Subgroups — C₆₁×#12988 #29516421

Order 15616.

466 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3407	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#2932	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2117	C₂	1	2	yes	maximal, complemented
4	C₆₁×#2115	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1375	C₂	1	2	yes	maximal, complemented
6	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
7	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
8	C₆₁×#532	C₄	1	1	yes	characteristic
9	C₆₁×#504	—	2	4	no	—
10	C₂×M₃₂×C₆₁	—	2	2	no	—
11	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
12	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
13	C₆₁×#435	—	2	2	no	—
14	C₆₁×#435	—	2	2	no	—
15	C₆₁×#423	C₄	1	1	yes	characteristic
16	C₆₁×#408	C₄	1	1	yes	characteristic
17	C₆₁×#408	C₄	1	1	yes	characteristic
18	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
19	C₆₁×#350	—	2	4	no	—
20	C₆₁×#349	—	2	2	no	—
21	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
23	#131×C₆₁	—	2	2	no	—
24	#131×C₆₁	—	2	2	no	—
25	#131×C₆₁	—	2	2	no	—
26	#131×C₆₁	—	2	2	no	—
27	C₂×M₁₆×C₆₁	—	2	1	no	—
28	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂×M₁₆×C₆₁	D₄	1	2	yes	—
31	C₂²×C₈×C₆₁	—	2	1	no	—
32	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₁	D₄	1	2	yes	—
34	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
35	C₄⋊Q₈×C₆₁	C₈	1	1	yes	characteristic
36	#127×C₆₁	C₈	1	1	yes	characteristic
37	#124×C₆₁	C₈	1	1	yes	characteristic
38	#124×C₆₁	C₈	1	1	yes	characteristic
39	#121×C₆₁	—	2	2	no	—
40	C₄×D₄×C₆₁	—	2	2	no	—
41	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
42	M₃₂×C₆₁	—	2	4	no	—
43	M₃₂×C₆₁	—	2	2	no	—
44	M₃₂×C₆₁	—	2	2	no	—
45	M₃₂×C₆₁	—	2	2	no	—
46	M₃₂×C₆₁	—	2	2	no	—
47	C₂×C₁₆×C₆₁	—	2	4	no	—
48	C₂×C₁₆×C₆₁	—	2	2	no	—
49	C₂×C₁₆×C₆₁	D₄	1	4	yes	complemented
50	C₄⋊C₈×C₆₁	—	2	2	no	—
51	C₂²⋊C₈×C₆₁	—	2	2	no	—
52	C₂²⋊C₈×C₆₁	—	2	2	no	—
53	#97×C₆₁	—	2	1	no	—
54	#97×C₆₁	—	2	1	no	—
55	#41×C₆₁	—	2	2	no	—
56	#41×C₆₁	—	2	2	no	—
57	#41×C₆₁	—	2	2	no	—
58	#41×C₆₁	—	2	2	no	—
59	C₂×Q₈×C₆₁	C₂×C₈	1	1	yes	characteristic
60	C₂×Q₈×C₆₁	C₂×C₈	1	1	yes	characteristic
61	C₂×D₄×C₆₁	—	2	1	no	—
62	C₂×D₄×C₆₁	—	2	2	no	—
63	C₂×D₄×C₆₁	C₂×C₈	1	1	yes	characteristic
64	C₂×D₄×C₆₁	C₂×C₈	1	1	yes	characteristic
65	C₂×D₄×C₆₁	M₁₆	1	2	yes	—
66	C₂²×C₄×C₆₁	—	2	1	no	—
67	C₂²×C₄×C₆₁	—	2	1	no	—
68	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
70	M₁₆×C₆₁	—	2	2	no	—
71	M₁₆×C₆₁	—	2	4	no	—
72	M₁₆×C₆₁	—	2	2	no	—
73	M₁₆×C₆₁	—	4	1	no	—
74	M₁₆×C₆₁	—	4	1	no	—
75	C₂×C₈×C₆₁	—	2	2	no	—
76	C₂×C₈×C₆₁	—	2	2	no	—
77	C₂×C₈×C₆₁	—	2	1	no	—
78	C₂×C₈×C₆₁	—	2	1	no	—
79	C₂×C₈×C₆₁	—	2	2	no	—
80	C₂×C₈×C₆₁	—	4	2	no	—
81	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₆₁	#41	1	2	yes	—
86	C₄⋊C₄×C₆₁	—	2	2	no	—
87	C₂²⋊C₄×C₆₁	—	2	2	no	—
88	C₂²⋊C₄×C₆₁	—	2	2	no	—
89	C₄²×C₆₁	—	2	1	no	—
90	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
91	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
92	C₁₆×C₆₁	—	2	4	no	—
93	C₁₆×C₆₁	—	4	2	no	—
94	C₂³×C₆₁	—	2	1	no	—
95	C₂³×C₆₁	C₂²⋊C₈	1	2	yes	—
