Finite Group Databasebeta

← #38768162

Subgroups — #38768162

Order 25472.

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#27164768C211yesmaximal, characteristic
2#27164427C212yesmaximal, complemented
3#27164413C211yesmaximal, characteristic
4#27163759C212yesmaximal, complemented
5#329×C199C211yesmaximal, characteristic
6C2×#15691876—21no—
7C2×#15691874—21no—
8#20263955C2211yescharacteristic
9#20263953—21no—
10#20263948—22no—
11#20263947C2212yes—
12#20263936—21no—
13#20263932—22no—
14#20263889—22no—
15#20263887—22no—
16#20263866C2211yescharacteristic
17#124×C199C2211yescharacteristic
18#97×C199C2212yes—
19C2×Dic398—21no—
20C2×Dic398—21no—
21C22×Dic199—21no—
22#15691876—42no—
23#15691874—42no—
24C2×C199⋊C8—21no—
25C4⋊Dic199—22no—
26C4⋊Dic199D412yes—
27C4⋊Dic199D412yes—
28Dic199⋊C4—22no—
29Dic199⋊C4—22no—
30Dic199⋊C4—22no—
31C22⋊Dic199—21no—
32C22⋊Dic199—22no—
33C4×Dic199—21no—
34C2×Q8×C199D411yescharacteristic
35C2×D4×C199D411yescharacteristic
36C2×C8×C199—22no—
37C22⋊C4×C199—22no—
38C42×C199C2311yescharacteristic
39Dic398—41no—
40Dic398—41no—
41Dic398—81no—
42C2×Dic199—22no—
43C2×Dic199—22no—
44C2×Dic199—41no—
45C2×Dic199—81no—
46C199⋊C8—22no—
47C23×C199—21no—
48Q8×C199—41no—
49D4×C199—41no—
50C2×C4×C199—21no—
51C2×C4×C199C2×D412yes—
52C2×C4×C199C2×D411yescharacteristic
53C8×C199—24no—
54C22×C199—41no—
55C22×C199—42no—
56C22×C199#12011yescharacteristic
57C4×C199—21no—
58C4×C199—22no—
59C4×C199—41no—
60Dic199—42no—
61Dic199—42no—
62Dic199—81no—
63C2×C199—81no—
64C2×C199#45311yescharacteristic
65C2×C199#45612yes—
66C199#166011yescharacteristic, complemented, Sylow
67#1660—1991nomaximal, Sylow
68#563—1991no—
69#501—1992no—
70#497—1991no—
71#330—1992no—
72#329D19911yescharacteristic
73C2×Q16—3981no—
74C2×SD16—3981no—
75C4⋊Q8—3981no—
76C4⋊Q8—1991no—
77#126—3982no—
78#125—1992no—
79#124C2×D19911yescharacteristic
80#123—3981no—
81C22⋊Q8—3982no—
82#107—3982no—
83#106—3982no—
84#97—1991no—
85#97C2×D19912yes—
86C2×Q8—3981no—
87C2×Q8—3981no—
88C2×Q8D79611yescharacteristic
89C2×D4D79611yescharacteristic
90C22×C4—3981no—
91Q16—7962no—
92SD16—7962no—
93C2×C8—22no—
94C2×C8—3981no—
95C4⋊C4—3982no—
96C4⋊C4—3982no—
97C4⋊C4—3982no—
98C4⋊C4—3982no—
99C4⋊C4—1992no—
100C4⋊C4—1992no—
101C22⋊C4—22no—
102C22⋊C4—3982no—
103C22⋊C4—3981no—
104C42—3981no—
105C42C22×D19911yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—7961no—
109Q8—7961no—
110Q8—15921no—
111D4—41no—
112C2×C4—21no—
113C2×C4—7961no—
114C2×C4—15921no—
115C2×C4—3982no—
116C2×C4—3982no—
117C2×C4D4×D19912yes—
118C2×C4C2×D79611yescharacteristic
119C8—24no—
120C8—3982no—
121C22—41no—
122C22—42no—
123C22#2026392511yescharacteristic
124C4—21no—
125C4—22no—
126C4—41no—
127C4—7962no—
128C4—7962no—
129C4—15921no—
130C2—81no—
131C2#2716418811yescharacteristic
132C2#2716420212yes—
subgroupG/H
centerC22#20263925
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#329×C199C2
socleC22×C199#120
prime radicalC2×D4×C199D4
hypercenter#329D199
hypocenterC199#1660
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #27164427C2semidirect
4. #27163759C2semidirect
66. C199#1660semidirect

Maximal subgroups

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

subgroupindex
#271647682
#271644272
#271644132
#271637592
#329×C1992
#1660199