Subgroups — #27537467

Order 13568.

646 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#20283578	C₂	1	2	yes	maximal
3	#20276587	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276586	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276585	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13133163	C₂	1	2	yes	maximal, complemented
7	#20276571	C₂	1	2	yes	maximal, complemented
8	#20276498	C₂	1	1	yes	maximal, characteristic
9	#20276433	C₂	1	2	yes	maximal
10	#20276313	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#2176	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₅₃	C₂²	1	2	yes	—
13	#16811099	—	2	2	no	—
14	C₂×Dic₄₂₄	C₂²	1	1	yes	characteristic
15	C₂×#13133163	C₂²	1	2	yes	—
16	Q₃₂×D₅₃	C₄	1	8	yes	complemented
17	C₂×#13133156	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
19	#16810728	—	2	2	no	—
20	#16810727	—	2	2	no	—
21	C₄×Dic₂₁₂	C₂²	1	2	yes	—
22	Q₁₆×Dic₅₃	C₂²	1	2	yes	—
23	#16810717	C₂²	1	2	yes	—
24	#16810660	C₂²	1	2	yes	—
25	#16810630	—	2	2	no	—
26	#16810577	C₂²	1	1	yes	characteristic
27	#16810451	C₂²	1	1	yes	characteristic
28	#16810448	C₂²	1	2	yes	—
29	#16810428	C₂²	1	2	yes	—
30	#16810427	C₂²	1	2	yes	—
31	#16810426	C₂²	1	2	yes	—
32	#16810398	C₂²	1	1	yes	characteristic
33	C₄×#11719908	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
35	C₂×Q₃₂×C₅₃	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
37	C₅₃×#367	C₂²	1	1	yes	characteristic
38	C₅₃×#358	C₂²	1	2	yes	—
39	C₅₃×#346	C₂²	1	1	yes	characteristic
40	C₂×Q₈×D₅₃	—	2	2	no	—
41	C₂×Dic₂₁₂	C₂³	1	2	yes	—
42	C₂×#11719925	—	2	2	no	—
43	Q₁₆×D₅₃	—	2	4	no	—
44	Q₁₆×D₅₃	C₂×C₄	1	8	yes	—
45	C₂×#11719912	D₄	1	1	yes	characteristic
46	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
47	Q₈×Dic₅₃	—	2	2	no	—
48	C₄×Dic₁₀₆	—	2	2	no	—
49	C₄×Dic₁₀₆	—	4	2	no	—
50	#13133182	—	2	2	no	—
51	#13133182	—	4	2	no	—
52	#13133175	—	2	2	no	—
53	#13133175	—	4	2	no	—
54	#13133174	—	4	2	no	—
55	#13133174	D₄	1	1	yes	characteristic
56	Dic₄₂₄	C₂×C₄	1	4	yes	—
57	#13133163	C₂×C₄	1	8	yes	—
58	#13133156	—	2	2	no	—
59	C₂×#11719908	C₂³	1	1	yes	characteristic
60	C₂×#11719908	C₂³	1	1	yes	characteristic
61	C₁₆×D₅₃	C₂×C₄	1	4	yes	—
62	#13133151	C₂³	1	2	yes	—
63	#13133148	—	2	2	no	—
64	#13133142	—	2	2	no	—
65	#13133141	—	2	2	no	—
66	#13133140	—	2	2	no	—
67	#13133128	D₄	1	1	yes	characteristic
68	#13133127	D₄	1	1	yes	characteristic
69	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
70	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
71	C₄×Q₈×C₅₃	—	2	2	no	—
72	Q₃₂×C₅₃	C₂×C₄	1	4	yes	—
73	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
75	#107×C₅₃	C₂³	1	2	yes	—
76	#103×C₅₃	—	2	2	no	—
77	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₂×Q₈×C₅₃	—	2	2	no	—
79	Q₈×D₅₃	—	2	8	no	—
80	Q₈×D₅₃	—	4	4	no	—
81	C₂×C₄×D₅₃	—	4	2	no	—
82	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂×Dic₁₀₆	—	2	2	no	—
84	C₂×Dic₁₀₆	—	4	2	no	—
85	Q₁₆×C₅₃	—	2	2	no	—
86	Q₁₆×C₅₃	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄×C₅₃	—	2	2	no	—
90	C₄⋊C₄×C₅₃	—	4	2	no	—
91	C₄²×C₅₃	—	4	2	no	—
92	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
93	#11719925	—	2	8	no	—
94	#11719921	—	4	2	no	—
95	#11719921	D₈	1	2	yes	—
96	C₄⋊Dic₅₃	—	2	2	no	—
97	C₄⋊Dic₅₃	—	4	2	no	—
