Subgroups — #27289782

Order 13568.

612 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810393	C₂	1	2	yes	maximal
2	#20274441	C₂	1	2	yes	maximal
3	C₂×#16810381	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810381	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#2873	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	2	yes	—
7	C₂²×#11719917	C₂²	1	2	yes	—
8	#16810393	—	2	4	no	—
9	#16810387	C₄	1	8	yes	complemented
10	#16810381	—	2	2	no	—
11	#16810381	—	2	2	no	—
12	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#405	—	2	2	no	—
16	C₅₃×#405	C₄	1	4	yes	—
17	C₄×M₁₆×C₅₃	—	2	2	no	—
18	C₂×#11719917	—	2	2	no	—
19	C₂×#11719917	—	2	2	no	—
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719917	—	2	2	no	—
22	C₂×#11719917	—	2	2	no	—
23	C₂×#11719917	—	2	2	no	—
24	C₂×#11719917	—	2	2	no	—
25	C₂×#11719917	—	2	2	no	—
26	C₂×#11719917	—	2	2	no	—
27	C₂×#11719917	—	2	2	no	—
28	C₂×#11719917	—	2	2	no	—
29	C₂×#11719917	C₂×C₄	1	4	yes	—
30	C₂²×#88322	—	2	2	no	—
31	C₂²×#88322	C₂×C₄	1	4	yes	—
32	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
33	C₂×M₁₆×C₅₃	—	2	1	no	—
34	C₂×M₁₆×C₅₃	—	2	2	no	—
35	C₂×M₁₆×C₅₃	—	2	2	no	—
36	C₂×M₁₆×C₅₃	—	2	2	no	—
37	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
38	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
40	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
41	C₂²×C₈×C₅₃	—	2	1	no	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
43	#117×C₅₃	—	2	1	no	—
44	#117×C₅₃	—	2	1	no	—
45	#117×C₅₃	C₂×C₄	1	2	yes	—
46	#117×C₅₃	D₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
48	C₂×C₂²⋊C₄×C₅₃	Q₈	1	2	yes	—
49	C₂×C₄²×C₅₃	—	2	1	no	—
50	#97×C₅₃	—	2	4	no	—
51	C₄×C₈×C₅₃	—	2	4	no	—
52	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	—	2	2	no	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	#41	1	2	yes	—
62	C₂²×C₄×C₅₃	#41	1	2	yes	—
63	C₂²×C₄×C₅₃	#41	1	2	yes	—
64	M₁₆×C₅₃	—	2	2	no	—
65	M₁₆×C₅₃	—	2	4	no	—
66	M₁₆×C₅₃	—	2	4	no	—
67	M₁₆×C₅₃	—	2	4	no	—
68	M₁₆×C₅₃	C₄²	1	4	yes	—
69	C₂×C₈×C₅₃	—	2	2	no	—
70	C₂×C₈×C₅₃	—	2	2	no	—
71	C₂×C₈×C₅₃	—	2	2	no	—
72	C₂×C₈×C₅₃	—	2	4	no	—
73	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
74	C₄⋊C₄×C₅₃	—	2	2	no	—
75	C₄⋊C₄×C₅₃	—	2	2	no	—
76	C₂²⋊C₄×C₅₃	—	2	2	no	—
77	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
78	C₄²×C₅₃	—	2	2	no	—
79	C₄²×C₅₃	—	2	2	no	—
80	#11719917	—	2	4	no	—
81	#11719917	—	2	4	no	—
82	#11719917	—	4	2	no	—
83	#11719917	—	4	2	no	—
84	#11719917	—	4	2	no	—
85	#11719917	—	4	2	no	—
86	#11719917	—	4	2	no	—
87	#11719917	—	4	2	no	—
88	C₂×#88322	—	2	4	no	—
89	C₂×#88322	—	2	4	no	—
90	C₂×#88322	—	2	4	no	—
91	C₂×#88322	—	2	4	no	—
92	C₂×#88322	—	4	2	no	—
93	C₂×#88322	—	4	2	no	—
94	C₂³×C₅₃	—	8	1	no	—
95	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
97	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂³×C₅₃	#125	1	2	yes	—
100	C₂×C₄×C₅₃	—	2	4	no	—
101	C₂×C₄×C₅₃	—	8	2	no	—
