Finite Group Databasebeta

← #50303638

Subgroups — #50303638

Order 38220.

144 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 43 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
1C2×C5×C13×#3861C211yesmaximal, characteristic
2C7×Dic455C311yesmaximal, characteristic, complemented, Hall
3C5×C13×#3861C411yescharacteristic, complemented, Hall
4#21439536—51nomaximal, Hall
5C2×C5×C72×C13C2×C311yescharacteristic
6ΓL(1,𝔽8)⋊Dic65—71nomaximal
7ΓL(1,𝔽8)×Dic65—71nomaximal
8C2×C13×#3861D511yescharacteristic
9C5×C72×C13C3×C411yescharacteristic, complemented, Hall
10#15668736—131nomaximal, Hall
11C2×C5×C13×ΓL(1,𝔽8)—71no—
12C2×C5×C13×ΓL(1,𝔽8)—71no—
13C2×C5×C13×ΓL(1,𝔽8)—143no—
14C7×Dic91—51noHall
15C13×#3861Dic511yescharacteristic, complemented, Hall
16Dic455ΓL(1,𝔽8)11yescharacteristic, complemented
17C7×Dic65—71no—
18C2×C5×#3861D1311yescharacteristic
19C5×C13×ΓL(1,𝔽8)—71no—
20C5×C13×ΓL(1,𝔽8)—71no—
21C5×C13×ΓL(1,𝔽8)—143no—
22C2×C72×C13C3×D511yescharacteristic
23ΓL(1,𝔽8)⋊Dic13—351no—
24ΓL(1,𝔽8)×Dic13—351no—
25C7×Dic35—131noHall
26C2×C5×C7×C13—23no—
27C2×C5×C7×C13AGL(1,𝔽7)11yescharacteristic
28C2×C5×C7×C13C2×ΓL(1,𝔽8)11yescharacteristic
29C3×Dic65—491noHall
30C5×#3861Dic1311yescharacteristic, complemented, Hall
31C72×C13C3×Dic511yescharacteristic, complemented, Hall
32#10596873—651noHall
33C2×C13×ΓL(1,𝔽8)—71no—
34C2×C13×ΓL(1,𝔽8)—71no—
35C2×C13×ΓL(1,𝔽8)—143no—
36C2×C5×C72C3×D1311yescharacteristic
37C5×C7×C13—23no—
38C5×C7×C13ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
39C5×C7×C13C4×ΓL(1,𝔽8)11yescharacteristic, complemented
40ΓL(1,𝔽8)⋊Dic5—911no—
41Dic5×ΓL(1,𝔽8)—911no—
42C2×C3×C5×C13—491no—
43Dic91—51no—
44C7×Dic13—351no—
45C2×#3861D6511yescharacteristic
46C13×ΓL(1,𝔽8)—71no—
47C13×ΓL(1,𝔽8)—71no—
48C13×ΓL(1,𝔽8)—143no—
49Dic65—71noHall
50C5×C72C3×Dic1311yescharacteristic, complemented, Hall
51C2×C5×ΓL(1,𝔽8)—71no—
52C2×C5×ΓL(1,𝔽8)—71no—
53C2×C5×ΓL(1,𝔽8)—143no—
54C7×Dic7—651noHall
55C3×C5×C13—491noHall
56C2×C7×C13—23no—
57C2×C7×C13D5×ΓL(1,𝔽8)11yescharacteristic
58C2×C7×C13C5⋊AGL(1,𝔽7)11yescharacteristic
59C3×Dic13—2451noHall
60#3861Dic6511yescharacteristic, complemented, Hall
61Dic35—131no—
62C7×Dic5—911no—
63C2×C5×C13#6442311yescharacteristic
64C5×ΓL(1,𝔽8)—71no—
65C5×ΓL(1,𝔽8)—71no—
66C5×ΓL(1,𝔽8)—143no—
67C2×C72C3×D6511yescharacteristic
68C7×C13—23no—
