Subgroups — C₆₁×#12942 #29516375

Order 15616.

380 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2957	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2161	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#2148	C₂	1	1	yes	maximal, characteristic
4	C₂×C₆₁×#359	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1953	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#1381	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₆₁	C₂²	1	1	yes	characteristic
9	C₆₁×#568	—	2	1	no	—
10	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×#444	—	2	2	no	—
13	C₆₁×#443	—	2	2	no	—
14	C₆₁×#430	—	2	1	no	—
15	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#421	—	2	1	no	—
19	C₆₁×#405	C₂²	1	1	yes	characteristic
20	C₆₁×#368	—	2	2	no	—
21	C₆₁×#362	C₄	1	2	yes	complemented
22	C₆₁×#361	C₄	1	2	yes	complemented
23	C₆₁×#359	—	2	2	no	—
24	C₆₁×#344	—	2	1	no	—
25	C₂×#41×C₆₁	—	2	1	no	—
26	#135×C₆₁	—	2	1	no	—
27	#135×C₆₁	—	2	1	no	—
28	#135×C₆₁	D₄	1	2	yes	—
29	#135×C₆₁	Q₈	1	2	yes	—
30	C₂×Q₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂×SD₁₆×C₆₁	—	2	1	no	—
32	C₂×D₈×C₆₁	C₂×C₄	1	1	yes	characteristic
33	#131×C₆₁	—	2	4	no	—
34	#131×C₆₁	—	4	2	no	—
35	C₂×M₁₆×C₆₁	—	2	1	no	—
36	C₂×M₁₆×C₆₁	—	2	1	no	—
37	C₂×M₁₆×C₆₁	—	2	1	no	—
38	C₂×M₁₆×C₆₁	—	4	2	no	—
39	C₂×M₁₆×C₆₁	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
41	#117×C₆₁	D₄	1	1	yes	characteristic
42	M₃₂×C₆₁	—	2	1	no	—
43	M₃₂×C₆₁	C₂×C₄	1	2	yes	—
44	C₂×C₁₆×C₆₁	—	2	1	no	—
45	C₂×C₁₆×C₆₁	—	2	1	no	—
46	C₂×C₁₆×C₆₁	—	2	1	no	—
47	C₂×C₁₆×C₆₁	—	2	1	no	—
48	#108×C₆₁	—	2	1	no	—
49	#108×C₆₁	—	2	1	no	—
50	#108×C₆₁	—	2	1	no	—
51	#108×C₆₁	—	2	1	no	—
52	#108×C₆₁	—	4	2	no	—
53	#108×C₆₁	C₂×C₄	1	2	yes	—
54	#108×C₆₁	D₄	1	2	yes	—
55	#104×C₆₁	—	2	4	no	—
56	#97×C₆₁	—	2	1	no	—
57	C₄×C₈×C₆₁	—	2	1	no	—
58	#41×C₆₁	—	2	2	no	—
59	#41×C₆₁	—	2	2	no	—
60	#41×C₆₁	—	4	1	no	—
61	#41×C₆₁	—	4	1	no	—
62	C₂×Q₈×C₆₁	—	2	1	no	—
63	C₂×D₄×C₆₁	—	2	1	no	—
64	C₂×D₄×C₆₁	—	4	1	no	—
65	C₂²×C₄×C₆₁	—	4	1	no	—
66	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
67	Q₁₆×C₆₁	—	2	1	no	—
68	Q₁₆×C₆₁	C₄⋊C₄	1	2	yes	—
69	SD₁₆×C₆₁	—	2	2	no	—
70	SD₁₆×C₆₁	—	2	2	no	—
71	D₈×C₆₁	—	2	1	no	—
72	D₈×C₆₁	C₄⋊C₄	1	2	yes	—
73	M₁₆×C₆₁	—	2	1	no	—
74	M₁₆×C₆₁	—	2	2	no	—
75	M₁₆×C₆₁	—	2	2	no	—
76	M₁₆×C₆₁	—	4	4	no	—
77	M₁₆×C₆₁	—	4	1	no	—
78	M₁₆×C₆₁	—	4	1	no	—
79	M₁₆×C₆₁	—	4	1	no	—
80	M₁₆×C₆₁	—	4	1	no	—
81	M₁₆×C₆₁	D₈	1	2	yes	—
82	C₂×C₈×C₆₁	—	2	1	no	—
83	C₂×C₈×C₆₁	—	4	4	no	—
84	C₂×C₈×C₆₁	—	4	1	no	—
85	C₂×C₈×C₆₁	—	4	1	no	—
86	C₂×C₈×C₆₁	—	4	1	no	—
87	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
91	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
92	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
93	C₄⋊C₄×C₆₁	—	2	1	no	—
94	C₂²⋊C₄×C₆₁	—	2	1	no	—
95	C₄²×C₆₁	—	2	1	no	—
96	C₁₆×C₆₁	—	2	2	no	—
97	C₁₆×C₆₁	—	4	1	no	—
98	C₂³×C₆₁	—	4	1	no	—
99	C₂³×C₆₁	#137	1	1	yes	characteristic
100	Q₈×C₆₁	—	2	2	no	—
101	Q₈×C₆₁	—	4	1	no	—
102	D₄×C₆₁	—	2	2	no	—
103	D₄×C₆₁	—	4	2	no	—
104	D₄×C₆₁	—	4	2	no	—
105	D₄×C₆₁	—	4	1	no	—
106	C₂×C₄×C₆₁	—	4	1	no	—
107	C₂×C₄×C₆₁	—	4	2	no	—
