Subgroups — C₆₁×#8663 #29512096

Order 15616.

374 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	C₂×C₆₁×#446	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#1848	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1564	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1487	C₂	1	2	yes	maximal
5	C₆₁×#1301	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
7	C₆₁×#446	—	2	1	no	—
8	C₆₁×#446	—	2	1	no	—
9	C₆₁×#446	C₄	1	2	yes	—
10	C₆₁×#446	C₄	1	2	yes	—
11	C₆₁×#431	C₂²	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	C₂×C₄⋊C₈×C₆₁	—	2	1	no	—
16	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×#97×C₆₁	—	2	1	no	—
18	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₆₁×#335	—	2	1	no	—
20	C₆₁×#334	—	2	1	no	—
21	C₆₁×Q₈⋊C₈	—	2	4	no	—
22	C₂²×Q₈×C₆₁	Q₈	1	1	yes	characteristic
23	C₂×M₁₆×C₆₁	—	2	2	no	—
24	C₂²×C₈×C₆₁	—	2	1	no	—
25	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
27	C₄×Q₈×C₆₁	—	2	1	no	—
28	C₄×Q₈×C₆₁	—	2	1	no	—
29	C₄×Q₈×C₆₁	C₂×C₄	1	2	yes	—
30	C₄×Q₈×C₆₁	D₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
32	C₂×C₄⋊C₄×C₆₁	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₆₁	—	2	1	no	—
34	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
35	#108×C₆₁	—	4	2	no	—
36	#108×C₆₁	—	4	2	no	—
37	C₄⋊C₈×C₆₁	—	2	2	no	—
38	C₄⋊C₈×C₆₁	—	2	2	no	—
39	C₄⋊C₈×C₆₁	—	2	1	no	—
40	C₄⋊C₈×C₆₁	—	2	2	no	—
41	C₄⋊C₈×C₆₁	—	2	2	no	—
42	C₄⋊C₈×C₆₁	—	2	2	no	—
43	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
44	C₂²⋊C₈×C₆₁	—	2	2	no	—
45	#97×C₆₁	—	2	2	no	—
46	#97×C₆₁	—	2	2	no	—
47	C₄×C₈×C₆₁	—	2	2	no	—
48	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
51	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
52	C₂×Q₈×C₆₁	—	2	1	no	—
53	C₂×Q₈×C₆₁	—	4	2	no	—
54	C₂×Q₈×C₆₁	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₆₁	—	2	1	no	—
56	C₂²×C₄×C₆₁	—	2	1	no	—
57	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
60	M₁₆×C₆₁	—	4	2	no	—
61	M₁₆×C₆₁	—	4	2	no	—
62	C₂×C₈×C₆₁	—	2	1	no	—
63	C₂×C₈×C₆₁	—	2	1	no	—
64	C₂×C₈×C₆₁	—	2	2	no	—
65	C₂×C₈×C₆₁	—	4	1	no	—
66	C₂×C₈×C₆₁	—	4	1	no	—
67	C₂×C₈×C₆₁	—	4	2	no	—
68	C₂×C₈×C₆₁	—	4	2	no	—
69	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
74	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
75	C₄⋊C₄×C₆₁	—	2	2	no	—
76	C₄⋊C₄×C₆₁	—	2	2	no	—
77	C₄⋊C₄×C₆₁	—	2	1	no	—
78	C₄⋊C₄×C₆₁	C₄⋊C₄	1	2	yes	—
79	C₄²×C₆₁	—	2	2	no	—
80	C₄²×C₆₁	—	2	2	no	—
81	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₆₁	#41	1	1	yes	characteristic
85	C₂³×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
86	Q₈×C₆₁	—	2	4	no	—
87	Q₈×C₆₁	—	4	2	no	—
88	C₂×C₄×C₆₁	—	2	2	no	—
89	C₂×C₄×C₆₁	—	2	2	no	—
90	C₂×C₄×C₆₁	—	2	2	no	—
91	C₂×C₄×C₆₁	—	4	1	no	—
92	C₂×C₄×C₆₁	—	4	2	no	—
93	C₂×C₄×C₆₁	—	4	2	no	—
94	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
97	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
98	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
99	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
102	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
104	C₈×C₆₁	—	2	1	no	—
105	C₈×C₆₁	—	2	1	no	—
106	C₈×C₆₁	—	4	2	no	—
107	C₈×C₆₁	—	8	2	no	—
108	C₈×C₆₁	—	8	2	no	—
109	C₈×C₆₁	#103	1	2	yes	—
110	C₈×C₆₁	#103	1	2	yes	—
111	#8663	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂²×C₆₁	—	2	1	no	—
113	C₂²×C₆₁	—	2	1	no	—
114	C₂²×C₆₁	#380	1	1	yes	characteristic
115	C₂²×C₆₁	#475	1	1	yes	characteristic
116	C₂²×C₆₁	#477	1	1	yes	characteristic
117	C₄×C₆₁	—	2	1	no	—
118	C₄×C₆₁	—	2	1	no	—
119	C₄×C₆₁	—	2	1	no	—
120	C₄×C₆₁	—	4	2	no	—
121	C₄×C₆₁	—	4	2	no	—
122	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
123	C₄×C₆₁	#417	1	1	yes	characteristic
124	C₄×C₆₁	#430	1	2	yes	—
125	C₄×C₆₁	#460	1	1	yes	characteristic
126	C₄×C₆₁	#462	1	1	yes	characteristic
127	C₂×#446	C₂×C₆₁	1	1	yes	characteristic
128	#1848	C₂×C₆₁	1	1	yes	characteristic
129	#1564	C₂×C₆₁	1	1	yes	characteristic
130	#1487	C₂×C₆₁	1	2	yes	—
131	#1301	C₂×C₆₁	1	2	yes	—
132	C₂×C₆₁	—	2	1	no	—
133	C₂×C₆₁	—	2	1	no	—
134	C₂×C₆₁	#1803	1	1	yes	characteristic
135	C₂×C₆₁	#1811	1	1	yes	characteristic
136	C₂×C₆₁	#1814	1	1	yes	characteristic
