Subgroups — C₆₁×#13269 #29516702

Order 15616.

308 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2961	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2153	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2110	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1378	C₂	1	2	yes	maximal
5	C₆₁×#1375	C₂	1	2	yes	maximal
6	C₆₁×#533	C₄	1	1	yes	characteristic
7	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
8	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
9	C₆₁×#446	—	2	2	no	—
10	C₆₁×#446	—	2	2	no	—
11	C₆₁×#424	C₄	1	1	yes	characteristic
12	C₆₁×#424	C₄	1	1	yes	characteristic
13	C₆₁×#423	C₄	1	1	yes	characteristic
14	C₆₁×#405	C₂²	1	1	yes	characteristic
15	C₆₁×#405	C₂²	1	1	yes	characteristic
16	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#364	—	2	2	no	—
18	C₆₁×#347	—	2	2	no	—
19	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
20	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
23	C₄⋊Q₈×C₆₁	—	2	1	no	—
24	#125×C₆₁	—	2	1	no	—
25	C₂²⋊Q₈×C₆₁	—	2	2	no	—
26	C₄×Q₈×C₆₁	—	2	2	no	—
27	#117×C₆₁	C₂×C₄	1	1	yes	characteristic
28	#117×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
30	M₃₂×C₆₁	—	2	2	no	—
31	M₃₂×C₆₁	—	2	2	no	—
32	M₃₂×C₆₁	—	2	2	no	—
33	M₃₂×C₆₁	—	2	2	no	—
34	C₂×C₁₆×C₆₁	—	2	2	no	—
35	C₂×C₁₆×C₆₁	Q₈	1	4	yes	complemented
36	C₄⋊C₈×C₆₁	—	2	2	no	—
37	C₄⋊C₈×C₆₁	—	2	2	no	—
38	C₄⋊C₈×C₆₁	—	2	2	no	—
39	#97×C₆₁	—	2	1	no	—
40	#97×C₆₁	—	2	1	no	—
41	#97×C₆₁	—	2	1	no	—
42	#97×C₆₁	D₄	1	2	yes	—
43	C₄×C₈×C₆₁	—	2	1	no	—
44	C₄×C₈×C₆₁	D₄	1	2	yes	—
45	C₂×Q₈×C₆₁	—	2	2	no	—
46	C₂²×C₄×C₆₁	—	2	1	no	—
47	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
48	M₁₆×C₆₁	—	2	2	no	—
49	M₁₆×C₆₁	—	2	2	no	—
50	C₂×C₈×C₆₁	—	2	1	no	—
51	C₂×C₈×C₆₁	—	2	1	no	—
52	C₂×C₈×C₆₁	—	2	2	no	—
53	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
57	C₂×C₈×C₆₁	#41	1	2	yes	—
58	C₄⋊C₄×C₆₁	—	2	4	no	—
59	C₄⋊C₄×C₆₁	—	2	2	no	—
60	C₄⋊C₄×C₆₁	—	2	1	no	—
61	C₄⋊C₄×C₆₁	—	2	1	no	—
62	C₂²⋊C₄×C₆₁	—	2	1	no	—
63	C₂²⋊C₄×C₆₁	M₁₆	1	2	yes	—
64	C₄²×C₆₁	—	2	1	no	—
65	C₄²×C₆₁	—	2	1	no	—
66	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
67	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
68	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
69	C₁₆×C₆₁	—	2	4	no	—
70	C₁₆×C₆₁	—	4	2	no	—
71	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
72	Q₈×C₆₁	—	8	2	no	—
73	C₂×C₄×C₆₁	—	2	1	no	—
74	C₂×C₄×C₆₁	—	2	1	no	—
75	C₂×C₄×C₆₁	—	2	2	no	—
76	C₂×C₄×C₆₁	—	2	2	no	—
77	C₂×C₄×C₆₁	—	8	1	no	—
78	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
80	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₆₁	#131	1	2	yes	—
82	C₈×C₆₁	—	2	2	no	—
83	C₈×C₆₁	—	4	1	no	—
84	C₈×C₆₁	—	4	1	no	—
85	C₈×C₆₁	C₂²⋊Q₈	1	2	yes	—
86	C₈×C₆₁	#124	1	1	yes	characteristic
87	C₈×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
88	#13269	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂²×C₆₁	—	4	1	no	—
90	C₂²×C₆₁	#407	1	1	yes	characteristic
91	C₂²×C₆₁	#446	1	2	yes	—
92	C₄×C₆₁	—	4	1	no	—
93	C₄×C₆₁	—	4	1	no	—
94	C₄×C₆₁	—	4	2	no	—
95	C₄×C₆₁	—	4	2	no	—
96	C₄×C₆₁	#391	1	1	yes	characteristic
97	C₄×C₆₁	#408	1	1	yes	characteristic
98	C₄×C₆₁	#446	1	2	yes	—
99	#2961	C₂×C₆₁	1	1	yes	characteristic
100	#2153	C₂×C₆₁	1	1	yes	characteristic
101	#2110	C₂×C₆₁	1	1	yes	characteristic
102	#1378	C₂×C₆₁	1	2	yes	—
103	#1375	C₂×C₆₁	1	2	yes	—
104	C₂×C₆₁	—	2	1	no	—
105	C₂×C₆₁	—	2	2	no	—
106	C₂×C₆₁	#1985	1	1	yes	characteristic
