Subgroups — #20283614

Order 6784.

456 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	#16811421	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133225	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13133224	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810742	C₂	1	4	yes	maximal, complemented
5	#16810737	C₂	1	4	yes	maximal, complemented
6	#16810671	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810641	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810579	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#421	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11719945	C₂²	1	1	yes	characteristic
11	#13133248	C₄	1	2	yes	complemented
12	#13133247	C₄	1	2	yes	complemented
13	#13133243	C₄	1	2	yes	complemented
14	#13133242	C₄	1	2	yes	complemented
15	#13133237	C₂²	1	4	yes	—
16	#13133225	—	2	1	no	—
17	#13133225	—	2	1	no	—
18	#13133225	C₂²	1	2	yes	complemented
19	#13133225	C₂²	1	2	yes	complemented
20	#13133224	—	2	1	no	—
21	#13133224	—	2	1	no	—
22	#13133224	C₂²	1	2	yes	complemented
23	#13133224	C₂²	1	2	yes	complemented
24	C₂×#11719917	—	2	1	no	—
25	C₂×#11719917	C₂²	1	1	yes	characteristic
26	C₂×#11719917	C₂²	1	1	yes	characteristic
27	C₂×#11719912	—	2	1	no	—
28	C₂²×#88322	—	2	1	no	—
29	C₂²×#88322	C₂²	1	1	yes	characteristic
30	C₂×C₈×D₅₃	—	2	1	no	—
31	#13133152	C₂²	1	4	yes	complemented
32	#13133145	C₂²	1	4	yes	complemented
33	#13133128	C₂²	1	2	yes	complemented
34	C₄×#88322	C₂²	1	2	yes	complemented
35	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
37	#117×C₅₃	C₂²	1	1	yes	characteristic
38	#104×C₅₃	C₂²	1	4	yes	complemented
39	#41×C₅₃	—	2	1	no	—
40	#41×C₅₃	—	2	1	no	—
41	#41×C₅₃	C₂×C₄	1	2	yes	—
42	#41×C₅₃	C₂³	1	2	yes	—
43	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×D₄×C₅₃	—	2	1	no	—
45	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	—	2	1	no	—
47	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊D₅₃	—	2	1	no	—
49	#11719945	—	2	1	no	—
50	#11719945	—	2	1	no	—
51	#11719945	C₂×C₄	1	2	yes	—
52	#11719945	C₂³	1	2	yes	—
53	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₅₃	—	2	1	no	—
55	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
56	M₁₆×C₅₃	—	2	1	no	—
57	M₁₆×C₅₃	C₂³	1	2	yes	—
58	C₂×C₈×C₅₃	—	2	1	no	—
59	C₄⋊C₄×C₅₃	—	2	1	no	—
60	C₂²⋊C₄×C₅₃	—	2	1	no	—
61	C₄²×C₅₃	C₂³	1	2	yes	—
62	#11719925	C₂×C₄	1	4	yes	—
63	#11719924	C₂×C₄	1	4	yes	—
64	#11719923	C₂×C₄	1	4	yes	—
65	#11719922	C₂×C₄	1	4	yes	—
66	#11719917	—	2	1	no	—
67	#11719917	—	2	2	no	—
68	#11719917	—	2	2	no	—
69	#11719917	C₂×C₄	1	2	yes	—
70	#11719917	C₂×C₄	1	2	yes	—
71	#11719917	C₂³	1	2	yes	—
72	C₂×#88322	—	2	1	no	—
73	C₂×#88322	—	2	2	no	—
74	C₂×#88322	—	2	2	no	—
75	C₂×#88322	D₄	1	1	yes	characteristic
76	C₂×#88322	D₄	1	1	yes	characteristic
77	C₂×#88322	D₄	1	1	yes	characteristic
78	C₂×#88322	D₄	1	1	yes	characteristic
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	2	no	—
83	#11719912	—	2	2	no	—
84	C₈×D₅₃	—	2	2	no	—
85	C₈×D₅₃	—	2	2	no	—
86	C₂³×C₅₃	—	2	1	no	—
87	C₂³×C₅₃	#41	1	1	yes	characteristic
88	C₂²×D₅₃	—	2	1	no	—
