Subgroups — #21454719

Order 7808.

456 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16840559	C₂	1	1	yes	maximal, characteristic
2	#131×D₆₁	C₂	1	2	yes	maximal, complemented
3	#16839885	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839882	C₂	1	2	yes	maximal, complemented
5	#16839879	C₂	1	1	yes	maximal, characteristic, complemented
6	#16839878	C₂	1	1	yes	maximal, characteristic, complemented
7	#16839876	C₂	1	2	yes	maximal, complemented
8	#16839809	C₂	1	1	yes	maximal, characteristic, complemented
9	#16839777	C₂	1	2	yes	maximal, complemented
10	#16839719	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#444	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₆₁	C₂²	1	2	yes	—
13	#14519809	C₄	1	2	yes	complemented
14	#14519806	C₄	1	2	yes	complemented
15	#14519804	C₄	1	2	yes	complemented
16	#14519801	C₄	1	2	yes	complemented
17	#14519799	C₂²	1	1	yes	characteristic
18	#14519797	C₂²	1	2	yes	—
19	#14519785	—	2	2	no	—
20	#14519785	C₂²	1	2	yes	complemented
21	#14519784	C₂²	1	2	yes	complemented
22	#14519783	—	2	2	no	—
23	#14519783	C₂²	1	2	yes	complemented
24	C₂×#11758189	C₂²	1	1	yes	characteristic
25	M₁₆×D₆₁	—	2	2	no	—
26	M₁₆×D₆₁	C₂²	1	2	yes	—
27	C₂×C₈×D₆₁	—	2	2	no	—
28	C₂×C₈×D₆₁	C₂²	1	1	yes	characteristic
29	#14519734	C₂²	1	1	yes	characteristic
30	#14519713	C₂²	1	1	yes	characteristic, complemented
31	#14519712	C₂²	1	2	yes	complemented
32	#14519705	C₂²	1	2	yes	complemented
33	#14519704	C₂²	1	2	yes	complemented
34	#14519703	C₂²	1	2	yes	complemented
35	#14519687	C₂²	1	1	yes	characteristic, complemented
36	C₄×#92418	C₂²	1	1	yes	characteristic, complemented
37	C₈×Dic₆₁	C₂²	1	1	yes	characteristic, complemented
38	#135×C₆₁	C₂²	1	1	yes	characteristic
39	#131×C₆₁	C₂²	1	2	yes	complemented
40	#108×C₆₁	C₂²	1	1	yes	characteristic, complemented
41	#104×C₆₁	C₂²	1	2	yes	complemented
42	#97×C₆₁	C₂²	1	1	yes	characteristic, complemented
43	#41×C₆₁	C₂³	1	2	yes	—
44	#11758226	C₂×C₄	1	2	yes	—
45	Q₈×D₆₁	C₂×C₄	1	2	yes	—
46	#11758224	—	2	2	no	—
47	#11758224	C₂×C₄	1	2	yes	—
48	D₄×D₆₁	—	2	2	no	—
49	D₄×D₆₁	C₂×C₄	1	2	yes	—
50	#11758222	—	2	2	no	—
51	#11758222	C₂³	1	2	yes	—
52	C₂×C₄×D₆₁	—	2	2	no	—
53	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
54	Q₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
55	SD₁₆×C₆₁	C₂×C₄	1	2	yes	—
56	D₈×C₆₁	C₂×C₄	1	1	yes	characteristic
57	M₁₆×C₆₁	—	2	2	no	—
58	M₁₆×C₆₁	C₂³	1	2	yes	—
59	C₂×C₈×C₆₁	—	2	2	no	—
60	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
62	C₄²×C₆₁	C₂³	1	1	yes	characteristic
63	#11758202	C₂×C₄	1	2	yes	—
64	#11758201	C₂×C₄	1	2	yes	—
65	#11758200	C₂×C₄	1	2	yes	—
66	#11758199	C₂×C₄	1	2	yes	—
67	#11758198	—	2	1	no	—
68	#11758196	—	2	1	no	—
69	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
70	#11758194	—	2	2	no	—
71	#11758194	C₂³	1	2	yes	—
72	C₂×#92418	—	2	2	no	—
73	C₂×#92418	C₂³	1	1	yes	characteristic
74	C₂×#92418	C₂³	1	1	yes	characteristic
75	Dic₂₄₄	C₂×C₄	1	1	yes	characteristic
76	D₄₈₈	C₂×C₄	1	1	yes	characteristic
77	#11758190	C₂×C₄	1	2	yes	—
78	#11758189	—	2	2	no	—
79	#11758189	—	2	2	no	—
80	#11758189	—	2	2	no	—
81	#11758189	C₂×C₄	1	2	yes	—
82	#11758189	C₂×C₄	1	2	yes	—
83	C₈×D₆₁	—	2	2	no	—
84	C₈×D₆₁	—	2	2	no	—
85	C₈×D₆₁	—	2	2	no	—
86	C₈×D₆₁	D₄	1	2	yes	—
87	C₈×D₆₁	D₄	1	2	yes	—
88	C₂²×D₆₁	—	2	2	no	—
89	C₂²×D₆₁	#41	1	1	yes	characteristic
90	Q₈×C₆₁	C₂²×C₄	1	2	yes	—
91	D₄×C₆₁	—	2	2	no	—
92	D₄×C₆₁	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₆₁	—	2	2	no	—
94	C₂×C₄×C₆₁	—	2	1	no	—
95	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
96	C₂²⋊D₆₁	—	2	2	no	—
97	C₂²⋊D₆₁	—	2	2	no	—
98	C₂²⋊D₆₁	—	2	2	no	—
99	C₂×Dic₆₁	—	2	2	no	—
100	C₂×Dic₆₁	—	2	1	no	—
101	C₂×Dic₆₁	#41	1	1	yes	characteristic
102	D₂₄₄	—	2	2	no	—
103	D₂₄₄	C₂²×C₄	1	2	yes	—
104	C₄×D₆₁	—	2	2	no	—
105	C₄×D₆₁	—	2	2	no	—
106	C₄×D₆₁	—	2	2	no	—
107	C₄×D₆₁	C₂²×C₄	1	2	yes	—
108	C₄×D₆₁	C₂×D₄	1	2	yes	—
109	Dic₁₂₂	—	2	2	no	—
110	Dic₁₂₂	C₂²×C₄	1	2	yes	—
111	C₈×C₆₁	—	2	2	no	—
112	C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
113	C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
114	C₈×C₆₁	C₂×D₄	1	2	yes	—
115	#92418	—	2	2	no	—
116	#92418	C₂²×C₄	1	2	yes	—
117	#92418	C₂×D₄	1	1	yes	characteristic
118	#92418	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₆₁	—	2	2	no	—
