Subgroups — C₆₁×#8264 #29511697

Order 15616.

462 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₆₁×#322	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈²×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₆₁×#1273	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#1273	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊C₈×C₆₁	C₄	1	2	yes	—
9	C₂×C₄⋊C₈×C₆₁	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₄³×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#336	C₄	1	2	yes	—
18	C₆₁×#336	C₄	1	2	yes	—
19	C₆₁×#336	C₄	1	2	yes	—
20	C₆₁×#336	C₄	1	2	yes	—
21	C₆₁×#322	—	2	2	no	—
22	C₈²×C₆₁	—	2	2	no	—
23	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₆₁	—	2	4	no	—
37	C₄⋊C₈×C₆₁	C₈	1	8	yes	complemented
38	C₄×C₈×C₆₁	—	2	1	no	—
39	C₄×C₈×C₆₁	—	2	1	no	—
40	C₄×C₈×C₆₁	—	2	1	no	—
41	C₄×C₈×C₆₁	—	2	1	no	—
42	C₄×C₈×C₆₁	—	2	1	no	—
43	C₄×C₈×C₆₁	—	2	1	no	—
44	C₄×C₈×C₆₁	—	2	1	no	—
45	C₄×C₈×C₆₁	—	2	1	no	—
46	C₄×C₈×C₆₁	—	2	1	no	—
47	C₄×C₈×C₆₁	—	2	1	no	—
48	C₄×C₈×C₆₁	D₄	1	2	yes	—
49	C₄×C₈×C₆₁	Q₈	1	2	yes	—
50	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₆₁	—	2	2	no	—
58	C₂×C₈×C₆₁	—	2	2	no	—
59	C₂×C₈×C₆₁	—	2	2	no	—
60	C₂×C₈×C₆₁	—	2	2	no	—
61	C₂×C₈×C₆₁	—	2	2	no	—
62	C₂×C₈×C₆₁	—	2	2	no	—
63	C₂×C₈×C₆₁	—	2	2	no	—
64	C₂×C₈×C₆₁	—	2	2	no	—
65	C₂×C₈×C₆₁	—	2	2	no	—
66	C₂×C₈×C₆₁	—	2	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	—	2	2	no	—
71	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
72	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
73	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
74	C₂×C₈×C₆₁	M₁₆	1	4	yes	—
75	C₄²×C₆₁	—	2	2	no	—
76	C₄²×C₆₁	—	2	2	no	—
77	C₄²×C₆₁	—	2	2	no	—
78	C₄²×C₆₁	—	2	2	no	—
79	C₄²×C₆₁	—	2	2	no	—
80	C₄²×C₆₁	C₂×C₈	1	4	yes	—
81	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
83	C₄²×C₆₁	#41	1	1	yes	characteristic
84	C₄²×C₆₁	#41	1	1	yes	characteristic
85	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
86	C₂×C₄×C₆₁	—	2	1	no	—
87	C₂×C₄×C₆₁	—	2	1	no	—
88	C₂×C₄×C₆₁	—	2	1	no	—
89	C₂×C₄×C₆₁	—	2	1	no	—
90	C₂×C₄×C₆₁	—	2	1	no	—
91	C₂×C₄×C₆₁	—	2	1	no	—
92	C₂×C₄×C₆₁	—	2	1	no	—
93	C₂×C₄×C₆₁	—	2	1	no	—
94	C₂×C₄×C₆₁	—	2	1	no	—
95	C₂×C₄×C₆₁	—	2	1	no	—
96	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
97	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
98	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
105	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
106	C₂×C₄×C₆₁	#131	1	2	yes	—
107	C₂×C₄×C₆₁	#131	1	2	yes	—
108	C₂×C₄×C₆₁	#131	1	2	yes	—
109	C₂×C₄×C₆₁	#131	1	2	yes	—
110	C₈×C₆₁	—	2	4	no	—
111	C₈×C₆₁	—	2	4	no	—
112	C₈×C₆₁	—	2	4	no	—
113	C₈×C₆₁	—	2	4	no	—
114	C₈×C₆₁	—	4	2	no	—
115	C₈×C₆₁	—	4	2	no	—
116	#8264	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
