Subgroups — C₄×#6857 #11757342

Order 972.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#6857	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×3₋¹⁺²	C₃	1	2	yes	maximal, complemented
3	C₃×C₄×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
4	C₄×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
5	#6857	C₄	1	1	yes	characteristic, direct product factor, Sylow
6	C₂×C₃×3₋¹⁺²	C₂×C₃	1	2	yes	—
7	C₂×C₃×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
8	C₂×C₃²⋊C₉	C₂×C₃	1	1	yes	characteristic
9	C₃³×C₄	—	3	1	no	—
10	C₃³×C₄	C₃²	1	1	yes	characteristic
11	C₄×3₋¹⁺²	—	3	6	no	—
12	C₄×H(3,𝔽₃)	—	3	3	no	—
13	C₃×C₄×C₉	—	3	2	no	—
14	C₃×C₄×C₉	—	3	1	no	—
15	C₃×3₋¹⁺²	C₃×C₄	1	2	yes	complemented
16	C₃×H(3,𝔽₃)	C₃×C₄	1	1	yes	characteristic
17	C₃²⋊C₉	C₃×C₄	1	1	yes	characteristic, complemented
18	C₂×C₃³	—	3	1	no	—
19	C₂×C₃³	C₂×C₃²	1	1	yes	characteristic
20	C₂×3₋¹⁺²	—	3	6	no	—
21	C₂×H(3,𝔽₃)	—	3	3	no	—
22	C₂×C₃×C₉	—	3	2	no	—
23	C₂×C₃×C₉	—	3	1	no	—
24	C₃²×C₄	—	3	1	no	—
25	C₃²×C₄	—	3	3	no	—
26	C₃²×C₄	—	3	2	no	—
27	C₃²×C₄	—	3	1	no	—
28	C₃²×C₄	—	9	2	no	—
29	C₃²×C₄	—	9	1	no	—
30	C₃²×C₄	H(3,𝔽₃)	1	1	yes	characteristic
31	C₄×C₉	—	3	6	no	—
32	C₄×C₉	—	9	1	no	—
33	C₃³	—	3	1	no	—
34	C₃³	C₃²×C₄	1	1	yes	characteristic
35	3₋¹⁺²	—	3	6	no	—
36	H(3,𝔽₃)	—	3	3	no	—
37	C₃×C₉	—	3	2	no	—
38	C₃×C₉	—	3	1	no	—
39	C₂×C₃²	—	3	1	no	—
40	C₂×C₃²	—	3	3	no	—
41	C₂×C₃²	—	3	2	no	—
42	C₂×C₃²	—	3	1	no	—
43	C₂×C₃²	—	9	2	no	—
44	C₂×C₃²	—	9	1	no	—
45	C₂×C₃²	C₂×H(3,𝔽₃)	1	1	yes	characteristic
46	C₂×C₉	—	3	6	no	—
47	C₂×C₉	—	9	1	no	—
48	C₃×C₄	—	9	1	no	—
49	C₃×C₄	—	9	3	no	—
50	C₃×C₄	AΣL(1,𝔽₂₇)	1	2	yes	—
51	C₃×C₄	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
52	C₃×C₄	#731	1	1	yes	characteristic
53	C₃²	—	3	1	no	—
54	C₃²	—	3	3	no	—
55	C₃²	—	3	2	no	—
56	C₃²	—	3	1	no	—
57	C₃²	—	9	2	no	—
58	C₃²	—	9	1	no	—
59	C₃²	C₄×H(3,𝔽₃)	1	1	yes	characteristic
60	C₉	—	3	6	no	—
61	C₉	—	9	1	no	—
62	C₂×C₃	—	9	1	no	—
63	C₂×C₃	—	9	3	no	—
64	C₂×C₃	C₂×AΣL(1,𝔽₂₇)	1	2	yes	—
65	C₂×C₃	C₂×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
66	C₂×C₃	C₂×#731	1	1	yes	characteristic
67	C₄	#6857	1	1	yes	characteristic, direct product factor, Sylow
68	C₃	—	9	1	no	—
69	C₃	—	9	3	no	—
70	C₃	C₄×AΣL(1,𝔽₂₇)	1	2	yes	—
71	C₃	C₄×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
72	C₃	C₄×#731	1	1	yes	characteristic
73	C₂	C₂×#6857	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₄	H(3,𝔽₃)
commutator	C₃³	C₃²×C₄
Frattini, cosocle	C₂×C₃³	C₂×C₃²
socle	C₂×C₃²	C₂×H(3,𝔽₃)
prime radical	C₂×C₃×H(3,𝔽₃)	C₂×C₃
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₄ ◁ C₃³×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×3₋¹⁺²	C₃	semidirect
4. C₄×C₃²⋊C₉	C₃	semidirect
5. #6857	C₄	direct (N × H)
15. C₃×3₋¹⁺²	C₃×C₄	semidirect
17. C₃²⋊C₉	C₃×C₄	semidirect
67. C₄	#6857	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×#6857	2
C₃×C₄×3₋¹⁺²	3
C₃×C₄×H(3,𝔽₃)	3
C₄×C₃²⋊C₉	3

Subgroups — C4×#6857 #11757342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#6857 #11757342