Subgroups — C₆₁×#12933 #29516366

Order 15616.

380 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2916	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₁×#365	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₆₁×#365	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2117	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1815	C₂	1	1	yes	maximal, characteristic
6	C₆₁×#1388	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1385	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#568	—	2	1	no	—
9	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
10	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
11	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×#430	—	2	1	no	—
13	C₆₁×#430	—	2	2	no	—
14	C₂×#108×C₆₁	—	2	2	no	—
15	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
19	C₆₁×#417	—	2	1	no	—
20	C₆₁×#365	—	2	2	no	—
21	C₆₁×#365	—	2	2	no	—
22	C₆₁×#350	C₄	1	4	yes	complemented
23	C₆₁×#343	—	2	1	no	—
24	C₂×#41×C₆₁	D₄	1	1	yes	characteristic
25	#131×C₆₁	—	2	1	no	—
26	#131×C₆₁	—	2	1	no	—
27	#131×C₆₁	—	4	2	no	—
28	#131×C₆₁	—	4	2	no	—
29	#131×C₆₁	C₂×C₄	1	2	yes	—
30	#131×C₆₁	Q₈	1	2	yes	—
31	C₂×M₁₆×C₆₁	—	2	1	no	—
32	C₂×M₁₆×C₆₁	—	2	4	no	—
33	C₂×M₁₆×C₆₁	—	2	1	no	—
34	C₂×M₁₆×C₆₁	—	2	1	no	—
35	C₂×M₁₆×C₆₁	—	2	1	no	—
36	C₂×M₁₆×C₆₁	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₁	—	2	1	no	—
38	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
40	M₃₂×C₆₁	—	2	1	no	—
41	M₃₂×C₆₁	—	2	1	no	—
42	M₃₂×C₆₁	—	2	1	no	—
43	M₃₂×C₆₁	C₂×C₄	1	2	yes	—
44	C₂×C₁₆×C₆₁	—	2	1	no	—
45	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
46	#108×C₆₁	—	2	4	no	—
47	#108×C₆₁	—	2	1	no	—
48	#108×C₆₁	—	2	1	no	—
49	#108×C₆₁	—	2	1	no	—
50	#108×C₆₁	—	2	1	no	—
51	#108×C₆₁	D₄	1	2	yes	—
52	#108×C₆₁	D₄	1	2	yes	—
53	#106×C₆₁	—	2	1	no	—
54	#106×C₆₁	—	2	1	no	—
55	#103×C₆₁	—	4	1	no	—
56	#102×C₆₁	—	4	1	no	—
57	#41×C₆₁	—	2	1	no	—
58	#41×C₆₁	—	2	1	no	—
59	#41×C₆₁	C₄⋊C₄	1	2	yes	—
60	#41×C₆₁	SD₁₆	1	2	yes	—
61	C₂×Q₈×C₆₁	SD₁₆	1	1	yes	characteristic
62	C₂×D₄×C₆₁	—	2	1	no	—
63	C₂×D₄×C₆₁	SD₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₆₁	—	2	1	no	—
65	C₂²×C₄×C₆₁	—	4	1	no	—
66	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
67	M₁₆×C₆₁	—	2	1	no	—
68	M₁₆×C₆₁	—	2	2	no	—
69	M₁₆×C₆₁	—	2	2	no	—
70	M₁₆×C₆₁	—	2	2	no	—
71	M₁₆×C₆₁	—	2	2	no	—
72	M₁₆×C₆₁	—	4	4	no	—
73	M₁₆×C₆₁	—	4	1	no	—
74	M₁₆×C₆₁	—	4	1	no	—
75	M₁₆×C₆₁	—	4	1	no	—
76	M₁₆×C₆₁	—	4	1	no	—
77	M₁₆×C₆₁	C₄⋊C₄	1	2	yes	—
78	C₂×C₈×C₆₁	—	2	1	no	—
79	C₂×C₈×C₆₁	—	2	2	no	—
80	C₂×C₈×C₆₁	—	2	2	no	—
81	C₂×C₈×C₆₁	—	4	4	no	—
82	C₂×C₈×C₆₁	—	4	1	no	—
83	C₂×C₈×C₆₁	—	4	1	no	—
84	C₂×C₈×C₆₁	—	4	1	no	—
85	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
90	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
91	C₄⋊C₄×C₆₁	—	4	1	no	—
92	C₄⋊C₄×C₆₁	—	4	1	no	—
93	C₁₆×C₆₁	—	2	2	no	—
94	C₁₆×C₆₁	—	2	2	no	—
95	C₂³×C₆₁	—	2	1	no	—
96	C₂³×C₆₁	#137	1	1	yes	characteristic
97	Q₈×C₆₁	—	2	1	no	—
98	Q₈×C₆₁	#106	1	2	yes	—
99	D₄×C₆₁	—	2	2	no	—
100	D₄×C₆₁	—	2	2	no	—
101	D₄×C₆₁	—	2	1	no	—
102	D₄×C₆₁	#106	1	2	yes	—
103	C₂×C₄×C₆₁	—	2	1	no	—
104	C₂×C₄×C₆₁	—	2	2	no	—
