Subgroups — C₆₁×#12890 #29516323

Order 15616.

318 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#346	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#1846	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1845	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1380	C₂	1	4	yes	maximal
5	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
6	C₂×#107×C₆₁	—	2	1	no	—
7	C₂×#107×C₆₁	C₂²	1	2	yes	—
8	C₂×#107×C₆₁	C₂²	1	2	yes	—
9	C₂×#106×C₆₁	—	2	1	no	—
10	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
11	C₆₁×#384	—	2	1	no	—
12	C₆₁×#382	—	2	1	no	—
13	C₆₁×#346	C₄	1	4	yes	complemented
14	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
17	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
18	C₂×C₄⋊C₄×C₆₁	—	4	2	no	—
19	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
20	C₂×C₁₆×C₆₁	—	2	2	no	—
21	C₂×C₁₆×C₆₁	—	2	2	no	—
22	C₂×C₁₆×C₆₁	C₂×C₄	1	4	yes	—
23	#107×C₆₁	—	2	4	no	—
24	#107×C₆₁	—	2	4	no	—
25	#107×C₆₁	—	4	2	no	—
26	#106×C₆₁	—	4	2	no	—
27	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₆₁	Q₈	1	2	yes	—
29	#95×C₆₁	—	4	2	no	—
30	C₂²×C₄×C₆₁	—	4	2	no	—
31	C₂²×C₄×C₆₁	—	4	2	no	—
32	C₂²×C₄×C₆₁	D₈	1	2	yes	—
33	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
36	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
37	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₆₁	#41	1	2	yes	—
40	C₂×C₈×C₆₁	#41	1	2	yes	—
41	C₄⋊C₄×C₆₁	—	4	2	no	—
42	C₄⋊C₄×C₆₁	—	4	2	no	—
43	C₄⋊C₄×C₆₁	—	4	2	no	—
44	C₄⋊C₄×C₆₁	—	4	2	no	—
45	C₄⋊C₄×C₆₁	—	8	4	no	—
46	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
47	C₄²×C₆₁	Q₁₆	1	2	yes	—
48	C₁₆×C₆₁	—	2	4	no	—
49	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₆₁	—	2	2	no	—
51	C₂×C₄×C₆₁	—	2	2	no	—
52	C₂×C₄×C₆₁	—	8	2	no	—
53	C₂×C₄×C₆₁	—	8	2	no	—
54	C₂×C₄×C₆₁	—	8	4	no	—
55	C₂×C₄×C₆₁	—	8	2	no	—
56	C₂×C₄×C₆₁	—	8	2	no	—
57	C₂×C₄×C₆₁	#107	1	4	yes	—
58	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
60	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
61	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
64	C₈×C₆₁	#106	1	4	yes	—
65	C₈×C₆₁	#117	1	2	yes	—
66	C₈×C₆₁	#125	1	2	yes	—
67	#12890	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
69	C₂²×C₆₁	#486	1	1	yes	characteristic
70	C₂²×C₆₁	#486	1	1	yes	characteristic
71	C₂²×C₆₁	#509	1	2	yes	—
72	C₂²×C₆₁	#510	1	2	yes	—
73	C₄×C₆₁	—	2	4	no	—
74	C₄×C₆₁	—	16	2	no	—
75	C₄×C₆₁	—	16	2	no	—
76	C₄×C₆₁	#382	1	1	yes	characteristic
77	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
78	C₄×C₆₁	#487	1	1	yes	characteristic
79	C₄×C₆₁	#487	1	1	yes	characteristic
80	C₂×#346	C₂×C₆₁	1	1	yes	characteristic
81	#1846	C₂×C₆₁	1	1	yes	characteristic
82	#1845	C₂×C₆₁	1	1	yes	characteristic
83	#1380	C₂×C₆₁	1	4	yes	—
84	C₂×C₆₁	#1834	1	1	yes	characteristic
85	C₂×C₆₁	#2159	1	2	yes	—
86	C₂×C₆₁	#2243	1	2	yes	—
87	C₂×C₆₁	#2244	1	1	yes	characteristic
88	C₂×C₆₁	#2245	1	1	yes	characteristic
89	C₂²×C₁₆	C₂²×C₆₁	1	2	yes	—
90	C₂×#107	—	2	1	no	—
