Finite Group Databasebeta

← #72222016

Subgroups — #72222016

Order 3528.

224 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 51 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×ΓL(1,𝔽8)2C211yesmaximal, characteristic, complemented
2C2×ΓL(1,𝔽8)⋊AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
3ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4)C211yesmaximal, characteristic, complemented
4#11924222C311yesmaximal, characteristic, complemented
5#11924217C311yesmaximal, characteristic, complemented
6#11924202C312yesmaximal, complemented
7C2×ΓL(1,𝔽8)2—21no—
8C2×ΓL(1,𝔽8)2C2211yescharacteristic
9ΓL(1,𝔽8)⋊AGL(1,𝔽7)—21no—
10C22×#3862C2×C311yescharacteristic, complemented
11C22×#3861C2×C311yescharacteristic, complemented
12C22×C7×ΓL(1,𝔽8)C2×C312yescomplemented
13C2×#64425C2×C311yescharacteristic, complemented
14C2×#64424C2×C311yescharacteristic, complemented
15C2×ΓL(1,𝔽8)⋊D7C2×C312yescomplemented
16#10596875C2×C311yescharacteristic, complemented
17#10596874C2×C311yescharacteristic, complemented
18ΓL(1,𝔽8)⋊Dic7C2×C312yescomplemented
19C3×(C22⋊D7)⋊C3—72nomaximal
20ΓL(1,𝔽8)2D411yescharacteristic, complemented, Hall
21C22⋊(C7⋊D7)C3211yescharacteristic, complemented, Hall
22C2×#3862—21no—
23C2×#3862C22×C311yescharacteristic
24C2×#3861—21no—
25C2×#3861C22×C311yescharacteristic
26C2×C7×ΓL(1,𝔽8)—22no—
27C2×C7×ΓL(1,𝔽8)C22×C312yes—
28#64425—21no—
29#64424—21no—
30ΓL(1,𝔽8)⋊D7—22no—
31C22×C3×ΓL(1,𝔽8)—72no—
32C2×C3×AGL(1,𝔽7)—72no—
33C3×ΓL(1,𝔽8)⋊C4—72no—
34C22×C72C2×C3211yescharacteristic, complemented
35C2×C7⋊D7C2×C3211yescharacteristic, complemented
36C7⋊Dic7C2×C3211yescharacteristic, complemented
37C3×C22⋊D7—72no—
38(C22⋊D7)⋊C3—72no—
39(C22⋊D7)⋊C3—72no—
40(C22⋊D7)⋊C3—72no—
41(C22⋊D7)⋊C3—212no—
42#3862C3×D411yescharacteristic, complemented
43#3861C3×D411yescharacteristic, complemented
44C7×ΓL(1,𝔽8)C3×D412yescomplemented
45C2×C3×ΓL(1,𝔽8)—72no—
46C2×C3×ΓL(1,𝔽8)—142no—
47C3×AGL(1,𝔽7)—142no—
48C2×C72—21no—
49C2×C72C22×C3211yescharacteristic
50C7⋊D7—21no—
51C22×C3×C7—72no—
52C2×C3×D7—72no—
53C22×ΓL(1,𝔽8)—72no—
54C22×ΓL(1,𝔽8)—72no—
55C22×ΓL(1,𝔽8)—212no—
56C22×ΓL(1,𝔽8)AGL(1,𝔽7)12yescomplemented
57C2×AGL(1,𝔽7)—72no—
58C2×AGL(1,𝔽7)—72no—
59C2×AGL(1,𝔽7)—72no—
60C2×AGL(1,𝔽7)—212no—
61C3×Dic7—72no—
62ΓL(1,𝔽8)⋊C4—72no—
63ΓL(1,𝔽8)⋊C4—72no—
64ΓL(1,𝔽8)⋊C4—72no—
65ΓL(1,𝔽8)⋊C4—212no—
66C32×D4—491noHall
67C3×ΓL(1,𝔽8)—72no—
68C22⋊D7—72no—
69C22⋊D7—212no—
70C72C32×D411yescharacteristic, complemented, Sylow
71C2×C3×C7—72no—
72C2×C3×C7—142no—
73C3×D7—142no—
74C2×ΓL(1,𝔽8)—22no—
75C2×ΓL(1,𝔽8)—72no—
76C2×ΓL(1,𝔽8)—72no—
77C2×ΓL(1,𝔽8)—142no—
