Finite Group Databasebeta

← #89961⋊C3 #13052090

Subgroups — #89961⋊C3 #13052090

Order 1344.

220 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C2×Q16×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2#6509⋊C3C212yesmaximal
3C2×#1131⋊C3C211yesmaximal, characteristic
4#6409⋊C3C212yesmaximal, complemented
5#6391⋊C3C211yesmaximal, characteristic, complemented
6#89961C311yesmaximal, characteristic, complemented, Hall
7C2×Q8×ΓL(1,𝔽8)C2212yes—
8C2×C4×AGL(1,𝔽7)C2211yescharacteristic
9Q16×ΓL(1,𝔽8)—22no—
10C2×C8×ΓL(1,𝔽8)C2211yescharacteristic
11(C2×AGL(1,𝔽7))⋊C4—22no—
12C4⋊(ΓL(1,𝔽8)⋊C4)C2212yes—
13(ΓL(1,𝔽8)⋊C4)⋊C4—22no—
14C2×ΓL(1,𝔽8)⋊C8C2211yescharacteristic
15#1131⋊C3—22no—
16C2×C7×Q16C2×C311yescharacteristic, complemented
17#6509C2×C312yes—
18C2×#1131C2×C311yescharacteristic
19#6409C2×C312yescomplemented
20#6391C2×C311yescharacteristic, complemented
21C3×#470—71nomaximal, Hall
22C22×AGL(1,𝔽7)D411yescharacteristic
23Q8×ΓL(1,𝔽8)—22no—
24Q8×ΓL(1,𝔽8)—22no—
25C2×C4×ΓL(1,𝔽8)—22no—
26C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
27C2×ΓL(1,𝔽8)⋊C4—22no—
28C2×ΓL(1,𝔽8)⋊C4D411yescharacteristic
29C4×AGL(1,𝔽7)—21no—
30C4×AGL(1,𝔽7)—21no—
31C8×ΓL(1,𝔽8)D412yes—
32ΓL(1,𝔽8)⋊C8—21no—
33C2×C7×Q8C22×C312yes—
34C2×C4×D7C22×C311yescharacteristic
35C7×Q16—22no—
36C2×C7×C8C22×C311yescharacteristic
37(C2×D7)⋊C4—22no—
38C4⋊Dic7C22×C312yes—
39Dic7⋊C4—22no—
40C2×C7⋊C8C22×C311yescharacteristic
41#1131—22no—
42C2×C3×Q16—71no—
43C2×C3×M16—71no—
44C3×C22⋊Q8—72no—
45C3×#106—71no—
46C3×#103—72no—
47C22×ΓL(1,𝔽8)C2×D411yescharacteristic
48C2×AGL(1,𝔽7)—21no—
49C2×AGL(1,𝔽7)—22no—
50C4×ΓL(1,𝔽8)—42no—
51C4×ΓL(1,𝔽8)C2×D411yescharacteristic
52C4×ΓL(1,𝔽8)#4111yescharacteristic
53ΓL(1,𝔽8)⋊C4—21no—
54ΓL(1,𝔽8)⋊C4—42no—
55#470—71noSylow
56C22×D7C3×D411yescharacteristic
57C7×Q8—22no—
58C7×Q8—22no—
59C2×C4×C7—22no—
60C2×C4×C7C23×C311yescharacteristic
61C2×Dic7—22no—
62C2×Dic7C3×D411yescharacteristic
63C4×D7—21no—
64C4×D7—21no—
65C7×C8C3×D412yes—
66C7⋊C8—21no—
67C2×C3×Q8—72no—
68C22×C3×C4—71no—
69C3×Q16—142no—
70C3×M16—142no—
71C2×C3×C8—71no—
72C2×C3×C8—71no—
73C3×C4⋊C4—72no—
74C3×C4⋊C4—142no—
75C3×C22⋊C4—142no—
76C2×ΓL(1,𝔽8)#12111yescharacteristic
77C2×ΓL(1,𝔽8)#13712yes—
78AGL(1,𝔽7)—41no—
79C2×Q16AGL(1,𝔽7)11yescharacteristic, complemented
80C2×M16—71no—
81C22⋊Q8—72no—
82#106—71no—
83#103—72no—
84C22×C7C2×C3×D411yescharacteristic
85C2×D7—21no—
86C2×D7—22no—
87C4×C7—42no—
88C4×C7C2×C3×D411yescharacteristic
89C4×C7C3×#4111yescharacteristic
90Dic7—21no—
91Dic7—42no—
92C23×C3—71no—
93C3×Q8—142no—
94C3×Q8—142no—
95C2×C3×C4—71no—
96C2×C3×C4—71no—
97C2×C3×C4—142no—
98C2×C3×C4—141no—
99C2×C3×C4—141no—
100C2×C3×C4—142no—
101C3×C8—72no—
102C3×C8—141no—
103ΓL(1,𝔽8)#47011yescharacteristic, complemented, Hall
104C2×Q8C2×AGL(1,𝔽7)12yes—
105C22×C4—71no—
106Q16—22no—
107M16—142no—
108C2×C8—71no—
109C2×C8C2×AGL(1,𝔽7)11yescharacteristic
110C4⋊C4—72no—
111C4⋊C4—142no—
112C22⋊C4—142no—
113C2×C7C3×#12111yescharacteristic
114C2×C7C3×#13712yes—
115D7—41no—
116C22×C3—71no—
117C22×C3—141no—
118C22×C3—142no—
119C3×C4—71no—
120C3×C4—71no—
121C3×C4—141no—
122C3×C4—282no—
123C3×C4—282no—
124C23—71no—
125Q8—22no—
126Q8—22no—
127C2×C4—22no—
128C2×C4—71no—
129C2×C4—141no—
130C2×C4—141no—
131C2×C4—142no—
132C2×C4C22×AGL(1,𝔽7)11yescharacteristic
133C8—141no—
134C8(C22⋊D7)⋊C312yes—
135C7C3×#47011yescharacteristic, complemented, Sylow
136C2×C3—72no—
137C2×C3—71no—
138C2×C3—281no—
139C22—142no—
140C22—141no—
141C22D4×AGL(1,𝔽7)11yescharacteristic
142C4—42no—
143C4—141no—
144C4—282no—
145C4#1159⋊C311yescharacteristic
146C4C2×(C22⋊D7)⋊C311yescharacteristic
147C3—71noSylow
148C2—281no—
149C2#6481⋊C312yes—
150C2#6501⋊C311yescharacteristic
subgroupG/H
centerC22D4×AGL(1,𝔽7)
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×AGL(1,𝔽7)
FittingC2×C7×Q16C2×C3
socleC22×C7C2×C3×D4
prime radicalC22×AGL(1,𝔽7)D4
hypercenterC2×Q16AGL(1,𝔽7)
hypocenterC7C3×#470
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q16

Complements

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

NH ≅ G/Ntype
1. C2×Q16×ΓL(1,𝔽8)C2semidirect
4. #6409⋊C3C2semidirect
5. #6391⋊C3C2semidirect
6. #89961C3semidirect
16. C2×C7×Q16C2×C3semidirect
19. #6409C2×C3semidirect
20. #6391C2×C3semidirect
79. C2×Q16AGL(1,𝔽7)semidirect
103. ΓL(1,𝔽8)#470semidirect
135. C7C3×#470semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×ΓL(1,𝔽8)2
#6509⋊C32
C2×#1131⋊C32
#6409⋊C32
#6391⋊C32
#899613
C3×#4707