Subgroups — C₆₁×#12934 #29516367

Order 15616.

394 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2912	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2155	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×#1814	C₂	1	1	yes	maximal, characteristic
6	C₆₁×#1386	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
9	C₆₁×#520	—	2	1	no	—
10	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
11	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
12	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
13	C₆₁×#428	—	2	1	no	—
14	C₆₁×#428	—	2	2	no	—
15	C₆₁×#427	—	2	1	no	—
16	C₆₁×#427	C₂²	1	1	yes	characteristic
17	C₆₁×#427	C₂²	1	1	yes	characteristic
18	C₂×#106×C₆₁	—	2	1	no	—
19	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
20	C₆₁×#421	—	2	1	no	—
21	C₆₁×#365	—	2	2	no	—
22	C₆₁×#365	—	2	1	no	—
23	C₆₁×#365	—	2	1	no	—
24	C₆₁×#350	C₄	1	4	yes	complemented
25	C₆₁×#344	—	2	1	no	—
26	C₂×#41×C₆₁	D₄	1	1	yes	characteristic
27	#131×C₆₁	—	2	1	no	—
28	#131×C₆₁	—	2	1	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	1	no	—
33	C₂×M₁₆×C₆₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₁	—	2	1	no	—
35	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
36	C₄×Q₈×C₆₁	—	4	1	no	—
37	C₄×D₄×C₆₁	—	4	2	no	—
38	C₄×D₄×C₆₁	—	4	1	no	—
39	#117×C₆₁	—	2	2	no	—
40	#117×C₆₁	—	2	1	no	—
41	#117×C₆₁	—	2	1	no	—
42	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
44	M₃₂×C₆₁	—	2	1	no	—
45	M₃₂×C₆₁	—	2	1	no	—
46	M₃₂×C₆₁	—	2	1	no	—
47	M₃₂×C₆₁	C₂×C₄	1	2	yes	complemented
48	C₂×C₁₆×C₆₁	—	2	1	no	—
49	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	complemented
50	#108×C₆₁	—	2	1	no	—
51	#108×C₆₁	—	2	1	no	—
52	#107×C₆₁	—	2	2	no	—
53	#107×C₆₁	—	2	1	no	—
54	#107×C₆₁	—	2	1	no	—
55	#106×C₆₁	—	2	2	no	—
56	#106×C₆₁	—	2	1	no	—
57	#106×C₆₁	—	2	1	no	—
58	#106×C₆₁	D₄	1	1	yes	characteristic
59	#106×C₆₁	D₄	1	2	yes	—
60	#106×C₆₁	D₄	1	1	yes	characteristic
61	#104×C₆₁	—	4	2	no	—
62	#41×C₆₁	—	2	1	no	—
63	#41×C₆₁	—	2	1	no	—
64	#41×C₆₁	C₄⋊C₄	1	2	yes	—
65	#41×C₆₁	SD₁₆	1	2	yes	—
66	C₂×Q₈×C₆₁	SD₁₆	1	1	yes	characteristic
67	C₂×D₄×C₆₁	—	2	1	no	—
68	C₂×D₄×C₆₁	SD₁₆	1	1	yes	characteristic
69	C₂²×C₄×C₆₁	—	2	1	no	—
70	C₂²×C₄×C₆₁	—	4	2	no	—
71	C₂²×C₄×C₆₁	—	4	1	no	—
72	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
73	M₁₆×C₆₁	—	2	1	no	—
74	M₁₆×C₆₁	—	2	2	no	—
75	M₁₆×C₆₁	—	2	2	no	—
76	M₁₆×C₆₁	—	4	1	no	—
77	M₁₆×C₆₁	—	4	1	no	—
78	M₁₆×C₆₁	C₄⋊C₄	1	2	yes	—
79	C₂×C₈×C₆₁	—	2	1	no	—
80	C₂×C₈×C₆₁	—	2	2	no	—
81	C₂×C₈×C₆₁	—	2	2	no	—
82	C₂×C₈×C₆₁	—	4	1	no	—
83	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
88	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
89	C₄⋊C₄×C₆₁	—	2	1	no	—
90	C₄⋊C₄×C₆₁	—	2	2	no	—
91	C₄⋊C₄×C₆₁	—	2	1	no	—
92	C₄⋊C₄×C₆₁	—	4	2	no	—
93	C₄⋊C₄×C₆₁	—	4	1	no	—
94	C₄⋊C₄×C₆₁	—	4	1	no	—
95	C₂²⋊C₄×C₆₁	—	4	2	no	—
96	C₂²⋊C₄×C₆₁	—	4	1	no	—
97	C₂²⋊C₄×C₆₁	—	4	1	no	—
98	C₄²×C₆₁	—	4	2	no	—
99	C₄²×C₆₁	—	4	1	no	—
100	C₄²×C₆₁	—	4	1	no	—
101	C₁₆×C₆₁	—	2	2	no	—
102	C₁₆×C₆₁	—	2	2	no	—
103	C₂³×C₆₁	—	2	1	no	—
104	C₂³×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
105	Q₈×C₆₁	—	2	1	no	—
106	Q₈×C₆₁	#106	1	2	yes	—
107	D₄×C₆₁	—	2	2	no	—
108	D₄×C₆₁	—	2	2	no	—
109	D₄×C₆₁	—	2	1	no	—
