Subgroups — #27539456

Order 13568.

598 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285229	C₂	1	1	yes	maximal, characteristic, complemented
2	#20276951	C₂	1	4	yes	maximal, complemented
3	#20276946	C₂	1	4	yes	maximal, complemented
4	#20276435	C₂	1	2	yes	maximal, complemented
5	#20276431	C₂	1	2	yes	maximal
6	C₄×#13133157	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₃×#495	C₂	1	1	yes	maximal, characteristic
8	#16811180	—	2	1	no	—
9	C₂×#13133157	C₂²	1	2	yes	—
10	#16811028	C₂²	1	2	yes	complemented
11	#16811025	—	2	1	no	—
12	#16811025	C₂²	1	2	yes	complemented
13	#16811021	—	2	1	no	—
14	#16810661	—	2	1	no	—
15	#16810661	C₂²	1	2	yes	—
16	#16810427	C₂²	1	8	yes	complemented
17	#16810424	C₂²	1	8	yes	—
18	#16810399	C₂²	1	2	yes	complemented
19	C₄×#11719908	C₂²	1	2	yes	complemented
20	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×SD₁₆×C₅₃	—	2	1	no	—
22	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#124×C₅₃	—	2	1	no	—
24	C₅₃×#495	—	2	1	no	—
25	C₅₃×#495	—	2	1	no	—
26	C₅₃×#495	C₂²	1	2	yes	—
27	C₅₃×#495	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	#13133217	—	2	2	no	—
30	#13133209	—	2	2	no	—
31	#13133196	—	4	2	no	—
32	C₄×Dic₁₀₆	—	4	2	no	—
33	#13133176	—	2	2	no	—
34	#13133157	—	2	4	no	—
35	C₂×#11719908	C₂³	1	4	yes	—
36	#13133151	—	2	2	no	—
37	#13133151	C₂³	1	4	yes	—
38	#13133149	—	2	2	no	—
39	#13133149	—	2	2	no	—
40	C₂²×Q₈×C₅₃	—	2	1	no	—
41	C₂²×D₄×C₅₃	—	2	1	no	—
42	C₂×Q₁₆×C₅₃	—	2	1	no	—
43	C₂×Q₁₆×C₅₃	—	2	4	no	—
44	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
45	C₂×SD₁₆×C₅₃	—	2	2	no	—
46	C₂×SD₁₆×C₅₃	—	2	2	no	—
47	C₂×SD₁₆×C₅₃	—	4	4	no	—
48	C₂×D₈×C₅₃	—	2	1	no	—
49	C₂×D₈×C₅₃	—	2	4	no	—
50	C₂×D₈×C₅₃	C₂³	1	2	yes	—
51	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	#124×C₅₃	—	2	2	no	—
54	#124×C₅₃	—	2	2	no	—
55	#124×C₅₃	—	4	2	no	—
56	C₂×C₂²⋊C₄×C₅₃	—	4	2	no	—
57	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
58	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
59	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
60	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₂⁴×C₅₃	—	4	1	no	—
63	C₂×Q₈×C₅₃	—	2	2	no	—
64	C₂×Q₈×C₅₃	—	4	1	no	—
65	C₂×Q₈×C₅₃	—	4	2	no	—
66	C₂×Q₈×C₅₃	—	4	2	no	—
67	C₂×D₄×C₅₃	—	2	2	no	—
68	C₂×D₄×C₅₃	—	4	2	no	—
69	C₂×D₄×C₅₃	—	4	1	no	—
70	C₂×D₄×C₅₃	—	4	2	no	—
71	C₂²×C₄×C₅₃	—	4	1	no	—
72	C₂²×C₄×C₅₃	D₈	1	2	yes	—
73	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂×Dic₁₀₆	—	4	2	no	—
75	Q₁₆×C₅₃	—	2	4	no	—
76	Q₁₆×C₅₃	—	4	2	no	—
77	SD₁₆×C₅₃	—	4	4	no	—
78	SD₁₆×C₅₃	—	4	4	no	—
79	D₈×C₅₃	—	2	4	no	—
80	D₈×C₅₃	—	4	2	no	—
81	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
82	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	#41	1	4	yes	—
86	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
87	C₂²⋊C₄×C₅₃	—	4	4	no	—
88	C₂²⋊C₄×C₅₃	—	4	4	no	—
89	C₄²×C₅₃	D₈	1	2	yes	—
90	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂²⋊Dic₅₃	—	4	2	no	—
93	C₄⋊Dic₅₃	—	2	4	no	—
94	C₄⋊Dic₅₃	—	4	2	no	—
95	#11719919	—	4	4	no	—
96	C₄×Dic₅₃	—	4	2	no	—
97	#11719908	—	2	4	no	—
98	C₂³×C₅₃	—	4	2	no	—
