Subgroups — C₆₁×#9864 #29513297

Order 15616.

528 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1279	C₂	1	4	yes	maximal
5	C₂×C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₆₁	—	2	1	no	—
7	C₂×C₄⋊C₈×C₆₁	—	2	1	no	—
8	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₆₁	—	2	1	no	—
10	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
11	C₆₁×#391	—	2	1	no	—
12	C₆₁×#389	—	2	1	no	—
13	C₆₁×#389	C₄	1	2	yes	—
14	C₆₁×#384	C₄	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₆₁	—	2	1	no	—
16	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
17	C₄³×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#336	—	2	1	no	—
19	C₆₁×#336	—	2	1	no	—
20	C₂²×Q₈×C₆₁	—	2	1	no	—
21	C₂²×C₈×C₆₁	—	2	2	no	—
22	C₂²×C₈×C₆₁	—	2	2	no	—
23	C₄⋊Q₈×C₆₁	—	2	2	no	—
24	C₄⋊Q₈×C₆₁	C₈	1	4	yes	complemented
25	C₄×Q₈×C₆₁	—	4	4	no	—
26	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
27	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
28	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
29	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
30	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₆₁	—	2	2	no	—
33	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₆₁	—	2	4	no	—
41	C₄⋊C₈×C₆₁	—	2	2	no	—
42	C₄⋊C₈×C₆₁	Q₈	1	4	yes	complemented
43	C₄×C₈×C₆₁	—	2	4	no	—
44	C₄×C₈×C₆₁	—	4	2	no	—
45	#95×C₆₁	—	2	2	no	—
46	#95×C₆₁	—	2	2	no	—
47	C₂×Q₈×C₆₁	—	2	4	no	—
48	C₂×Q₈×C₆₁	—	4	2	no	—
49	C₂×Q₈×C₆₁	—	4	2	no	—
50	C₂²×C₄×C₆₁	—	2	2	no	—
51	C₂²×C₄×C₆₁	—	2	2	no	—
52	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	C₂²⋊C₄	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	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₆₁	—	2	4	no	—
60	C₂×C₈×C₆₁	—	2	4	no	—
61	C₂×C₈×C₆₁	—	4	2	no	—
62	C₂×C₈×C₆₁	—	4	2	no	—
63	C₂×C₈×C₆₁	—	4	2	no	—
64	C₂×C₈×C₆₁	—	4	2	no	—
65	C₄⋊C₄×C₆₁	—	2	4	no	—
66	C₄⋊C₄×C₆₁	—	2	2	no	—
67	C₄⋊C₄×C₆₁	—	2	4	no	—
68	C₄⋊C₄×C₆₁	—	4	2	no	—
69	C₄⋊C₄×C₆₁	—	4	4	no	—
70	C₄⋊C₄×C₆₁	—	4	2	no	—
71	C₄⋊C₄×C₆₁	C₂×C₈	1	4	yes	—
72	C₄²×C₆₁	—	2	2	no	—
73	C₄²×C₆₁	—	2	2	no	—
74	C₄²×C₆₁	—	2	2	no	—
75	C₄²×C₆₁	—	4	1	no	—
76	C₄²×C₆₁	—	4	4	no	—
77	C₄²×C₆₁	C₂×C₈	1	2	yes	—
78	C₄²×C₆₁	M₁₆	1	2	yes	—
79	C₄²×C₆₁	SD₁₆	1	2	yes	—
80	C₄²×C₆₁	Q₁₆	1	2	yes	—
81	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
82	C₄²×C₆₁	#41	1	2	yes	—
83	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	Q₈×C₆₁	—	4	8	no	—
85	C₂×C₄×C₆₁	—	2	1	no	—
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	4	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	—	4	2	no	—
96	C₂×C₄×C₆₁	—	4	2	no	—
97	C₂×C₄×C₆₁	—	4	2	no	—
98	C₂×C₄×C₆₁	—	4	2	no	—
99	C₂×C₄×C₆₁	—	4	2	no	—
100	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
101	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
102	C₂×C₄×C₆₁	#103	1	2	yes	—
103	C₂×C₄×C₆₁	#103	1	2	yes	—
104	C₂×C₄×C₆₁	#104	1	2	yes	—
105	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