118	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
119	C₂²×C₆₁	C₄×M₁₆	1	1	yes	characteristic
120	C₂²×C₆₁	#405	1	1	yes	characteristic
121	C₂²×C₆₁	#405	1	1	yes	characteristic
122	C₂²×C₆₁	#405	1	1	yes	characteristic
123	C₂²×C₆₁	#405	1	1	yes	characteristic
124	C₄×C₆₁	—	2	2	no	—
125	C₄×C₆₁	—	2	2	no	—
126	C₄×C₆₁	—	4	1	no	—
127	C₄×C₆₁	—	4	1	no	—
128	C₄×C₆₁	#424	1	2	yes	—
129	C₄×C₆₁	#433	1	2	yes	—
130	C₄×C₆₁	C₈×D₄	1	2	yes	—
131	C₄×C₆₁	#435	1	2	yes	—
132	C₄×C₆₁	C₈×Q₈	1	2	yes	—
133	C₄×C₆₁	#446	1	2	yes	—
134	C₄×C₄⋊C₈	C₂×C₆₁	1	1	yes	characteristic
135	C₄×C₄⋊C₈	C₂×C₆₁	1	1	yes	characteristic
136	#1744	C₂×C₆₁	1	1	yes	characteristic
137	C₂×#322	C₂×C₆₁	1	1	yes	characteristic
138	C₂×C₈²	C₂×C₆₁	1	1	yes	characteristic
139	#1273	C₂×C₆₁	1	1	yes	characteristic
140	#1273	C₂×C₆₁	1	1	yes	characteristic
141	C₂×C₆₁	C₈×M₁₆	1	1	yes	characteristic
142	C₂×C₆₁	#1450	1	1	yes	characteristic
143	C₂×C₆₁	#1453	1	1	yes	characteristic
144	C₂×C₆₁	#1454	1	1	yes	characteristic
145	C₂×C₆₁	#1768	1	1	yes	characteristic
146	C₂×C₆₁	C₈×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₆₁	#1771	1	1	yes	characteristic
148	C₂×C₄⋊C₈	C₄×C₆₁	1	2	yes	—
149	C₂×C₄⋊C₈	C₄×C₆₁	1	2	yes	—
150	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
151	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
154	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
156	C₄³	C₂²×C₆₁	1	1	yes	characteristic
157	#336	C₄×C₆₁	1	2	yes	—
158	#336	C₄×C₆₁	1	2	yes	—
159	#336	C₄×C₆₁	1	2	yes	—
160	#336	C₄×C₆₁	1	2	yes	—
161	#322	—	2	2	no	—
162	C₈²	—	2	2	no	—
163	C₆₁	#8264	1	1	yes	characteristic, direct product factor, Sylow
164	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
165	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
166	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
167	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
168	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
169	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
170	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
171	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
172	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
173	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
174	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
175	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
176	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
177	C₄⋊C₈	—	2	4	no	—
178	C₄⋊C₈	C₈×C₆₁	1	8	yes	complemented
179	C₄×C₈	—	2	1	no	—
180	C₄×C₈	—	2	1	no	—
181	C₄×C₈	—	2	1	no	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	—	2	1	no	—
185	C₄×C₈	—	2	1	no	—
186	C₄×C₈	—	2	1	no	—
187	C₄×C₈	—	2	1	no	—
188	C₄×C₈	—	2	1	no	—
189	C₄×C₈	D₄×C₆₁	1	2	yes	—
190	C₄×C₈	Q₈×C₆₁	1	2	yes	—
191	C₂²×C₄	C₄²×C₆₁	1	1	yes	characteristic