96	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
97	Q₈×C₆₁	—	2	2	no	—
98	Q₈×C₆₁	—	2	2	no	—
99	D₄×C₆₁	—	2	4	no	—
100	D₄×C₆₁	—	2	2	no	—
101	D₄×C₆₁	—	2	2	no	—
102	D₄×C₆₁	—	4	2	no	—
103	D₄×C₆₁	—	8	2	no	—
104	C₂×C₄×C₆₁	—	2	2	no	—
105	C₂×C₄×C₆₁	—	2	2	no	—
106	C₂×C₄×C₆₁	—	2	1	no	—
107	C₂×C₄×C₆₁	—	2	1	no	—
108	C₂×C₄×C₆₁	—	2	1	no	—
109	C₂×C₄×C₆₁	—	4	1	no	—
110	C₂×C₄×C₆₁	—	4	1	no	—
111	C₂×C₄×C₆₁	—	8	1	no	—
112	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
113	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₆₁	#131	1	2	yes	—
117	C₈×C₆₁	—	2	2	no	—
118	C₈×C₆₁	—	4	1	no	—
119	C₈×C₆₁	—	4	1	no	—
120	C₈×C₆₁	#120	1	1	yes	characteristic
121	C₈×C₆₁	#121	1	2	yes	—
122	C₈×C₆₁	#123	1	1	yes	characteristic
123	#12988	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂²×C₆₁	—	2	2	no	—
125	C₂²×C₆₁	—	4	1	no	—
126	C₂²×C₆₁	—	4	2	no	—
127	C₂²×C₆₁	—	4	2	no	—
128	C₂²×C₆₁	—	4	1	no	—
129	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
130	C₂²×C₆₁	#435	1	2	yes	—
131	C₄×C₆₁	—	2	2	no	—
132	C₄×C₆₁	—	4	1	no	—
133	C₄×C₆₁	—	4	1	no	—
134	C₄×C₆₁	—	4	1	no	—
135	C₄×C₆₁	#386	1	1	yes	characteristic
136	C₄×C₆₁	#408	1	1	yes	characteristic
137	C₄×C₆₁	C₈×D₄	1	2	yes	—
138	#3407	C₂×C₆₁	1	1	yes	characteristic, complemented
139	#2932	C₂×C₆₁	1	1	yes	characteristic
140	#2117	C₂×C₆₁	1	2	yes	complemented
141	#2115	C₂×C₆₁	1	1	yes	characteristic, complemented
142	#1375	C₂×C₆₁	1	2	yes	complemented
143	C₂×C₆₁	—	2	1	no	—
144	C₂×C₆₁	—	2	2	no	—
145	C₂×C₆₁	—	4	2	no	—
146	C₂×C₆₁	—	8	1	no	—
147	C₂×C₆₁	#1869	1	1	yes	characteristic
148	C₂×#131	C₂²×C₆₁	1	1	yes	characteristic
149	C₂×#131	C₂²×C₆₁	1	1	yes	characteristic
150	#532	C₄×C₆₁	1	1	yes	characteristic
151	#504	—	2	4	no	—
152	C₂×M₃₂	—	2	2	no	—
153	C₂×M₃₂	C₂²×C₆₁	1	2	yes	—
154	C₂×M₃₂	C₂²×C₆₁	1	2	yes	—
155	#435	—	2	2	no	—
156	#435	—	2	2	no	—
157	#423	C₄×C₆₁	1	1	yes	characteristic
158	#408	C₄×C₆₁	1	1	yes	characteristic
159	#408	C₄×C₆₁	1	1	yes	characteristic
160	C₂×#97	C₂²×C₆₁	1	1	yes	characteristic
161	#350	—	2	4	no	—
162	#349	—	2	2	no	—
163	C₆₁	#12988	1	1	yes	characteristic, direct product factor, Sylow
164	C₂×#41	C₂×C₄×C₆₁	1	1	yes	characteristic
165	C₂×#41	C₂×C₄×C₆₁	1	1	yes	characteristic
166	#131	—	2	2	no	—
167	#131	—	2	2	no	—
168	#131	—	2	2	no	—
169	#131	—	2	2	no	—
170	C₂×M₁₆	—	2	1	no	—
171	C₂×M₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
172	C₂×M₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
173	C₂×M₁₆	D₄×C₆₁	1	2	yes	—
174	C₂²×C₈	—	2	1	no	—
175	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
176	C₂²×C₈	D₄×C₆₁	1	2	yes	—
177	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
178	C₄⋊Q₈	C₈×C₆₁	1	1	yes	characteristic
179	#127	C₈×C₆₁	1	1	yes	characteristic
180	#124	C₈×C₆₁	1	1	yes	characteristic
181	#124	C₈×C₆₁	1	1	yes	characteristic
182	#121	—	2	2	no	—
183	C₄×D₄	—	2	2	no	—
184	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
185	M₃₂	—	2	4	no	—
186	M₃₂	—	2	2	no	—
187	M₃₂	—	2	2	no	—
188	M₃₂	—	2	2	no	—
189	M₃₂	—	2	2	no	—
190	C₂×C₁₆	—	2	4	no	—
191	C₂×C₁₆	—	2	2	no	—
192	C₂×C₁₆	D₄×C₆₁	1	4	yes	complemented
193	C₄⋊C₈	—	2	2	no	—