105	C₂×C₄×C₆₁	—	2	2	no	—
106	C₂×C₄×C₆₁	—	8	1	no	—
107	C₂×C₄×C₆₁	—	16	1	no	—
108	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
110	C₂×C₄×C₆₁	#123	1	1	yes	characteristic
111	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₈×C₆₁	—	2	2	no	—
115	C₈×C₆₁	—	4	2	no	—
116	C₈×C₆₁	—	4	2	no	—
117	C₈×C₆₁	—	8	1	no	—
118	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
119	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
120	C₈×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
121	C₈×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
122	#12933	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂²×C₆₁	—	2	2	no	—
124	C₂²×C₆₁	—	4	1	no	—
125	C₂²×C₆₁	—	8	1	no	—
126	C₂²×C₆₁	#428	1	1	yes	characteristic
127	C₂²×C₆₁	#448	1	1	yes	characteristic
128	C₂²×C₆₁	#483	1	1	yes	characteristic
129	C₄×C₆₁	—	2	2	no	—
130	C₄×C₆₁	—	16	1	no	—
131	C₄×C₆₁	#385	1	1	yes	characteristic
132	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
133	C₄×C₆₁	#450	1	1	yes	characteristic
134	C₄×C₆₁	#481	1	1	yes	characteristic
135	#2916	C₂×C₆₁	1	1	yes	characteristic
136	C₂×#365	C₂×C₆₁	1	1	yes	characteristic, complemented
137	C₂×#365	C₂×C₆₁	1	1	yes	characteristic, complemented
138	#2117	C₂×C₆₁	1	1	yes	characteristic
139	#1815	C₂×C₆₁	1	1	yes	characteristic
140	#1388	C₂×C₆₁	1	1	yes	characteristic, complemented
141	#1385	C₂×C₆₁	1	1	yes	characteristic, complemented
142	C₂×C₆₁	—	2	1	no	—
143	C₂×C₆₁	—	2	1	no	—
144	C₂×C₆₁	—	2	1	no	—
145	C₂×C₆₁	—	4	2	no	—
146	C₂×C₆₁	#1853	1	1	yes	characteristic
147	#568	—	2	1	no	—
148	C₂×#131	C₂²×C₆₁	1	1	yes	characteristic
149	C₂×M₃₂	C₂²×C₆₁	1	1	yes	characteristic
150	C₂×M₃₂	C₂²×C₆₁	1	1	yes	characteristic
151	#430	—	2	1	no	—
152	#430	—	2	2	no	—
153	C₂×#108	—	2	2	no	—
154	C₂×#108	C₂²×C₆₁	1	1	yes	characteristic
155	C₂×#108	C₂²×C₆₁	1	1	yes	characteristic
156	C₂×#108	C₂²×C₆₁	1	1	yes	characteristic
157	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
158	#417	—	2	1	no	—
159	#365	—	2	2	no	—
160	#365	—	2	2	no	—
161	#350	C₄×C₆₁	1	4	yes	complemented
162	#343	—	2	1	no	—
163	C₆₁	#12933	1	1	yes	characteristic, direct product factor, Sylow
164	C₂×#41	D₄×C₆₁	1	1	yes	characteristic
165	#131	—	2	1	no	—
166	#131	—	2	1	no	—
167	#131	—	4	2	no	—
168	#131	—	4	2	no	—
169	#131	C₂×C₄×C₆₁	1	2	yes	—
170	#131	Q₈×C₆₁	1	2	yes	—
171	C₂×M₁₆	—	2	1	no	—
172	C₂×M₁₆	—	2	4	no	—
173	C₂×M₁₆	—	2	1	no	—
174	C₂×M₁₆	—	2	1	no	—
175	C₂×M₁₆	—	2	1	no	—
176	C₂×M₁₆	D₄×C₆₁	1	1	yes	characteristic
177	C₂²×C₈	—	2	1	no	—
178	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
179	C₂×C₄⋊C₄	—	2	1	no	—
180	M₃₂	—	2	1	no	—
181	M₃₂	—	2	1	no	—
182	M₃₂	—	2	1	no	—
183	M₃₂	C₂×C₄×C₆₁	1	2	yes	—
184	C₂×C₁₆	—	2	1	no	—
185	C₂×C₁₆	C₂×C₄×C₆₁	1	2	yes	—
186	#108	—	2	4	no	—
187	#108	—	2	1	no	—
188	#108	—	2	1	no	—
189	#108	—	2	1	no	—
190	#108	—	2	1	no	—
191	#108	D₄×C₆₁	1	2	yes	—
192	#108	D₄×C₆₁	1	2	yes	—
193	#106	—	2	1	no	—
194	#106	—	2	1	no	—
195	#103	—	4	1	no	—
196	#102	—	4	1	no	—
197	#41	—	2	1	no	—
198	#41	—	2	1	no	—
199	#41	C₄⋊C₄×C₆₁	1	2	yes	—
200	#41	SD₁₆×C₆₁	1	2	yes	—
201	C₂×Q₈	SD₁₆×C₆₁	1	1	yes	characteristic
202	C₂×D₄	—	2	1	no	—