108	C₂×C₄×C₆₁	—	4	2	no	—
109	C₂×C₄×C₆₁	—	4	1	no	—
110	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
112	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
115	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₈×C₆₁	—	2	2	no	—
117	C₈×C₆₁	—	4	2	no	—
118	C₈×C₆₁	—	4	2	no	—
119	C₈×C₆₁	—	8	1	no	—
120	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
122	C₈×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
123	C₈×C₆₁	#123	1	1	yes	characteristic
124	#12942	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂²×C₆₁	—	4	1	no	—
126	C₂²×C₆₁	—	4	2	no	—
127	C₂²×C₆₁	—	16	1	no	—
128	C₂²×C₆₁	#440	1	1	yes	characteristic
129	C₂²×C₆₁	#452	1	1	yes	characteristic
130	C₂²×C₆₁	#474	1	1	yes	characteristic
131	C₄×C₆₁	—	4	2	no	—
132	C₄×C₆₁	—	8	1	no	—
133	C₄×C₆₁	#385	1	1	yes	characteristic
134	C₄×C₆₁	C₄×D₈	1	1	yes	characteristic
135	C₄×C₆₁	#447	1	1	yes	characteristic
136	C₄×C₆₁	#478	1	1	yes	characteristic
137	#2957	C₂×C₆₁	1	1	yes	characteristic
138	#2161	C₂×C₆₁	1	1	yes	characteristic, complemented
139	#2148	C₂×C₆₁	1	1	yes	characteristic
140	C₂×#359	C₂×C₆₁	1	1	yes	characteristic
141	#1953	C₂×C₆₁	1	1	yes	characteristic, complemented
142	#1381	C₂×C₆₁	1	1	yes	characteristic, complemented
143	#1379	C₂×C₆₁	1	1	yes	characteristic, complemented
144	C₂×C₆₁	—	2	1	no	—
145	C₂×C₆₁	—	2	1	no	—
146	C₂×C₆₁	—	2	1	no	—
147	C₂×C₆₁	—	8	2	no	—
148	C₂×C₆₁	#1862	1	1	yes	characteristic
149	C₂×#135	C₂²×C₆₁	1	1	yes	characteristic
150	#568	—	2	1	no	—
151	C₂×M₃₂	C₂²×C₆₁	1	1	yes	characteristic
152	C₂²×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
153	#444	—	2	2	no	—
154	#443	—	2	2	no	—
155	#430	—	2	1	no	—
156	C₂×#108	C₂²×C₆₁	1	1	yes	characteristic
157	C₂×#108	C₂²×C₆₁	1	1	yes	characteristic
158	C₂×#108	C₂²×C₆₁	1	1	yes	characteristic
159	#421	—	2	1	no	—
160	#405	C₂²×C₆₁	1	1	yes	characteristic
161	#368	—	2	2	no	—
162	#362	C₄×C₆₁	1	2	yes	complemented
163	#361	C₄×C₆₁	1	2	yes	complemented
164	#359	—	2	2	no	—
165	#344	—	2	1	no	—
166	C₆₁	#12942	1	1	yes	characteristic, direct product factor, Sylow
167	C₂×#41	—	2	1	no	—
168	#135	—	2	1	no	—
169	#135	—	2	1	no	—
170	#135	D₄×C₆₁	1	2	yes	—
171	#135	Q₈×C₆₁	1	2	yes	—
172	C₂×Q₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
173	C₂×SD₁₆	—	2	1	no	—
174	C₂×D₈	C₂×C₄×C₆₁	1	1	yes	characteristic
175	#131	—	2	4	no	—
176	#131	—	4	2	no	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	—	4	2	no	—
181	C₂×M₁₆	D₄×C₆₁	1	1	yes	characteristic
182	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
183	#117	D₄×C₆₁	1	1	yes	characteristic
184	M₃₂	—	2	1	no	—
185	M₃₂	C₂×C₄×C₆₁	1	2	yes	—
186	C₂×C₁₆	—	2	1	no	—
187	C₂×C₁₆	—	2	1	no	—
188	C₂×C₁₆	—	2	1	no	—
189	C₂×C₁₆	—	2	1	no	—
190	#108	—	2	1	no	—
191	#108	—	2	1	no	—
192	#108	—	2	1	no	—
193	#108	—	2	1	no	—
194	#108	—	4	2	no	—
195	#108	C₂×C₄×C₆₁	1	2	yes	—
196	#108	D₄×C₆₁	1	2	yes	—
197	#104	—	2	4	no	—
198	#97	—	2	1	no	—
199	C₄×C₈	—	2	1	no	—
200	#41	—	2	2	no	—
201	#41	—	2	2	no	—
202	#41	—	4	1	no	—
203	#41	—	4	1	no	—
204	C₂×Q₈	—	2	1	no	—
