Subgroups — C₆₁×#3011 #21443052

Order 7808.

700 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×2₋¹⁺⁴×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#137×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×Q₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#536	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#452	C₂	1	4	yes	maximal, complemented
6	C₆₁×#451	C₂	1	4	yes	maximal, complemented
7	C₆₁×#419	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#416	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#408	C₂	1	1	yes	maximal, characteristic, complemented
10	2₋¹⁺⁴×C₆₁	—	2	4	no	—
11	2₋¹⁺⁴×C₆₁	—	2	4	no	—
12	C₂×#41×C₆₁	—	2	2	no	—
13	C₂×#41×C₆₁	—	2	1	no	—
14	C₂×#41×C₆₁	C₂²	1	2	yes	—
15	C₂×#41×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×#41×C₆₁	C₂²	1	1	yes	characteristic
17	C₂²×Q₈×C₆₁	—	2	1	no	—
18	C₂²×Q₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₂²×Q₈×C₆₁	C₂²	1	2	yes	—
20	C₂²×Q₈×C₆₁	C₂²	1	1	yes	characteristic
21	#137×C₆₁	—	2	4	no	—
22	C₂×Q₁₆×C₆₁	—	2	4	no	—
23	C₂×Q₁₆×C₆₁	C₂²	1	4	yes	—
24	C₂×Q₁₆×C₆₁	C₂²	1	2	yes	—
25	C₂×SD₁₆×C₆₁	C₂²	1	2	yes	—
26	C₂×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
27	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
28	C₄⋊Q₈×C₆₁	—	2	1	no	—
29	#124×C₆₁	—	2	1	no	—
30	#123×C₆₁	—	2	2	no	—
31	C₂²⋊Q₈×C₆₁	C₂²	1	4	yes	—
32	#117×C₆₁	C₂²	1	1	yes	characteristic
33	#103×C₆₁	C₂²	1	2	yes	—
34	#103×C₆₁	C₂²	1	4	yes	—
35	#102×C₆₁	C₂²	1	2	yes	—
36	C₂²⋊C₈×C₆₁	C₂²	1	4	yes	—
37	#41×C₆₁	—	2	16	no	—
38	#41×C₆₁	—	2	8	no	—
39	#41×C₆₁	—	2	4	no	—
40	#41×C₆₁	—	2	4	no	—
41	#41×C₆₁	—	2	2	no	—
42	#41×C₆₁	—	2	2	no	—
43	#41×C₆₁	—	2	2	no	—
44	#41×C₆₁	D₄	1	4	yes	—
45	C₂×Q₈×C₆₁	—	2	8	no	—
46	C₂×Q₈×C₆₁	—	2	4	no	—
47	C₂×Q₈×C₆₁	—	2	4	no	—
48	C₂×Q₈×C₆₁	—	2	2	no	—
49	C₂×Q₈×C₆₁	—	2	1	no	—
50	C₂×Q₈×C₆₁	—	2	2	no	—
51	C₂×Q₈×C₆₁	—	2	1	no	—
52	C₂×Q₈×C₆₁	—	2	2	no	—
53	C₂×Q₈×C₆₁	—	2	2	no	—
54	C₂×Q₈×C₆₁	D₄	1	4	yes	—
55	C₂×Q₈×C₆₁	D₄	1	1	yes	characteristic
56	C₂×Q₈×C₆₁	C₂³	1	2	yes	—
57	C₂×Q₈×C₆₁	C₂³	1	1	yes	characteristic
58	C₂×Q₈×C₆₁	C₂³	1	2	yes	—
59	C₂×D₄×C₆₁	—	2	2	no	—
60	C₂×D₄×C₆₁	—	2	1	no	—
61	C₂×D₄×C₆₁	D₄	1	2	yes	—
62	C₂×D₄×C₆₁	D₄	1	1	yes	characteristic
63	C₂×D₄×C₆₁	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₆₁	—	2	2	no	—
65	C₂²×C₄×C₆₁	—	2	1	no	—
66	C₂²×C₄×C₆₁	—	2	2	no	—
67	C₂²×C₄×C₆₁	—	2	1	no	—
68	C₂²×C₄×C₆₁	—	2	1	no	—
69	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
70	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
71	Q₁₆×C₆₁	—	2	8	no	—
72	Q₁₆×C₆₁	—	2	4	no	—
73	SD₁₆×C₆₁	—	2	4	no	—
74	M₁₆×C₆₁	—	2	2	no	—
75	C₂×C₈×C₆₁	—	2	2	no	—
76	C₂×C₈×C₆₁	C₂³	1	2	yes	—
77	C₂×C₈×C₆₁	C₂³	1	2	yes	—
78	C₄⋊C₄×C₆₁	—	2	4	no	—
79	C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
80	C₂²⋊C₄×C₆₁	—	2	1	no	—
81	C₂²⋊C₄×C₆₁	—	2	4	no	—
82	C₄²×C₆₁	—	2	1	no	—
83	C₂³×C₆₁	—	2	1	no	—
84	C₂³×C₆₁	C₂×D₄	1	2	yes	—
85	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
86	Q₈×C₆₁	—	2	8	no	—
87	Q₈×C₆₁	—	2	4	no	—
88	Q₈×C₆₁	—	2	2	no	—
89	Q₈×C₆₁	—	2	1	no	—
90	Q₈×C₆₁	—	2	2	no	—
91	Q₈×C₆₁	—	2	2	no	—
