Subgroups — C₆₁×#12617 #29516050

Order 15616.

382 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	C₂×C₆₁×#504	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2115	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2110	C₂	1	2	yes	maximal, complemented
4	C₆₁×#1757	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1379	C₂	1	2	yes	maximal, complemented
6	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
7	C₆₁×#504	—	2	4	no	—
8	C₂×M₃₂×C₆₁	C₄	1	2	yes	complemented
9	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
10	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₆₁	C₄	1	2	yes	complemented
12	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
13	C₆₁×#413	C₄	1	1	yes	characteristic
14	C₆₁×#410	C₄	1	1	yes	characteristic
15	C₆₁×#405	C₂²	1	2	yes	—
16	C₆₁×#349	C₄	1	4	yes	complemented
17	C₆₁×#347	—	2	4	no	—
18	C₆₁×#344	C₄	1	2	yes	—
19	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
20	#131×C₆₁	—	2	2	no	—
21	#131×C₆₁	—	2	2	no	—
22	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
23	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
27	#117×C₆₁	C₂×C₄	1	2	yes	—
28	M₃₂×C₆₁	—	2	2	no	—
29	M₃₂×C₆₁	—	2	4	no	—
30	M₃₂×C₆₁	C₂×C₄	1	4	yes	—
31	M₃₂×C₆₁	D₄	1	4	yes	—
32	C₂×C₁₆×C₆₁	—	2	2	no	—
33	C₂×C₁₆×C₆₁	—	2	2	no	—
34	C₂×C₁₆×C₆₁	—	2	4	no	—
35	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
38	#101×C₆₁	—	2	1	no	—
39	#100×C₆₁	—	2	1	no	—
40	#99×C₆₁	—	2	2	no	—
41	#97×C₆₁	—	2	2	no	—
42	C₄×C₈×C₆₁	—	2	2	no	—
43	#41×C₆₁	—	2	2	no	—
44	#41×C₆₁	—	2	2	no	—
45	C₂×Q₈×C₆₁	M₁₆	1	1	yes	characteristic
46	C₂×D₄×C₆₁	M₁₆	1	2	yes	—
47	C₂×D₄×C₆₁	M₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
49	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
50	M₁₆×C₆₁	—	2	2	no	—
51	M₁₆×C₆₁	—	2	4	no	—
52	M₁₆×C₆₁	—	2	2	no	—
53	M₁₆×C₆₁	—	2	2	no	—
54	C₂×C₈×C₆₁	—	2	2	no	—
55	C₂×C₈×C₆₁	—	2	2	no	—
56	C₂×C₈×C₆₁	—	2	2	no	—
57	C₂×C₈×C₆₁	C₄²	1	2	yes	—
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	2	yes	—
61	C₂×C₈×C₆₁	#41	1	2	yes	—
62	C₄⋊C₄×C₆₁	—	2	2	no	—
63	C₂²⋊C₄×C₆₁	—	2	2	no	—
64	C₄²×C₆₁	—	2	2	no	—
65	C₁₆×C₆₁	—	2	1	no	—
66	C₁₆×C₆₁	—	2	1	no	—
67	C₁₆×C₆₁	—	2	4	no	—
68	C₁₆×C₆₁	C₂²⋊C₄	1	4	yes	—
69	C₂³×C₆₁	#97	1	2	yes	—
70	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
71	Q₈×C₆₁	—	2	2	no	—
72	D₄×C₆₁	—	2	4	no	—
73	D₄×C₆₁	—	2	2	no	—
74	C₂×C₄×C₆₁	—	2	2	no	—
75	C₂×C₄×C₆₁	—	2	2	no	—
76	C₂×C₄×C₆₁	—	4	2	no	—
77	C₂×C₄×C₆₁	#97	1	2	yes	—
78	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₆₁	C₄×D₄	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	2	no	—
84	C₈×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₈×C₆₁	#117	1	1	yes	characteristic
86	C₈×C₆₁	C₄×D₄	1	2	yes	—
87	#12617	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂²×C₆₁	—	2	2	no	—
89	C₂²×C₆₁	—	4	1	no	—
90	C₂²×C₆₁	—	4	2	no	—
91	C₂²×C₆₁	C₂×#97	1	1	yes	characteristic
92	C₂²×C₆₁	#435	1	2	yes	—
93	C₄×C₆₁	—	2	2	no	—
94	C₄×C₆₁	—	8	2	no	—
95	C₄×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
96	C₄×C₆₁	#405	1	1	yes	characteristic
97	C₄×C₆₁	#435	1	2	yes	—
98	C₂×#504	C₂×C₆₁	1	1	yes	characteristic
99	#2115	C₂×C₆₁	1	1	yes	characteristic
100	#2110	C₂×C₆₁	1	2	yes	complemented
101	#1757	C₂×C₆₁	1	1	yes	characteristic
102	#1379	C₂×C₆₁	1	2	yes	complemented
103	C₂×C₆₁	—	2	2	no	—
104	C₂×C₆₁	—	2	1	no	—
105	C₂×C₆₁	—	4	2	no	—
106	C₂×C₆₁	#1752	1	1	yes	characteristic
107	C₂×#131	C₂²×C₆₁	1	1	yes	characteristic
108	#504	—	2	4	no	—
109	C₂×M₃₂	C₄×C₆₁	1	2	yes	complemented
110	C₂×M₃₂	C₂²×C₆₁	1	2	yes	—
111	C₂×M₃₂	C₂²×C₆₁	1	1	yes	characteristic
112	C₂²×C₁₆	C₄×C₆₁	1	2	yes	complemented
