Subgroups — C₆₁×#17336 #29520769

Order 15616.

782 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2965	C₂	1	2	yes	maximal
2	C₄×#131×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1455	C₂	1	2	yes	maximal, complemented
4	C₆₁×#1453	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1451	C₂	1	4	yes	maximal, complemented
6	C₆₁×#1451	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1450	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₆₁×#322	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#131×C₆₁	C₄	1	2	yes	—
10	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
11	C₆₁×#446	C₄	1	4	yes	—
12	C₆₁×#436	C₄	1	4	yes	—
13	C₆₁×#435	C₄	1	8	yes	—
14	C₆₁×#431	C₂²	1	2	yes	—
15	C₆₁×#431	C₂²	1	4	yes	—
16	C₆₁×#405	C₄	1	4	yes	—
17	C₆₁×#405	C₄	1	2	yes	—
18	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
19	C₄×M₁₆×C₆₁	C₂²	1	2	yes	—
20	C₄×M₁₆×C₆₁	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
24	C₆₁×#322	C₂²	1	2	yes	—
25	C₆₁×#322	C₂²	1	1	yes	characteristic
26	C₆₁×#322	C₂²	1	1	yes	characteristic
27	C₆₁×#322	C₂²	1	2	yes	—
28	C₆₁×#322	C₂²	1	4	yes	—
29	C₆₁×#322	C₂²	1	4	yes	—
30	C₈²×C₆₁	C₂²	1	2	yes	—
31	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
32	#131×C₆₁	—	2	8	no	—
33	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
34	C₂×M₁₆×C₆₁	C₂×C₄	1	4	yes	—
35	C₂×M₁₆×C₆₁	C₂×C₄	1	8	yes	—
36	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
38	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
39	C₄×Q₈×C₆₁	C₂×C₄	1	2	yes	—
40	C₄×D₄×C₆₁	C₂×C₄	1	4	yes	—
41	C₄×D₄×C₆₁	C₂×C₄	1	2	yes	—
42	#117×C₆₁	C₂×C₄	1	2	yes	—
43	#117×C₆₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
46	C₄⋊C₈×C₆₁	C₂×C₄	1	8	yes	—
47	C₄⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
48	C₂²⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
49	C₂²⋊C₈×C₆₁	C₂×C₄	1	8	yes	—
50	#97×C₆₁	C₂×C₄	1	2	yes	—
51	#97×C₆₁	C₂×C₄	1	4	yes	—
52	#97×C₆₁	C₂×C₄	1	8	yes	—
53	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
54	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
56	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₆₁	C₂³	1	4	yes	—
58	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₆₁	C₂³	1	2	yes	—
60	C₄×C₈×C₆₁	C₂³	1	2	yes	—
61	C₄×C₈×C₆₁	C₂³	1	2	yes	—
62	#41×C₆₁	—	2	4	no	—
63	C₂×Q₈×C₆₁	C₄²	1	1	yes	characteristic
64	C₂×D₄×C₆₁	C₄²	1	2	yes	—
65	C₂×D₄×C₆₁	C₄²	1	1	yes	characteristic
66	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
70	M₁₆×C₆₁	—	2	4	no	—
71	M₁₆×C₆₁	—	2	8	no	—
72	M₁₆×C₆₁	—	4	8	no	—
73	C₂×C₈×C₆₁	—	2	8	no	—
74	C₂×C₈×C₆₁	—	4	4	no	—
75	C₂×C₈×C₆₁	M₁₆	1	8	yes	complemented
76	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
77	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
78	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₆₁	C₂²×C₄	1	8	yes	—
80	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
82	C₄⋊C₄×C₆₁	C₄²	1	4	yes	—
83	C₄⋊C₄×C₆₁	C₄²	1	2	yes	—
84	C₂²⋊C₄×C₆₁	C₄²	1	4	yes	—
85	C₂²⋊C₄×C₆₁	C₄²	1	2	yes	—
86	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
88	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
89	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
90	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
91	C₂³×C₆₁	C₂×C₄²	1	2	yes	—
92	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
93	Q₈×C₆₁	—	2	2	no	—
94	D₄×C₆₁	—	2	4	no	—
95	D₄×C₆₁	—	2	2	no	—
96	C₂×C₄×C₆₁	—	2	4	no	—
97	C₂×C₄×C₆₁	—	2	4	no	—
98	C₂×C₄×C₆₁	—	2	8	no	—
99	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
100	C₂×C₄×C₆₁	C₂×C₄²	1	4	yes	—
101	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₆₁	C₂×M₁₆	1	4	yes	—
104	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
106	C₈×C₆₁	—	2	4	no	—
107	C₈×C₆₁	—	2	4	no	—
108	C₈×C₆₁	—	2	4	no	—
109	C₈×C₆₁	—	2	8	no	—
110	C₈×C₆₁	C₂×M₁₆	1	4	yes	—
111	C₈×C₆₁	C₂×M₁₆	1	4	yes	—
112	#17336	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂²×C₆₁	—	2	2	no	—
114	C₂²×C₆₁	—	2	2	no	—
115	C₂²×C₆₁	—	2	4	no	—
116	C₂²×C₆₁	C₄×M₁₆	1	2	yes	—
117	C₂²×C₆₁	C₂²×C₄²	1	1	yes	characteristic
118	C₄×C₆₁	—	2	2	no	—
119	C₄×C₆₁	—	2	2	no	—
120	C₄×C₆₁	—	2	4	no	—
121	C₄×C₆₁	C₄×M₁₆	1	2	yes	—
122	C₄×C₆₁	C₂²×M₁₆	1	2	yes	—
123	C₄×C₆₁	#568	1	4	yes	—
124	#2965	C₂×C₆₁	1	2	yes	—
125	C₄×#131	C₂×C₆₁	1	1	yes	characteristic
126	#1455	C₂×C₆₁	1	2	yes	complemented
127	#1453	C₂×C₆₁	1	2	yes	complemented
128	#1451	C₂×C₆₁	1	4	yes	complemented
