Subgroups — #27298971

Order 13568.

606 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719908	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810661	C₂	1	1	yes	maximal, characteristic
3	#20272806	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#427	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133176	—	2	1	no	—
6	C₂²×#11719908	—	2	2	no	—
7	C₂²×#11719908	—	2	2	no	—
8	C₂²×#11719908	C₄	1	4	yes	complemented
9	C₂²×#11719908	C₂²	1	2	yes	—
10	#16810661	—	2	2	no	—
11	#16810661	C₄	1	4	yes	complemented
12	C₂×#13133151	C₂²	1	2	yes	—
13	C₂×#13133149	—	2	1	no	—
14	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#117×C₅₃	—	2	1	no	—
16	C₅₃×#427	—	2	2	no	—
17	C₅₃×#427	C₄	1	4	yes	complemented
18	C₂×#107×C₅₃	C₂²	1	2	yes	—
19	C₂×#106×C₅₃	—	2	1	no	—
20	#13133176	—	2	4	no	—
21	#13133176	—	4	2	no	—
22	C₂×C₄⋊Dic₅₃	—	2	2	no	—
23	C₂×C₂²⋊Dic₅₃	—	4	1	no	—
24	C₂×C₄×Dic₅₃	—	4	1	no	—
25	C₂×#11719908	—	2	8	no	—
26	C₂×#11719908	—	2	2	no	—
27	C₂×#11719908	—	2	2	no	—
28	C₂×#11719908	C₂×C₄	1	4	yes	—
29	#13133151	—	2	2	no	—
30	#13133151	C₂×C₄	1	4	yes	—
31	#13133149	—	2	4	no	—
32	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₅₃	D₄	1	2	yes	—
37	C₂²×C₈×C₅₃	D₄	1	2	yes	—
38	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
40	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	#117×C₅₃	—	2	4	no	—
42	#117×C₅₃	—	4	2	no	—
43	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
44	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
45	C₂×C₄²×C₅₃	—	4	1	no	—
46	#107×C₅₃	—	2	2	no	—
47	#107×C₅₃	C₂×C₄	1	4	yes	—
48	#106×C₅₃	—	2	4	no	—
49	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	—	4	2	no	—
51	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
56	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
57	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₅₃	—	4	2	no	—
60	C₂×C₈×C₅₃	C₄²	1	4	yes	—
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
64	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
65	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
66	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
67	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
73	C₄⋊C₄×C₅₃	—	2	4	no	—
74	C₄⋊C₄×C₅₃	—	4	2	no	—
75	C₂²⋊C₄×C₅₃	—	4	4	no	—
76	C₄²×C₅₃	—	4	4	no	—
77	C₂²⋊Dic₅₃	—	4	4	no	—
78	C₄⋊Dic₅₃	—	2	4	no	—
79	C₄⋊Dic₅₃	—	4	2	no	—
80	C₄×Dic₅₃	—	4	4	no	—
81	#11719908	—	2	4	no	—
82	C₂³×C₅₃	—	2	4	no	—
83	C₂³×C₅₃	#102	1	2	yes	—
84	C₂³×C₅₃	#102	1	2	yes	—
85	C₂³×C₅₃	#107	1	2	yes	—
86	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	—	4	4	no	—
88	C₂×C₄×C₅₃	—	8	4	no	—
89	C₂×C₄×C₅₃	#95	1	4	yes	—
90	C₂×C₄×C₅₃	#102	1	2	yes	—
91	C₂×C₄×C₅₃	#102	1	2	yes	—
92	C₂×C₄×C₅₃	#103	1	4	yes	—
93	C₂×C₄×C₅₃	#103	1	4	yes	—
94	C₂×C₄×C₅₃	#106	1	4	yes	—
95	C₂×C₄×C₅₃	#107	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
102	C₂×Dic₅₃	—	4	4	no	—
103	C₂×Dic₅₃	—	8	4	no	—
104	C₈×C₅₃	#95	1	4	yes	—
105	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
106	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	#12556	—	53	1	no	maximal, Sylow
110	C₂²×C₅₃	—	2	4	no	—
111	C₂²×C₅₃	—	2	4	no	—
112	C₂²×C₅₃	—	2	4	no	—
113	C₂²×C₅₃	#340	1	4	yes	—
114	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
115	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
116	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
117	C₂²×C₅₃	#508	1	4	yes	—
118	C₄×C₅₃	—	8	4	no	—
119	C₄×C₅₃	#340	1	4	yes	—
120	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
121	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
122	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
123	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
124	Dic₅₃	—	8	4	no	—
125	C₂³×C₁₆	—	53	1	no	—
126	C₂×#427	—	53	1	no	—
127	C₂×#427	D₅₃	1	1	yes	characteristic
128	#1380	—	53	4	no	—
129	C₂×C₅₃	—	2	4	no	—
130	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
131	C₂×C₅₃	#2138	1	2	yes	—
132	C₂×C₅₃	#2138	1	2	yes	—
133	C₂×C₅₃	#2158	1	2	yes	—
134	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
135	C₂×#117	—	2	1	no	—
136	C₂×#117	—	106	1	no	—
137	C₂²×C₁₆	—	53	4	no	—
138	C₂²×C₁₆	—	53	2	no	—
139	C₂²×C₁₆	—	106	2	no	—
140	C₂²×C₁₆	—	106	2	no	—
141	#427	—	2	2	no	—
142	#427	—	53	4	no	—
143	#427	—	106	2	no	—
144	#427	Dic₅₃	1	4	yes	complemented
145	C₂×#107	—	53	2	no	—
146	C₂×#107	C₂×D₅₃	1	2	yes	—
147	C₂×#106	—	2	1	no	—
148	C₂×#106	—	106	1	no	—
149	C₅₃	#12556	1	1	yes	characteristic, complemented, Sylow
150	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