110	D₄×C₆₁	#106	1	2	yes	—
111	C₂×C₄×C₆₁	—	2	1	no	—
112	C₂×C₄×C₆₁	—	2	2	no	—
113	C₂×C₄×C₆₁	—	2	2	no	—
114	C₂×C₄×C₆₁	—	4	1	no	—
115	C₂×C₄×C₆₁	—	4	2	no	—
116	C₂×C₄×C₆₁	—	4	1	no	—
117	C₂×C₄×C₆₁	—	8	1	no	—
118	C₂×C₄×C₆₁	—	16	1	no	—
119	C₂×C₄×C₆₁	—	16	2	no	—
120	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
122	C₂×C₄×C₆₁	#123	1	1	yes	characteristic
123	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
126	C₈×C₆₁	—	2	2	no	—
127	C₈×C₆₁	—	8	1	no	—
128	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
129	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
130	C₈×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
131	C₈×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
132	#12934	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂²×C₆₁	—	2	2	no	—
134	C₂²×C₆₁	—	4	1	no	—
135	C₂²×C₆₁	—	4	2	no	—
136	C₂²×C₆₁	#428	1	1	yes	characteristic
137	C₂²×C₆₁	#450	1	1	yes	characteristic
138	C₂²×C₆₁	#481	1	1	yes	characteristic
139	C₄×C₆₁	—	2	2	no	—
140	C₄×C₆₁	—	8	1	no	—
141	C₄×C₆₁	—	8	2	no	—
142	C₄×C₆₁	—	8	1	no	—
143	C₄×C₆₁	—	16	1	no	—
144	C₄×C₆₁	#385	1	1	yes	characteristic
145	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
146	C₄×C₆₁	#448	1	1	yes	characteristic
147	C₄×C₆₁	#483	1	1	yes	characteristic
148	#2912	C₂×C₆₁	1	1	yes	characteristic
149	#2155	C₂×C₆₁	1	1	yes	characteristic, complemented
150	C₂×#365	C₂×C₆₁	1	1	yes	characteristic, complemented
151	#2117	C₂×C₆₁	1	1	yes	characteristic
152	#1814	C₂×C₆₁	1	1	yes	characteristic
153	#1386	C₂×C₆₁	1	1	yes	characteristic, complemented
154	#1384	C₂×C₆₁	1	1	yes	characteristic, complemented
155	C₂×C₆₁	—	2	1	no	—
156	C₂×C₆₁	—	2	1	no	—
157	C₂×C₆₁	—	2	1	no	—
158	C₂×C₆₁	—	4	2	no	—
159	C₂×C₆₁	#1853	1	1	yes	characteristic
160	C₂×#131	C₂²×C₆₁	1	1	yes	characteristic
161	#520	—	2	1	no	—
162	C₂×M₃₂	C₂²×C₆₁	1	1	yes	characteristic
163	C₂×M₃₂	C₂²×C₆₁	1	1	yes	characteristic
164	C₂×#108	C₂²×C₆₁	1	1	yes	characteristic
165	#428	—	2	1	no	—
166	#428	—	2	2	no	—
167	#427	—	2	1	no	—
168	#427	C₂²×C₆₁	1	1	yes	characteristic
169	#427	C₂²×C₆₁	1	1	yes	characteristic
170	C₂×#106	—	2	1	no	—
171	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
172	#421	—	2	1	no	—
173	#365	—	2	2	no	—
174	#365	—	2	1	no	—
175	#365	—	2	1	no	—
176	#350	C₄×C₆₁	1	4	yes	complemented
177	#344	—	2	1	no	—
178	C₆₁	#12934	1	1	yes	characteristic, direct product factor, Sylow
179	C₂×#41	D₄×C₆₁	1	1	yes	characteristic
180	#131	—	2	1	no	—
181	#131	—	2	1	no	—
182	#131	C₂×C₄×C₆₁	1	2	yes	—
183	#131	Q₈×C₆₁	1	2	yes	—
184	C₂×M₁₆	—	2	1	no	—
185	C₂×M₁₆	—	2	1	no	—
186	C₂×M₁₆	D₄×C₆₁	1	1	yes	characteristic
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
189	C₄×Q₈	—	4	1	no	—
190	C₄×D₄	—	4	2	no	—
191	C₄×D₄	—	4	1	no	—
192	#117	—	2	2	no	—
193	#117	—	2	1	no	—
194	#117	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	M₃₂	—	2	1	no	—
198	M₃₂	—	2	1	no	—
199	M₃₂	—	2	1	no	—
200	M₃₂	C₂×C₄×C₆₁	1	2	yes	complemented
201	C₂×C₁₆	—	2	1	no	—
202	C₂×C₁₆	C₂×C₄×C₆₁	1	2	yes	complemented
203	#108	—	2	1	no	—
204	#108	—	2	1	no	—
205	#107	—	2	2	no	—
206	#107	—	2	1	no	—
207	#107	—	2	1	no	—
208	#106	—	2	2	no	—
209	#106	—	2	1	no	—
210	#106	—	2	1	no	—
211	#106	D₄×C₆₁	1	1	yes	characteristic
212	#106	D₄×C₆₁	1	2	yes	—
