Finite Group Databasebeta

← ΓL(1,𝔽8)×ΓL(1,𝔽8)⋊D37 #45657429

Subgroups — ΓL(1,𝔽8)×ΓL(1,𝔽8)⋊D37 #45657429

Order 32634.

100 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 100 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1ΓL(1,𝔽8)2×C37C211yesmaximal, characteristic, complemented, Hall
2ΓL(1,𝔽8)×D259C311yesmaximal, characteristic, complemented
3#24925967C311yesmaximal, characteristic, complemented
4#24925961C311yesmaximal, characteristic, complemented
5C7×ΓL(1,𝔽8)⋊D37C311yesmaximal, characteristic, complemented
6C37×#3862C2×C311yescharacteristic, complemented
7C37×#3861C2×C311yescharacteristic, complemented
8C7×ΓL(1,𝔽8)×C37C2×C311yescharacteristic, complemented
9C7×ΓL(1,𝔽8)×C37C2×C311yescharacteristic, complemented
10C3×ΓL(1,𝔽8)×D37—71nomaximal
11C3×ΓL(1,𝔽8)⋊D37—71nomaximal
12C7×D259C3211yescharacteristic, complemented, Hall
13C3×ΓL(1,𝔽8)×C37—71no—
14C3×ΓL(1,𝔽8)×C37—71no—
15C72×C37C2×C3211yescharacteristic, complemented, Hall
16C3×D259—71no—
17C3×C7×D37—71no—
18ΓL(1,𝔽8)⋊D37—71no—
19ΓL(1,𝔽8)⋊D37—71no—
20ΓL(1,𝔽8)⋊D37ΓL(1,𝔽8)11yescharacteristic, direct product factor
21ΓL(1,𝔽8)×D37—71no—
22ΓL(1,𝔽8)×D37—71no—
23ΓL(1,𝔽8)×D37—71no—
24ΓL(1,𝔽8)×AGL(1,𝔽7)—371nomaximal, Hall
25C3×C7×C37—71no—
26C3×C7×C37—71no—
27ΓL(1,𝔽8)×C37—71no—
28ΓL(1,𝔽8)×C37—71no—
29ΓL(1,𝔽8)×C37—71no—
30ΓL(1,𝔽8)×C37—71no—
31ΓL(1,𝔽8)×C37—421no—
32ΓL(1,𝔽8)×C37AGL(1,𝔽7)11yescharacteristic, complemented
33ΓL(1,𝔽8)×C37C2×ΓL(1,𝔽8)11yescharacteristic, complemented
34C32×D37—491noHall
35D259C3×ΓL(1,𝔽8)11yescharacteristic, complemented
36C7×D37—71no—
37ΓL(1,𝔽8)2D3711yescharacteristic, complemented, Hall
38C32×C37—491noHall
39#64423—371no—
40#64422—371no—
41D7×ΓL(1,𝔽8)—371no—
42C7×AGL(1,𝔽7)—371no—
43C7×C37—61no—
44C7×C37C3×AGL(1,𝔽7)11yescharacteristic, complemented
45C7×C37C2×C3×ΓL(1,𝔽8)11yescharacteristic, complemented
46C3×D37—71no—
47C3×D37—491no—
48C3×D37—491no—
49C3×D37—491no—
50#3862C3×D3711yescharacteristic, complemented
51#3861C3×D3711yescharacteristic, complemented
52C7×ΓL(1,𝔽8)C3×D3711yescharacteristic, complemented
53C7×ΓL(1,𝔽8)C3×D3711yescharacteristic, complemented
54C2×C3×ΓL(1,𝔽8)—2591no—
55C3×AGL(1,𝔽7)—2591no—
56C3×C37—71no—
57C3×C37—71no—
58C3×C37—491no—
59C3×C37—491no—
60C7×D7—371noHall
61D37—71noHall
62C3×ΓL(1,𝔽8)—71no—
63C3×ΓL(1,𝔽8)—71no—
64C72C32×D3711yescharacteristic, complemented, Sylow
65C2×C3×C7—2591no—
66C3×D7—2591no—
67C2×ΓL(1,𝔽8)—2591no—
68C2×ΓL(1,𝔽8)—2591no—
69C2×ΓL(1,𝔽8)—2591no—
70AGL(1,𝔽7)—2591no—
71AGL(1,𝔽7)—2591no—
72AGL(1,𝔽7)—371no—
73C37ΓL(1,𝔽8)×AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
74C3×C7—71no—
75C3×C7—71no—
76ΓL(1,𝔽8)—71no—
77ΓL(1,𝔽8)—71no—
78ΓL(1,𝔽8)—71no—
79ΓL(1,𝔽8)—71no—
80ΓL(1,𝔽8)—421no—
81ΓL(1,𝔽8)ΓL(1,𝔽8)×D3711yescharacteristic, complemented
82ΓL(1,𝔽8)ΓL(1,𝔽8)⋊D3711yescharacteristic, direct product factor
83C2×C32—18131noHall
84C2×C7—2591no—
85D7—371no—
86C32—491noSylow
87C7—61no—
88C7C3×ΓL(1,𝔽8)⋊D3711yescharacteristic, complemented
89C7C3×ΓL(1,𝔽8)×D3711yescharacteristic, complemented
90C2×C3—2591no—
91C2×C3—18131no—
92C2×C3—18131no—
93C2×C3—18131no—
94C3—71no—
95C3—71no—
96C3—491no—
97C3—491no—
98C2—2591noSylow
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC72×C37C2×C32
LCSG ▷ C72×C37
UCSC1

