Subgroups — C₃×B(4,𝔽₂) #5331

Order 192.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#120	C₂	1	3	yes	maximal, complemented
3	C₃×#99	C₂	1	3	yes	maximal, complemented
4	B(4,𝔽₂)	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂⁴×C₃	C₂²	1	1	yes	characteristic, complemented
6	C₃×#41	—	2	3	no	—
7	C₂×C₃×D₄	—	2	3	no	—
8	C₂×C₃×D₄	—	2	3	no	—
9	C₂×C₃×D₄	C₂²	1	3	yes	—
10	C₃×C₂²⋊C₄	—	2	3	no	—
11	C₃×C₂²⋊C₄	C₂²	1	3	yes	complemented
12	H(5,𝔽₂)	C₂×C₃	1	1	yes	characteristic, complemented
13	#120	C₂×C₃	1	3	yes	complemented
14	#99	C₂×C₃	1	3	yes	complemented
15	C₂³×C₃	—	2	1	no	—
16	C₂³×C₃	—	2	3	no	—
17	C₂³×C₃	—	4	2	no	—
18	C₂³×C₃	D₄	1	3	yes	complemented
19	C₂³×C₃	C₂³	1	1	yes	characteristic
20	C₃×Q₈	—	2	1	no	—
21	C₃×D₄	—	2	3	no	—
22	C₃×D₄	—	2	6	no	—
23	C₃×D₄	—	4	6	no	—
24	C₂×C₃×C₄	—	2	3	no	—
25	C₂×C₃×C₄	—	2	3	no	—
26	C₂×C₃×C₄	D₄	1	3	yes	complemented
27	C₂⁴	C₂²×C₃	1	1	yes	characteristic, complemented
28	#41	—	2	3	no	—
29	C₂×D₄	—	2	3	no	—
30	C₂×D₄	—	2	3	no	—
31	C₂×D₄	C₂²×C₃	1	3	yes	—
32	C₂²⋊C₄	—	2	3	no	—
33	C₂²⋊C₄	C₂²×C₃	1	3	yes	complemented
34	C₂²×C₃	—	2	6	no	—
35	C₂²×C₃	—	2	3	no	—
36	C₂²×C₃	—	4	1	no	—
37	C₂²×C₃	—	4	3	no	—
38	C₂²×C₃	—	4	3	no	—
39	C₂²×C₃	—	4	1	no	—
40	C₂²×C₃	—	8	1	no	—
41	C₂²×C₃	C₂×D₄	1	3	yes	—
42	C₃×C₄	—	2	3	no	—
43	C₃×C₄	—	4	3	no	—
44	C₂³	—	2	1	no	—
45	C₂³	—	2	3	no	—
46	C₂³	—	4	2	no	—
47	C₂³	C₃×D₄	1	3	yes	complemented
48	C₂³	C₂³×C₃	1	1	yes	characteristic
49	Q₈	—	2	1	no	—
50	D₄	—	2	3	no	—
51	D₄	—	2	6	no	—
52	D₄	—	4	6	no	—
53	C₂×C₄	—	2	3	no	—
54	C₂×C₄	—	2	3	no	—
55	C₂×C₄	C₃×D₄	1	3	yes	complemented
56	C₂×C₃	—	2	3	no	—
57	C₂×C₃	—	4	2	no	—
58	C₂×C₃	—	4	3	no	—
59	C₂×C₃	#120	1	1	yes	characteristic
60	C₂²	—	2	6	no	—
61	C₂²	—	2	3	no	—
62	C₂²	—	4	1	no	—
63	C₂²	—	4	3	no	—
64	C₂²	—	4	3	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	8	1	no	—
67	C₂²	C₂×C₃×D₄	1	3	yes	—
68	C₄	—	2	3	no	—
69	C₄	—	4	3	no	—
70	C₃	B(4,𝔽₂)	1	1	yes	characteristic, direct product factor, Sylow
71	C₂	—	2	3	no	—
72	C₂	—	4	2	no	—
73	C₂	—	4	3	no	—
74	C₂	C₃×#120	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	#120
commutator, Frattini, cosocle	C₂³	C₂³×C₃
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×H(5,𝔽₂)	C₂	semidirect
2. C₃×#120	C₂	semidirect
3. C₃×#99	C₂	semidirect
4. B(4,𝔽₂)	C₃	direct (N × H)
5. C₂⁴×C₃	C₂²	semidirect
11. C₃×C₂²⋊C₄	C₂²	semidirect
12. H(5,𝔽₂)	C₂×C₃	semidirect
13. #120	C₂×C₃	semidirect
14. #99	C₂×C₃	semidirect
18. C₂³×C₃	D₄	semidirect
26. C₂×C₃×C₄	D₄	semidirect
27. C₂⁴	C₂²×C₃	semidirect
33. C₂²⋊C₄	C₂²×C₃	semidirect
47. C₂³	C₃×D₄	semidirect
55. C₂×C₄	C₃×D₄	semidirect
70. C₃	B(4,𝔽₂)	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×H(5,𝔽₂)	2
C₃×#120	2
C₃×#99	2
B(4,𝔽₂)	3

Subgroups — C3×B(4,𝔽2) #5331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×B(4,𝔽₂) #5331