Subgroups — C₆₁×#13061 #29516494

Order 15616.

558 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₆₁×#466	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#357	C₂	1	2	yes	maximal, complemented
4	C₆₁×#1808	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1380	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₆₁	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
8	C₂×#121×C₆₁	—	2	1	no	—
9	C₂×M₃₂×C₆₁	—	2	2	no	—
10	C₂×M₃₂×C₆₁	—	2	4	no	—
11	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
12	C₆₁×#466	—	2	2	no	—
13	C₆₁×#466	C₄	1	4	yes	complemented
14	C₂×#107×C₆₁	C₂²	1	2	yes	—
15	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×#102×C₆₁	—	2	1	no	—
17	C₆₁×#380	—	2	1	no	—
18	C₆₁×#357	—	2	4	no	—
19	C₆₁×#340	—	2	1	no	—
20	C₂²×D₄×C₆₁	—	2	1	no	—
21	C₂²×D₄×C₆₁	—	4	1	no	—
22	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
23	C₂×D₈×C₆₁	—	2	2	no	—
24	C₂×D₈×C₆₁	—	2	1	no	—
25	C₂×D₈×C₆₁	—	2	2	no	—
26	C₂×D₈×C₆₁	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₆₁	—	2	2	no	—
28	C₂²×C₈×C₆₁	—	2	2	no	—
29	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₁	D₄	1	2	yes	—
32	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
33	#121×C₆₁	—	2	4	no	—
34	#121×C₆₁	—	4	2	no	—
35	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
36	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
37	C₂×C₂²⋊C₄×C₆₁	—	4	1	no	—
38	C₂×C₂²⋊C₄×C₆₁	—	4	1	no	—
39	M₃₂×C₆₁	—	2	8	no	—
40	C₂×C₁₆×C₆₁	—	2	2	no	—
41	C₂×C₁₆×C₆₁	—	2	2	no	—
42	C₂×C₁₆×C₆₁	D₄	1	4	yes	complemented
43	#107×C₆₁	—	2	1	no	—
44	#107×C₆₁	—	2	2	no	—
45	#107×C₆₁	—	2	2	no	—
46	#107×C₆₁	C₂×C₄	1	2	yes	—
47	#102×C₆₁	—	2	4	no	—
48	#95×C₆₁	—	4	1	no	—
49	C₂⁴×C₆₁	—	4	1	no	—
50	C₂⁴×C₆₁	D₈	1	1	yes	characteristic
51	C₂×D₄×C₆₁	—	2	2	no	—
52	C₂×D₄×C₆₁	—	4	4	no	—
53	C₂×D₄×C₆₁	—	4	2	no	—
54	C₂×D₄×C₆₁	—	4	1	no	—
55	C₂×D₄×C₆₁	—	4	2	no	—
56	C₂×D₄×C₆₁	—	8	4	no	—
57	C₂²×C₄×C₆₁	—	2	2	no	—
58	C₂²×C₄×C₆₁	—	2	2	no	—
59	C₂²×C₄×C₆₁	—	4	1	no	—
60	C₂²×C₄×C₆₁	—	4	2	no	—
61	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₆₁	SD₁₆	1	2	yes	—
64	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
65	D₈×C₆₁	—	2	4	no	—
66	D₈×C₆₁	—	4	1	no	—
67	D₈×C₆₁	—	4	1	no	—
68	C₂×C₈×C₆₁	—	2	2	no	—
69	C₂×C₈×C₆₁	—	2	4	no	—
70	C₂×C₈×C₆₁	—	2	2	no	—
71	C₂×C₈×C₆₁	—	2	2	no	—
72	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₆₁	#41	1	2	yes	—
77	C₄⋊C₄×C₆₁	—	2	2	no	—
78	C₄⋊C₄×C₆₁	—	4	1	no	—
79	C₄⋊C₄×C₆₁	—	4	2	no	—
80	C₄⋊C₄×C₆₁	—	4	2	no	—
81	C₂²⋊C₄×C₆₁	—	4	4	no	—
82	C₂²⋊C₄×C₆₁	—	8	2	no	—
83	C₁₆×C₆₁	—	2	4	no	—
84	C₂³×C₆₁	—	2	2	no	—
85	C₂³×C₆₁	—	2	1	no	—
86	C₂³×C₆₁	—	2	2	no	—
87	C₂³×C₆₁	—	2	1	no	—
88	C₂³×C₆₁	—	4	2	no	—
89	C₂³×C₆₁	—	8	2	no	—
90	C₂³×C₆₁	—	8	2	no	—
91	C₂³×C₆₁	—	8	1	no	—
92	C₂³×C₆₁	—	8	1	no	—
93	C₂³×C₆₁	#102	1	2	yes	—
94	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
95	D₄×C₆₁	—	4	2	no	—
96	D₄×C₆₁	—	4	2	no	—
97	D₄×C₆₁	—	8	8	no	—
98	D₄×C₆₁	—	8	2	no	—
99	C₂×C₄×C₆₁	—	2	2	no	—
100	C₂×C₄×C₆₁	—	2	2	no	—
101	C₂×C₄×C₆₁	—	2	2	no	—
102	C₂×C₄×C₆₁	—	2	4	no	—
103	C₂×C₄×C₆₁	—	4	2	no	—
104	C₂×C₄×C₆₁	—	8	2	no	—
