Subgroups — C₆₁×#11583 #29515016

Order 15616.

450 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×M₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×D₄⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₆₁	—	2	1	no	—
6	C₂×C₄×D₄×C₆₁	C₄	1	1	yes	characteristic
7	C₂×M₃₂×C₆₁	—	2	1	no	—
8	C₂×M₃₂×C₆₁	—	2	1	no	—
9	C₈×D₄×C₆₁	—	2	1	no	—
10	C₈×D₄×C₆₁	—	2	1	no	—
11	C₈×D₄×C₆₁	C₄	1	2	yes	—
12	C₈×D₄×C₆₁	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₆₁	C₄	1	1	yes	characteristic
14	C₂×C₂²⋊C₈×C₆₁	—	2	1	no	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×C₄⋊C₁₆	—	2	1	no	—
17	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	complemented
18	C₆₁×#346	—	2	1	no	—
19	C₄×C₁₆×C₆₁	C₂²	1	2	yes	complemented
20	C₂²×D₄×C₆₁	C₈	1	1	yes	characteristic
21	C₂³×C₄×C₆₁	—	2	1	no	—
22	C₂²×C₈×C₆₁	—	2	1	no	—
23	C₂²×C₈×C₆₁	—	2	2	no	—
24	C₂²×C₈×C₆₁	—	2	2	no	—
25	C₂²×C₈×C₆₁	—	4	2	no	—
26	C₂²×C₈×C₆₁	—	4	2	no	—
27	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
29	C₄×D₄×C₆₁	—	2	1	no	—
30	C₄×D₄×C₆₁	—	2	1	no	—
31	C₄×D₄×C₆₁	C₈	1	2	yes	—
32	C₄×D₄×C₆₁	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₆₁	C₈	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
35	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
36	M₃₂×C₆₁	—	2	4	no	—
37	M₃₂×C₆₁	—	4	2	no	—
38	C₂×C₁₆×C₆₁	—	2	2	no	—
39	C₂×C₁₆×C₆₁	—	2	2	no	—
40	C₄⋊C₈×C₆₁	—	2	1	no	—
41	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
42	C₂²⋊C₈×C₆₁	—	2	2	no	—
43	C₂²⋊C₈×C₆₁	—	2	2	no	—
44	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
46	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
47	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
48	C₂⁴×C₆₁	—	2	1	no	—
49	C₂×D₄×C₆₁	—	2	4	no	—
50	C₂×D₄×C₆₁	—	2	1	no	—
51	C₂×D₄×C₆₁	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₆₁	—	2	1	no	—
53	C₂²×C₄×C₆₁	—	2	2	no	—
54	C₂²×C₄×C₆₁	—	2	2	no	—
55	C₂²×C₄×C₆₁	—	4	2	no	—
56	C₂²×C₄×C₆₁	—	4	2	no	—
57	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
60	C₂×C₈×C₆₁	—	2	2	no	—
61	C₂×C₈×C₆₁	—	2	2	no	—
62	C₂×C₈×C₆₁	—	4	2	no	—
63	C₂×C₈×C₆₁	—	4	4	no	—
64	C₂×C₈×C₆₁	—	4	2	no	—
65	C₂×C₈×C₆₁	—	4	2	no	—
66	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₆₁	D₈	1	2	yes	—
69	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
70	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₆₁	—	2	1	no	—
73	C₄⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
74	C₂²⋊C₄×C₆₁	—	2	2	no	—
75	C₂²⋊C₄×C₆₁	—	2	2	no	—
76	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
78	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
79	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
80	C₁₆×C₆₁	—	2	4	no	—
81	C₁₆×C₆₁	—	4	2	no	—
82	C₂³×C₆₁	—	2	4	no	—
83	C₂³×C₆₁	—	2	2	no	—
84	C₂³×C₆₁	—	4	2	no	—
85	C₂³×C₆₁	—	4	2	no	—
86	C₂³×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
87	D₄×C₆₁	—	2	4	no	—
88	D₄×C₆₁	—	2	2	no	—
89	D₄×C₆₁	M₃₂	1	4	yes	complemented
90	C₂×C₄×C₆₁	—	2	2	no	—
