Subgroups — #21454720

Order 7808.

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	#16840560	C₂	1	1	yes	maximal, characteristic
2	C₂×#14519783	C₂	1	1	yes	maximal, characteristic, complemented
3	#16839882	C₂	1	4	yes	maximal, complemented
4	#16839877	C₂	1	4	yes	maximal, complemented
5	#16839810	C₂	1	1	yes	maximal, characteristic, complemented
6	#16839779	C₂	1	1	yes	maximal, characteristic, complemented
7	#16839778	C₂	1	1	yes	maximal, characteristic, complemented
8	#16839717	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#131×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11758222	C₂²	1	1	yes	characteristic
11	#14519808	C₄	1	2	yes	complemented
12	#14519807	C₄	1	2	yes	complemented
13	#14519803	C₄	1	2	yes	complemented
14	#14519802	C₄	1	2	yes	complemented
15	#14519797	C₂²	1	4	yes	—
16	#14519783	—	2	1	no	—
17	#14519783	—	2	1	no	—
18	#14519783	C₂²	1	2	yes	complemented
19	#14519783	C₂²	1	2	yes	complemented
20	C₂×#11758194	C₂²	1	1	yes	characteristic
21	C₂×#11758194	C₂²	1	1	yes	characteristic
22	C₂×#11758189	—	2	1	no	—
23	C₂×C₈×D₆₁	—	2	1	no	—
24	#14519736	C₂²	1	1	yes	characteristic
25	#14519712	C₂²	1	4	yes	complemented
26	#14519704	C₂²	1	4	yes	complemented
27	#14519703	C₂²	1	4	yes	complemented
28	#14519687	C₂²	1	2	yes	complemented
29	C₈×Dic₆₁	C₂²	1	2	yes	complemented
30	C₂×#41×C₆₁	C₂²	1	1	yes	characteristic
31	#131×C₆₁	—	2	1	no	—
32	#131×C₆₁	—	2	1	no	—
33	#131×C₆₁	C₂²	1	2	yes	complemented
34	#131×C₆₁	C₂²	1	2	yes	complemented
35	C₂×M₁₆×C₆₁	—	2	1	no	—
36	C₂×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₆₁	—	2	1	no	—
38	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
39	#41×C₆₁	—	2	1	no	—
40	#41×C₆₁	—	2	1	no	—
41	#41×C₆₁	C₂×C₄	1	2	yes	—
42	#41×C₆₁	C₂³	1	2	yes	—
43	C₂×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
44	C₂×D₄×C₆₁	—	2	1	no	—
45	C₂×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₁	—	2	1	no	—
47	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊D₆₁	—	2	1	no	—
49	#11758222	—	2	1	no	—
50	#11758222	—	2	1	no	—
51	#11758222	C₂×C₄	1	2	yes	—
52	#11758222	C₂³	1	2	yes	—
53	C₂×D₂₄₄	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₆₁	—	2	1	no	—
55	C₂×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
56	M₁₆×C₆₁	—	2	1	no	—
57	M₁₆×C₆₁	—	2	2	no	—
58	M₁₆×C₆₁	—	2	2	no	—
59	M₁₆×C₆₁	C₂³	1	2	yes	—
60	C₂×C₈×C₆₁	—	2	1	no	—
61	C₂×C₈×C₆₁	—	2	2	no	—
62	C₂×C₈×C₆₁	—	2	2	no	—
63	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
64	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
65	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
66	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
67	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
69	C₂²⋊Dic₆₁	—	2	1	no	—
70	#11758202	C₂×C₄	1	4	yes	—
71	#11758201	C₂×C₄	1	4	yes	—
72	#11758200	C₂×C₄	1	4	yes	—
73	#11758199	C₂×C₄	1	4	yes	—
74	C₄⋊Dic₆₁	—	2	1	no	—
75	C₄×Dic₆₁	C₂³	1	2	yes	—
76	#11758194	—	2	1	no	—
77	#11758194	C₂×C₄	1	2	yes	—
78	#11758194	C₂×C₄	1	2	yes	—
79	#11758194	C₂³	1	2	yes	—
80	C₂×#92418	—	2	1	no	—
81	C₂×#92418	C₂³	1	2	yes	—
82	#11758189	—	2	2	no	—
83	#11758189	—	2	2	no	—
84	C₈×D₆₁	—	2	2	no	—
85	C₈×D₆₁	—	2	2	no	—
86	C₂³×C₆₁	—	2	1	no	—
87	C₂³×C₆₁	#41	1	1	yes	characteristic
88	C₂²×D₆₁	—	2	1	no	—
89	Q₈×C₆₁	—	2	1	no	—
90	Q₈×C₆₁	C₂²×C₄	1	2	yes	—
91	D₄×C₆₁	—	2	2	no	—
92	D₄×C₆₁	—	2	2	no	—
