Finite Group Databasebeta

← C3×ΓL(1,𝔽8)×D175 #36347740

Subgroups — C3×ΓL(1,𝔽8)×D175 #36347740

Order 22050.

150 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 56 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
2C3×C7×D175C311yesmaximal, characteristic, complemented
3ΓL(1,𝔽8)×D175C313yesmaximal, direct product factor
4C3×ΓL(1,𝔽8)×D35—51nomaximal
5C7×ΓL(1,𝔽8)×C25C2×C313yescomplemented
6C3×C72×C25C2×C311yescharacteristic, complemented
7C32×D175—71nomaximal
8C3×ΓL(1,𝔽8)×D25—71nomaximal
9C7×D175C3211yescharacteristic, complemented, Hall
10C3×C5×C7×ΓL(1,𝔽8)D511yescharacteristic
11C32×C7×C25—71no—
12C3×ΓL(1,𝔽8)×C25D711yescharacteristic, complemented
13C3×C7×D35—51no—
14ΓL(1,𝔽8)×D35—53no—
15C72×C25C2×C3211yescharacteristic, complemented, Hall
16C3×D175—73no—
17C3×D175ΓL(1,𝔽8)11yescharacteristic, direct product factor
18C3×C7×D25—71no—
19ΓL(1,𝔽8)×D25—73no—
20C3×D7×ΓL(1,𝔽8)—251noHall
21C3×C5×C72C3×D511yescharacteristic
22C5×C7×ΓL(1,𝔽8)C3×D513yes—
23C32×D35—351no—
24C3×D5×ΓL(1,𝔽8)—351no—
25C3×C7×C25—61no—
26C3×C7×C25—73no—
27C3×C7×C25C2×ΓL(1,𝔽8)11yescharacteristic, complemented
28C3×C7×C25C3×D711yescharacteristic, complemented
29ΓL(1,𝔽8)×C25C3×D713yescomplemented
30C7×D35—51no—
31C32×D25—491noHall
32C3×C7×ΓL(1,𝔽8)D2511yescharacteristic, complemented, Hall
33D175C3×ΓL(1,𝔽8)11yescharacteristic, direct product factor
34C7×D25—71no—
35C32×C5×C7—71no—
36C3×C5×ΓL(1,𝔽8)D3511yescharacteristic
37C3×C7×D7—251no—
38D7×ΓL(1,𝔽8)—253no—
39C5×C72C32×D511yescharacteristic
40C32×C25—71noHall
41C3×D35—51no—
42C3×D35—353no—
43C3×C7×D5—351no—
44D5×ΓL(1,𝔽8)—353no—
45C7×C25—61no—
46C7×C25C2×C3×ΓL(1,𝔽8)11yescharacteristic, complemented
47C7×C25C32×D711yescharacteristic, complemented
48C3×D25—71no—
49C3×D25—493no—
50C3×C72C3×D2511yescharacteristic, complemented
51C7×ΓL(1,𝔽8)C3×D2513yescomplemented
52C32×D7—1751no—
53C2×C3×ΓL(1,𝔽8)—1751no—
54C3×C5×C7—61no—
55C3×C5×C7—73no—
56C3×C5×C7D5×ΓL(1,𝔽8)11yescharacteristic
57C3×C5×C7C3×D3511yescharacteristic
58C5×ΓL(1,𝔽8)C3×D3513yes—
59C7×D7—251noHall
60C32×D5—2451no—
61C3×C25—73no—
62C3×C25D7×ΓL(1,𝔽8)11yescharacteristic, complemented
63D35—51no—
64C7×D5—351no—
65C32×C7—71no—
66C3×ΓL(1,𝔽8)D17511yescharacteristic, direct product factor
67D25—71noHall
68C72C32×D2511yescharacteristic, complemented, Sylow
69C32×C5—71no—
70C2×C3×C7—1751no—
71C3×D7—251no—
72C3×D7—1753no—
73C2×ΓL(1,𝔽8)—1753no—
74C5×C7—61no—
75C5×C7C3×D5×ΓL(1,𝔽8)11yescharacteristic
76C5×C7C32×D3511yescharacteristic
77C3×D5—351no—
78C3×D5—2453no—
79C25C3×D7×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
80C3×C7—61no—
81C3×C7—73no—
82C3×C7ΓL(1,𝔽8)×D2511yescharacteristic, complemented
83C3×C7C3×D17511yescharacteristic, complemented
84ΓL(1,𝔽8)C3×D17513yesdirect product factor
85C2×C32—12251noHall
86C3×C5—73no—
87C3×C5ΓL(1,𝔽8)×D3511yescharacteristic
88C2×C7—1751no—
89D7—251no—
90D5—351no—
91C32—71noSylow
92C7—61no—
93C7C3×ΓL(1,𝔽8)×D2511yescharacteristic, complemented
94C7C32×D17511yescharacteristic, complemented
95C2×C3—1751no—
96C2×C3—12253no—
97C5C3×ΓL(1,𝔽8)×D3511yescharacteristic
98C3—73no—
99C3ΓL(1,𝔽8)×D17511yescharacteristic, direct product factor
100C2—1751noSylow
subgroupG/H
center, hypercenterC3ΓL(1,𝔽8)×D175
commutator, cosocle, hypocenterC72×C25C2×C32
FrattiniC5C3×ΓL(1,𝔽8)×D35
FittingC3×C72×C25C2×C3
socleC3×C5×C72C3×D5
LCSG ▷ C72×C25
UCSC1 ◁ C3

Complements

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

NH ≅ G/Ntype
1. C3×C7×ΓL(1,𝔽8)×C25C2semidirect
2. C3×C7×D175C3semidirect
3. ΓL(1,𝔽8)×D175C3direct (N × H)
5. C7×ΓL(1,𝔽8)×C25C2×C3semidirect
6. C3×C72×C25C2×C3semidirect
9. C7×D175C32semidirect
12. C3×ΓL(1,𝔽8)×C25D7semidirect
15. C72×C25C2×C32semidirect
17. C3×D175ΓL(1,𝔽8)direct (N × H)
27. C3×C7×C25C2×ΓL(1,𝔽8)semidirect
28. C3×C7×C25C3×D7semidirect
29. ΓL(1,𝔽8)×C25C3×D7semidirect
32. C3×C7×ΓL(1,𝔽8)D25semidirect
33. D175C3×ΓL(1,𝔽8)direct (N × H)
46. C7×C25C2×C3×ΓL(1,𝔽8)semidirect
47. C7×C25C32×D7semidirect
50. C3×C72C3×D25semidirect
51. C7×ΓL(1,𝔽8)C3×D25semidirect
62. C3×C25D7×ΓL(1,𝔽8)semidirect
66. C3×ΓL(1,𝔽8)D175direct (N × H)
68. C72C32×D25semidirect
79. C25C3×D7×ΓL(1,𝔽8)semidirect
82. C3×C7ΓL(1,𝔽8)×D25semidirect
83. C3×C7C3×D175semidirect
84. ΓL(1,𝔽8)C3×D175direct (N × H)
93. C7C3×ΓL(1,𝔽8)×D25semidirect
94. C7C32×D175semidirect
99. C3ΓL(1,𝔽8)×D175direct (N × H)

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
C3×C7×D1753
ΓL(1,𝔽8)×D1753
C3×ΓL(1,𝔽8)×D355
C32×D1757
C3×ΓL(1,𝔽8)×D257