92	Q₈×C₆₁	—	2	2	no	—
93	Q₈×C₆₁	—	4	2	no	—
94	Q₈×C₆₁	C₂×D₄	1	2	yes	—
95	Q₈×C₆₁	C₂×D₄	1	4	yes	—
96	D₄×C₆₁	—	2	8	no	—
97	D₄×C₆₁	—	2	4	no	—
98	D₄×C₆₁	—	2	4	no	—
99	D₄×C₆₁	—	2	2	no	—
100	D₄×C₆₁	—	2	1	no	—
101	D₄×C₆₁	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₆₁	—	2	8	no	—
103	C₂×C₄×C₆₁	—	2	8	no	—
104	C₂×C₄×C₆₁	—	2	4	no	—
105	C₂×C₄×C₆₁	—	2	2	no	—
106	C₂×C₄×C₆₁	—	2	1	no	—
107	C₂×C₄×C₆₁	—	2	4	no	—
108	C₂×C₄×C₆₁	—	2	2	no	—
109	C₂×C₄×C₆₁	—	2	2	no	—
110	C₂×C₄×C₆₁	—	2	2	no	—
111	C₂×C₄×C₆₁	—	2	2	no	—
112	C₂×C₄×C₆₁	—	4	2	no	—
113	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
114	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
116	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
117	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
118	C₈×C₆₁	—	2	2	no	—
119	C₈×C₆₁	—	2	2	no	—
120	C₂²×C₆₁	—	2	2	no	—
121	C₂²×C₆₁	—	2	2	no	—
122	C₂²×C₆₁	—	2	2	no	—
123	C₂²×C₆₁	—	2	4	no	—
124	C₂²×C₆₁	—	4	2	no	—
125	C₂²×C₆₁	#120	1	2	yes	—
126	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
127	C₄×C₆₁	—	2	4	no	—
128	C₄×C₆₁	—	2	2	no	—
129	C₄×C₆₁	—	2	2	no	—
130	C₄×C₆₁	—	4	2	no	—
131	C₄×C₆₁	—	4	2	no	—
132	C₄×C₆₁	#120	1	2	yes	—
133	C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
134	C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
135	#3011	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂×C₆₁	—	2	2	no	—
137	C₂×C₆₁	—	4	2	no	—
138	C₂×C₆₁	—	4	2	no	—
139	C₂×C₆₁	C₂×#120	1	1	yes	characteristic
140	C₂×C₆₁	#578	1	2	yes	—
141	C₂×2₋¹⁺⁴	C₂×C₆₁	1	1	yes	characteristic
142	C₂×#137	C₂×C₆₁	1	1	yes	characteristic, complemented
143	C₂²×Q₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
144	#536	C₂×C₆₁	1	1	yes	characteristic, complemented
145	#452	C₂×C₆₁	1	4	yes	complemented
146	#451	C₂×C₆₁	1	4	yes	complemented
147	#419	C₂×C₆₁	1	1	yes	characteristic, complemented
148	#416	C₂×C₆₁	1	1	yes	characteristic, complemented
149	#408	C₂×C₆₁	1	1	yes	characteristic, complemented
150	C₆₁	#3011	1	1	yes	characteristic, direct product factor, Sylow
151	2₋¹⁺⁴	—	2	4	no	—
152	2₋¹⁺⁴	—	2	4	no	—
153	C₂×#41	—	2	2	no	—
154	C₂×#41	—	2	1	no	—
155	C₂×#41	C₂²×C₆₁	1	2	yes	—
156	C₂×#41	C₂²×C₆₁	1	1	yes	characteristic
157	C₂×#41	C₂²×C₆₁	1	1	yes	characteristic
158	C₂²×Q₈	—	2	1	no	—
159	C₂²×Q₈	C₂²×C₆₁	1	1	yes	characteristic
160	C₂²×Q₈	C₂²×C₆₁	1	2	yes	—
161	C₂²×Q₈	C₂²×C₆₁	1	1	yes	characteristic
162	#137	—	2	4	no	—
163	C₂×Q₁₆	—	2	4	no	—
164	C₂×Q₁₆	C₂²×C₆₁	1	4	yes	—
165	C₂×Q₁₆	C₂²×C₆₁	1	2	yes	—
166	C₂×SD₁₆	C₂²×C₆₁	1	2	yes	—
167	C₂×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
168	C₂²×C₈	C₂²×C₆₁	1	1	yes	characteristic
169	C₄⋊Q₈	—	2	1	no	—
170	#124	—	2	1	no	—
171	#123	—	2	2	no	—
172	C₂²⋊Q₈	C₂²×C₆₁	1	4	yes	—
173	#117	C₂²×C₆₁	1	1	yes	characteristic
174	#103	C₂²×C₆₁	1	2	yes	—
175	#103	C₂²×C₆₁	1	4	yes	—
176	#102	C₂²×C₆₁	1	2	yes	—
177	C₂²⋊C₈	C₂²×C₆₁	1	4	yes	—
178	#41	—	2	16	no	—
179	#41	—	2	8	no	—
180	#41	—	2	4	no	—
181	#41	—	2	4	no	—
182	#41	—	2	2	no	—
183	#41	—	2	2	no	—
184	#41	—	2	2	no	—
185	#41	D₄×C₆₁	1	4	yes	—
186	C₂×Q₈	—	2	8	no	—