129	#1451	C₂×C₆₁	1	2	yes	complemented
130	#1450	C₂×C₆₁	1	1	yes	characteristic, complemented
131	C₂×#322	C₂×C₆₁	1	1	yes	characteristic, complemented
132	C₂×C₆₁	—	2	2	no	—
133	C₂×C₆₁	—	4	2	no	—
134	C₂×C₆₁	C₂×C₄×M₁₆	1	1	yes	characteristic
135	C₂×C₆₁	#2875	1	2	yes	—
136	C₂×#131	C₄×C₆₁	1	2	yes	—
137	C₄×#41	C₂²×C₆₁	1	1	yes	characteristic
138	#446	C₄×C₆₁	1	4	yes	—
139	#436	C₄×C₆₁	1	4	yes	—
140	#435	C₄×C₆₁	1	8	yes	—
141	#431	C₂²×C₆₁	1	2	yes	—
142	#431	C₂²×C₆₁	1	4	yes	—
143	#405	C₄×C₆₁	1	4	yes	—
144	#405	C₄×C₆₁	1	2	yes	—
145	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
146	C₄×M₁₆	C₂²×C₆₁	1	2	yes	—
147	C₄×M₁₆	C₂²×C₆₁	1	4	yes	—
148	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
149	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
150	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
151	#322	C₂²×C₆₁	1	2	yes	—
152	#322	C₂²×C₆₁	1	1	yes	characteristic
153	#322	C₂²×C₆₁	1	1	yes	characteristic
154	#322	C₂²×C₆₁	1	2	yes	—
155	#322	C₂²×C₆₁	1	4	yes	—
156	#322	C₂²×C₆₁	1	4	yes	—
157	C₈²	C₂²×C₆₁	1	2	yes	—
158	C₆₁	#17336	1	1	yes	characteristic, direct product factor, Sylow
159	C₂×#41	C₂×C₄×C₆₁	1	1	yes	characteristic
160	#131	—	2	8	no	—
161	C₂×M₁₆	C₂×C₄×C₆₁	1	2	yes	—
162	C₂×M₁₆	C₂×C₄×C₆₁	1	4	yes	—
163	C₂×M₁₆	C₂×C₄×C₆₁	1	8	yes	—
164	C₂²×C₈	C₂×C₄×C₆₁	1	4	yes	—
165	C₂²×C₈	C₂×C₄×C₆₁	1	4	yes	—
166	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
167	C₄×Q₈	C₂×C₄×C₆₁	1	2	yes	—
168	C₄×D₄	C₂×C₄×C₆₁	1	4	yes	—
169	C₄×D₄	C₂×C₄×C₆₁	1	2	yes	—
170	#117	C₂×C₄×C₆₁	1	2	yes	—
171	#117	C₂×C₄×C₆₁	1	1	yes	characteristic
172	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
173	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
174	C₄⋊C₈	C₂×C₄×C₆₁	1	8	yes	—
175	C₄⋊C₈	C₂×C₄×C₆₁	1	4	yes	—
176	C₂²⋊C₈	C₂×C₄×C₆₁	1	4	yes	—
177	C₂²⋊C₈	C₂×C₄×C₆₁	1	8	yes	—
178	#97	C₂×C₄×C₆₁	1	2	yes	—
179	#97	C₂×C₄×C₆₁	1	4	yes	—
180	#97	C₂×C₄×C₆₁	1	8	yes	—
181	C₄×C₈	C₂×C₄×C₆₁	1	4	yes	—
182	C₄×C₈	C₂×C₄×C₆₁	1	2	yes	—
183	C₄×C₈	C₂×C₄×C₆₁	1	4	yes	—
184	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
185	C₄×C₈	C₂³×C₆₁	1	4	yes	—
186	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
187	C₄×C₈	C₂³×C₆₁	1	2	yes	—
188	C₄×C₈	C₂³×C₆₁	1	2	yes	—
189	C₄×C₈	C₂³×C₆₁	1	2	yes	—
190	#41	—	2	4	no	—
191	C₂×Q₈	C₄²×C₆₁	1	1	yes	characteristic
192	C₂×D₄	C₄²×C₆₁	1	2	yes	—
193	C₂×D₄	C₄²×C₆₁	1	1	yes	characteristic
194	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
195	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
196	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
197	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
198	M₁₆	—	2	4	no	—
199	M₁₆	—	2	8	no	—
200	M₁₆	—	4	8	no	—
201	C₂×C₈	—	2	8	no	—
202	C₂×C₈	—	4	4	no	—
203	C₂×C₈	M₁₆×C₆₁	1	8	yes	complemented
204	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
205	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
206	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
207	C₂×C₈	C₂²×C₄×C₆₁	1	8	yes	—
208	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
209	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
210	C₄⋊C₄	C₄²×C₆₁	1	4	yes	—
211	C₄⋊C₄	C₄²×C₆₁	1	2	yes	—
212	C₂²⋊C₄	C₄²×C₆₁	1	4	yes	—
213	C₂²⋊C₄	C₄²×C₆₁	1	2	yes	—
214	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
215	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
216	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
217	C₄²	C₂²×C₄×C₆₁	1	4	yes	—
218	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
219	C₂³	C₂×C₄²×C₆₁	1	2	yes	—
220	C₂³	C₂×C₄²×C₆₁	1	1	yes	characteristic
221	Q₈	—	2	2	no	—
222	D₄	—	2	4	no	—
223	D₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	C₂×C₄²×C₆₁	1	2	yes	—
228	C₂×C₄	C₂×C₄²×C₆₁	1	4	yes	—
229	C₂×C₄	C₂×C₄²×C₆₁	1	2	yes	—
230	C₂×C₄	C₂×C₄²×C₆₁	1	1	yes	characteristic
231	C₂×C₄	C₂×M₁₆×C₆₁	1	4	yes	—
232	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
233	C₂×C₄	C₂³×C₄×C₆₁	1	2	yes	—
234	C₈	—	2	4	no	—
235	C₈	—	2	4	no	—
236	C₈	—	2	4	no	—
237	C₈	—	2	8	no	—
238	C₈	C₂×M₁₆×C₆₁	1	4	yes	—
239	C₈	C₂×M₁₆×C₆₁	1	4	yes	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	4	no	—
243	C₂²	C₄×M₁₆×C₆₁	1	2	yes	—
244	C₂²	C₂²×C₄²×C₆₁	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	4	no	—
248	C₄	C₄×M₁₆×C₆₁	1	2	yes	—
249	C₄	C₂²×M₁₆×C₆₁	1	2	yes	—
250	C₄	C₆₁×#568	1	4	yes	—
251	C₂	—	2	2	no	—
252	C₂	—	4	2	no	—
253	C₂	C₂×C₄×M₁₆×C₆₁	1	1	yes	characteristic
254	C₂	C₆₁×#2875	1	2	yes	—

