Subgroups — #21454727

Order 7808.

458 conjugacy classes of subgroups.

The subgroup lattice has 278 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	#16840527	C₂	1	2	yes	maximal, complemented
3	#16839885	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839881	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	#16839877	C₂	1	2	yes	maximal, complemented
8	#16839809	C₂	1	1	yes	maximal, characteristic, complemented
9	#104×D₆₁	C₂	1	2	yes	maximal, complemented
10	#16839714	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	—	2	2	no	—
25	C₂×#11758189	C₂²	1	1	yes	characteristic
26	C₂×#11758189	C₂²	1	1	yes	characteristic
27	M₁₆×D₆₁	—	2	2	no	—
28	M₁₆×D₆₁	C₂²	1	2	yes	—
29	C₄²×D₆₁	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	#14519688	C₂²	1	1	yes	characteristic, complemented
36	C₈×Dic₆₁	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	—	2	1	no	—
54	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
55	Q₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
56	SD₁₆×C₆₁	C₂×C₄	1	2	yes	complemented
57	D₈×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
58	M₁₆×C₆₁	—	2	2	no	—
59	M₁₆×C₆₁	C₂³	1	2	yes	—
60	C₂×C₈×C₆₁	—	2	2	no	—
61	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
63	C₄²×C₆₁	C₂³	1	1	yes	characteristic
64	#11758202	C₂×C₄	1	2	yes	complemented
65	#11758201	C₂×C₄	1	2	yes	complemented
66	#11758200	C₂×C₄	1	2	yes	complemented
67	#11758199	C₂×C₄	1	2	yes	complemented
68	C₄×Dic₆₁	—	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, complemented
76	D₄₈₈	C₂×C₄	1	1	yes	characteristic, complemented
77	#11758190	C₂×C₄	1	2	yes	complemented
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	#11758189	D₄	1	2	yes	—
84	#11758189	D₄	1	2	yes	—
85	C₈×D₆₁	—	2	2	no	—
86	C₈×D₆₁	—	2	2	no	—
87	C₈×D₆₁	—	2	2	no	—
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	1	no	—
100	C₂×Dic₆₁	—	2	2	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	—	4	2	no	—
108	C₄×D₆₁	C₂²×C₄	1	2	yes	—
109	C₄×D₆₁	C₂×D₄	1	2	yes	—
110	Dic₁₂₂	—	2	2	no	—
111	Dic₁₂₂	C₂²×C₄	1	2	yes	—
112	C₈×C₆₁	—	2	2	no	—
113	C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
114	C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
115	C₈×C₆₁	C₂×D₄	1	2	yes	—
116	#92418	—	2	2	no	—
117	#92418	C₂²×C₄	1	2	yes	—
118	#92418	C₂×D₄	1	1	yes	characteristic
119	#92418	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₆₁	—	2	2	no	—
121	C₂²×C₆₁	C₂×#41	1	1	yes	characteristic
122	C₂×D₆₁	—	2	2	no	—
123	C₂×D₆₁	—	4	1	no	—
124	C₂×D₆₁	—	8	2	no	—
125	C₂×D₆₁	C₄×D₄	1	2	yes	—
126	C₄×C₆₁	—	2	2	no	—
127	C₄×C₆₁	—	4	1	no	—
128	C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
129	C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
130	Dic₆₁	—	2	2	no	—
131	Dic₆₁	—	2	2	no	—
132	Dic₆₁	C₄×D₄	1	2	yes	—
133	#2956	—	61	1	no	maximal, Sylow
134	C₂×C₆₁	—	2	1	no	—
135	C₂×C₆₁	—	4	2	no	—
136	C₂×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
137	D₆₁	—	2	2	no	—
138	D₆₁	—	4	2	no	—
139	#575	—	61	1	no	—
140	#568	—	61	2	no	—
141	#444	—	61	2	no	—
142	#444	—	61	1	no	—
143	#444	D₆₁	1	1	yes	characteristic, complemented
144	#443	—	61	1	no	—
145	#443	—	61	2	no	—
146	#443	—	61	1	no	—
147	#430	—	61	1	no	—
148	C₂×#104	—	61	2	no	—
149	C₄×M₁₆	—	61	1	no	—
150	C₆₁	#2956	1	1	yes	characteristic, complemented, Sylow
151	C₂×#41	—	61	2	no	—
152	#137	—	61	2	no	—
153	#137	—	61	2	no	—
154	AGL(1,ℤ/8)	—	61	2	no	—
155	AGL(1,ℤ/8)	—	61	2	no	—
156	#135	—	61	2	no	—
157	#135	—	61	1	no	—
158	#135	C₂×D₆₁	1	1	yes	characteristic
159	#131	—	61	2	no	—
160	#131	—	61	2	no	—
161	#131	—	61	2	no	—
162	#131	—	122	2	no	—
163	#131	—	122	2	no	—
164	#131	C₂×D₆₁	1	2	yes	complemented
165	C₂×M₁₆	—	61	1	no	—
166	C₂×M₁₆	—	61	1	no	—
167	C₂×M₁₆	—	61	2	no	—
168	C₂×M₁₆	—	122	2	no	—
