Subgroups — C₆₁×#19699 #29523132

Order 15616.

744 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3229	C₂	1	2	yes	maximal, complemented
2	C₆₁×#3226	C₂	1	2	yes	maximal
3	C₂×C₆₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#3131	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1723	C₂	1	4	yes	maximal, complemented
6	C₆₁×#1720	C₂	1	4	yes	maximal, complemented
7	C₆₁×#1452	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₁₆×C₆₁	—	2	2	no	—
9	C₆₁×#533	C₂²	1	1	yes	characteristic
10	C₆₁×#532	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
12	C₆₁×#500	C₂²	1	4	yes	—
13	C₆₁×#498	C₂²	1	4	yes	—
14	C₆₁×#494	—	2	2	no	—
15	C₆₁×#494	C₂²	1	4	yes	—
16	C₆₁×#488	—	2	2	no	—
17	C₆₁×#487	C₂²	1	2	yes	—
18	C₆₁×#487	C₂²	1	4	yes	—
19	C₆₁×#486	C₂²	1	2	yes	—
20	C₆₁×#486	C₂²	1	4	yes	—
21	C₆₁×#483	—	2	4	no	—
22	C₆₁×#480	—	2	4	no	—
23	C₆₁×#475	—	2	4	no	—
24	C₆₁×#474	—	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	—
29	C₂×#41×C₆₁	—	2	1	no	—
30	C₂²×Q₈×C₆₁	—	2	2	no	—
31	C₂×Q₁₆×C₆₁	—	2	8	no	—
32	C₂×Q₁₆×C₆₁	—	4	8	no	—
33	C₂²×C₈×C₆₁	D₄	1	2	yes	—
34	C₄⋊Q₈×C₆₁	—	2	2	no	—
35	C₄⋊Q₈×C₆₁	—	2	2	no	—
36	C₄⋊Q₈×C₆₁	—	2	2	no	—
37	C₄⋊Q₈×C₆₁	C₂³	1	4	yes	—
38	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
39	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
40	#127×C₆₁	C₂³	1	1	yes	characteristic
41	#125×C₆₁	—	2	1	no	—
42	#124×C₆₁	—	2	1	no	—
43	C₂²⋊Q₈×C₆₁	—	2	2	no	—
44	#121×C₆₁	—	2	2	no	—
45	C₄×Q₈×C₆₁	—	2	2	no	—
46	C₄×D₄×C₆₁	—	2	2	no	—
47	#117×C₆₁	—	2	1	no	—
48	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
49	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
50	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
51	#107×C₆₁	—	2	8	no	—
52	#106×C₆₁	—	2	8	no	—
53	C₄⋊C₈×C₆₁	D₄	1	4	yes	—
54	#103×C₆₁	—	2	4	no	—
55	#103×C₆₁	—	2	8	no	—
56	#102×C₆₁	—	2	4	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	#41×C₆₁	—	4	4	no	—
63	C₂×Q₈×C₆₁	—	2	2	no	—
64	C₂×Q₈×C₆₁	—	2	1	no	—
65	C₂×Q₈×C₆₁	—	2	4	no	—
66	C₂×Q₈×C₆₁	—	4	2	no	—
67	C₂×Q₈×C₆₁	—	4	8	no	—
68	C₂×D₄×C₆₁	—	2	2	no	—
69	C₂×D₄×C₆₁	—	2	1	no	—
70	C₂×D₄×C₆₁	—	4	1	no	—
71	C₂²×C₄×C₆₁	—	4	2	no	—
72	C₂²×C₄×C₆₁	—	4	1	no	—
73	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
75	Q₁₆×C₆₁	—	4	16	no	—
76	C₂×C₈×C₆₁	—	2	4	no	—
77	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
78	C₂×C₈×C₆₁	C₂×D₄	1	8	yes	—
79	C₄⋊C₄×C₆₁	—	2	2	no	—
80	C₄⋊C₄×C₆₁	—	2	4	no	—
81	C₄⋊C₄×C₆₁	—	2	4	no	—
82	C₄⋊C₄×C₆₁	—	2	2	no	—
83	C₄⋊C₄×C₆₁	—	2	2	no	—
84	C₄⋊C₄×C₆₁	—	4	2	no	—
85	C₄⋊C₄×C₆₁	—	4	2	no	—
86	C₄⋊C₄×C₆₁	—	4	2	no	—
87	C₂²⋊C₄×C₆₁	—	4	1	no	—
88	C₂²⋊C₄×C₆₁	—	4	2	no	—
89	C₄²×C₆₁	—	4	1	no	—
90	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₆₁	C₂×D₄	1	2	yes	—
92	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
