Subgroups — C₆₁×#12983 #29516416

Order 15616.

508 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#2929	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#350	C₂	1	2	yes	maximal, complemented
4	C₆₁×#2112	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1379	C₂	1	2	yes	maximal, complemented
6	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
8	C₆₁×#525	C₄	1	1	yes	characteristic
9	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
10	C₂²×C₁₆×C₆₁	—	2	2	no	—
11	C₂²×C₁₆×C₆₁	—	2	4	no	—
12	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
13	C₆₁×#435	—	2	2	no	—
14	C₈×D₄×C₆₁	—	2	2	no	—
15	C₆₁×#424	C₄	1	1	yes	characteristic
16	C₆₁×#408	C₄	1	1	yes	characteristic
17	C₆₁×#407	C₄	1	1	yes	characteristic
18	C₆₁×#405	C₂²	1	1	yes	characteristic
19	C₆₁×#350	—	2	4	no	—
20	C₆₁×C₂²⋊C₁₆	—	2	2	no	—
21	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
23	#131×C₆₁	—	2	2	no	—
24	#131×C₆₁	—	2	2	no	—
25	C₂×M₁₆×C₆₁	—	2	1	no	—
26	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₁	—	2	2	no	—
29	C₂²×C₈×C₆₁	—	2	2	no	—
30	C₂²×C₈×C₆₁	—	2	1	no	—
31	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₁	D₄	1	2	yes	—
33	C₂²×C₈×C₆₁	D₄	1	2	yes	—
34	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
35	#123×C₆₁	—	2	1	no	—
36	C₂²⋊Q₈×C₆₁	C₈	1	2	yes	—
37	#121×C₆₁	C₈	1	2	yes	—
38	#120×C₆₁	—	2	1	no	—
39	C₄×D₄×C₆₁	—	2	2	no	—
40	#117×C₆₁	C₂×C₄	1	1	yes	characteristic
41	M₃₂×C₆₁	—	2	2	no	—
42	M₃₂×C₆₁	—	2	2	no	—
43	C₂×C₁₆×C₆₁	—	2	8	no	—
44	C₂×C₁₆×C₆₁	—	2	2	no	—
45	C₂×C₁₆×C₆₁	—	2	2	no	—
46	C₂×C₁₆×C₆₁	—	2	2	no	—
47	C₂×C₁₆×C₆₁	D₄	1	4	yes	complemented
48	C₄⋊C₈×C₆₁	—	2	2	no	—
49	C₂²⋊C₈×C₆₁	—	2	2	no	—
50	C₂²⋊C₈×C₆₁	—	2	2	no	—
51	#97×C₆₁	—	2	1	no	—
52	C₄×C₈×C₆₁	—	2	1	no	—
53	C₂⁴×C₆₁	M₁₆	1	1	yes	characteristic
54	#41×C₆₁	—	2	2	no	—
55	#41×C₆₁	—	2	2	no	—
56	C₂×Q₈×C₆₁	C₂×C₈	1	1	yes	characteristic
57	C₂×D₄×C₆₁	—	2	2	no	—
58	C₂×D₄×C₆₁	—	2	1	no	—
59	C₂×D₄×C₆₁	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	—	2	2	no	—
61	C₂²×C₄×C₆₁	—	2	2	no	—
62	C₂²×C₄×C₆₁	—	2	1	no	—
63	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
65	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
67	M₁₆×C₆₁	—	2	2	no	—
68	M₁₆×C₆₁	—	2	2	no	—
69	M₁₆×C₆₁	—	4	1	no	—
70	M₁₆×C₆₁	—	4	1	no	—
71	C₂×C₈×C₆₁	—	2	2	no	—
72	C₂×C₈×C₆₁	—	2	2	no	—
73	C₂×C₈×C₆₁	—	2	4	no	—
74	C₂×C₈×C₆₁	—	2	2	no	—
75	C₂×C₈×C₆₁	—	2	1	no	—
76	C₂×C₈×C₆₁	—	2	1	no	—
77	C₂×C₈×C₆₁	—	4	2	no	—
78	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₆₁	#41	1	2	yes	—
83	C₄⋊C₄×C₆₁	—	2	2	no	—
84	C₄⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
85	C₂²⋊C₄×C₆₁	—	2	2	no	—
86	C₂²⋊C₄×C₆₁	—	2	1	no	—
87	C₂²⋊C₄×C₆₁	—	2	2	no	—
88	C₄²×C₆₁	—	2	1	no	—
89	C₁₆×C₆₁	—	2	4	no	—
90	C₁₆×C₆₁	—	4	2	no	—
91	C₂³×C₆₁	—	2	4	no	—
92	C₂³×C₆₁	—	2	2	no	—
93	C₂³×C₆₁	—	2	1	no	—
94	C₂³×C₆₁	C₂²⋊C₈	1	2	yes	—
95	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
96	Q₈×C₆₁	—	2	2	no	—
97	D₄×C₆₁	—	2	2	no	—
