Subgroups — D₆₁×#2158 #29833038

Order 15616.

694 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₆₁	C₂	1	1	yes	maximal, characteristic
2	#427×D₆₁	C₂	1	2	yes	maximal
3	#21447603	C₂	1	1	yes	maximal, characteristic, complemented
4	#21447602	C₂	1	2	yes	maximal, complemented
5	#21447601	C₂	1	2	yes	maximal, complemented
6	#21447600	C₂	1	2	yes	maximal, complemented
7	#367×D₆₁	C₂	1	2	yes	maximal, complemented
8	#366×D₆₁	C₂	1	2	yes	maximal, complemented
9	C₆₁×#2158	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₈×D₆₁	C₂²	1	1	yes	characteristic
11	#117×D₆₁	—	2	2	no	—
12	C₂²×#11758185	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×D₆₁	C₄	1	8	yes	complemented
14	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
18	#16839800	C₂²	1	2	yes	—
19	#16839799	C₂²	1	4	yes	—
20	#16839798	—	2	2	no	—
21	#16839797	—	2	2	no	—
22	#107×D₆₁	C₂²	1	4	yes	—
23	#106×D₆₁	—	2	2	no	—
24	#16839590	C₂²	1	2	yes	complemented
25	#16839589	C₂²	1	4	yes	complemented
26	#16839588	C₂²	1	2	yes	complemented
27	#16839587	C₂²	1	4	yes	complemented
28	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
29	C₆₁×#427	C₂²	1	2	yes	—
30	C₆₁×#367	C₂²	1	2	yes	complemented
31	C₆₁×#366	C₂²	1	2	yes	complemented
32	C₂²×C₄×D₆₁	D₄	1	1	yes	characteristic
33	C₂²×#92418	D₄	1	1	yes	characteristic
34	C₂×C₈×D₆₁	C₂×C₄	1	4	yes	—
35	C₂×C₈×D₆₁	D₄	1	1	yes	characteristic
36	C₂×C₈×D₆₁	D₄	1	1	yes	characteristic
37	C₂×C₈×D₆₁	Q₈	1	4	yes	—
38	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
39	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
40	#14519736	—	2	2	no	—
41	#14519735	—	2	4	no	—
42	#14519734	—	4	2	no	—
43	#14519733	—	4	2	no	—
44	C₄⋊C₄×D₆₁	—	2	4	no	—
45	C₂²⋊C₄×D₆₁	—	4	2	no	—
46	C₄²×D₆₁	—	4	2	no	—
47	C₂×#11758185	C₂×C₄	1	4	yes	complemented
48	C₂×#11758185	C₂³	1	1	yes	characteristic
49	C₂×#11758185	C₂³	1	1	yes	characteristic
50	C₁₆×D₆₁	C₂×C₄	1	4	yes	—
51	C₁₆×D₆₁	C₂×C₄	1	4	yes	—
52	C₁₆×D₆₁	C₂×C₄	1	4	yes	complemented
53	C₁₆×D₆₁	C₂×C₄	1	4	yes	complemented
54	#14519711	C₂³	1	4	yes	—
55	#14519710	—	2	4	no	—
56	#14519709	—	2	2	no	—
57	#14519708	—	2	4	no	—
58	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
59	#117×C₆₁	—	2	2	no	—
60	C₂×C₁₆×C₆₁	C₂×C₄	1	4	yes	complemented
61	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
62	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
63	#107×C₆₁	C₂³	1	4	yes	—
64	#106×C₆₁	—	2	2	no	—
65	C₂³×D₆₁	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×Dic₆₁	D₈	1	1	yes	characteristic
68	C₂×C₄×D₆₁	—	4	4	no	—
69	C₂×C₄×D₆₁	C₄⋊C₄	1	4	yes	—
70	C₂×C₄×D₆₁	D₈	1	1	yes	characteristic
