Subgroups — #21454731

Order 7808.

452 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×D₆₁	C₂	1	1	yes	maximal, characteristic
2	#16840527	C₂	1	2	yes	maximal, complemented
3	#16839882	C₂	1	2	yes	maximal, complemented
4	#16839881	C₂	1	2	yes	maximal, complemented
5	#16839880	C₂	1	1	yes	maximal, characteristic, complemented
6	#16839879	C₂	1	1	yes	maximal, characteristic, complemented
7	#16839878	C₂	1	1	yes	maximal, characteristic, complemented
8	#108×D₆₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#16839777	C₂	1	2	yes	maximal, complemented
10	#16839716	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#443	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₆₁	C₂²	1	2	yes	—
13	#14519800	C₄	1	1	yes	characteristic, complemented
14	#14519799	C₂²	1	1	yes	characteristic
15	#14519798	C₄	1	2	yes	complemented
16	#14519797	C₂²	1	2	yes	—
17	#14519796	C₄	1	1	yes	characteristic, complemented
18	Q₁₆×D₆₁	C₄	1	1	yes	characteristic, complemented
19	SD₁₆×D₆₁	C₄	1	2	yes	complemented
20	D₈×D₆₁	C₄	1	1	yes	characteristic, complemented
21	#14519785	—	2	2	no	—
22	#14519785	C₂²	1	2	yes	complemented
23	#14519784	C₂²	1	2	yes	complemented
24	#14519783	—	2	2	no	—
25	#14519783	C₂²	1	2	yes	complemented
26	C₂×#11758189	—	2	2	no	—
27	C₂×#11758189	C₂²	1	1	yes	characteristic
28	M₁₆×D₆₁	—	2	2	no	—
29	M₁₆×D₆₁	C₂²	1	2	yes	—
30	C₂×C₈×D₆₁	C₂²	1	1	yes	characteristic
31	#14519734	C₂²	1	1	yes	characteristic
32	#14519713	C₂²	1	1	yes	characteristic, complemented
33	#14519712	C₂²	1	2	yes	complemented
34	#14519705	C₂²	1	2	yes	complemented
35	#14519704	C₂²	1	2	yes	complemented
36	#14519703	C₂²	1	2	yes	complemented
37	#14519688	C₂²	1	1	yes	characteristic, complemented
38	#14519687	C₂²	1	1	yes	characteristic, complemented
39	C₈×Dic₆₁	C₂²	1	1	yes	characteristic, complemented
40	#135×C₆₁	C₂²	1	1	yes	characteristic
41	#131×C₆₁	C₂²	1	2	yes	complemented
42	#108×C₆₁	C₂²	1	1	yes	characteristic, complemented
43	#104×C₆₁	C₂²	1	2	yes	complemented
44	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic, complemented
45	#41×C₆₁	C₂³	1	2	yes	—
46	#11758226	C₂×C₄	1	2	yes	—
47	Q₈×D₆₁	C₂×C₄	1	2	yes	—
48	#11758224	—	2	2	no	—
49	#11758224	C₂×C₄	1	2	yes	—
50	D₄×D₆₁	—	2	2	no	—
51	D₄×D₆₁	C₂×C₄	1	2	yes	—
52	#11758222	—	2	2	no	—
53	#11758222	C₂³	1	2	yes	—
54	C₂×C₄×D₆₁	—	2	2	no	—
55	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
56	Q₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
57	SD₁₆×C₆₁	C₂×C₄	1	2	yes	—
58	D₈×C₆₁	C₂×C₄	1	1	yes	characteristic
59	M₁₆×C₆₁	—	2	2	no	—
60	M₁₆×C₆₁	C₂³	1	2	yes	—
61	C₂×C₈×C₆₁	—	2	2	no	—
62	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
64	C₄²×C₆₁	C₂³	1	1	yes	characteristic