169	C₂×M₁₆	—	122	2	no	—
170	C₂×C₄²	—	61	1	no	—
171	#108	—	61	2	no	—
172	#108	—	61	1	no	—
173	#108	C₂×D₆₁	1	1	yes	characteristic, complemented
174	#104	—	61	2	no	—
175	#104	—	61	2	no	—
176	#104	—	61	2	no	—
177	#104	C₂×D₆₁	1	2	yes	complemented
178	#97	—	61	1	no	—
179	#97	C₂×D₆₁	1	1	yes	characteristic, complemented
180	C₄×C₈	—	61	1	no	—
181	C₄×C₈	—	61	1	no	—
182	#41	—	61	2	no	—
183	#41	—	61	2	no	—
184	#41	—	61	2	no	—
185	#41	—	122	2	no	—
186	#41	—	122	2	no	—
187	#41	C₂²×D₆₁	1	2	yes	—
188	C₂×Q₈	—	61	2	no	—
189	C₂×D₄	—	61	2	no	—
190	C₂×D₄	—	122	2	no	—
191	C₂²×C₄	—	61	1	no	—
192	C₂²×C₄	—	122	2	no	—
193	C₂²×C₄	—	122	1	no	—
194	Q₁₆	—	61	2	no	—
195	Q₁₆	—	61	1	no	—
196	Q₁₆	C₄×D₆₁	1	1	yes	characteristic, complemented
197	SD₁₆	—	61	2	no	—
198	SD₁₆	—	61	2	no	—
199	SD₁₆	—	61	2	no	—
200	SD₁₆	C₄×D₆₁	1	2	yes	complemented
201	D₈	—	61	2	no	—
202	D₈	—	61	1	no	—
203	D₈	C₄×D₆₁	1	1	yes	characteristic, complemented
204	M₁₆	—	2	2	no	—
205	M₁₆	—	61	2	no	—
206	M₁₆	—	61	2	no	—
207	M₁₆	—	61	2	no	—
208	M₁₆	—	61	2	no	—
209	M₁₆	—	61	2	no	—
210	M₁₆	—	122	2	no	—
211	M₁₆	—	122	2	no	—
212	M₁₆	—	122	2	no	—
213	M₁₆	—	122	2	no	—
214	M₁₆	C₂²×D₆₁	1	2	yes	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	61	1	no	—
217	C₂×C₈	—	61	1	no	—
218	C₂×C₈	—	122	2	no	—
219	C₂×C₈	—	122	2	no	—
220	C₂×C₈	—	122	2	no	—
221	C₂×C₈	—	122	2	no	—
222	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
223	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
224	C₄²	—	61	1	no	—
225	C₄²	—	122	1	no	—
226	C₄²	C₂²×D₆₁	1	1	yes	characteristic
227	C₂³	—	61	1	no	—
228	C₂³	—	122	2	no	—
229	Q₈	—	61	2	no	—
230	Q₈	—	122	2	no	—
231	Q₈	C₂×C₄×D₆₁	1	2	yes	—
232	D₄	—	2	2	no	—
233	D₄	—	61	2	no	—
234	D₄	—	122	2	no	—
235	D₄	—	122	2	no	—
236	D₄	—	122	2	no	—
237	D₄	—	122	2	no	—
238	D₄	C₂×C₄×D₆₁	1	2	yes	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	61	1	no	—
242	C₂×C₄	—	61	2	no	—
243	C₂×C₄	—	61	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	—	122	2	no	—
249	C₂×C₄	—	244	2	no	—
250	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	61	2	no	—
253	C₈	—	61	1	no	—
254	C₈	—	61	1	no	—
255	C₈	—	122	2	no	—
256	C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
257	C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
258	C₈	D₄×D₆₁	1	2	yes	—
259	C₂²	—	2	2	no	—
260	C₂²	—	61	2	no	—
261	C₂²	—	122	2	no	—
262	C₂²	—	488	2	no	—
263	C₂²	—	244	1	no	—
264	C₂²	#41×D₆₁	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	4	1	no	—
267	C₄	—	61	2	no	—
268	C₄	—	122	2	no	—
269	C₄	—	122	2	no	—
270	C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
271	C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	4	2	no	—
274	C₂	—	122	2	no	—
275	C₂	—	244	2	no	—
276	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	#2956
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. #16840527	C₂	semidirect
3. #16839885	C₂	semidirect
4. #16839881	C₂	semidirect
5. #16839879	C₂	semidirect
6. #16839878	C₂	semidirect
7. #16839877	C₂	semidirect
8. #16839809	C₂	semidirect
9. #104×D₆₁	C₂	semidirect
10. #16839714	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. #14519688	C₂²	semidirect
36. C₈×Dic₆₁	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
55. Q₁₆×C₆₁	C₂×C₄	semidirect
56. SD₁₆×C₆₁	C₂×C₄	semidirect
57. D₈×C₆₁	C₂×C₄	semidirect
64. #11758202	C₂×C₄	semidirect
65. #11758201	C₂×C₄	semidirect
66. #11758200	C₂×C₄	semidirect
67. #11758199	C₂×C₄	semidirect
75. Dic₂₄₄	C₂×C₄	semidirect
76. D₄₈₈	C₂×C₄	semidirect
77. #11758190	C₂×C₄	semidirect
143. #444	D₆₁	semidirect
150. C₆₁	#2956	semidirect
164. #131	C₂×D₆₁	semidirect
173. #108	C₂×D₆₁	semidirect
177. #104	C₂×D₆₁	semidirect
179. #97	C₂×D₆₁	semidirect
196. Q₁₆	C₄×D₆₁	semidirect
200. SD₁₆	C₄×D₆₁	semidirect
203. D₈	C₄×D₆₁	semidirect