194	C₂²⋊C₈	—	2	2	no	—
195	C₂²⋊C₈	—	2	2	no	—
196	#97	—	2	1	no	—
197	#97	—	2	1	no	—
198	#41	—	2	2	no	—
199	#41	—	2	2	no	—
200	#41	—	2	2	no	—
201	#41	—	2	2	no	—
202	C₂×Q₈	C₂×C₈×C₆₁	1	1	yes	characteristic
203	C₂×Q₈	C₂×C₈×C₆₁	1	1	yes	characteristic
204	C₂×D₄	—	2	1	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	C₂×C₈×C₆₁	1	1	yes	characteristic
207	C₂×D₄	C₂×C₈×C₆₁	1	1	yes	characteristic
208	C₂×D₄	M₁₆×C₆₁	1	2	yes	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
212	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
213	M₁₆	—	2	2	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	2	2	no	—
216	M₁₆	—	4	1	no	—
217	M₁₆	—	4	1	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
225	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
226	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
227	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
228	C₂×C₈	#41×C₆₁	1	2	yes	—
229	C₄⋊C₄	—	2	2	no	—
230	C₂²⋊C₄	—	2	2	no	—
231	C₂²⋊C₄	—	2	2	no	—
232	C₄²	—	2	1	no	—
233	C₄²	C₂×C₈×C₆₁	1	1	yes	characteristic
234	C₄²	C₂×C₈×C₆₁	1	1	yes	characteristic
235	C₁₆	—	2	4	no	—
236	C₁₆	—	4	2	no	—
237	C₂³	—	2	1	no	—
238	C₂³	C₂²⋊C₈×C₆₁	1	2	yes	—
239	C₂³	C₂×M₁₆×C₆₁	1	1	yes	characteristic
240	Q₈	—	2	2	no	—
241	Q₈	—	2	2	no	—
242	D₄	—	2	4	no	—
243	D₄	—	2	2	no	—
244	D₄	—	2	2	no	—
245	D₄	—	4	2	no	—
246	D₄	—	8	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	8	1	no	—
255	C₂×C₄	C₂²⋊C₈×C₆₁	1	2	yes	—
256	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
257	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
258	C₂×C₄	C₂²×C₈×C₆₁	1	1	yes	characteristic
259	C₂×C₄	#131×C₆₁	1	2	yes	—
260	C₈	—	2	2	no	—
261	C₈	—	4	1	no	—
262	C₈	—	4	1	no	—
263	C₈	#120×C₆₁	1	1	yes	characteristic
264	C₈	#121×C₆₁	1	2	yes	—
265	C₈	#123×C₆₁	1	1	yes	characteristic
266	C₂²	—	2	2	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	1	no	—
271	C₂²	C₂×C₂²⋊C₈×C₆₁	1	1	yes	characteristic
272	C₂²	C₆₁×#435	1	2	yes	—
273	C₄	—	2	2	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	4	1	no	—
277	C₄	C₆₁×#386	1	1	yes	characteristic
278	C₄	C₆₁×#408	1	1	yes	characteristic
279	C₄	C₈×D₄×C₆₁	1	2	yes	—
280	C₂	—	2	1	no	—
281	C₂	—	2	2	no	—
282	C₂	—	4	2	no	—
283	C₂	—	8	1	no	—
284	C₂	C₆₁×#1869	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₆₁	#386
commutator	C₂×C₄	C₂²×C₈×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
socle	C₂×C₆₁	#1869
prime radical	C₆₁×#532	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₁ ◁ C₂²×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3407	C₂	semidirect
3. C₆₁×#2117	C₂	semidirect
4. C₆₁×#2115	C₂	semidirect
5. C₆₁×#1375	C₂	semidirect
49. C₂×C₁₆×C₆₁	D₄	semidirect
123. #12988	C₆₁	direct (N × H)
138. #3407	C₂×C₆₁	semidirect
140. #2117	C₂×C₆₁	semidirect
141. #2115	C₂×C₆₁	semidirect
142. #1375	C₂×C₆₁	semidirect
163. C₆₁	#12988	direct (N × H)
192. 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
C₆₁×#3407	2
C₆₁×#2932	2
C₆₁×#2117	2
C₆₁×#2115	2
C₆₁×#1375	2
#12988	61

Subgroups — C61×#12988 #29516421

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#12988 #29516421