Subgroups — C₆₁×#11672 #29515105

Order 15616.

334 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3170	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1567	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1332	C₂	1	4	yes	maximal, complemented
5	C₆₁×#533	—	2	1	no	—
6	C₂²×C₁₆×C₆₁	—	2	1	no	—
7	C₆₁×#500	C₄	1	1	yes	characteristic
8	C₆₁×#494	C₄	1	1	yes	characteristic
9	C₆₁×#487	—	2	1	no	—
10	C₆₁×#467	—	2	1	no	—
11	C₆₁×#467	C₄	1	2	yes	—
12	C₄×Q₁₆×C₆₁	—	2	1	no	—
13	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
14	C₆₁×#423	—	2	1	no	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₄×C₁₆×C₆₁	—	2	1	no	—
17	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
18	C₆₁×#335	—	2	1	no	—
19	C₆₁×#335	C₂²	1	2	yes	—
20	C₂×Q₁₆×C₆₁	—	2	1	no	—
21	C₂×Q₁₆×C₆₁	C₂×C₄	1	2	yes	—
22	C₂×M₁₆×C₆₁	—	4	1	no	—
23	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
25	C₄⋊Q₈×C₆₁	—	2	1	no	—
26	C₄⋊Q₈×C₆₁	—	2	1	no	—
27	#125×C₆₁	—	4	1	no	—
28	C₂²⋊Q₈×C₆₁	—	2	2	no	—
29	C₂²⋊Q₈×C₆₁	—	4	1	no	—
30	C₄×Q₈×C₆₁	—	2	2	no	—
31	C₄×Q₈×C₆₁	—	4	1	no	—
32	#117×C₆₁	—	4	1	no	—
33	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₆₁	—	2	4	no	—
35	C₂×C₁₆×C₆₁	—	2	2	no	—
36	#107×C₆₁	—	2	1	no	—
37	#107×C₆₁	C₂×C₄	1	2	yes	—
38	C₄⋊C₈×C₆₁	—	2	2	no	—
39	C₄⋊C₈×C₆₁	—	4	1	no	—
40	#103×C₆₁	—	2	2	no	—
41	#103×C₆₁	—	2	2	no	—
42	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
44	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
45	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₆₁	—	2	2	no	—
47	C₂×Q₈×C₆₁	—	4	1	no	—
48	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	SD₁₆	1	1	yes	characteristic
51	Q₁₆×C₆₁	—	2	4	no	—
52	Q₁₆×C₆₁	—	4	2	no	—
53	M₁₆×C₆₁	—	8	2	no	—
54	C₂×C₈×C₆₁	—	2	2	no	—
55	C₂×C₈×C₆₁	—	2	2	no	—
56	C₂×C₈×C₆₁	—	4	2	no	—
57	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₆₁	—	2	2	no	—
62	C₄⋊C₄×C₆₁	—	4	1	no	—
63	C₄⋊C₄×C₆₁	—	4	1	no	—
64	C₄⋊C₄×C₆₁	—	4	2	no	—
65	C₄⋊C₄×C₆₁	—	4	1	no	—
66	C₄⋊C₄×C₆₁	—	4	2	no	—
67	C₂²⋊C₄×C₆₁	—	4	2	no	—
68	C₄²×C₆₁	—	4	2	no	—
69	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₆₁	D₈	1	1	yes	characteristic
71	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
72	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
73	C₁₆×C₆₁	—	2	4	no	—
74	C₂³×C₆₁	#102	1	1	yes	characteristic
75	Q₈×C₆₁	—	4	2	no	—
76	Q₈×C₆₁	—	4	2	no	—
77	Q₈×C₆₁	—	8	2	no	—
78	C₂×C₄×C₆₁	—	2	2	no	—
79	C₂×C₄×C₆₁	—	2	1	no	—
80	C₂×C₄×C₆₁	—	2	1	no	—
81	C₂×C₄×C₆₁	—	4	2	no	—
82	C₂×C₄×C₆₁	—	4	2	no	—
83	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
84	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
85	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
86	C₂×C₄×C₆₁	SD₃₂	1	2	yes	—
87	C₂×C₄×C₆₁	Q₃₂	1	2	yes	—
88	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
91	C₈×C₆₁	—	2	1	no	—
92	C₈×C₆₁	—	2	1	no	—
93	C₈×C₆₁	—	8	2	no	—
94	C₈×C₆₁	#104	1	2	yes	—
95	C₈×C₆₁	#131	1	2	yes	—
96	#11672	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂²×C₆₁	—	2	1	no	—
98	C₂²×C₆₁	—	2	1	no	—
99	C₂²×C₆₁	#358	1	2	yes	—
100	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
101	C₄×C₆₁	—	2	1	no	—
102	C₄×C₆₁	—	2	1	no	—
103	C₄×C₆₁	—	8	2	no	—
104	C₄×C₆₁	—	8	2	no	—
105	C₄×C₆₁	#358	1	2	yes	—
106	C₄×C₆₁	#359	1	2	yes	—
107	C₄×C₆₁	#408	1	1	yes	characteristic
108	C₄×C₆₁	C₂×#104	1	1	yes	characteristic
109	C₄×C₆₁	C₂×SD₃₂	1	1	yes	characteristic
110	C₄×C₆₁	C₂×Q₃₂	1	1	yes	characteristic
111	#3170	C₂×C₆₁	1	1	yes	characteristic
112	C₂×C₄×C₁₆	C₂×C₆₁	1	1	yes	characteristic
113	#1567	C₂×C₆₁	1	1	yes	characteristic
114	#1332	C₂×C₆₁	1	4	yes	complemented
115	C₂×C₆₁	—	2	2	no	—
116	C₂×C₆₁	#1492	1	1	yes	characteristic
117	C₂×C₆₁	C₂×#358	1	1	yes	characteristic
118	C₂×C₆₁	C₂×#359	1	1	yes	characteristic
119	#533	—	2	1	no	—
120	C₂²×C₁₆	—	2	1	no	—
121	#500	C₄×C₆₁	1	1	yes	characteristic