187	C₂×Q₈	—	2	4	no	—
188	C₂×Q₈	—	2	4	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	2	1	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×Q₈	—	2	1	no	—
193	C₂×Q₈	—	2	2	no	—
194	C₂×Q₈	—	2	2	no	—
195	C₂×Q₈	D₄×C₆₁	1	4	yes	—
196	C₂×Q₈	D₄×C₆₁	1	1	yes	characteristic
197	C₂×Q₈	C₂³×C₆₁	1	2	yes	—
198	C₂×Q₈	C₂³×C₆₁	1	1	yes	characteristic
199	C₂×Q₈	C₂³×C₆₁	1	2	yes	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	D₄×C₆₁	1	2	yes	—
203	C₂×D₄	D₄×C₆₁	1	1	yes	characteristic
204	C₂×D₄	C₂³×C₆₁	1	1	yes	characteristic
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	C₂³×C₆₁	1	2	yes	—
211	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
212	Q₁₆	—	2	8	no	—
213	Q₁₆	—	2	4	no	—
214	SD₁₆	—	2	4	no	—
215	M₁₆	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	C₂³×C₆₁	1	2	yes	—
218	C₂×C₈	C₂³×C₆₁	1	2	yes	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
221	C₂²⋊C₄	—	2	1	no	—
222	C₂²⋊C₄	—	2	4	no	—
223	C₄²	—	2	1	no	—
224	C₂³	—	2	1	no	—
225	C₂³	C₂×D₄×C₆₁	1	2	yes	—
226	C₂³	C₂×D₄×C₆₁	1	1	yes	characteristic
227	Q₈	—	2	8	no	—
228	Q₈	—	2	4	no	—
229	Q₈	—	2	2	no	—
230	Q₈	—	2	1	no	—
231	Q₈	—	2	2	no	—
232	Q₈	—	2	2	no	—
233	Q₈	—	2	2	no	—
234	Q₈	—	4	2	no	—
235	Q₈	C₂×D₄×C₆₁	1	2	yes	—
236	Q₈	C₂×D₄×C₆₁	1	4	yes	—
237	D₄	—	2	8	no	—
238	D₄	—	2	4	no	—
239	D₄	—	2	4	no	—
240	D₄	—	2	2	no	—
241	D₄	—	2	1	no	—
242	D₄	C₂×D₄×C₆₁	1	2	yes	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	C₂×D₄×C₆₁	1	2	yes	—
255	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
256	C₂×C₄	C₂×D₄×C₆₁	1	2	yes	—
257	C₂×C₄	C₂×D₄×C₆₁	1	2	yes	—
258	C₂×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	4	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	#120×C₆₁	1	2	yes	—
267	C₂²	C₂²×D₄×C₆₁	1	1	yes	characteristic
268	C₄	—	2	4	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	2	no	—
273	C₄	#120×C₆₁	1	2	yes	—
274	C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
275	C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
276	C₂	—	2	2	no	—
277	C₂	—	4	2	no	—
278	C₂	—	4	2	no	—
279	C₂	C₂×#120×C₆₁	1	1	yes	characteristic
280	C₂	C₆₁×#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₆₁
prime radical	C₂×2₋¹⁺⁴×C₆₁	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×#41×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#137×C₆₁	C₂	semidirect
3. C₂²×Q₁₆×C₆₁	C₂	semidirect
4. C₆₁×#536	C₂	semidirect
5. C₆₁×#452	C₂	semidirect
6. C₆₁×#451	C₂	semidirect
7. C₆₁×#419	C₂	semidirect
8. C₆₁×#416	C₂	semidirect
9. C₆₁×#408	C₂	semidirect
135. #3011	C₆₁	direct (N × H)
142. C₂×#137	C₂×C₆₁	semidirect
143. C₂²×Q₁₆	C₂×C₆₁	semidirect
144. #536	C₂×C₆₁	semidirect
145. #452	C₂×C₆₁	semidirect
146. #451	C₂×C₆₁	semidirect
147. #419	C₂×C₆₁	semidirect
148. #416	C₂×C₆₁	semidirect
149. #408	C₂×C₆₁	semidirect
150. C₆₁	#3011	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×2₋¹⁺⁴×C₆₁	2
C₂×#137×C₆₁	2
C₂²×Q₁₆×C₆₁	2
C₆₁×#536	2
C₆₁×#452	2
C₆₁×#451	2
C₆₁×#419	2
C₆₁×#416	2
C₆₁×#408	2
#3011	61

Subgroups — C61×#3011 #21443052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#3011 #21443052