102	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#117	1	2	yes	—
105	C₂×C₄×C₅₃	#117	1	2	yes	—
106	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
108	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
115	C₂×C₄×C₅₃	#123	1	2	yes	—
116	C₂×C₄×C₅₃	#123	1	2	yes	—
117	C₂×C₄×C₅₃	#124	1	2	yes	—
118	C₂×C₄×C₅₃	#126	1	2	yes	—
119	C₂×C₄×C₅₃	#126	1	2	yes	—
120	C₈×C₅₃	—	2	4	no	—
121	C₈×C₅₃	—	2	4	no	—
122	#88322	—	4	4	no	—
123	#88322	—	4	4	no	—
124	#10812	—	53	1	no	maximal, Sylow
125	C₂²×C₅₃	—	4	1	no	—
126	C₂²×C₅₃	—	4	1	no	—
127	C₂²×C₅₃	—	4	2	no	—
128	C₂²×C₅₃	—	4	1	no	—
129	C₂²×C₅₃	—	4	2	no	—
130	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
131	C₂²×C₅₃	#380	1	1	yes	characteristic
132	C₂²×C₅₃	#382	1	1	yes	characteristic
133	C₂²×C₅₃	#387	1	2	yes	—
134	C₂²×C₅₃	#387	1	2	yes	—
135	C₄×C₅₃	—	4	4	no	—
136	C₄×C₅₃	#376	1	1	yes	characteristic
137	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
138	C₄×C₅₃	#381	1	1	yes	characteristic
139	C₄×C₅₃	#383	1	1	yes	characteristic
140	C₄×C₅₃	#388	1	2	yes	—
141	C₄×C₅₃	#388	1	2	yes	—
142	#2873	D₅₃	1	1	yes	characteristic
143	C₂×#344	—	53	2	no	—
144	#1738	—	53	2	no	—
145	C₂×#338	—	53	1	no	—
146	C₂×#338	—	53	1	no	—
147	C₂×C₅₃	—	2	1	no	—
148	C₂×C₅₃	—	2	1	no	—
149	C₂×C₅₃	—	2	1	no	—
150	C₂×C₅₃	—	2	2	no	—
151	C₂×C₅₃	—	2	2	no	—
152	C₂×C₅₃	#1434	1	1	yes	characteristic
153	C₂²×M₁₆	—	53	2	no	—
154	C₂²×M₁₆	—	53	2	no	—
155	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
156	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
157	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
158	#405	—	2	2	no	—
159	#405	Dic₅₃	1	4	yes	—
160	C₄×M₁₆	—	2	2	no	—
161	#344	—	106	4	no	—
162	#341	—	53	8	no	—
163	#338	—	106	2	no	—
164	#338	—	106	2	no	—
165	C₅₃	#10812	1	1	yes	characteristic, complemented, Sylow
166	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	1	no	—
171	C₂×M₁₆	—	53	4	no	—
172	C₂×M₁₆	—	106	2	no	—
173	C₂×M₁₆	—	106	2	no	—
174	C₂×M₁₆	—	106	2	no	—
175	C₂×M₁₆	—	106	2	no	—
176	C₂×M₁₆	—	106	2	no	—
177	C₂×M₁₆	—	106	2	no	—
178	C₂×M₁₆	—	106	2	no	—
179	C₂×M₁₆	—	106	2	no	—
180	C₂×M₁₆	—	106	2	no	—
181	C₂×M₁₆	—	106	2	no	—
182	C₂×M₁₆	—	106	2	no	—
183	C₂×M₁₆	C₄×D₅₃	1	4	yes	—
184	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
185	C₂×M₁₆	C₂²⋊D₅₃	1	2	yes	—
186	C₂×M₁₆	C₂²⋊D₅₃	1	2	yes	—
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	—	53	4	no	—
189	C₂²×C₈	—	106	2	no	—
190	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
191	#117	—	2	1	no	—
192	#117	—	2	1	no	—
193	#117	D₂₁₂	1	2	yes	—
194	#117	C₂×Dic₅₃	1	2	yes	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₂²⋊C₄	Dic₁₀₆	1	2	yes	—
197	C₂×C₄²	—	2	1	no	—
198	#97	—	2	4	no	—
199	C₄×C₈	—	2	4	no	—
200	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
203	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
204	C₂²×C₄	#11719945	1	1	yes	characteristic
205	C₂²×C₄	#11719945	1	1	yes	characteristic
206	C₂²×C₄	#11719945	1	2	yes	—
207	C₂²×C₄	#11719945	1	2	yes	—