113	C₂²×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
114	#413	C₄×C₆₁	1	1	yes	characteristic
115	#410	C₄×C₆₁	1	1	yes	characteristic
116	#405	C₂²×C₆₁	1	2	yes	—
117	#349	C₄×C₆₁	1	4	yes	complemented
118	#347	—	2	4	no	—
119	#344	C₄×C₆₁	1	2	yes	—
120	C₆₁	#12617	1	1	yes	characteristic, direct product factor, Sylow
121	C₂×#41	C₂×C₄×C₆₁	1	1	yes	characteristic
122	#131	—	2	2	no	—
123	#131	—	2	2	no	—
124	C₂×M₁₆	C₂×C₄×C₆₁	1	2	yes	—
125	C₂×M₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
126	C₂×M₁₆	C₂×C₄×C₆₁	1	2	yes	—
127	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
128	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
129	#117	C₂×C₄×C₆₁	1	2	yes	—
130	M₃₂	—	2	2	no	—
131	M₃₂	—	2	4	no	—
132	M₃₂	C₂×C₄×C₆₁	1	4	yes	—
133	M₃₂	D₄×C₆₁	1	4	yes	—
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	2	4	no	—
137	C₂×C₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
138	C₂×C₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
139	C₂×C₁₆	C₂×C₄×C₆₁	1	2	yes	—
140	#101	—	2	1	no	—
141	#100	—	2	1	no	—
142	#99	—	2	2	no	—
143	#97	—	2	2	no	—
144	C₄×C₈	—	2	2	no	—
145	#41	—	2	2	no	—
146	#41	—	2	2	no	—
147	C₂×Q₈	M₁₆×C₆₁	1	1	yes	characteristic
148	C₂×D₄	M₁₆×C₆₁	1	2	yes	—
149	C₂×D₄	M₁₆×C₆₁	1	1	yes	characteristic
150	C₂²×C₄	C₄²×C₆₁	1	2	yes	—
151	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
152	M₁₆	—	2	2	no	—
153	M₁₆	—	2	4	no	—
154	M₁₆	—	2	2	no	—
155	M₁₆	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	C₄²×C₆₁	1	2	yes	—
160	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
161	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
162	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
163	C₂×C₈	#41×C₆₁	1	2	yes	—
164	C₄⋊C₄	—	2	2	no	—
165	C₂²⋊C₄	—	2	2	no	—
166	C₄²	—	2	2	no	—
167	C₁₆	—	2	1	no	—
168	C₁₆	—	2	1	no	—
169	C₁₆	—	2	4	no	—
170	C₁₆	C₂²⋊C₄×C₆₁	1	4	yes	—
171	C₂³	#97×C₆₁	1	2	yes	—
172	C₂³	C₂×M₁₆×C₆₁	1	1	yes	characteristic
173	Q₈	—	2	2	no	—
174	D₄	—	2	4	no	—
175	D₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	#97×C₆₁	1	2	yes	—
180	C₂×C₄	C₂×C₄²×C₆₁	1	1	yes	characteristic
181	C₂×C₄	C₄×D₄×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	2	no	—
186	C₈	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
187	C₈	#117×C₆₁	1	1	yes	characteristic
188	C₈	C₄×D₄×C₆₁	1	2	yes	—
189	C₂²	—	2	2	no	—
190	C₂²	—	4	1	no	—
191	C₂²	—	4	2	no	—
192	C₂²	C₂×#97×C₆₁	1	1	yes	characteristic
193	C₂²	C₆₁×#435	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	8	2	no	—
196	C₄	C₄×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
197	C₄	C₆₁×#405	1	1	yes	characteristic
198	C₄	C₆₁×#435	1	2	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	2	1	no	—
201	C₂	—	4	2	no	—
202	C₂	C₆₁×#1752	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₆₁	#117
commutator	C₂×C₄	C₂×C₄²×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
socle	C₂×C₆₁	#1752
prime radical	C₂×#41×C₆₁	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₆₁ ◁ C₂×M₃₂×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₁×#2110	C₂	semidirect
5. C₆₁×#1379	C₂	semidirect
8. C₂×M₃₂×C₆₁	C₄	semidirect
11. C₂²×C₁₆×C₆₁	C₄	semidirect
16. C₆₁×#349	C₄	semidirect
87. #12617	C₆₁	direct (N × H)
100. #2110	C₂×C₆₁	semidirect
102. #1379	C₂×C₆₁	semidirect
109. C₂×M₃₂	C₄×C₆₁	semidirect
112. C₂²×C₁₆	C₄×C₆₁	semidirect
117. #349	C₄×C₆₁	semidirect
120. C₆₁	#12617	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#504	2
C₆₁×#2115	2
C₆₁×#2110	2
C₆₁×#1757	2
C₆₁×#1379	2
#12617	61

Subgroups — C61×#12617 #29516050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#12617 #29516050