71	C₂×C₄×D₆₁	D₈	1	1	yes	characteristic
72	C₂×C₄×D₆₁	Q₁₆	1	4	yes	—
73	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
78	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
79	C₄⋊C₄×C₆₁	—	2	4	no	—
80	C₂²⋊C₄×C₆₁	—	4	2	no	—
81	C₄²×C₆₁	—	4	2	no	—
82	C₂²⋊Dic₆₁	—	4	2	no	—
83	#11758198	—	4	4	no	—
84	C₄⋊Dic₆₁	—	2	4	no	—
85	#11758196	—	4	4	no	—
86	C₄×Dic₆₁	—	4	2	no	—
87	C₄×Dic₆₁	—	4	4	no	—
88	C₂×#92418	C₄⋊C₄	1	4	yes	—
89	C₂×#92418	C₂×D₄	1	1	yes	characteristic
90	C₂×#92418	C₂×D₄	1	1	yes	characteristic
91	C₈×D₆₁	C₄⋊C₄	1	4	yes	—
92	C₈×D₆₁	C₄⋊C₄	1	4	yes	—
93	C₈×D₆₁	C₂²×C₄	1	2	yes	—
94	C₈×D₆₁	C₂²×C₄	1	2	yes	—
95	C₈×D₆₁	C₂×Q₈	1	2	yes	—
96	C₈×D₆₁	C₂×Q₈	1	2	yes	—
97	C₁₆×C₆₁	C₂²×C₄	1	2	yes	—
98	C₁₆×C₆₁	C₂²×C₄	1	2	yes	—
99	#11758185	C₂²×C₄	1	2	yes	—
100	#11758185	C₂²×C₄	1	2	yes	—
101	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
102	C₂²×D₆₁	—	2	4	no	—
103	C₂²×D₆₁	#107	1	4	yes	—
104	C₂²×D₆₁	C₂×D₈	1	1	yes	characteristic
105	C₂²×D₆₁	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₆₁	—	4	4	no	—
107	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
110	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
111	C₂×Dic₆₁	—	4	4	no	—
112	C₂×Dic₆₁	#107	1	4	yes	—
113	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
114	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
115	C₄×D₆₁	—	8	8	no	—
116	C₄×D₆₁	#107	1	4	yes	—
117	C₄×D₆₁	#107	1	4	yes	—
118	C₄×D₆₁	C₂×C₄⋊C₄	1	2	yes	—
119	C₄×D₆₁	C₂×C₄⋊C₄	1	2	yes	—
120	C₄×D₆₁	C₂×Q₁₆	1	2	yes	—
121	C₄×D₆₁	C₂×Q₁₆	1	2	yes	—
122	C₈×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
123	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
124	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
125	#92418	C₂×C₄⋊C₄	1	2	yes	—
126	#92418	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×#2158	—	61	1	no	maximal, Sylow
128	C₂²×C₆₁	—	2	2	no	—
129	C₂²×C₆₁	C₂×#107	1	2	yes	—
130	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
131	C₂×D₆₁	—	2	2	no	—
132	C₂×D₆₁	—	2	8	no	—
133	C₂×D₆₁	C₂×#107	1	2	yes	—
134	C₂×D₆₁	C₂×#107	1	2	yes	—
135	C₂×D₆₁	#508	1	4	yes	—
136	C₄×C₆₁	—	8	4	no	—
137	C₄×C₆₁	C₂×#107	1	2	yes	—
138	C₄×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
139	C₄×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
140	Dic₆₁	—	8	4	no	—
141	Dic₆₁	C₂×#107	1	2	yes	—
142	Dic₆₁	C₂×#107	1	2	yes	—
143	C₂³×C₁₆	—	61	1	no	—
144	C₂×#427	—	61	2	no	—
145	#2158	—	61	1	no	—
146	#2158	—	61	2	no	—
147	#2158	—	61	2	no	—
148	#2158	—	61	2	no	—
149	#2158	D₆₁	1	1	yes	characteristic, direct product factor
150	C₂×#367	—	61	2	no	—
151	C₂×#366	—	61	2	no	—
152	C₂×C₆₁	—	2	2	no	—
