Finite Group Databasebeta

← #16818816

Subgroups — #16818816

Order 3712.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#14271911C211yesmaximal, characteristic
2#14271533C211yesmaximal, characteristic, complemented
3#14271523C212yesmaximal, complemented
4#14270902C212yesmaximal, complemented
5C29×#330C211yesmaximal, characteristic, complemented
6C29×#125C2211yescharacteristic
7#11745258—22no—
8#11745257—22no—
9#11745256—22no—
10#11745237C2212yes—
11#11745231C2211yescharacteristic
12C29×#97C2212yes—
13#11745169—22no—
14#11745164—22no—
15#11745158—22no—
16#11745151C2211yescharacteristic
17C2×D116—22no—
18C2×D116D412yes—
19C2×C4×D29—22no—
20C2×Dic58—21no—
21C2×Dic58D412yes—
22C2×C8×C29—22no—
23C4⋊C4×C29—22no—
24C4⋊C4×C29D412yes—
25C42×C29C2311yescharacteristic
26#90781—22no—
27#90781—22no—
28#90781—22no—
29C4⋊Dic29—22no—
30#90779—22no—
31C2×#6597—21no—
32C22×D29—22no—
33C2×C4×C29—22no—
34C2×C4×C29C2×D411yescharacteristic
35C2×C4×C29C2×D412yes—
36C2×Dic29—22no—
37C2×Dic29—41no—
38D116—42no—
39D116—82no—
40C4×D29—82no—
41Dic58—42no—
42Dic58—42no—
43C8×C29—42no—
44#6597—41no—
45#1662—291nomaximal, Sylow
46C22×C29#12011yescharacteristic
47C2×D29—42no—
48C2×D29—42no—
49C2×D29—42no—
50C4×C29—21no—
51C4×C29—22no—
52C4×C29—42no—
53Dic29—42no—
54Dic29—81no—
55#562—291no—
56#491—291no—
57#489—292no—
58#330D2911yescharacteristic, complemented
59#329—292no—
60C2×C29#45411yescharacteristic
61C2×C29#45512yes—
62D29—82no—
63C4⋊Q8—291no—
64#125C2×D2911yescharacteristic
65#124—292no—
66#123—582no—
67C22⋊Q8—582no—
68#121—582no—
69#103—582no—
70#103—582no—
71#102—582no—
72#97—291no—
73#97C2×D2912yes—
74C29#166211yescharacteristic, complemented, Sylow
75C2×Q8—292no—
76C2×Q8—581no—
77C2×D4—292no—
78C2×D4—582no—
79C22×C4—582no—
80C2×C8—22no—
81C2×C8—581no—
82C4⋊C4—22no—
83C4⋊C4—582no—
84C4⋊C4—582no—
85C4⋊C4D11612yes—
86C22⋊C4—582no—
87C22⋊C4—582no—
88C22⋊C4—582no—
89C42C22×D2911yescharacteristic
90C23—582no—
91Q8—1162no—
92Q8—1162no—
93D4—1162no—
94D4—2322no—
95C2×C4—22no—
96C2×C4—582no—
97C2×C4—1161no—
98C2×C4—2322no—
99C2×C4C2×D11611yescharacteristic
100C2×C4D4×D2912yes—
101C8—42no—
102C8—1161no—
103C22—1162no—
104C22—1162no—
105C22—1162no—
106C22#1174524411yescharacteristic
107C4—21no—
108C4—22no—
109C4—42no—
110C4—1162no—
111C4—2321no—
112C2—2322no—
113C2#1427133811yescharacteristic
114C2#1427134212yes—
subgroupG/H
centerC22#11745244
commutator, cosocleC42×C29C23
FrattiniC42C22×D29
FittingC29×#330C2
socleC22×C29#120
prime radical#14271911C2
hypercenter#330D29
hypocenterC29#1662
LCSG ▷ C42×C29 ▷ C22×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
2. #14271533C2semidirect
3. #14271523C2semidirect
4. #14270902C2semidirect
5. C29×#330C2semidirect
58. #330D29semidirect
74. C29#1662semidirect

Maximal subgroups

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

subgroupindex
#142719112
#142715332
#142715232
#142709022
C29×#3302
#166229