122	#494	C₄×C₆₁	1	1	yes	characteristic
123	#487	—	2	1	no	—
124	#467	—	2	1	no	—
125	#467	C₄×C₆₁	1	2	yes	—
126	C₄×Q₁₆	—	2	1	no	—
127	C₄×Q₁₆	C₂²×C₆₁	1	2	yes	—
128	#423	—	2	1	no	—
129	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
130	C₄×C₁₆	—	2	1	no	—
131	C₄×C₁₆	C₂²×C₆₁	1	2	yes	—
132	#335	—	2	1	no	—
133	#335	C₂²×C₆₁	1	2	yes	—
134	C₆₁	#11672	1	1	yes	characteristic, direct product factor, Sylow
135	C₂×Q₁₆	—	2	1	no	—
136	C₂×Q₁₆	C₂×C₄×C₆₁	1	2	yes	—
137	C₂×M₁₆	—	4	1	no	—
138	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
139	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
140	C₄⋊Q₈	—	2	1	no	—
141	C₄⋊Q₈	—	2	1	no	—
142	#125	—	4	1	no	—
143	C₂²⋊Q₈	—	2	2	no	—
144	C₂²⋊Q₈	—	4	1	no	—
145	C₄×Q₈	—	2	2	no	—
146	C₄×Q₈	—	4	1	no	—
147	#117	—	4	1	no	—
148	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
149	C₂×C₁₆	—	2	4	no	—
150	C₂×C₁₆	—	2	2	no	—
151	#107	—	2	1	no	—
152	#107	C₂×C₄×C₆₁	1	2	yes	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	4	1	no	—
155	#103	—	2	2	no	—
156	#103	—	2	2	no	—
157	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
158	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
159	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
160	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
161	C₂×Q₈	—	2	2	no	—
162	C₂×Q₈	—	4	1	no	—
163	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
164	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
165	C₂²×C₄	SD₁₆×C₆₁	1	1	yes	characteristic
166	Q₁₆	—	2	4	no	—
167	Q₁₆	—	4	2	no	—
168	M₁₆	—	8	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
173	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
174	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
175	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	4	1	no	—
178	C₄⋊C₄	—	4	1	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	1	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₂²⋊C₄	—	4	2	no	—
183	C₄²	—	4	2	no	—
184	C₄²	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
185	C₄²	D₈×C₆₁	1	1	yes	characteristic
186	C₄²	SD₁₆×C₆₁	1	1	yes	characteristic
187	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
188	C₁₆	—	2	4	no	—
189	C₂³	#102×C₆₁	1	1	yes	characteristic
190	Q₈	—	4	2	no	—
191	Q₈	—	4	2	no	—
192	Q₈	—	8	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	#102×C₆₁	1	1	yes	characteristic
199	C₂×C₄	#102×C₆₁	1	1	yes	characteristic
200	C₂×C₄	#102×C₆₁	1	1	yes	characteristic
201	C₂×C₄	SD₃₂×C₆₁	1	2	yes	—
202	C₂×C₄	Q₃₂×C₆₁	1	2	yes	—
203	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
204	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
205	C₂×C₄	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	8	2	no	—
209	C₈	#104×C₆₁	1	2	yes	—
210	C₈	#131×C₆₁	1	2	yes	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	C₆₁×#358	1	2	yes	—
214	C₂²	C₂×#102×C₆₁	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	8	2	no	—
218	C₄	—	8	2	no	—
219	C₄	C₆₁×#358	1	2	yes	—
220	C₄	C₆₁×#359	1	2	yes	—
221	C₄	C₆₁×#408	1	1	yes	characteristic
222	C₄	C₂×#104×C₆₁	1	1	yes	characteristic
223	C₄	C₂×SD₃₂×C₆₁	1	1	yes	characteristic
224	C₄	C₂×Q₃₂×C₆₁	1	1	yes	characteristic
225	C₂	—	2	2	no	—
226	C₂	C₆₁×#1492	1	1	yes	characteristic
227	C₂	C₂×C₆₁×#358	1	1	yes	characteristic
228	C₂	C₂×C₆₁×#359	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₆₁
Frattini, cosocle	C₄×C₈	C₂³×C₆₁
socle	C₂²×C₆₁	C₂×#102
prime radical	C₂³×C₆₁	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₁×#1332	C₂	semidirect
96. #11672	C₆₁	direct (N × H)
114. #1332	C₂×C₆₁	semidirect
134. C₆₁	#11672	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×#3170	2
C₂×C₄×C₁₆×C₆₁	2
C₆₁×#1567	2
C₆₁×#1332	2
#11672	61

Subgroups — C61×#11672 #29515105

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#11672 #29515105