65	#11758202	C₂×C₄	1	2	yes	—
66	#11758201	C₂×C₄	1	2	yes	—
67	#11758200	C₂×C₄	1	2	yes	—
68	#11758199	C₂×C₄	1	2	yes	—
69	#11758198	—	2	1	no	—
70	#11758196	—	2	1	no	—
71	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
72	#11758194	—	2	2	no	—
73	#11758194	C₂³	1	2	yes	—
74	C₂×#92418	—	2	2	no	—
75	C₂×#92418	C₂³	1	1	yes	characteristic
76	C₂×#92418	C₂³	1	1	yes	characteristic
77	Dic₂₄₄	C₂×C₄	1	1	yes	characteristic
78	D₄₈₈	C₂×C₄	1	1	yes	characteristic
79	#11758190	C₂×C₄	1	2	yes	—
80	#11758189	—	2	2	no	—
81	#11758189	—	2	2	no	—
82	#11758189	—	2	2	no	—
83	#11758189	D₄	1	1	yes	characteristic
84	#11758189	D₄	1	1	yes	characteristic
85	#11758189	D₄	1	1	yes	characteristic
86	#11758189	D₄	1	1	yes	characteristic
87	C₈×D₆₁	—	2	2	no	—
88	C₈×D₆₁	—	2	2	no	—
89	C₈×D₆₁	—	2	2	no	—
90	C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
91	C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
92	C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
93	C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
94	C₂²×D₆₁	—	2	2	no	—
95	C₂²×D₆₁	#41	1	1	yes	characteristic
96	Q₈×C₆₁	C₂²×C₄	1	2	yes	—
97	D₄×C₆₁	—	2	2	no	—
98	D₄×C₆₁	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₆₁	—	2	2	no	—
100	C₂×C₄×C₆₁	—	2	1	no	—
101	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
102	C₂²⋊D₆₁	—	2	2	no	—
103	C₂²⋊D₆₁	—	2	2	no	—
104	C₂²⋊D₆₁	—	2	2	no	—
105	C₂×Dic₆₁	—	2	2	no	—
106	C₂×Dic₆₁	—	2	1	no	—
107	C₂×Dic₆₁	#41	1	1	yes	characteristic
108	D₂₄₄	—	2	2	no	—
109	D₂₄₄	C₂²×C₄	1	2	yes	—
110	C₄×D₆₁	—	2	2	no	—
111	C₄×D₆₁	—	2	2	no	—
112	C₄×D₆₁	—	2	2	no	—
113	C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
114	C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
115	C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
116	C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
117	Dic₁₂₂	—	2	2	no	—
118	Dic₁₂₂	C₂²×C₄	1	2	yes	—
119	C₈×C₆₁	—	2	2	no	—
120	C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
121	C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
122	C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
123	C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
124	#92418	—	2	2	no	—
125	#92418	C₂²×C₄	1	1	yes	characteristic
126	#92418	C₂²×C₄	1	1	yes	characteristic
127	#92418	C₂×D₄	1	1	yes	characteristic
128	#92418	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₆₁	—	2	2	no	—
130	C₂²×C₆₁	C₂×#41	1	1	yes	characteristic
131	C₂×D₆₁	—	2	2	no	—
132	C₂×D₆₁	—	4	1	no	—
133	C₂×D₆₁	—	8	2	no	—
134	C₂×D₆₁	C₄×D₄	1	1	yes	characteristic
135	C₂×D₆₁	C₄×D₄	1	1	yes	characteristic
136	C₄×C₆₁	—	2	2	no	—
137	C₄×C₆₁	—	4	1	no	—
138	C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
