Subgroups — #20283457

Order 6784.

492 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133225	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811169	C₂	1	1	yes	maximal, characteristic
3	#16810699	C₂	1	2	yes	maximal, complemented
4	#16810694	C₂	1	2	yes	maximal, complemented
5	#16810692	C₂	1	2	yes	maximal, complemented
6	C₈×C₂²⋊D₅₃	C₂	1	2	yes	maximal, complemented
7	C₂×#13133146	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810590	C₂	1	1	yes	maximal, characteristic, complemented
9	#16810589	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#131×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719945	C₂²	1	1	yes	characteristic
13	#13133225	—	2	4	no	—
14	C₂×#11719917	C₂²	1	1	yes	characteristic
15	C₂×#11719912	C₂²	1	2	yes	—
16	C₂²×#88322	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₅₃	C₂²	1	2	yes	—
18	#13133215	C₄	1	1	yes	characteristic
19	#13133213	C₄	1	1	yes	characteristic
20	#13133206	C₄	1	1	yes	characteristic
21	#13133204	C₄	1	1	yes	characteristic
22	#13133189	C₄	1	2	yes	—
23	#13133189	C₄	1	2	yes	—
24	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
25	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
26	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
27	#13133146	C₂²	1	1	yes	characteristic
28	#13133146	C₂²	1	1	yes	characteristic
29	#13133146	C₂²	1	1	yes	characteristic
30	#13133146	C₂²	1	1	yes	characteristic
31	C₂²⋊#88322	C₂²	1	2	yes	—
32	C₂²⋊#88322	C₂²	1	2	yes	—
33	#13133129	C₂²	1	2	yes	—
34	#13133129	C₂²	1	2	yes	—
35	#13133127	C₂²	1	2	yes	—
36	C₈×Dic₅₃	C₂²	1	2	yes	—
37	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
38	#131×C₅₃	—	2	4	no	—
39	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
40	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
43	#41×C₅₃	—	2	2	no	—
44	#41×C₅₃	—	2	2	no	—
45	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
47	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
50	C₂×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
51	C₂×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
53	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
55	#11719945	—	2	2	no	—
56	#11719945	—	2	2	no	—
57	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×D₅₃	C₂³	1	2	yes	—
59	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
60	M₁₆×C₅₃	—	2	4	no	—
61	M₁₆×C₅₃	D₄	1	4	yes	complemented
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	—	2	4	no	—
64	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂³	1	2	yes	—
67	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
68	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
69	#11719921	C₂×C₄	1	2	yes	—
70	#11719921	C₂×C₄	1	2	yes	—
71	#11719919	C₂×C₄	1	2	yes	—
72	#11719919	C₂×C₄	1	2	yes	—
73	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
74	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
75	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
76	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
77	#11719917	—	2	2	no	—
78	C₂×#88322	—	2	2	no	—
