Finite Group Databasebeta

← #13057186

Subgroups — #13057186

Order 1344.

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

normal subgroups only

This group has 160 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
1C22×C7×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×C7⋊#226C211yesmaximal, characteristic, complemented
3C7⋊#859C211yesmaximal, characteristic, complemented
4#90000—31nomaximal, Hall
5C2×C7×SL(2,𝔽3)—21no—
6C2×C7×SL(2,𝔽3)C2211yescharacteristic
7C7⋊#226—21no—
8#66636—41nomaximal
9C22×C7×Q8D311yescharacteristic
10C2×#1144—31no—
11C2×#1144—31no—
12#6487—31no—
13#6409—31no—
14#6409—31no—
15C22⋊(C7⋊C8)—31no—
16#5419—71nomaximal, Hall
17C23×C3×C7—41no—
18C2×Dic21—41no—
19C2×Dic21—41no—
20C7×SL(2,𝔽3)D411yescharacteristic
21C2×C7×Q8—21no—
22C2×C7×Q8—31no—
23C2×C7×Q8—61no—
24C2×C7×Q8C2×D311yescharacteristic
25C22×C4×C7—31no—
26C2×Dic14—31no—
27C22⋊Dic7—61no—
28#1144—61no—
29#1144—62no—
30#1144—61no—
31C4⋊Dic7—31no—
32Dic7⋊C4—61no—
33C2×C7⋊C8—31no—
34C2×C7⋊C8—31no—
35C22×SL(2,𝔽3)D711yescharacteristic
36C2×#226—71no—
37#859—71no—
38C22×C3×C7—42no—
39C22×C3×C7—41no—
40C22×C3×C7—81no—
41C22×C3×C7—81no—
42Dic21—81no—
43Dic21—81no—
44#451—211noSylow
45C23×C7S411yescharacteristic
46C7×Q8—31no—
47C7×Q8—32no—
48C7×Q8—61no—
49C7×Q8#6711yescharacteristic
50C2×C4×C7—31no—
51C2×C4×C7—31no—
52C2×C4×C7—61no—
53C2×C4×C7—61no—
54C2×Dic7—61no—
55C2×Dic7—61no—
56Dic14—61no—
57Dic14—61no—
58C7⋊C8—61no—
59C7⋊C8—61no—
60C2×SL(2,𝔽3)—21no—
61C2×SL(2,𝔽3)C2×D711yescharacteristic
62#226—141no—
63#217—281no—
64C2×C3×C7—41no—
65C2×C3×C7—41no—
66C2×C3×C7—41no—
67C2×C3×C7—82no—
68C22×Q8D2111yescharacteristic
69C2×Q16—211no—
70C2×Q16—211no—
71C22⋊Q8—211no—
72#103—211no—
73#103—211no—
74C22⋊C8—211no—
75C22×C7—21no—
76C22×C7—21no—
77C22×C7#22612yescomplemented
78C22×C7C2×S411yescharacteristic
79C4×C7—31no—
80C4×C7—31no—
81C4×C7—61no—
82Dic7—121no—
83Dic7—121no—
84C23×C3—41no—
85C2×Dic3—281no—
86C2×Dic3—281no—
87SL(2,𝔽3)C22⋊D711yescharacteristic
88C3×C7—41noHall
89C2×Q8—21no—
90C2×Q8—31no—
91C2×Q8—61no—
92C2×Q8—211no—
93C2×Q8C2×D2111yescharacteristic
94C22×C4—31no—
95Q16—421no—
96Q16—422no—
97Q16—421no—
98C2×C8—211no—
99C2×C8—211no—
100C4⋊C4—211no—
101C4⋊C4—421no—
102C22⋊C4—421no—
103C2×C7—22no—
104C2×C7C2×#22611yescharacteristic
105C2×C7#98311yescharacteristic
106C2×C7GL(2,ℤ/4)11yescharacteristic
107C22×C3—42no—
108C22×C3—41no—
109C22×C3—81no—
110C22×C3—81no—
111Dic3—561no—
112Dic3—561no—
113C23C7⋊S411yescharacteristic
114Q8—31no—
115Q8—32no—
116Q8—61no—
117Q8—421no—
118Q8—421no—
119Q8#426811yescharacteristic
120C2×C4—31no—
121C2×C4—31no—
122C2×C4—61no—
123C2×C4—61no—
124C2×C4—421no—
125C2×C4—421no—
126C8—421no—
127C8—421no—
128C7#541911yescharacteristic, complemented, Sylow
129C2×C3—41no—
130C2×C3—41no—
131C2×C3—41no—
132C2×C3—82no—
133C22—21no—
134C22—21no—
135C22C7⋊#22612yescomplemented
136C22C2×C7⋊S411yescharacteristic
137C4—31no—
138C4—31no—
139C4—61no—
140C4—841no—
141C4—841no—
142C3—41noSylow
143C2—22no—
144C2C2×C7⋊#22611yescharacteristic
145C2#1062150111yescharacteristic
146C2#1062150611yescharacteristic
subgroupG/H
center, FrattiniC22C2×C7⋊S4
commutator, cosocleC2×C7×SL(2,𝔽3)C22
FittingC22×C7×Q8D3
socleC22×C7C2×S4
prime radicalC22×C7×SL(2,𝔽3)C2
hypercenterC23C7⋊S4
hypocenterC7×SL(2,𝔽3)D4
LCSG ▷ C2×C7×SL(2,𝔽3) ▷ C7×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×C7⋊#226C2semidirect
3. C7⋊#859C2semidirect
77. C22×C7#226semidirect
128. C7#5419semidirect
135. C22C7⋊#226semidirect

Maximal subgroups

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

subgroupindex
C22×C7×SL(2,𝔽3)2
C2×C7⋊#2262
C7⋊#8592
#900003
#666364
#54197