99	C₂³×C₅₃	—	8	4	no	—
100	C₂³×C₅₃	—	8	2	no	—
101	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
102	Q₈×C₅₃	—	4	2	no	—
103	Q₈×C₅₃	—	4	2	no	—
104	Q₈×C₅₃	—	8	2	no	—
105	D₄×C₅₃	—	4	2	no	—
106	D₄×C₅₃	—	4	2	no	—
107	D₄×C₅₃	—	8	2	no	—
108	C₂×C₄×C₅₃	—	4	2	no	—
109	C₂×C₄×C₅₃	—	4	2	no	—
110	C₂×C₄×C₅₃	—	8	2	no	—
111	C₂×C₄×C₅₃	#124	1	2	yes	—
112	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
116	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×Dic₅₃	—	4	4	no	—
118	Dic₁₀₆	—	8	4	no	—
119	C₈×C₅₃	#124	1	2	yes	—
120	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
121	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
122	C₈×C₅₃	C₂×#41	1	2	yes	—
123	#33706	—	53	1	no	maximal, Sylow
124	C₂²×C₅₃	—	2	2	no	—
125	C₂²×C₅₃	—	8	2	no	—
126	C₂²×C₅₃	—	8	4	no	—
127	C₂²×C₅₃	—	16	4	no	—
128	C₂²×C₅₃	#486	1	2	yes	—
129	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
130	C₄×C₅₃	—	2	2	no	—
131	C₄×C₅₃	—	2	2	no	—
132	C₄×C₅₃	—	8	2	no	—
133	C₄×C₅₃	#486	1	2	yes	—
134	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
135	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
136	Dic₅₃	—	8	4	no	—
137	#3160	—	53	1	no	—
138	C₂×#495	D₅₃	1	1	yes	characteristic
139	#2243	—	53	4	no	—
140	#2242	—	53	4	no	—
141	#2141	—	53	2	no	—
142	#2140	—	53	2	no	—
143	C₄×M₃₂	—	53	1	no	—
144	C₂×C₅₃	—	2	2	no	—
145	C₂×C₅₃	—	16	2	no	—
146	C₂×C₅₃	C₂×#486	1	1	yes	characteristic
147	C₂×C₅₃	#3414	1	2	yes	—
148	C₂²×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
149	C₂²×SD₁₆	—	2	1	no	—
150	C₂²×D₈	C₂×D₅₃	1	1	yes	characteristic
151	#533	—	106	1	no	—
152	C₂×#124	—	2	1	no	—
153	C₂×M₃₂	—	53	2	no	—
154	#500	—	53	2	no	—
155	#499	—	53	2	no	—
156	#499	—	106	1	no	—
157	#498	—	106	1	no	—
158	#495	—	2	1	no	—
159	#495	—	2	1	no	—
160	#495	C₂×D₅₃	1	2	yes	—
161	#495	C₂×D₅₃	1	2	yes	—
162	#427	—	53	2	no	—
163	#427	—	106	1	no	—
164	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
165	#358	—	53	8	no	—
166	#357	—	53	8	no	—
167	#346	—	53	2	no	—
168	C₄×C₁₆	—	53	2	no	—
169	C₅₃	#33706	1	1	yes	characteristic, complemented, Sylow
170	C₂²×Q₈	—	2	1	no	—
171	C₂²×D₄	—	2	1	no	—
172	C₂×Q₁₆	—	2	4	no	—
173	C₂×Q₁₆	—	2	1	no	—
174	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
175	C₂×SD₁₆	—	2	2	no	—
176	C₂×SD₁₆	—	2	2	no	—
177	C₂×SD₁₆	—	4	4	no	—
178	C₂×D₈	—	2	4	no	—
179	C₂×D₈	—	2	1	no	—
180	C₂×D₈	C₂²×D₅₃	1	2	yes	—
181	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
183	C₄⋊Q₈	—	106	2	no	—
184	#125	—	106	2	no	—
185	#124	—	2	2	no	—
186	#124	—	2	2	no	—
187	#124	—	4	2	no	—
188	C₂²⋊Q₈	—	212	2	no	—
189	C₄×Q₈	—	212	2	no	—
190	#117	—	106	2	no	—
191	C₂×C₂²⋊C₄	—	4	2	no	—
192	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
193	M₃₂	—	106	4	no	—
194	C₂×C₁₆	—	53	4	no	—
195	#107	—	53	4	no	—
196	#107	—	106	2	no	—
197	#106	—	106	2	no	—
198	#106	—	106	2	no	—
199	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
200	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
201	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
202	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