153	C₂×C₆₁	C₂²×#107	1	1	yes	characteristic
154	C₂×C₆₁	C₂×#508	1	2	yes	—
155	D₆₁	—	2	2	no	—
156	D₆₁	#2158	1	4	yes	direct product factor
157	C₂³×C₈	—	61	1	no	—
158	C₂×#117	—	122	2	no	—
159	C₂²×C₁₆	—	61	1	no	—
160	C₂²×C₁₆	—	61	8	no	—
161	C₂²×C₁₆	—	61	1	no	—
162	C₂²×C₁₆	—	61	1	no	—
163	C₂²×C₁₆	—	61	1	no	—
164	C₂²×C₁₆	—	61	1	no	—
165	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
166	#427	—	61	2	no	—
167	#427	—	61	4	no	—
168	#427	—	122	2	no	—
169	#427	—	122	2	no	—
170	#427	C₂×D₆₁	1	2	yes	—
171	C₂×#107	—	61	4	no	—
172	C₂×#106	—	122	2	no	—
173	#367	—	61	2	no	—
174	#367	—	61	4	no	—
175	#367	C₂×D₆₁	1	2	yes	complemented
176	#366	—	61	2	no	—
177	#366	—	61	4	no	—
178	#366	C₂×D₆₁	1	2	yes	complemented
179	C₆₁	C₂×#2158	1	1	yes	characteristic, complemented, Sylow
180	C₂³×C₄	—	61	1	no	—
181	C₂²×C₈	—	61	1	no	—
182	C₂²×C₈	—	61	4	no	—
183	C₂²×C₈	—	61	1	no	—
184	C₂²×C₈	—	61	1	no	—
185	C₂²×C₈	—	61	4	no	—
186	C₂²×C₈	—	61	1	no	—
187	C₂²×C₈	—	61	1	no	—
188	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
189	#117	—	2	2	no	—
190	#117	—	122	4	no	—
191	#117	—	122	2	no	—
192	#117	—	244	2	no	—
193	#117	—	244	2	no	—
194	C₂×C₄⋊C₄	—	122	4	no	—
195	C₂×C₂²⋊C₄	—	244	2	no	—
196	C₂×C₄²	—	244	2	no	—
197	C₂×C₁₆	—	61	4	no	—
198	C₂×C₁₆	—	61	1	no	—
199	C₂×C₁₆	—	61	1	no	—
200	C₂×C₁₆	—	61	4	no	—
201	C₂×C₁₆	—	61	4	no	—
202	C₂×C₁₆	—	61	4	no	—
203	C₂×C₁₆	—	61	4	no	—
204	C₂×C₁₆	C₄×D₆₁	1	4	yes	complemented
205	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
206	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
207	#107	—	61	4	no	—
208	#107	—	122	4	no	—
209	#107	C₂²×D₆₁	1	4	yes	—
210	#106	—	2	2	no	—
211	#106	—	122	4	no	—
212	#106	—	122	2	no	—
213	C₂⁴	—	61	1	no	—
214	C₂²×C₄	—	61	1	no	—
215	C₂²×C₄	—	61	1	no	—
216	C₂²×C₄	—	61	1	no	—
217	C₂²×C₄	—	61	4	no	—
218	C₂²×C₄	—	61	1	no	—
219	C₂²×C₄	—	61	1	no	—
220	C₂²×C₄	—	61	4	no	—
221	C₂²×C₄	—	244	4	no	—
222	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
223	C₂×C₈	—	61	1	no	—
224	C₂×C₈	—	61	1	no	—
225	C₂×C₈	—	61	4	no	—
226	C₂×C₈	—	61	2	no	—
227	C₂×C₈	—	61	2	no	—
228	C₂×C₈	—	61	2	no	—
229	C₂×C₈	—	61	2	no	—
230	C₂×C₈	—	61	4	no	—
231	C₂×C₈	—	61	4	no	—
232	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
233	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
234	C₂×C₈	Q₈×D₆₁	1	2	yes	—
235	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	122	4	no	—
238	C₄⋊C₄	—	244	4	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	244	4	no	—
241	C₂²⋊C₄	—	244	2	no	—
242	C₄²	—	4	2	no	—
243	C₄²	—	244	4	no	—
244	C₄²	—	244	2	no	—
245	C₁₆	—	61	2	no	—
246	C₁₆	—	61	2	no	—
