Finite Group Databasebeta

← #13057069

Subgroups — #13057069

Order 1344.

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

normal subgroups only

This group has 160 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×D7×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×C7×#226C211yesmaximal, characteristic, complemented
3C7⋊#859C211yesmaximal, characteristic, complemented
4#89958—31nomaximal, Hall
5C2×C7×SL(2,𝔽3)C2211yescharacteristic
6D7×SL(2,𝔽3)—21no—
7C7×#226—21no—
8(C2×D7)⋊Dic3—41nomaximal
9C2×Q8×D7D311yescharacteristic
10C2×C7×Q16—31no—
11#6509—31no—
12C2×#1144—31no—
13#6409—31no—
14(C2×D7)⋊C8—31no—
15#6390—31no—
16#5419—71nomaximal, Hall
17C22×C3×D7—41no—
18C2×Dic21—41no—
19C2×C7×Dic3—41no—
20C7×SL(2,𝔽3)D411yescharacteristic
21C2×C7×Q8—31no—
22C2×C7×Q8C2×D311yescharacteristic
23Q8×D7—21no—
24Q8×D7—61no—
25C2×C4×D7—31no—
26C2×Dic14—31no—
27C7×Q16—61no—
28C7×Q16—61no—
29C2×C7×C8—31no—
30#1144—62no—
31(C2×D7)⋊C4—61no—
32C4⋊Dic7—31no—
33Dic7⋊C4—61no—
34C2×C7⋊C8—31no—
35C22×SL(2,𝔽3)—71no—
36C2×#226D711yescharacteristic, complemented
37#859—71no—
38C22×C3×C7—41no—
39C2×C3×D7—42no—
40C2×C3×D7—81no—
41C2×C3×D7—81no—
42Dic21—81no—
43C7×Dic3—81no—
44#451—211noSylow
45C22×D7S411yescharacteristic
46C7×Q8—31no—
47C7×Q8—61no—
48C7×Q8—61no—
49C7×Q8#6711yescharacteristic
50C2×C4×C7—31no—
51C2×C4×C7—61no—
52C2×Dic7—31no—
53C2×Dic7—61no—
54C4×D7—61no—
55C4×D7—61no—
56Dic14—32no—
57Dic14—61no—
58C7×C8—61no—
59C7⋊C8—61no—
60C2×SL(2,𝔽3)—141no—
61C2×SL(2,𝔽3)C2×D711yescharacteristic
62#226—21no—
63#217—281no—
64C2×C3×C7—41no—
65C2×C3×C7—41no—
66C2×C3×C7—41no—
67C3×D7—82no—
68C22×Q8—71no—
69C2×Q16—31no—
70C2×Q16—211no—
71C22⋊Q8—211no—
72#103—211no—
73#103—211no—
74C22⋊C8—211no—
75C22×C7C2×S411yescharacteristic
76C2×D7—21no—
77C2×D7—21no—
78C2×D7#22612yescomplemented
79C4×C7—31no—
80C4×C7—31no—
81C4×C7—121no—
82Dic7—61no—
83Dic7—121no—
84C23×C3—281no—
85C2×Dic3—41no—
86C2×Dic3—281no—
87SL(2,𝔽3)C22⋊D711yescharacteristic
88C3×C7—41noHall
89C2×Q8—31no—
90C2×Q8—141no—
91C2×Q8—211no—
92C2×Q8—421no—
93C2×Q8D3×D711yescharacteristic
94C22×C4—211no—
95Q16—61no—
96Q16—61no—
97Q16—422no—
98C2×C8—31no—
99C2×C8—211no—
100C4⋊C4—211no—
101C4⋊C4—421no—
102C22⋊C4—421no—
103C2×C7C2×#22611yescharacteristic
104C2×C7#98311yescharacteristic
105C2×C7GL(2,ℤ/4)11yescharacteristic
106D7—22no—
107C22×C3—41no—
108C22×C3—282no—
109C22×C3—561no—
110C22×C3—561no—
111Dic3—81no—
112Dic3—561no—
113C23—71no—
114Q8—31no—
115Q8—61no—
116Q8—61no—
117Q8—212no—
118Q8—421no—
119Q8#424511yescharacteristic
120C2×C4—31no—
121C2×C4—61no—
122C2×C4—211no—
123C2×C4—421no—
124C2×C4—421no—
125C2×C4—421no—
126C8—61no—
127C8—421no—
128C7#541911yescharacteristic, complemented, Sylow
129C2×C3—41no—
130C2×C3—41no—
131C2×C3—41no—
132C2×C3—562no—
133C22—72no—
134C22—141no—
135C22—141no—
136C22D7×S411yescharacteristic
137C4—31no—
138C4—31no—
139C4—121no—
140C4—421no—
141C4—841no—
142C3—41noSylow
143C2—142no—
144C2D7×#22611yescharacteristic
145C2#1062147511yescharacteristic
146C2#1062148211yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D7×S4
commutator, cosocleC2×C7×SL(2,𝔽3)C22
FittingC2×C7×Q8C2×D3
socleC22×C7C2×S4
prime radicalC2×D7×SL(2,𝔽3)C2
hypocenterC7×SL(2,𝔽3)D4
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
2. C2×C7×#226C2semidirect
3. C7⋊#859C2semidirect
36. C2×#226D7semidirect
78. C2×D7#226semidirect
128. C7#5419semidirect

Maximal subgroups

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

subgroupindex
C2×D7×SL(2,𝔽3)2
C2×C7×#2262
C7⋊#8592
#899583
(C2×D7)⋊Dic34
#54197