89	Q₈×C₅₃	—	2	1	no	—
90	Q₈×C₅₃	C₂²×C₄	1	2	yes	—
91	D₄×C₅₃	—	2	2	no	—
92	D₄×C₅₃	—	2	2	no	—
93	D₄×C₅₃	—	2	1	no	—
94	D₄×C₅₃	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₅₃	—	2	1	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
105	C₂²⋊D₅₃	—	2	2	no	—
106	C₂²⋊D₅₃	—	2	2	no	—
107	C₂×Dic₅₃	—	2	1	no	—
108	D₂₁₂	—	2	1	no	—
109	D₂₁₂	C₂²×C₄	1	2	yes	—
110	C₄×D₅₃	—	2	2	no	—
111	C₄×D₅₃	—	2	2	no	—
112	Dic₁₀₆	—	2	1	no	—
113	Dic₁₀₆	C₂²×C₄	1	2	yes	—
114	C₈×C₅₃	—	2	2	no	—
115	#88322	—	2	2	no	—
116	#88322	C₂²×C₄	1	4	yes	—
117	#88322	C₂×D₄	1	1	yes	characteristic
118	#88322	C₂×D₄	1	1	yes	characteristic
119	#88322	C₂×D₄	1	1	yes	characteristic
120	#88322	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₅₃	—	2	2	no	—
122	C₂²×C₅₃	—	4	1	no	—
123	C₂²×C₅₃	—	4	2	no	—
124	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
125	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
126	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
127	C₂×D₅₃	—	2	2	no	—
128	C₂×D₅₃	—	4	2	no	—
129	C₄×C₅₃	—	2	2	no	—
130	C₄×C₅₃	—	4	2	no	—
131	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
132	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
133	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
134	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
135	Dic₅₃	—	2	2	no	—
136	#2955	—	53	1	no	maximal, Sylow
137	C₂×C₅₃	—	2	1	no	—
138	C₂×C₅₃	—	2	1	no	—
139	C₂×C₅₃	—	2	1	no	—
140	C₂×C₅₃	—	4	2	no	—
141	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
142	D₅₃	—	4	2	no	—
143	#575	—	53	1	no	—
144	C₂×#131	—	53	1	no	—
145	C₂×#131	—	53	1	no	—
146	#444	—	53	4	no	—
147	#443	—	53	4	no	—
148	#430	—	53	1	no	—
149	#421	—	53	1	no	—
150	#421	D₅₃	1	1	yes	characteristic, complemented
151	#405	—	53	1	no	—
152	C₅₃	#2955	1	1	yes	characteristic, complemented, Sylow
153	C₂×#41	—	53	1	no	—
154	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
155	#137	—	53	2	no	—
156	#137	—	53	2	no	—
157	AGL(1,ℤ/8)	—	53	2	no	—
158	AGL(1,ℤ/8)	—	53	2	no	—
159	#135	—	53	4	no	—
160	#131	—	53	2	no	—
161	#131	—	53	2	no	—
162	#131	—	53	2	no	—
163	#131	—	53	2	no	—
164	#131	—	106	1	no	—
165	#131	—	106	1	no	—
166	#131	—	106	1	no	—
167	#131	—	106	1	no	—
168	C₂×M₁₆	—	53	1	no	—
169	C₂×M₁₆	—	53	1	no	—
170	C₂×M₁₆	—	106	1	no	—
171	C₂×M₁₆	—	106	1	no	—
172	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
173	C₂²×C₈	—	53	1	no	—
174	C₂²×C₈	—	106	1	no	—
175	C₂²×C₈	—	106	1	no	—
176	#117	C₂×D₅₃	1	1	yes	characteristic
177	#108	—	53	4	no	—
178	#104	—	53	4	no	—
179	#104	C₂×D₅₃	1	4	yes	complemented
180	#97	—	53	2	no	—
181	C₄×C₈	—	53	2	no	—
182	#41	—	2	1	no	—
183	#41	—	2	1	no	—
184	#41	—	53	2	no	—
185	#41	—	53	2	no	—
186	#41	—	106	1	no	—
187	#41	—	106	1	no	—
188	#41	C₄×D₅₃	1	2	yes	—
189	#41	C₂²×D₅₃	1	2	yes	—
190	C₂×Q₈	—	53	1	no	—
191	C₂×Q₈	C₄×D₅₃	1	1	yes	characteristic
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	—	53	1	no	—
194	C₂×D₄	—	106	1	no	—
195	C₂×D₄	C₄×D₅₃	1	1	yes	characteristic
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	106	1	no	—
198	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
