Subgroups — #27539645

Order 13568.

526 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	#20285285	C₂	1	2	yes	maximal, complemented
2	#20276994	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276976	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276947	C₂	1	2	yes	maximal, complemented
5	#20276811	C₂	1	2	yes	maximal, complemented
6	#20276806	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276801	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13133162	C₂	1	2	yes	maximal, complemented
10	C₄×D₈₄₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811157	—	2	2	no	—
13	C₂×Dic₄₂₄	C₂²	1	1	yes	characteristic
14	C₂×#13133162	C₂²	1	2	yes	—
15	C₂×D₈₄₈	C₂²	1	1	yes	characteristic
16	#16811025	C₂²	1	2	yes	—
17	#16811007	C₂²	1	2	yes	—
18	#16810978	—	2	2	no	—
19	#16810864	C₂²	1	2	yes	—
20	#16810859	C₂²	1	2	yes	—
21	#16810852	—	2	2	no	—
22	C₄×Dic₂₁₂	C₂²	1	2	yes	—
23	C₄×#11719913	—	2	2	no	—
24	C₄×D₄₂₄	C₂²	1	2	yes	—
25	#16810451	C₂²	1	2	yes	—
26	#16810449	C₂²	1	1	yes	characteristic
27	#16810449	C₂²	1	1	yes	characteristic
28	#16810428	C₂²	1	2	yes	—
29	#16810428	C₂²	1	2	yes	—
30	#16810425	C₂²	1	2	yes	—
31	#16810425	C₂²	1	2	yes	—
32	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
33	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
36	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
37	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
39	C₂×Dic₂₁₂	C₂³	1	2	yes	—
40	C₂×#11719913	—	2	2	no	—
41	C₂×D₄₂₄	C₂³	1	2	yes	—
42	#13133211	—	4	2	no	—
43	#13133209	—	2	2	no	—
44	#13133205	—	2	2	no	—
45	#13133200	—	4	2	no	—
46	#13133196	—	2	2	no	—
47	#13133191	—	2	2	no	—
48	C₄×Dic₁₀₆	—	2	2	no	—
49	C₄×D₂₁₂	—	2	2	no	—
50	C₄×C₂²⋊D₅₃	—	4	2	no	—
51	#13133174	—	4	2	no	—
52	Dic₄₂₄	—	2	2	no	—
53	#13133162	—	2	4	no	—
54	D₈₄₈	—	2	2	no	—
55	#13133151	C₂³	1	2	yes	—
56	#13133151	C₂³	1	2	yes	—
57	#13133149	—	2	2	no	—
58	#13133142	—	2	2	no	—
59	#13133142	—	2	2	no	—
60	#13133139	—	2	2	no	—
61	#13133139	—	2	2	no	—
62	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
65	C₂×C₁₆×C₅₃	—	2	2	no	—
66	C₂×C₁₆×C₅₃	—	2	2	no	—
67	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
69	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
70	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
72	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
73	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₂²⋊D₅₃	—	4	2	no	—
79	C₂×D₂₁₂	—	2	2	no	—
80	C₂×C₄×D₅₃	—	4	2	no	—
81	C₂×Dic₁₀₆	—	2	2	no	—
82	C₂×C₈×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	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
89	C₄²×C₅₃	D₈	1	1	yes	characteristic
90	C₄²×C₅₃	D₈	1	1	yes	characteristic
91	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂²⋊Dic₅₃	—	4	2	no	—
94	#11719921	—	4	2	no	—
95	#11719921	—	4	2	no	—
96	#11719921	—	4	2	no	—
97	#11719921	—	4	2	no	—
98	C₄⋊Dic₅₃	—	2	2	no	—
99	C₄⋊Dic₅₃	—	2	2	no	—
100	#11719919	—	4	2	no	—
101	#11719919	—	4	2	no	—
102	#11719919	—	4	2	no	—
103	#11719919	—	4	2	no	—
104	C₄×Dic₅₃	—	4	2	no	—
105	Dic₂₁₂	—	2	2	no	—
106	Dic₂₁₂	—	2	2	no	—
107	D₄₂₄	—	2	2	no	—
108	D₄₂₄	—	2	2	no	—
109	#11719913	—	4	4	no	—
110	C₁₆×C₅₃	—	2	1	no	—
111	C₁₆×C₅₃	—	2	1	no	—
112	C₁₆×C₅₃	#41	1	2	yes	—
113	C₁₆×C₅₃	#41	1	2	yes	—
114	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
115	C₂²×D₅₃	—	4	2	no	—
116	C₂×C₄×C₅₃	—	2	2	no	—
117	C₂×C₄×C₅₃	—	2	1	no	—
