Finite Group Databasebeta

← ΓL(1,𝔽8)×C25×D21 #36347751

Subgroups — ΓL(1,𝔽8)×C25×D21 #36347751

Order 22050.

114 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 114 conjugacy classes of subgroups — too many to draw. Only its 45 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
1C3×C7×ΓL(1,𝔽8)×C25C211yesmaximal, characteristic, complemented, Hall
2C7×C25×D21C311yesmaximal, characteristic, complemented
3D7×ΓL(1,𝔽8)×C25—31nomaximal
4C5×ΓL(1,𝔽8)×D21C511yesmaximal, characteristic
5C7×ΓL(1,𝔽8)×C25—21no—
6C7×ΓL(1,𝔽8)×C25D311yescharacteristic, complemented
7C3×C72×C25C2×C311yescharacteristic, complemented
8C3×C25×D21—71nomaximal
9D3×ΓL(1,𝔽8)×C25—71nomaximal
10C7×D7×C25—31noHall
11C3×C5×C7×ΓL(1,𝔽8)C2×C511yescharacteristic
12C32×C7×C25—71no—
13C3×ΓL(1,𝔽8)×C25D711yescharacteristic, complemented
14C5×C7×D21C3×C511yescharacteristic
15C5×D7×ΓL(1,𝔽8)—31no—
16C72×C25C3×D311yescharacteristic, complemented, Hall
17C25×D21ΓL(1,𝔽8)11yescharacteristic, direct product factor
18D3×C7×C25—71no—
19C3×D7×C25—211no—
20C2×ΓL(1,𝔽8)×C25—211no—
21ΓL(1,𝔽8)×D21C2511yescharacteristic, direct product factor, Hall
22C3×C5×C72C2×C3×C511yescharacteristic
23C5×C7×ΓL(1,𝔽8)—21no—
24C5×C7×ΓL(1,𝔽8)C5×D311yescharacteristic
25C3×C5×D21—71no—
26C5×D3×ΓL(1,𝔽8)—71no—
27C3×C7×C25—61no—
28C3×C7×C25—71no—
29C3×C7×C25—141no—
30C3×C7×C25C2×ΓL(1,𝔽8)11yescharacteristic, complemented
31C3×C7×C25C3×D711yescharacteristic, complemented
32ΓL(1,𝔽8)×C25—21no—
33ΓL(1,𝔽8)×C25D2111yescharacteristic, direct product factor
34C5×C7×D7—31no—
35C3×D3×C25—491noHall
36C3×C7×ΓL(1,𝔽8)C2×C2511yescharacteristic, complemented, Hall
37C2×C7×C25—211no—
38D7×C25—31no—
39C32×C5×C7—71no—
40C3×C5×ΓL(1,𝔽8)C5×D711yescharacteristic
41C7×D21C3×C2511yescharacteristic, complemented
42D7×ΓL(1,𝔽8)—31no—
43C5×C72C3×C5×D311yescharacteristic
44C32×C25—71noHall
45C5×D21C5×ΓL(1,𝔽8)11yescharacteristic
46C5×D3×C7—71no—
47C3×C5×D7—211no—
48C2×C5×ΓL(1,𝔽8)—211no—
49C7×C25—61no—
50C7×C25D3×ΓL(1,𝔽8)11yescharacteristic, complemented
51C7×C25C3×D2111yescharacteristic, complemented
52C2×C3×C25—1471no—
53D3×C25—71no—
54C3×C72C2×C3×C2511yescharacteristic, complemented
55C7×ΓL(1,𝔽8)—21no—
56C7×ΓL(1,𝔽8)D3×C2511yescharacteristic, complemented
57C3×D21—71no—
58D3×ΓL(1,𝔽8)—71no—
59C3×C5×C7—61no—
60C3×C5×C7—71no—
61C3×C5×C7—141no—
62C3×C5×C7C2×C5×ΓL(1,𝔽8)11yescharacteristic
63C3×C5×C7C3×C5×D711yescharacteristic
64C5×ΓL(1,𝔽8)—21no—
65C5×ΓL(1,𝔽8)C5×D2111yescharacteristic
66C7×D7—31noHall
67C3×C5×D3—491no—
68C3×C25—71no—
69C3×C25—141no—
70C3×C25D7×ΓL(1,𝔽8)11yescharacteristic, complemented
71C2×C5×C7—211no—
72C5×D7—31no—
73C32×C7—71no—
74C3×ΓL(1,𝔽8)D7×C2511yescharacteristic, complemented
75C2×C25—211noHall
76C72C3×D3×C2511yescharacteristic, complemented, Sylow
77C32×C5—71no—
78D21ΓL(1,𝔽8)×C2511yescharacteristic, direct product factor
79C3×D7—211no—
80D3×C7—71no—
81C2×ΓL(1,𝔽8)—211no—
82C5×C7—61no—
83C5×C7C5×D3×ΓL(1,𝔽8)11yescharacteristic
84C5×C7C3×C5×D2111yescharacteristic
85C2×C3×C5—1471no—
86C5×D3—71no—
87C25ΓL(1,𝔽8)×D2111yescharacteristic, direct product factor, Sylow
88C3×C7—61no—
89C3×C7—71no—
90C3×C7—141no—
91C3×C7C2×ΓL(1,𝔽8)×C2511yescharacteristic, complemented
92C3×C7C3×D7×C2511yescharacteristic, complemented
93ΓL(1,𝔽8)—21no—
94ΓL(1,𝔽8)C25×D2111yescharacteristic, direct product factor
95C3×D3—491noHall
96C3×C5—71no—
97C3×C5—141no—
98C3×C5C5×D7×ΓL(1,𝔽8)11yescharacteristic
99C2×C7—211no—
100D7—31no—
101C2×C5—211no—
102C32—71noSylow
103C7—61no—
104C7D3×ΓL(1,𝔽8)×C2511yescharacteristic, complemented
105C7C3×C25×D2111yescharacteristic, complemented
106C2×C3—1471no—
107D3—71no—
108C5C5×ΓL(1,𝔽8)×D2111yescharacteristic
109C3—71no—
110C3—141no—
111C3D7×ΓL(1,𝔽8)×C2511yescharacteristic, complemented
112C2—211noSylow
subgroupG/H
center, hypercenterC25ΓL(1,𝔽8)×D21
commutator, hypocenterC3×C72C2×C3×C25
FrattiniC5C5×ΓL(1,𝔽8)×D21
FittingC3×C72×C25C2×C3
socle, cosocleC3×C5×C72C2×C3×C5
prime radicalC5×ΓL(1,𝔽8)×D21C5
LCSG ▷ C3×C72
UCSC1 ◁ C25

