Finite Group Databasebeta

← #37565433

Subgroups — #37565433

Order 24256.

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#27131376C211yesmaximal, characteristic
2#27131314C212yesmaximal, complemented
3#27131246C211yesmaximal, characteristic, complemented
4#27131245C212yesmaximal, complemented
5#123×C379C211yesmaximal, characteristic, complemented
6#20258606—21no—
7#20258605—22no—
8C2×C22⋊D379—21no—
9C2×C22⋊D379C2212yes—
10D4×D379—22no—
11Dic379⋊C4—22no—
12C22⋊Dic379C2212yescomplemented
13(C2×D379)⋊C4—22no—
14C2×D4×C379C2211yescharacteristic
15C22×C4×C379C2211yescharacteristic, complemented
16C4⋊C4×C379—21no—
17C22⋊C4×C379—21no—
18C22⋊C4×C379C2211yescharacteristic, complemented
19C22×D379—21no—
20C22×D379D412yes—
21Dic758—21no—
22D1516—21no—
23C22⋊D379—22no—
24C22⋊D379—22no—
25C22⋊D379—22no—
26C2×Dic379—21no—
27C2×Dic379—22no—
28C2×Dic379D412yes—
29C4×D379—22no—
30C23×C379D411yescharacteristic
31C23×C379C2311yescharacteristic
32D4×C379—22no—
33C2×C4×C379—21no—
34C2×C4×C379—21no—
35C2×C4×C379—22no—
36C2×C4×C379D411yescharacteristic
37C22×C379—21no—
38C22×C379—41no—
39C22×C379—41no—
40C22×C379C2×D412yes—
41C22×C379C2×D411yescharacteristic
42C2×D379—22no—
43C2×D379—42no—
44C2×D379—81no—
45C4×C379—21no—
46C4×C379—41no—
47C4×C379—41no—
48Dic379—22no—
49Dic379—42no—
50C2×C379—22no—
51C2×C379—21no—
52C2×C379—41no—
53C2×C379#12011yescharacteristic
54D379—42no—
55C379#45811yescharacteristic, complemented, Sylow
56#458—3791nomaximal, Sylow
57H(5,𝔽2)—3791no—
58#123—3792no—
59#123D37911yescharacteristic, complemented
60#99—3791no—
61#99—3792no—
62#41—7582no—
63#41—7581no—
64C2×D4—3792no—
65C2×D4—7582no—
66C2×D4—7581no—
67C2×D4C2×D37911yescharacteristic
68C22×C4C2×D37911yescharacteristic, complemented
69C4⋊C4—21no—
70C4⋊C4—7582no—
71C22⋊C4—21no—
72C22⋊C4—3792no—
73C22⋊C4—7582no—
74C22⋊C4C2×D37911yescharacteristic, complemented
75C23—3792no—
76C23—7581no—
77C23C22⋊D37911yescharacteristic
78C23C22×D37911yescharacteristic
79Q8—7581no—
80D4—22no—
81D4—7581no—
82D4—7582no—
83D4—7582no—
84D4—7582no—
85C2×C4—22no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—3792no—
89C2×C4—7582no—
90C2×C4—7582no—
91C2×C4—7581no—
92C2×C4C22⋊D37911yescharacteristic
93C22—21no—
94C22—41no—
95C22—41no—
96C22—30321no—
97C22—15162no—
98C22—7582no—
99C22D4×D37912yes—
100C22C2×C22⋊D37911yescharacteristic
101C4—21no—
102C4—41no—
103C4—41no—
104C4—15162no—
105C4—7582no—
106C2—21no—
107C2—22no—
108C2—41no—
109C2—15162no—
110C2#2713129911yescharacteristic
subgroupG/H
centerC2#27131299
commutator, cosocleC23×C379C23
FrattiniC23C22×D379
Fitting#123×C379C2
socleC2×C379#120
prime radical#27131376C2
hypercenter#123D379
hypocenterC379#458
LCSG ▷ C23×C379 ▷ C2×C379 ▷ C379
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
2. #27131314C2semidirect
3. #27131246C2semidirect
4. #27131245C2semidirect
5. #123×C379C2semidirect
12. C22⋊Dic379C22semidirect
15. C22×C4×C379C22semidirect
18. C22⋊C4×C379C22semidirect
55. C379#458semidirect
59. #123D379semidirect
68. C22×C4C2×D379semidirect
74. C22⋊C4C2×D379semidirect

Maximal subgroups

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

subgroupindex
#271313762
#271313142
#271312462
#271312452
#123×C3792
#458379