Subgroups — C₆₁×#19698 #29523131

Order 15616.

896 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3208	C₂	1	2	yes	maximal, complemented
2	C₆₁×#3205	C₂	1	2	yes	maximal, complemented
3	C₂×C₆₁×#493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#3131	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1721	C₂	1	4	yes	maximal, complemented
6	C₆₁×#1717	C₂	1	4	yes	maximal, complemented
7	C₆₁×#1452	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₆₁	—	2	2	no	—
9	C₆₁×#533	C₂²	1	1	yes	characteristic
10	C₆₁×#532	C₂²	1	1	yes	characteristic
11	C₂×#127×C₆₁	C₂²	1	2	yes	—
12	C₆₁×#493	—	2	2	no	—
13	C₆₁×#493	C₂²	1	4	yes	—
14	C₆₁×#493	C₂²	1	4	yes	complemented
15	C₆₁×#492	C₂²	1	4	yes	complemented
16	C₆₁×#488	—	2	2	no	—
17	C₆₁×#487	C₂²	1	2	yes	complemented
18	C₆₁×#486	C₂²	1	2	yes	complemented
19	C₆₁×#486	C₂²	1	4	yes	complemented
20	C₆₁×#486	C₂²	1	4	yes	complemented
21	C₆₁×#460	—	2	4	no	—
22	C₆₁×#459	—	2	4	no	—
23	C₆₁×#450	—	2	4	no	—
24	C₆₁×#447	—	2	4	no	—
25	C₆₁×#431	C₂²	1	2	yes	—
26	C₆₁×#416	—	2	2	no	—
27	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
28	C₆₁×#322	C₂²	1	4	yes	complemented
29	C₂×#41×C₆₁	—	2	1	no	—
30	C₂²×D₄×C₆₁	—	2	2	no	—
31	C₂²×D₄×C₆₁	—	2	2	no	—
32	C₂²×D₄×C₆₁	—	2	2	no	—
33	C₂×SD₁₆×C₆₁	—	2	8	no	—
34	C₂×D₈×C₆₁	—	2	8	no	—
35	C₂×D₈×C₆₁	—	2	8	no	—
36	C₂×D₈×C₆₁	—	4	8	no	—
37	C₂²×C₈×C₆₁	D₄	1	2	yes	—
38	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
39	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
40	#127×C₆₁	—	2	2	no	—
41	#127×C₆₁	—	2	2	no	—
42	#127×C₆₁	—	2	2	no	—
43	#127×C₆₁	C₂³	1	4	yes	—
44	#127×C₆₁	C₂³	1	1	yes	characteristic
45	#125×C₆₁	—	2	1	no	—
46	#124×C₆₁	—	2	1	no	—
47	C₂²⋊Q₈×C₆₁	—	2	2	no	—
48	#121×C₆₁	—	2	2	no	—
49	C₄×Q₈×C₆₁	—	2	2	no	—
50	C₄×D₄×C₆₁	—	2	2	no	—
51	#117×C₆₁	—	2	1	no	—
52	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₆₁	D₄	1	4	yes	—
54	#103×C₆₁	—	2	4	no	—
55	#102×C₆₁	—	2	4	no	—
56	#102×C₆₁	—	2	8	no	—
57	#102×C₆₁	—	2	8	no	—
58	C₂²⋊C₈×C₆₁	D₄	1	4	yes	—
59	C₄×C₈×C₆₁	D₄	1	2	yes	—
60	C₄×C₈×C₆₁	C₂³	1	2	yes	—
61	C₄×C₈×C₆₁	C₂³	1	4	yes	—
62	C₂⁴×C₆₁	—	4	2	no	—
63	#41×C₆₁	—	4	4	no	—
64	C₂×Q₈×C₆₁	—	2	1	no	—
65	C₂×Q₈×C₆₁	—	2	2	no	—
66	C₂×D₄×C₆₁	—	2	1	no	—
67	C₂×D₄×C₆₁	—	2	4	no	—
68	C₂×D₄×C₆₁	—	2	4	no	—
69	C₂×D₄×C₆₁	—	2	2	no	—
70	C₂×D₄×C₆₁	—	2	4	no	—
71	C₂×D₄×C₆₁	—	4	1	no	—
72	C₂×D₄×C₆₁	—	4	2	no	—
73	C₂×D₄×C₆₁	—	4	2	no	—
74	C₂×D₄×C₆₁	—	4	2	no	—
75	C₂×D₄×C₆₁	—	4	8	no	—
