Finite Group Databasebeta

← #13057672

Subgroups — #13057672

Order 1344.

172 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 40 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
2Dic7⋊SL(2,𝔽3)C212yesmaximal, complemented
3#90503C311yesmaximal, characteristic, complemented, Hall
4C2×C7⋊SL(2,𝔽3)C412yescomplemented
5C2×C7⋊SL(2,𝔽3)C2211yescharacteristic
6(C22⋊Dic7)⋊C3—41nomaximal
7C22×C7×Q8C2×C311yescharacteristic
8Q8×Dic7—32no—
9Q8×Dic7C2×C312yescomplemented
10#6489—31no—
11#6489—31no—
12#519⋊C3—71nomaximal, Hall
13C23×ΓL(1,𝔽8)—41no—
14C7⋊SL(2,𝔽3)C2×C411yescharacteristic
15C2×ΓL(1,𝔽8)⋊C4—42no—
16C2×C7×Q8—31no—
17C2×C7×Q8—32no—
18C2×C7×Q8C3×C412yescomplemented
19C2×C7×Q8C22×C311yescharacteristic
20C22×C4×C7—31no—
21C22⋊Dic7—31no—
22C22⋊Dic7A411yescharacteristic
23C4⋊Dic7—32no—
24C4⋊Dic7—32no—
25C4×Dic7—32no—
26C4×Dic7—32no—
27C22×SL(2,𝔽3)—71no—
28C4×SL(2,𝔽3)—72no—
29C22×ΓL(1,𝔽8)—42no—
30C22×ΓL(1,𝔽8)—41no—
31C22×ΓL(1,𝔽8)—82no—
32ΓL(1,𝔽8)⋊C4—82no—
33#519—71noSylow
34C23×C7C2×A411yescharacteristic
35C7×Q8—32no—
36C7×Q8—31no—
37C7×Q8—32no—
38C7×Q8C2×C3×C411yescharacteristic
39C2×C4×C7—32no—
40C2×C4×C7—32no—
41C2×C4×C7—31no—
42C2×C4×C7—31no—
43C2×Dic7—32no—
44C2×Dic7C2×A412yes—
45C2×SL(2,𝔽3)—72no—
46C2×SL(2,𝔽3)—71no—
47C3×C22⋊C4—281no—
48C2×ΓL(1,𝔽8)—41no—
49C2×ΓL(1,𝔽8)—42no—
50C2×ΓL(1,𝔽8)—82no—
51C22×Q8AGL(1,𝔽7)11yescharacteristic
52C4×Q8—72no—
53C4×Q8—212no—
54#117—211no—
55#117—211no—
56C22×C7—22no—
57C22×C7C4×A412yes—
58C22×C7C22×A411yescharacteristic
59C4×C7—31no—
60C4×C7—32no—
61C4×C7—31no—
62Dic7—22no—
63Dic7—62no—
64C23×C3—281no—
65C2×C3×C4—282no—
66SL(2,𝔽3)—71no—
67ΓL(1,𝔽8)—41noHall
68C2×Q8—31no—
69C2×Q8—32no—
70C2×Q8ΓL(1,𝔽8)⋊C412yescomplemented
71C2×Q8C2×AGL(1,𝔽7)11yescharacteristic
72C22×C4—31no—
73C4⋊C4—212no—
74C4⋊C4—212no—
75C22⋊C4—71no—
76C22⋊C4—211no—
77C42—212no—
78C42—212no—
79C2×C7—22no—
80C2×C7C2×C4×A411yescharacteristic
81C2×C72−1+4⋊C312yes—
82C22×C3—282no—
83C22×C3—281no—
84C22×C3—562no—
85C3×C4—562no—
86C23D7⋊A411yescharacteristic
87Q8—32no—
88Q8—31no—
89Q8—32no—
90Q8C2×ΓL(1,𝔽8)⋊C411yescharacteristic
91C2×C4—32no—
92C2×C4—32no—
93C2×C4—31no—
94C2×C4—31no—
95C2×C4—72no—
96C2×C4—212no—
97C7#519⋊C311yescharacteristic, complemented, Sylow
98C2×C3—282no—
99C2×C3—281no—
100C2×C3—562no—
101C22—22no—
102C22Dic7⋊A412yes—
103C22C2×D7⋊A411yescharacteristic
104C4—31no—
105C4—31no—
106C4—32no—
107C4—142no—
108C4—422no—
109C3—281noSylow
110C2—22no—
111C2#1062153812yes—
112C2C2×Dic7⋊A411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D7⋊A4
commutator, hypocenterC7×Q8C2×C3×C4
FittingC22×C7×Q8C2×C3
socleC22×C7C22×A4
prime radicalC22×C7⋊SL(2,𝔽3)C2
cosocleC2×C7×Q8C22×C3
hypercenterC23D7⋊A4
LCSG ▷ C7×Q8
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. Dic7⋊SL(2,𝔽3)C2semidirect
3. #90503C3semidirect
4. C2×C7⋊SL(2,𝔽3)C4semidirect
9. Q8×Dic7C2×C3semidirect
18. C2×C7×Q8C3×C4semidirect
70. C2×Q8ΓL(1,𝔽8)⋊C4semidirect
97. C7#519⋊C3semidirect

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
Dic7⋊SL(2,𝔽3)2
#905033
(C22⋊Dic7)⋊C34
#519⋊C37