Subgroups — #27273970

Order 13568.

714 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20277122	C₂	1	2	yes	maximal
2	#20272583	C₂	1	4	yes	maximal
3	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133176	—	2	2	no	—
5	#16810493	—	2	2	no	—
6	#16810489	C₄	1	4	yes	complemented
7	#16810483	—	2	2	no	—
8	#16810483	C₂²	1	4	yes	—
9	C₄×C₄⋊Dic₅₃	—	2	2	no	—
10	C₄×C₄⋊Dic₅₃	C₄	1	4	yes	complemented
11	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#431	—	2	2	no	—
13	C₅₃×#431	—	2	2	no	—
14	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	complemented
15	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	complemented
17	#13133176	—	4	4	no	—
18	#13133176	—	4	4	no	—
19	C₂×C₄⋊Dic₅₃	—	2	4	no	—
20	C₂×C₄⋊Dic₅₃	—	2	4	no	—
21	C₂×C₄⋊Dic₅₃	—	2	4	no	—
22	C₂×C₂²⋊Dic₅₃	—	2	4	no	—
23	C₂×C₄×Dic₅₃	—	2	4	no	—
24	#13133124	—	2	4	no	—
25	#13133124	C₂×C₄	1	8	yes	—
26	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	D₄	1	2	yes	—
35	C₂×C₄²×C₅₃	D₄	1	2	yes	—
36	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
38	C₄⋊C₈×C₅₃	—	2	4	no	—
39	C₂²⋊C₈×C₅₃	—	2	4	no	—
40	C₄×C₈×C₅₃	—	2	2	no	—
41	C₄×C₈×C₅₃	—	2	2	no	—
42	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	—	4	2	no	—
44	C₂²×C₄×C₅₃	—	4	2	no	—
45	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	D₈	1	2	yes	—
54	C₂²×C₄×C₅₃	SD₁₆	1	4	yes	—
55	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
60	C₂²×Dic₅₃	—	2	8	no	—
61	C₂×C₈×C₅₃	—	2	4	no	—
62	C₂×C₈×C₅₃	—	4	4	no	—
63	C₄²×C₅₃	C₄²	1	2	yes	—
64	C₄²×C₅₃	C₄²	1	1	yes	characteristic
65	C₄²×C₅₃	C₄²	1	1	yes	characteristic
66	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
70	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
71	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
72	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
73	C₂²⋊Dic₅₃	—	4	4	no	—
74	C₂²⋊Dic₅₃	—	4	4	no	—
75	C₄⋊Dic₅₃	—	4	8	no	—
76	C₄⋊Dic₅₃	—	4	8	no	—
77	C₄⋊Dic₅₃	—	4	8	no	—
78	C₄×Dic₅₃	—	4	8	no	—
79	C₂³×C₅₃	—	4	1	no	—
80	C₂³×C₅₃	—	4	1	no	—
81	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
82	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
84	C₂³×C₅₃	C₂×SD₁₆	1	2	yes	—
85	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
86	C₂×C₄×C₅₃	—	2	4	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	4	no	—
90	C₂×C₄×C₅₃	—	4	2	no	—
91	C₂×C₄×C₅₃	—	4	1	no	—
92	C₂×C₄×C₅₃	—	4	2	no	—
93	C₂×C₄×C₅₃	—	4	1	no	—
94	C₂×C₄×C₅₃	#95	1	2	yes	—
95	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#95	1	2	yes	—
98	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#102	1	4	yes	—
101	C₂×C₄×C₅₃	#102	1	4	yes	—
102	C₂×C₄×C₅₃	#103	1	4	yes	—
103	C₂×C₄×C₅₃	#103	1	4	yes	—
104	C₂×C₄×C₅₃	#106	1	2	yes	—
105	C₂×C₄×C₅₃	#106	1	2	yes	—
106	C₂×C₄×C₅₃	#107	1	2	yes	—
107	C₂×C₄×C₅₃	#107	1	2	yes	—
108	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×Dic₅₃	—	4	8	no	—
112	C₂×Dic₅₃	—	4	8	no	—
113	C₂×Dic₅₃	—	4	8	no	—
114	C₈×C₅₃	—	4	4	no	—
115	#7981	—	53	1	no	maximal, Sylow