118	C₂×C₄×C₅₃	—	2	1	no	—
119	C₂×C₄×C₅₃	—	2	2	no	—
120	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
126	C₂²⋊D₅₃	—	8	4	no	—
127	C₂×Dic₅₃	—	4	2	no	—
128	C₂×Dic₅₃	—	4	2	no	—
129	C₂×Dic₅₃	—	4	2	no	—
130	D₂₁₂	—	4	2	no	—
131	D₂₁₂	—	4	2	no	—
132	C₄×D₅₃	—	8	4	no	—
133	Dic₁₀₆	—	4	2	no	—
134	Dic₁₀₆	—	4	2	no	—
135	C₈×C₅₃	—	2	1	no	—
136	C₈×C₅₃	—	2	1	no	—
137	C₈×C₅₃	#135	1	2	yes	—
138	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
139	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
140	#33735	—	53	1	no	maximal, Sylow
141	C₂²×C₅₃	—	2	1	no	—
142	C₂²×C₅₃	—	2	1	no	—
143	C₂²×C₅₃	#508	1	2	yes	—
144	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
145	C₂×D₅₃	—	8	2	no	—
146	C₂×D₅₃	—	8	2	no	—
147	C₂×D₅₃	—	8	2	no	—
148	C₄×C₅₃	—	2	1	no	—
149	C₄×C₅₃	—	2	1	no	—
150	C₄×C₅₃	—	2	1	no	—
151	C₄×C₅₃	—	2	1	no	—
152	C₄×C₅₃	#508	1	2	yes	—
153	C₄×C₅₃	#532	1	1	yes	characteristic
154	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
155	Dic₅₃	—	8	2	no	—
156	Dic₅₃	—	8	2	no	—
157	Dic₅₃	—	8	2	no	—
158	#3168	—	53	2	no	—
159	#2253	—	53	1	no	—
160	#2249	—	53	1	no	—
161	#2242	—	53	2	no	—
162	#2217	—	53	2	no	—
163	#2216	—	53	1	no	—
164	#2215	—	53	1	no	—
165	C₄×Q₃₂	—	53	1	no	—
166	C₄×SD₃₂	—	53	2	no	—
167	C₄×D₁₆	—	53	1	no	—
168	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic, complemented
169	C₂×C₅₃	—	2	2	no	—
170	C₂×C₅₃	#3169	1	1	yes	characteristic
171	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
172	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
173	D₅₃	—	16	2	no	—
174	#529	—	106	2	no	—
175	C₂×Q₃₂	—	53	1	no	—
176	C₂×SD₃₂	—	53	2	no	—
177	C₂×D₁₆	—	53	1	no	—
178	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
179	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
180	#499	—	53	2	no	—
181	#495	—	53	2	no	—
182	#488	—	106	2	no	—
183	#467	—	53	2	no	—
184	#466	—	53	2	no	—
185	#465	—	106	2	no	—
186	C₄×Q₁₆	—	53	2	no	—
187	C₄×SD₁₆	—	106	2	no	—
188	C₄×D₈	—	53	2	no	—
189	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
190	#367	—	53	2	no	—
191	#366	—	53	1	no	—
192	#366	—	53	1	no	—
193	#358	—	53	2	no	—
194	#358	—	53	2	no	—
195	#357	—	53	2	no	—
196	#357	—	53	2	no	—
197	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
198	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
199	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
200	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
201	C₅₃	#33735	1	1	yes	characteristic, complemented, Sylow
202	C₂×Q₁₆	—	53	2	no	—
203	C₂×SD₁₆	—	106	2	no	—
204	C₂×D₈	—	53	2	no	—
205	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
206	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
207	#126	—	212	2	no	—
208	#125	—	106	2	no	—
209	#124	—	106	2	no	—
210	#123	—	212	2	no	—
211	C₂²⋊Q₈	—	106	2	no	—
212	#121	—	106	2	no	—
213	C₄×Q₈	—	106	2	no	—
214	C₄×D₄	—	106	2	no	—
215	C₄×D₄	—	212	2	no	—
216	#117	—	212	2	no	—
217	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
218	Q₃₂	—	106	2	no	—
219	SD₃₂	—	106	4	no	—
220	D₁₆	—	106	2	no	—
221	C₂×C₁₆	—	2	2	no	—
222	C₂×C₁₆	—	2	2	no	—
223	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
224	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
225	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
226	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
227	#107	—	53	2	no	—
228	#107	—	53	2	no	—
229	#106	—	106	2	no	—
230	#103	—	106	2	no	—
