Subgroups — C₂²×C₃×#401 #10653155

Order 768.

2942 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#401	C₂	1	24	yes	maximal, direct product factor
2	C₂²×C₃×#95	C₂	1	7	yes	maximal
3	C₂²×#401	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₂⁴×C₃×C₄	C₂²	1	7	yes	—
5	C₃×#401	C₂²	1	64	yes	direct product factor
6	C₂×C₃×#95	C₂²	1	84	yes	—
7	C₂×#401	C₂×C₃	1	24	yes	direct product factor
8	C₂²×#95	C₂×C₃	1	7	yes	—
9	C₂⁵×C₃	C₂³	1	1	yes	characteristic
10	C₂³×C₃×C₄	—	2	84	no	—
11	C₂³×C₃×C₄	C₂³	1	42	yes	—
12	C₃×#95	C₂³	1	112	yes	—
13	C₂⁴×C₄	C₂²×C₃	1	7	yes	—
14	#401	C₂²×C₃	1	64	yes	direct product factor
15	C₂×#95	C₂²×C₃	1	84	yes	—
16	C₂⁴×C₃	#41	1	28	yes	—
17	C₂⁴×C₃	C₂⁴	1	3	yes	—
18	C₂²×C₃×C₄	—	2	252	no	—
19	C₂²×C₃×C₄	—	4	112	no	—
20	C₂²×C₃×C₄	C₂⁴	1	28	yes	—
21	C₂⁵	C₂³×C₃	1	1	yes	characteristic
22	C₂³×C₄	—	2	84	no	—
23	C₂³×C₄	C₂³×C₃	1	42	yes	—
24	#95	C₂³×C₃	1	112	yes	—
25	C₂³×C₃	#126	1	112	yes	—
26	C₂³×C₃	C₂×#41	1	42	yes	—
27	C₂³×C₃	C₂⁵	1	1	yes	characteristic
28	C₂×C₃×C₄	—	2	84	no	—
29	C₂×C₃×C₄	—	4	168	no	—
30	C₂⁴	C₃×#41	1	28	yes	—
31	C₂⁴	C₂⁴×C₃	1	3	yes	—
32	C₂²×C₄	—	2	252	no	—
33	C₂²×C₄	—	4	112	no	—
34	C₂²×C₄	C₂⁴×C₃	1	28	yes	—
35	C₂²×C₃	#401	1	64	yes	direct product factor
36	C₂²×C₃	C₂×#126	1	84	yes	—
37	C₂²×C₃	C₂²×#41	1	7	yes	—
38	C₃×C₄	—	4	28	no	—
39	C₂³	C₃×#126	1	112	yes	—
40	C₂³	C₂×C₃×#41	1	42	yes	—
41	C₂³	C₂⁵×C₃	1	1	yes	characteristic
42	C₂×C₄	—	2	84	no	—
43	C₂×C₄	—	4	168	no	—
44	C₂×C₃	C₂×#401	1	24	yes	direct product factor
45	C₂×C₃	C₂²×#126	1	7	yes	—
46	C₂²	C₃×#401	1	64	yes	direct product factor
47	C₂²	C₂×C₃×#126	1	84	yes	—
48	C₂²	C₂²×C₃×#41	1	7	yes	—
49	C₄	—	4	28	no	—
50	C₃	C₂²×#401	1	1	yes	characteristic, direct product factor, Sylow
51	C₂	C₂×C₃×#401	1	24	yes	direct product factor
52	C₂	C₂²×C₃×#126	1	7	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₃	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#401	C₂	direct (N × H)
3. C₂²×#401	C₃	direct (N × H)
5. C₃×#401	C₂²	direct (N × H)
7. C₂×#401	C₂×C₃	direct (N × H)
14. #401	C₂²×C₃	direct (N × H)
35. C₂²×C₃	#401	direct (N × H)
44. C₂×C₃	C₂×#401	direct (N × H)
46. C₂²	C₃×#401	direct (N × H)
50. C₃	C₂²×#401	direct (N × H)
51. C₂	C₂×C₃×#401	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×C₃×#401	2
C₂²×C₃×#95	2
C₂²×#401	3

Subgroups — C22×C3×#401 #10653155

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#401 #10653155