Finite Group Databasebeta

← 2−1+4×AGL(1,𝔽7) #13062157

Subgroups — 2−1+4×AGL(1,𝔽7) #13062157

Order 1344.

1588 conjugacy classes of subgroups; only the 822 normal subgroups are shown.

normal subgroups only

This group has 1588 conjugacy classes of subgroups — too many to draw. Only its 822 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)×2−1+4C211yesmaximal, characteristic, complemented
2#10621519C2110yesmaximal, complemented
3#41×AGL(1,𝔽7)C2110yesmaximal, complemented
4#10621516C215yesmaximal, complemented
5C2×Q8×AGL(1,𝔽7)C215yesmaximal, complemented
6D7×2−1+4C311yesmaximal, characteristic, complemented, Hall
7#41×ΓL(1,𝔽8)C22110yescomplemented
8C2×Q8×ΓL(1,𝔽8)C2215yescomplemented
9#66659C22110yescomplemented
10Q8×AGL(1,𝔽7)C22110yes—
11Q8×AGL(1,𝔽7)C22120yescomplemented
12#1159⋊C3C22130yescomplemented
13D4×AGL(1,𝔽7)C22110yes—
14#1157⋊C3C22130yescomplemented
15C2×C4×AGL(1,𝔽7)C22115yes—
16C2×ΓL(1,𝔽8)⋊Q8C22115yescomplemented
17C7×2−1+4C2×C311yescharacteristic, complemented
18#6554C2×C3110yescomplemented
19D7×#41C2×C3110yescomplemented
20#6551C2×C315yescomplemented
21C2×Q8×D7C2×C315yescomplemented
22C2×C3×2−1+4—71nomaximal, Hall
23C22×AGL(1,𝔽7)C2315yes—
24Q8×ΓL(1,𝔽8)C23110yes—
25D4×ΓL(1,𝔽8)C23110yes—
26C2×C4×ΓL(1,𝔽8)C23115yes—
27(C22⋊D7)⋊C3C23120yes—
28C2×ΓL(1,𝔽8)⋊C4C23115yes—
29C4⋊AGL(1,𝔽7)C23110yes—
30C4×AGL(1,𝔽7)C23110yes—
31C4×AGL(1,𝔽7)C23130yes—
32ΓL(1,𝔽8)⋊Q8C23130yes—
33C7×#41C22×C3110yescomplemented
34C2×C7×Q8C22×C315yescomplemented
35#1161C22×C3110yescomplemented
36Q8×D7C22×C3110yes—
37Q8×D7C22×C3120yescomplemented
38#1159C22×C3130yescomplemented
39D4×D7C22×C3110yes—
40#1157C22×C3130yescomplemented
41C2×C4×D7C22×C3115yes—
42C2×Dic14C22×C3115yescomplemented
43C3×2−1+4—75no—
44C3×2−1+4—710no—
45C3×2−1+4—71no—
46C2×C3×#41—710no—
47C22×C3×Q8—75no—
48C22×ΓL(1,𝔽8)C2415yes—
49C2×AGL(1,𝔽7)—210no—
50C2×AGL(1,𝔽7)C2415yes—
51C2×AGL(1,𝔽7)C2411yescharacteristic
52C4×ΓL(1,𝔽8)C24110yes—
53ΓL(1,𝔽8)⋊C4C24110yes—
54C2×2−1+4—71noSylow
55C22×D7C23×C315yes—
56C7×Q8C23×C3110yes—
57C7×D4C23×C3110yes—
58C2×C4×C7C23×C3115yes—
59C22⋊D7C23×C3120yes—
60C2×Dic7C23×C3115yes—
61D28C23×C3110yes—
62C4×D7C23×C3110yes—
63C4×D7C23×C3130yes—
64Dic14C23×C3130yes—
65C3×#41—730no—
66C3×#41—730no—
67C3×#41—710no—
68C3×#41—710no—
69C2×C3×Q8—715no—
70C2×C3×Q8—710no—
71C2×C3×Q8—720no—
72C2×C3×Q8—75no—
73C2×C3×D4—710no—
74C22×C3×C4—715no—
75C2×ΓL(1,𝔽8)—25no—
76C2×ΓL(1,𝔽8)C2511yescharacteristic
77AGL(1,𝔽7)—25no—
78AGL(1,𝔽7)2−1+412yesdirect product factor
792−1+4—710no—
802−1+4—75no—
812−1+4AGL(1,𝔽7)11yescharacteristic, direct product factor
82C2×#41—710no—
83C22×Q8—75no—
84C22×C7C24×C315yes—
85C2×D7—210no—
86C2×D7C24×C315yes—
87C2×D7C24×C311yescharacteristic
88C4×C7C24×C3110yes—
89Dic7C24×C3110yes—
90C23×C3—75no—
91C3×Q8—730no—
92C3×Q8—710no—
93C3×D4—710no—
94C3×D4—720no—
95C3×D4—710no—
96C2×C3×C4—710no—
97C2×C3×C4—730no—
98C2×C3×C4—715no—
99C2×C3×C4—715no—
100ΓL(1,𝔽8)C2×2−1+411yescharacteristic, complemented, Hall
101#41—710no—
102#41—730no—
103#41—730no—
104#41C2×AGL(1,𝔽7)110yescomplemented
105C2×Q8—710no—
106C2×Q8—720no—
107C2×Q8—715no—
108C2×Q8C2×AGL(1,𝔽7)15yescomplemented
109C2×D4—710no—
110C22×C4—715no—
111C2×C7—25no—
112C2×C7C25×C311yescharacteristic
113D7—25no—
114D7C3×2−1+412yescomplemented
115C22×C3—71no—
116C22×C3—75no—
117C22×C3—75no—
118C22×C3—1410no—
119C3×C4—710no—
120C3×C4—710no—
121C23—75no—
122Q8—730no—
123Q8C22×AGL(1,𝔽7)110yes—
124D4—720no—
125D4—710no—
126D4C22×AGL(1,𝔽7)110yes—
127C2×C4—715no—
128C2×C4—710no—
129C2×C4—730no—
130C2×C4C22×AGL(1,𝔽7)115yes—
131C7C2×C3×2−1+411yescharacteristic, complemented, Sylow
132C2×C3—72no—
133C2×C3—71no—
134C2×C3—145no—
135C2×C3—145no—
136C22—71no—
137C22—75no—
138C22—1410no—
139C22C23×AGL(1,𝔽7)15yes—
140C4—710no—
141C4C23×AGL(1,𝔽7)110yes—
142C3—71noSylow
143C2—25no—
144C2—72no—
145C2—145no—
146C2C24×AGL(1,𝔽7)11yescharacteristic
subgroupG/H
center, FrattiniC2C24×AGL(1,𝔽7)
commutator, socle, cosocleC2×C7C25×C3
FittingC7×2−1+4C2×C3
hypercenter2−1+4AGL(1,𝔽7)
hypocenterC7C2×C3×2−1+4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ 2−1+4