137	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
138	#446	—	2	1	no	—
139	#446	—	2	1	no	—
140	#446	C₄×C₆₁	1	2	yes	—
141	#446	C₄×C₆₁	1	2	yes	—
142	#431	C₂²×C₆₁	1	2	yes	—
143	C₂×#108	—	2	1	no	—
144	C₂×#108	—	2	1	no	—
145	#423	C₂²×C₆₁	1	2	yes	—
146	C₂×C₄⋊C₈	—	2	1	no	—
147	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
148	C₂×#97	—	2	1	no	—
149	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
150	#335	—	2	1	no	—
151	#334	—	2	1	no	—
152	Q₈⋊C₈	—	2	4	no	—
153	C₆₁	#8663	1	1	yes	characteristic, direct product factor, Sylow
154	C₂²×Q₈	Q₈×C₆₁	1	1	yes	characteristic
155	C₂×M₁₆	—	2	2	no	—
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
158	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
159	C₄×Q₈	—	2	1	no	—
160	C₄×Q₈	—	2	1	no	—
161	C₄×Q₈	C₂×C₄×C₆₁	1	2	yes	—
162	C₄×Q₈	D₄×C₆₁	1	2	yes	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	Q₈×C₆₁	1	1	yes	characteristic
165	C₂×C₄²	—	2	1	no	—
166	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
167	#108	—	4	2	no	—
168	#108	—	4	2	no	—
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	1	no	—
172	C₄⋊C₈	—	2	2	no	—
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	C₂×C₄×C₆₁	1	2	yes	—
176	C₂²⋊C₈	—	2	2	no	—
177	#97	—	2	2	no	—
178	#97	—	2	2	no	—
179	C₄×C₈	—	2	2	no	—
180	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
181	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
182	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
183	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
184	C₂×Q₈	—	2	1	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	C₄⋊C₄×C₆₁	1	2	yes	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
190	C₂²×C₄	C₂×Q₈×C₆₁	1	1	yes	characteristic
191	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
192	M₁₆	—	4	2	no	—
193	M₁₆	—	4	2	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
203	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
205	C₂×C₈	SD₁₆×C₆₁	1	2	yes	—
206	C₂×C₈	Q₁₆×C₆₁	1	2	yes	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	C₄⋊C₄×C₆₁	1	2	yes	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
214	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
215	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
216	C₄²	#41×C₆₁	1	1	yes	characteristic
217	C₂³	C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
218	Q₈	—	2	4	no	—
219	Q₈	—	4	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
227	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
228	C₂×C₄	#117×C₆₁	1	1	yes	characteristic
229	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
230	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
231	C₂×C₄	C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
232	C₂×C₄	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
233	C₂×C₄	C₂×Q₁₆×C₆₁	1	1	yes	characteristic
234	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
235	C₂×C₄	#137×C₆₁	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	4	2	no	—
239	C₈	—	8	2	no	—
240	C₈	—	8	2	no	—
241	C₈	#103×C₆₁	1	2	yes	—
242	C₈	#103×C₆₁	1	2	yes	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	C₆₁×#380	1	1	yes	characteristic
246	C₂²	C₆₁×#475	1	1	yes	characteristic
247	C₂²	C₆₁×#477	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	4	2	no	—
252	C₄	—	4	2	no	—
253	C₄	C₂×#103×C₆₁	1	1	yes	characteristic
254	C₄	C₆₁×#417	1	1	yes	characteristic
255	C₄	C₆₁×#430	1	2	yes	—
256	C₄	C₆₁×#460	1	1	yes	characteristic
257	C₄	C₆₁×#462	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	C₆₁×#1803	1	1	yes	characteristic
261	C₂	C₆₁×#1811	1	1	yes	characteristic
262	C₂	C₆₁×#1814	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₁	#121
commutator	C₄²	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
socle	C₂²×C₆₁	#380
prime radical	C₂³×C₆₁	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
111. #8663	C₆₁	direct (N × H)
153. C₆₁	#8663	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#446	2
C₆₁×#1848	2
C₆₁×#1564	2
C₆₁×#1487	2
C₆₁×#1301	2
#8663	61

Subgroups — C61×#8663 #29512096

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#8663 #29512096