91	C₂×C₄×C₆₁	—	2	2	no	—
92	C₂×C₄×C₆₁	—	4	1	no	—
93	C₂×C₄×C₆₁	—	4	2	no	—
94	C₂×C₄×C₆₁	—	4	4	no	—
95	C₂×C₄×C₆₁	—	4	2	no	—
96	C₂×C₄×C₆₁	—	4	2	no	—
97	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
98	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
99	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
100	C₂×C₄×C₆₁	#102	1	2	yes	—
101	C₂×C₄×C₆₁	#104	1	2	yes	—
102	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
105	C₈×C₆₁	—	2	1	no	—
106	C₈×C₆₁	—	2	1	no	—
107	C₈×C₆₁	—	4	2	no	—
108	C₈×C₆₁	#102	1	2	yes	—
109	C₈×C₆₁	C₂×D₈	1	1	yes	characteristic
110	C₈×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
111	#11583	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂²×C₆₁	—	2	1	no	—
113	C₂²×C₆₁	—	2	1	no	—
114	C₂²×C₆₁	—	2	4	no	—
115	C₂²×C₆₁	—	4	8	no	—
116	C₂²×C₆₁	—	4	2	no	—
117	C₂²×C₆₁	—	4	2	no	—
118	C₂²×C₆₁	D₄⋊C₈	1	2	yes	—
119	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₄×C₆₁	—	2	1	no	—
121	C₄×C₆₁	—	2	1	no	—
122	C₄×C₆₁	—	2	1	no	—
123	C₄×C₆₁	—	4	2	no	—
124	C₄×C₆₁	D₄⋊C₈	1	2	yes	—
125	C₄×C₆₁	C₂×#102	1	1	yes	characteristic
126	C₄×C₆₁	C₂×#104	1	1	yes	characteristic
127	C₄×C₆₁	C₂×M₃₂	1	2	yes	—
128	C₂×C₈×D₄	C₂×C₆₁	1	1	yes	characteristic
129	#2150	C₂×C₆₁	1	1	yes	characteristic, complemented
130	C₄×M₃₂	C₂×C₆₁	1	1	yes	characteristic, complemented
131	D₄⋊C₁₆	C₂×C₆₁	1	4	yes	complemented
132	C₂×C₆₁	—	2	2	no	—
133	C₂×C₆₁	—	4	4	no	—
134	C₂×C₆₁	C₂×D₄⋊C₈	1	1	yes	characteristic
135	C₂×C₆₁	#2112	1	1	yes	characteristic
136	C₂×C₆₁	#2117	1	1	yes	characteristic
137	C₂³×C₈	—	2	1	no	—
138	C₂×C₄×D₄	C₄×C₆₁	1	1	yes	characteristic
139	C₂×M₃₂	—	2	1	no	—
140	C₂×M₃₂	—	2	1	no	—
141	C₈×D₄	—	2	1	no	—
142	C₈×D₄	—	2	1	no	—
143	C₈×D₄	C₄×C₆₁	1	2	yes	—
144	C₈×D₄	C₂²×C₆₁	1	2	yes	—
145	C₂×C₄⋊C₈	C₄×C₆₁	1	1	yes	characteristic
146	C₂×C₂²⋊C₈	—	2	1	no	—
147	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
148	C₄⋊C₁₆	—	2	1	no	—
149	C₄⋊C₁₆	C₂²×C₆₁	1	2	yes	complemented
150	#346	—	2	1	no	—
151	C₄×C₁₆	C₂²×C₆₁	1	2	yes	complemented
152	C₆₁	#11583	1	1	yes	characteristic, direct product factor, Sylow
153	C₂²×D₄	C₈×C₆₁	1	1	yes	characteristic
154	C₂³×C₄	—	2	1	no	—
155	C₂²×C₈	—	2	1	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	4	2	no	—
159	C₂²×C₈	—	4	2	no	—
160	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
161	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
162	C₄×D₄	—	2	1	no	—
163	C₄×D₄	—	2	1	no	—
164	C₄×D₄	C₈×C₆₁	1	2	yes	—
165	C₄×D₄	C₂×C₄×C₆₁	1	2	yes	—
166	C₂×C₄⋊C₄	C₈×C₆₁	1	1	yes	characteristic
167	C₂×C₂²⋊C₄	—	2	1	no	—
168	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
169	M₃₂	—	2	4	no	—
170	M₃₂	—	4	2	no	—
171	C₂×C₁₆	—	2	2	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₄⋊C₈	—	2	1	no	—
174	C₄⋊C₈	C₂×C₄×C₆₁	1	2	yes	—
175	C₂²⋊C₈	—	2	2	no	—
176	C₂²⋊C₈	—	2	2	no	—
177	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
178	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
179	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