Complements

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

NH ≅ G/Ntype
1. ΓL(1,𝔽8)×2−1+4C2semidirect
2. #10621519C2semidirect
3. #41×AGL(1,𝔽7)C2semidirect
4. #10621516C2semidirect
5. C2×Q8×AGL(1,𝔽7)C2semidirect
6. D7×2−1+4C3semidirect
7. #41×ΓL(1,𝔽8)C22semidirect
8. C2×Q8×ΓL(1,𝔽8)C22semidirect
9. #66659C22semidirect
11. Q8×AGL(1,𝔽7)C22semidirect
12. #1159⋊C3C22semidirect
14. #1157⋊C3C22semidirect
16. C2×ΓL(1,𝔽8)⋊Q8C22semidirect
17. C7×2−1+4C2×C3semidirect
18. #6554C2×C3semidirect
19. D7×#41C2×C3semidirect
20. #6551C2×C3semidirect
21. C2×Q8×D7C2×C3semidirect
33. C7×#41C22×C3semidirect
34. C2×C7×Q8C22×C3semidirect
35. #1161C22×C3semidirect
37. Q8×D7C22×C3semidirect
38. #1159C22×C3semidirect
40. #1157C22×C3semidirect
42. C2×Dic14C22×C3semidirect
78. AGL(1,𝔽7)2−1+4direct (N × H)
81. 2−1+4AGL(1,𝔽7)direct (N × H)
100. ΓL(1,𝔽8)C2×2−1+4semidirect
104. #41C2×AGL(1,𝔽7)semidirect
108. C2×Q8C2×AGL(1,𝔽7)semidirect
114. D7C3×2−1+4semidirect
131. C7C2×C3×2−1+4semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)×2−1+42
#106215192
#41×AGL(1,𝔽7)2
#106215162
C2×Q8×AGL(1,𝔽7)2
D7×2−1+43
C2×C3×2−1+47