247	C₁₆	C₂×C₄×D₆₁	1	2	yes	—
248	C₁₆	C₂×C₄×D₆₁	1	2	yes	—
249	C₂³	—	61	1	no	—
250	C₂³	—	61	1	no	—
251	C₂³	—	61	4	no	—
252	C₂³	—	122	4	no	—
253	C₂³	D₈×D₆₁	1	1	yes	characteristic
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	61	1	no	—
256	C₂×C₄	—	61	1	no	—
257	C₂×C₄	—	61	4	no	—
258	C₂×C₄	—	61	2	no	—
259	C₂×C₄	—	61	2	no	—
260	C₂×C₄	—	61	2	no	—
261	C₂×C₄	—	61	2	no	—
262	C₂×C₄	—	61	4	no	—
263	C₂×C₄	—	61	4	no	—
264	C₂×C₄	—	488	8	no	—
265	C₂×C₄	—	244	4	no	—
266	C₂×C₄	C₄⋊C₄×D₆₁	1	2	yes	—
267	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
268	C₂×C₄	Q₁₆×D₆₁	1	2	yes	—
269	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
270	C₈	—	61	2	no	—
271	C₈	—	61	2	no	—
272	C₈	C₄⋊C₄×D₆₁	1	2	yes	—
273	C₈	C₂²×C₄×D₆₁	1	1	yes	characteristic
274	C₈	C₂×Q₈×D₆₁	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	61	2	no	—
277	C₂²	—	61	4	no	—
278	C₂²	—	61	2	no	—
279	C₂²	—	122	8	no	—
280	C₂²	—	122	2	no	—
281	C₂²	#107×D₆₁	1	2	yes	—
282	C₂²	C₂×D₈×D₆₁	1	1	yes	characteristic
283	C₄	—	8	4	no	—
284	C₄	—	61	2	no	—
285	C₄	—	61	2	no	—
286	C₄	—	488	4	no	—
287	C₄	#107×D₆₁	1	2	yes	—
288	C₄	C₂×C₄⋊C₄×D₆₁	1	1	yes	characteristic
289	C₄	C₂×Q₁₆×D₆₁	1	1	yes	characteristic
290	C₂	—	2	2	no	—
291	C₂	—	61	4	no	—
292	C₂	—	122	2	no	—
293	C₂	C₂×#107×D₆₁	1	1	yes	characteristic
294	C₂	#508×D₆₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₆₁
commutator	C₈×C₆₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₆₁×#2158	C₂
socle	C₂²×C₆₁	C₂²×D₈
prime radical	C₂³×D₆₁	D₈
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	#2158	D₆₁
hypocenter	C₆₁	C₂×#2158
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
3. #21447603	C₂	semidirect
4. #21447602	C₂	semidirect
5. #21447601	C₂	semidirect
6. #21447600	C₂	semidirect
7. #367×D₆₁	C₂	semidirect
8. #366×D₆₁	C₂	semidirect
9. C₆₁×#2158	C₂	semidirect
13. C₂×C₁₆×D₆₁	C₄	semidirect
24. #16839590	C₂²	semidirect
25. #16839589	C₂²	semidirect
26. #16839588	C₂²	semidirect
27. #16839587	C₂²	semidirect
30. C₆₁×#367	C₂²	semidirect
31. C₆₁×#366	C₂²	semidirect
47. C₂×#11758185	C₂×C₄	semidirect
52. C₁₆×D₆₁	C₂×C₄	semidirect
53. C₁₆×D₆₁	C₂×C₄	semidirect
60. C₂×C₁₆×C₆₁	C₂×C₄	semidirect
149. #2158	D₆₁	direct (N × H)
156. D₆₁	#2158	direct (N × H)
175. #367	C₂×D₆₁	semidirect
178. #366	C₂×D₆₁	semidirect
179. C₆₁	C₂×#2158	semidirect
204. C₂×C₁₆	C₄×D₆₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₁₆×D₆₁	2
#427×D₆₁	2
#21447603	2
#21447602	2
#21447601	2
#21447600	2
#367×D₆₁	2
#366×D₆₁	2
C₆₁×#2158	2
C₂×#2158	61

Subgroups — D61×#2158 #29833038

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₆₁×#2158 #29833038