Finite Group Databasebeta

← #6446⋊C3 #10620719

Subgroups — #6446⋊C3 #10620719

Order 672.

156 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 53 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×(C22⋊D7)⋊C3C211yesmaximal, characteristic
2C2×C4×AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
3C22⋊C4×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
4(C22⋊Dic7)⋊C3C211yesmaximal, characteristic, complemented
5(C2×AGL(1,𝔽7))⋊C4C211yesmaximal, characteristic, complemented
6C4⋊(ΓL(1,𝔽8)⋊C4)C211yesmaximal, characteristic, complemented
7(ΓL(1,𝔽8)⋊C4)⋊C4C211yesmaximal, characteristic, complemented
8#6446C311yesmaximal, characteristic, complemented, Hall
9C23×ΓL(1,𝔽8)C2211yescharacteristic
10C22×AGL(1,𝔽7)C2211yescharacteristic
11C2×C4×ΓL(1,𝔽8)C2211yescharacteristic
12C2×C4×ΓL(1,𝔽8)C2211yescharacteristic
13(C22⋊D7)⋊C3—22no—
14C2×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
15C2×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
16C2×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
17C4×AGL(1,𝔽7)—22no—
18C2×C22⋊D7C2×C311yescharacteristic
19C2×C4×D7C2×C311yescharacteristic, complemented
20C7×C22⋊C4C2×C311yescharacteristic, complemented
21C22⋊Dic7C2×C311yescharacteristic, complemented
22(C2×D7)⋊C4C2×C311yescharacteristic, complemented
23C4⋊Dic7C2×C311yescharacteristic, complemented
24Dic7⋊C4C2×C311yescharacteristic, complemented
25C3×#123—71nomaximal, Hall
26C22×ΓL(1,𝔽8)—21no—
27C22×ΓL(1,𝔽8)—21no—
28C22×ΓL(1,𝔽8)—21no—
29C22×ΓL(1,𝔽8)C2311yescharacteristic
30C2×AGL(1,𝔽7)—21no—
31C2×AGL(1,𝔽7)—21no—
32C2×AGL(1,𝔽7)D412yes—
33C4×ΓL(1,𝔽8)—21no—
34C4×ΓL(1,𝔽8)—21no—
35ΓL(1,𝔽8)⋊C4—21no—
36ΓL(1,𝔽8)⋊C4—21no—
37ΓL(1,𝔽8)⋊C4—21no—
38C23×C7C22×C311yescharacteristic
39C22×D7C22×C311yescharacteristic
40C2×C4×C7C22×C311yescharacteristic
41C2×C4×C7C22×C311yescharacteristic
42C22⋊D7—22no—
43C2×Dic7C22×C311yescharacteristic
44C2×Dic7C22×C311yescharacteristic
45C2×Dic7C22×C311yescharacteristic
46C4×D7—22no—
47C2×C3×D4—71no—
48C22×C3×C4—71no—
49C3×C4⋊C4—71no—
50C3×C4⋊C4—71no—
51C3×C22⋊C4—71no—
52C3×C22⋊C4—71no—
53C3×C22⋊C4—71no—
54C2×ΓL(1,𝔽8)—41no—
55C2×ΓL(1,𝔽8)C2×D411yescharacteristic
56C2×ΓL(1,𝔽8)#4111yescharacteristic
57C2×ΓL(1,𝔽8)#4111yescharacteristic
58AGL(1,𝔽7)—22no—
59#123—71noSylow
60C22×C7—21no—
61C22×C7—21no—
62C22×C7—21no—
63C22×C7C23×C311yescharacteristic
64C2×D7—21no—
65C2×D7—21no—
66C2×D7C3×D412yes—
67C4×C7—21no—
68C4×C7—21no—
69Dic7—21no—
70Dic7—21no—
71Dic7—21no—
72C23×C3—71no—
73C23×C3—71no—
74C3×D4—142no—
75C2×C3×C4—71no—
76C2×C3×C4—71no—
77C2×C3×C4—71no—
78C2×C3×C4—71no—
79C2×C3×C4—71no—
80C2×C3×C4—142no—
81ΓL(1,𝔽8)#12311yescharacteristic, complemented, Hall
82C2×D4—71no—
83C22×C4—71no—
84C4⋊C4—71no—
85C4⋊C4—71no—
86C22⋊C4—71no—
87C22⋊C4—71no—
88C22⋊C4AGL(1,𝔽7)11yescharacteristic, complemented
89C2×C7—41no—
90C2×C7C2×C3×D411yescharacteristic
91C2×C7C3×#4111yescharacteristic
92C2×C7C3×#4111yescharacteristic
93D7—22no—
94C22×C3—72no—
95C22×C3—71no—
96C22×C3—141no—
97C22×C3—141no—
98C22×C3—141no—
99C22×C3—141no—
100C22×C3—141no—
101C3×C4—141no—
102C3×C4—141no—
103C3×C4—141no—
104C3×C4—141no—
105C3×C4—141no—
106C23—71no—
107C23C2×AGL(1,𝔽7)11yescharacteristic
108D4—142no—
109C2×C4—71no—
110C2×C4—71no—
111C2×C4—71no—
112C2×C4—142no—
113C2×C4C2×AGL(1,𝔽7)11yescharacteristic
114C2×C4C2×AGL(1,𝔽7)11yescharacteristic
115C7C3×#12311yescharacteristic, complemented, Sylow
116C2×C3—71no—
117C2×C3—71no—
118C2×C3—71no—
119C2×C3—142no—
120C2×C3—281no—
121C22—21no—
122C22—21no—
123C22—21no—
124C22—72no—
125C22—141no—
126C22—141no—
127C22C22×AGL(1,𝔽7)11yescharacteristic
128C4—21no—
129C4—21no—
130C4—141no—
131C4—141no—
132C4—141no—
133C3—71noSylow
134C2—41no—
135C2—142no—
136C2#1157⋊C311yescharacteristic
137C2D4×AGL(1,𝔽7)11yescharacteristic
138C2#1159⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×AGL(1,𝔽7)
commutator, socle, cosocleC22×C7C23×C3
FittingC7×C22⋊C4C2×C3
prime radicalC2×(C22⋊D7)⋊C3C2
hypercenterC22⋊C4AGL(1,𝔽7)
hypocenterC7C3×#123
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×AGL(1,𝔽7)C2semidirect
3. C22⋊C4×ΓL(1,𝔽8)C2semidirect
4. (C22⋊Dic7)⋊C3C2semidirect
5. (C2×AGL(1,𝔽7))⋊C4C2semidirect
6. C4⋊(ΓL(1,𝔽8)⋊C4)C2semidirect
7. (ΓL(1,𝔽8)⋊C4)⋊C4C2semidirect
8. #6446C3semidirect
19. C2×C4×D7C2×C3semidirect
20. C7×C22⋊C4C2×C3semidirect
21. C22⋊Dic7C2×C3semidirect
22. (C2×D7)⋊C4C2×C3semidirect
23. C4⋊Dic7C2×C3semidirect
24. Dic7⋊C4C2×C3semidirect
81. ΓL(1,𝔽8)#123semidirect
88. C22⋊C4AGL(1,𝔽7)semidirect
115. C7C3×#123semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D7)⋊C32
C2×C4×AGL(1,𝔽7)2
C22⋊C4×ΓL(1,𝔽8)2
(C22⋊Dic7)⋊C32
(C2×AGL(1,𝔽7))⋊C42
C4⋊(ΓL(1,𝔽8)⋊C4)2
(ΓL(1,𝔽8)⋊C4)⋊C42
#64463
C3×#1237