Subgroups — C₆₁×#23847 #29527280

Order 15616.

604 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2104	C₂	1	8	yes	maximal
2	C₆₁×#1740	C₂	1	7	yes	maximal
3	C₂×#117×C₆₁	C₂²	1	7	yes	—
4	C₆₁×#342	C₂²	1	28	yes	—
5	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
6	#117×C₆₁	—	2	14	no	—
7	#117×C₆₁	—	2	14	no	—
8	C₂×C₄⋊C₄×C₆₁	—	2	7	no	—
9	C₂×C₂²⋊C₄×C₆₁	C₂³	1	14	yes	—
10	C₂×C₄²×C₆₁	—	2	7	no	—
11	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
12	C₂²×C₄×C₆₁	—	2	14	no	—
13	C₂²×C₄×C₆₁	#41	1	7	yes	—
14	C₂²×C₄×C₆₁	#41	1	7	yes	—
15	C₄⋊C₄×C₆₁	—	4	14	no	—
16	C₂²⋊C₄×C₆₁	—	2	28	no	—
17	C₄²×C₆₁	—	4	7	no	—
18	C₄²×C₆₁	—	4	7	no	—
19	C₂³×C₆₁	—	8	1	no	—
20	C₂³×C₆₁	C₂×#41	1	7	yes	—
21	C₂×C₄×C₆₁	—	4	14	no	—
22	C₂×C₄×C₆₁	—	8	14	no	—
23	C₂×C₄×C₆₁	#126	1	7	yes	—
24	C₂×C₄×C₆₁	#126	1	14	yes	—
25	C₂×C₄×C₆₁	#126	1	7	yes	—
26	#23847	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂²×C₆₁	—	4	7	no	—
28	C₂²×C₆₁	C₂×#126	1	7	yes	—
29	C₄×C₆₁	—	8	14	no	—
30	C₄×C₆₁	#401	1	7	yes	—
31	C₄×C₆₁	#401	1	1	yes	characteristic
32	#2104	C₂×C₆₁	1	8	yes	—
33	#1740	C₂×C₆₁	1	7	yes	—
34	C₂×C₆₁	—	2	7	no	—
35	C₂×C₆₁	C₂×#401	1	1	yes	characteristic
36	C₂×#117	C₂²×C₆₁	1	7	yes	—
37	#342	C₂²×C₆₁	1	28	yes	—
38	C₆₁	#23847	1	1	yes	characteristic, direct product factor, Sylow
39	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
40	#117	—	2	14	no	—
41	#117	—	2	14	no	—
42	C₂×C₄⋊C₄	—	2	7	no	—
43	C₂×C₂²⋊C₄	C₂³×C₆₁	1	14	yes	—
44	C₂×C₄²	—	2	7	no	—
45	C₂⁴	C₂⁴×C₆₁	1	1	yes	characteristic
46	C₂²×C₄	—	2	14	no	—
47	C₂²×C₄	#41×C₆₁	1	7	yes	—
48	C₂²×C₄	#41×C₆₁	1	7	yes	—
49	C₄⋊C₄	—	4	14	no	—
50	C₂²⋊C₄	—	2	28	no	—
51	C₄²	—	4	7	no	—
52	C₄²	—	4	7	no	—
53	C₂³	—	8	1	no	—
54	C₂³	C₂×#41×C₆₁	1	7	yes	—
55	C₂×C₄	—	4	14	no	—
56	C₂×C₄	—	8	14	no	—
57	C₂×C₄	#126×C₆₁	1	7	yes	—
58	C₂×C₄	#126×C₆₁	1	7	yes	—
59	C₂×C₄	#126×C₆₁	1	14	yes	—
60	C₂²	—	4	7	no	—
61	C₂²	C₂×#126×C₆₁	1	7	yes	—
62	C₄	—	8	14	no	—
63	C₄	C₆₁×#401	1	7	yes	—
64	C₄	C₆₁×#401	1	1	yes	characteristic
65	C₂	—	2	7	no	—
66	C₂	C₂×C₆₁×#401	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₆₁	#401
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₆₁
socle	C₂×C₆₁	C₂×#401
prime radical	C₂⁴×C₆₁	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₁ ◁ C₂³×C₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
26. #23847	C₆₁	direct (N × H)
38. C₆₁	#23847	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×#2104	2
C₆₁×#1740	2
#23847	61

Subgroups — C61×#23847 #29527280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#23847 #29527280