Finite Group Databasebeta

← #13062208

Subgroups — #13062208

Order 1344.

350 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 27 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
1C22⋊(C7⋊SL(2,𝔽3))C211yesmaximal, characteristic, complemented
2C22×#1161C311yesmaximal, characteristic, complemented, Hall
3Dic7⋊A4—41nomaximal
4#66665—44nomaximal
5C22×C7×Q8C2×C311yescharacteristic
6C2×#1161—34no—
7C2×#1161—34no—
8C22×D28—31no—
9C22×C4×D7—31no—
10#5948—71nomaximal, Hall
11C2×C22⋊ΓL(1,𝔽8)—41no—
12C7⋊SL(2,𝔽3)—44no—
13C23×D7—31no—
14C2×C7×Q8—34no—
15C22×C4×C7—31no—
16C22×Dic7A411yescharacteristic
17#1161—312no—
18#1161—34no—
19#1161—34no—
20#1161A414yescomplemented
21C2×D28—312no—
22C2×C4×D7—36no—
23C2×C4×D7—36no—
24C22⋊SL(2,𝔽3)—71no—
25C22⋊ΓL(1,𝔽8)—41no—
26ΓL(1,𝔽8)⋊C4—161no—
27C22×#41—71noSylow
28C23×C7C2×A411yescharacteristic
29C22×D7—36no—
30C22×D7—63no—
31C22×D7—61no—
32C7×Q8—34no—
33C7×Q8C2×A414yes—
34C2×C4×C7—36no—
35C2×Dic7—31no—
36C2×Dic7—31no—
37D28—34no—
38D28—312no—
39C4×D7—312no—
40C4×D7—34no—
41#41⋊C3—284no—
42C4×A4—281no—
43C2×ΓL(1,𝔽8)—161no—
44C2×#41—214no—
45C2×#41—214no—
46C22×Q8AGL(1,𝔽7)11yescharacteristic
47C22×D4—211no—
48C23×C4—211no—
49C22×C7—31no—
50C22×C7—31no—
51C22×C7#41⋊C311yescharacteristic, complemented
52C2×D7—34no—
53C2×D7—66no—
54C2×D7—66no—
55C4×C7—34no—
56Dic7—31no—
57Dic7C22⋊A411yescharacteristic
58C2×A4—281no—
59SL(2,𝔽3)—284no—
60ΓL(1,𝔽8)—161noHall
61C24—211no—
62#41—74no—
63#41—214no—
64#41—2112no—
65#41—214no—
66C2×Q8—34no—
67C2×D4—2112no—
68C22×C4—31no—
69C22×C4—71no—
70C22×C4—216no—
71C22×C4—216no—
72C2×C7—31no—
73C2×C7—31no—
74C2×C7C2×C22⋊A411yescharacteristic
75D7—64no—
76A4—281no—
77C3×C4—1121no—
78C23—216no—
79C23—423no—
80C23—421no—
81C23D7⋊A411yescharacteristic
82Q8—34no—
83Q8D7⋊A414yes—
84D4—214no—
85D4—2112no—
86C2×C4—36no—
87C2×C4—214no—
88C2×C4—2112no—
89C2×C4—211no—
90C2×C4—211no—
91C7#594811yescharacteristic, complemented, Sylow
92C2×C3—1121no—
93C22—31no—
94C22—31no—
95C22—214no—
96C22—426no—
97C22—426no—
98C22#6666511yescharacteristic, complemented
99C4—34no—
100C4—71no—
101C4—211no—
102C3—1121noSylow
103C2—31no—
104C2—31no—
105C2—424no—
106C2D7⋊(C22⋊A4)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D7⋊(C22⋊A4)
commutator, Fitting, cosocle, hypocenterC22×C7×Q8C2×C3
socleC23×C7C2×A4
LCSG ▷ C22×C7×Q8
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C22⋊(C7⋊SL(2,𝔽3))C2semidirect
2. C22×#1161C3semidirect
20. #1161A4semidirect
51. C22×C7#41⋊C3semidirect
91. C7#5948semidirect
98. C22#66665semidirect

Maximal subgroups

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

subgroupindex
C22⋊(C7⋊SL(2,𝔽3))2
C22×#11613
Dic7⋊A44
#666654
#59487