116	C₂²×C₅₃	—	2	2	no	—
117	C₂²×C₅₃	—	2	1	no	—
118	C₂²×C₅₃	—	2	2	no	—
119	C₂²×C₅₃	—	2	1	no	—
120	C₂²×C₅₃	—	4	1	no	—
121	C₂²×C₅₃	—	4	2	no	—
122	C₂²×C₅₃	—	4	1	no	—
123	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
124	C₂²×C₅₃	C₂×#102	1	2	yes	—
125	C₂²×C₅₃	C₂×#103	1	2	yes	—
126	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
127	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
128	C₄×C₅₃	—	2	2	no	—
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	—	2	1	no	—
131	C₄×C₅₃	—	2	2	no	—
132	C₄×C₅₃	H(3,ℤ/4)	1	2	yes	—
133	C₄×C₅₃	#340	1	4	yes	—
134	C₄×C₅₃	#340	1	2	yes	—
135	C₄×C₅₃	#340	1	2	yes	—
136	C₄×C₅₃	#410	1	2	yes	—
137	Dic₅₃	—	8	8	no	—
138	C₂×#431	D₅₃	1	1	yes	characteristic
139	#2285	—	53	2	no	—
140	#1289	—	53	4	no	—
141	C₂×C₅₃	—	2	2	no	—
142	C₂×C₅₃	—	2	2	no	—
143	C₂×C₅₃	—	2	1	no	—
144	C₂×C₅₃	—	2	1	no	—
145	C₂×C₅₃	C₂×H(3,ℤ/4)	1	1	yes	characteristic
146	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
147	C₂×C₅₃	#1740	1	1	yes	characteristic
148	C₂×#117	—	106	2	no	—
149	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
150	#431	—	2	2	no	—
151	#431	—	2	2	no	—
152	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	complemented
153	C₂×C₂²⋊C₈	C₂×D₅₃	1	2	yes	—
154	C₂×C₄×C₈	Dic₅₃	1	2	yes	complemented
155	#384	—	106	2	no	—
156	#382	—	53	4	no	—
157	#380	—	53	4	no	—
158	#380	—	106	2	no	—
159	C₄×C₄⋊C₄	—	53	4	no	—
160	C₄×C₄⋊C₄	—	106	2	no	—
161	C₅₃	#7981	1	1	yes	characteristic, complemented, Sylow
162	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
163	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
164	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
165	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
166	#117	—	212	4	no	—
167	#117	—	212	4	no	—
168	C₂×C₄⋊C₄	—	106	4	no	—
169	C₂×C₄⋊C₄	—	106	4	no	—
170	C₂×C₄⋊C₄	—	106	4	no	—
171	C₂×C₂²⋊C₄	—	106	4	no	—
172	C₂×C₄²	—	106	4	no	—
173	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
174	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
175	C₂×C₄²	C₄×D₅₃	1	2	yes	—
176	C₂×C₄²	C₄×D₅₃	1	2	yes	—
177	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
178	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
179	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
180	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
181	C₄⋊C₈	—	2	4	no	—
182	C₂²⋊C₈	—	2	4	no	—
183	C₄×C₈	—	2	2	no	—
184	C₄×C₈	—	2	2	no	—
185	#95	—	53	8	no	—
186	#95	—	106	4	no	—
187	C₂⁴	C₂×D₂₁₂	1	1	yes	characteristic
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	106	8	no	—
191	C₂²×C₄	#11719913	1	4	yes	—
192	C₂²×C₄	D₄₂₄	1	2	yes	—
193	C₂²×C₄	Dic₂₁₂	1	2	yes	—
194	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
195	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
196	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
197	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
198	C₂²×C₄	#11719921	1	2	yes	—
199	C₂²×C₄	#11719921	1	2	yes	—
200	C₂²×C₄	#11719921	1	2	yes	—
