Finite Group Databasebeta

← #13057059

Subgroups — #13057059

Order 1344.

158 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 31 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×#66662C211yesmaximal, characteristic
2C7⋊#859C211yesmaximal, characteristic, complemented
3C7×#859C211yesmaximal, characteristic, complemented
4#89575—31nomaximal, Hall
5C2×C7×SL(2,𝔽3)C2211yescharacteristic
6#66662C412yescomplemented
7Dic7⋊Dic3—41nomaximal
8C2×#1161D311yescharacteristic
9C2×#1131—31no—
10D7×C4⋊C4—31no—
11C7×#103—31no—
12#6409—31no—
13#6395—31no—
14#6383—31no—
15#5426—71nomaximal, Hall
16C2×Dic21—41no—
17C2×C7×Dic3—41no—
18C2×C3×Dic7—41no—
19C7×SL(2,𝔽3)C2×C411yescharacteristic, complemented
20C2×C7×Q8C2×D311yescharacteristic
21#1161—32no—
22#1161Dic312yescomplemented
23C2×D28—31no—
24C2×C4×D7—31no—
25C2×C4×D7—61no—
26C2×C7×C8—31no—
27C7×C4⋊C4—31no—
28C4⋊Dic7—31no—
29Dic7⋊C4—61no—
30C2×C7⋊C8—31no—
31#1131—62no—
32C2×#41⋊C3—71no—
33#859—71no—
34#859D711yescharacteristic, complemented
35C22×C3×C7—41no—
36Dic21—81no—
37C3×Dic7—42no—
38C7×Dic3—81no—
39#417—211noSylow
40C22×D7—31no—
41C7×Q8—31no—
42C7×Q8C2×Dic311yescharacteristic
43C2×C4×C7—31no—
44C2×C4×C7—61no—
45C2×Dic7—61no—
46C2×Dic7S411yescharacteristic
47D28—32no—
48D28—61no—
49C4×D7—32no—
50C4×D7—32no—
51C4×D7—122no—
52C7×C8—61no—
53C7⋊C8—61no—
54#41⋊C3—72no—
55C2×SL(2,𝔽3)C2×D711yescharacteristic
56C4⋊Dic3—281no—
57C2×C3×C7—41no—
58C2×C3×C7—41no—
59C2×C3×C7—41no—
60C2×#41—71no—
61C2×M16—211no—
62C2×C4⋊C4—211no—
63#103—31no—
64#103—211no—
65#102—211no—
66#102—211no—
67C22×C7C2×S411yescharacteristic
68C2×D7—32no—
69C2×D7—61no—
70C2×D7—61no—
71C4×C7—31no—
72C4×C7—31no—
73C4×C7—121no—
74Dic7—121no—
75Dic7SL(2,ℤ/4)12yes—
76C2×C3×C4—281no—
77C2×Dic3—41no—
78C2×Dic3—281no—
79SL(2,𝔽3)C4×D711yescharacteristic, complemented
80C3×C7—41noHall
81#41—72no—
82#41—212no—
83C2×Q8D3×D711yescharacteristic
84C2×D4—211no—
85C22×C4—211no—
86C22×C4—421no—
87M16—422no—
88C2×C8—31no—
89C2×C8—211no—
90C4⋊C4—31no—
91C4⋊C4—211no—
92C4⋊C4—421no—
93C2×C7#98411yescharacteristic
94C2×C7#98611yescharacteristic
95C2×C7C2×SL(2,ℤ/4)11yescharacteristic
96D7—62no—
97C22×C3—41no—
98C3×C4—282no—
99Dic3—81no—
100Dic3—561no—
101C23—211no—
102Q8—31no—
103Q8Dic3×D711yescharacteristic
104D4—212no—
105D4—421no—
106C2×C4—31no—
107C2×C4—61no—
108C2×C4—71no—
109C2×C4—212no—
110C2×C4—212no—
111C2×C4—421no—
112C2×C4—842no—
113C8—61no—
114C8—421no—
115C7#542611yescharacteristic, complemented, Sylow
116C2×C3—41no—
117C2×C3—41no—
118C2×C3—41no—
119C22—212no—
120C22—421no—
121C22—421no—
122C22D7×S411yescharacteristic
123C4—31no—
124C4—31no—
125C4—72no—
126C4—121no—
127C4—841no—
128C3—41noSylow
129C2—422no—
130C2#1062146911yescharacteristic
131C2#1062147211yescharacteristic
132C2D7×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D7×S4
commutator, hypocenterC7×SL(2,𝔽3)C2×C4
FittingC2×C7×Q8C2×D3
socleC22×C7C2×S4
prime radicalC2×#66662C2
cosocleC2×C7×SL(2,𝔽3)C22
LCSG ▷ C7×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C7⋊#859C2semidirect
3. C7×#859C2semidirect
6. #66662C4semidirect
19. C7×SL(2,𝔽3)C2×C4semidirect
22. #1161Dic3semidirect
34. #859D7semidirect
79. SL(2,𝔽3)C4×D7semidirect
115. C7#5426semidirect

Maximal subgroups

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

subgroupindex
C2×#666622
C7⋊#8592
C7×#8592
#895753
Dic7⋊Dic34
#54267