120	C₂²×C₆₁	C₂×#41	1	1	yes	characteristic
121	C₂×D₆₁	—	2	2	no	—
122	C₂×D₆₁	—	4	1	no	—
123	C₂×D₆₁	—	4	4	no	—
124	C₂×D₆₁	C₄×D₄	1	2	yes	—
125	C₄×C₆₁	—	2	2	no	—
126	C₄×C₆₁	—	4	1	no	—
127	C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
128	C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
129	Dic₆₁	—	2	2	no	—
130	Dic₆₁	—	4	1	no	—
131	Dic₆₁	C₄×D₄	1	2	yes	—
132	#2955	—	61	1	no	maximal, Sylow
133	C₂×C₆₁	—	2	1	no	—
134	C₂×C₆₁	—	4	2	no	—
135	C₂×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
136	D₆₁	—	2	2	no	—
137	D₆₁	—	4	2	no	—
138	#575	—	61	1	no	—
139	C₂×#131	—	61	2	no	—
140	#444	—	61	2	no	—
141	#444	—	61	1	no	—
142	#444	D₆₁	1	1	yes	characteristic, complemented
143	#443	—	61	1	no	—
144	#443	—	61	1	no	—
145	#443	—	61	2	no	—
146	#430	—	61	1	no	—
147	#421	—	61	2	no	—
148	#405	—	61	1	no	—
149	C₆₁	#2955	1	1	yes	characteristic, complemented, Sylow
150	C₂×#41	—	61	2	no	—
151	#137	—	61	2	no	—
152	#137	—	61	2	no	—
153	AGL(1,ℤ/8)	—	61	2	no	—
154	AGL(1,ℤ/8)	—	61	2	no	—
155	#135	—	61	2	no	—
156	#135	—	61	1	no	—
157	#135	C₂×D₆₁	1	1	yes	characteristic
158	#131	—	61	2	no	—
159	#131	—	61	2	no	—
160	#131	—	61	2	no	—
161	#131	—	122	2	no	—
162	#131	—	122	2	no	—
163	#131	C₂×D₆₁	1	2	yes	complemented
164	C₂×M₁₆	—	61	1	no	—
165	C₂×M₁₆	—	61	2	no	—
166	C₂×M₁₆	—	122	2	no	—
167	C₂²×C₈	—	61	1	no	—
168	C₂²×C₈	—	122	2	no	—
169	#117	—	61	1	no	—
170	#108	—	61	2	no	—
171	#108	—	61	1	no	—
172	#108	C₂×D₆₁	1	1	yes	characteristic, complemented
173	#104	—	61	2	no	—
174	#104	—	61	2	no	—
175	#104	—	61	2	no	—
176	#104	C₂×D₆₁	1	2	yes	complemented
177	#97	—	61	1	no	—
178	#97	C₂×D₆₁	1	1	yes	characteristic, complemented
179	C₄×C₈	—	61	1	no	—
180	C₄×C₈	—	61	1	no	—
181	#41	—	61	2	no	—
182	#41	—	61	2	no	—
183	#41	—	61	2	no	—
184	#41	—	122	2	no	—
185	#41	—	122	2	no	—
186	#41	C₂²×D₆₁	1	2	yes	—
187	C₂×Q₈	—	61	2	no	—
188	C₂×D₄	—	61	2	no	—
189	C₂×D₄	—	122	2	no	—
190	C₂²×C₄	—	61	1	no	—
191	C₂²×C₄	—	122	2	no	—
192	Q₁₆	—	61	2	no	—
193	Q₁₆	—	61	1	no	—
194	Q₁₆	C₄×D₆₁	1	1	yes	characteristic
195	SD₁₆	—	61	2	no	—
196	SD₁₆	—	61	2	no	—
197	SD₁₆	—	61	2	no	—
198	SD₁₆	C₄×D₆₁	1	2	yes	—
199	D₈	—	61	2	no	—
200	D₈	—	61	1	no	—
201	D₈	C₄×D₆₁	1	1	yes	characteristic
202	M₁₆	—	2	2	no	—
203	M₁₆	—	61	2	no	—
204	M₁₆	—	61	2	no	—
205	M₁₆	—	61	2	no	—
206	M₁₆	—	122	2	no	—
207	M₁₆	—	122	2	no	—
208	M₁₆	—	122	2	no	—
209	M₁₆	—	122	2	no	—
210	M₁₆	C₂²×D₆₁	1	2	yes	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	61	1	no	—
213	C₂×C₈	—	61	1	no	—
214	C₂×C₈	—	61	2	no	—
215	C₂×C₈	—	61	2	no	—
216	C₂×C₈	—	122	2	no	—
217	C₂×C₈	—	122	2	no	—
218	C₂×C₈	—	122	2	no	—
219	C₂×C₈	—	122	2	no	—
220	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
221	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
222	C₄⋊C₄	—	122	1	no	—
223	C₂²⋊C₄	—	122	1	no	—
224	C₄²	—	61	1	no	—
225	C₄²	C₂²×D₆₁	1	1	yes	characteristic
226	C₂³	—	61	1	no	—
227	C₂³	—	122	2	no	—
228	Q₈	—	61	2	no	—
229	Q₈	—	122	2	no	—
230	Q₈	C₂×C₄×D₆₁	1	2	yes	—
231	D₄	—	2	2	no	—
232	D₄	—	61	2	no	—
233	D₄	—	122	2	no	—
234	D₄	—	122	2	no	—
235	D₄	—	122	2	no	—
236	D₄	—	122	2	no	—
237	D₄	C₂×C₄×D₆₁	1	2	yes	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	61	1	no	—
241	C₂×C₄	—	61	2	no	—
242	C₂×C₄	—	61	2	no	—
243	C₂×C₄	—	122	2	no	—
244	C₂×C₄	—	122	2	no	—
245	C₂×C₄	—	122	2	no	—
246	C₂×C₄	—	122	2	no	—
247	C₂×C₄	—	122	1	no	—
248	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	61	2	no	—
251	C₈	—	61	1	no	—
252	C₈	—	61	1	no	—
253	C₈	—	122	2	no	—
254	C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
255	C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
256	C₈	D₄×D₆₁	1	2	yes	—
257	C₂²	—	2	2	no	—
258	C₂²	—	61	2	no	—
259	C₂²	—	122	2	no	—
260	C₂²	—	244	1	no	—
261	C₂²	—	244	4	no	—
262	C₂²	#41×D₆₁	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	61	2	no	—
266	C₄	—	122	2	no	—
267	C₄	—	244	1	no	—
268	C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
269	C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	4	2	no	—
272	C₂	—	122	2	no	—
273	C₂	—	244	2	no	—
274	C₂	C₄×D₄×D₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₆₁
commutator	C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#444	C₂
socle	C₂×C₆₁	C₂×C₄×D₄
hypercenter	#444	D₆₁
hypocenter	C₆₁	#2955
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #444