107	#533	C₄×C₆₁	1	1	yes	characteristic
108	C₂×M₃₂	C₂²×C₆₁	1	2	yes	—
109	C₂×M₃₂	C₂²×C₆₁	1	2	yes	—
110	#446	—	2	2	no	—
111	#446	—	2	2	no	—
112	#424	C₄×C₆₁	1	1	yes	characteristic
113	#424	C₄×C₆₁	1	1	yes	characteristic
114	#423	C₄×C₆₁	1	1	yes	characteristic
115	#405	C₂²×C₆₁	1	1	yes	characteristic
116	#405	C₂²×C₆₁	1	1	yes	characteristic
117	C₂×#97	C₂²×C₆₁	1	1	yes	characteristic
118	#364	—	2	2	no	—
119	#347	—	2	2	no	—
120	C₆₁	#13269	1	1	yes	characteristic, direct product factor, Sylow
121	C₂×M₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
122	C₂×M₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
123	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
124	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
125	C₄⋊Q₈	—	2	1	no	—
126	#125	—	2	1	no	—
127	C₂²⋊Q₈	—	2	2	no	—
128	C₄×Q₈	—	2	2	no	—
129	#117	C₂×C₄×C₆₁	1	1	yes	characteristic
130	#117	C₂×C₄×C₆₁	1	1	yes	characteristic
131	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
132	M₃₂	—	2	2	no	—
133	M₃₂	—	2	2	no	—
134	M₃₂	—	2	2	no	—
135	M₃₂	—	2	2	no	—
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	Q₈×C₆₁	1	4	yes	complemented
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	—	2	2	no	—
141	#97	—	2	1	no	—
142	#97	—	2	1	no	—
143	#97	—	2	1	no	—
144	#97	D₄×C₆₁	1	2	yes	—
145	C₄×C₈	—	2	1	no	—
146	C₄×C₈	D₄×C₆₁	1	2	yes	—
147	C₂×Q₈	—	2	2	no	—
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
150	M₁₆	—	2	2	no	—
151	M₁₆	—	2	2	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
156	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
157	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
158	C₂×C₈	C₂×Q₈×C₆₁	1	2	yes	—
159	C₂×C₈	#41×C₆₁	1	2	yes	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	2	1	no	—
163	C₄⋊C₄	—	2	1	no	—
164	C₂²⋊C₄	—	2	1	no	—
165	C₂²⋊C₄	M₁₆×C₆₁	1	2	yes	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	1	no	—
168	C₄²	C₂²⋊C₄×C₆₁	1	2	yes	—
169	C₄²	M₁₆×C₆₁	1	1	yes	characteristic
170	C₄²	M₁₆×C₆₁	1	1	yes	characteristic
171	C₁₆	—	2	4	no	—
172	C₁₆	—	4	2	no	—
173	C₂³	C₂×M₁₆×C₆₁	1	1	yes	characteristic
174	Q₈	—	8	2	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	8	1	no	—
180	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
181	C₂×C₄	C₄×Q₈×C₆₁	1	2	yes	—
182	C₂×C₄	C₂×M₁₆×C₆₁	1	1	yes	characteristic
183	C₂×C₄	#131×C₆₁	1	2	yes	—
184	C₈	—	2	2	no	—
185	C₈	—	4	1	no	—
186	C₈	—	4	1	no	—
187	C₈	C₂²⋊Q₈×C₆₁	1	2	yes	—
188	C₈	#124×C₆₁	1	1	yes	characteristic
189	C₈	C₄⋊Q₈×C₆₁	1	1	yes	characteristic
190	C₂²	—	4	1	no	—
191	C₂²	C₆₁×#407	1	1	yes	characteristic
192	C₂²	C₆₁×#446	1	2	yes	—
193	C₄	—	4	1	no	—
194	C₄	—	4	1	no	—
195	C₄	—	4	2	no	—
196	C₄	—	4	2	no	—
197	C₄	C₆₁×#391	1	1	yes	characteristic
198	C₄	C₆₁×#408	1	1	yes	characteristic
199	C₄	C₆₁×#446	1	2	yes	—
200	C₂	—	2	1	no	—
201	C₂	—	2	2	no	—
202	C₂	C₆₁×#1985	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₆₁	#391
commutator	C₂²×C₄	C₂²×C₄×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
socle	C₂×C₆₁	#1985
prime radical	C₂³×C₆₁	C₂×M₁₆
LCS	G ▷ C₂²×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
35. C₂×C₁₆×C₆₁	Q₈	semidirect
88. #13269	C₆₁	direct (N × H)
120. C₆₁	#13269	direct (N × H)
137. C₂×C₁₆	Q₈×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#2961	2
C₆₁×#2153	2
C₆₁×#2110	2
C₆₁×#1378	2
C₆₁×#1375	2
#13269	61

Subgroups — C61×#13269 #29516702

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#13269 #29516702