79	C₂×#88322	C₂³	1	2	yes	—
80	C₂×#88322	C₂³	1	1	yes	characteristic
81	C₂×#88322	C₂³	1	1	yes	characteristic
82	#11719912	—	2	4	no	—
83	C₈×D₅₃	—	2	4	no	—
84	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
85	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
87	Q₈×C₅₃	—	2	2	no	—
88	D₄×C₅₃	—	2	4	no	—
89	D₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₅₃	#41	1	2	yes	—
96	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₅₃	—	2	4	no	—
98	C₂²⋊D₅₃	—	2	4	no	—
99	C₂²⋊D₅₃	—	4	2	no	—
100	C₂×Dic₅₃	—	2	2	no	—
101	C₂×Dic₅₃	—	4	1	no	—
102	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
103	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
107	D₂₁₂	—	2	2	no	—
108	C₄×D₅₃	—	2	2	no	—
109	C₄×D₅₃	—	2	2	no	—
110	Dic₁₀₆	—	2	2	no	—
111	C₈×C₅₃	—	2	1	no	—
112	C₈×C₅₃	—	2	1	no	—
113	C₈×C₅₃	C₂×D₄	1	4	yes	—
114	#88322	—	2	1	no	—
115	#88322	—	2	1	no	—
116	#88322	—	2	2	no	—
117	C₂²×C₅₃	—	2	1	no	—
118	C₂²×C₅₃	—	2	1	no	—
119	C₂²×C₅₃	—	2	2	no	—
120	C₂²×C₅₃	—	2	2	no	—
121	C₂²×C₅₃	—	2	2	no	—
122	C₂²×C₅₃	C₄×D₄	1	2	yes	—
123	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×D₅₃	—	2	2	no	—
125	C₂×D₅₃	—	2	2	no	—
126	C₂×D₅₃	—	2	2	no	—
127	C₄×C₅₃	—	2	2	no	—
128	C₄×C₅₃	C₄×D₄	1	2	yes	—
129	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
130	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
131	Dic₅₃	—	2	2	no	—
132	Dic₅₃	—	2	1	no	—
133	Dic₅₃	—	2	1	no	—
134	Dic₅₃	—	2	1	no	—
135	Dic₅₃	#131	1	2	yes	complemented
136	#2935	—	53	1	no	maximal, Sylow
137	C₂×C₅₃	—	2	2	no	—
138	C₂×C₅₃	—	4	2	no	—
139	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
141	C₂×C₅₃	#568	1	1	yes	characteristic
142	D₅₃	—	4	2	no	—
143	C₂×#131	—	53	1	no	—
144	C₂×#131	D₅₃	1	1	yes	characteristic, complemented
145	#532	—	53	1	no	—
146	#436	—	53	2	no	—
147	#435	—	53	2	no	—
148	#435	—	53	2	no	—
149	C₈×D₄	—	53	2	no	—
150	C₂×C₄⋊C₈	—	53	1	no	—
151	#408	—	53	1	no	—
152	#408	—	53	1	no	—
153	C₄×M₁₆	—	53	1	no	—
154	C₅₃	#2935	1	1	yes	characteristic, complemented, Sylow
155	C₂×#41	—	53	1	no	—
156	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
157	#131	—	2	4	no	—
158	#131	—	106	4	no	—
159	C₂×M₁₆	—	53	1	no	—
160	C₂×M₁₆	—	53	2	no	—
161	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
162	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
163	C₂²×C₈	—	53	1	no	—
164	C₂²×C₈	—	53	2	no	—
165	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
166	C₂²×C₈	C₂×D₅₃	1	2	yes	—
167	C₄⋊Q₈	—	53	1	no	—
168	#127	—	53	1	no	—
169	#124	—	53	1	no	—
170	#124	—	53	1	no	—
171	#121	—	53	2	no	—
172	#121	—	53	2	no	—
173	C₄×D₄	—	53	2	no	—
174	C₄×D₄	—	53	2	no	—
175	C₂×C₄²	—	53	1	no	—
176	C₄⋊C₈	—	53	1	no	—
177	C₄⋊C₈	—	53	1	no	—
178	C₄⋊C₈	—	53	1	no	—
179	C₄⋊C₈	—	53	1	no	—
180	C₂²⋊C₈	—	53	2	no	—
181	C₂²⋊C₈	—	53	2	no	—
182	C₂²⋊C₈	—	53	2	no	—
183	C₂²⋊C₈	—	53	2	no	—
184	#97	—	53	2	no	—
185	C₄×C₈	—	53	2	no	—
186	#41	—	2	2	no	—
187	#41	—	2	2	no	—
188	#41	—	106	2	no	—
189	#41	—	106	2	no	—
190	C₂×Q₈	—	53	1	no	—
191	C₂×Q₈	C₄×D₅₃	1	1	yes	characteristic
192	C₂×D₄	—	53	2	no	—
