Finite Group Databasebeta

← ΓL(1,𝔽8)×#13095535 #45657423

Subgroups — ΓL(1,𝔽8)×#13095535 #45657423

Order 32634.

100 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 100 conjugacy classes of subgroups — too many to draw. Only its 28 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)×#11715913C211yesmaximal, characteristic, complemented, Hall
2D7×ΓL(1,𝔽8)×C37C311yesmaximal, characteristic, complemented
3#24925965C311yesmaximal, characteristic, complemented
4#24925959C311yesmaximal, characteristic, complemented
5C7×#13095535C311yesmaximal, characteristic, complemented
6C37⋊#3862C2×C311yescharacteristic, complemented
7#19129588C2×C311yescharacteristic, complemented
8C7×#11715913C2×C311yescharacteristic, complemented
9C7×ΓL(1,𝔽8)×C37C2×C311yescharacteristic, complemented
10C3×#13095535—71nomaximal
11C2×ΓL(1,𝔽8)×C37⋊C3—71nomaximal
12C7×D7×C37C3211yescharacteristic, complemented, Hall
13C3×#11715913—71no—
14ΓL(1,𝔽8)×C37⋊C3—71no—
15C72×C37C2×C3211yescharacteristic, complemented, Hall
16C3×D7×C37—71no—
17C2×#11715912—71no—
18C2×#11715913—71no—
19C2×ΓL(1,𝔽8)×C37—71no—
20C2×C7×C37⋊C3—71no—
21#13095535—71no—
22#13095535—71no—
23#13095535ΓL(1,𝔽8)11yescharacteristic, direct product factor
24ΓL(1,𝔽8)×AGL(1,𝔽7)—371nomaximal, Hall
25C3×C7×C37—71no—
26#11715913—71no—
27#11715913—71no—
28#11715913—71no—
29#11715913—421no—
30#11715913C2×ΓL(1,𝔽8)11yescharacteristic, complemented
31#11715912—71no—
32C7×C37⋊C3—71no—
33ΓL(1,𝔽8)×C37AGL(1,𝔽7)11yescharacteristic, complemented
34C2×C3×C37⋊C3—491noHall
35C2×C7×C37—71no—
36D7×C37C3×ΓL(1,𝔽8)11yescharacteristic, complemented
37ΓL(1,𝔽8)2—371noHall
38C3×C37⋊C3—491noHall
39#64423—371no—
40#64422—371no—
41D7×ΓL(1,𝔽8)C37⋊C311yescharacteristic, complemented
42C7×AGL(1,𝔽7)—371no—
43C7×C37—61no—
44C7×C37C3×AGL(1,𝔽7)11yescharacteristic, complemented
45C7×C37C2×C3×ΓL(1,𝔽8)11yescharacteristic, complemented
46C2×C3×C37—491no—
47C2×C37⋊C3—71no—
48C2×C37⋊C3—491no—
49C2×C37⋊C3—491no—
50#3862—371no—
51#3861—371no—
52C7×ΓL(1,𝔽8)—371no—
53C7×ΓL(1,𝔽8)C2×C37⋊C311yescharacteristic, complemented
54C2×C3×ΓL(1,𝔽8)—2591no—
55C3×AGL(1,𝔽7)—2591no—
56C3×C37—71no—
57C37⋊C3—71no—
58C37⋊C3—491no—
59C37⋊C3—491no—
60C7×D7C3×C37⋊C311yescharacteristic, complemented, Hall
61C2×C37—71noHall
62C3×ΓL(1,𝔽8)—2591no—
63C3×ΓL(1,𝔽8)—2591no—
64C72C2×C3×C37⋊C311yescharacteristic, complemented, Sylow
65C2×C3×C7—2591no—
66C3×D7—71no—
67C2×ΓL(1,𝔽8)—2591no—
68C2×ΓL(1,𝔽8)—2591no—
69C2×ΓL(1,𝔽8)—71no—
70AGL(1,𝔽7)—2591no—
71AGL(1,𝔽7)—2591no—
72AGL(1,𝔽7)—371no—
73C37ΓL(1,𝔽8)×AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
74C3×C7—2591no—
75C3×C7—71no—
76ΓL(1,𝔽8)—2591no—
77ΓL(1,𝔽8)—2591no—
78ΓL(1,𝔽8)—2591no—
79ΓL(1,𝔽8)—2591no—
80ΓL(1,𝔽8)—15541no—
81ΓL(1,𝔽8)—371no—
82ΓL(1,𝔽8)#1309553511yescharacteristic, direct product factor
83C2×C32—18131noHall
84C2×C7—71no—
85D7ΓL(1,𝔽8)×C37⋊C311yescharacteristic, complemented
86C32—18131noSylow
87C7—61no—
88C7C2×ΓL(1,𝔽8)×C37⋊C311yescharacteristic, complemented
89C7C3×#1309553511yescharacteristic, complemented
90C2×C3—2591no—
91C2×C3—18131no—
92C2×C3—18131no—
93C2×C3—491no—
94C3—2591no—
95C3—71no—
96C3—18131no—
97C3—18131no—
98C2—71noSylow
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)×#11715913C2semidirect
2. D7×ΓL(1,𝔽8)×C37C3semidirect
3. #24925965C3semidirect
4. #24925959C3semidirect
5. C7×#13095535C3semidirect
6. C37⋊#3862C2×C3semidirect
7. #19129588C2×C3semidirect
8. C7×#11715913C2×C3semidirect
9. C7×ΓL(1,𝔽8)×C37C2×C3semidirect
12. C7×D7×C37C32semidirect
15. C72×C37C2×C32semidirect
23. #13095535ΓL(1,𝔽8)direct (N × H)
30. #11715913C2×ΓL(1,𝔽8)semidirect
33. ΓL(1,𝔽8)×C37AGL(1,𝔽7)semidirect
36. D7×C37C3×ΓL(1,𝔽8)semidirect
41. D7×ΓL(1,𝔽8)C37⋊C3semidirect
44. C7×C37C3×AGL(1,𝔽7)semidirect
45. C7×C37C2×C3×ΓL(1,𝔽8)semidirect
53. C7×ΓL(1,𝔽8)C2×C37⋊C3semidirect
60. C7×D7C3×C37⋊C3semidirect
64. C72C2×C3×C37⋊C3semidirect
73. C37ΓL(1,𝔽8)×AGL(1,𝔽7)semidirect
82. ΓL(1,𝔽8)#13095535direct (N × H)
85. D7ΓL(1,𝔽8)×C37⋊C3semidirect
88. C7C2×ΓL(1,𝔽8)×C37⋊C3semidirect
89. C7C3×#13095535semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)×#117159132
D7×ΓL(1,𝔽8)×C373
#249259653
#249259593
C7×#130955353
C3×#130955357
C2×ΓL(1,𝔽8)×C37⋊C37
ΓL(1,𝔽8)×AGL(1,𝔽7)37