Subgroups — C₆₁×#8682 #29512115

Order 15616.

430 conjugacy classes of subgroups.

The subgroup lattice has 300 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×#1848	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	C₂×#108×C₆₁	—	2	1	no	—
13	C₂×#108×C₆₁	—	2	1	no	—
14	C₆₁×#423	C₂²	1	2	yes	—
15	C₆₁×#423	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
17	C₆₁×#408	—	2	1	no	—
18	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
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	2	no	—
25	#131×C₆₁	—	4	2	no	—
26	C₂×M₁₆×C₆₁	—	2	1	no	—
27	C₂×M₁₆×C₆₁	—	2	2	no	—
28	C₂×M₁₆×C₆₁	—	2	2	no	—
29	C₂²×C₈×C₆₁	—	2	2	no	—
30	C₂²×C₈×C₆₁	—	2	2	no	—
31	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
33	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
34	#127×C₆₁	D₄	1	1	yes	characteristic
35	#124×C₆₁	—	2	1	no	—
36	#121×C₆₁	—	2	1	no	—
37	#121×C₆₁	Q₈	1	2	yes	—
38	C₄×D₄×C₆₁	—	2	1	no	—
39	C₄×D₄×C₆₁	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
41	#108×C₆₁	—	4	2	no	—
42	#108×C₆₁	—	4	2	no	—
43	C₄⋊C₈×C₆₁	—	2	1	no	—
44	C₄⋊C₈×C₆₁	—	2	2	no	—
45	C₄⋊C₈×C₆₁	—	2	2	no	—
46	C₄⋊C₈×C₆₁	—	2	2	no	—
47	C₄⋊C₈×C₆₁	—	2	2	no	—
48	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
49	C₂²⋊C₈×C₆₁	—	2	2	no	—
50	C₂²⋊C₈×C₆₁	—	2	2	no	—
51	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
54	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
55	#41×C₆₁	—	4	1	no	—
56	#41×C₆₁	—	4	1	no	—
57	#41×C₆₁	—	4	1	no	—
58	#41×C₆₁	—	4	1	no	—
59	C₂×Q₈×C₆₁	—	2	1	no	—
60	C₂×D₄×C₆₁	—	2	2	no	—
61	C₂×D₄×C₆₁	—	2	1	no	—
62	C₂×D₄×C₆₁	—	2	1	no	—
63	C₂×D₄×C₆₁	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₆₁	—	2	2	no	—
65	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
68	M₁₆×C₆₁	—	4	1	no	—
69	M₁₆×C₆₁	—	4	1	no	—
70	M₁₆×C₆₁	—	4	4	no	—
71	M₁₆×C₆₁	—	4	2	no	—
72	M₁₆×C₆₁	—	4	2	no	—
73	C₂×C₈×C₆₁	—	2	1	no	—
74	C₂×C₈×C₆₁	—	2	1	no	—
75	C₂×C₈×C₆₁	—	2	2	no	—
76	C₂×C₈×C₆₁	—	4	4	no	—
77	C₂×C₈×C₆₁	—	4	2	no	—
78	C₂×C₈×C₆₁	—	4	2	no	—
79	C₂×C₈×C₆₁	—	8	2	no	—
80	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₆₁	D₈	1	2	yes	—
85	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
86	C₄⋊C₄×C₆₁	—	2	1	no	—
87	C₄⋊C₄×C₆₁	C₄⋊C₄	1	2	yes	—
88	C₂²⋊C₄×C₆₁	—	2	2	no	—
89	C₂²⋊C₄×C₆₁	—	2	2	no	—
90	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₆₁	#41	1	1	yes	characteristic
94	C₂³×C₆₁	—	2	2	no	—
95	C₂³×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
96	Q₈×C₆₁	—	4	1	no	—
97	Q₈×C₆₁	—	4	1	no	—
98	D₄×C₆₁	—	4	2	no	—
99	D₄×C₆₁	—	4	2	no	—
100	D₄×C₆₁	—	4	2	no	—
101	D₄×C₆₁	—	4	1	no	—
102	D₄×C₆₁	—	4	1	no	—
103	D₄×C₆₁	—	4	2	no	—
104	C₂×C₄×C₆₁	—	2	2	no	—
105	C₂×C₄×C₆₁	—	2	2	no	—
106	C₂×C₄×C₆₁	—	4	1	no	—
107	C₂×C₄×C₆₁	—	4	2	no	—
108	C₂×C₄×C₆₁	—	4	2	no	—
109	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
112	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
113	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
114	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