93	D₄×C₆₁	—	2	1	no	—
94	D₄×C₆₁	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₆₁	—	2	1	no	—
96	C₂×C₄×C₆₁	—	2	2	no	—
97	C₂×C₄×C₆₁	—	2	2	no	—
98	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
103	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
104	C₂²⋊D₆₁	—	2	2	no	—
105	C₂²⋊D₆₁	—	2	2	no	—
106	C₂×Dic₆₁	—	2	1	no	—
107	C₂×Dic₆₁	—	2	2	no	—
108	D₂₄₄	—	2	1	no	—
109	D₂₄₄	C₂²×C₄	1	2	yes	—
110	C₄×D₆₁	—	2	2	no	—
111	C₄×D₆₁	—	2	2	no	—
112	Dic₁₂₂	—	2	1	no	—
113	Dic₁₂₂	C₂²×C₄	1	2	yes	—
114	C₈×C₆₁	—	2	2	no	—
115	C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
116	C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
117	C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
118	C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
119	#92418	—	2	2	no	—
120	#92418	C₂²×C₄	1	4	yes	—
121	C₂²×C₆₁	—	2	2	no	—
122	C₂²×C₆₁	—	4	1	no	—
123	C₂²×C₆₁	—	4	2	no	—
124	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
125	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
126	C₂²×C₆₁	C₂×#41	1	1	yes	characteristic
127	C₂×D₆₁	—	2	2	no	—
128	C₂×D₆₁	—	4	2	no	—
129	C₄×C₆₁	—	2	2	no	—
130	C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
131	C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
132	C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
133	C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
134	Dic₆₁	—	2	2	no	—
135	Dic₆₁	—	4	2	no	—
136	#2955	—	61	1	no	maximal, Sylow
137	C₂×C₆₁	—	2	1	no	—
138	C₂×C₆₁	—	2	1	no	—
139	C₂×C₆₁	—	2	1	no	—
140	C₂×C₆₁	—	4	2	no	—
141	C₂×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
142	D₆₁	—	4	2	no	—
143	#575	—	61	1	no	—
144	C₂×#131	—	61	1	no	—
145	C₂×#131	D₆₁	1	1	yes	characteristic, complemented
146	#444	—	61	4	no	—
147	#443	—	61	4	no	—
148	#430	—	61	1	no	—
149	#421	—	61	1	no	—
150	#421	—	61	1	no	—
151	#405	—	61	1	no	—
152	C₆₁	#2955	1	1	yes	characteristic, complemented, Sylow
153	C₂×#41	—	61	1	no	—
154	C₂×#41	C₂×D₆₁	1	1	yes	characteristic
155	#137	—	61	2	no	—
156	#137	—	61	2	no	—
157	AGL(1,ℤ/8)	—	61	2	no	—
158	AGL(1,ℤ/8)	—	61	2	no	—
159	#135	—	61	4	no	—
160	#131	—	2	1	no	—
161	#131	—	2	1	no	—
162	#131	—	61	2	no	—
163	#131	—	61	2	no	—
164	#131	—	122	1	no	—
165	#131	—	122	1	no	—
166	#131	C₂×D₆₁	1	2	yes	complemented
167	#131	C₂×D₆₁	1	2	yes	complemented
168	C₂×M₁₆	—	2	1	no	—
169	C₂×M₁₆	—	61	1	no	—
170	C₂×M₁₆	—	61	1	no	—
171	C₂×M₁₆	—	122	1	no	—
172	C₂×M₁₆	C₂×D₆₁	1	1	yes	characteristic
173	C₂²×C₈	—	2	1	no	—
174	C₂²×C₈	—	122	1	no	—
175	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
176	#117	—	61	1	no	—
177	#108	—	61	4	no	—
178	#104	—	61	4	no	—
179	#104	—	61	4	no	—
180	#97	—	61	2	no	—
181	C₄×C₈	—	61	2	no	—
182	#41	—	2	1	no	—
183	#41	—	2	1	no	—
184	#41	—	61	2	no	—
185	#41	—	61	2	no	—
186	#41	—	122	1	no	—
187	#41	—	122	1	no	—
188	#41	C₄×D₆₁	1	2	yes	—
189	#41	C₂²×D₆₁	1	2	yes	—
190	C₂×Q₈	—	61	1	no	—
191	C₂×Q₈	C₄×D₆₁	1	1	yes	characteristic
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	—	61	1	no	—
194	C₂×D₄	—	122	1	no	—
195	C₂×D₄	C₄×D₆₁	1	1	yes	characteristic
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	122	1	no	—
198	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
199	Q₁₆	—	61	4	no	—
200	SD₁₆	—	61	4	no	—
201	SD₁₆	—	61	4	no	—
202	D₈	—	61	4	no	—
203	M₁₆	—	2	1	no	—
204	M₁₆	—	2	2	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	61	2	no	—
