Finite Group Databasebeta

← #13057071

Subgroups — #13057071

Order 1344.

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

normal subgroups only

This group has 130 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
1SL(2,𝔽3)×Dic7C211yesmaximal, characteristic
2C7⋊#859C211yesmaximal, characteristic
3C7×#859C211yesmaximal, characteristic
4#89563—31nomaximal, Hall
5C2×C7×SL(2,𝔽3)C2211yescharacteristic
6C2×#4248—41nomaximal
7Q8×Dic7D311yescharacteristic
8#6451—31no—
9C7×#103—31no—
10#6409—31no—
11#6391—31no—
12Dic7⋊C8—31no—
13#6382—31no—
14#5386—71nomaximal, Hall
15C2×Dic21—41no—
16C2×C7×Dic3—41no—
17C2×C3×Dic7—41no—
18C7×SL(2,𝔽3)Q811yescharacteristic, complemented
19#4248—44no—
20C2×C7×Q8C2×D311yescharacteristic
21C2×Dic14—61no—
22C2×C7×C8—31no—
23C7×C4⋊C4—31no—
24C4⋊Dic7—31no—
25C4⋊Dic7—31no—
26Dic7⋊C4—61no—
27C4×Dic7—31no—
28C2×C7⋊C8—31no—
29C4×SL(2,𝔽3)—71no—
30#859—71no—
31#859D711yescharacteristic
32C22×C3×C7—41no—
33Dic21—42no—
34C3×Dic7—42no—
35C7×Dic3—42no—
36#475—211noSylow
37C7×Q8—31no—
38C7×Q8Dic611yescharacteristic, complemented
39C2×C4×C7—31no—
40C2×C4×C7—61no—
41C2×Dic7—31no—
42C2×Dic7—61no—
43C2×Dic7S411yescharacteristic
44Dic14—122no—
45C7×C8—61no—
46C7⋊C8—61no—
47C2×Dic6—281no—
48C2×SL(2,𝔽3)C2×D711yescharacteristic
49C2×C3×C7—41no—
50C2×C3×C7—41no—
51C2×C3×C7—41no—
52C4⋊Q8—211no—
53C4×Q8—71no—
54#106—211no—
55#106—211no—
56C4⋊C8—211no—
57#103—31no—
58#103—211no—
59C22×C7C2×S411yescharacteristic
60C4×C7—31no—
61C4×C7—31no—
62C4×C7—121no—
63Dic7—61no—
64Dic7—121no—
65Dic7GL(2,𝔽3)12yes—
66C2×C3×C4—281no—
67C2×Dic3—41no—
68C2×Dic3—281no—
69Dic6—284no—
70SL(2,𝔽3)Dic1411yescharacteristic, complemented
71C3×C7—41noHall
72C2×Q8—421no—
73C2×Q8D3×D711yescharacteristic
74C2×C8—31no—
75C2×C8—211no—
76C4⋊C4—31no—
77C4⋊C4—211no—
78C4⋊C4—211no—
79C4⋊C4—421no—
80C42—211no—
81C2×C7A4⋊Q811yescharacteristic
82C2×C7C2×GL(2,𝔽3)11yescharacteristic
83C2×C7#98411yescharacteristic
84C22×C3—41no—
85C3×C4—282no—
86Dic3—42no—
87Dic3—282no—
88Q8—31no—
89Q8—842no—
90Q8#424811yescharacteristic, complemented
91C2×C4—31no—
92C2×C4—61no—
93C2×C4—71no—
94C2×C4—211no—
95C2×C4—421no—
96C8—61no—
97C8—421no—
98C7#538611yescharacteristic, complemented, Sylow
99C2×C3—41no—
100C2×C3—41no—
101C2×C3—41no—
102C22D7×S411yescharacteristic
103C4—31no—
104C4—31no—
105C4—72no—
106C4—121no—
107C4—421no—
108C4—841no—
109C3—41noSylow
110C2#1062146911yescharacteristic
111C2D7×GL(2,𝔽3)11yescharacteristic
112C2#1062147711yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D7×S4
commutator, prime radical, cosocleC2×C7×SL(2,𝔽3)C22
FittingC2×C7×Q8C2×D3
socleC22×C7C2×S4
hypocenterC7×SL(2,𝔽3)Q8
LCSG ▷ C2×C7×SL(2,𝔽3) ▷ C7×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
18. C7×SL(2,𝔽3)Q8semidirect
38. C7×Q8Dic6semidirect
70. SL(2,𝔽3)Dic14semidirect
90. Q8#4248semidirect
98. C7#5386semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×Dic72
C7⋊#8592
C7×#8592
#895633
C2×#42484
#53867