Finite Group Databasebeta

← #13062156

Subgroups — #13062156

Order 1344.

1620 conjugacy classes of subgroups; only the 818 normal subgroups are shown.

normal subgroups only

This group has 1620 conjugacy classes of subgroups — too many to draw. Only its 818 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
1ΓL(1,𝔽8)×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2#10621519C216yesmaximal, complemented
3#41×AGL(1,𝔽7)C216yesmaximal, complemented
4#6548⋊C3C219yesmaximal, complemented
5C2×#1159⋊C3C219yesmaximal, complemented
6#90618C311yesmaximal, characteristic, complemented, Hall
7#41×ΓL(1,𝔽8)C2216yescomplemented
8C2×D4×ΓL(1,𝔽8)C2219yescomplemented
9C2×(C22⋊D7)⋊C3C22118yescomplemented
10C22×ΓL(1,𝔽8)⋊C4C2216yescomplemented
11#66659C2212yescomplemented
12Q8×AGL(1,𝔽7)C2216yescomplemented
13#1159⋊C3C22118yescomplemented
14#1159⋊C3C22136yescomplemented
15D4×AGL(1,𝔽7)C22118yescomplemented
16#1157⋊C3C22118yescomplemented
17C2×C4×AGL(1,𝔽7)C2219yescomplemented
18C2×ΓL(1,𝔽8)⋊Q8C2219yescomplemented
19C7×H(5,𝔽2)C2×C311yescharacteristic, complemented
20#6554C2×C316yescomplemented
21D7×#41C2×C316yescomplemented
22#6548C2×C319yescomplemented
23C2×#1159C2×C319yescomplemented
24C3×#585—71nomaximal, Hall
25C23×ΓL(1,𝔽8)C2316yes—
26C22×AGL(1,𝔽7)C2319yes—
27Q8×ΓL(1,𝔽8)C2312yes—
28D4×ΓL(1,𝔽8)C23118yes—
29C2×C4×ΓL(1,𝔽8)C2319yes—
30(C22⋊D7)⋊C3C23136yes—
31C2×ΓL(1,𝔽8)⋊C4C23118yes—
32C2×ΓL(1,𝔽8)⋊C4C2319yes—
33C4⋊AGL(1,𝔽7)C2316yes—
34C4×AGL(1,𝔽7)C2316yes—
35C4×AGL(1,𝔽7)C23118yes—
36ΓL(1,𝔽8)⋊Q8C23118yes—
37C7×#41C22×C316yescomplemented
38C2×C7×D4C22×C319yescomplemented
39C2×C22⋊D7C22×C3118yescomplemented
40C22×Dic7C22×C316yescomplemented
41#1161C22×C312yescomplemented
42Q8×D7C22×C316yescomplemented
43#1159C22×C3118yescomplemented
44#1159C22×C3136yescomplemented
45D4×D7C22×C3118yescomplemented
46#1157C22×C3118yescomplemented
47C2×C4×D7C22×C319yescomplemented
48C2×Dic14C22×C319yescomplemented
49C3×2−1+4—76no—
50C3×H(5,𝔽2)—79no—
51C3×H(5,𝔽2)—71no—
52C2×C3×#41—79no—
53C2×C3×#41—76no—
54C22×ΓL(1,𝔽8)—46no—
55C22×ΓL(1,𝔽8)C2419yes—
56C2×AGL(1,𝔽7)—49no—
57C2×AGL(1,𝔽7)C2416yes—
58C4×ΓL(1,𝔽8)C2416yes—
59ΓL(1,𝔽8)⋊C4C2419yes—
60ΓL(1,𝔽8)⋊C4C2411yescharacteristic
61#585—71noSylow
62C23×C7C23×C316yes—
63C22×D7C23×C319yes—
64C7×Q8C23×C312yes—
65C7×D4C23×C3118yes—
66C2×C4×C7C23×C319yes—
67C22⋊D7C23×C3136yes—
68C2×Dic7C23×C3118yes—
69C2×Dic7C23×C319yes—
70D28C23×C316yes—
71C4×D7C23×C316yes—
72C4×D7C23×C3118yes—
73Dic14C23×C3118yes—
74C3×#41—718no—
75C3×#41—718no—
76C3×#41—736no—
77C3×#41—72no—
78C3×#41—76no—
79C2×C3×Q8—79no—
80C2×C3×Q8—76no—
81C2×C3×D4—718no—
82C2×C3×D4—718no—
83C2×C3×D4—79no—
84C22×C3×C4—79no—
85C22×C3×C4—76no—
86C2×ΓL(1,𝔽8)—29no—
87C2×ΓL(1,𝔽8)C2511yescharacteristic
88AGL(1,𝔽7)—26no—
892−1+4—76no—
90H(5,𝔽2)—79no—
91H(5,𝔽2)AGL(1,𝔽7)11yescharacteristic, complemented