139	C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
140	Dic₆₁	—	2	2	no	—
141	Dic₆₁	—	4	1	no	—
142	Dic₆₁	C₄×D₄	1	1	yes	characteristic
143	Dic₆₁	C₄×D₄	1	1	yes	characteristic
144	#2957	—	61	1	no	maximal, Sylow
145	C₂×C₆₁	—	2	1	no	—
146	C₂×C₆₁	—	4	2	no	—
147	C₂×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
148	D₆₁	—	2	1	no	—
149	D₆₁	—	2	1	no	—
150	D₆₁	—	4	2	no	—
151	C₂×#135	—	61	1	no	—
152	#568	—	61	2	no	—
153	#444	—	61	2	no	—
154	#444	—	61	2	no	—
155	#443	—	61	1	no	—
156	#443	—	61	1	no	—
157	#443	—	61	1	no	—
158	#443	D₆₁	1	1	yes	characteristic, complemented
159	C₂×#108	—	61	1	no	—
160	#421	—	61	2	no	—
161	#405	—	61	1	no	—
162	C₆₁	#2957	1	1	yes	characteristic, complemented, Sylow
163	C₂×#41	—	61	2	no	—
164	#135	—	61	2	no	—
165	#135	—	61	1	no	—
166	#135	—	61	2	no	—
167	#135	—	61	1	no	—
168	#135	—	61	1	no	—
169	#135	C₂×D₆₁	1	1	yes	characteristic
170	C₂×Q₁₆	—	61	1	no	—
171	C₂×SD₁₆	—	61	2	no	—
172	C₂×D₈	—	61	1	no	—
173	#131	—	61	2	no	—
174	#131	—	61	2	no	—
175	#131	—	61	2	no	—
176	#131	—	122	2	no	—
177	#131	—	122	2	no	—
178	#131	C₂×D₆₁	1	2	yes	complemented
179	C₂×M₁₆	—	61	1	no	—
180	C₂×M₁₆	—	61	2	no	—
181	C₂×M₁₆	—	122	2	no	—
182	C₂×M₁₆	—	122	2	no	—
183	C₂²×C₈	—	61	1	no	—
184	#117	—	61	1	no	—
185	#108	—	61	2	no	—
186	#108	—	61	1	no	—
187	#108	C₂×D₆₁	1	1	yes	characteristic, complemented
188	#104	—	61	2	no	—
189	#104	—	61	2	no	—
190	#104	—	61	2	no	—
191	#104	C₂×D₆₁	1	2	yes	complemented
192	#97	—	61	1	no	—
193	#97	—	61	1	no	—
194	C₄×C₈	—	61	1	no	—
195	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic, complemented
196	#41	—	61	2	no	—
197	#41	—	61	2	no	—
198	#41	—	61	2	no	—
199	#41	—	122	2	no	—
200	#41	—	122	2	no	—
201	#41	C₂²×D₆₁	1	2	yes	—
202	C₂×Q₈	—	61	2	no	—
203	C₂×D₄	—	61	2	no	—
204	C₂×D₄	—	122	2	no	—
205	C₂²×C₄	—	61	1	no	—
206	C₂²×C₄	—	122	2	no	—
207	Q₁₆	—	61	2	no	—
208	Q₁₆	—	61	1	no	—
209	Q₁₆	C₄×D₆₁	1	1	yes	characteristic
210	SD₁₆	—	61	2	no	—
211	SD₁₆	—	61	2	no	—
212	SD₁₆	—	61	2	no	—
213	SD₁₆	C₄×D₆₁	1	2	yes	—
214	D₈	—	61	2	no	—
215	D₈	—	61	1	no	—
216	D₈	C₄×D₆₁	1	1	yes	characteristic
217	M₁₆	—	2	2	no	—
218	M₁₆	—	61	1	no	—
219	M₁₆	—	61	1	no	—
220	M₁₆	—	61	1	no	—
221	M₁₆	—	61	1	no	—
222	M₁₆	—	61	2	no	—
223	M₁₆	—	122	2	no	—
224	M₁₆	—	122	2	no	—
225	M₁₆	—	122	2	no	—
226	M₁₆	—	122	2	no	—
227	M₁₆	C₂²×D₆₁	1	2	yes	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	61	1	no	—
230	C₂×C₈	—	61	1	no	—
231	C₂×C₈	—	61	1	no	—
232	C₂×C₈	—	61	1	no	—
233	C₂×C₈	—	61	1	no	—
234	C₂×C₈	—	61	1	no	—
235	C₂×C₈	—	122	2	no	—
236	C₂×C₈	—	122	2	no	—
237	C₂×C₈	—	122	2	no	—
238	C₂×C₈	—	122	2	no	—
239	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
241	C₄⋊C₄	—	122	1	no	—
242	C₂²⋊C₄	—	122	1	no	—