119	C₈×C₆₁	—	2	1	no	—
120	C₈×C₆₁	—	2	1	no	—
121	C₈×C₆₁	—	4	2	no	—
122	C₈×C₆₁	—	8	2	no	—
123	C₈×C₆₁	—	8	2	no	—
124	C₈×C₆₁	#102	1	2	yes	—
125	C₈×C₆₁	#102	1	2	yes	—
126	#8682	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂²×C₆₁	—	2	1	no	—
128	C₂²×C₆₁	—	2	1	no	—
129	C₂²×C₆₁	—	4	2	no	—
130	C₂²×C₆₁	—	4	2	no	—
131	C₂²×C₆₁	—	4	2	no	—
132	C₂²×C₆₁	#380	1	1	yes	characteristic
133	C₂²×C₆₁	#474	1	1	yes	characteristic
134	C₂²×C₆₁	#476	1	1	yes	characteristic
135	C₄×C₆₁	—	2	1	no	—
136	C₄×C₆₁	—	2	1	no	—
137	C₄×C₆₁	—	2	1	no	—
138	C₄×C₆₁	—	4	2	no	—
139	C₄×C₆₁	C₂×#102	1	1	yes	characteristic
140	C₄×C₆₁	#417	1	1	yes	characteristic
141	C₄×C₆₁	#459	1	1	yes	characteristic
142	C₄×C₆₁	#461	1	1	yes	characteristic
143	C₄×C₆₁	#471	1	2	yes	—
144	#2931	C₂×C₆₁	1	1	yes	characteristic
145	#1848	C₂×C₆₁	1	1	yes	characteristic, complemented
146	C₂×#334	C₂×C₆₁	1	1	yes	characteristic, complemented
147	#1547	C₂×C₆₁	1	2	yes	—
148	#1292	C₂×C₆₁	1	2	yes	complemented
149	C₂×C₆₁	—	2	1	no	—
150	C₂×C₆₁	—	2	1	no	—
151	C₂×C₆₁	—	8	2	no	—
152	C₂×C₆₁	#1802	1	1	yes	characteristic
153	C₂×C₆₁	#1813	1	1	yes	characteristic
154	C₂×C₆₁	#1815	1	1	yes	characteristic
155	C₂×#131	—	2	1	no	—
156	#532	C₂²×C₆₁	1	1	yes	characteristic
157	#436	—	2	1	no	—
158	#436	C₄×C₆₁	1	2	yes	—
159	C₈×D₄	—	2	1	no	—
160	C₈×D₄	C₄×C₆₁	1	2	yes	—
161	C₂×#108	—	2	1	no	—
162	C₂×#108	—	2	1	no	—
163	#423	C₂²×C₆₁	1	2	yes	—
164	#423	C₂²×C₆₁	1	1	yes	characteristic
165	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
166	#408	—	2	1	no	—
167	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
168	#334	—	2	1	no	—
169	#334	—	2	1	no	—
170	#332	—	2	2	no	—
171	#331	—	2	2	no	—
172	C₆₁	#8682	1	1	yes	characteristic, direct product factor, Sylow
173	C₂×#41	—	2	1	no	—
174	#131	—	4	2	no	—
175	#131	—	4	2	no	—
176	C₂×M₁₆	—	2	1	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
182	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
183	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
184	#127	D₄×C₆₁	1	1	yes	characteristic
185	#124	—	2	1	no	—
186	#121	—	2	1	no	—
187	#121	Q₈×C₆₁	1	2	yes	—
188	C₄×D₄	—	2	1	no	—
189	C₄×D₄	C₂×C₄×C₆₁	1	2	yes	—
190	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
191	#108	—	4	2	no	—
192	#108	—	4	2	no	—
193	C₄⋊C₈	—	2	1	no	—
194	C₄⋊C₈	—	2	2	no	—
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	—	2	2	no	—
198	C₄⋊C₈	C₂×C₄×C₆₁	1	2	yes	—
199	C₂²⋊C₈	—	2	2	no	—
200	C₂²⋊C₈	—	2	2	no	—
201	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
202	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
203	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
204	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
205	#41	—	4	1	no	—
206	#41	—	4	1	no	—
207	#41	—	4	1	no	—
208	#41	—	4	1	no	—
209	C₂×Q₈	—	2	1	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	1	no	—
212	C₂×D₄	—	2	1	no	—
213	C₂×D₄	C₄⋊C₄×C₆₁	1	2	yes	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
