Subgroups — #27537752

Order 13568.

460 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283644	C₂	1	1	yes	maximal, characteristic
2	#20276624	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276623	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276622	C₂	1	2	yes	maximal, complemented
5	C₁₆×Dic₁₀₆	C₂	1	2	yes	maximal, complemented
6	#20276617	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276521	C₂	1	2	yes	maximal
8	#20276520	C₂	1	2	yes	maximal
9	#20276513	C₂	1	1	yes	maximal, characteristic
10	#20276512	C₂	1	1	yes	maximal, characteristic
11	Q₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811177	C₄	1	1	yes	characteristic
13	#16810754	C₄	1	1	yes	characteristic
14	#16810752	C₄	1	2	yes	—
15	#16810750	C₄	1	1	yes	characteristic
16	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
17	C₈×Dic₁₀₆	C₂²	1	2	yes	—
18	#16810648	C₄	1	2	yes	—
19	#16810645	C₄	1	1	yes	characteristic
20	#16810577	C₂²	1	2	yes	—
21	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
22	C₄⋊#11719908	C₂²	1	1	yes	characteristic
23	C₄⋊#11719908	C₂²	1	2	yes	—
24	#16810441	C₂²	1	1	yes	characteristic
25	#16810441	C₂²	1	1	yes	characteristic
26	#16810441	C₂²	1	2	yes	—
27	#16810441	C₂²	1	2	yes	—
28	#16810402	C₂²	1	1	yes	characteristic
29	#16810402	C₂²	1	1	yes	characteristic
30	#16810402	C₂²	1	2	yes	—
31	#16810399	C₂²	1	2	yes	—
32	#16810398	C₂²	1	2	yes	—
33	C₄×#11719908	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₅₃	C₂²	1	2	yes	—
35	C₁₆×Dic₅₃	C₂²	1	2	yes	—
36	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
38	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
39	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
41	C₂×#11719912	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
44	#13133216	C₈	1	1	yes	characteristic
45	#13133215	C₈	1	1	yes	characteristic
46	#13133207	C₈	1	2	yes	—
47	#13133197	C₈	1	1	yes	characteristic
48	#13133195	C₈	1	2	yes	—
49	#13133193	C₈	1	1	yes	characteristic
50	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
51	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
52	#13133174	C₂×C₄	1	2	yes	—
53	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×#11719908	C₂³	1	1	yes	characteristic
55	C₂×#11719908	C₂³	1	2	yes	—
56	C₂×#11719908	C₂³	1	1	yes	characteristic
57	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
58	C₄⋊#88322	C₂×C₄	1	2	yes	—
59	#13133146	C₂×C₄	1	2	yes	—
60	#13133146	C₂×C₄	1	2	yes	—
61	#13133146	C₂×C₄	1	2	yes	—
62	#13133128	C₂×C₄	1	2	yes	—
63	#13133127	C₂×C₄	1	2	yes	—
64	C₄×#88322	C₂×C₄	1	1	yes	characteristic
65	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
66	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
67	C₈×Dic₅₃	C₂³	1	2	yes	—
68	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
70	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
71	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
72	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
73	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₅₃	C₂³	1	2	yes	—
76	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
77	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
78	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
79	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
81	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
82	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
85	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
86	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
87	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
89	#11719921	C₂×C₈	1	2	yes	—
90	#11719921	C₂×C₈	1	2	yes	—
91	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
