Finite Group Databasebeta

← #37565430

Subgroups — #37565430

Order 24256.

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

normal subgroups only

This group has 198 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, complemented
2#27131300C212yesmaximal, complemented
3#27131246C211yesmaximal, characteristic, complemented
4#27131245C212yesmaximal, complemented
5#120×C379C211yesmaximal, characteristic, complemented
6#20258606—21no—
7#20258605—22no—
8C2×C22⋊D379—22no—
9C2×C22⋊D379—21no—
10C2×C22⋊D379C2212yes—
11D4×D379—22no—
12C22⋊Dic379—22no—
13C22⋊Dic379C2212yescomplemented
14C24×C379C2211yescharacteristic, complemented
15C2×D4×C379—21no—
16C2×D4×C379C2211yescharacteristic
17C22⋊C4×C379—21no—
18C22⋊C4×C379C2211yescharacteristic, complemented
19C22×D379—21no—
20C22×D379D412yescomplemented
21Dic758—21no—
22D1516—21no—
23C22⋊D379—22no—
24C22⋊D379—22no—
25C22⋊D379—22no—
26C22⋊D379—44no—
27C2×Dic379—21no—
28C2×Dic379—22no—
29C2×Dic379D412yescomplemented
30C4×D379—22no—
31C23×C379—21no—
32C23×C379—22no—
33C23×C379—42no—
34C23×C379D411yescharacteristic, complemented
35C23×C379C2311yescharacteristic
36D4×C379—22no—
37D4×C379—42no—
38C2×C4×C379—21no—
39C2×C4×C379D411yescharacteristic, complemented
40C22×C379—22no—
41C22×C379—24no—
42C22×C379—21no—
43C22×C379—41no—
44C22×C379—41no—
45C22×C379—42no—
46C22×C379—41no—
47C22×C379—41no—
48C22×C379C2×D412yes—
49C22×C379C2×D411yescharacteristic
50C2×D379—22no—
51C2×D379—42no—
52C2×D379—81no—
53C4×C379—21no—
54C4×C379—41no—
55Dic379—22no—
56Dic379—42no—
57C2×C379—22no—
58C2×C379—21no—
59C2×C379—41no—
60C2×C379—42no—
61C2×C379#12011yescharacteristic
62D379—42no—
63C379B(4,𝔽2)11yescharacteristic, complemented, Sylow
64B(4,𝔽2)—3791nomaximal, Sylow
65H(5,𝔽2)—3791no—
66#120—3792no—
67#120D37911yescharacteristic, complemented
68#99—3791no—
69#99—3792no—
70C24C2×D37911yescharacteristic, complemented
71#41—7582no—
72#41—7581no—
73C2×D4—21no—
74C2×D4—3792no—
75C2×D4—7582no—
76C2×D4—7582no—
77C2×D4—7581no—
78C2×D4C2×D37911yescharacteristic
79C22⋊C4—21no—
80C22⋊C4—3792no—
81C22⋊C4—7582no—
82C22⋊C4C2×D37911yescharacteristic, complemented
83C23—22no—
84C23—21no—
85C23—42no—
86C23—3792no—
87C23—7581no—
88C23C22⋊D37911yescharacteristic, complemented
89C23C22×D37911yescharacteristic
90Q8—7581no—
91D4—22no—
92D4—42no—
93D4—15164no—
94D4—7581no—
95D4—7582no—
96D4—7582no—
97D4—7582no—
98C2×C4—21no—
99C2×C4—3792no—
100C2×C4—7582no—
101C2×C4—7582no—
102C2×C4—7581no—
103C2×C4C22⋊D37911yescharacteristic, complemented
104C22—22no—
105C22—24no—
106C22—21no—
107C22—41no—
108C22—41no—
109C22—41no—
110C22—42no—
111C22—41no—
112C22—30321no—
113C22—15162no—
114C22—7582no—
115C22D4×D37912yes—
116C22C2×C22⋊D37911yescharacteristic
117C4—21no—
118C4—41no—
119C4—15162no—
120C4—7582no—
121C2—21no—
122C2—22no—
123C2—42no—
124C2—41no—
125C2—15162no—
126C2#2713129911yescharacteristic
subgroupG/H
centerC2#27131299
commutator, cosocleC23×C379C23
FrattiniC23C22×D379
Fitting#120×C379C2
socleC2×C379#120
hypercenter#120D379
hypocenterC379B(4,𝔽2)
LCSG ▷ C23×C379 ▷ C2×C379 ▷ C379
UCSC1 ◁ C2 ◁ C23 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #27131376C2semidirect
2. #27131300C2semidirect
3. #27131246C2semidirect
4. #27131245C2semidirect
5. #120×C379C2semidirect
13. C22⋊Dic379C22semidirect
14. C24×C379C22semidirect
18. C22⋊C4×C379C22semidirect
20. C22×D379D4semidirect
29. C2×Dic379D4semidirect
34. C23×C379D4semidirect
39. C2×C4×C379D4semidirect
63. C379B(4,𝔽2)semidirect
67. #120D379semidirect
70. C24C2×D379semidirect
82. C22⋊C4C2×D379semidirect
88. C23C22⋊D379semidirect
103. C2×C4C22⋊D379semidirect

Maximal subgroups

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

subgroupindex
#271313762
#271313002
#271312462
#271312452
#120×C3792
B(4,𝔽2)379