93	C₂³×C₆₁	—	4	1	no	—
94	C₂³×C₆₁	#127	1	1	yes	characteristic
95	Q₈×C₆₁	—	4	8	no	—
96	Q₈×C₆₁	—	4	2	no	—
97	Q₈×C₆₁	—	4	4	no	—
98	Q₈×C₆₁	—	8	4	no	—
99	D₄×C₆₁	—	4	2	no	—
100	D₄×C₆₁	—	4	4	no	—
101	D₄×C₆₁	—	8	2	no	—
102	C₂×C₄×C₆₁	—	2	4	no	—
103	C₂×C₄×C₆₁	—	4	2	no	—
104	C₂×C₄×C₆₁	—	4	1	no	—
105	C₂×C₄×C₆₁	—	4	4	no	—
106	C₂×C₄×C₆₁	—	8	4	no	—
107	C₂×C₄×C₆₁	—	8	2	no	—
108	C₂×C₄×C₆₁	#127	1	2	yes	—
109	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
110	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	4	yes	—
111	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
113	C₈×C₆₁	—	2	4	no	—
114	C₈×C₆₁	—	2	8	no	—
115	#19699	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂²×C₆₁	—	2	2	no	—
117	C₂²×C₆₁	—	8	1	no	—
118	C₂²×C₆₁	—	8	2	no	—
119	C₂²×C₆₁	#496	1	2	yes	—
120	C₂²×C₆₁	C₂×#127	1	1	yes	characteristic
121	C₄×C₆₁	—	2	2	no	—
122	C₄×C₆₁	—	8	1	no	—
123	C₄×C₆₁	—	8	4	no	—
124	C₄×C₆₁	—	8	2	no	—
125	C₄×C₆₁	#496	1	2	yes	—
126	C₄×C₆₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
127	C₄×C₆₁	#578	1	4	yes	—
128	#3229	C₂×C₆₁	1	2	yes	complemented
129	#3226	C₂×C₆₁	1	2	yes	—
130	C₂×#494	C₂×C₆₁	1	1	yes	characteristic, complemented
131	#3131	C₂×C₆₁	1	1	yes	characteristic
132	#1723	C₂×C₆₁	1	4	yes	complemented
133	#1720	C₂×C₆₁	1	4	yes	complemented
134	#1452	C₂×C₆₁	1	1	yes	characteristic, complemented
135	C₂×C₆₁	—	2	2	no	—
136	C₂×C₆₁	—	16	1	no	—
137	C₂×C₆₁	C₂×#496	1	1	yes	characteristic
138	C₂×C₆₁	#3156	1	2	yes	—
139	C₂²×Q₁₆	—	2	2	no	—
140	#533	C₂²×C₆₁	1	1	yes	characteristic
141	#532	C₂²×C₆₁	1	1	yes	characteristic
142	C₂×C₄⋊Q₈	C₂²×C₆₁	1	2	yes	—
143	#500	C₂²×C₆₁	1	4	yes	—
144	#498	C₂²×C₆₁	1	4	yes	—
145	#494	—	2	2	no	—
146	#494	C₂²×C₆₁	1	4	yes	—
147	#488	—	2	2	no	—
148	#487	C₂²×C₆₁	1	2	yes	—
149	#487	C₂²×C₆₁	1	4	yes	—
150	#486	C₂²×C₆₁	1	2	yes	—
151	#486	C₂²×C₆₁	1	4	yes	—
152	#483	—	2	4	no	—
153	#480	—	2	4	no	—
154	#475	—	2	4	no	—
155	#474	—	2	4	no	—
156	#431	C₂²×C₆₁	1	2	yes	—
157	#416	—	2	2	no	—
158	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
159	#322	C₂²×C₆₁	1	4	yes	—
160	C₆₁	#19699	1	1	yes	characteristic, direct product factor, Sylow
161	C₂×#41	—	2	1	no	—
162	C₂²×Q₈	—	2	2	no	—
163	C₂×Q₁₆	—	2	8	no	—
164	C₂×Q₁₆	—	4	8	no	—
165	C₂²×C₈	D₄×C₆₁	1	2	yes	—
166	C₄⋊Q₈	—	2	2	no	—
167	C₄⋊Q₈	—	2	2	no	—
168	C₄⋊Q₈	—	2	2	no	—
169	C₄⋊Q₈	C₂³×C₆₁	1	4	yes	—
170	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
171	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
172	#127	C₂³×C₆₁	1	1	yes	characteristic
173	#125	—	2	1	no	—
174	#124	—	2	1	no	—
175	C₂²⋊Q₈	—	2	2	no	—
176	#121	—	2	2	no	—
177	C₄×Q₈	—	2	2	no	—
178	C₄×D₄	—	2	2	no	—
179	#117	—	2	1	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
183	#107	—	2	8	no	—