193	C₂×D₄	—	53	2	no	—
194	C₂×D₄	—	53	2	no	—
195	C₂×D₄	—	53	1	no	—
196	C₂×D₄	C₄×D₅₃	1	2	yes	—
197	C₂×D₄	C₄×D₅₃	1	1	yes	characteristic
198	C₂²×C₄	—	53	1	no	—
199	C₂²×C₄	—	53	1	no	—
200	C₂²×C₄	—	53	2	no	—
201	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
202	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
203	M₁₆	—	2	4	no	—
204	M₁₆	—	106	2	no	—
205	M₁₆	—	106	4	no	—
206	M₁₆	C₂²⋊D₅₃	1	4	yes	complemented
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	53	2	no	—
210	C₂×C₈	—	53	1	no	—
211	C₂×C₈	—	53	1	no	—
212	C₂×C₈	—	106	2	no	—
213	C₂×C₈	—	106	4	no	—
214	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
215	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
216	C₂×C₈	C₂²×D₅₃	1	2	yes	—
217	C₄⋊C₄	—	53	2	no	—
218	C₄⋊C₄	—	53	2	no	—
219	C₂²⋊C₄	—	53	2	no	—
220	C₂²⋊C₄	—	53	2	no	—
221	C₂²⋊C₄	—	53	2	no	—
222	C₂²⋊C₄	—	53	2	no	—
223	C₄²	—	53	1	no	—
224	C₄²	—	53	1	no	—
225	C₄²	—	53	1	no	—
226	C₄²	—	53	1	no	—
227	C₂³	—	53	2	no	—
228	C₂³	C₂×C₄×D₅₃	1	2	yes	—
229	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
230	Q₈	—	2	2	no	—
231	Q₈	—	106	2	no	—
232	D₄	—	2	4	no	—
233	D₄	—	2	2	no	—
234	D₄	—	106	4	no	—
235	D₄	—	106	4	no	—
236	D₄	—	106	2	no	—
237	D₄	—	212	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	53	1	no	—
241	C₂×C₄	—	53	1	no	—
242	C₂×C₄	—	53	1	no	—
243	C₂×C₄	—	53	1	no	—
244	C₂×C₄	—	53	2	no	—
245	C₂×C₄	—	106	2	no	—
246	C₂×C₄	—	106	2	no	—
247	C₂×C₄	—	106	2	no	—
248	C₂×C₄	—	212	1	no	—
249	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
250	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
251	C₂×C₄	#11719945	1	2	yes	—
252	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
253	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	106	1	no	—
257	C₈	—	106	1	no	—
258	C₈	—	106	2	no	—
259	C₈	C₂×C₂²⋊D₅₃	1	4	yes	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	106	2	no	—
266	C₂²	—	106	2	no	—
267	C₂²	—	106	2	no	—
268	C₂²	C₄×C₂²⋊D₅₃	1	2	yes	—
269	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	53	2	no	—
272	C₄	—	106	2	no	—
273	C₄	—	106	1	no	—
274	C₄	—	106	1	no	—
275	C₄	—	106	1	no	—
276	C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
277	C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
278	C₄	C₂×#11719945	1	1	yes	characteristic
279	C₂	—	2	2	no	—
280	C₂	—	4	2	no	—
281	C₂	—	212	2	no	—
282	C₂	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
283	C₂	#131×D₅₃	1	1	yes	characteristic
284	C₂	#16811388	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₂×#131×C₅₃	C₂
hypercenter	C₂×#131	D₅₃
hypocenter	C₅₃	#2935
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#13133225	C₂	semidirect
3. #16810699	C₂	semidirect
4. #16810694	C₂	semidirect
5. #16810692	C₂	semidirect
6. C₈×C₂²⋊D₅₃	C₂	semidirect
7. C₂×#13133146	C₂	semidirect
8. #16810590	C₂	semidirect
9. #16810589	C₂	semidirect
10. M₁₆×Dic₅₃	C₂	semidirect
11. C₂×#131×C₅₃	C₂	semidirect
61. M₁₆×C₅₃	D₄	semidirect
135. Dic₅₃	#131	semidirect
144. C₂×#131	D₅₃	semidirect
154. C₅₃	#2935	semidirect
206. M₁₆	C₂²⋊D₅₃	semidirect