Subgroups — C₂×C₃×#125 #5857

Order 192.

226 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#125	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₃×C₄⋊C₄	C₂	1	4	yes	maximal
3	C₂×C₃×C₄⋊C₄	C₂	1	2	yes	maximal
4	C₂×C₃×C₄²	C₂	1	1	yes	maximal, characteristic
5	C₂×#125	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂²×C₃×C₄	C₂²	1	2	yes	—
7	C₂²×C₃×C₄	C₂²	1	4	yes	—
8	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
9	C₃×C₄⋊C₄	C₂²	1	16	yes	—
10	C₃×C₄⋊C₄	C₂²	1	8	yes	—
11	C₃×C₄²	C₂²	1	4	yes	—
12	#125	C₂×C₃	1	8	yes	direct product factor
13	C₂×C₄⋊C₄	C₂×C₃	1	4	yes	—
14	C₂×C₄⋊C₄	C₂×C₃	1	2	yes	—
15	C₂×C₄²	C₂×C₃	1	1	yes	characteristic
16	C₂³×C₃	C₂³	1	1	yes	characteristic
17	C₂×C₃×C₄	—	2	4	no	—
18	C₂×C₃×C₄	—	2	8	no	—
19	C₂×C₃×C₄	Q₈	1	4	yes	—
20	C₂×C₃×C₄	C₂³	1	4	yes	—
21	C₂×C₃×C₄	C₂³	1	8	yes	—
22	C₂×C₃×C₄	C₂³	1	2	yes	—
23	C₂²×C₄	C₂²×C₃	1	2	yes	—
24	C₂²×C₄	C₂²×C₃	1	4	yes	—
25	C₂²×C₄	C₂²×C₃	1	1	yes	characteristic
26	C₄⋊C₄	C₂²×C₃	1	16	yes	—
27	C₄⋊C₄	C₂²×C₃	1	8	yes	—
28	C₄²	C₂²×C₃	1	4	yes	—
29	C₂²×C₃	C₂×Q₈	1	2	yes	—
30	C₂²×C₃	#41	1	4	yes	—
31	C₂²×C₃	C₂⁴	1	1	yes	characteristic
32	C₃×C₄	—	2	4	no	—
33	C₃×C₄	—	2	8	no	—
34	C₃×C₄	C₂×Q₈	1	4	yes	—
35	C₂³	C₂³×C₃	1	1	yes	characteristic
36	C₂×C₄	—	2	4	no	—
37	C₂×C₄	—	2	8	no	—
38	C₂×C₄	C₃×Q₈	1	4	yes	—
39	C₂×C₄	C₂³×C₃	1	4	yes	—
40	C₂×C₄	C₂³×C₃	1	8	yes	—
41	C₂×C₄	C₂³×C₃	1	2	yes	—
42	C₂×C₃	#125	1	4	yes	direct product factor
43	C₂×C₃	C₂²×Q₈	1	1	yes	characteristic
44	C₂×C₃	C₂×#41	1	2	yes	—
45	C₂²	C₂×C₃×Q₈	1	2	yes	—
46	C₂²	C₃×#41	1	4	yes	—
47	C₂²	C₂⁴×C₃	1	1	yes	characteristic
48	C₄	—	2	4	no	—
49	C₄	—	2	8	no	—
50	C₄	C₂×C₃×Q₈	1	4	yes	—
51	C₃	C₂×#125	1	1	yes	characteristic, direct product factor, Sylow
52	C₂	C₃×#125	1	4	yes	direct product factor
53	C₂	C₂²×C₃×Q₈	1	1	yes	characteristic
54	C₂	C₂×C₃×#41	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#125	C₂	direct (N × H)
5. C₂×#125	C₃	direct (N × H)
12. #125	C₂×C₃	direct (N × H)
42. C₂×C₃	#125	direct (N × H)
51. C₃	C₂×#125	direct (N × H)
52. C₂	C₃×#125	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#125	2
C₂×C₃×C₄⋊C₄	2
C₂×C₃×C₄²	2
C₂×#125	3

Subgroups — C2×C3×#125 #5857

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#125 #5857