	4	1	no	—
106	#11719919	—	4	2	no	—
107	#11719919	—	4	2	no	—
108	C₄×Dic₅₃	—	4	1	no	—
109	C₄×Dic₅₃	—	4	2	no	—
110	C₂×#88322	C₂×D₄	1	2	yes	—
111	Dic₂₁₂	—	2	1	no	—
112	Dic₂₁₂	—	2	1	no	—
113	C₈×D₅₃	—	2	2	no	—
114	C₈×D₅₃	—	2	2	no	—
115	C₁₆×C₅₃	—	2	2	no	—
116	#11719908	—	2	1	no	—
117	#11719908	#41	1	2	yes	—
118	C₂²×D₅₃	C₂×D₈	1	2	yes	—
119	Q₈×C₅₃	—	4	1	no	—
120	Q₈×C₅₃	—	4	1	no	—
121	Q₈×C₅₃	—	16	1	no	—
122	C₂×C₄×C₅₃	—	4	2	no	—
123	C₂×C₄×C₅₃	—	4	1	no	—
124	C₂×C₄×C₅₃	—	4	1	no	—
125	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×Dic₅₃	—	4	1	no	—
129	C₂×Dic₅₃	—	4	2	no	—
130	C₂×Dic₅₃	—	4	1	no	—
131	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
132	D₂₁₂	—	2	2	no	—
133	C₄×D₅₃	—	2	2	no	—
134	C₄×D₅₃	—	2	2	no	—
135	C₄×D₅₃	—	16	2	no	—
136	Dic₁₀₆	—	4	1	no	—
137	Dic₁₀₆	—	4	1	no	—
138	Dic₁₀₆	—	8	2	no	—
139	C₈×C₅₃	—	2	1	no	—
140	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
141	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
142	#88322	—	2	2	no	—
143	#33923	—	53	1	no	maximal, Sylow
144	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
145	C₂×D₅₃	—	2	2	no	—
146	C₂×D₅₃	—	2	2	no	—
147	C₂×D₅₃	—	2	2	no	—
148	C₄×C₅₃	—	2	1	no	—
149	C₄×C₅₃	—	8	1	no	—
150	C₄×C₅₃	—	8	2	no	—
151	C₄×C₅₃	—	8	1	no	—
152	C₄×C₅₃	#510	1	2	yes	—
153	C₄×C₅₃	#547	1	1	yes	characteristic
154	C₄×C₅₃	#578	1	1	yes	characteristic
155	Dic₅₃	—	2	2	no	—
156	Dic₅₃	—	8	1	no	—
157	Dic₅₃	—	8	2	no	—
158	Dic₅₃	—	8	1	no	—
159	#3342	—	53	1	no	—
160	#3338	—	53	1	no	—
161	#2254	—	53	1	no	—
162	#2237	—	53	2	no	—
163	#2235	—	53	2	no	—
164	#2227	D₅₃	1	1	yes	characteristic, complemented
165	#2225	—	53	1	no	—
166	#2219	—	53	2	no	—
167	C₄×Q₃₂	—	53	1	no	—
168	#2169	—	53	1	no	—
169	#2159	—	53	2	no	—
170	C₂×C₅₃	#3334	1	1	yes	characteristic
171	C₂×C₅₃	C₂×#510	1	1	yes	characteristic
172	C₂×C₅₃	#3416	1	1	yes	characteristic
173	D₅₃	—	4	2	no	—
174	#556	—	106	1	no	—
175	#554	—	106	1	no	—
176	C₂×Q₃₂	—	53	1	no	—
177	C₂×M₃₂	—	53	2	no	—
178	#500	—	53	1	no	—
179	#500	C₂×D₅₃	1	1	yes	characteristic
180	#485	—	106	2	no	—
181	#484	—	106	2	no	—
182	#477	—	106	1	no	—
183	#475	—	106	1	no	—
184	#467	—	53	2	no	—
185	#467	—	53	2	no	—
186	C₄×Q₁₆	—	53	1	no	—
187	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
188	C₈×D₄	—	53	1	no	—
189	#427	—	53	2	no	—
190	C₂×#107	—	53	2	no	—
191	#367	—	53	2	no	—
192	#367	—	53	2	no	—
193	#367	—	53	2	no	—
194	#366	—	53	1	no	—
195	#366	C₂×D₅₃	1	2	yes	—
196	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
197	#358	—	53	1	no	—
198	#358	—	53	2	no	—
199	#358	—	53	1	no	—
200	#358	C₂×D₅₃	1	2	yes	—
201	C₂²⋊C₁₆	—	53	2	no	—
202	C₄×C₁₆	—	53	1	no	—
203	C₅₃	#33923	1	1	yes	characteristic, complemented, Sylow
204	C₂×Q₁₆	—	53	1	no	—
205	C₂×Q₁₆	C₂²×D₅₃	1	1	yes	characteristic
206	C₂²×C₈	—	53	2	no	—
207	C₄⋊Q₈	—	2	1	no	—
208	C₄⋊Q₈	—	106	1	no	—
209	#126	—	212	2	no	—
210	#125	—	212	1	no	—
211	#124	—	212	1	no	—
212	C₂²⋊Q₈	—	106	2	no	—
213	C₂²⋊Q₈	—	106	2	no	—
214	C₂²⋊Q₈	—	212	2	no	—
215	C₄×Q₈	—	2	1	no	—
216	C₄×Q₈	—	106	1	no	—
217	C₄×Q₈	—	212	1	no	—
218	C₄×D₄	—	53	1	no	—
219	C₄×D₄	—	212	1	no	—
220	#117	—	106	2	no	—
221	C₂×C₄⋊C₄	—	106	2	no	—
222	Q₃₂	—	106	2	no	—
223	M₃₂	—	106	4	no	—
224	C₂×C₁₆	—	53	1	no	—
225	C₂×C₁₆	—	53	1	no	—