98	#11719919	—	4	2	no	—
99	#11719919	—	4	2	no	—
100	#11719919	D₈	1	2	yes	—
101	C₄×Dic₅₃	—	4	2	no	—
102	C₄×Dic₅₃	—	4	2	no	—
103	C₄×Dic₅₃	—	4	2	no	—
104	C₄×Dic₅₃	—	4	2	no	—
105	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
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	C₂²×C₄	1	4	yes	—
110	#11719912	—	2	2	no	—
111	C₈×D₅₃	C₂²×C₄	1	2	yes	—
112	C₈×D₅₃	#41	1	2	yes	—
113	C₁₆×C₅₃	—	2	1	no	—
114	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
115	#11719908	—	2	1	no	—
116	#11719908	C₂²×C₄	1	2	yes	—
117	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
118	Q₈×C₅₃	—	2	4	no	—
119	Q₈×C₅₃	—	4	2	no	—
120	C₂×C₄×C₅₃	—	4	2	no	—
121	C₂×C₄×C₅₃	—	4	2	no	—
122	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
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	C₂×D₈	1	2	yes	—
127	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
128	C₄×D₅₃	—	4	8	no	—
129	C₄×D₅₃	C₄×D₄	1	2	yes	—
130	C₄×D₅₃	#135	1	2	yes	—
131	Dic₁₀₆	—	2	4	no	—
132	Dic₁₀₆	—	4	2	no	—
133	Dic₁₀₆	—	4	8	no	—
134	C₈×C₅₃	—	2	1	no	—
135	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
136	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
137	#88322	—	2	1	no	—
138	#88322	C₄×D₄	1	2	yes	—
139	#33458	—	53	1	no	maximal, Sylow
140	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
141	C₂×D₅₃	—	2	2	no	—
142	C₂×D₅₃	C₄×D₈	1	2	yes	—
143	C₄×C₅₃	—	2	2	no	—
144	C₄×C₅₃	—	4	4	no	—
145	C₄×C₅₃	—	8	2	no	—
146	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
147	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
148	Dic₅₃	—	2	2	no	—
149	Dic₅₃	—	4	4	no	—
150	Dic₅₃	—	8	2	no	—
151	Dic₅₃	C₄×D₈	1	2	yes	—
152	C₂²×Q₃₂	—	53	1	no	—
153	#2950	—	53	2	no	—
154	#2176	—	53	1	no	—
155	#2176	—	53	1	no	—
156	#2176	—	53	1	no	—
157	#2176	D₅₃	1	1	yes	characteristic, complemented
158	C₄×Q₃₂	—	53	2	no	—
159	C₄×Q₃₂	—	53	2	no	—
160	#2158	—	53	1	no	—
161	#2141	—	53	2	no	—
162	#2108	—	53	1	no	—
163	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
164	C₂×C₅₃	#3417	1	2	yes	—
165	D₅₃	—	2	2	no	—
166	C₂²×Q₁₆	—	53	2	no	—
167	#519	—	106	2	no	—
168	C₂×Q₃₂	—	53	2	no	—
169	C₂×Q₃₂	—	53	8	no	—
170	C₂×Q₃₂	—	53	1	no	—
171	C₂×Q₃₂	C₂×D₅₃	1	1	yes	characteristic
172	C₂×M₃₂	—	53	1	no	—
173	C₂²×C₁₆	—	53	1	no	—
174	#441	—	106	2	no	—
175	#441	—	106	2	no	—
176	C₄×Q₁₆	—	53	2	no	—
177	C₄×Q₁₆	—	53	2	no	—
178	C₄×Q₁₆	—	53	2	no	—
179	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
180	#427	—	53	2	no	—
181	#419	—	106	2	no	—
182	#405	—	53	1	no	—
183	#367	—	53	1	no	—
184	#367	C₂×D₅₃	1	1	yes	characteristic
185	#366	—	53	2	no	—
186	#358	—	53	2	no	—
187	#358	—	53	2	no	—
188	#358	—	53	2	no	—
189	#358	C₂×D₅₃	1	2	yes	—
190	#346	—	53	1	no	—
191	#346	C₂×D₅₃	1	1	yes	characteristic
192	C₄×C₁₆	—	53	1	no	—
193	C₄×C₁₆	—	53	1	no	—
194	C₅₃	#33458	1	1	yes	characteristic, complemented, Sylow
195	C₂²×Q₈	—	106	2	no	—
196	C₂×Q₁₆	—	53	8	no	—
197	C₂×Q₁₆	—	53	2	no	—
198	C₂×Q₁₆	—	106	4	no	—
199	C₂×Q₁₆	—	106	2	no	—
200	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
201	C₂×M₁₆	—	53	1	no	—
202	C₂²×C₈	—	53	1	no	—
203	C₄×Q₈	—	2	2	no	—
204	C₄×Q₈	—	106	2	no	—
205	C₄×Q₈	—	106	2	no	—
206	C₄×Q₈	—	106	2	no	—
207	C₄×Q₈	—	212	2	no	—
208	C₄×Q₈	—	212	2	no	—
209	#117	—	53	1	no	—
