Finite Group Databasebeta

← #35203412

Subgroups — #35203412

Order 20288.

160 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 160 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#23778825C211yesmaximal, characteristic
2#23778763C212yesmaximal, complemented
3#23778695C211yesmaximal, characteristic, complemented
4#23778694C212yesmaximal, complemented
5#123×C317C211yesmaximal, characteristic, complemented
6#19099574—21no—
7#19099573—22no—
8C2×C22⋊D317—21no—
9C2×C22⋊D317C2212yes—
10D4×D317—22no—
11#19099553—22no—
12C22⋊Dic317C2212yescomplemented
13#19099551—22no—
14C2×D4×C317C2211yescharacteristic
15C22×C4×C317C2211yescharacteristic, complemented
16C4⋊C4×C317—21no—
17C22⋊C4×C317—21no—
18C22⋊C4×C317C2211yescharacteristic, complemented
19C22×D317—21no—
20C22×D317D412yes—
21Dic634—21no—
22D1268—21no—
23C22⋊D317—22no—
24C22⋊D317—22no—
25C22⋊D317—22no—
26C2×Dic317—21no—
27C2×Dic317—22no—
28C2×Dic317D412yes—
29C4×D317—22no—
30C23×C317D411yescharacteristic
31C23×C317C2311yescharacteristic
32D4×C317—22no—
33C2×C4×C317—21no—
34C2×C4×C317—21no—
35C2×C4×C317—22no—
36C2×C4×C317D411yescharacteristic
37C22×C317—21no—
38C22×C317—41no—
39C22×C317—41no—
40C22×C317C2×D412yes—
41C22×C317C2×D411yescharacteristic
42C2×D317—22no—
43C2×D317—42no—
44C2×D317—81no—
45C4×C317—21no—
46C4×C317—41no—
47C4×C317—41no—
48Dic317—22no—
49Dic317—42no—
50C2×C317—22no—
51C2×C317—21no—
52C2×C317—41no—
53C2×C317#12011yescharacteristic
54D317—42no—
55C317#45811yescharacteristic, complemented, Sylow
56#458—3171nomaximal, Sylow
57H(5,𝔽2)—3171no—
58#123—3172no—
59#123D31711yescharacteristic, complemented
60#99—3171no—
61#99—3172no—
62#41—6342no—
63#41—6341no—
64C2×D4—3172no—
65C2×D4—6342no—
66C2×D4—6341no—
67C2×D4C2×D31711yescharacteristic
68C22×C4C2×D31711yescharacteristic, complemented
69C4⋊C4—21no—
70C4⋊C4—6342no—
71C22⋊C4—21no—
72C22⋊C4—3172no—
73C22⋊C4—6342no—
74C22⋊C4C2×D31711yescharacteristic, complemented
75C23—3172no—
76C23—6341no—
77C23C22⋊D31711yescharacteristic
78C23C22×D31711yescharacteristic
79Q8—6341no—
80D4—22no—
81D4—6341no—
82D4—6342no—
83D4—6342no—
84D4—6342no—
85C2×C4—22no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—3172no—
89C2×C4—6342no—
90C2×C4—6341no—
91C2×C4—6342no—
92C2×C4C22⋊D31711yescharacteristic
93C22—21no—
94C22—41no—
95C22—41no—
96C22—6342no—
97C22—25361no—
98C22—12682no—
99C22D4×D31712yes—
100C22C2×C22⋊D31711yescharacteristic
101C4—21no—
102C4—41no—
103C4—41no—
104C4—6342no—
105C4—12682no—
106C2—21no—
107C2—22no—
108C2—41no—
109C2—12682no—
110C2#2377874811yescharacteristic
subgroupG/H
centerC2#23778748
commutator, cosocleC23×C317C23
FrattiniC23C22×D317
Fitting#123×C317C2
socleC2×C317#120
prime radical#23778825C2
hypercenter#123D317
hypocenterC317#458
LCSG ▷ C23×C317 ▷ C2×C317 ▷ C317
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
2. #23778763C2semidirect
3. #23778695C2semidirect
4. #23778694C2semidirect
5. #123×C317C2semidirect
12. C22⋊Dic317C22semidirect
15. C22×C4×C317C22semidirect
18. C22⋊C4×C317C22semidirect
55. C317#458semidirect
59. #123D317semidirect
68. C22×C4C2×D317semidirect
74. C22⋊C4C2×D317semidirect

Maximal subgroups

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

subgroupindex
#237788252
#237787632
#237786952
#237786942
#123×C3172
#458317