Finite Group Databasebeta

← #13057064

Subgroups — #13057064

Order 1344.

146 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 25 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⋊#226C211yesmaximal, characteristic, complemented
2C2×C7×GL(2,𝔽3)C211yesmaximal, characteristic
3SL(2,𝔽3)×Dic7C211yesmaximal, characteristic, complemented
4#89938—31nomaximal, Hall
5C2×C7×SL(2,𝔽3)C2211yescharacteristic
6C7⋊#226—21no—
7C7×GL(2,𝔽3)—21no—
8(C2×D3)⋊Dic7—41nomaximal
9C2×C7×SD16—31no—
10Q8×Dic7D311yescharacteristic, complemented
11C2×#1144—31no—
12#6499—31no—
13#6406—31no—
14#6390—31no—
15Dic7⋊C8—31no—
16#5395—71nomaximal, Hall
17C22×D3×C7—41no—
18C2×Dic21—41no—
19C2×C3×Dic7—41no—
20C7×SL(2,𝔽3)D411yescharacteristic
21C2×C7×Q8C2×D311yescharacteristic
22C2×C7×D4—31no—
23C2×Dic14—31no—
24C7×SD16—61no—
25C7×SD16—61no—
26C2×C7×C8—31no—
27C22⋊Dic7—61no—
28C22⋊Dic7—61no—
29#1144—61no—
30#1144—61no—
31C4⋊Dic7—31no—
32C4×Dic7—31no—
33C2×C7⋊C8—31no—
34C2×GL(2,𝔽3)D711yescharacteristic
35C2×#226—71no—
36C4×SL(2,𝔽3)—71no—
37C22×C3×C7—41no—
38C2×D3×C7—42no—
39C2×D3×C7—81no—
40C2×D3×C7—81no—
41Dic21—81no—
42C3×Dic7—81no—
43#464—211noSylow
44C23×C7—61no—
45C7×Q8—31no—
46C7×Q8D1211yescharacteristic
47C7×D4—61no—
48C7×D4—61no—
49C2×C4×C7—31no—
50C2×Dic7—31no—
51C2×Dic7—61no—
52C2×Dic7S411yescharacteristic
53Dic14—61no—
54Dic14—61no—
55C7×C8—61no—
56C7⋊C8—61no—
57C2×SL(2,𝔽3)C2×D711yescharacteristic
58GL(2,𝔽3)—21no—
59#226—141no—
60(C2×D3)⋊C4—281no—
61C2×C3×C7—41no—
62C2×C3×C7—41no—
63C2×C3×C7—41no—
64D3×C7—82no—
65C2×Q16—211no—
66C2×SD16—31no—
67#124—211no—
68C4×Q8—71no—
69C4⋊C8—211no—
70#103—211no—
71#102—211no—
72C22×C7—121no—
73C22×C7—121no—
74C22×C7—121no—
75C22×C7C2×S411yescharacteristic
76C4×C7—31no—
77C4×C7—31no—
78Dic7—21no—
79Dic7—61no—
80Dic7—121no—
81C22×D3—41no—
82C2×C3×C4—281no—
83C2×Dic3—281no—
84SL(2,𝔽3)C22⋊D711yescharacteristic
85C3×C7—41noHall
86C2×Q8—211no—
87C2×Q8D3×D711yescharacteristic
88C2×D4—31no—
89Q16—421no—
90Q16—421no—
91SD16—61no—
92SD16—61no—
93C2×C8—31no—
94C2×C8—211no—
95C4⋊C4—211no—
96C22⋊C4—421no—
97C22⋊C4—421no—
98C42—211no—
99C2×C7—241no—
100C2×C7C4⋊S411yescharacteristic
101C2×C7#98311yescharacteristic
102C2×C7#98511yescharacteristic
103C22×C3—41no—
104C2×D3—42no—
105C2×D3—81no—
106C2×D3—81no—
107C3×C4—561no—
108Dic3—561no—
109C23—61no—
110Q8—31no—
111Q8—421no—
112Q8—421no—
113Q8#424711yescharacteristic
114D4—61no—
115D4—61no—
116C2×C4—31no—
117C2×C4—71no—
118C2×C4—211no—
119C2×C4—421no—
120C8—61no—
121C8—421no—
122C7#539511yescharacteristic, complemented, Sylow
123C2×C3—41no—
124C2×C3—41no—
125C2×C3—41no—
126D3—82no—
127C22—121no—
128C22—121no—
129C22—121no—
130C22D7×S411yescharacteristic
131C4—31no—
132C4—31no—
133C4—141no—
134C4—421no—
135C4—841no—
136C3—41noSylow
137C2—241no—
138C2#1062147011yescharacteristic
139C2#1062147411yescharacteristic
140C2#1062148011yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D7×S4
commutator, cosocleC2×C7×SL(2,𝔽3)C22
FittingC2×C7×Q8C2×D3
socleC22×C7C2×S4
prime radicalC2×C7×GL(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
1. C2×C7⋊#226C2semidirect
3. SL(2,𝔽3)×Dic7C2semidirect
10. Q8×Dic7D3semidirect
122. C7#5395semidirect

Maximal subgroups

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

subgroupindex
C2×C7⋊#2262
C2×C7×GL(2,𝔽3)2
SL(2,𝔽3)×Dic72
#899383
(C2×D3)⋊Dic74
#53957