Finite Group Databasebeta

← #38767967

Subgroups — #38767967

Order 25472.

314 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 57 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#27164653C211yesmaximal, characteristic
2#27164390C211yesmaximal, characteristic, complemented
3#27164309C212yesmaximal, complemented
4#27163745C212yesmaximal, complemented
5#331×C199C211yesmaximal, characteristic, complemented
6C2×D4×D199—22no—
7C2×#15691874—22no—
8#20263955C2211yescharacteristic, complemented
9#20263926—22no—
10#20263923—22no—
11C4×D796C2212yes—
12#20263887—22no—
13#20263877—22no—
14C4×C199⋊C8C2211yescharacteristic, complemented
15#127×C199C2211yescharacteristic
16C4⋊C8×C199C2212yescomplemented
17C23×D199—22no—
18C2×Dic398—21no—
19C2×D796D412yes—
20C2×C22⋊D199—24no—
21D4×D199—44no—
22D4×D199—44no—
23C2×C4×D199—22no—
24#15691874—44no—
25C2×C199⋊C8—21no—
26C4⋊Dic199—22no—
27C4⋊Dic199D412yes—
28C22⋊Dic199—22no—
29(C2×D199)⋊C4—22no—
30C2×D4×C199—22no—
31C2×D4×C199D412yes—
32C2×C8×C199—22no—
33C42×C199C2311yescharacteristic
34C22×D199—22no—
35C22×D199—48no—
36C22×D199—82no—
37Dic398—42no—
38D796—22no—
39D796SD1614yescomplemented
40C22⋊D199—48no—
41C2×Dic199—22no—
42C2×Dic199—41no—
43C4×D199—42no—
44C4×D199—42no—
45C199⋊C8—22no—
46C23×C199—22no—
47D4×C199—42no—
48D4×C199—42no—
49D4×C199—42no—
50C2×C4×C199C2×D412yes—
51C2×C4×C199C2×D411yescharacteristic
52C8×C199—42no—
53C22×C199—44no—
54C22×C199—42no—
55C22×C199#12011yescharacteristic
56C2×D199—24no—
57C2×D199—42no—
58C2×D199—42no—
59C2×D199—88no—
60C4×C199—21no—
61C4×C199C2×SD1612yes—
62C4×C199AGL(1,ℤ/8)12yes—
63Dic199—42no—
64Dic199—81no—
65C2×C199—82no—
66C2×C199#45012yes—
67C2×C199#45311yescharacteristic
68D199—44no—
69C199#162911yescharacteristic, complemented, Sylow
70#1629—1991nomaximal, Sylow
71D42—1991no—
72#492—1991no—
73#476—1992no—
74#331D19911yescharacteristic, complemented
75D4⋊C8—1992no—
76C22×D4—3982no—
77C2×SD16—3982no—
78C4⋊Q8—1991no—
79#127C2×D19911yescharacteristic
80#121—3982no—
81#120—3982no—
82C4×D4—1992no—
83#106—3982no—
84C4⋊C8C2×D19912yescomplemented
85#102—3982no—
86C4×C8—1991no—
87C24—3982no—
88C2×Q8—3981no—
89C2×D4—22no—
90C2×D4—7964no—
91C2×D4—7964no—
92C2×D4—3984no—
93C2×D4—1992no—
94C2×D4D79612yes—
95C22×C4—3982no—
96SD16—7964no—
97C2×C8—22no—
98C2×C8—3981no—
99C4⋊C4—3982no—
100C4⋊C4—1992no—
101C22⋊C4—3982no—
102C22⋊C4—3982no—
103C42C22×D19911yescharacteristic
104C23—22no—
105C23—7968no—
106C23—15922no—
107C23—3982no—
108Q8—7962no—
109D4—42no—
110D4—42no—
111D4—42no—
112D4—7968no—
113D4—3982no—
114D4—1994no—
115C2×C4—7962no—
116C2×C4—7962no—
117C2×C4—7961no—
118C2×C4—3982no—
119C2×C4D4×D19912yes—
120C2×C4C2×D79611yescharacteristic
121C8—42no—
122C8—3982no—
123C22—44no—
124C22—42no—
125C22—7962no—
126C22—7962no—
127C22—15928no—
128C22—3984no—
129C22#2026392511yescharacteristic
130C4—21no—
131C4—7962no—
132C4—15921no—
133C4SD16×D19912yes—
134C4#2026397912yes—
135C2—82no—
136C2—7964no—
137C2#2716417012yes—
138C2#2716418811yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#331×C199C2
socleC22×C199#120
prime radical#27164653C2
hypercenter#331D199
hypocenterC199#1629
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #27164390C2semidirect
3. #27164309C2semidirect
4. #27163745C2semidirect
5. #331×C199C2semidirect
8. #20263955C22semidirect
14. C4×C199⋊C8C22semidirect
16. C4⋊C8×C199C22semidirect
39. D796SD16semidirect
69. C199#1629semidirect
74. #331D199semidirect
84. C4⋊C8C2×D199semidirect

Maximal subgroups

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

subgroupindex
#271646532
#271643902
#271643092
#271637452
#331×C1992
#1629199