Complements

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

NH ≅ G/Ntype
1. ΓL(1,𝔽8)2×C37C2semidirect
2. ΓL(1,𝔽8)×D259C3semidirect
3. #24925967C3semidirect
4. #24925961C3semidirect
5. C7×ΓL(1,𝔽8)⋊D37C3semidirect
6. C37×#3862C2×C3semidirect
7. C37×#3861C2×C3semidirect
8. C7×ΓL(1,𝔽8)×C37C2×C3semidirect
9. C7×ΓL(1,𝔽8)×C37C2×C3semidirect
12. C7×D259C32semidirect
15. C72×C37C2×C32semidirect
20. ΓL(1,𝔽8)⋊D37ΓL(1,𝔽8)direct (N × H)
32. ΓL(1,𝔽8)×C37AGL(1,𝔽7)semidirect
33. ΓL(1,𝔽8)×C37C2×ΓL(1,𝔽8)semidirect
35. D259C3×ΓL(1,𝔽8)semidirect
37. ΓL(1,𝔽8)2D37semidirect
44. C7×C37C3×AGL(1,𝔽7)semidirect
45. C7×C37C2×C3×ΓL(1,𝔽8)semidirect
50. #3862C3×D37semidirect
51. #3861C3×D37semidirect
52. C7×ΓL(1,𝔽8)C3×D37semidirect
53. C7×ΓL(1,𝔽8)C3×D37semidirect
64. C72C32×D37semidirect
73. C37ΓL(1,𝔽8)×AGL(1,𝔽7)semidirect
81. ΓL(1,𝔽8)ΓL(1,𝔽8)×D37semidirect
82. ΓL(1,𝔽8)ΓL(1,𝔽8)⋊D37direct (N × H)
88. C7C3×ΓL(1,𝔽8)⋊D37semidirect
89. C7C3×ΓL(1,𝔽8)×D37semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)2×C372
ΓL(1,𝔽8)×D2593
#249259673
#249259613
C7×ΓL(1,𝔽8)⋊D373
C3×ΓL(1,𝔽8)×D377
C3×ΓL(1,𝔽8)⋊D377
ΓL(1,𝔽8)×AGL(1,𝔽7)37