192	C₂²×C₄	C₄²×C₆₁	1	1	yes	characteristic
193	C₂²×C₄	C₄²×C₆₁	1	1	yes	characteristic
194	C₂²×C₄	C₄²×C₆₁	1	1	yes	characteristic
195	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
196	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
197	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	C₄⋊C₄×C₆₁	1	4	yes	—
213	C₂×C₈	C₂×C₈×C₆₁	1	4	yes	—
214	C₂×C₈	C₂×C₈×C₆₁	1	4	yes	—
215	C₂×C₈	M₁₆×C₆₁	1	4	yes	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	C₂×C₈×C₆₁	1	4	yes	—
222	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
223	C₄²	C₂×Q₈×C₆₁	1	1	yes	characteristic
224	C₄²	#41×C₆₁	1	1	yes	characteristic
225	C₄²	#41×C₆₁	1	1	yes	characteristic
226	C₂³	C₂×C₄²×C₆₁	1	1	yes	characteristic
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	C₄×C₈×C₆₁	1	2	yes	—
238	C₂×C₄	C₄×C₈×C₆₁	1	2	yes	—
239	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
240	C₂×C₄	#117×C₆₁	1	1	yes	characteristic
241	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
242	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
243	C₂×C₄	C₄×Q₈×C₆₁	1	1	yes	characteristic
244	C₂×C₄	C₄×Q₈×C₆₁	1	1	yes	characteristic
245	C₂×C₄	C₂²×C₈×C₆₁	1	2	yes	—
246	C₂×C₄	C₂×M₁₆×C₆₁	1	2	yes	—
247	C₂×C₄	#131×C₆₁	1	2	yes	—
248	C₂×C₄	#131×C₆₁	1	2	yes	—
249	C₂×C₄	#131×C₆₁	1	2	yes	—
250	C₂×C₄	#131×C₆₁	1	2	yes	—
251	C₈	—	2	4	no	—
252	C₈	—	2	4	no	—
253	C₈	—	2	4	no	—
254	C₈	—	2	4	no	—
255	C₈	—	4	2	no	—
256	C₈	—	4	2	no	—
257	C₂²	C₄×C₄⋊C₄×C₆₁	1	1	yes	characteristic
258	C₂²	C₂×C₄×C₈×C₆₁	1	1	yes	characteristic
259	C₂²	C₄×M₁₆×C₆₁	1	1	yes	characteristic
260	C₂²	C₆₁×#405	1	1	yes	characteristic
261	C₂²	C₆₁×#405	1	1	yes	characteristic
262	C₂²	C₆₁×#405	1	1	yes	characteristic
263	C₂²	C₆₁×#405	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	4	1	no	—
267	C₄	—	4	1	no	—
268	C₄	C₆₁×#424	1	2	yes	—
269	C₄	C₆₁×#433	1	2	yes	—
270	C₄	C₈×D₄×C₆₁	1	2	yes	—
271	C₄	C₆₁×#435	1	2	yes	—
272	C₄	C₈×Q₈×C₆₁	1	2	yes	—
273	C₄	C₆₁×#446	1	2	yes	—
274	C₂	C₈×M₁₆×C₆₁	1	1	yes	characteristic
275	C₂	C₆₁×#1450	1	1	yes	characteristic
276	C₂	C₆₁×#1453	1	1	yes	characteristic
277	C₂	C₆₁×#1454	1	1	yes	characteristic
278	C₂	C₆₁×#1768	1	1	yes	characteristic
279	C₂	C₈×C₄⋊C₄×C₆₁	1	1	yes	characteristic
280	C₂	C₆₁×#1771	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²×C₆₁	C₂³
commutator	C₂²	C₂×C₄×C₈×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
socle, prime radical	C₂³×C₆₁	C₂×C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
37. C₄⋊C₈×C₆₁	C₈	semidirect
116. #8264	C₆₁	direct (N × H)
163. C₆₁	#8264	direct (N × H)
178. C₄⋊C₈	C₈×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊C₈×C₆₁	2
C₆₁×#1744	2
C₂×C₆₁×#322	2
C₂×C₈²×C₆₁	2
C₆₁×#1273	2
#8264	61

Subgroups — C61×#8264 #29511697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#8264 #29511697