98	D₄×C₆₁	—	4	4	no	—
99	D₄×C₆₁	—	4	2	no	—
100	C₂×C₄×C₆₁	—	2	2	no	—
101	C₂×C₄×C₆₁	—	2	2	no	—
102	C₂×C₄×C₆₁	—	2	2	no	—
103	C₂×C₄×C₆₁	—	2	4	no	—
104	C₂×C₄×C₆₁	—	2	2	no	—
105	C₂×C₄×C₆₁	—	2	1	no	—
106	C₂×C₄×C₆₁	—	4	1	no	—
107	C₂×C₄×C₆₁	—	4	1	no	—
108	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
109	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₆₁	#131	1	2	yes	—
113	C₈×C₆₁	—	2	2	no	—
114	C₈×C₆₁	—	4	1	no	—
115	C₈×C₆₁	—	4	1	no	—
116	C₈×C₆₁	#120	1	1	yes	characteristic
117	C₈×C₆₁	#121	1	2	yes	—
118	C₈×C₆₁	#123	1	1	yes	characteristic
119	#12983	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂²×C₆₁	—	2	8	no	—
121	C₂²×C₆₁	—	2	2	no	—
122	C₂²×C₆₁	—	2	2	no	—
123	C₂²×C₆₁	—	2	2	no	—
124	C₂²×C₆₁	—	4	2	no	—
125	C₂²×C₆₁	—	4	1	no	—
126	C₂²×C₆₁	#350	1	4	yes	complemented
127	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
128	C₂²×C₆₁	#435	1	2	yes	—
129	C₄×C₆₁	—	2	2	no	—
130	C₄×C₆₁	—	4	2	no	—
131	C₄×C₆₁	—	4	1	no	—
132	C₄×C₆₁	#386	1	1	yes	characteristic
133	C₄×C₆₁	#408	1	1	yes	characteristic
134	C₄×C₆₁	C₈×D₄	1	2	yes	—
135	C₂³×C₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
136	#2929	C₂×C₆₁	1	1	yes	characteristic
137	C₂×#350	C₂×C₆₁	1	2	yes	complemented
138	#2112	C₂×C₆₁	1	1	yes	characteristic, complemented
139	#1379	C₂×C₆₁	1	2	yes	complemented
140	C₂×C₆₁	—	2	4	no	—
141	C₂×C₆₁	—	2	2	no	—
142	C₂×C₆₁	—	8	1	no	—
143	C₂×C₆₁	#1869	1	1	yes	characteristic
144	C₂×C₆₁	C₂×#350	1	2	yes	—
145	C₂×#131	C₂²×C₆₁	1	1	yes	characteristic
146	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
147	#525	C₄×C₆₁	1	1	yes	characteristic
148	C₂×M₃₂	C₂²×C₆₁	1	2	yes	—
149	C₂²×C₁₆	—	2	2	no	—
150	C₂²×C₁₆	—	2	4	no	—
151	C₂²×C₁₆	C₂²×C₆₁	1	2	yes	—
152	#435	—	2	2	no	—
153	C₈×D₄	—	2	2	no	—
154	#424	C₄×C₆₁	1	1	yes	characteristic
155	#408	C₄×C₆₁	1	1	yes	characteristic
156	#407	C₄×C₆₁	1	1	yes	characteristic
157	#405	C₂²×C₆₁	1	1	yes	characteristic
158	#350	—	2	4	no	—
159	C₂²⋊C₁₆	—	2	2	no	—
160	C₆₁	#12983	1	1	yes	characteristic, direct product factor, Sylow
161	C₂×#41	C₂×C₄×C₆₁	1	1	yes	characteristic
162	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
163	#131	—	2	2	no	—
164	#131	—	2	2	no	—
165	C₂×M₁₆	—	2	1	no	—
166	C₂×M₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
167	C₂×M₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	2	1	no	—
171	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
172	C₂²×C₈	D₄×C₆₁	1	2	yes	—
173	C₂²×C₈	D₄×C₆₁	1	2	yes	—
174	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
175	#123	—	2	1	no	—
176	C₂²⋊Q₈	C₈×C₆₁	1	2	yes	—
177	#121	C₈×C₆₁	1	2	yes	—
178	#120	—	2	1	no	—
179	C₄×D₄	—	2	2	no	—
180	#117	C₂×C₄×C₆₁	1	1	yes	characteristic
181	M₃₂	—	2	2	no	—
182	M₃₂	—	2	2	no	—
183	C₂×C₁₆	—	2	8	no	—
184	C₂×C₁₆	—	2	2	no	—
185	C₂×C₁₆	—	2	2	no	—
186	C₂×C₁₆	—	2	2	no	—
187	C₂×C₁₆	D₄×C₆₁	1	4	yes	complemented
188	C₄⋊C₈	—	2	2	no	—
189	C₂²⋊C₈	—	2	2	no	—
190	C₂²⋊C₈	—	2	2	no	—
191	#97	—	2	1	no	—