201	C₂²×C₄	#11719921	1	2	yes	—
202	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
204	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	4	4	no	—
208	C₄⋊C₄	—	212	8	no	—
209	C₄⋊C₄	—	212	8	no	—
210	C₄⋊C₄	—	212	8	no	—
211	C₂²⋊C₄	—	212	4	no	—
212	C₂²⋊C₄	—	212	4	no	—
213	C₄²	—	212	8	no	—
214	C₄²	C₄×Dic₅₃	1	2	yes	—
215	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
216	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
217	C₄²	#11719919	1	2	yes	—
218	C₄²	#11719919	1	2	yes	—
219	C₄²	#11719919	1	2	yes	—
220	C₄²	#11719919	1	2	yes	—
221	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
222	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
223	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
224	C₂³	—	4	1	no	—
225	C₂³	—	4	1	no	—
226	C₂³	C₂×#11719921	1	2	yes	—
227	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
228	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
229	C₂³	C₂×#11719913	1	2	yes	—
230	C₂³	C₂×Dic₂₁₂	1	1	yes	characteristic
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	212	8	no	—
240	C₂×C₄	—	212	8	no	—
241	C₂×C₄	—	212	8	no	—
242	C₂×C₄	#13133124	1	1	yes	characteristic
243	C₂×C₄	#13133124	1	2	yes	—
244	C₂×C₄	#13133124	1	1	yes	characteristic
245	C₂×C₄	#13133124	1	1	yes	characteristic
246	C₂×C₄	#13133124	1	2	yes	—
247	C₂×C₄	#13133124	1	1	yes	characteristic
248	C₂×C₄	#13133139	1	4	yes	—
249	C₂×C₄	#13133139	1	4	yes	—
250	C₂×C₄	#13133142	1	4	yes	—
251	C₂×C₄	#13133142	1	4	yes	—
252	C₂×C₄	#13133149	1	2	yes	—
253	C₂×C₄	#13133149	1	2	yes	—
254	C₂×C₄	#13133151	1	2	yes	—
255	C₂×C₄	#13133151	1	2	yes	—
256	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
258	C₂×C₄	C₂×#11719919	1	2	yes	—
259	C₈	—	4	4	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	C₂×#13133124	1	1	yes	characteristic
268	C₂²	C₂×#13133139	1	2	yes	—
269	C₂²	C₂×#13133142	1	2	yes	—
270	C₂²	C₂×#13133149	1	1	yes	characteristic
271	C₂²	C₂×#13133151	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	1	no	—
276	C₄	—	424	8	no	—
277	C₄	#16810380	1	2	yes	—
278	C₄	#16810386	1	4	yes	—
279	C₄	#16810386	1	2	yes	—
280	C₄	#16810386	1	2	yes	—
281	C₄	#16810597	1	2	yes	—
282	C₂	—	2	2	no	—
283	C₂	—	2	2	no	—
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	C₂×#16810380	1	1	yes	characteristic
287	C₂	C₂×#16810386	1	1	yes	characteristic
288	C₂	#20274448	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133124
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#431	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#431	D₅₃
hypocenter	C₅₃	#7981
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
6. #16810489	C₄	semidirect
10. C₄×C₄⋊Dic₅₃	C₄	semidirect
14. C₂×C₄⋊C₈×C₅₃	C₄	semidirect
16. C₂×C₄×C₈×C₅₃	C₄	semidirect
152. C₂×C₄⋊C₈	Dic₅₃	semidirect
154. C₂×C₄×C₈	Dic₅₃	semidirect
161. C₅₃	#7981	semidirect

Maximal subgroups

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

subgroup	index
#20277122	2
#20272583	2
C₂×C₅₃×#431	2
#7981	53