Finite Group Databasebeta

← #89960⋊C3 #13052089

Subgroups — #89960⋊C3 #13052089

Order 1344.

228 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 72 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×C8×AGL(1,𝔽7)C211yesmaximal, characteristic
4#6409⋊C3C212yesmaximal, complemented
5#6392⋊C3C211yesmaximal, characteristic, complemented
6#89960C311yesmaximal, 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
15C8×AGL(1,𝔽7)—22no—
16C2×C7×Q16C2×C311yescharacteristic, complemented
17#6509C2×C312yes—
18C2×C8×D7C2×C311yescharacteristic
19#6409C2×C312yescomplemented
20#6392C2×C311yescharacteristic, complemented
21C3×#467—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
41C8×D7—22no—
42C2×C3×Q16—71no—
43C22×C3×C8—71no—
44C3×C22⋊Q8—72no—
45C3×#107—71no—
46C3×#103—72no—
47C22×ΓL(1,𝔽8)C2×D411yescharacteristic
48C2×AGL(1,𝔽7)—21no—
49C2×AGL(1,𝔽7)—21no—
50C2×AGL(1,𝔽7)Q1612yescomplemented
51C4×ΓL(1,𝔽8)—42no—
52C4×ΓL(1,𝔽8)C2×D411yescharacteristic
53C4×ΓL(1,𝔽8)#4111yescharacteristic
54ΓL(1,𝔽8)⋊C4—21no—
55ΓL(1,𝔽8)⋊C4—42no—
56#467—71noSylow
57C22×D7C3×D411yescharacteristic
58C7×Q8—22no—
59C7×Q8—22no—
60C2×C4×C7—22no—
61C2×C4×C7C23×C311yescharacteristic
62C2×Dic7—22no—
63C2×Dic7C3×D411yescharacteristic
64C4×D7—21no—
65C4×D7—21no—
66C7×C8C3×D412yes—
67C7⋊C8—21no—
68C2×C3×Q8—72no—
69C22×C3×C4—71no—
70C3×Q16—142no—
71C2×C3×C8—71no—
72C2×C3×C8—71no—
73C2×C3×C8—142no—
74C3×C4⋊C4—72no—
75C3×C4⋊C4—142no—
76C3×C22⋊C4—142no—
77C2×ΓL(1,𝔽8)#12111yescharacteristic
78C2×ΓL(1,𝔽8)C2×Q1611yescharacteristic
79C2×ΓL(1,𝔽8)#13511yescharacteristic
80AGL(1,𝔽7)—22no—
81C2×Q16AGL(1,𝔽7)11yescharacteristic, complemented
82C22×C8—71no—
83C22⋊Q8—72no—
84#107—71no—
85#103—72no—
86C22×C7C2×C3×D411yescharacteristic
87C2×D7—21no—
88C2×D7—21no—
89C2×D7C3×Q1612yescomplemented
90C4×C7—42no—
91C4×C7C2×C3×D411yescharacteristic
92C4×C7C3×#4111yescharacteristic
93Dic7—21no—
94Dic7—42no—
95C23×C3—71no—
96C3×Q8—142no—
97C3×Q8—142no—
98C2×C3×C4—71no—
99C2×C3×C4—71no—
100C2×C3×C4—142no—
101C2×C3×C4—141no—
102C2×C3×C4—141no—
103C2×C3×C4—142no—
104C3×C8—72no—
105C3×C8—141no—
106ΓL(1,𝔽8)#46711yescharacteristic, complemented, Hall
107C2×Q8C2×AGL(1,𝔽7)12yes—
108C22×C4—71no—
109Q16—22no—
110C2×C8—71no—
111C2×C8—142no—
112C2×C8C2×AGL(1,𝔽7)11yescharacteristic
113C4⋊C4—72no—
114C4⋊C4—142no—
115C22⋊C4—142no—
116C2×C7C3×#12111yescharacteristic
117C2×C7C2×C3×Q1611yescharacteristic
118C2×C7C3×#13511yescharacteristic
119D7—22no—
120C22×C3—72no—
121C22×C3—71no—
122C22×C3—141no—
123C22×C3—141no—
124C3×C4—71no—
125C3×C4—71no—
126C3×C4—141no—
127C3×C4—282no—
128C3×C4—282no—
129C23—71no—
130Q8—22no—
131Q8—22no—
132C2×C4—22no—
133C2×C4—71no—
134C2×C4—141no—
135C2×C4—141no—
136C2×C4—142no—
137C2×C4C22×AGL(1,𝔽7)11yescharacteristic
138C8—141no—
139C8(C22⋊D7)⋊C312yes—
140C7C3×#46711yescharacteristic, complemented, Sylow
141C2×C3—71no—
142C2×C3—71no—
143C2×C3—71no—
144C2×C3—142no—
145C22—72no—
146C22—141no—
147C22—141no—
148C22D4×AGL(1,𝔽7)11yescharacteristic
149C4—42no—
150C4—141no—
151C4—282no—
152C4#1159⋊C311yescharacteristic
153C4C2×(C22⋊D7)⋊C311yescharacteristic
154C3—71noSylow
155C2—142no—
156C2Q16×AGL(1,𝔽7)11yescharacteristic
157C2#6482⋊C311yescharacteristic
158C2#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×#467
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. #6392⋊C3C2semidirect
6. #89960C3semidirect
16. C2×C7×Q16C2×C3semidirect
19. #6409C2×C3semidirect
20. #6392C2×C3semidirect
50. C2×AGL(1,𝔽7)Q16semidirect
81. C2×Q16AGL(1,𝔽7)semidirect
89. C2×D7C3×Q16semidirect
106. ΓL(1,𝔽8)#467semidirect
140. C7C3×#467semidirect

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×C8×AGL(1,𝔽7)2
#6409⋊C32
#6392⋊C32
#899603
C3×#4677