199	Q₁₆	—	53	4	no	—
200	SD₁₆	—	53	4	no	—
201	SD₁₆	—	53	4	no	—
202	D₈	—	53	4	no	—
203	M₁₆	—	2	1	no	—
204	M₁₆	—	53	2	no	—
205	M₁₆	—	53	2	no	—
206	M₁₆	—	53	2	no	—
207	M₁₆	—	106	2	no	—
208	M₁₆	—	106	2	no	—
209	M₁₆	—	106	2	no	—
210	M₁₆	—	106	2	no	—
211	M₁₆	—	106	1	no	—
212	M₁₆	C₂²×D₅₃	1	2	yes	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	53	1	no	—
215	C₂×C₈	—	53	1	no	—
216	C₂×C₈	—	53	1	no	—
217	C₂×C₈	—	53	1	no	—
218	C₂×C₈	—	53	1	no	—
219	C₂×C₈	—	53	2	no	—
220	C₂×C₈	—	53	1	no	—
221	C₂×C₈	—	106	2	no	—
222	C₂×C₈	—	106	2	no	—
223	C₂×C₈	—	106	2	no	—
224	C₂×C₈	—	106	2	no	—
225	C₂×C₈	—	106	1	no	—
226	C₄⋊C₄	—	2	1	no	—
227	C₂²⋊C₄	—	2	1	no	—
228	C₄²	C₂²×D₅₃	1	2	yes	—
229	C₂³	—	2	1	no	—
230	C₂³	—	106	1	no	—
231	C₂³	#11719947	1	1	yes	characteristic
232	Q₈	—	2	1	no	—
233	Q₈	—	53	2	no	—
234	Q₈	—	106	1	no	—
235	Q₈	C₂×C₄×D₅₃	1	2	yes	—
236	D₄	—	2	2	no	—
237	D₄	—	2	2	no	—
238	D₄	—	2	1	no	—
239	D₄	—	53	2	no	—
240	D₄	—	106	2	no	—
241	D₄	—	106	2	no	—
242	D₄	—	106	1	no	—
243	D₄	C₂×C₄×D₅₃	1	2	yes	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	106	1	no	—
249	C₂×C₄	—	106	2	no	—
250	C₂×C₄	—	106	2	no	—
251	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
252	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
253	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
254	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
255	C₂×C₄	#11719947	1	1	yes	characteristic
256	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	53	4	no	—
259	C₈	—	53	1	no	—
260	C₈	—	53	1	no	—
261	C₈	—	53	1	no	—
262	C₈	—	53	1	no	—
263	C₈	—	106	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	106	2	no	—
268	C₂²	—	212	2	no	—
269	C₂²	#13133177	1	1	yes	characteristic
270	C₂²	#13133177	1	1	yes	characteristic
271	C₂²	C₂×#11719947	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	106	2	no	—
275	C₄	#13133177	1	1	yes	characteristic
276	C₄	#13133177	1	1	yes	characteristic
277	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
278	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	4	2	no	—
283	C₂	—	212	2	no	—
284	C₂	C₂×#13133177	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#421	C₂
socle	C₂×C₅₃	C₂×C₄×D₄
hypercenter	#421	D₅₃
hypocenter	C₅₃	#2955
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #421

Complements

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

N	H ≅ G/N	type
2. C₂×#13133225	C₂	semidirect
3. C₂×#13133224	C₂	semidirect
4. #16810742	C₂	semidirect
5. #16810737	C₂	semidirect
6. #16810671	C₂	semidirect
7. #16810641	C₂	semidirect
8. #16810579	C₂	semidirect
9. C₅₃×#421	C₂	semidirect
11. #13133248	C₄	semidirect
12. #13133247	C₄	semidirect
13. #13133243	C₄	semidirect
14. #13133242	C₄	semidirect
18. #13133225	C₂²	semidirect
19. #13133225	C₂²	semidirect
22. #13133224	C₂²	semidirect
23. #13133224	C₂²	semidirect
31. #13133152	C₂²	semidirect
32. #13133145	C₂²	semidirect
33. #13133128	C₂²	semidirect
34. C₄×#88322	C₂²	semidirect
38. #104×C₅₃	C₂²	semidirect
150. #421	D₅₃	semidirect
152. C₅₃	#2955	semidirect
179. #104	C₂×D₅₃	semidirect