92	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
93	#11719919	C₂×C₈	1	2	yes	—
94	#11719919	C₂×C₈	1	2	yes	—
95	#11719919	C₂×C₈	1	2	yes	—
96	#11719919	C₂×C₈	1	2	yes	—
97	#11719919	C₂×C₈	1	2	yes	—
98	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
99	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
100	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
102	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
103	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
104	C₂×#88322	C₂²×C₄	1	2	yes	—
105	#11719912	—	4	2	no	—
106	C₈×D₅₃	—	2	1	no	—
107	C₈×D₅₃	—	2	1	no	—
108	C₈×D₅₃	—	4	1	no	—
109	C₁₆×C₅₃	—	2	2	no	—
110	C₁₆×C₅₃	—	2	1	no	—
111	C₁₆×C₅₃	—	2	1	no	—
112	#11719908	—	2	1	no	—
113	#11719908	—	2	1	no	—
114	#11719908	—	2	2	no	—
115	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
116	Q₈×C₅₃	—	2	2	no	—
117	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
118	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
120	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
123	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
124	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
125	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
127	C₄×D₅₃	—	2	1	no	—
128	C₄×D₅₃	—	2	1	no	—
129	C₄×D₅₃	—	2	2	no	—
130	C₄×D₅₃	—	2	2	no	—
131	Dic₁₀₆	—	2	2	no	—
132	Dic₁₀₆	—	2	4	no	—
133	C₈×C₅₃	—	2	1	no	—
134	C₈×C₅₃	—	2	2	no	—
135	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
136	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
137	#88322	—	2	1	no	—
138	#88322	—	2	1	no	—
139	#88322	—	2	2	no	—
140	#33500	—	53	1	no	maximal, Sylow
141	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
142	C₂×D₅₃	—	2	1	no	—
143	C₂×D₅₃	—	2	1	no	—
144	C₂×D₅₃	—	2	1	no	—
145	C₄×C₅₃	—	2	2	no	—
146	C₄×C₅₃	—	2	1	no	—
147	C₄×C₅₃	—	2	2	no	—
148	C₄×C₅₃	#504	1	2	yes	—
149	C₄×C₅₃	#519	1	1	yes	characteristic
150	C₄×C₅₃	#568	1	1	yes	characteristic
151	Dic₅₃	—	2	2	no	—
152	Dic₅₃	—	2	2	no	—
153	Dic₅₃	—	2	1	no	—
154	Dic₅₃	—	2	1	no	—
155	Dic₅₃	#504	1	2	yes	—
156	#2960	—	53	1	no	—
157	#2183	—	53	1	no	—
158	#2183	—	53	2	no	—
159	#2183	—	53	1	no	—
160	Q₈×C₁₆	—	53	1	no	—
161	Q₈×C₁₆	—	53	2	no	—
162	Q₈×C₁₆	D₅₃	1	1	yes	characteristic, complemented
163	#2164	—	53	2	no	—
164	#2164	—	53	2	no	—
165	#2162	—	53	1	no	—
166	#2162	—	53	1	no	—
167	C₂×C₅₃	#2982	1	1	yes	characteristic
168	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
169	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
170	D₅₃	—	4	1	no	—
171	#533	—	53	1	no	—
172	#446	—	53	1	no	—
173	#446	—	53	2	no	—
174	#446	—	53	1	no	—
175	C₈×Q₈	—	53	2	no	—
176	C₈×Q₈	—	53	1	no	—
177	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
178	#424	—	53	2	no	—
179	#423	—	53	1	no	—
180	#405	—	53	2	no	—
181	C₂×C₄×C₈	—	53	1	no	—
182	C₄⋊C₁₆	—	53	1	no	—
183	C₄⋊C₁₆	—	53	1	no	—
184	C₄⋊C₁₆	—	53	2	no	—
185	C₄⋊C₁₆	—	53	2	no	—
186	C₄⋊C₁₆	—	53	1	no	—
187	C₄⋊C₁₆	—	53	2	no	—
188	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
189	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
190	C₂²⋊C₁₆	—	53	1	no	—
191	C₂²⋊C₁₆	—	53	2	no	—
192	C₂²⋊C₁₆	—	53	1	no	—
193	#346	—	53	2	no	—
194	#346	—	53	2	no	—
195	C₄×C₁₆	—	53	2	no	—
196	C₄×C₁₆	—	53	2	no	—
197	C₄×C₁₆	—	53	1	no	—
198	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
199	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
200	C₅₃	#33500	1	1	yes	characteristic, complemented, Sylow
201	C₂×M₁₆	—	53	2	no	—
202	C₂²×C₈	—	53	1	no	—
203	C₂²×C₈	—	53	1	no	—
204	C₄⋊Q₈	—	53	1	no	—
205	C₄⋊Q₈	—	53	1	no	—
206	#125	—	53	2	no	—
207	C₂²⋊Q₈	—	53	1	no	—
208	C₂²⋊Q₈	—	53	2	no	—
209	C₂²⋊Q₈	—	53	1	no	—
210	C₄×Q₈	—	53	2	no	—
211	C₄×Q₈	—	53	1	no	—
