Finite Group Databasebeta

← ΓL(1,𝔽8)×C7⋊(C11⋊AGL(1,𝔽5)) #44568656

Subgroups — ΓL(1,𝔽8)×C7⋊(C11⋊AGL(1,𝔽5)) #44568656

Order 32340.

104 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 39 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
1C7×D5×C11×ΓL(1,𝔽8)C211yesmaximal, characteristic
2C7×C7⋊(C11⋊AGL(1,𝔽5))C311yesmaximal, characteristic, complemented, Hall
3C5×C7×C11×ΓL(1,𝔽8)C411yescharacteristic, complemented, Hall
4ΓL(1,𝔽8)×Dic77—51nomaximal, Hall
5C72×D5×C11C2×C311yescharacteristic
6C3×C7⋊(C11⋊AGL(1,𝔽5))—71nomaximal
7ΓL(1,𝔽8)×C11⋊AGL(1,𝔽5)—71nomaximal
8C2×C7×C11×ΓL(1,𝔽8)—51no—
9ΓL(1,𝔽8)×C7⋊AGL(1,𝔽5)—111nomaximal, Hall
10C5×C72×C11C3×C411yescharacteristic, complemented, Hall
11D5×C11×ΓL(1,𝔽8)D711yescharacteristic
12C3×C7×D5×C11—71no—
13C7×Dic77—51noHall
14C7×C11×ΓL(1,𝔽8)AGL(1,𝔽5)11yescharacteristic, complemented, Hall
15C7⋊(C11⋊AGL(1,𝔽5))ΓL(1,𝔽8)11yescharacteristic, direct product factor
16C7×C11⋊AGL(1,𝔽5)—71no—
17C7×D5×ΓL(1,𝔽8)D1111yescharacteristic
18C3×C5×C7×C11—71no—
19C5×C11×ΓL(1,𝔽8)Dic711yescharacteristic, complemented
20C2×C72×C11—51no—
21C7×C7⋊AGL(1,𝔽5)—111noHall
22C3×Dic77—351no—
23ΓL(1,𝔽8)×Dic11—351no—
24C7×D5×C11—61no—
25C7×D5×C11C2×ΓL(1,𝔽8)11yescharacteristic
26C7×D5×C11C3×D711yescharacteristic
27C5×C7×ΓL(1,𝔽8)Dic1111yescharacteristic, complemented, Hall
28C3×C11⋊AGL(1,𝔽5)—491noHall
29ΓL(1,𝔽8)×Dic7—551noHall
30C72×C11C3×AGL(1,𝔽5)11yescharacteristic, complemented, Hall
31C72×D5C3×D1111yescharacteristic
32C2×C3×C7×C11—351no—
33C2×C11×ΓL(1,𝔽8)—51no—
34C3×C7⋊AGL(1,𝔽5)—771no—
35AGL(1,𝔽5)×ΓL(1,𝔽8)—771no—
36C5×C7×C11—61no—
37C5×C7×C11C4×ΓL(1,𝔽8)11yescharacteristic, complemented
38C5×C7×C11C3×Dic711yescharacteristic, complemented
39C3×D5×C11—71no—
40Dic77—51no—
41C7×Dic11—351no—
42C2×C7×ΓL(1,𝔽8)—51no—
43C5×C72C3×Dic1111yescharacteristic, complemented, Hall
44C3×C7×C11—71no—
45C11×ΓL(1,𝔽8)C7⋊AGL(1,𝔽5)11yescharacteristic, complemented
46C11⋊AGL(1,𝔽5)—71noHall
47C3×C7×D5—71no—
48D5×ΓL(1,𝔽8)D7711yescharacteristic
49C7×Dic7—551noHall
50C3×C5×C11—71noHall
51C2×C7×C11—51no—
52C2×C7×C11—51no—
53C2×C7×C11—301no—
54C7×ΓL(1,𝔽8)C11⋊AGL(1,𝔽5)11yescharacteristic, complemented, Hall
55C7⋊AGL(1,𝔽5)—111no—
56C7×AGL(1,𝔽5)—771no—
57C3×Dic11—2451noHall
58D5×C11D7×ΓL(1,𝔽8)11yescharacteristic
59C3×C5×C7—71no—
60C5×ΓL(1,𝔽8)Dic7711yescharacteristic, complemented
61C2×C72—51no—
62C3×Dic7—3851no—
63C4×ΓL(1,𝔽8)—3851no—
64C7×C11—61no—
65C7×C11AGL(1,𝔽5)×ΓL(1,𝔽8)11yescharacteristic, complemented
66C7×C11C3×C7⋊AGL(1,𝔽5)11yescharacteristic, complemented
67C7×D5—61no—
68C7×D5ΓL(1,𝔽8)×D1111yescharacteristic
69C7×D5C3×D7711yescharacteristic
70C2×C3×C11—351no—
71C3×AGL(1,𝔽5)—5391noHall
72C5×C11ΓL(1,𝔽8)×Dic711yescharacteristic, complemented, Hall
73C72C3×C11⋊AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
74Dic11—351noHall
75C2×C3×C7—351no—
76C2×ΓL(1,𝔽8)—51no—
77C5×C7—61no—
78C5×C7ΓL(1,𝔽8)×Dic1111yescharacteristic, complemented
79C5×C7C3×Dic7711yescharacteristic, complemented
80C3×C11—71noHall
81C3×D5—71no—
82C4×C7—3851no—
83Dic7—551no—
84C2×C11—51no—
85C3×C7—71no—
86ΓL(1,𝔽8)C7⋊(C11⋊AGL(1,𝔽5))11yescharacteristic, direct product factor
87AGL(1,𝔽5)—771noHall
88C3×C5—71noHall
89C2×C7—51no—
90C2×C7—51no—
91C2×C7—301no—
92C3×C4—26951noHall
93C11ΓL(1,𝔽8)×C7⋊AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
94D5ΓL(1,𝔽8)×D7711yescharacteristic
95C7—61no—
96C7ΓL(1,𝔽8)×C11⋊AGL(1,𝔽5)11yescharacteristic, complemented
97C7C3×C7⋊(C11⋊AGL(1,𝔽5))11yescharacteristic, complemented
98C2×C3—351no—
99C5ΓL(1,𝔽8)×Dic7711yescharacteristic, complemented, Sylow
100C4—3851noSylow
101C3—71noSylow
102C2—51no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC5×C72×C11C3×C4
prime radicalC7×D5×C11×ΓL(1,𝔽8)C2
cosocleC72×D5×C11C2×C3
LCSG ▷ C5×C72×C11
UCSC1