243	C₄²	—	61	1	no	—
244	C₄²	C₂²×D₆₁	1	1	yes	characteristic
245	C₂³	—	61	1	no	—
246	C₂³	—	122	2	no	—
247	Q₈	—	61	2	no	—
248	Q₈	—	122	2	no	—
249	Q₈	C₂×C₄×D₆₁	1	2	yes	—
250	D₄	—	2	2	no	—
251	D₄	—	61	2	no	—
252	D₄	—	122	2	no	—
253	D₄	—	122	2	no	—
254	D₄	—	122	2	no	—
255	D₄	—	122	2	no	—
256	D₄	C₂×C₄×D₆₁	1	2	yes	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	61	1	no	—
260	C₂×C₄	—	61	1	no	—
261	C₂×C₄	—	61	1	no	—
262	C₂×C₄	—	61	1	no	—
263	C₂×C₄	—	61	1	no	—
264	C₂×C₄	—	122	2	no	—
265	C₂×C₄	—	122	2	no	—
266	C₂×C₄	—	122	2	no	—
267	C₂×C₄	—	122	2	no	—
268	C₂×C₄	—	122	1	no	—
269	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	61	1	no	—
272	C₈	—	61	1	no	—
273	C₈	—	61	1	no	—
274	C₈	—	61	1	no	—
275	C₈	—	122	2	no	—
276	C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
277	C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
278	C₈	D₄×D₆₁	1	1	yes	characteristic
279	C₈	D₄×D₆₁	1	1	yes	characteristic
280	C₂²	—	2	2	no	—
281	C₂²	—	61	1	no	—
282	C₂²	—	61	1	no	—
283	C₂²	—	122	2	no	—
284	C₂²	—	488	2	no	—
285	C₂²	—	244	1	no	—
286	C₂²	#41×D₆₁	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	4	1	no	—
289	C₄	—	61	1	no	—
290	C₄	—	61	1	no	—
291	C₄	—	122	2	no	—
292	C₄	—	244	1	no	—
293	C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
294	C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
295	C₂	—	2	1	no	—
296	C₂	—	4	2	no	—
297	C₂	—	122	1	no	—
298	C₂	—	122	1	no	—
299	C₂	—	244	2	no	—
300	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×#443	C₂
socle	C₂×C₆₁	C₂×C₄×D₄
hypercenter	#443	D₆₁
hypocenter	C₆₁	#2957
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
2. #16840527	C₂	semidirect
3. #16839882	C₂	semidirect
4. #16839881	C₂	semidirect
5. #16839880	C₂	semidirect
6. #16839879	C₂	semidirect
7. #16839878	C₂	semidirect
8. #108×D₆₁	C₂	semidirect
9. #16839777	C₂	semidirect
10. #16839716	C₂	semidirect
11. C₆₁×#443	C₂	semidirect
13. #14519800	C₄	semidirect
15. #14519798	C₄	semidirect
17. #14519796	C₄	semidirect
18. Q₁₆×D₆₁	C₄	semidirect
19. SD₁₆×D₆₁	C₄	semidirect
20. D₈×D₆₁	C₄	semidirect
22. #14519785	C₂²	semidirect
23. #14519784	C₂²	semidirect
25. #14519783	C₂²	semidirect
32. #14519713	C₂²	semidirect
33. #14519712	C₂²	semidirect
34. #14519705	C₂²	semidirect
35. #14519704	C₂²	semidirect
36. #14519703	C₂²	semidirect
37. #14519688	C₂²	semidirect
38. #14519687	C₂²	semidirect
39. C₈×Dic₆₁	C₂²	semidirect
41. #131×C₆₁	C₂²	semidirect
42. #108×C₆₁	C₂²	semidirect
43. #104×C₆₁	C₂²	semidirect
44. C₄×C₈×C₆₁	C₂²	semidirect
158. #443	D₆₁	semidirect
162. C₆₁	#2957	semidirect
178. #131	C₂×D₆₁	semidirect
187. #108	C₂×D₆₁	semidirect
191. #104	C₂×D₆₁	semidirect
195. C₄×C₈	C₂×D₆₁	semidirect