Complements

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

NH ≅ G/Ntype
1. C3×C7×ΓL(1,𝔽8)×C25C2semidirect
2. C7×C25×D21C3semidirect
6. C7×ΓL(1,𝔽8)×C25D3semidirect
7. C3×C72×C25C2×C3semidirect
13. C3×ΓL(1,𝔽8)×C25D7semidirect
16. C72×C25C3×D3semidirect
17. C25×D21ΓL(1,𝔽8)direct (N × H)
21. ΓL(1,𝔽8)×D21C25direct (N × H)
30. C3×C7×C25C2×ΓL(1,𝔽8)semidirect
31. C3×C7×C25C3×D7semidirect
33. ΓL(1,𝔽8)×C25D21direct (N × H)
36. C3×C7×ΓL(1,𝔽8)C2×C25semidirect
41. C7×D21C3×C25semidirect
50. C7×C25D3×ΓL(1,𝔽8)semidirect
51. C7×C25C3×D21semidirect
54. C3×C72C2×C3×C25semidirect
56. C7×ΓL(1,𝔽8)D3×C25semidirect
70. C3×C25D7×ΓL(1,𝔽8)semidirect
74. C3×ΓL(1,𝔽8)D7×C25semidirect
76. C72C3×D3×C25semidirect
78. D21ΓL(1,𝔽8)×C25direct (N × H)
87. C25ΓL(1,𝔽8)×D21direct (N × H)
91. C3×C7C2×ΓL(1,𝔽8)×C25semidirect
92. C3×C7C3×D7×C25semidirect
94. ΓL(1,𝔽8)C25×D21direct (N × H)
104. C7D3×ΓL(1,𝔽8)×C25semidirect
105. C7C3×C25×D21semidirect
111. C3D7×ΓL(1,𝔽8)×C25semidirect

Maximal subgroups

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

subgroupindex
C3×C7×ΓL(1,𝔽8)×C252
C7×C25×D213
D7×ΓL(1,𝔽8)×C253
C5×ΓL(1,𝔽8)×D215
C3×C25×D217
D3×ΓL(1,𝔽8)×C257