76	C₂×D₄×C₆₁	—	8	4	no	—
77	C₂×D₄×C₆₁	—	8	4	no	—
78	C₂²×C₄×C₆₁	—	4	1	no	—
79	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
81	SD₁₆×C₆₁	—	8	8	no	—
82	D₈×C₆₁	—	4	16	no	—
83	D₈×C₆₁	—	8	8	no	—
84	C₂×C₈×C₆₁	—	2	4	no	—
85	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
86	C₂×C₈×C₆₁	C₂×D₄	1	8	yes	—
87	C₄⋊C₄×C₆₁	—	2	2	no	—
88	C₄⋊C₄×C₆₁	—	2	2	no	—
89	C₄⋊C₄×C₆₁	—	2	2	no	—
90	C₄⋊C₄×C₆₁	—	4	2	no	—
91	C₂²⋊C₄×C₆₁	—	4	1	no	—
92	C₂²⋊C₄×C₆₁	—	4	2	no	—
93	C₄²×C₆₁	—	4	1	no	—
94	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₆₁	C₂×D₄	1	2	yes	—
96	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
97	C₂³×C₆₁	—	4	1	no	—
98	C₂³×C₆₁	—	4	4	no	—
99	C₂³×C₆₁	—	8	4	no	—
100	C₂³×C₆₁	—	8	4	no	—
101	C₂³×C₆₁	—	8	4	no	—
102	C₂³×C₆₁	#127	1	1	yes	characteristic
103	Q₈×C₆₁	—	4	2	no	—
104	Q₈×C₆₁	—	4	4	no	—
105	D₄×C₆₁	—	4	8	no	—
106	D₄×C₆₁	—	4	2	no	—
107	D₄×C₆₁	—	4	4	no	—
108	D₄×C₆₁	—	4	8	no	—
109	D₄×C₆₁	—	4	8	no	—
110	D₄×C₆₁	—	8	4	no	—
111	D₄×C₆₁	—	8	4	no	—
112	D₄×C₆₁	—	8	2	no	—
113	D₄×C₆₁	—	8	4	no	—
114	C₂×C₄×C₆₁	—	2	4	no	—
115	C₂×C₄×C₆₁	—	4	2	no	—
116	C₂×C₄×C₆₁	—	4	1	no	—
117	C₂×C₄×C₆₁	—	8	2	no	—
118	C₂×C₄×C₆₁	#127	1	2	yes	—
119	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
120	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	4	yes	—
121	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
123	C₈×C₆₁	—	2	4	no	—
124	C₈×C₆₁	—	2	8	no	—
125	#19698	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂²×C₆₁	—	2	2	no	—
127	C₂²×C₆₁	—	8	1	no	—
128	C₂²×C₆₁	—	8	4	no	—
129	C₂²×C₆₁	—	8	4	no	—
130	C₂²×C₆₁	—	8	4	no	—
131	C₂²×C₆₁	—	8	2	no	—
132	C₂²×C₆₁	—	16	8	no	—
133	C₂²×C₆₁	#496	1	2	yes	—
134	C₂²×C₆₁	C₂×#127	1	1	yes	characteristic
135	C₄×C₆₁	—	2	2	no	—
136	C₄×C₆₁	—	8	1	no	—
137	C₄×C₆₁	—	8	2	no	—
138	C₄×C₆₁	#496	1	2	yes	—
139	C₄×C₆₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
140	C₄×C₆₁	#576	1	4	yes	—
141	#3208	C₂×C₆₁	1	2	yes	complemented
142	#3205	C₂×C₆₁	1	2	yes	complemented
143	C₂×#493	C₂×C₆₁	1	1	yes	characteristic, complemented
144	#3131	C₂×C₆₁	1	1	yes	characteristic, complemented
145	#1721	C₂×C₆₁	1	4	yes	complemented
146	#1717	C₂×C₆₁	1	4	yes	complemented
147	#1452	C₂×C₆₁	1	1	yes	characteristic, complemented
148	C₂×C₆₁	—	2	2	no	—
149	C₂×C₆₁	—	16	1	no	—
150	C₂×C₆₁	—	16	4	no	—
151	C₂×C₆₁	C₂×#496	1	1	yes	characteristic
152	C₂×C₆₁	#3155	1	2	yes	—
153	C₂²×D₈	—	2	2	no	—
154	#533	C₂²×C₆₁	1	1	yes	characteristic