91	C₂×#107	C₂²×C₆₁	1	2	yes	—
92	C₂×#107	C₂²×C₆₁	1	2	yes	—
93	C₂×#106	—	2	1	no	—
94	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
95	#384	—	2	1	no	—
96	#382	—	2	1	no	—
97	#346	C₄×C₆₁	1	4	yes	complemented
98	C₆₁	#12890	1	1	yes	characteristic, direct product factor, Sylow
99	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
100	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
101	C₂×C₄⋊C₄	—	2	2	no	—
102	C₂×C₄⋊C₄	—	2	2	no	—
103	C₂×C₄⋊C₄	—	4	2	no	—
104	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	—	2	2	no	—
107	C₂×C₁₆	C₂×C₄×C₆₁	1	4	yes	—
108	#107	—	2	4	no	—
109	#107	—	2	4	no	—
110	#107	—	4	2	no	—
111	#106	—	4	2	no	—
112	C₄×C₈	C₂×C₄×C₆₁	1	2	yes	—
113	C₄×C₈	Q₈×C₆₁	1	2	yes	—
114	#95	—	4	2	no	—
115	C₂²×C₄	—	4	2	no	—
116	C₂²×C₄	—	4	2	no	—
117	C₂²×C₄	D₈×C₆₁	1	2	yes	—
118	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
119	C₂×C₈	C₄⋊C₄×C₆₁	1	2	yes	—
120	C₂×C₈	SD₁₆×C₆₁	1	2	yes	—
121	C₂×C₈	SD₁₆×C₆₁	1	2	yes	—
122	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
123	C₂×C₈	C₂×Q₈×C₆₁	1	1	yes	characteristic
124	C₂×C₈	#41×C₆₁	1	2	yes	—
125	C₂×C₈	#41×C₆₁	1	2	yes	—
126	C₄⋊C₄	—	4	2	no	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	4	2	no	—
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	—	8	4	no	—
131	C₄²	C₄⋊C₄×C₆₁	1	2	yes	—
132	C₄²	Q₁₆×C₆₁	1	2	yes	—
133	C₁₆	—	2	4	no	—
134	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	8	2	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	8	4	no	—
140	C₂×C₄	—	8	2	no	—
141	C₂×C₄	—	8	2	no	—
142	C₂×C₄	#107×C₆₁	1	4	yes	—
143	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
144	C₂×C₄	#124×C₆₁	1	1	yes	characteristic
145	C₂×C₄	#124×C₆₁	1	1	yes	characteristic
146	C₂×C₄	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
147	C₂×C₄	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
148	C₂×C₄	C₂×Q₁₆×C₆₁	1	1	yes	characteristic
149	C₈	#106×C₆₁	1	4	yes	—
150	C₈	#117×C₆₁	1	2	yes	—
151	C₈	#125×C₆₁	1	2	yes	—
152	C₂²	C₂×#107×C₆₁	1	1	yes	characteristic
153	C₂²	C₆₁×#486	1	1	yes	characteristic
154	C₂²	C₆₁×#486	1	1	yes	characteristic
155	C₂²	C₆₁×#509	1	2	yes	—
156	C₂²	C₆₁×#510	1	2	yes	—
157	C₄	—	2	4	no	—
158	C₄	—	16	2	no	—
159	C₄	—	16	2	no	—
160	C₄	C₆₁×#382	1	1	yes	characteristic
161	C₄	C₂×#106×C₆₁	1	1	yes	characteristic
162	C₄	C₆₁×#487	1	1	yes	characteristic
163	C₄	C₆₁×#487	1	1	yes	characteristic
164	C₂	C₆₁×#1834	1	1	yes	characteristic
165	C₂	C₆₁×#2159	1	2	yes	—
166	C₂	C₆₁×#2243	1	2	yes	—
167	C₂	C₆₁×#2244	1	1	yes	characteristic
168	C₂	C₆₁×#2245	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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
13. C₆₁×#346	C₄	semidirect
67. #12890	C₆₁	direct (N × H)
97. #346	C₄×C₆₁	semidirect
98. C₆₁	#12890	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×#346	2
C₆₁×#1846	2
C₆₁×#1845	2
C₆₁×#1380	2
#12890	61

Subgroups — C61×#12890 #29516323

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#12890 #29516323