105	C₂×C₄×C₆₁	—	8	2	no	—
106	C₂×C₄×C₆₁	—	8	2	no	—
107	C₂×C₄×C₆₁	—	8	2	no	—
108	C₂×C₄×C₆₁	#102	1	2	yes	—
109	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₆₁	#121	1	2	yes	—
111	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×C₄×C₆₁	#135	1	2	yes	—
113	C₈×C₆₁	—	2	2	no	—
114	C₈×C₆₁	C₄×D₄	1	2	yes	—
115	C₈×C₆₁	#120	1	1	yes	characteristic
116	C₈×C₆₁	#123	1	1	yes	characteristic
117	#13061	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂²×C₆₁	—	2	2	no	—
119	C₂²×C₆₁	—	2	2	no	—
120	C₂²×C₆₁	—	2	2	no	—
121	C₂²×C₆₁	—	4	2	no	—
122	C₂²×C₆₁	—	4	2	no	—
123	C₂²×C₆₁	—	8	2	no	—
124	C₂²×C₆₁	—	8	2	no	—
125	C₂²×C₆₁	—	8	2	no	—
126	C₂²×C₆₁	—	16	2	no	—
127	C₂²×C₆₁	—	16	2	no	—
128	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
129	C₂²×C₆₁	#466	1	2	yes	—
130	C₂²×C₆₁	#509	1	1	yes	characteristic
131	C₂²×C₆₁	#509	1	2	yes	—
132	C₂²×C₆₁	#510	1	1	yes	characteristic
133	C₄×C₆₁	—	2	2	no	—
134	C₄×C₆₁	—	8	2	no	—
135	C₄×C₆₁	—	16	2	no	—
136	C₄×C₆₁	#386	1	1	yes	characteristic
137	C₄×C₆₁	#416	1	1	yes	characteristic
138	C₄×C₆₁	#465	1	2	yes	—
139	C₂²×M₃₂	C₂×C₆₁	1	1	yes	characteristic, complemented
140	C₂×#466	C₂×C₆₁	1	1	yes	characteristic
141	C₂×#357	C₂×C₆₁	1	2	yes	complemented
142	#1808	C₂×C₆₁	1	1	yes	characteristic, complemented
143	#1380	C₂×C₆₁	1	2	yes	complemented
144	C₂×C₆₁	—	4	2	no	—
145	C₂×C₆₁	—	16	2	no	—
146	C₂×C₆₁	#1893	1	1	yes	characteristic
147	C₂×C₆₁	#2139	1	1	yes	characteristic
148	C₂×C₆₁	#2140	1	1	yes	characteristic
149	C₂×C₆₁	#2218	1	2	yes	—
150	C₂×C₆₁	#2220	1	2	yes	—
151	C₂²×D₈	C₂²×C₆₁	1	1	yes	characteristic
152	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
153	C₂×#121	—	2	1	no	—
154	C₂×M₃₂	—	2	2	no	—
155	C₂×M₃₂	—	2	4	no	—
156	C₂²×C₁₆	C₂²×C₆₁	1	2	yes	—
157	#466	—	2	2	no	—
158	#466	C₄×C₆₁	1	4	yes	complemented
159	C₂×#107	C₂²×C₆₁	1	2	yes	—
160	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
161	C₂×#102	—	2	1	no	—
162	#380	—	2	1	no	—
163	#357	—	2	4	no	—
164	#340	—	2	1	no	—
165	C₆₁	#13061	1	1	yes	characteristic, direct product factor, Sylow
166	C₂²×D₄	—	2	1	no	—
167	C₂²×D₄	—	4	1	no	—
168	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
169	C₂×D₈	—	2	2	no	—
170	C₂×D₈	—	2	1	no	—
171	C₂×D₈	—	2	2	no	—
172	C₂×D₈	C₂×C₄×C₆₁	1	2	yes	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
176	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
177	C₂²×C₈	D₄×C₆₁	1	2	yes	—
178	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
179	#121	—	2	4	no	—
180	#121	—	4	2	no	—
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₂²⋊C₄	—	4	1	no	—
184	C₂×C₂²⋊C₄	—	4	1	no	—
185	M₃₂	—	2	8	no	—
186	C₂×C₁₆	—	2	2	no	—
187	C₂×C₁₆	—	2	2	no	—
188	C₂×C₁₆	D₄×C₆₁	1	4	yes	complemented
189	#107	—	2	1	no	—
190	#107	—	2	2	no	—
191	#107	—	2	2	no	—
192	#107	C₂×C₄×C₆₁	1	2	yes	—
193	#102	—	2	4	no	—
194	#95	—	4	1	no	—
195	C₂⁴	—	4	1	no	—
196	C₂⁴	D₈×C₆₁	1	1	yes	characteristic
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	4	4	no	—
199	C₂×D₄	—	4	2	no	—