151	C₂²×C₈	Dic₁₀₆	1	2	yes	—
152	C₂²×C₈	C₄×D₅₃	1	2	yes	—
153	C₂²×C₈	C₄×D₅₃	1	2	yes	—
154	C₂²×C₈	D₂₁₂	1	2	yes	—
155	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
156	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
157	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
158	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
159	#117	—	2	4	no	—
160	#117	—	4	2	no	—
161	#117	—	106	4	no	—
162	#117	—	212	2	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	106	2	no	—
165	C₂×C₂²⋊C₄	—	4	1	no	—
166	C₂×C₂²⋊C₄	—	212	1	no	—
167	C₂×C₄²	—	4	1	no	—
168	C₂×C₄²	—	212	1	no	—
169	C₂×C₁₆	—	53	4	no	—
170	C₂×C₁₆	—	106	2	no	—
171	C₂×C₁₆	—	106	2	no	—
172	C₂×C₁₆	—	106	8	no	—
173	#107	—	2	2	no	—
174	#107	—	53	4	no	—
175	#107	—	106	2	no	—
176	#107	C₂×Dic₅₃	1	4	yes	—
177	#106	—	2	4	no	—
178	#106	—	106	4	no	—
179	C₂⁴	#11719922	1	1	yes	characteristic
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	—	212	2	no	—
182	C₂²×C₄	#11719919	1	2	yes	—
183	C₂²×C₄	#11719921	1	2	yes	—
184	C₂²×C₄	#11719922	1	1	yes	characteristic
185	C₂²×C₄	#11719923	1	2	yes	—
186	C₂²×C₄	#11719924	1	2	yes	—
187	C₂²×C₄	#11719925	1	2	yes	—
188	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
189	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
190	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
191	C₂×C₈	#11719919	1	2	yes	—
192	C₂×C₈	#11719919	1	4	yes	—
193	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
194	C₂×C₈	#11719921	1	2	yes	—
195	C₂×C₈	#11719921	1	4	yes	—
196	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
197	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
198	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
199	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
200	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
201	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
202	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	106	4	no	—
206	C₄⋊C₄	—	212	2	no	—
207	C₂²⋊C₄	—	4	4	no	—
208	C₂²⋊C₄	—	212	4	no	—
209	C₄²	—	4	4	no	—
210	C₄²	—	212	4	no	—
211	C₁₆	—	106	4	no	—
212	C₂³	—	2	4	no	—
213	C₂³	#13133137	1	2	yes	—
214	C₂³	#13133138	1	2	yes	—
215	C₂³	#13133150	1	2	yes	—
216	C₂³	C₂×#11719922	1	1	yes	characteristic
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	8	4	no	—
219	C₂×C₄	—	424	4	no	—
220	C₂×C₄	—	212	4	no	—
221	C₂×C₄	#13133124	1	4	yes	—
222	C₂×C₄	#13133137	1	2	yes	—
223	C₂×C₄	#13133138	1	2	yes	—
224	C₂×C₄	#13133140	1	4	yes	—
225	C₂×C₄	#13133141	1	4	yes	—
226	C₂×C₄	#13133148	1	4	yes	—
227	C₂×C₄	#13133150	1	2	yes	—
228	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
229	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
230	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
231	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
232	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
233	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
234	C₈	#13133124	1	4	yes	—
235	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
236	C₈	C₂×#11719921	1	1	yes	characteristic
237	C₈	C₂×#11719919	1	1	yes	characteristic
238	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
239	C₂²	—	2	4	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	4	no	—
242	C₂²	#16810385	1	4	yes	—
243	C₂²	C₂×#13133137	1	1	yes	characteristic
244	C₂²	C₂×#13133138	1	1	yes	characteristic
245	C₂²	C₂×#13133150	1	1	yes	characteristic
246	C₂²	#16811053	1	4	yes	—
247	C₄	—	8	4	no	—
248	C₄	—	424	4	no	—
249	C₄	#16810385	1	4	yes	—
250	C₄	C₂×#13133124	1	1	yes	characteristic
251	C₄	C₂×#13133140	1	1	yes	characteristic
252	C₄	C₂×#13133141	1	1	yes	characteristic
253	C₄	C₂×#13133148	1	1	yes	characteristic
254	C₂	—	2	4	no	—
255	C₂	C₂×#16810385	1	1	yes	characteristic
256	C₂	#20276420	1	2	yes	—
257	C₂	#20276422	1	2	yes	—
258	C₂	#20276497	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	#11719922
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#427	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂⁴×C₅₃	D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#427	D₅₃
hypocenter	C₅₃	#12556
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
3. #20272806	C₂	semidirect
8. C₂²×#11719908	C₄	semidirect
11. #16810661	C₄	semidirect
17. C₅₃×#427	C₄	semidirect
144. #427	Dic₅₃	semidirect
149. C₅₃	#12556	semidirect

Maximal subgroups

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

subgroup	index
C₂³×#11719908	2
C₂×#16810661	2
#20272806	2
C₂×C₅₃×#427	2
#12556	53