Finite Group Databasebeta

← #89964⋊C3 #13052092

Subgroups — #89964⋊C3 #13052092

Order 1344.

200 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 62 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#6505⋊C3C212yesmaximal
3#6464⋊C3C211yesmaximal, characteristic
4#6410⋊C3C212yesmaximal, complemented
5#6393⋊C3C211yesmaximal, characteristic, complemented
6#89964C311yesmaximal, characteristic, complemented, Hall
7C2×Q8×ΓL(1,𝔽8)C2212yes—
8#1157⋊C3C2211yescharacteristic
9Q16×ΓL(1,𝔽8)—21no—
10Q16×ΓL(1,𝔽8)—21no—
11C2×C8×ΓL(1,𝔽8)C2211yescharacteristic
12#1144⋊C3—22no—
13#1143⋊C3—22no—
14#1136⋊C3C2212yes—
15#1136⋊C3C2211yescharacteristic
16#1131⋊C3—21no—
17C8×AGL(1,𝔽7)—21no—
18C2×C7×Q16C2×C311yescharacteristic, complemented
19#6505C2×C312yes—
20#6464C2×C311yescharacteristic
21#6410C2×C312yescomplemented
22#6393C2×C311yescharacteristic, complemented
23C3×#473—71nomaximal, Hall
24Q8×ΓL(1,𝔽8)—22no—
25Q8×ΓL(1,𝔽8)—22no—
26C2×C4×ΓL(1,𝔽8)—22no—
27C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
28(C22⋊D7)⋊C3—21no—
29C4⋊AGL(1,𝔽7)D411yescharacteristic
30C4×AGL(1,𝔽7)—21no—
31ΓL(1,𝔽8)⋊Q8D411yescharacteristic
32C8×ΓL(1,𝔽8)D411yescharacteristic
33C8×ΓL(1,𝔽8)D411yescharacteristic
34ΓL(1,𝔽8)⋊C8—21no—
35ΓL(1,𝔽8)⋊C8—22no—
36C2×C7×Q8C22×C312yes—
37#1157C22×C311yescharacteristic
38C7×Q16—21no—
39C7×Q16—21no—
40C2×C7×C8C22×C311yescharacteristic
41#1144—22no—
42#1143—22no—
43#1136C22×C311yescharacteristic
44#1136C22×C312yes—
45#1131—21no—
46C8×D7—21no—
47C3×#137—72no—
48C2×C3×Q16—71no—
49C3×#131—71no—
50C3×#108—71no—
51C3×#101—72no—
52C22×ΓL(1,𝔽8)#4111yescharacteristic
53C2×AGL(1,𝔽7)—21no—
54C4×ΓL(1,𝔽8)—42no—
55C4×ΓL(1,𝔽8)C2×D411yescharacteristic
56C4×ΓL(1,𝔽8)C2×D411yescharacteristic
57ΓL(1,𝔽8)⋊C4—21no—
58#473—71noSylow
59C7×Q8—22no—
60C7×Q8—22no—
61C2×C4×C7—22no—
62C2×C4×C7C23×C311yescharacteristic
63C22⋊D7—21no—
64D28C3×D411yescharacteristic
65C4×D7—21no—
66Dic14C3×D411yescharacteristic
67C7×C8C3×D411yescharacteristic
68C7×C8C3×D411yescharacteristic
69C7⋊C8—21no—
70C7⋊C8—22no—
71C3×#41—71no—
72C2×C3×Q8—72no—
73C3×Q16—141no—
74C3×Q16—141no—
75C3×Q16—142no—
76C3×SD16—142no—
77C3×M16—71no—
78C3×M16—72no—
79C3×M16—141no—
80C2×C3×C8—71no—
81C2×C3×C8—141no—
82C2×ΓL(1,𝔽8)—21no—
83C2×ΓL(1,𝔽8)#12111yescharacteristic
84AGL(1,𝔽7)—41no—
85#137—72no—
86C2×Q16AGL(1,𝔽7)11yescharacteristic, complemented
87#131—71no—
88#108—71no—
89#101—72no—
90C22×C7C3×#4111yescharacteristic
91C2×D7—21no—
92C4×C7—42no—
93C4×C7C2×C3×D411yescharacteristic
94C4×C7C2×C3×D411yescharacteristic
95Dic7—21no—
96C3×Q8—71no—
97C3×Q8—142no—
98C3×Q8—142no—
99C3×D4—71no—
100C3×D4—141no—
101C2×C3×C4—71no—
102C2×C3×C4—142no—
103C2×C3×C4—141no—
104C3×C8—71no—
105C3×C8—71no—
106C3×C8—141no—
107C3×C8—142no—
108ΓL(1,𝔽8)#47311yescharacteristic, complemented, Hall
109#41—71no—
110C2×Q8C2×AGL(1,𝔽7)12yes—
111Q16—21no—
112Q16—21no—
113Q16—142no—
114SD16—142no—
115M16—71no—
116M16—72no—
117M16—141no—
118C2×C8—141no—
119C2×C8C2×AGL(1,𝔽7)11yescharacteristic
120C2×C7—21no—
121C2×C7C3×#12111yescharacteristic
122D7—41no—
123C22×C3—71no—
124C22×C3—141no—
125C3×C4—71no—
126C3×C4—71no—
127C3×C4—141no—
128C3×C4—282no—
129Q8—22no—
130Q8—22no—
131Q8—71no—
132D4—71no—
133D4—141no—
134C2×C4—22no—
135C2×C4—141no—
136C2×C4C22×AGL(1,𝔽7)11yescharacteristic
137C8—142no—
138C8—141no—
139C8(C22⋊D7)⋊C311yescharacteristic
140C8(C22⋊D7)⋊C311yescharacteristic
141C7C3×#47311yescharacteristic, complemented, Sylow
142C2×C3—71no—
143C2×C3—141no—
144C2×C3—281no—
145C22—141no—
146C22#1159⋊C311yescharacteristic
147C4—42no—
148C4—141no—
149C4D4×AGL(1,𝔽7)11yescharacteristic
150C4C2×(C22⋊D7)⋊C311yescharacteristic
151C3—71noSylow
152C2—21no—
153C2—281no—
154C2#6501⋊C311yescharacteristic
subgroupG/H
centerC2#6501⋊C3
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×AGL(1,𝔽7)
FittingC2×C7×Q16C2×C3
socleC2×C7C3×#121
prime radical#1157⋊C3C22
hypercenterC2×Q16AGL(1,𝔽7)
hypocenterC7C3×#473
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ 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. #6410⋊C3C2semidirect
5. #6393⋊C3C2semidirect
6. #89964C3semidirect
18. C2×C7×Q16C2×C3semidirect
21. #6410C2×C3semidirect
22. #6393C2×C3semidirect
86. C2×Q16AGL(1,𝔽7)semidirect
108. ΓL(1,𝔽8)#473semidirect
141. C7C3×#473semidirect

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
#6505⋊C32
#6464⋊C32
#6410⋊C32
#6393⋊C32
#899643
C3×#4737