Finite Group Databasebeta

← #89973

Subgroups — #89973

Order 448.

146 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 39 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
1#6554C211yesmaximal, characteristic, complemented
2C7×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
3#6499C211yesmaximal, characteristic, complemented
4#6495C211yesmaximal, characteristic, complemented
5#6411C211yesmaximal, characteristic, complemented
6#6399C211yesmaximal, characteristic, complemented
7#6398C211yesmaximal, characteristic, complemented
8C7×#41C2211yescharacteristic
9C2×C7×D4C2211yescharacteristic
10Q8×D7—21no—
11#1159—21no—
12#1159—22no—
13#1157—21no—
14#1157C2211yescharacteristic
15C2×Dic14—21no—
16C2×Dic14C2211yescharacteristic
17C7×SD16—21no—
18C7×D8—21no—
19C7×M16C2211yescharacteristic
20C22⋊Dic7—22no—
21#1142—21no—
22#1141—21no—
23C4×Dic7C2211yescharacteristic
24#1136C2211yescharacteristic
25#454—71nomaximal, Sylow
26C23×C7—21no—
27C7×Q8D411yescharacteristic
28C7×D4—21no—
29C7×D4—21no—
30C7×D4—21no—
31C7×D4D411yescharacteristic
32C2×C4×C7—21no—
33C2×C4×C7C2311yescharacteristic
34C22⋊D7—22no—
35C22⋊D7—21no—
36C2×Dic7—22no—
37C2×Dic7—21no—
38C2×Dic7D412yes—
39D28D411yescharacteristic
40C4×D7—22no—
41C4×D7—21no—
42Dic14—22no—
43Dic14—21no—
44Dic14—21no—
45Dic14D411yescharacteristic
46C7×C8—21no—
47C7⋊C8—21no—
482−1+4—71no—
49AGL(1,ℤ/8)—71no—
50AGL(1,ℤ/8)D711yescharacteristic, complemented
51#124—71no—
52#104—71no—
53#104—71no—
54#101—71no—
55C22×C7—21no—
56C22×C7—42no—
57C22×C7—41no—
58C22×C7C2×D411yescharacteristic
59C2×D7—21no—
60C4×C7—21no—
61C4×C7C2×D411yescharacteristic
62C4×C7C2×D411yescharacteristic
63Dic7—22no—
64Dic7—21no—
65Dic7—41no—
66#41—71no—
67#41—141no—
68#41—142no—
69#41—141no—
70#41C2×D711yescharacteristic
71C2×Q8—71no—
72C2×Q8—141no—
73C2×Q8—141no—
74C2×D4C2×D711yescharacteristic
75SD16—21no—
76SD16—141no—
77D8—21no—
78D8—141no—
79M16—71no—
80M16C2×D711yescharacteristic
81C22⋊C4—142no—
82C42—71no—
83C2×C7—21no—
84C2×C7—41no—
85C2×C7—81no—
86C2×C7#12011yescharacteristic
87D7—41no—
88C23—21no—
89Q8—71no—
90Q8—142no—
91Q8—141no—
92Q8—141no—
93Q8C22⋊D711yescharacteristic
94D4—21no—
95D4—21no—
96D4—21no—
97D4—71no—
98D4—142no—
99D4—141no—
100D4C22⋊D711yescharacteristic
101C2×C4—21no—
102C2×C4—72no—
103C2×C4—142no—
104C2×C4—142no—
105C2×C4—141no—
106C2×C4—141no—
107C2×C4C22×D711yescharacteristic
108C8—21no—
109C8—141no—
110C7#45411yescharacteristic, complemented, Sylow
111C22—21no—
112C22—42no—
113C22—41no—
114C22—141no—
115C22D4×D711yescharacteristic
116C4—21no—
117C4—142no—
118C4—141no—
119C4—281no—
120C4D4×D711yescharacteristic
121C4C2×C22⋊D711yescharacteristic
122C2—21no—
123C2—41no—
124C2—81no—
125C2—281no—
126C2#650011yescharacteristic
subgroupG/H
centerC2#6500
commutator, cosocleC2×C4×C7C23
FrattiniC2×C4C22×D7
FittingC7×AGL(1,ℤ/8)C2
socleC2×C7#120
hypercenterAGL(1,ℤ/8)D7
hypocenterC7#454
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #6554C2semidirect
2. C7×AGL(1,ℤ/8)C2semidirect
3. #6499C2semidirect
4. #6495C2semidirect
5. #6411C2semidirect
6. #6399C2semidirect
7. #6398C2semidirect
50. AGL(1,ℤ/8)D7semidirect
110. C7#454semidirect

Maximal subgroups

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

subgroupindex
#65542
C7×AGL(1,ℤ/8)2
#64992
#64952
#64112
#63992
#63982
#4547