Subgroups — C₆₁×#8686 #29512119

Order 15616.

414 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2931	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#1837	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₆₁×#334	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1547	C₂	1	2	yes	maximal
5	C₆₁×#1292	C₂	1	2	yes	maximal, complemented
6	C₂×#131×C₆₁	—	2	1	no	—
7	C₆₁×#532	C₂²	1	1	yes	characteristic
8	C₆₁×#436	—	2	1	no	—
9	C₆₁×#436	C₄	1	2	yes	—
10	C₈×D₄×C₆₁	—	2	1	no	—
11	C₈×D₄×C₆₁	C₄	1	2	yes	—
12	C₆₁×#423	C₂²	1	2	yes	—
13	C₆₁×#423	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
15	C₆₁×#408	—	2	1	no	—
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#341	—	2	1	no	—
18	C₆₁×#341	—	2	1	no	—
19	C₆₁×#334	—	2	1	no	—
20	C₆₁×#334	—	2	1	no	—
21	C₆₁×#332	—	2	2	no	—
22	C₆₁×#331	—	2	2	no	—
23	C₂×#41×C₆₁	—	2	1	no	—
24	#131×C₆₁	—	4	4	no	—
25	C₂×M₁₆×C₆₁	—	2	1	no	—
26	C₂×M₁₆×C₆₁	—	2	2	no	—
27	C₂×M₁₆×C₆₁	—	2	2	no	—
28	C₂²×C₈×C₆₁	—	2	2	no	—
29	C₂²×C₈×C₆₁	—	2	2	no	—
30	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
32	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
33	#127×C₆₁	D₄	1	1	yes	characteristic
34	#124×C₆₁	—	2	1	no	—
35	#121×C₆₁	—	2	1	no	—
36	#121×C₆₁	Q₈	1	2	yes	—
37	C₄×D₄×C₆₁	—	2	1	no	—
38	C₄×D₄×C₆₁	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₆₁	—	2	1	no	—
41	C₄⋊C₈×C₆₁	—	2	2	no	—
42	C₄⋊C₈×C₆₁	—	2	2	no	—
43	C₄⋊C₈×C₆₁	—	2	2	no	—
44	C₄⋊C₈×C₆₁	—	2	2	no	—
45	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
46	C₂²⋊C₈×C₆₁	—	2	2	no	—
47	C₂²⋊C₈×C₆₁	—	2	2	no	—
48	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
51	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
52	#41×C₆₁	—	4	2	no	—
53	#41×C₆₁	—	4	2	no	—
54	C₂×Q₈×C₆₁	—	2	1	no	—
55	C₂×D₄×C₆₁	—	2	2	no	—
56	C₂×D₄×C₆₁	—	2	1	no	—
57	C₂×D₄×C₆₁	—	2	1	no	—
58	C₂×D₄×C₆₁	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₆₁	—	2	2	no	—
60	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
63	M₁₆×C₆₁	—	4	4	no	—
64	M₁₆×C₆₁	—	4	2	no	—
65	M₁₆×C₆₁	—	4	2	no	—
66	M₁₆×C₆₁	—	4	2	no	—
67	C₂×C₈×C₆₁	—	2	2	no	—
68	C₂×C₈×C₆₁	—	2	2	no	—
69	C₂×C₈×C₆₁	—	2	2	no	—
70	C₂×C₈×C₆₁	—	4	4	no	—
71	C₂×C₈×C₆₁	—	4	2	no	—
72	C₂×C₈×C₆₁	—	4	2	no	—
73	C₂×C₈×C₆₁	—	8	2	no	—
74	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₄⋊C₄×C₆₁	—	2	1	no	—
79	C₄⋊C₄×C₆₁	C₄⋊C₄	1	2	yes	—
80	C₂²⋊C₄×C₆₁	—	2	2	no	—
81	C₂²⋊C₄×C₆₁	—	2	2	no	—
82	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₆₁	#41	1	1	yes	characteristic
86	C₂³×C₆₁	—	2	2	no	—
87	C₂³×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
88	Q₈×C₆₁	—	4	2	no	—
89	D₄×C₆₁	—	4	2	no	—
90	D₄×C₆₁	—	4	4	no	—
91	D₄×C₆₁	—	4	2	no	—
92	D₄×C₆₁	—	4	2	no	—
93	C₂×C₄×C₆₁	—	2	2	no	—
94	C₂×C₄×C₆₁	—	2	2	no	—
95	C₂×C₄×C₆₁	—	4	1	no	—
96	C₂×C₄×C₆₁	—	4	2	no	—
97	C₂×C₄×C₆₁	—	4	2	no	—
98	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
101	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
102	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
103	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₆₁	#135	1	2	yes	—
105	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
106	C₈×C₆₁	—	2	1	no	—
107	C₈×C₆₁	—	2	1	no	—
108	C₈×C₆₁	—	2	1	no	—
109	C₈×C₆₁	—	2	1	no	—
110	C₈×C₆₁	—	4	2	no	—
111	C₈×C₆₁	—	8	2	no	—
112	C₈×C₆₁	—	8	2	no	—
113	#8686	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂²×C₆₁	—	2	1	no	—
115	C₂²×C₆₁	—	2	1	no	—
116	C₂²×C₆₁	—	4	2	no	—
117	C₂²×C₆₁	—	4	2	no	—
118	C₂²×C₆₁	—	4	2	no	—
119	C₂²×C₆₁	#380	1	1	yes	characteristic
120	C₂²×C₆₁	#478	1	2	yes	—
121	C₄×C₆₁	—	2	1	no	—
122	C₄×C₆₁	—	2	1	no	—
123	C₄×C₆₁	—	2	1	no	—
124	C₄×C₆₁	—	4	2	no	—
125	C₄×C₆₁	#416	1	1	yes	characteristic
126	C₄×C₆₁	#418	1	1	yes	characteristic
127	C₄×C₆₁	#463	1	2	yes	—
128	C₄×C₆₁	#471	1	2	yes	—
129	#2931	C₂×C₆₁	1	1	yes	characteristic
130	#1837	C₂×C₆₁	1	1	yes	characteristic, complemented
131	C₂×#334	C₂×C₆₁	1	1	yes	characteristic, complemented
132	#1547	C₂×C₆₁	1	2	yes	—
133	#1292	C₂×C₆₁	1	2	yes	complemented
134	C₂×C₆₁	—	2	2	no	—
135	C₂×C₆₁	—	8	2	no	—
136	C₂×C₆₁	#1804	1	1	yes	characteristic
137	C₂×C₆₁	#1813	1	1	yes	characteristic
138	C₂×C₆₁	#1815	1	1	yes	characteristic
139	C₂×#131	—	2	1	no	—
140	#532	C₂²×C₆₁	1	1	yes	characteristic