155	#532	C₂²×C₆₁	1	1	yes	characteristic
156	C₂×#127	C₂²×C₆₁	1	2	yes	—
157	#493	—	2	2	no	—
158	#493	C₂²×C₆₁	1	4	yes	—
159	#493	C₂²×C₆₁	1	4	yes	complemented
160	#492	C₂²×C₆₁	1	4	yes	complemented
161	#488	—	2	2	no	—
162	#487	C₂²×C₆₁	1	2	yes	complemented
163	#486	C₂²×C₆₁	1	2	yes	complemented
164	#486	C₂²×C₆₁	1	4	yes	complemented
165	#486	C₂²×C₆₁	1	4	yes	complemented
166	#460	—	2	4	no	—
167	#459	—	2	4	no	—
168	#450	—	2	4	no	—
169	#447	—	2	4	no	—
170	#431	C₂²×C₆₁	1	2	yes	—
171	#416	—	2	2	no	—
172	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
173	#322	C₂²×C₆₁	1	4	yes	complemented
174	C₆₁	#19698	1	1	yes	characteristic, direct product factor, Sylow
175	C₂×#41	—	2	1	no	—
176	C₂²×D₄	—	2	2	no	—
177	C₂²×D₄	—	2	2	no	—
178	C₂²×D₄	—	2	2	no	—
179	C₂×SD₁₆	—	2	8	no	—
180	C₂×D₈	—	2	8	no	—
181	C₂×D₈	—	2	8	no	—
182	C₂×D₈	—	4	8	no	—
183	C₂²×C₈	D₄×C₆₁	1	2	yes	—
184	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
185	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
186	#127	—	2	2	no	—
187	#127	—	2	2	no	—
188	#127	—	2	2	no	—
189	#127	C₂³×C₆₁	1	4	yes	—
190	#127	C₂³×C₆₁	1	1	yes	characteristic
191	#125	—	2	1	no	—
192	#124	—	2	1	no	—
193	C₂²⋊Q₈	—	2	2	no	—
194	#121	—	2	2	no	—
195	C₄×Q₈	—	2	2	no	—
196	C₄×D₄	—	2	2	no	—
197	#117	—	2	1	no	—
198	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
199	C₄⋊C₈	D₄×C₆₁	1	4	yes	—
200	#103	—	2	4	no	—
201	#102	—	2	4	no	—
202	#102	—	2	8	no	—
203	#102	—	2	8	no	—
204	C₂²⋊C₈	D₄×C₆₁	1	4	yes	—
205	C₄×C₈	D₄×C₆₁	1	2	yes	—
206	C₄×C₈	C₂³×C₆₁	1	2	yes	—
207	C₄×C₈	C₂³×C₆₁	1	4	yes	—
208	C₂⁴	—	4	2	no	—
209	#41	—	4	4	no	—
210	C₂×Q₈	—	2	1	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×D₄	—	2	1	no	—
213	C₂×D₄	—	2	4	no	—
214	C₂×D₄	—	2	4	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	2	4	no	—
217	C₂×D₄	—	4	1	no	—
218	C₂×D₄	—	4	2	no	—
219	C₂×D₄	—	4	2	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂×D₄	—	4	8	no	—
222	C₂×D₄	—	8	4	no	—
223	C₂×D₄	—	8	4	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
227	SD₁₆	—	8	8	no	—
228	D₈	—	4	16	no	—
229	D₈	—	8	8	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
232	C₂×C₈	C₂×D₄×C₆₁	1	8	yes	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₂²⋊C₄	—	4	1	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₄²	—	4	1	no	—
240	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
241	C₄²	C₂×D₄×C₆₁	1	2	yes	—