212	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
213	#117	—	53	2	no	—
214	C₂×C₄²	—	53	1	no	—
215	C₂×C₁₆	—	53	1	no	—
216	C₂×C₁₆	—	53	1	no	—
217	C₂×C₁₆	—	53	2	no	—
218	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
219	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
220	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
221	C₄⋊C₈	—	53	2	no	—
222	C₄⋊C₈	—	53	2	no	—
223	C₄⋊C₈	—	53	2	no	—
224	C₄⋊C₈	—	53	1	no	—
225	C₄⋊C₈	—	53	2	no	—
226	C₄⋊C₈	C₄×D₅₃	1	1	yes	characteristic
227	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
228	#97	—	53	2	no	—
229	#97	—	53	2	no	—
230	C₄×C₈	—	53	1	no	—
231	C₄×C₈	—	53	1	no	—
232	C₄×C₈	—	53	2	no	—
233	C₄×C₈	—	53	1	no	—
234	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
235	C₄×C₈	C₂²×D₅₃	1	2	yes	—
236	C₂×Q₈	—	53	2	no	—
237	C₂×Q₈	—	53	1	no	—
238	C₂×Q₈	C₈×D₅₃	1	1	yes	characteristic
239	C₂²×C₄	—	53	1	no	—
240	C₂²×C₄	—	53	1	no	—
241	C₂²×C₄	—	53	1	no	—
242	M₁₆	—	212	2	no	—
243	C₂×C₈	—	53	1	no	—
244	C₂×C₈	—	53	1	no	—
245	C₂×C₈	—	53	2	no	—
246	C₂×C₈	—	106	1	no	—
247	C₂×C₈	—	106	1	no	—
248	C₂×C₈	—	212	1	no	—
249	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
250	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
251	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
252	C₄⋊C₄	—	53	2	no	—
253	C₄⋊C₄	—	53	2	no	—
254	C₄⋊C₄	—	53	2	no	—
255	C₄⋊C₄	—	53	2	no	—
256	C₄⋊C₄	—	53	2	no	—
257	C₄⋊C₄	—	53	2	no	—
258	C₄⋊C₄	—	53	1	no	—
259	C₄⋊C₄	C₈×D₅₃	1	2	yes	—
260	C₄⋊C₄	C₈×D₅₃	1	1	yes	characteristic
261	C₂²⋊C₄	—	53	2	no	—
262	C₂²⋊C₄	—	53	2	no	—
263	C₄²	—	53	1	no	—
264	C₄²	—	53	1	no	—
265	C₄²	—	53	1	no	—
266	C₄²	—	53	2	no	—
267	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
268	C₄²	C₂×C₄×D₅₃	1	2	yes	—
269	C₁₆	—	2	2	no	—
270	C₁₆	—	2	1	no	—
271	C₁₆	—	2	1	no	—
272	C₁₆	—	106	1	no	—
273	C₁₆	—	106	1	no	—
274	C₁₆	—	106	2	no	—
275	C₂³	—	53	1	no	—
276	Q₈	—	2	2	no	—
277	Q₈	—	106	4	no	—
278	Q₈	—	106	2	no	—
279	C₂×C₄	—	53	1	no	—
280	C₂×C₄	—	53	1	no	—
281	C₂×C₄	—	53	1	no	—
282	C₂×C₄	—	53	2	no	—
283	C₂×C₄	—	53	2	no	—
284	C₂×C₄	—	106	2	no	—
285	C₂×C₄	—	106	2	no	—
286	C₂×C₄	—	106	1	no	—
287	C₂×C₄	—	106	1	no	—
288	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
289	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
290	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
291	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	2	2	no	—
295	C₈	—	106	1	no	—
296	C₈	—	106	1	no	—
297	C₈	—	106	2	no	—
298	C₈	#13133264	1	1	yes	characteristic
299	C₈	#13133265	1	1	yes	characteristic
300	C₂²	—	106	1	no	—
301	C₂²	—	106	1	no	—
302	C₂²	—	106	1	no	—
303	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	53	2	no	—
308	C₄	—	106	2	no	—
309	C₄	—	106	1	no	—
310	C₄	—	106	2	no	—
311	C₄	—	106	1	no	—
312	C₄	#16811039	1	2	yes	—
313	C₄	#16811099	1	1	yes	characteristic
314	C₄	#16811388	1	1	yes	characteristic
315	C₂	—	212	1	no	—
316	C₂	#20283794	1	1	yes	characteristic
317	C₂	#504×D₅₃	1	1	yes	characteristic
318	C₂	C₂×#16811039	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	Q₈×C₁₆×C₅₃	C₂
prime radical	C₂²×D₅₃	C₂²×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	Q₈×C₁₆	D₅₃
hypocenter	C₅₃	#33500
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20276624	C₂	semidirect
3. #20276623	C₂	semidirect
4. #20276622	C₂	semidirect
5. C₁₆×Dic₁₀₆	C₂	semidirect
6. #20276617	C₂	semidirect
11. Q₈×C₁₆×C₅₃	C₂	semidirect
162. Q₈×C₁₆	D₅₃	semidirect
200. C₅₃	#33500	semidirect

Maximal subgroups

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

subgroup	index
#20283644	2
#20276624	2
#20276623	2
#20276622	2
C₁₆×Dic₁₀₆	2
#20276617	2
#20276521	2
#20276520	2
#20276513	2
#20276512	2
Q₈×C₁₆×C₅₃	2
#33500	53