184	#106	—	2	8	no	—
185	C₄⋊C₈	D₄×C₆₁	1	4	yes	—
186	#103	—	2	4	no	—
187	#103	—	2	8	no	—
188	#102	—	2	4	no	—
189	#102	—	2	8	no	—
190	C₂²⋊C₈	D₄×C₆₁	1	4	yes	—
191	C₄×C₈	D₄×C₆₁	1	2	yes	—
192	C₄×C₈	C₂³×C₆₁	1	2	yes	—
193	C₄×C₈	C₂³×C₆₁	1	4	yes	—
194	#41	—	4	4	no	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	—	2	1	no	—
197	C₂×Q₈	—	2	4	no	—
198	C₂×Q₈	—	4	2	no	—
199	C₂×Q₈	—	4	8	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	—	4	1	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	—	4	1	no	—
205	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
206	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
207	Q₁₆	—	4	16	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
210	C₂×C₈	C₂×D₄×C₆₁	1	8	yes	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₂²⋊C₄	—	4	1	no	—
220	C₂²⋊C₄	—	4	2	no	—
221	C₄²	—	4	1	no	—
222	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
223	C₄²	C₂×D₄×C₆₁	1	2	yes	—
224	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
225	C₂³	—	4	1	no	—
226	C₂³	#127×C₆₁	1	1	yes	characteristic
227	Q₈	—	4	8	no	—
228	Q₈	—	4	2	no	—
229	Q₈	—	4	4	no	—
230	Q₈	—	8	4	no	—
231	D₄	—	4	2	no	—
232	D₄	—	4	4	no	—
233	D₄	—	8	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	8	4	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	#127×C₆₁	1	2	yes	—
241	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
242	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	4	yes	—
243	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
244	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
245	C₈	—	2	4	no	—
246	C₈	—	2	8	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	8	1	no	—
249	C₂²	—	8	2	no	—
250	C₂²	C₆₁×#496	1	2	yes	—
251	C₂²	C₂×#127×C₆₁	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	8	1	no	—
254	C₄	—	8	4	no	—
255	C₄	—	8	2	no	—
256	C₄	C₆₁×#496	1	2	yes	—
257	C₄	C₂×AGL(1,ℤ/8)×C₆₁	1	2	yes	—
258	C₄	C₆₁×#578	1	4	yes	—
259	C₂	—	2	2	no	—
260	C₂	—	16	1	no	—
261	C₂	C₂×C₆₁×#496	1	1	yes	characteristic
262	C₂	C₆₁×#3156	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×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
1. C₆₁×#3229	C₂	semidirect
3. C₂×C₆₁×#494	C₂	semidirect
5. C₆₁×#1723	C₂	semidirect
6. C₆₁×#1720	C₂	semidirect
7. C₆₁×#1452	C₂	semidirect
115. #19699	C₆₁	direct (N × H)
128. #3229	C₂×C₆₁	semidirect
130. C₂×#494	C₂×C₆₁	semidirect
132. #1723	C₂×C₆₁	semidirect
133. #1720	C₂×C₆₁	semidirect
134. #1452	C₂×C₆₁	semidirect
160. C₆₁	#19699	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×#3229	2
C₆₁×#3226	2
C₂×C₆₁×#494	2
C₆₁×#3131	2
C₆₁×#1723	2
C₆₁×#1720	2
C₆₁×#1452	2
#19699	61

Subgroups — C61×#19699 #29523132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#19699 #29523132