106	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
108	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₆₁	#131	1	2	yes	—
112	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
114	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
115	C₈×C₆₁	—	4	4	no	—
116	C₈×C₆₁	—	4	4	no	—
117	#9864	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂²×C₆₁	#391	1	1	yes	characteristic
119	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₂²×C₆₁	#408	1	1	yes	characteristic
121	C₂²×C₆₁	C₂×#103	1	1	yes	characteristic
122	C₂²×C₆₁	#418	1	1	yes	characteristic
123	C₂²×C₆₁	C₂×#104	1	1	yes	characteristic
124	C₂²×C₆₁	#421	1	1	yes	characteristic
125	C₄×C₆₁	—	2	1	no	—
126	C₄×C₆₁	—	2	1	no	—
127	C₄×C₆₁	—	2	2	no	—
128	C₄×C₆₁	—	2	2	no	—
129	C₄×C₆₁	—	4	4	no	—
130	C₄×C₆₁	—	4	1	no	—
131	C₄×C₆₁	—	8	2	no	—
132	C₄×C₆₁	Q₈⋊C₈	1	4	yes	—
133	C₄×C₆₁	C₈×Q₈	1	2	yes	—
134	C₄×C₆₁	#446	1	2	yes	—
135	C₄×C₆₁	#476	1	2	yes	—
136	C₄×C₆₁	#477	1	2	yes	—
137	C₄×C₄⋊Q₈	C₂×C₆₁	1	1	yes	characteristic
138	C₄×C₄⋊C₈	C₂×C₆₁	1	1	yes	characteristic
139	C₄×C₄⋊C₈	C₂×C₆₁	1	1	yes	characteristic
140	#1279	C₂×C₆₁	1	4	yes	—
141	C₂×C₆₁	C₂×Q₈⋊C₈	1	1	yes	characteristic
142	C₂×C₆₁	#1478	1	1	yes	characteristic
143	C₂×C₆₁	#1481	1	1	yes	characteristic
144	C₂×C₆₁	#1484	1	1	yes	characteristic
145	C₂×C₆₁	#1984	1	1	yes	characteristic
146	C₂×C₆₁	#1986	1	1	yes	characteristic
147	C₂×C₆₁	#1987	1	1	yes	characteristic
148	C₂×C₄⋊Q₈	C₄×C₆₁	1	1	yes	characteristic
149	C₂×C₄×Q₈	—	2	1	no	—
150	C₂×C₄⋊C₈	—	2	1	no	—
151	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
152	C₂×C₄×C₈	—	2	1	no	—
153	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
154	#391	—	2	1	no	—
155	#389	—	2	1	no	—
156	#389	C₄×C₆₁	1	2	yes	—
157	#384	C₄×C₆₁	1	1	yes	characteristic
158	C₄×C₄⋊C₄	—	2	1	no	—
159	C₄×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
160	C₄³	C₂²×C₆₁	1	1	yes	characteristic
161	#336	—	2	1	no	—
162	#336	—	2	1	no	—
163	C₆₁	#9864	1	1	yes	characteristic, direct product factor, Sylow
164	C₂²×Q₈	—	2	1	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	2	2	no	—
167	C₄⋊Q₈	—	2	2	no	—
168	C₄⋊Q₈	C₈×C₆₁	1	4	yes	complemented
169	C₄×Q₈	—	4	4	no	—
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	—	2	1	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
175	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
176	C₂×C₄²	—	2	2	no	—
177	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
178	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
179	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
180	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
181	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
182	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
183	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
184	C₄⋊C₈	—	2	4	no	—
185	C₄⋊C₈	—	2	2	no	—
186	C₄⋊C₈	Q₈×C₆₁	1	4	yes	complemented
187	C₄×C₈	—	2	4	no	—
188	C₄×C₈	—	4	2	no	—