203	C₂⁴	—	4	1	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	—	4	1	no	—
206	C₂×Q₈	—	4	2	no	—
207	C₂×Q₈	—	4	2	no	—
208	C₂×Q₈	—	212	2	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂×D₄	—	4	1	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	#11719922	1	2	yes	—
215	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
216	Q₁₆	—	2	4	no	—
217	Q₁₆	—	4	2	no	—
218	SD₁₆	—	4	4	no	—
219	SD₁₆	—	4	4	no	—
220	D₈	—	2	4	no	—
221	D₈	—	4	2	no	—
222	C₂×C₈	#11719923	1	4	yes	—
223	C₂×C₈	#11719947	1	4	yes	—
224	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
225	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
226	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
227	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
228	C₄⋊C₄	—	106	4	no	—
229	C₄⋊C₄	—	212	4	no	—
230	C₄⋊C₄	—	212	2	no	—
231	C₂²⋊C₄	—	4	4	no	—
232	C₂²⋊C₄	—	4	4	no	—
233	C₂²⋊C₄	—	212	2	no	—
234	C₄²	—	212	2	no	—
235	C₄²	#11719922	1	2	yes	—
236	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
237	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
238	C₁₆	—	106	4	no	—
239	C₂³	—	4	2	no	—
240	C₂³	—	8	4	no	—
241	C₂³	—	8	2	no	—
242	C₂³	C₂×#11719922	1	1	yes	characteristic
243	Q₈	—	4	2	no	—
244	Q₈	—	4	2	no	—
245	Q₈	—	8	2	no	—
246	Q₈	—	424	4	no	—
247	D₄	—	4	2	no	—
248	D₄	—	4	2	no	—
249	D₄	—	8	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	8	2	no	—
253	C₂×C₄	—	212	4	no	—
254	C₂×C₄	#13133204	1	2	yes	—
255	C₂×C₄	C₂×#11719922	1	2	yes	—
256	C₂×C₄	C₂×#11719922	1	2	yes	—
257	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
258	C₂×C₄	C₂×#11719923	1	2	yes	—
259	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
260	C₈	#13133204	1	2	yes	—
261	C₈	C₂×#11719923	1	2	yes	—
262	C₈	C₂×#11719923	1	2	yes	—
263	C₈	C₂×#11719947	1	2	yes	—
264	C₂²	—	2	2	no	—
265	C₂²	—	8	2	no	—
266	C₂²	—	8	4	no	—
267	C₂²	—	16	4	no	—
268	C₂²	#16810969	1	2	yes	—
269	C₂²	C₂²×#11719922	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	8	2	no	—
273	C₄	—	424	4	no	—
274	C₄	#16810969	1	2	yes	—
275	C₄	C₂×#13133204	1	1	yes	characteristic
276	C₄	C₂²×#11719923	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	16	2	no	—
279	C₂	C₂×#16810969	1	1	yes	characteristic
280	C₂	#20287849	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#495	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂²×D₈×C₅₃	C₂²
hypercenter	C₂×#495	D₅₃
hypocenter	C₅₃	#33706
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#495

Complements

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

N	H ≅ G/N	type
1. #20285229	C₂	semidirect
2. #20276951	C₂	semidirect
3. #20276946	C₂	semidirect
4. #20276435	C₂	semidirect
6. C₄×#13133157	C₂	semidirect
10. #16811028	C₂²	semidirect
12. #16811025	C₂²	semidirect
16. #16810427	C₂²	semidirect
18. #16810399	C₂²	semidirect
19. C₄×#11719908	C₂²	semidirect
169. C₅₃	#33706	semidirect

Maximal subgroups

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

subgroup	index
#20285229	2
#20276951	2
#20276946	2
#20276435	2
#20276431	2
C₄×#13133157	2
C₂×C₅₃×#495	2
#33706	53