207	M₁₆	—	61	2	no	—
208	M₁₆	—	61	2	no	—
209	M₁₆	—	122	1	no	—
210	M₁₆	—	122	2	no	—
211	M₁₆	—	122	2	no	—
212	M₁₆	C₂²×D₆₁	1	2	yes	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	61	2	no	—
217	C₂×C₈	—	122	1	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	C₂×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
223	C₂×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
224	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
225	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
226	C₄⋊C₄	—	122	1	no	—
227	C₂²⋊C₄	—	122	1	no	—
228	C₄²	—	61	2	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	122	1	no	—
231	C₂³	#11758222	1	1	yes	characteristic
232	Q₈	—	2	1	no	—
233	Q₈	—	61	2	no	—
234	Q₈	—	122	1	no	—
235	Q₈	C₂×C₄×D₆₁	1	2	yes	—
236	D₄	—	2	2	no	—
237	D₄	—	2	2	no	—
238	D₄	—	2	1	no	—
239	D₄	—	61	2	no	—
240	D₄	—	122	2	no	—
241	D₄	—	122	2	no	—
242	D₄	—	122	1	no	—
243	D₄	C₂×C₄×D₆₁	1	2	yes	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	122	1	no	—
248	C₂×C₄	—	122	2	no	—
249	C₂×C₄	—	122	2	no	—
250	C₂×C₄	—	122	2	no	—
251	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
252	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
253	C₂×C₄	#11758222	1	1	yes	characteristic
254	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
255	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
256	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	61	4	no	—
259	C₈	—	122	2	no	—
260	C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
261	C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
262	C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
263	C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
264	C₂²	—	2	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	122	2	no	—
268	C₂²	—	244	2	no	—
269	C₂²	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
270	C₂²	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
271	C₂²	C₂×#11758222	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	122	2	no	—
274	C₄	—	244	2	no	—
275	C₄	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
276	C₄	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
277	C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
278	C₄	C₂²×C₂²⋊D₆₁	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	—	244	2	no	—
284	C₂	C₂×C₄×C₂²⋊D₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×C₂²⋊D₆₁
commutator	C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₂×#131×C₆₁	C₂
socle	C₂×C₆₁	C₂×C₄×D₄
hypercenter	C₂×#131	D₆₁
hypocenter	C₆₁	#2955
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. C₂×#14519783	C₂	semidirect
3. #16839882	C₂	semidirect
4. #16839877	C₂	semidirect
5. #16839810	C₂	semidirect
6. #16839779	C₂	semidirect
7. #16839778	C₂	semidirect
8. #16839717	C₂	semidirect
9. C₂×#131×C₆₁	C₂	semidirect
11. #14519808	C₄	semidirect
12. #14519807	C₄	semidirect
13. #14519803	C₄	semidirect
14. #14519802	C₄	semidirect
18. #14519783	C₂²	semidirect
19. #14519783	C₂²	semidirect
25. #14519712	C₂²	semidirect
26. #14519704	C₂²	semidirect
27. #14519703	C₂²	semidirect
28. #14519687	C₂²	semidirect
29. C₈×Dic₆₁	C₂²	semidirect
33. #131×C₆₁	C₂²	semidirect
34. #131×C₆₁	C₂²	semidirect
145. C₂×#131	D₆₁	semidirect
152. C₆₁	#2955	semidirect
166. #131	C₂×D₆₁	semidirect
167. #131	C₂×D₆₁	semidirect