213	#106	D₄×C₆₁	1	1	yes	characteristic
214	#104	—	4	2	no	—
215	#41	—	2	1	no	—
216	#41	—	2	1	no	—
217	#41	C₄⋊C₄×C₆₁	1	2	yes	—
218	#41	SD₁₆×C₆₁	1	2	yes	—
219	C₂×Q₈	SD₁₆×C₆₁	1	1	yes	characteristic
220	C₂×D₄	—	2	1	no	—
221	C₂×D₄	SD₁₆×C₆₁	1	1	yes	characteristic
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
226	M₁₆	—	2	1	no	—
227	M₁₆	—	2	2	no	—
228	M₁₆	—	2	2	no	—
229	M₁₆	—	4	1	no	—
230	M₁₆	—	4	1	no	—
231	M₁₆	C₄⋊C₄×C₆₁	1	2	yes	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
237	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
238	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
239	C₂×C₈	C₂×Q₈×C₆₁	1	1	yes	characteristic
240	C₂×C₈	#41×C₆₁	1	1	yes	characteristic
241	C₂×C₈	#41×C₆₁	1	1	yes	characteristic
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	1	no	—
247	C₄⋊C₄	—	4	1	no	—
248	C₂²⋊C₄	—	4	2	no	—
249	C₂²⋊C₄	—	4	1	no	—
250	C₂²⋊C₄	—	4	1	no	—
251	C₄²	—	4	2	no	—
252	C₄²	—	4	1	no	—
253	C₄²	—	4	1	no	—
254	C₁₆	—	2	2	no	—
255	C₁₆	—	2	2	no	—
256	C₂³	—	2	1	no	—
257	C₂³	AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
258	Q₈	—	2	1	no	—
259	Q₈	#106×C₆₁	1	2	yes	—
260	D₄	—	2	2	no	—
261	D₄	—	2	2	no	—
262	D₄	—	2	1	no	—
263	D₄	#106×C₆₁	1	2	yes	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	—	16	1	no	—
272	C₂×C₄	—	16	2	no	—
273	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
274	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
275	C₂×C₄	#123×C₆₁	1	1	yes	characteristic
276	C₂×C₄	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
277	C₂×C₄	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
278	C₂×C₄	#137×C₆₁	1	1	yes	characteristic
279	C₈	—	2	2	no	—
280	C₈	—	8	1	no	—
281	C₈	C₄×D₄×C₆₁	1	1	yes	characteristic
282	C₈	C₄×D₄×C₆₁	1	1	yes	characteristic
283	C₈	C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
284	C₈	C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
285	C₂²	—	2	2	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	4	2	no	—
288	C₂²	C₆₁×#428	1	1	yes	characteristic
289	C₂²	C₆₁×#450	1	1	yes	characteristic
290	C₂²	C₆₁×#481	1	1	yes	characteristic
291	C₄	—	2	2	no	—
292	C₄	—	8	1	no	—
293	C₄	—	8	2	no	—
294	C₄	—	8	1	no	—
295	C₄	—	16	1	no	—
296	C₄	C₆₁×#385	1	1	yes	characteristic
297	C₄	C₂×#106×C₆₁	1	1	yes	characteristic
298	C₄	C₆₁×#448	1	1	yes	characteristic
299	C₄	C₆₁×#483	1	1	yes	characteristic
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	4	2	no	—
304	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. C₆₁×#2155	C₂	semidirect
3. C₂×C₆₁×#365	C₂	semidirect
6. C₆₁×#1386	C₂	semidirect
7. C₆₁×#1384	C₂	semidirect
24. C₆₁×#350	C₄	semidirect
47. M₃₂×C₆₁	C₂×C₄	semidirect
49. C₂×C₁₆×C₆₁	C₂×C₄	semidirect
132. #12934	C₆₁	direct (N × H)
149. #2155	C₂×C₆₁	semidirect
150. C₂×#365	C₂×C₆₁	semidirect
153. #1386	C₂×C₆₁	semidirect
154. #1384	C₂×C₆₁	semidirect
176. #350	C₄×C₆₁	semidirect
178. C₆₁	#12934	direct (N × H)
200. M₃₂	C₂×C₄×C₆₁	semidirect
202. C₂×C₁₆	C₂×C₄×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#2912	2
C₆₁×#2155	2
C₂×C₆₁×#365	2
C₆₁×#2117	2
C₆₁×#1814	2
C₆₁×#1386	2
C₆₁×#1384	2
#12934	61

Subgroups — C61×#12934 #29516367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#12934 #29516367