141	#436	—	2	1	no	—
142	#436	C₄×C₆₁	1	2	yes	—
143	C₈×D₄	—	2	1	no	—
144	C₈×D₄	C₄×C₆₁	1	2	yes	—
145	#423	C₂²×C₆₁	1	2	yes	—
146	#423	C₂²×C₆₁	1	1	yes	characteristic
147	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
148	#408	—	2	1	no	—
149	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
150	#341	—	2	1	no	—
151	#341	—	2	1	no	—
152	#334	—	2	1	no	—
153	#334	—	2	1	no	—
154	#332	—	2	2	no	—
155	#331	—	2	2	no	—
156	C₆₁	#8686	1	1	yes	characteristic, direct product factor, Sylow
157	C₂×#41	—	2	1	no	—
158	#131	—	4	4	no	—
159	C₂×M₁₆	—	2	1	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
165	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
166	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
167	#127	D₄×C₆₁	1	1	yes	characteristic
168	#124	—	2	1	no	—
169	#121	—	2	1	no	—
170	#121	Q₈×C₆₁	1	2	yes	—
171	C₄×D₄	—	2	1	no	—
172	C₄×D₄	C₂×C₄×C₆₁	1	2	yes	—
173	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
174	C₄⋊C₈	—	2	1	no	—
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	—	2	2	no	—
177	C₄⋊C₈	—	2	2	no	—
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	C₂×C₄×C₆₁	1	2	yes	—
180	C₂²⋊C₈	—	2	2	no	—
181	C₂²⋊C₈	—	2	2	no	—
182	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
183	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
184	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
185	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
186	#41	—	4	2	no	—
187	#41	—	4	2	no	—
188	C₂×Q₈	—	2	1	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	C₄⋊C₄×C₆₁	1	2	yes	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
195	C₂²×C₄	C₂×Q₈×C₆₁	1	1	yes	characteristic
196	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
197	M₁₆	—	4	4	no	—
198	M₁₆	—	4	2	no	—
199	M₁₆	—	4	2	no	—
200	M₁₆	—	4	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	4	4	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	8	2	no	—
208	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
209	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
210	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
211	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	C₄⋊C₄×C₆₁	1	2	yes	—
214	C₂²⋊C₄	—	2	2	no	—
215	C₂²⋊C₄	—	2	2	no	—
216	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
217	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
218	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
219	C₄²	#41×C₆₁	1	1	yes	characteristic
220	C₂³	—	2	2	no	—
221	C₂³	C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
222	Q₈	—	4	2	no	—
223	D₄	—	4	2	no	—
224	D₄	—	4	4	no	—
225	D₄	—	4	2	no	—
226	D₄	—	4	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
234	C₂×C₄	#117×C₆₁	1	1	yes	characteristic
235	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
236	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
237	C₂×C₄	C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
238	C₂×C₄	#135×C₆₁	1	2	yes	—
239	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	2	yes	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	4	2	no	—
245	C₈	—	8	2	no	—
246	C₈	—	8	2	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	C₆₁×#380	1	1	yes	characteristic
253	C₂²	C₆₁×#478	1	2	yes	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	4	2	no	—
258	C₄	C₆₁×#416	1	1	yes	characteristic
259	C₄	C₆₁×#418	1	1	yes	characteristic
260	C₄	C₆₁×#463	1	2	yes	—
261	C₄	C₆₁×#471	1	2	yes	—
262	C₂	—	2	2	no	—
263	C₂	—	8	2	no	—
264	C₂	C₆₁×#1804	1	1	yes	characteristic
265	C₂	C₆₁×#1813	1	1	yes	characteristic
266	C₂	C₆₁×#1815	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	#380
commutator	C₄²	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
prime radical	C₆₁×#532	C₂²
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
2. C₆₁×#1837	C₂	semidirect
3. C₂×C₆₁×#334	C₂	semidirect
5. C₆₁×#1292	C₂	semidirect
113. #8686	C₆₁	direct (N × H)
130. #1837	C₂×C₆₁	semidirect
131. C₂×#334	C₂×C₆₁	semidirect
133. #1292	C₂×C₆₁	semidirect
156. C₆₁	#8686	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×#2931	2
C₆₁×#1837	2
C₂×C₆₁×#334	2
C₆₁×#1547	2
C₆₁×#1292	2
#8686	61

Subgroups — C61×#8686 #29512119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#8686 #29512119