92C2×#41—76no—
93C2×#41—79no—
94C22×C7—46no—
95C22×C7C24×C319yes—
96C2×D7—49no—
97C2×D7C24×C316yes—
98C4×C7C24×C316yes—
99Dic7C24×C319yes—
100Dic7C24×C311yescharacteristic
101C23×C3—79no—
102C23×C3—76no—
103C3×Q8—718no—
104C3×Q8—72no—
105C3×D4—76no—
106C3×D4—736no—
107C3×D4—718no—
108C2×C3×C4—76no—
109C2×C3×C4—718no—
110C2×C3×C4—79no—
111C2×C3×C4—718no—
112C2×C3×C4—79no—
113ΓL(1,𝔽8)#58511yescharacteristic, complemented, Hall
114#41—72no—
115#41—718no—
116#41—736no—
117#41—718no—
118#41C2×AGL(1,𝔽7)16yescomplemented
119C2×Q8—76no—
120C2×Q8—79no—
121C2×D4—718no—
122C2×D4—718no—
123C2×D4C2×AGL(1,𝔽7)19yescomplemented
124C22×C4—76no—
125C22×C4—79no—
126C2×C7—29no—
127C2×C7C25×C311yescharacteristic
128D7—26no—
129C22×C3—76no—
130C22×C3—79no—
131C22×C3—286no—
132C22×C3—289no—
133C3×C4—71no—
134C3×C4—79no—
135C3×C4—76no—
136C23—79no—
137C23C22×AGL(1,𝔽7)16yes—
138Q8—718no—
139Q8C22×AGL(1,𝔽7)12yes—
140D4—736no—
141D4—76no—
142D4C22×AGL(1,𝔽7)118yes—
143C2×C4—79no—
144C2×C4—718no—
145C2×C4—76no—
146C2×C4—718no—
147C2×C4C22×AGL(1,𝔽7)19yes—
148C7C3×#58511yescharacteristic, complemented, Sylow
149C2×C3—71no—
150C2×C3—149no—
151C2×C3—146no—
152C22—46no—
153C22—76no—
154C22—289no—
155C22C23×AGL(1,𝔽7)19yes—
156C4—71no—
157C4—79no—
158C4C23×AGL(1,𝔽7)16yes—
159C3—71noSylow
160C2—29no—
161C2—146no—
162C2C24×AGL(1,𝔽7)11yescharacteristic
subgroupG/H
center, FrattiniC2C24×AGL(1,𝔽7)
commutator, socle, cosocleC2×C7C25×C3
FittingC7×H(5,𝔽2)C2×C3
hypercenterH(5,𝔽2)AGL(1,𝔽7)
hypocenterC7C3×#585
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. ΓL(1,𝔽8)×H(5,𝔽2)C2semidirect
2. #10621519C2semidirect
3. #41×AGL(1,𝔽7)C2semidirect
4. #6548⋊C3C2semidirect
5. C2×#1159⋊C3C2semidirect
6. #90618C3semidirect
7. #41×ΓL(1,𝔽8)C22semidirect
8. C2×D4×ΓL(1,𝔽8)C22semidirect
9. C2×(C22⋊D7)⋊C3C22semidirect
10. C22×ΓL(1,𝔽8)⋊C4C22semidirect
11. #66659C22semidirect
12. Q8×AGL(1,𝔽7)C22semidirect
13. #1159⋊C3C22semidirect
14. #1159⋊C3C22semidirect
15. D4×AGL(1,𝔽7)C22semidirect
16. #1157⋊C3C22semidirect
17. C2×C4×AGL(1,𝔽7)C22semidirect
18. C2×ΓL(1,𝔽8)⋊Q8C22semidirect
19. C7×H(5,𝔽2)C2×C3semidirect
20. #6554C2×C3semidirect
21. D7×#41C2×C3semidirect
22. #6548C2×C3semidirect
23. C2×#1159C2×C3semidirect
37. C7×#41C22×C3semidirect
38. C2×C7×D4C22×C3semidirect
39. C2×C22⋊D7C22×C3semidirect
40. C22×Dic7C22×C3semidirect
41. #1161C22×C3semidirect
42. Q8×D7C22×C3semidirect
43. #1159C22×C3semidirect
44. #1159C22×C3semidirect
45. D4×D7C22×C3semidirect
46. #1157C22×C3semidirect
47. C2×C4×D7C22×C3semidirect
48. C2×Dic14C22×C3semidirect
91. H(5,𝔽2)AGL(1,𝔽7)semidirect
113. ΓL(1,𝔽8)#585semidirect
118. #41C2×AGL(1,𝔽7)semidirect
123. C2×D4C2×AGL(1,𝔽7)semidirect
148. C7C3×#585semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)×H(5,𝔽2)2
#106215192
#41×AGL(1,𝔽7)2
#6548⋊C32
C2×#1159⋊C32
#906183
C3×#5857