210	#117	—	106	2	no	—
211	#117	—	212	2	no	—
212	#117	—	212	2	no	—
213	Q₃₂	—	53	8	no	—
214	Q₃₂	—	53	4	no	—
215	Q₃₂	C₄×D₅₃	1	4	yes	—
216	M₃₂	—	106	2	no	—
217	C₂×C₁₆	—	53	1	no	—
218	C₂×C₁₆	—	53	4	no	—
219	C₂×C₁₆	—	53	1	no	—
220	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
221	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
222	#107	—	53	2	no	—
223	#107	C₂²×D₅₃	1	2	yes	—
224	#106	—	106	2	no	—
225	#103	—	2	2	no	—
226	#103	—	106	2	no	—
227	#103	—	106	2	no	—
228	#103	—	106	2	no	—
229	#97	—	53	1	no	—
230	#97	—	53	1	no	—
231	C₄×C₈	—	53	1	no	—
232	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	106	8	no	—
235	C₂×Q₈	—	106	2	no	—
236	C₂×Q₈	—	212	2	no	—
237	C₂×Q₈	—	212	4	no	—
238	C₂²×C₄	—	53	1	no	—
239	C₂²×C₄	—	212	2	no	—
240	Q₁₆	—	2	2	no	—
241	Q₁₆	—	53	4	no	—
242	Q₁₆	—	106	8	no	—
243	Q₁₆	—	106	2	no	—
244	Q₁₆	C₂×C₄×D₅₃	1	4	yes	—
245	M₁₆	—	106	2	no	—
246	C₂×C₈	—	53	1	no	—
247	C₂×C₈	—	53	2	no	—
248	C₂×C₈	—	53	2	no	—
249	C₂×C₈	—	53	1	no	—
250	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
251	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	53	2	no	—
255	C₄⋊C₄	—	106	2	no	—
256	C₄⋊C₄	—	212	2	no	—
257	C₄⋊C₄	—	212	2	no	—
258	C₄⋊C₄	—	212	2	no	—
259	C₂²⋊C₄	—	53	2	no	—
260	C₂²⋊C₄	—	212	2	no	—
261	C₄²	—	4	2	no	—
262	C₄²	—	53	1	no	—
263	C₄²	—	212	2	no	—
264	C₄²	—	212	2	no	—
265	C₄²	—	212	2	no	—
266	C₄²	—	212	2	no	—
267	C₄²	D₄×D₅₃	1	1	yes	characteristic
268	C₁₆	—	2	1	no	—
269	C₁₆	—	53	2	no	—
270	C₁₆	—	106	1	no	—
271	C₁₆	C₂×C₄×D₅₃	1	2	yes	—
272	C₂³	—	53	1	no	—
273	Q₈	—	2	4	no	—
274	Q₈	—	4	2	no	—
275	Q₈	—	106	4	no	—
276	Q₈	—	212	8	no	—
277	Q₈	—	212	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	53	1	no	—
281	C₂×C₄	—	53	2	no	—
282	C₂×C₄	—	53	2	no	—
283	C₂×C₄	—	53	2	no	—
284	C₂×C₄	—	212	8	no	—
285	C₂×C₄	—	212	2	no	—
286	C₂×C₄	—	212	2	no	—
287	C₂×C₄	D₈×D₅₃	1	2	yes	—
288	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	53	2	no	—
291	C₈	—	106	1	no	—
292	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
293	C₈	#41×D₅₃	1	1	yes	characteristic
294	C₂²	—	53	2	no	—
295	C₂²	—	106	2	no	—
296	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
297	C₄	—	2	2	no	—
298	C₄	—	4	4	no	—
299	C₄	—	8	2	no	—
300	C₄	—	53	2	no	—
301	C₄	—	106	2	no	—
302	C₄	—	424	2	no	—
303	C₄	—	212	4	no	—
304	C₄	C₄×D₄×D₅₃	1	1	yes	characteristic
305	C₄	#135×D₅₃	1	1	yes	characteristic
306	C₂	—	106	2	no	—
307	C₂	C₄×D₈×D₅₃	1	1	yes	characteristic
308	C₂	#20287867	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2176	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂²×D₅₃	C₂×D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2176	D₅₃
hypocenter	C₅₃	#33458
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2176

Complements

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

N	H ≅ G/N	type
3. #20276587	C₂	semidirect
4. #20276586	C₂	semidirect
5. #20276585	C₂	semidirect
6. C₄×#13133163	C₂	semidirect
7. #20276571	C₂	semidirect
11. C₅₃×#2176	C₂	semidirect
16. Q₃₂×D₅₃	C₄	semidirect
157. #2176	D₅₃	semidirect
194. C₅₃	#33458	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₃₂×D₅₃	2
#20283578	2
#20276587	2
#20276586	2
#20276585	2
C₄×#13133163	2
#20276571	2
#20276498	2
#20276433	2
#20276313	2
C₅₃×#2176	2
#33458	53