189	#95	—	2	2	no	—
190	#95	—	2	2	no	—
191	C₂×Q₈	—	2	4	no	—
192	C₂×Q₈	—	4	2	no	—
193	C₂×Q₈	—	4	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
197	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
198	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
199	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
200	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	4	2	no	—
209	C₄⋊C₄	—	2	4	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	C₂×C₈×C₆₁	1	4	yes	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	4	1	no	—
220	C₄²	—	4	4	no	—
221	C₄²	C₂×C₈×C₆₁	1	2	yes	—
222	C₄²	M₁₆×C₆₁	1	2	yes	—
223	C₄²	SD₁₆×C₆₁	1	2	yes	—
224	C₄²	Q₁₆×C₆₁	1	2	yes	—
225	C₄²	C₂×Q₈×C₆₁	1	2	yes	—
226	C₄²	#41×C₆₁	1	2	yes	—
227	C₂³	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
228	Q₈	—	4	8	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	4	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	C₂²⋊C₈×C₆₁	1	2	yes	—
245	C₂×C₄	C₂²⋊C₈×C₆₁	1	2	yes	—
246	C₂×C₄	#103×C₆₁	1	2	yes	—
247	C₂×C₄	#103×C₆₁	1	2	yes	—
248	C₂×C₄	#104×C₆₁	1	2	yes	—
249	C₂×C₄	C₄×Q₈×C₆₁	1	2	yes	—
250	C₂×C₄	C₂²⋊Q₈×C₆₁	1	2	yes	—
251	C₂×C₄	#124×C₆₁	1	1	yes	characteristic
252	C₂×C₄	C₄⋊Q₈×C₆₁	1	1	yes	characteristic
253	C₂×C₄	C₂²×C₈×C₆₁	1	1	yes	characteristic
254	C₂×C₄	C₂×M₁₆×C₆₁	1	1	yes	characteristic
255	C₂×C₄	#131×C₆₁	1	2	yes	—
256	C₂×C₄	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
257	C₂×C₄	C₂×Q₁₆×C₆₁	1	1	yes	characteristic
258	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	2	yes	—
259	C₈	—	4	4	no	—
260	C₈	—	4	4	no	—
261	C₂²	C₆₁×#391	1	1	yes	characteristic
262	C₂²	C₂×C₂²⋊C₈×C₆₁	1	1	yes	characteristic
263	C₂²	C₆₁×#408	1	1	yes	characteristic
264	C₂²	C₂×#103×C₆₁	1	1	yes	characteristic
265	C₂²	C₆₁×#418	1	1	yes	characteristic
266	C₂²	C₂×#104×C₆₁	1	1	yes	characteristic
267	C₂²	C₆₁×#421	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	4	4	no	—
273	C₄	—	4	1	no	—
274	C₄	—	8	2	no	—
275	C₄	C₆₁×Q₈⋊C₈	1	4	yes	—
276	C₄	C₈×Q₈×C₆₁	1	2	yes	—
277	C₄	C₆₁×#446	1	2	yes	—
278	C₄	C₆₁×#476	1	2	yes	—
279	C₄	C₆₁×#477	1	2	yes	—
280	C₂	C₂×C₆₁×Q₈⋊C₈	1	1	yes	characteristic
281	C₂	C₆₁×#1478	1	1	yes	characteristic
282	C₂	C₆₁×#1481	1	1	yes	characteristic
283	C₂	C₆₁×#1484	1	1	yes	characteristic
284	C₂	C₆₁×#1984	1	1	yes	characteristic
285	C₂	C₆₁×#1986	1	1	yes	characteristic
286	C₂	C₆₁×#1987	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
socle, prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₁ ◁ C₄³×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
24. C₄⋊Q₈×C₆₁	C₈	semidirect
42. C₄⋊C₈×C₆₁	Q₈	semidirect
117. #9864	C₆₁	direct (N × H)
163. C₆₁	#9864	direct (N × H)
168. C₄⋊Q₈	C₈×C₆₁	semidirect
186. C₄⋊C₈	Q₈×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⋊Q₈×C₆₁	2
C₄×C₄⋊C₈×C₆₁	2
C₆₁×#1279	2
#9864	61

Subgroups — C61×#9864 #29513297

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#9864 #29513297