Finite Group Databasebeta

← #16818575

Subgroups — #16818575

Order 3712.

240 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 57 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#14271537C211yesmaximal, characteristic, complemented
2#14271358C212yesmaximal, complemented
3#14270912C211yesmaximal, characteristic, complemented
4D4⋊#6597C212yesmaximal, complemented
5C29×#555C211yesmaximal, characteristic, complemented
6C29×C4⋊Q8C2211yescharacteristic
7C29×#124—22no—
8C29×#121—22no—
9C4×D4×C29C2212yes—
10C29×#117—22no—
11C2×#90784—22no—
12C2×#90782—22no—
13#11745236C2211yescharacteristic
14#11745157—22no—
15C4⋊#6597C2212yes—
16C4×#6597C2211yescharacteristic
17C2×Q8×C29D412yes—
18C2×D4×C29—22no—
19C2×D4×C29D412yes—
20C22×C4×C29—22no—
21C2×D116—22no—
22C2×D116—21no—
23C4⋊C4×C29—22no—
24C4⋊C4×C29D412yes—
25C22⋊C4×C29—24no—
26C22⋊C4×C29—22no—
27C42×C29—22no—
28C42×C29C2311yescharacteristic
29#90784—44no—
30#90782—44no—
31C2×#6597—21no—
32C2×#6597—22no—
33C23×C29—22no—
34C22×D29—41no—
35Q8×C29—42no—
36D4×C29—22no—
37D4×C29—22no—
38D4×C29—82no—
39C2×C4×C29—22no—
40C2×C4×C29—22no—
41C2×C4×C29—42no—
42C2×C4×C29—42no—
43C2×C4×C29C2×D411yescharacteristic
44C2×C4×C29C2×D412yes—
45D116—42no—
46D116—42no—
47D116—42no—
48#6597—22no—
49#6597—42no—
50#1624—291nomaximal, Sylow
51C22×C29—42no—
52C22×C29—42no—
53C22×C29—42no—
54C22×C29#12011yescharacteristic
55C2×D29—82no—
56C2×D29—81no—
57C4×C29—21no—
58C4×C29—42no—
59C4×C29—42no—
60C4×C29#13512yes—
61C4×C29AGL(1,ℤ/8)12yes—
62#555D2911yescharacteristic, complemented
63#492—291no—
64#459—292no—
65#333—291no—
66D4⋊C8—292no—
67C2×C29—82no—
68C2×C29#44912yes—
69C2×C29#45311yescharacteristic
70D29—161no—
71C2×SD16—582no—
72C2×D8—582no—
73C4⋊Q8C2×D2911yescharacteristic
74#127—291no—
75#124—22no—
76#121—22no—
77C4×D4C2×D2912yes—
78#117—22no—
79C4⋊C8—292no—
80#102—582no—
81C4×C8—291no—
82C29#162411yescharacteristic, complemented, Sylow
83C2×Q8C22⋊D2912yes—
84C2×D4—22no—
85C2×D4—582no—
86C2×D4—581no—
87C2×D4C22⋊D2912yes—
88C22×C4—22no—
89SD16—1164no—
90D8—1164no—
91C2×C8—581no—
92C2×C8—582no—
93C4⋊C4—22no—
94C4⋊C4C22⋊D2912yes—
95C22⋊C4—24no—
96C22⋊C4—22no—
97C42—22no—
98C42C22×D2911yescharacteristic
99C23—22no—
100C23—1161no—
101Q8—42no—
102D4—22no—
103D4—22no—
104D4—82no—
105D4—1162no—
106D4—1162no—
107D4—1162no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4C2×C22⋊D2911yescharacteristic
113C2×C4C2×C22⋊D2912yes—
114C8—582no—
115C8—1162no—
116C22—42no—
117C22—42no—
118C22—42no—
119C22—2322no—
120C22—2321no—
121C22#1174531511yescharacteristic
122C4—21no—
123C4—42no—
124C4—42no—
125C4#1174531112yes—
126C4#1174531212yes—
127C2—82no—
128C2—4641no—
129C2#1427130812yes—
130C2#1427133111yescharacteristic
subgroupG/H
centerC22#11745315
commutator, cosocleC42×C29C23
FrattiniC42C22×D29
FittingC29×#555C2
socleC22×C29#120
hypercenter#555D29
hypocenterC29#1624
LCSG ▷ C42×C29 ▷ C22×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #14271537C2semidirect
2. #14271358C2semidirect
3. #14270912C2semidirect
4. D4⋊#6597C2semidirect
5. C29×#555C2semidirect
62. #555D29semidirect
82. C29#1624semidirect

Maximal subgroups

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

subgroupindex
#142715372
#142713582
#142709122
D4⋊#65972
C29×#5552
#162429