Subgroups — C₂³×#6875 #14516713

Order 1944.

320 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#6875	C₂	1	7	yes	maximal, direct product factor
2	C₂³×C₂₇⋊C₃	C₃	1	2	yes	maximal
3	C₂³×C₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic
4	C₂³×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic
5	C₂×#6875	C₂²	1	7	yes	direct product factor
6	C₂²×C₂₇⋊C₃	C₂×C₃	1	14	yes	—
7	C₂²×C₂₇⋊C₃	C₂×C₃	1	7	yes	—
8	C₂²×C₉⋊C₉	C₂×C₃	1	7	yes	—
9	#6875	C₂³	1	1	yes	characteristic, direct product factor, Sylow
10	C₂³×C₃×C₉	—	3	1	no	—
11	C₂³×C₃×C₉	C₃²	1	1	yes	characteristic
12	C₂³×C₂₇	—	3	2	no	—
13	C₂³×C₂₇	C₉	1	3	yes	complemented
14	C₂×C₂₇⋊C₃	C₂²×C₃	1	14	yes	—
15	C₂×C₂₇⋊C₃	C₂²×C₃	1	7	yes	—
16	C₂×C₉⋊C₉	C₂²×C₃	1	7	yes	—
17	C₂²×C₃×C₉	—	3	7	no	—
18	C₂²×C₃×C₉	C₂×C₃²	1	7	yes	—
19	C₂²×C₂₇	—	3	14	no	—
20	C₂²×C₂₇	C₂×C₉	1	21	yes	complemented
21	C₂₇⋊C₃	C₂³×C₃	1	2	yes	—
22	C₂₇⋊C₃	C₂³×C₃	1	1	yes	characteristic
23	C₉⋊C₉	C₂³×C₃	1	1	yes	characteristic
24	C₂³×C₃²	3₋¹⁺²	1	1	yes	characteristic
25	C₂³×C₉	—	9	1	no	—
26	C₂³×C₉	C₃×C₉	1	1	yes	characteristic
27	C₂³×C₉	3₋¹⁺²	1	2	yes	—
28	C₂×C₃×C₉	—	3	7	no	—
29	C₂×C₃×C₉	C₂²×C₃²	1	7	yes	—
30	C₂×C₂₇	—	3	14	no	—
31	C₂×C₂₇	C₂²×C₉	1	21	yes	complemented
32	C₂²×C₃²	C₂×3₋¹⁺²	1	7	yes	—
33	C₂²×C₉	—	9	7	no	—
34	C₂²×C₉	C₂×C₃×C₉	1	7	yes	—
35	C₂²×C₉	C₂×3₋¹⁺²	1	14	yes	—
36	C₃×C₉	—	3	1	no	—
37	C₃×C₉	C₂³×C₃²	1	1	yes	characteristic
38	C₂₇	—	3	2	no	—
39	C₂₇	C₂³×C₉	1	3	yes	complemented
40	C₂³×C₃	—	3	1	no	—
41	C₂³×C₃	C₉⋊C₉	1	1	yes	characteristic
42	C₂×C₃²	C₂²×3₋¹⁺²	1	7	yes	—
43	C₂×C₉	—	9	7	no	—
44	C₂×C₉	C₂²×C₃×C₉	1	7	yes	—
45	C₂×C₉	C₂²×3₋¹⁺²	1	14	yes	—
46	C₂²×C₃	—	3	7	no	—
47	C₂²×C₃	C₂×C₉⋊C₉	1	7	yes	—
48	C₃²	C₂³×3₋¹⁺²	1	1	yes	characteristic
49	C₉	—	9	1	no	—
50	C₉	C₂³×C₃×C₉	1	1	yes	characteristic
51	C₉	C₂³×3₋¹⁺²	1	2	yes	—
52	C₂³	#6875	1	1	yes	characteristic, direct product factor, Sylow
53	C₂×C₃	—	3	7	no	—
54	C₂×C₃	C₂²×C₉⋊C₉	1	7	yes	—
55	C₂²	C₂×#6875	1	7	yes	direct product factor
56	C₃	—	3	1	no	—
57	C₃	C₂³×C₉⋊C₉	1	1	yes	characteristic
58	C₂	C₂²×#6875	1	7	yes	direct product factor

	subgroup	G/H
center, socle	C₂³×C₃	C₉⋊C₉
commutator	C₉	C₂³×C₃×C₉
Frattini, cosocle	C₃×C₉	C₂³×C₃²
prime radical	C₂³×C₃²	3₋¹⁺²
LCS	G ▷ C₉ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂³×C₃×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#6875	C₂	direct (N × H)
5. C₂×#6875	C₂²	direct (N × H)
9. #6875	C₂³	direct (N × H)
13. C₂³×C₂₇	C₉	semidirect
20. C₂²×C₂₇	C₂×C₉	semidirect
31. C₂×C₂₇	C₂²×C₉	semidirect
39. C₂₇	C₂³×C₉	semidirect
52. C₂³	#6875	direct (N × H)
55. C₂²	C₂×#6875	direct (N × H)
58. C₂	C₂²×#6875	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#6875	2
C₂³×C₂₇⋊C₃	3
C₂³×C₉⋊C₉	3

Subgroups — C23×#6875 #14516713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#6875 #14516713