69C7×C13Dic5×ΓL(1,𝔽8)11yescharacteristic, complemented
70C7×C13ΓL(1,𝔽8)⋊Dic511yescharacteristic, complemented
71C4×ΓL(1,𝔽8)—4551no—
72ΓL(1,𝔽8)⋊C4—4551no—
73C2×C3×C13—491no—
74C2×C5×C7—23no—
75C2×C5×C7ΓL(1,𝔽8)×D1311yescharacteristic
76C2×C5×C7ΓL(1,𝔽8)⋊D1311yescharacteristic
77C5×C13#1059687311yescharacteristic, complemented, Hall
78C3×Dic5—6371noHall
79Dic13—351noHall
80C72C3×Dic6511yescharacteristic, complemented, Sylow
81C2×ΓL(1,𝔽8)—71no—
82C2×ΓL(1,𝔽8)—71no—
83C2×ΓL(1,𝔽8)—143no—
84C3×C13—491noHall
85C5×C7—23no—
86C5×C7ΓL(1,𝔽8)×Dic1311yescharacteristic, complemented
87C5×C7ΓL(1,𝔽8)⋊Dic1311yescharacteristic, complemented
88C2×C3×C5—491no—
89C4×C7—4551no—
90Dic7—651no—
91C2×C13C5⋊#6442311yescharacteristic
92ΓL(1,𝔽8)—71no—
93ΓL(1,𝔽8)—71no—
94ΓL(1,𝔽8)—143no—
95Dic5—911noHall
96C3×C5—491noHall
97C2×C7—23no—
98C2×C7ΓL(1,𝔽8)×D6511yescharacteristic
99C2×C7C5⋊(ΓL(1,𝔽8)⋊D13)11yescharacteristic
100C13#1566873611yescharacteristic, complemented, Sylow
101C3×C4—31851noHall
102C2×C5#1683846011yescharacteristic
103C7—23no—
104C7ΓL(1,𝔽8)×Dic6511yescharacteristic, complemented
105C7ΓL(1,𝔽8)⋊Dic6511yescharacteristic, complemented
106C2×C3—491no—
107C5#2143953611yescharacteristic, complemented, Sylow
108C4—4551noSylow
109C3—491noSylow
110C2C5⋊#1683846011yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2C5⋊#16838460
commutator, hypocenterC5×C72×C13C3×C4
Fitting, socle, cosocleC2×C5×C72×C13C2×C3
prime radicalC2×C5×C13×#3861C2
LCSG ▷ C5×C72×C13
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. C7×Dic455C3semidirect
3. C5×C13×#3861C4semidirect
9. C5×C72×C13C3×C4semidirect
15. C13×#3861Dic5semidirect
16. Dic455ΓL(1,𝔽8)semidirect
30. C5×#3861Dic13semidirect
31. C72×C13C3×Dic5semidirect
38. C5×C7×C13ΓL(1,𝔽8)⋊C4semidirect
39. C5×C7×C13C4×ΓL(1,𝔽8)semidirect
50. C5×C72C3×Dic13semidirect
60. #3861Dic65semidirect
69. C7×C13Dic5×ΓL(1,𝔽8)semidirect
70. C7×C13ΓL(1,𝔽8)⋊Dic5semidirect
77. C5×C13#10596873semidirect
80. C72C3×Dic65semidirect
86. C5×C7ΓL(1,𝔽8)×Dic13semidirect
87. C5×C7ΓL(1,𝔽8)⋊Dic13semidirect
100. C13#15668736semidirect
104. C7ΓL(1,𝔽8)×Dic65semidirect
105. C7ΓL(1,𝔽8)⋊Dic65semidirect
107. C5#21439536semidirect

Maximal subgroups

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

subgroupindex
C2×C5×C13×#38612
C7×Dic4553
#214395365
ΓL(1,𝔽8)⋊Dic657
ΓL(1,𝔽8)×Dic657
#1566873613