203	C₂×D₄	SD₁₆×C₆₁	1	1	yes	characteristic
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	4	1	no	—
206	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
207	M₁₆	—	2	1	no	—
208	M₁₆	—	2	2	no	—
209	M₁₆	—	2	2	no	—
210	M₁₆	—	2	2	no	—
211	M₁₆	—	2	2	no	—
212	M₁₆	—	4	4	no	—
213	M₁₆	—	4	1	no	—
214	M₁₆	—	4	1	no	—
215	M₁₆	—	4	1	no	—
216	M₁₆	—	4	1	no	—
217	M₁₆	C₄⋊C₄×C₆₁	1	2	yes	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	4	4	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
226	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
227	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
228	C₂×C₈	C₂×Q₈×C₆₁	1	1	yes	characteristic
229	C₂×C₈	#41×C₆₁	1	1	yes	characteristic
230	C₂×C₈	#41×C₆₁	1	1	yes	characteristic
231	C₄⋊C₄	—	4	1	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₁₆	—	2	2	no	—
234	C₁₆	—	2	2	no	—
235	C₂³	—	2	1	no	—
236	C₂³	#137×C₆₁	1	1	yes	characteristic
237	Q₈	—	2	1	no	—
238	Q₈	#106×C₆₁	1	2	yes	—
239	D₄	—	2	2	no	—
240	D₄	—	2	2	no	—
241	D₄	—	2	1	no	—
242	D₄	#106×C₆₁	1	2	yes	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	8	1	no	—
247	C₂×C₄	—	16	1	no	—
248	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
249	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
250	C₂×C₄	#123×C₆₁	1	1	yes	characteristic
251	C₂×C₄	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
252	C₂×C₄	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
253	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
254	C₈	—	2	2	no	—
255	C₈	—	4	2	no	—
256	C₈	—	4	2	no	—
257	C₈	—	8	1	no	—
258	C₈	C₄×D₄×C₆₁	1	1	yes	characteristic
259	C₈	C₄×D₄×C₆₁	1	1	yes	characteristic
260	C₈	C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
261	C₈	C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
262	C₂²	—	2	2	no	—
263	C₂²	—	4	1	no	—
264	C₂²	—	8	1	no	—
265	C₂²	C₆₁×#428	1	1	yes	characteristic
266	C₂²	C₆₁×#448	1	1	yes	characteristic
267	C₂²	C₆₁×#483	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	16	1	no	—
270	C₄	C₆₁×#385	1	1	yes	characteristic
271	C₄	C₂×#106×C₆₁	1	1	yes	characteristic
272	C₄	C₆₁×#450	1	1	yes	characteristic
273	C₄	C₆₁×#481	1	1	yes	characteristic
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	—	4	2	no	—
278	C₂	C₆₁×#1853	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₁	#1853
commutator	C₂×C₈	C₂²×C₄×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
prime radical	C₂×#41×C₆₁	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂×#131×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₆₁×#365	C₂	semidirect
3. C₂×C₆₁×#365	C₂	semidirect
6. C₆₁×#1388	C₂	semidirect
7. C₆₁×#1385	C₂	semidirect
22. C₆₁×#350	C₄	semidirect
122. #12933	C₆₁	direct (N × H)
136. C₂×#365	C₂×C₆₁	semidirect
137. C₂×#365	C₂×C₆₁	semidirect
140. #1388	C₂×C₆₁	semidirect
141. #1385	C₂×C₆₁	semidirect
161. #350	C₄×C₆₁	semidirect
163. C₆₁	#12933	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×#2916	2
C₂×C₆₁×#365	2
C₆₁×#2117	2
C₆₁×#1815	2
C₆₁×#1388	2
C₆₁×#1385	2
#12933	61

Subgroups — C61×#12933 #29516366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#12933 #29516366