Finite Group Databasebeta

← #89809

Subgroups — #89809

Order 448.

162 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 43 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×#1157C211yesmaximal, characteristic, complemented
2C7×#121C211yesmaximal, characteristic, complemented
3C2×#1142C211yesmaximal, characteristic, complemented
4C2×#1141C211yesmaximal, characteristic, complemented
5C22⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
6#6384C211yesmaximal, characteristic, complemented
7#6383C211yesmaximal, characteristic, complemented
8C2×C7×D4—21no—
9C2×C7×D4C2211yescharacteristic
10C22×C4×C7C2211yescharacteristic
11C2×C22⋊D7—21no—
12#1157—22no—
13#1157—22no—
14C2×D28C2211yescharacteristic
15C2×C4×D7—21no—
16C2×Dic14C2211yescharacteristic
17C7×C4⋊C4C2211yescharacteristic
18C7×C22⋊C4—21no—
19#1142—22no—
20#1141—22no—
21C2×C7⋊C8C2211yescharacteristic
22C2×C7⋊C8C2211yescharacteristic
23#449—71nomaximal, Sylow
24C23×C7—21no—
25C23×C7D411yescharacteristic
26C22×D7—21no—
27C7×D4—21no—
28C7×D4—21no—
29C7×D4—42no—
30C2×C4×C7—21no—
31C2×C4×C7—21no—
32C2×C4×C7—21no—
33C2×C4×C7D411yescharacteristic
34C2×C4×C7C2311yescharacteristic
35C22⋊D7—24no—
36C2×Dic7—21no—
37D28—21no—
38D28D412yescomplemented
39C4×D7—24no—
40Dic14—21no—
41Dic14D412yescomplemented
42C7⋊C8—21no—
43C7⋊C8—21no—
44C2×#41—71no—
45C2×SD16—71no—
46C2×D8—71no—
47#121D711yescharacteristic, complemented
48#103—71no—
49#102—71no—
50C22⋊C8—71no—
51C22×C7—21no—
52C22×C7—21no—
53C22×C7—21no—
54C22×C7—41no—
55C22×C7—41no—
56C22×C7—41no—
57C22×C7C2×D411yescharacteristic
58C2×D7—22no—
59C2×D7—41no—
60C2×D7—41no—
61C4×C7—21no—
62C4×C7—41no—
63C4×C7C2×D411yescharacteristic
64C4×C7C2×D411yescharacteristic
65Dic7—22no—
66#41—142no—
67#41—142no—
68C2×Q8—71no—
69C2×D4—21no—
70C2×D4—71no—
71C2×D4—141no—
72C2×D4C2×D711yescharacteristic
73C22×C4—141no—
74C22×C4C2×D711yescharacteristic
75SD16—142no—
76D8—142no—
77C2×C8—71no—
78C2×C8—71no—
79C4⋊C4C2×D711yescharacteristic
80C22⋊C4—21no—
81C2×C7—41no—
82C2×C7—81no—
83C2×C7#12011yescharacteristic
84C2×C7#13511yescharacteristic
85C2×C7AGL(1,ℤ/8)11yescharacteristic
86D7—42no—
87C23—21no—
88C23—141no—
89C23C22⋊D711yescharacteristic
90Q8—72no—
91Q8—141no—
92D4—21no—
93D4—21no—
94D4—42no—
95D4—72no—
96D4—144no—
97D4—141no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—144no—
102C2×C4—141no—
103C2×C4C22⋊D711yescharacteristic
104C2×C4C22×D711yescharacteristic
105C8—141no—
106C8—141no—
107C7#44911yescharacteristic, complemented, Sylow
108C22—21no—
109C22—21no—
110C22—21no—
111C22—41no—
112C22—41no—
113C22—41no—
114C22—142no—
115C22—281no—
116C22—281no—
117C22C2×C22⋊D711yescharacteristic
118C4—21no—
119C4—41no—
120C4—142no—
121C4D4×D711yescharacteristic
122C4D4×D711yescharacteristic
123C2—41no—
124C2—81no—
125C2—282no—
126C2#649511yescharacteristic
127C2#650011yescharacteristic
128C2#651311yescharacteristic
subgroupG/H
centerC22C2×C22⋊D7
commutator, cosocleC2×C4×C7C23
FrattiniC2×C4C22×D7
FittingC7×#121C2
socleC22×C7C2×D4
hypercenter#121D7
hypocenterC7#449
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#1157C2semidirect
2. C7×#121C2semidirect
3. C2×#1142C2semidirect
4. C2×#1141C2semidirect
5. C22⋊(C7⋊C8)C2semidirect
6. #6384C2semidirect
7. #6383C2semidirect
38. D28D4semidirect
41. Dic14D4semidirect
47. #121D7semidirect
107. C7#449semidirect

Maximal subgroups

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

subgroupindex
C2×#11572
C7×#1212
C2×#11422
C2×#11412
C22⋊(C7⋊C8)2
#63842
#63832
#4497