Finite Group Databasebeta

← #89553

Subgroups — #89553

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×#1159C211yesmaximal, characteristic, complemented
2C2×#1141C211yesmaximal, characteristic, complemented
3C2×#1132C211yesmaximal, characteristic, complemented
4#6458C211yesmaximal, characteristic, complemented
5C7×#102C211yesmaximal, characteristic, complemented
6(C2×D7)⋊C8C211yesmaximal, characteristic, complemented
7#6384C211yesmaximal, characteristic, complemented
8C2×C7×D4C2211yescharacteristic
9C2×C22⋊D7—21no—
10C22×Dic7—21no—
11#1159—22no—
12#1159—22no—
13C2×D28—21no—
14C2×D28C2211yescharacteristic
15C2×C4×D7C2211yescharacteristic
16C2×Dic14C2211yescharacteristic
17C2×C7×C8C2211yescharacteristic
18C7×C4⋊C4C2211yescharacteristic
19#1141—22no—
20(C2×D7)⋊C4—21no—
21C2×C7⋊C8C2211yescharacteristic
22#1132—22no—
23#449—71nomaximal, Sylow
24C23×C7—21no—
25C22×D7—21no—
26C22×D7D411yescharacteristic
27C7×D4—21no—
28C7×D4D412yescomplemented
29C2×C4×C7—21no—
30C2×C4×C7C2311yescharacteristic
31C22⋊D7—24no—
32C2×Dic7—24no—
33C2×Dic7—21no—
34C2×Dic7D411yescharacteristic
35D28—21no—
36D28—21no—
37D28—42no—
38C4×D7—21no—
39C4×D7—21no—
40Dic14—21no—
41Dic14D412yescomplemented
42C7×C8—21no—
43C7⋊C8—21no—
44C2×#41—71no—
45C2×SD16—71no—
46C2×D8—71no—
47#121—71no—
48#103—71no—
49#102D711yescharacteristic, complemented
50C22⋊C8—71no—
51C22×C7—22no—
52C22×C7—41no—
53C22×C7—41no—
54C22×C7C2×D411yescharacteristic
55C2×D7—21no—
56C2×D7—21no—
57C2×D7—21no—
58C2×D7—41no—
59C2×D7—41no—
60C2×D7—41no—
61C4×C7—41no—
62C4×C7C2×D411yescharacteristic
63C4×C7C2×D411yescharacteristic
64Dic7—22no—
65Dic7—21no—
66#41—142no—
67#41—142no—
68C2×Q8—71no—
69C2×D4—71no—
70C2×D4—141no—
71C2×D4—141no—
72C2×D4C2×D711yescharacteristic
73C22×C4—71no—
74C22×C4—141no—
75SD16—142no—
76D8—142no—
77C2×C8—71no—
78C2×C8C2×D711yescharacteristic
79C4⋊C4C2×D711yescharacteristic
80C22⋊C4—141no—
81C2×C7—42no—
82C2×C7#12011yescharacteristic
83C2×C7#13511yescharacteristic
84C2×C7AGL(1,ℤ/8)11yescharacteristic
85D7—41no—
86D7—81no—
87C23—21no—
88C23—71no—
89C23—141no—
90Q8—72no—
91Q8—141no—
92D4—21no—
93D4—144no—
94D4—141no—
95D4—141no—
96D4—282no—
97D4D2812yescomplemented
98C2×C4—21no—
99C2×C4—71no—
100C2×C4—144no—
101C2×C4—141no—
102C2×C4—141no—
103C2×C4—141no—
104C2×C4C22×D711yescharacteristic
105C8—21no—
106C8—141no—
107C7#44911yescharacteristic, complemented, Sylow
108C22—22no—
109C22—41no—
110C22—41no—
111C22—141no—
112C22—141no—
113C22—141no—
114C22—281no—
115C22—281no—
116C22—281no—
117C22D4×D711yescharacteristic
118C4—41no—
119C4—141no—
120C4—142no—
121C4C2×D2811yescharacteristic
122C4D4×D711yescharacteristic
123C2—42no—
124C2—281no—
125C2—561no—
126C2#644511yescharacteristic
127C2#647411yescharacteristic
128C2#647911yescharacteristic
subgroupG/H
centerC22D4×D7
commutator, cosocleC2×C4×C7C23
FrattiniC2×C4C22×D7
FittingC7×#102C2
socleC22×C7C2×D4
hypercenter#102D7
hypocenterC7#449
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
1. C2×#1159C2semidirect
2. C2×#1141C2semidirect
3. C2×#1132C2semidirect
4. #6458C2semidirect
5. C7×#102C2semidirect
6. (C2×D7)⋊C8C2semidirect
7. #6384C2semidirect
28. C7×D4D4semidirect
41. Dic14D4semidirect
49. #102D7semidirect
97. D4D28semidirect
107. C7#449semidirect

Maximal subgroups

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

subgroupindex
C2×#11592
C2×#11412
C2×#11322
#64582
C7×#1022
(C2×D7)⋊C82
#63842
#4497