200	C₂×D₄	—	4	1	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂×D₄	—	8	4	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	4	1	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
208	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
209	C₂²×C₄	SD₁₆×C₆₁	1	2	yes	—
210	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
211	D₈	—	2	4	no	—
212	D₈	—	4	1	no	—
213	D₈	—	4	1	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
219	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
220	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
221	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
222	C₂×C₈	#41×C₆₁	1	2	yes	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	4	1	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₂²⋊C₄	—	4	4	no	—
228	C₂²⋊C₄	—	8	2	no	—
229	C₁₆	—	2	4	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	1	no	—
234	C₂³	—	4	2	no	—
235	C₂³	—	8	2	no	—
236	C₂³	—	8	2	no	—
237	C₂³	—	8	1	no	—
238	C₂³	—	8	1	no	—
239	C₂³	#102×C₆₁	1	2	yes	—
240	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
241	D₄	—	4	2	no	—
242	D₄	—	4	2	no	—
243	D₄	—	8	8	no	—
244	D₄	—	8	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	8	2	no	—
251	C₂×C₄	—	8	2	no	—
252	C₂×C₄	—	8	2	no	—
253	C₂×C₄	—	8	2	no	—
254	C₂×C₄	#102×C₆₁	1	2	yes	—
255	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
256	C₂×C₄	#121×C₆₁	1	2	yes	—
257	C₂×C₄	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
258	C₂×C₄	#135×C₆₁	1	2	yes	—
259	C₈	—	2	2	no	—
260	C₈	C₄×D₄×C₆₁	1	2	yes	—
261	C₈	#120×C₆₁	1	1	yes	characteristic
262	C₈	#123×C₆₁	1	1	yes	characteristic
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	8	2	no	—
269	C₂²	—	8	2	no	—
270	C₂²	—	8	2	no	—
271	C₂²	—	16	2	no	—
272	C₂²	—	16	2	no	—
273	C₂²	C₂×#102×C₆₁	1	1	yes	characteristic
274	C₂²	C₆₁×#466	1	2	yes	—
275	C₂²	C₆₁×#509	1	1	yes	characteristic
276	C₂²	C₆₁×#509	1	2	yes	—
277	C₂²	C₆₁×#510	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	8	2	no	—
280	C₄	—	16	2	no	—
281	C₄	C₆₁×#386	1	1	yes	characteristic
282	C₄	C₆₁×#416	1	1	yes	characteristic
283	C₄	C₆₁×#465	1	2	yes	—
284	C₂	—	4	2	no	—
285	C₂	—	16	2	no	—
286	C₂	C₆₁×#1893	1	1	yes	characteristic
287	C₂	C₆₁×#2139	1	1	yes	characteristic
288	C₂	C₆₁×#2140	1	1	yes	characteristic
289	C₂	C₆₁×#2218	1	2	yes	—
290	C₂	C₆₁×#2220	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
prime radical	C₂×C₆₁×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂³×C₄×C₆₁ ◁ C₂³×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×M₃₂×C₆₁	C₂	semidirect
3. C₂×C₆₁×#357	C₂	semidirect
4. C₆₁×#1808	C₂	semidirect
5. C₆₁×#1380	C₂	semidirect
13. C₆₁×#466	C₄	semidirect
42. C₂×C₁₆×C₆₁	D₄	semidirect
117. #13061	C₆₁	direct (N × H)
139. C₂²×M₃₂	C₂×C₆₁	semidirect
141. C₂×#357	C₂×C₆₁	semidirect
142. #1808	C₂×C₆₁	semidirect
143. #1380	C₂×C₆₁	semidirect
158. #466	C₄×C₆₁	semidirect
165. C₆₁	#13061	direct (N × H)
188. C₂×C₁₆	D₄×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×M₃₂×C₆₁	2
C₂×C₆₁×#466	2
C₂×C₆₁×#357	2
C₆₁×#1808	2
C₆₁×#1380	2
#13061	61

Subgroups — C61×#13061 #29516494

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#13061 #29516494