180	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
181	C₂⁴	—	2	1	no	—
182	C₂×D₄	—	2	4	no	—
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	C₂×C₈×C₆₁	1	2	yes	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₈×C₆₁	1	1	yes	characteristic
192	C₂²×C₄	M₁₆×C₆₁	1	1	yes	characteristic
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	4	4	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
200	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
201	C₂×C₈	D₈×C₆₁	1	2	yes	—
202	C₂×C₈	SD₁₆×C₆₁	1	2	yes	—
203	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
204	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	C₂×C₈×C₆₁	1	2	yes	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	2	2	no	—
209	C₄²	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
210	C₄²	C₂×C₈×C₆₁	1	1	yes	characteristic
211	C₄²	M₁₆×C₆₁	1	1	yes	characteristic
212	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
213	C₁₆	—	2	4	no	—
214	C₁₆	—	4	2	no	—
215	C₂³	—	2	4	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	4	2	no	—
218	C₂³	—	4	2	no	—
219	C₂³	C₂²⋊C₈×C₆₁	1	1	yes	characteristic
220	D₄	—	2	4	no	—
221	D₄	—	2	2	no	—
222	D₄	M₃₂×C₆₁	1	4	yes	complemented
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	C₂²⋊C₈×C₆₁	1	1	yes	characteristic
231	C₂×C₄	C₂²⋊C₈×C₆₁	1	1	yes	characteristic
232	C₂×C₄	C₂²⋊C₈×C₆₁	1	1	yes	characteristic
233	C₂×C₄	#102×C₆₁	1	2	yes	—
234	C₂×C₄	#104×C₆₁	1	2	yes	—
235	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
236	C₂×C₄	C₂²×C₈×C₆₁	1	1	yes	characteristic
237	C₂×C₄	C₂×M₁₆×C₆₁	1	1	yes	characteristic
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	4	2	no	—
241	C₈	#102×C₆₁	1	2	yes	—
242	C₈	C₂×D₈×C₆₁	1	1	yes	characteristic
243	C₈	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	4	8	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	C₆₁×D₄⋊C₈	1	2	yes	—
251	C₂²	C₂×C₂²⋊C₈×C₆₁	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	4	2	no	—
256	C₄	C₆₁×D₄⋊C₈	1	2	yes	—
257	C₄	C₂×#102×C₆₁	1	1	yes	characteristic
258	C₄	C₂×#104×C₆₁	1	1	yes	characteristic
259	C₄	C₂×M₃₂×C₆₁	1	2	yes	—
260	C₂	—	2	2	no	—
261	C₂	—	4	4	no	—
262	C₂	C₂×C₆₁×D₄⋊C₈	1	1	yes	characteristic
263	C₂	C₆₁×#2112	1	1	yes	characteristic
264	C₂	C₆₁×#2117	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₆₁×#2150	C₂	semidirect
3. C₄×M₃₂×C₆₁	C₂	semidirect
4. C₆₁×D₄⋊C₁₆	C₂	semidirect
17. C₆₁×C₄⋊C₁₆	C₂²	semidirect
19. C₄×C₁₆×C₆₁	C₂²	semidirect
89. D₄×C₆₁	M₃₂	semidirect
111. #11583	C₆₁	direct (N × H)
129. #2150	C₂×C₆₁	semidirect
130. C₄×M₃₂	C₂×C₆₁	semidirect
131. D₄⋊C₁₆	C₂×C₆₁	semidirect
149. C₄⋊C₁₆	C₂²×C₆₁	semidirect
151. C₄×C₁₆	C₂²×C₆₁	semidirect
152. C₆₁	#11583	direct (N × H)
222. D₄	M₃₂×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₆₁	2
C₆₁×#2150	2
C₄×M₃₂×C₆₁	2
C₆₁×D₄⋊C₁₆	2
#11583	61

Subgroups — C61×#11583 #29515016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#11583 #29515016