208	C₂²×C₄	#11719945	1	2	yes	—
209	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
212	M₁₆	—	2	2	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	2	4	no	—
216	M₁₆	—	106	4	no	—
217	M₁₆	—	106	4	no	—
218	M₁₆	—	212	2	no	—
219	M₁₆	—	212	2	no	—
220	M₁₆	—	212	2	no	—
221	M₁₆	—	212	2	no	—
222	M₁₆	—	212	2	no	—
223	M₁₆	—	212	2	no	—
224	M₁₆	C₄×Dic₅₃	1	4	yes	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	106	4	no	—
230	C₂×C₈	—	106	4	no	—
231	C₂×C₈	—	106	4	no	—
232	C₂×C₈	—	106	4	no	—
233	C₂×C₈	—	212	2	no	—
234	C₂×C₈	—	212	2	no	—
235	C₂×C₈	C₂²⋊Dic₅₃	1	4	yes	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₂²⋊C₄	—	2	2	no	—
239	C₂²⋊C₄	C₄⋊Dic₅₃	1	4	yes	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₂³	—	8	1	no	—
243	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
244	C₂³	C₄×Dic₁₀₆	1	2	yes	—
245	C₂³	#13133196	1	1	yes	characteristic
246	C₂³	#13133196	1	1	yes	characteristic
247	C₂³	#13133209	1	2	yes	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	8	2	no	—
250	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
252	C₂×C₄	#13133174	1	2	yes	—
253	C₂×C₄	#13133174	1	2	yes	—
254	C₂×C₄	#13133176	1	1	yes	characteristic
255	C₂×C₄	#13133176	1	1	yes	characteristic
256	C₂×C₄	#13133176	1	1	yes	characteristic
257	C₂×C₄	#13133176	1	1	yes	characteristic
258	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
259	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
260	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
261	C₂×C₄	#13133191	1	1	yes	characteristic
262	C₂×C₄	#13133191	1	1	yes	characteristic
263	C₂×C₄	#13133200	1	2	yes	—
264	C₂×C₄	#13133200	1	2	yes	—
265	C₂×C₄	#13133205	1	2	yes	—
266	C₂×C₄	#13133211	1	2	yes	—
267	C₂×C₄	#13133211	1	2	yes	—
268	C₈	—	2	4	no	—
269	C₈	—	2	4	no	—
270	C₈	—	212	4	no	—
271	C₈	—	212	4	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	4	2	no	—
277	C₂²	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
278	C₂²	#16810483	1	1	yes	characteristic
279	C₂²	#16810489	1	1	yes	characteristic
280	C₂²	#16810507	1	2	yes	—
281	C₂²	#16810507	1	2	yes	—
282	C₄	—	4	4	no	—
283	C₄	#16810469	1	1	yes	characteristic
284	C₄	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
285	C₄	#16810486	1	1	yes	characteristic
286	C₄	#16810491	1	1	yes	characteristic
287	C₄	#16810514	1	2	yes	—
288	C₄	#16810514	1	2	yes	—
289	C₂	—	2	1	no	—
290	C₂	—	2	2	no	—
291	C₂	—	2	2	no	—
292	C₂	—	2	1	no	—
293	C₂	—	2	1	no	—
294	C₂	#20272942	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#2873	C₂
socle	C₂×C₅₃	#1434
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#2873	D₅₃
hypocenter	C₅₃	#10812
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ #2873

Complements

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

N	H ≅ G/N	type
9. #16810387	C₄	semidirect
165. C₅₃	#10812	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810393	2
#20274441	2
C₂×#16810381	2
C₅₃×#2873	2
#10812	53