Subgroups — C₆₁×#7846 #29511279

Order 15616.

582 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#97×C₆₁	C₂	1	3	yes	maximal
2	C₂×C₆₁×#322	C₂	1	3	yes	maximal
3	C₂×C₈²×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#97×C₆₁	C₄	1	12	yes	—
5	C₂×C₄×C₈×C₆₁	C₂²	1	3	yes	—
6	C₂×C₄×C₈×C₆₁	C₂²	1	3	yes	—
7	C₄³×C₆₁	C₂²	1	1	yes	characteristic
8	C₆₁×#322	C₄	1	12	yes	complemented
9	C₈²×C₆₁	C₄	1	4	yes	complemented
10	C₂²×C₈×C₆₁	C₂×C₄	1	6	yes	—
11	C₂²×C₈×C₆₁	C₂×C₄	1	6	yes	—
12	C₂×C₄²×C₆₁	C₂×C₄	1	6	yes	—
13	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
14	#97×C₆₁	—	2	24	no	—
15	C₄×C₈×C₆₁	C₂×C₄	1	6	yes	—
16	C₄×C₈×C₆₁	C₂×C₄	1	12	yes	—
17	C₄×C₈×C₆₁	C₂×C₄	1	6	yes	—
18	C₂²×C₄×C₆₁	C₄²	1	4	yes	—
19	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
20	C₂×C₈×C₆₁	—	2	12	no	—
21	C₂×C₈×C₆₁	C₄²	1	12	yes	—
22	C₂×C₈×C₆₁	C₄²	1	12	yes	—
23	C₂×C₈×C₆₁	M₁₆	1	24	yes	—
24	C₄²×C₆₁	—	2	12	no	—
25	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
26	C₄²×C₆₁	C₂²×C₄	1	3	yes	—
27	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
28	C₂×C₄×C₆₁	—	2	12	no	—
29	C₂×C₄×C₆₁	C₂×C₄²	1	3	yes	—
30	C₂×C₄×C₆₁	C₂×C₄²	1	3	yes	—
31	C₂×C₄×C₆₁	C₂×M₁₆	1	12	yes	—
32	C₈×C₆₁	—	2	12	no	—
33	C₈×C₆₁	#97	1	24	yes	complemented
34	#7846	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂²×C₆₁	C₄³	1	1	yes	characteristic
36	C₂²×C₆₁	C₄×M₁₆	1	6	yes	—
37	C₄×C₆₁	—	4	4	no	—
38	C₄×C₆₁	C₂×#97	1	6	yes	—
39	C₄×C₆₁	C₄×M₁₆	1	6	yes	—
40	C₄×#97	C₂×C₆₁	1	3	yes	—
41	C₂×#322	C₂×C₆₁	1	3	yes	—
42	C₂×C₈²	C₂×C₆₁	1	1	yes	characteristic
43	C₂×C₆₁	#1455	1	4	yes	—
44	C₂×C₆₁	C₄×#97	1	3	yes	—
45	C₂×#97	C₄×C₆₁	1	12	yes	—
46	C₂×C₄×C₈	C₂²×C₆₁	1	3	yes	—
47	C₂×C₄×C₈	C₂²×C₆₁	1	3	yes	—
48	C₄³	C₂²×C₆₁	1	1	yes	characteristic
49	#322	C₄×C₆₁	1	12	yes	complemented
50	C₈²	C₄×C₆₁	1	4	yes	complemented
51	C₆₁	#7846	1	1	yes	characteristic, direct product factor, Sylow
52	C₂²×C₈	C₂×C₄×C₆₁	1	6	yes	—
53	C₂²×C₈	C₂×C₄×C₆₁	1	6	yes	—
54	C₂×C₄²	C₂×C₄×C₆₁	1	6	yes	—
55	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
56	#97	—	2	24	no	—
57	C₄×C₈	C₂×C₄×C₆₁	1	6	yes	—
58	C₄×C₈	C₂×C₄×C₆₁	1	12	yes	—
59	C₄×C₈	C₂×C₄×C₆₁	1	6	yes	—
60	C₂²×C₄	C₄²×C₆₁	1	4	yes	—
61	C₂²×C₄	C₂²×C₄×C₆₁	1	3	yes	—
62	C₂×C₈	—	2	12	no	—
63	C₂×C₈	C₄²×C₆₁	1	12	yes	—
64	C₂×C₈	C₄²×C₆₁	1	12	yes	—
65	C₂×C₈	M₁₆×C₆₁	1	24	yes	—
66	C₄²	—	2	12	no	—
67	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
68	C₄²	C₂²×C₄×C₆₁	1	3	yes	—
69	C₂³	C₂×C₄²×C₆₁	1	1	yes	characteristic
70	C₂×C₄	—	2	12	no	—
71	C₂×C₄	C₂×C₄²×C₆₁	1	3	yes	—
72	C₂×C₄	C₂×C₄²×C₆₁	1	3	yes	—
73	C₂×C₄	C₂×M₁₆×C₆₁	1	12	yes	—
74	C₈	—	2	12	no	—
75	C₈	#97×C₆₁	1	24	yes	complemented
76	C₂²	C₄³×C₆₁	1	1	yes	characteristic
77	C₂²	C₄×M₁₆×C₆₁	1	6	yes	—
78	C₄	—	4	4	no	—
79	C₄	C₂×#97×C₆₁	1	6	yes	—
80	C₄	C₄×M₁₆×C₆₁	1	6	yes	—
81	C₂	C₆₁×#1455	1	4	yes	—
82	C₂	C₄×#97×C₆₁	1	3	yes	—

	subgroup	G/H
center	C₂×C₄²×C₆₁	C₂³
commutator	C₂²	C₄³×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
socle, prime radical	C₂³×C₆₁	C₂×C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₆₁×#322	C₄	semidirect
9. C₈²×C₆₁	C₄	semidirect
33. C₈×C₆₁	#97	semidirect
34. #7846	C₆₁	direct (N × H)
49. #322	C₄×C₆₁	semidirect
50. C₈²	C₄×C₆₁	semidirect
51. C₆₁	#7846	direct (N × H)
75. C₈	#97×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×#97×C₆₁	2
C₂×C₆₁×#322	2
C₂×C₈²×C₆₁	2
#7846	61

Subgroups — C61×#7846 #29511279

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#7846 #29511279