Finite Group Databasebeta

← #89259⋊C3 #13050737

Subgroups — #89259⋊C3 #13050737

Order 1344.

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

normal subgroups only

This group has 140 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
1#6438⋊C3C211yesmaximal, characteristic
2ΓL(1,𝔽8)×#97C211yesmaximal, characteristic, complemented
3#6377⋊C3C211yesmaximal, characteristic, complemented
4#89259C311yesmaximal, characteristic, complemented, Hall
5C2×C4⋊AGL(1,𝔽7)C411yescharacteristic
6C2×ΓL(1,𝔽8)⋊Q8C411yescharacteristic
7C2×C8×ΓL(1,𝔽8)—21no—
8C42×ΓL(1,𝔽8)C2211yescharacteristic
9(C2×AGL(1,𝔽7))⋊C4—21no—
10(C2×AGL(1,𝔽7))⋊C4—21no—
11C2×ΓL(1,𝔽8)⋊C8—21no—
12#6438C2×C311yescharacteristic
13C7×#97C2×C311yescharacteristic, complemented
14#6377C2×C311yescharacteristic, complemented
15C3×#329—71nomaximal, Hall
16C22×AGL(1,𝔽7)—21no—
17C2×C4×ΓL(1,𝔽8)C2×C411yescharacteristic
18C2×C4×ΓL(1,𝔽8)D411yescharacteristic
19C2×C4×ΓL(1,𝔽8)D411yescharacteristic
20C2×ΓL(1,𝔽8)⋊C4—21no—
21C4⋊AGL(1,𝔽7)—41no—
22ΓL(1,𝔽8)⋊Q8—41no—
23C8×ΓL(1,𝔽8)—22no—
24ΓL(1,𝔽8)⋊C8—22no—
25C2×D28C3×C411yescharacteristic
26C2×Dic14C3×C411yescharacteristic
27C2×C7×C8—21no—
28C42×C7C22×C311yescharacteristic
29(C2×D7)⋊C4—21no—
30(C2×D7)⋊C4—21no—
31C2×C7⋊C8—21no—
32C3×#124—71no—
33C3×#97—71no—
34C3×#97—71no—
35C22×ΓL(1,𝔽8)C22⋊C411yescharacteristic
36C2×AGL(1,𝔽7)—41no—
37C2×AGL(1,𝔽7)—41no—
38C2×AGL(1,𝔽7)—41no—
39C4×ΓL(1,𝔽8)—21no—
40C4×ΓL(1,𝔽8)—21no—
41C4×ΓL(1,𝔽8)—21no—
42ΓL(1,𝔽8)⋊C4—41no—
43#329—71noSylow
44C22×D7—21no—
45C2×C4×C7C2×C3×C411yescharacteristic
46C2×C4×C7C3×D411yescharacteristic
47C2×C4×C7C3×D411yescharacteristic
48C2×Dic7—21no—
49D28—41no—
50Dic14—41no—
51C7×C8—22no—
52C7⋊C8—22no—
53C2×C3×Q8—71no—
54C2×C3×D4—71no—
55C2×C3×C8—141no—
56C2×C3×C8—141no—
57C3×C22⋊C4—141no—
58C3×C22⋊C4—141no—
59C3×C42—71no—
60C2×ΓL(1,𝔽8)#10011yescharacteristic
61C2×ΓL(1,𝔽8)#10411yescharacteristic
62C2×ΓL(1,𝔽8)#10411yescharacteristic
63AGL(1,𝔽7)—81no—
64#124—71no—
65#97—71no—
66#97AGL(1,𝔽7)11yescharacteristic, complemented
67C22×C7C3×C22⋊C411yescharacteristic
68C2×D7—41no—
69C2×D7—41no—
70C2×D7—41no—
71C4×C7—21no—
72C4×C7—21no—
73C4×C7—21no—
74Dic7—41no—
75C23×C3—141no—
76C3×Q8—281no—
77C3×D4—281no—
78C2×C3×C4—71no—
79C2×C3×C4—71no—
80C2×C3×C4—71no—
81C2×C3×C4—141no—
82C3×C8—142no—
83C3×C8—142no—
84ΓL(1,𝔽8)#32911yescharacteristic, complemented, Hall
85C2×Q8—71no—
86C2×D4—71no—
87C2×C8—21no—
88C2×C8—141no—
89C22⋊C4—141no—
90C22⋊C4—141no—
91C42C2×AGL(1,𝔽7)11yescharacteristic
92C2×C7C3×#10011yescharacteristic
93C2×C7C3×#10411yescharacteristic
94C2×C7C3×#10411yescharacteristic
95D7—81no—
96C22×C3—71no—
97C22×C3—281no—
98C22×C3—281no—
99C22×C3—281no—
100C3×C4—141no—
101C3×C4—141no—
102C3×C4—141no—
103C3×C4—281no—
104C23—141no—
105Q8—281no—
106D4—281no—
107C2×C4—141no—
108C2×C4C4×AGL(1,𝔽7)11yescharacteristic
109C2×C4C4⋊AGL(1,𝔽7)11yescharacteristic
110C2×C4(C22⋊D7)⋊C311yescharacteristic
111C8—22no—
112C8—142no—
113C7C3×#32911yescharacteristic, complemented, Sylow
114C2×C3—71no—
115C2×C3—71no—
116C2×C3—71no—
117C2×C3—561no—
118C22—281no—
119C22—281no—
120C22—281no—
121C22(C2×AGL(1,𝔽7))⋊C411yescharacteristic
122C4—21no—
123C4—21no—
124C4—21no—
125C4—281no—
126C3—71noSylow
127C2—561no—
128C2#6379⋊C311yescharacteristic
129C2#6397⋊C311yescharacteristic
130C2#6399⋊C311yescharacteristic
subgroupG/H
centerC22(C2×AGL(1,𝔽7))⋊C4
commutatorC2×C4×C7C2×C3×C4
FrattiniC42C2×AGL(1,𝔽7)
FittingC7×#97C2×C3
socleC22×C7C3×C22⋊C4
prime radicalC2×C4⋊AGL(1,𝔽7)C4
cosocleC42×C7C22×C3
hypercenter#97AGL(1,𝔽7)
hypocenterC7C3×#329
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #97

Complements

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

NH ≅ G/Ntype
2. ΓL(1,𝔽8)×#97C2semidirect
3. #6377⋊C3C2semidirect
4. #89259C3semidirect
13. C7×#97C2×C3semidirect
14. #6377C2×C3semidirect
66. #97AGL(1,𝔽7)semidirect
84. ΓL(1,𝔽8)#329semidirect
113. C7C3×#329semidirect

Maximal subgroups

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

subgroupindex
#6438⋊C32
ΓL(1,𝔽8)×#972
#6377⋊C32
#892593
C3×#3297