Finite Group Databasebeta

← #13057058

Subgroups — #13057058

Order 1344.

112 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 25 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#89571—31nomaximal, Hall
5C2×C7×SL(2,𝔽3)C2211yescharacteristic
6Dic3⋊Dic7—41nomaximal
7Q8×Dic7D311yescharacteristic
8#6453—31no—
9C7×#103—31no—
10#6409—31no—
11#6391—31no—
12Dic7⋊C8—31no—
13#6381—31no—
14#5389—71nomaximal, Hall
15C2×Dic21—41no—
16C2×C7×Dic3—41no—
17C2×C3×Dic7—41no—
18C7×SL(2,𝔽3)Q811yescharacteristic
19C2×C7×Q8C2×D311yescharacteristic
20C2×C7×C8—31no—
21C7×C4⋊C4—31no—
22C4⋊Dic7—31no—
23C4⋊Dic7—31no—
24C4⋊Dic7—61no—
25Dic7⋊C4—61no—
26C4×Dic7—31no—
27C2×C7⋊C8—31no—
28C4×SL(2,𝔽3)—71no—
29#859—71no—
30#859D711yescharacteristic
31C22×C3×C7—41no—
32Dic21—81no—
33C3×Dic7—81no—
34C7×Dic3—42no—
35#479—211noSylow
36C7×Q8—31no—
37C7×Q8Dic611yescharacteristic
38C2×C4×C7—31no—
39C2×C4×C7—61no—
40C2×Dic7—31no—
41C2×Dic7—61no—
42C2×Dic7S411yescharacteristic
43C7×C8—61no—
44C7⋊C8—61no—
45C2×SL(2,𝔽3)C2×D711yescharacteristic
46Dic3⋊C4—281no—
47C2×C3×C7—41no—
48C2×C3×C7—41no—
49C2×C3×C7—41no—
50#125—211no—
51C4×Q8—71no—
52#107—211no—
53#106—211no—
54C4⋊C8—211no—
55#103—31no—
56#103—211no—
57C22×C7C2×S411yescharacteristic
58C4×C7—31no—
59C4×C7—31no—
60C4×C7—121no—
61Dic7—21no—
62Dic7—61no—
63Dic7—121no—
64C2×C3×C4—281no—
65C2×Dic3—41no—
66C2×Dic3—281no—
67SL(2,𝔽3)Dic1411yescharacteristic
68C3×C7—41noHall
69C2×Q8D3×D711yescharacteristic
70C2×C8—31no—
71C2×C8—211no—
72C4⋊C4—31no—
73C4⋊C4—211no—
74C4⋊C4—211no—
75C4⋊C4—421no—
76C4⋊C4—421no—
77C42—211no—
78C2×C7A4⋊Q811yescharacteristic
79C2×C7#98311yescharacteristic
80C2×C7#98511yescharacteristic
81C22×C3—41no—
82C3×C4—561no—
83Dic3—42no—
84Dic3—561no—
85Q8—31no—
86Q8#424811yescharacteristic
87C2×C4—31no—
88C2×C4—61no—
89C2×C4—71no—
90C2×C4—211no—
91C2×C4—421no—
92C8—61no—
93C8—421no—
94C7#538911yescharacteristic, complemented, Sylow
95C2×C3—41no—
96C2×C3—41no—
97C2×C3—41no—
98C22D7×S411yescharacteristic
99C4—31no—
100C4—31no—
101C4—121no—
102C4—141no—
103C4—421no—
104C4—841no—
105C3—41noSylow
106C2#1062147011yescharacteristic
107C2#1062147511yescharacteristic
108C2#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
94. C7#5389semidirect

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
#895713
Dic3⋊Dic74
#53897