Finite Group Databasebeta

← #13057158

Subgroups — #13057158

Order 1344.

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

normal subgroups only

This group has 112 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
1C4×C7×SL(2,𝔽3)C211yesmaximal, characteristic
2C7⋊#859C211yesmaximal, characteristic
3C7⋊#859C211yesmaximal, characteristic
4#89794—31nomaximal, Hall
5C2×C7×SL(2,𝔽3)C2211yescharacteristic
6#66629—41nomaximal
7C4×C7×Q8D311yescharacteristic
8#6433—31no—
9#6409—31no—
10#6409—31no—
11#6382—31no—
12#6381—31no—
13C4⋊(C7⋊C8)—31no—
14#5389—71nomaximal, Hall
15C2×C3×C4×C7—41no—
16C2×Dic21—41no—
17C2×Dic21—41no—
18C7×SL(2,𝔽3)Q811yescharacteristic
19C2×C7×Q8C2×D311yescharacteristic
20C7×C4⋊C4—31no—
21C42×C7—31no—
22C4⋊Dic7—31no—
23C4⋊Dic7—31no—
24Dic7⋊C4—61no—
25Dic7⋊C4—61no—
26C2×C7⋊C8—31no—
27C2×C7⋊C8—31no—
28C4×SL(2,𝔽3)D711yescharacteristic
29#859—71no—
30#859—71no—
31C22×C3×C7—41no—
32C3×C4×C7—81no—
33Dic21—42no—
34Dic21—81no—
35#479—211noSylow
36C7×Q8—31no—
37C7×Q8Dic611yescharacteristic
38C2×C4×C7—31no—
39C2×C4×C7—31no—
40C2×C4×C7S411yescharacteristic
41C2×Dic7—61no—
42C2×Dic7—61no—
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×Q8D2111yescharacteristic
52#107—211no—
53#106—211no—
54C4⋊C8—211no—
55#103—211no—
56#103—211no—
57C22×C7C2×S411yescharacteristic
58C4×C7—21no—
59C4×C7—31no—
60C4×C7—31no—
61C4×C7—61no—
62Dic7—121no—
63Dic7—121no—
64C2×C3×C4—41no—
65C2×Dic3—281no—
66C2×Dic3—281no—
67SL(2,𝔽3)Dic1411yescharacteristic
68C3×C7—41noHall
69C2×Q8C2×D2111yescharacteristic
70C2×C8—211no—
71C2×C8—211no—
72C4⋊C4—31no—
73C4⋊C4—211no—
74C4⋊C4—211no—
75C4⋊C4—421no—
76C4⋊C4—421no—
77C42—31no—
78C2×C7A4⋊Q811yescharacteristic
79C2×C7#98311yescharacteristic
80C2×C7#98511yescharacteristic
81C22×C3—41no—
82C3×C4—81no—
83Dic3—282no—
84Dic3—561no—
85Q8—31no—
86Q8Dic4211yescharacteristic
87C2×C4—31no—
88C2×C4—31no—
89C2×C4—421no—
90C2×C4—421no—
91C2×C4C7⋊S411yescharacteristic
92C8—421no—
93C8—421no—
94C7#538911yescharacteristic, complemented, Sylow
95C2×C3—41no—
96C2×C3—41no—
97C2×C3—41no—
98C22C2×C7⋊S411yescharacteristic
99C4—21no—
100C4—31no—
101C4—31no—
102C4—61no—
103C4—841no—
104C4—841no—
105C3—41noSylow
106C2#1062149611yescharacteristic
107C2#1062150111yescharacteristic
108C2#1062150311yescharacteristic
subgroupG/H
center, FrattiniC22C2×C7⋊S4
commutator, prime radical, cosocleC2×C7×SL(2,𝔽3)C22
FittingC4×C7×Q8D3
socleC22×C7C2×S4
hypercenterC2×C4C7⋊S4
hypocenterC7×SL(2,𝔽3)Q8
LCSG ▷ C2×C7×SL(2,𝔽3) ▷ C7×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

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
C4×C7×SL(2,𝔽3)2
C7⋊#8592
#897943
#666294
#53897