78C2×ΓL(1,𝔽8)—142no—
79C2×ΓL(1,𝔽8)—212no—
80C2×ΓL(1,𝔽8)—422no—
81C2×ΓL(1,𝔽8)C2×AGL(1,𝔽7)12yes—
82AGL(1,𝔽7)—142no—
83AGL(1,𝔽7)—142no—
84AGL(1,𝔽7)—142no—
85AGL(1,𝔽7)—422no—
86C22×C32—491no—
87C22×C32—491no—
88C32×C4—491no—
89C22×C7—32no—
90C22×C7C3×AGL(1,𝔽7)12yescomplemented
91C2×D7—72no—
92C2×D7—212no—
93Dic7—72no—
94Dic7—212no—
95C3×D4—492no—
96C3×D4—491no—
97C3×D4—491no—
98C3×C7—72no—
99ΓL(1,𝔽8)—72no—
100ΓL(1,𝔽8)—72no—
101ΓL(1,𝔽8)—212no—
102ΓL(1,𝔽8)(C22⋊D7)⋊C312yescomplemented
103C2×C32—491no—
104C2×C32—981no—
105C2×C32—981no—
106C2×C7—22no—
107C2×C7—32no—
108C2×C7—62no—
109C2×C7C2×C3×AGL(1,𝔽7)12yes—
110D7—142no—
111D7—422no—
112C22×C3—72no—
113C22×C3—492no—
114C22×C3—491no—
115C22×C3—491no—
116C22×C3—491no—
117C22×C3—491no—
118C3×C4—492no—
119C3×C4—491no—
120C3×C4—491no—
121C32—491noSylow
122D4—491noSylow
123C7—32no—
124C7C3×(C22⋊D7)⋊C312yescomplemented
125C2×C3—72no—
126C2×C3—142no—
127C2×C3—491no—
128C2×C3—491no—
129C2×C3—981no—
130C2×C3—981no—
131C2×C3—982no—
132C2×C3—981no—
133C2×C3—981no—
134C22—491no—
135C22ΓL(1,𝔽8)⋊AGL(1,𝔽7)11yescharacteristic, complemented
136C4—491no—
137C3—72no—
138C3—491no—
139C3—491no—
140C2—21no—
141C2—981no—
142C2C2×ΓL(1,𝔽8)⋊AGL(1,𝔽7)11yescharacteristic
subgroupG/H
center, FrattiniC2C2×ΓL(1,𝔽8)⋊AGL(1,𝔽7)
commutator, socle, cosocleC2×C72C22×C32
FittingC22×C72C2×C32
hypercenterC22ΓL(1,𝔽8)⋊AGL(1,𝔽7)
hypocenterC72C32×D4
LCSG ▷ C2×C72 ▷ C72
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×ΓL(1,𝔽8)2C2semidirect
2. C2×ΓL(1,𝔽8)⋊AGL(1,𝔽7)C2semidirect
3. ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4)C2semidirect
4. #11924222C3semidirect
5. #11924217C3semidirect
6. #11924202C3semidirect
10. C22×#3862C2×C3semidirect
11. C22×#3861C2×C3semidirect
12. C22×C7×ΓL(1,𝔽8)C2×C3semidirect
13. C2×#64425C2×C3semidirect
14. C2×#64424C2×C3semidirect
15. C2×ΓL(1,𝔽8)⋊D7C2×C3semidirect
16. #10596875C2×C3semidirect
17. #10596874C2×C3semidirect
18. ΓL(1,𝔽8)⋊Dic7C2×C3semidirect
20. ΓL(1,𝔽8)2D4semidirect
21. C22⋊(C7⋊D7)C32semidirect
34. C22×C72C2×C32semidirect
35. C2×C7⋊D7C2×C32semidirect
36. C7⋊Dic7C2×C32semidirect
42. #3862C3×D4semidirect
43. #3861C3×D4semidirect
44. C7×ΓL(1,𝔽8)C3×D4semidirect
56. C22×ΓL(1,𝔽8)AGL(1,𝔽7)semidirect
70. C72C32×D4semidirect
90. C22×C7C3×AGL(1,𝔽7)semidirect
102. ΓL(1,𝔽8)(C22⋊D7)⋊C3semidirect
124. C7C3×(C22⋊D7)⋊C3semidirect
135. C22ΓL(1,𝔽8)⋊AGL(1,𝔽7)semidirect

Maximal subgroups

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

subgroupindex
C22×ΓL(1,𝔽8)22
C2×ΓL(1,𝔽8)⋊AGL(1,𝔽7)2
ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4)2
#119242223
#119242173
#119242023
C3×(C22⋊D7)⋊C37