Finite Group Databasebeta

← #13057192

Subgroups — #13057192

Order 1344.

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

normal subgroups only

This group has 148 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
1C2×C7×#41⋊C3C211yesmaximal, characteristic
2C2×C7⋊#226C211yesmaximal, characteristic, complemented
3C7⋊#859C211yesmaximal, characteristic, complemented
4#90011—31nomaximal, Hall
5C7×#41⋊C3—21no—
6C2×C7×SL(2,𝔽3)C2211yescharacteristic
7C7⋊#226—21no—
8#66629—41nomaximal
9C2×C7×#41D311yescharacteristic
10C2×#1144—31no—
11C2×#1142—31no—
12#6487—31no—
13#6409—31no—
14#6406—31no—
15C22⋊(C7⋊C8)—31no—
16#5425—71nomaximal, Hall
17C2×C3×C4×C7—41no—
18C2×Dic21—41no—
19C2×Dic21—41no—
20C7×SL(2,𝔽3)D411yescharacteristic
21C7×#41—21no—
22C7×#41—61no—
23C2×C7×Q8C2×D311yescharacteristic
24C2×C7×D4—31no—
25C22×C4×C7—31no—
26C2×Dic14—31no—
27C22⋊Dic7—61no—
28#1144—61no—
29#1144—61no—
30#1142—62no—
31C4⋊Dic7—31no—
32Dic7⋊C4—61no—
33C2×C7⋊C8—31no—
34C2×C7⋊C8—31no—
35C2×#41⋊C3D711yescharacteristic
36C2×#226—71no—
37#859—71no—
38C22×C3×C7—41no—
39C3×C4×C7—81no—
40Dic21—42no—
41Dic21—81no—
42#452—211noSylow
43C23×C7—31no—
44C7×Q8—31no—
45C7×Q8#6711yescharacteristic
46C7×D4—32no—
47C7×D4—61no—
48C2×C4×C7—31no—
49C2×C4×C7—61no—
50C2×C4×C7—61no—
51C2×C4×C7S411yescharacteristic
52C2×Dic7—61no—
53C2×Dic7—61no—
54Dic14—61no—
55Dic14—61no—
56C7⋊C8—61no—
57C7⋊C8—61no—
58#41⋊C3—21no—
59C2×SL(2,𝔽3)C2×D711yescharacteristic
60#226—141no—
61Dic3⋊C4—281no—
62C2×C3×C7—41no—
63C2×C3×C7—41no—
64C2×C3×C7—41no—
65C2×#41D2111yescharacteristic
66C2×Q16—211no—
67C2×SD16—211no—
68C22⋊Q8—211no—
69#103—211no—
70#102—211no—
71C22⋊C8—211no—
72C22×C7—61no—
73C22×C7—61no—
74C22×C7—61no—
75C22×C7C2×S411yescharacteristic
76C4×C7—21no—
77C4×C7—31no—
78C4×C7—31no—
79Dic7—121no—
80Dic7—121no—
81C2×C3×C4—41no—
82C2×Dic3—281no—
83C2×Dic3—281no—
84SL(2,𝔽3)C22⋊D711yescharacteristic
85C3×C7—41noHall
86#41—21no—
87#41—61no—
88C2×Q8—211no—
89C2×Q8C2×D2111yescharacteristic
90C2×D4—31no—
91C22×C4—31no—
92Q16—421no—
93Q16—421no—
94SD16—422no—
95C2×C8—211no—
96C2×C8—211no—
97C4⋊C4—211no—
98C4⋊C4—421no—
99C22⋊C4—421no—
100C2×C7—121no—
101C2×C7#98411yescharacteristic
102C2×C7#98511yescharacteristic
103C2×C7GL(2,ℤ/4)11yescharacteristic
104C22×C3—41no—
105C3×C4—81no—
106Dic3—282no—
107Dic3—561no—
108C23—31no—
109Q8—31no—
110Q8—421no—
111Q8—421no—
112Q8#426811yescharacteristic
113D4—32no—
114D4—61no—
115C2×C4—31no—
116C2×C4—61no—
117C2×C4—61no—
118C2×C4—421no—
119C2×C4—421no—
120C2×C4C7⋊S411yescharacteristic
121C8—421no—
122C8—421no—
123C7#542511yescharacteristic, complemented, Sylow
124C2×C3—41no—
125C2×C3—41no—
126C2×C3—41no—
127C22—61no—
128C22—61no—
129C22—61no—
130C22C2×C7⋊S411yescharacteristic
131C4—21no—
132C4—31no—
133C4—31no—
134C4—841no—
135C4—841no—
136C3—41noSylow
137C2—121no—
138C2#1062150211yescharacteristic
139C2#1062150311yescharacteristic
140C2#1062150611yescharacteristic
subgroupG/H
center, FrattiniC22C2×C7⋊S4
commutator, cosocleC2×C7×SL(2,𝔽3)C22
FittingC2×C7×#41D3
socleC22×C7C2×S4
prime radicalC2×C7×#41⋊C3C2
hypercenterC2×C4C7⋊S4
hypocenterC7×SL(2,𝔽3)D4
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
2. C2×C7⋊#226C2semidirect
3. C7⋊#859C2semidirect
123. C7#5425semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#41⋊C32
C2×C7⋊#2262
C7⋊#8592
#900113
#666294
#54257