	subgroup	G/H
center	C₄²×C₆₁	C₂⁴
commutator	C₂²	C₂²×C₄²×C₆₁
Frattini, cosocle	C₄²	C₂⁴×C₆₁
socle	C₂²×C₆₁	C₂²×C₄²
prime radical	C₂×#41×C₆₁	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₁×#1455	C₂	semidirect
4. C₆₁×#1453	C₂	semidirect
5. C₆₁×#1451	C₂	semidirect
6. C₆₁×#1451	C₂	semidirect
7. C₆₁×#1450	C₂	semidirect
8. C₂×C₆₁×#322	C₂	semidirect
75. C₂×C₈×C₆₁	M₁₆	semidirect
112. #17336	C₆₁	direct (N × H)
126. #1455	C₂×C₆₁	semidirect
127. #1453	C₂×C₆₁	semidirect
128. #1451	C₂×C₆₁	semidirect
129. #1451	C₂×C₆₁	semidirect
130. #1450	C₂×C₆₁	semidirect
131. C₂×#322	C₂×C₆₁	semidirect
158. C₆₁	#17336	direct (N × H)
203. C₂×C₈	M₁₆×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#2965	2
C₄×#131×C₆₁	2
C₆₁×#1455	2
C₆₁×#1453	2
C₆₁×#1451	2
C₆₁×#1450	2
C₂×C₆₁×#322	2
#17336	61

Subgroups — C61×#17336 #29520769

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#17336 #29520769