216	C₂²×C₄	C₂×Q₈×C₆₁	1	1	yes	characteristic
217	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
218	M₁₆	—	4	1	no	—
219	M₁₆	—	4	1	no	—
220	M₁₆	—	4	4	no	—
221	M₁₆	—	4	2	no	—
222	M₁₆	—	4	2	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	4	4	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	8	2	no	—
230	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
231	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
232	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
233	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
234	C₂×C₈	D₈×C₆₁	1	2	yes	—
235	C₂×C₈	SD₁₆×C₆₁	1	2	yes	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	C₄⋊C₄×C₆₁	1	2	yes	—
238	C₂²⋊C₄	—	2	2	no	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
241	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
242	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
243	C₄²	#41×C₆₁	1	1	yes	characteristic
244	C₂³	—	2	2	no	—
245	C₂³	C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
246	Q₈	—	4	1	no	—
247	Q₈	—	4	1	no	—
248	D₄	—	4	2	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	2	no	—
251	D₄	—	4	1	no	—
252	D₄	—	4	1	no	—
253	D₄	—	4	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
260	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
261	C₂×C₄	#117×C₆₁	1	1	yes	characteristic
262	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
263	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
264	C₂×C₄	C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
265	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
266	C₂×C₄	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
267	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
268	C₂×C₄	#137×C₆₁	1	1	yes	characteristic
269	C₈	—	2	1	no	—
270	C₈	—	2	1	no	—
271	C₈	—	4	2	no	—
272	C₈	—	8	2	no	—
273	C₈	—	8	2	no	—
274	C₈	#102×C₆₁	1	2	yes	—
275	C₈	#102×C₆₁	1	2	yes	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	C₆₁×#380	1	1	yes	characteristic
282	C₂²	C₆₁×#474	1	1	yes	characteristic
283	C₂²	C₆₁×#476	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	4	2	no	—
288	C₄	C₂×#102×C₆₁	1	1	yes	characteristic
289	C₄	C₆₁×#417	1	1	yes	characteristic
290	C₄	C₆₁×#459	1	1	yes	characteristic
291	C₄	C₆₁×#461	1	1	yes	characteristic
292	C₄	C₆₁×#471	1	2	yes	—
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	8	2	no	—
296	C₂	C₆₁×#1802	1	1	yes	characteristic
297	C₂	C₆₁×#1813	1	1	yes	characteristic
298	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×#1848	C₂	semidirect
3. C₂×C₆₁×#334	C₂	semidirect
5. C₆₁×#1292	C₂	semidirect
126. #8682	C₆₁	direct (N × H)
145. #1848	C₂×C₆₁	semidirect
146. C₂×#334	C₂×C₆₁	semidirect
148. #1292	C₂×C₆₁	semidirect
172. C₆₁	#8682	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×#1848	2
C₂×C₆₁×#334	2
C₆₁×#1547	2
C₆₁×#1292	2
#8682	61

Subgroups — C61×#8682 #29512115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#8682 #29512115