231	#103	—	106	2	no	—
232	#102	—	106	2	no	—
233	#102	—	106	2	no	—
234	C₄×C₈	D₂₁₂	1	1	yes	characteristic
235	C₄×C₈	D₂₁₂	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
238	C₂×Q₈	—	106	2	no	—
239	C₂×D₄	—	106	2	no	—
240	C₂×D₄	—	212	2	no	—
241	C₂²×C₄	—	212	2	no	—
242	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
243	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
245	Q₁₆	—	106	2	no	—
246	Q₁₆	—	106	2	no	—
247	SD₁₆	—	212	4	no	—
248	D₈	—	106	2	no	—
249	D₈	—	106	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
254	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
255	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
256	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
257	C₄⋊C₄	—	106	2	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	C₄⋊C₄	—	212	2	no	—
263	C₂²⋊C₄	—	212	2	no	—
264	C₂²⋊C₄	—	212	2	no	—
265	C₂²⋊C₄	—	212	2	no	—
266	C₂²⋊C₄	—	212	2	no	—
267	C₂²⋊C₄	—	212	2	no	—
268	C₄²	—	212	2	no	—
269	C₄²	D₄₂₄	1	1	yes	characteristic
270	C₄²	D₄₂₄	1	1	yes	characteristic
271	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
272	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
273	C₁₆	—	2	1	no	—
274	C₁₆	—	2	1	no	—
275	C₁₆	#11719945	1	2	yes	—
276	C₁₆	#11719945	1	2	yes	—
277	C₂³	—	212	2	no	—
278	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
279	Q₈	—	212	2	no	—
280	Q₈	—	212	2	no	—
281	D₄	—	424	4	no	—
282	D₄	—	212	2	no	—
283	D₄	—	212	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	424	4	no	—
289	C₂×C₄	—	212	2	no	—
290	C₂×C₄	—	212	2	no	—
291	C₂×C₄	—	212	2	no	—
292	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
293	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
294	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
295	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
296	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
297	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	#13133239	1	2	yes	—
301	C₈	C₂×#11719945	1	1	yes	characteristic
302	C₈	C₂×#11719945	1	1	yes	characteristic
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	424	2	no	—
306	C₂²	—	424	2	no	—
307	C₂²	—	424	2	no	—
308	C₂²	#16811055	1	2	yes	—
309	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	424	2	no	—
315	C₄	—	424	2	no	—
316	C₄	—	424	2	no	—
317	C₄	#16811055	1	2	yes	—
318	C₄	#16811173	1	1	yes	characteristic
319	C₄	C₂×#13133239	1	1	yes	characteristic
320	C₂	—	2	2	no	—
321	C₂	—	848	2	no	—
322	C₂	#20285294	1	1	yes	characteristic
323	C₂	C₂×#16811055	1	1	yes	characteristic
324	C₂	C₂×#16811055	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄₂₄
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#20276801	C₂
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	#33735
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. #20285285	C₂	semidirect
2. #20276994	C₂	semidirect
3. #20276976	C₂	semidirect
4. #20276947	C₂	semidirect
5. #20276811	C₂	semidirect
6. #20276806	C₂	semidirect
8. C₄×Dic₄₂₄	C₂	semidirect
9. C₄×#13133162	C₂	semidirect
10. C₄×D₈₄₈	C₂	semidirect
11. C₂×C₄×C₁₆×C₅₃	C₂	semidirect
168. C₂×C₄×C₁₆	D₅₃	semidirect
201. C₅₃	#33735	semidirect

Maximal subgroups

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

subgroup	index
#20285285	2
#20276994	2
#20276976	2
#20276947	2
#20276811	2
#20276806	2
#20276801	2
C₄×Dic₄₂₄	2
C₄×#13133162	2
C₄×D₈₄₈	2
C₂×C₄×C₁₆×C₅₃	2
#33735	53