192	C₄×C₈	—	2	1	no	—
193	C₂⁴	M₁₆×C₆₁	1	1	yes	characteristic
194	#41	—	2	2	no	—
195	#41	—	2	2	no	—
196	C₂×Q₈	C₂×C₈×C₆₁	1	1	yes	characteristic
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	C₂×C₈×C₆₁	1	1	yes	characteristic
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
204	C₂²×C₄	C₂×C₈×C₆₁	1	2	yes	—
205	C₂²×C₄	M₁₆×C₆₁	1	1	yes	characteristic
206	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
207	M₁₆	—	2	2	no	—
208	M₁₆	—	2	2	no	—
209	M₁₆	—	4	1	no	—
210	M₁₆	—	4	1	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
219	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
220	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
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	C₂×C₈×C₆₁	1	2	yes	—
225	C₂²⋊C₄	—	2	2	no	—
226	C₂²⋊C₄	—	2	1	no	—
227	C₂²⋊C₄	—	2	2	no	—
228	C₄²	—	2	1	no	—
229	C₁₆	—	2	4	no	—
230	C₁₆	—	4	2	no	—
231	C₂³	—	2	4	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	1	no	—
234	C₂³	C₂²⋊C₈×C₆₁	1	2	yes	—
235	C₂³	C₂×M₁₆×C₆₁	1	1	yes	characteristic
236	Q₈	—	2	2	no	—
237	D₄	—	2	2	no	—
238	D₄	—	4	4	no	—
239	D₄	—	4	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	C₂²⋊C₈×C₆₁	1	2	yes	—
249	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
250	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
251	C₂×C₄	C₂²×C₈×C₆₁	1	1	yes	characteristic
252	C₂×C₄	#131×C₆₁	1	2	yes	—
253	C₈	—	2	2	no	—
254	C₈	—	4	1	no	—
255	C₈	—	4	1	no	—
256	C₈	#120×C₆₁	1	1	yes	characteristic
257	C₈	#121×C₆₁	1	2	yes	—
258	C₈	#123×C₆₁	1	1	yes	characteristic
259	C₂²	—	2	8	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	1	no	—
265	C₂²	C₆₁×#350	1	4	yes	complemented
266	C₂²	C₂×C₂²⋊C₈×C₆₁	1	1	yes	characteristic
267	C₂²	C₆₁×#435	1	2	yes	—
268	C₄	—	2	2	no	—
269	C₄	—	4	2	no	—
270	C₄	—	4	1	no	—
271	C₄	C₆₁×#386	1	1	yes	characteristic
272	C₄	C₆₁×#408	1	1	yes	characteristic
273	C₄	C₈×D₄×C₆₁	1	2	yes	—
274	C₂	—	2	4	no	—
275	C₂	—	2	2	no	—
276	C₂	—	8	1	no	—
277	C₂	C₆₁×#1869	1	1	yes	characteristic
278	C₂	C₂×C₆₁×#350	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₆₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
socle	C₂²×C₆₁	C₂×C₂²⋊C₈
prime radical	C₆₁×#525	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₆₁ ◁ C₂³×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₁₆×C₆₁	C₂	semidirect
3. C₂×C₆₁×#350	C₂	semidirect
4. C₆₁×#2112	C₂	semidirect
5. C₆₁×#1379	C₂	semidirect
47. C₂×C₁₆×C₆₁	D₄	semidirect
119. #12983	C₆₁	direct (N × H)
126. C₂²×C₆₁	#350	semidirect
135. C₂³×C₁₆	C₂×C₆₁	semidirect
137. C₂×#350	C₂×C₆₁	semidirect
138. #2112	C₂×C₆₁	semidirect
139. #1379	C₂×C₆₁	semidirect
160. C₆₁	#12983	direct (N × H)
187. C₂×C₁₆	D₄×C₆₁	semidirect
265. C₂²	C₆₁×#350	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₁₆×C₆₁	2
C₆₁×#2929	2
C₂×C₆₁×#350	2
C₆₁×#2112	2
C₆₁×#1379	2
#12983	61

Subgroups — C61×#12983 #29516416

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#12983 #29516416