Finite Group Databasebeta

← #89599

Subgroups — #89599

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
2#6512C211yesmaximal, characteristic, complemented
3#6471C211yesmaximal, characteristic, complemented
4#6438C211yesmaximal, characteristic, complemented
5C7×#104C211yesmaximal, characteristic, complemented
6#6398C211yesmaximal, characteristic, complemented
7#6379C211yesmaximal, characteristic, complemented
8C7×#41C2211yescharacteristic
9Q8×D7—21no—
10#1159—21no—
11#1159—22no—
12#1157—21no—
13#1157C2211yescharacteristic
14C2×D28C2211yescharacteristic
15C2×Dic14—21no—
16C2×Dic14C2211yescharacteristic
17C7×M16C2211yescharacteristic
18C42×C7C2211yescharacteristic
19#1143—21no—
20#1141—21no—
21(C2×D7)⋊C4—22no—
22#1136C2211yescharacteristic
23D56—21no—
24#1132—21no—
25#454—71nomaximal, Sylow
26C22×D7—21no—
27C7×Q8D411yescharacteristic
28C7×D4—21no—
29C7×D4D411yescharacteristic
30C2×C4×C7—21no—
31C2×C4×C7—21no—
32C2×C4×C7C2311yescharacteristic
33C22⋊D7—22no—
34C22⋊D7—21no—
35C2×Dic7—22no—
36C2×Dic7D412yes—
37D28—21no—
38D28—21no—
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)—71no—
51#124—71no—
52#104—71no—
53#104D711yescharacteristic, complemented
54#101—71no—
55C22×C7—21no—
56C22×C7C2×D411yescharacteristic
57C2×D7—21no—
58C2×D7—42no—
59C2×D7—41no—
60C4×C7—21no—
61C4×C7—41no—
62C4×C7C2×D411yescharacteristic
63C4×C7C2×D411yescharacteristic
64Dic7—22no—
65Dic7—21no—
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×D4—71no—
75SD16—141no—
76SD16—141no—
77D8—141no—
78D8—141no—
79M16—71no—
80M16C2×D711yescharacteristic
81C22⋊C4—142no—
82C42C2×D711yescharacteristic
83C2×C7—21no—
84C2×C7—41no—
85C2×C7#12011yescharacteristic
86D7—41no—
87D7—81no—
88C23—141no—
89Q8—71no—
90Q8—142no—
91Q8—141no—
92Q8—141no—
93Q8D2811yescharacteristic
94D4—21no—
95D4—71no—
96D4—142no—
97D4—141no—
98D4—141no—
99D4—141no—
100D4D2811yescharacteristic
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—72no—
104C2×C4—142no—
105C2×C4—142no—
106C2×C4—141no—
107C2×C4C22×D711yescharacteristic
108C8—21no—
109C8—141no—
110C7#45411yescharacteristic, complemented, Sylow
111C22—21no—
112C22—141no—
113C22—282no—
114C22—281no—
115C22D4×D711yescharacteristic
116C4—21no—
117C4—41no—
118C4—142no—
119C4—141no—
120C4C2×D2811yescharacteristic
121C4D4×D711yescharacteristic
122C2—21no—
123C2—41no—
124C2—281no—
125C2—561no—
126C2#644511yescharacteristic
subgroupG/H
centerC2#6445
commutator, cosocleC2×C4×C7C23
FrattiniC2×C4C22×D7
FittingC7×#104C2
socleC2×C7#120
hypercenter#104D7
hypocenterC7#454
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
1. #6554C2semidirect
2. #6512C2semidirect
3. #6471C2semidirect
4. #6438C2semidirect
5. C7×#104C2semidirect
6. #6398C2semidirect
7. #6379C2semidirect
53. #104D7semidirect
110. C7#454semidirect

Maximal subgroups

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

subgroupindex
#65542
#65122
#64712
#64382
C7×#1042
#63982
#63792
#4547