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	—	4	1	no	—
207	C₂²×C₄	—	4	1	no	—
208	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
209	Q₁₆	—	2	1	no	—
210	Q₁₆	C₄⋊C₄×C₆₁	1	2	yes	—
211	SD₁₆	—	2	2	no	—
212	SD₁₆	—	2	2	no	—
213	D₈	—	2	1	no	—
214	D₈	C₄⋊C₄×C₆₁	1	2	yes	—
215	M₁₆	—	2	1	no	—
216	M₁₆	—	2	2	no	—
217	M₁₆	—	2	2	no	—
218	M₁₆	—	4	4	no	—
219	M₁₆	—	4	1	no	—
220	M₁₆	—	4	1	no	—
221	M₁₆	—	4	1	no	—
222	M₁₆	—	4	1	no	—
223	M₁₆	D₈×C₆₁	1	2	yes	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	4	4	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
230	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
231	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
232	C₂×C₈	C₂×Q₈×C₆₁	1	1	yes	characteristic
233	C₂×C₈	#41×C₆₁	1	1	yes	characteristic
234	C₂×C₈	#41×C₆₁	1	1	yes	characteristic
235	C₄⋊C₄	—	2	1	no	—
236	C₂²⋊C₄	—	2	1	no	—
237	C₄²	—	2	1	no	—
238	C₁₆	—	2	2	no	—
239	C₁₆	—	4	1	no	—
240	C₂³	—	4	1	no	—
241	C₂³	#137×C₆₁	1	1	yes	characteristic
242	Q₈	—	2	2	no	—
243	Q₈	—	4	1	no	—
244	D₄	—	2	2	no	—
245	D₄	—	4	2	no	—
246	D₄	—	4	2	no	—
247	D₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
253	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
254	C₂×C₄	C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
255	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
256	C₂×C₄	#135×C₆₁	1	1	yes	characteristic
257	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	4	2	no	—
260	C₈	—	4	2	no	—
261	C₈	—	8	1	no	—
262	C₈	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
263	C₈	C₄×D₄×C₆₁	1	1	yes	characteristic
264	C₈	C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
265	C₈	#123×C₆₁	1	1	yes	characteristic
266	C₂²	—	4	1	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	16	1	no	—
269	C₂²	C₆₁×#440	1	1	yes	characteristic
270	C₂²	C₆₁×#452	1	1	yes	characteristic
271	C₂²	C₆₁×#474	1	1	yes	characteristic
272	C₄	—	4	2	no	—
273	C₄	—	8	1	no	—
274	C₄	C₆₁×#385	1	1	yes	characteristic
275	C₄	C₄×D₈×C₆₁	1	1	yes	characteristic
276	C₄	C₆₁×#447	1	1	yes	characteristic
277	C₄	C₆₁×#478	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	8	2	no	—
282	C₂	C₆₁×#1862	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₆₁	#447
commutator	C₂×C₈	C₂²×C₄×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
socle	C₂×C₆₁	#1862
prime radical	C₂×#135×C₆₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₆₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#2161	C₂	semidirect
5. C₆₁×#1953	C₂	semidirect
6. C₆₁×#1381	C₂	semidirect
7. C₆₁×#1379	C₂	semidirect
21. C₆₁×#362	C₄	semidirect
22. C₆₁×#361	C₄	semidirect
124. #12942	C₆₁	direct (N × H)
138. #2161	C₂×C₆₁	semidirect
141. #1953	C₂×C₆₁	semidirect
142. #1381	C₂×C₆₁	semidirect
143. #1379	C₂×C₆₁	semidirect
162. #362	C₄×C₆₁	semidirect
163. #361	C₄×C₆₁	semidirect
166. C₆₁	#12942	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₁×#2957	2
C₆₁×#2161	2
C₆₁×#2148	2
C₂×C₆₁×#359	2
C₆₁×#1953	2
C₆₁×#1381	2
C₆₁×#1379	2
#12942	61

Subgroups — C61×#12942 #29516375

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#12942 #29516375