Subgroups — C₂²×SL(2,𝔽₇) #13062016

Order 1344.

187 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×SL(2,𝔽₇)	C₂	1	3	yes	maximal, direct product factor
2	SL(2,𝔽₇)	C₂²	1	1	yes	characteristic, direct product factor
3	C₂²×#226	—	7	2	no	maximal, Hall
4	C₂³×ΓL(1,𝔽₈)	—	8	1	no	maximal
5	C₂²×SL(2,𝔽₃)	—	7	2	no	—
6	C₂×#226	—	7	6	no	—
7	C₂×#226	—	7	6	no	—
8	C₂²×ΓL(1,𝔽₈)	—	8	3	no	—
9	C₂²×ΓL(1,𝔽₈)	—	8	4	no	—
10	C₂²×Q₁₆	—	21	1	no	Sylow
11	C₂³×C₇	—	8	1	no	—
12	C₂²×Dic₃	—	28	1	no	—
13	C₂×SL(2,𝔽₃)	—	7	6	no	—
14	#226	—	7	2	no	—
15	#226	—	7	6	no	—
16	C₂×ΓL(1,𝔽₈)	—	8	1	no	—
17	C₂×ΓL(1,𝔽₈)	—	8	6	no	—
18	C₂²×Q₈	—	7	2	no	—
19	C₂×Q₁₆	—	21	3	no	—
20	C₂×Q₁₆	—	21	6	no	—
21	C₂×Q₁₆	—	21	3	no	—
22	C₂²×C₈	—	21	1	no	—
23	C₂²×C₇	—	8	4	no	—
24	C₂²×C₇	—	8	3	no	—
25	C₂³×C₃	—	28	1	no	—
26	C₂×Dic₃	—	28	3	no	—
27	C₂×Dic₃	—	28	3	no	—
28	SL(2,𝔽₃)	—	7	2	no	—
29	ΓL(1,𝔽₈)	—	8	1	no	Hall
30	C₂×Q₈	—	7	6	no	—
31	C₂×Q₈	—	21	6	no	—
32	C₂²×C₄	—	21	1	no	—
33	Q₁₆	—	21	1	no	—
34	Q₁₆	—	21	6	no	—
35	Q₁₆	—	21	3	no	—
36	Q₁₆	—	21	6	no	—
37	C₂×C₈	—	21	3	no	—
38	C₂×C₈	—	21	3	no	—
39	C₂×C₇	—	8	6	no	—
40	C₂×C₇	—	8	1	no	—
41	C₂²×C₃	—	28	4	no	—
42	C₂²×C₃	—	28	3	no	—
43	Dic₃	—	28	1	no	—
44	Dic₃	—	28	3	no	—
45	C₂³	GL(3,𝔽₂)	1	1	yes	maximal normal, characteristic
46	Q₈	—	7	2	no	—
47	Q₈	—	21	6	no	—
48	Q₈	—	42	2	no	—
49	C₂×C₄	—	21	3	no	—
50	C₂×C₄	—	21	3	no	—
51	C₈	—	21	1	no	—
52	C₈	—	21	3	no	—
53	C₇	—	8	1	no	Sylow
54	C₂×C₃	—	28	6	no	—
55	C₂×C₃	—	28	1	no	—
56	C₂²	SL(2,𝔽₇)	1	4	yes	direct product factor
57	C₂²	C₂×GL(3,𝔽₂)	1	3	yes	—
58	C₄	—	21	1	no	—
59	C₄	—	21	3	no	—
60	C₃	—	28	1	no	Sylow
61	C₂	C₂×SL(2,𝔽₇)	1	6	yes	direct product factor
62	C₂	C₂²×GL(3,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, Fitting, socle, hypercenter	C₂³	GL(3,𝔽₂)
commutator, hypocenter	SL(2,𝔽₇)	C₂²
Frattini, cosocle	C₂	C₂²×GL(3,𝔽₂)
LCS	G ▷ SL(2,𝔽₇)
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
1. C₂×SL(2,𝔽₇)	C₂	direct (N × H)
2. SL(2,𝔽₇)	C₂²	direct (N × H)
56. C₂²	SL(2,𝔽₇)	direct (N × H)
61. C₂	C₂×SL(2,𝔽₇)	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×SL(2,𝔽₇)	2
C₂²×#226	7
C₂³×ΓL(1,𝔽₈)	8

Subgroups — C22×SL(2,𝔽7) #13062016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×SL(2,𝔽₇) #13062016