226	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
227	#107	—	2	1	no	—
228	#107	—	53	1	no	—
229	#107	—	53	2	no	—
230	#107	—	106	2	no	—
231	#107	—	106	1	no	—
232	#107	C₂²×D₅₃	1	1	yes	characteristic
233	#107	C₂²×D₅₃	1	2	yes	—
234	#106	—	106	2	no	—
235	C₄⋊C₈	—	53	1	no	—
236	#103	—	2	1	no	—
237	#103	—	106	2	no	—
238	#103	—	106	2	no	—
239	#103	—	106	1	no	—
240	C₂²⋊C₈	—	53	2	no	—
241	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	—	106	1	no	—
245	C₂×Q₈	—	212	1	no	—
246	C₂×D₄	—	53	1	no	—
247	C₂²×C₄	—	53	2	no	—
248	C₂²×C₄	—	212	2	no	—
249	Q₁₆	—	2	1	no	—
250	Q₁₆	—	2	1	no	—
251	Q₁₆	—	106	1	no	—
252	Q₁₆	—	106	1	no	—
253	C₂×C₈	—	53	2	no	—
254	C₂×C₈	—	106	2	no	—
255	C₂×C₈	—	106	2	no	—
256	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
257	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	53	1	no	—
263	C₄⋊C₄	—	106	2	no	—
264	C₄⋊C₄	—	106	1	no	—
265	C₄⋊C₄	—	212	2	no	—
266	C₄⋊C₄	—	212	2	no	—
267	C₄⋊C₄	—	212	2	no	—
268	C₄⋊C₄	—	212	1	no	—
269	C₄⋊C₄	—	212	1	no	—
270	C₂²⋊C₄	—	53	2	no	—
271	C₂²⋊C₄	—	212	2	no	—
272	C₂²⋊C₄	—	212	2	no	—
273	C₂²⋊C₄	—	212	2	no	—
274	C₄²	—	4	1	no	—
275	C₄²	—	212	2	no	—
276	C₄²	—	212	1	no	—
277	C₄²	D₄×D₅₃	1	1	yes	characteristic
278	C₁₆	—	2	2	no	—
279	C₁₆	—	53	2	no	—
280	C₁₆	—	106	1	no	—
281	C₂³	—	53	2	no	—
282	Q₈	—	4	1	no	—
283	Q₈	—	4	1	no	—
284	Q₈	—	16	1	no	—
285	Q₈	—	424	2	no	—
286	Q₈	—	212	1	no	—
287	Q₈	—	212	1	no	—
288	D₄	—	106	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	53	2	no	—
293	C₂×C₄	—	848	2	no	—
294	C₂×C₄	—	106	2	no	—
295	C₂×C₄	—	106	2	no	—
296	C₂×C₄	—	212	2	no	—
297	C₂×C₄	—	212	1	no	—
298	C₂×C₄	—	212	1	no	—
299	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
300	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
301	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	106	2	no	—
304	C₈	#41×D₅₃	1	1	yes	characteristic
305	C₈	#13133265	1	1	yes	characteristic
306	C₂²	—	106	2	no	—
307	C₂²	—	106	2	no	—
308	C₂²	—	106	2	no	—
309	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	8	1	no	—
312	C₄	—	8	2	no	—
313	C₄	—	8	1	no	—
314	C₄	—	106	2	no	—
315	C₄	—	424	1	no	—
316	C₄	—	424	2	no	—
317	C₄	—	424	1	no	—
318	C₄	#16811065	1	2	yes	—
319	C₄	#16811275	1	1	yes	characteristic
320	C₄	#16811439	1	1	yes	characteristic
321	C₂	—	212	2	no	—
322	C₂	#20287061	1	1	yes	characteristic
323	C₂	C₂×#16811065	1	1	yes	characteristic
324	C₂	#20287861	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×#2227	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×D₂₁₂	D₈
hypercenter	#2227	D₅₃
hypocenter	C₅₃	#33923
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2227

Complements

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

N	H ≅ G/N	type
3. #20276995	C₂	semidirect
4. #20276925	C₂	semidirect
5. #20276911	C₂	semidirect
6. #20276858	C₂	semidirect
7. #20276822	C₂	semidirect
8. C₄×#13133163	C₂	semidirect
10. #20276500	C₂	semidirect
11. C₅₃×#2227	C₂	semidirect
164. #2227	D₅₃	semidirect
203. C₅₃	#33923	semidirect

Maximal subgroups

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

subgroup	index
#20287147	2
#20287107	2
#20276995	2
#20276925	2
#20276911	2
#20276858	2
#20276822	2
C₄×#13133163	2
#20276546	2
#20276500	2
C₅₃×#2227	2
#33923	53