Complements

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

NH ≅ G/Ntype
2. C7×C7⋊(C11⋊AGL(1,𝔽5))C3semidirect
3. C5×C7×C11×ΓL(1,𝔽8)C4semidirect
10. C5×C72×C11C3×C4semidirect
14. C7×C11×ΓL(1,𝔽8)AGL(1,𝔽5)semidirect
15. C7⋊(C11⋊AGL(1,𝔽5))ΓL(1,𝔽8)direct (N × H)
19. C5×C11×ΓL(1,𝔽8)Dic7semidirect
27. C5×C7×ΓL(1,𝔽8)Dic11semidirect
30. C72×C11C3×AGL(1,𝔽5)semidirect
37. C5×C7×C11C4×ΓL(1,𝔽8)semidirect
38. C5×C7×C11C3×Dic7semidirect
43. C5×C72C3×Dic11semidirect
45. C11×ΓL(1,𝔽8)C7⋊AGL(1,𝔽5)semidirect
54. C7×ΓL(1,𝔽8)C11⋊AGL(1,𝔽5)semidirect
60. C5×ΓL(1,𝔽8)Dic77semidirect
65. C7×C11AGL(1,𝔽5)×ΓL(1,𝔽8)semidirect
66. C7×C11C3×C7⋊AGL(1,𝔽5)semidirect
72. C5×C11ΓL(1,𝔽8)×Dic7semidirect
73. C72C3×C11⋊AGL(1,𝔽5)semidirect
78. C5×C7ΓL(1,𝔽8)×Dic11semidirect
79. C5×C7C3×Dic77semidirect
86. ΓL(1,𝔽8)C7⋊(C11⋊AGL(1,𝔽5))direct (N × H)
93. C11ΓL(1,𝔽8)×C7⋊AGL(1,𝔽5)semidirect
96. C7ΓL(1,𝔽8)×C11⋊AGL(1,𝔽5)semidirect
97. C7C3×C7⋊(C11⋊AGL(1,𝔽5))semidirect
99. C5ΓL(1,𝔽8)×Dic77semidirect

Maximal subgroups

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

subgroupindex
C7×D5×C11×ΓL(1,𝔽8)2
C7×C7⋊(C11⋊AGL(1,𝔽5))3
ΓL(1,𝔽8)×Dic775
C3×C7⋊(C11⋊AGL(1,𝔽5))7
ΓL(1,𝔽8)×C11⋊AGL(1,𝔽5)7
ΓL(1,𝔽8)×C7⋊AGL(1,𝔽5)11