242	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
243	C₂³	—	4	1	no	—
244	C₂³	—	4	4	no	—
245	C₂³	—	8	4	no	—
246	C₂³	—	8	4	no	—
247	C₂³	—	8	4	no	—
248	C₂³	#127×C₆₁	1	1	yes	characteristic
249	Q₈	—	4	2	no	—
250	Q₈	—	4	4	no	—
251	D₄	—	4	8	no	—
252	D₄	—	4	2	no	—
253	D₄	—	4	4	no	—
254	D₄	—	4	8	no	—
255	D₄	—	4	8	no	—
256	D₄	—	8	4	no	—
257	D₄	—	8	4	no	—
258	D₄	—	8	2	no	—
259	D₄	—	8	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	8	2	no	—
264	C₂×C₄	#127×C₆₁	1	2	yes	—
265	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
266	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	4	yes	—
267	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
268	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
269	C₈	—	2	4	no	—
270	C₈	—	2	8	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	8	1	no	—
273	C₂²	—	8	4	no	—
274	C₂²	—	8	4	no	—
275	C₂²	—	8	4	no	—
276	C₂²	—	8	2	no	—
277	C₂²	—	16	8	no	—
278	C₂²	C₆₁×#496	1	2	yes	—
279	C₂²	C₂×#127×C₆₁	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	8	1	no	—
282	C₄	—	8	2	no	—
283	C₄	C₆₁×#496	1	2	yes	—
284	C₄	C₂×AGL(1,ℤ/8)×C₆₁	1	2	yes	—
285	C₄	C₆₁×#576	1	4	yes	—
286	C₂	—	2	2	no	—
287	C₂	—	16	4	no	—
288	C₂	—	16	1	no	—
289	C₂	C₂×C₆₁×#496	1	1	yes	characteristic
290	C₂	C₆₁×#3155	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3208	C₂	semidirect
2. C₆₁×#3205	C₂	semidirect
3. C₂×C₆₁×#493	C₂	semidirect
4. C₆₁×#3131	C₂	semidirect
5. C₆₁×#1721	C₂	semidirect
6. C₆₁×#1717	C₂	semidirect
7. C₆₁×#1452	C₂	semidirect
14. C₆₁×#493	C₂²	semidirect
15. C₆₁×#492	C₂²	semidirect
17. C₆₁×#487	C₂²	semidirect
18. C₆₁×#486	C₂²	semidirect
19. C₆₁×#486	C₂²	semidirect
20. C₆₁×#486	C₂²	semidirect
28. C₆₁×#322	C₂²	semidirect
125. #19698	C₆₁	direct (N × H)
141. #3208	C₂×C₆₁	semidirect
142. #3205	C₂×C₆₁	semidirect
143. C₂×#493	C₂×C₆₁	semidirect
144. #3131	C₂×C₆₁	semidirect
145. #1721	C₂×C₆₁	semidirect
146. #1717	C₂×C₆₁	semidirect
147. #1452	C₂×C₆₁	semidirect
159. #493	C₂²×C₆₁	semidirect
160. #492	C₂²×C₆₁	semidirect
162. #487	C₂²×C₆₁	semidirect
163. #486	C₂²×C₆₁	semidirect
164. #486	C₂²×C₆₁	semidirect
165. #486	C₂²×C₆₁	semidirect
173. #322	C₂²×C₆₁	semidirect
174. C₆₁	#19698	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₁×#3208	2
C₆₁×#3205	2
C₂×C₆₁×#493	2
C₆₁×#3131	2
C₆₁×#1721	2
C₆₁×#1717	2
C₆₁×#1452	2
#19698	61

Subgroups — C61×#19698 #29523131

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#19698 #29523131