Complements

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

N	H ≅ G/N	type
2. #131×D₆₁	C₂	semidirect
3. #16839885	C₂	semidirect
4. #16839882	C₂	semidirect
5. #16839879	C₂	semidirect
6. #16839878	C₂	semidirect
7. #16839876	C₂	semidirect
8. #16839809	C₂	semidirect
9. #16839777	C₂	semidirect
10. #16839719	C₂	semidirect
11. C₆₁×#444	C₂	semidirect
13. #14519809	C₄	semidirect
14. #14519806	C₄	semidirect
15. #14519804	C₄	semidirect
16. #14519801	C₄	semidirect
20. #14519785	C₂²	semidirect
21. #14519784	C₂²	semidirect
23. #14519783	C₂²	semidirect
30. #14519713	C₂²	semidirect
31. #14519712	C₂²	semidirect
32. #14519705	C₂²	semidirect
33. #14519704	C₂²	semidirect
34. #14519703	C₂²	semidirect
35. #14519687	C₂²	semidirect
36. C₄×#92418	C₂²	semidirect
37. C₈×Dic₆₁	C₂²	semidirect
39. #131×C₆₁	C₂²	semidirect
40. #108×C₆₁	C₂²	semidirect
41. #104×C₆₁	C₂²	semidirect
42. #97×C₆₁	C₂²	semidirect
142. #444	D₆₁	semidirect
149. C₆₁	#2955	semidirect
163. #131	C₂×D₆₁	semidirect
172. #108	C₂×D₆₁	semidirect
176. #104	C₂×D₆₁	semidirect
178. #97	C₂×D₆₁	semidirect