Finite Group Databasebeta

← #38768174

Subgroups — #38768174

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#27164766C211yesmaximal, characteristic
2#27164388C211yesmaximal, characteristic, complemented
3#27164378C212yesmaximal, complemented
4#27163757C212yesmaximal, complemented
5#330×C199C211yesmaximal, characteristic, complemented
6#20263955C2211yescharacteristic
7#20263943C2212yes—
8#20263940—22no—
9#20263933—22no—
10#20263930—22no—
11#20263880—22no—
12#20263878—22no—
13#20263877—22no—
14#20263866C2211yescharacteristic
15#125×C199C2211yescharacteristic
16#97×C199C2212yes—
17C2×Dic398—21no—
18C2×Dic398D412yes—
19C2×D796—22no—
20C2×D796D412yes—
21C2×C4×D199—22no—
22C2×C199⋊C8—21no—
23C4⋊Dic199—22no—
24Dic199⋊C4—22no—
25(C2×D199)⋊C4—22no—
26(C2×D199)⋊C4—22no—
27(C2×D199)⋊C4—22no—
28C2×C8×C199—22no—
29C4⋊C4×C199—22no—
30C4⋊C4×C199D412yes—
31C42×C199C2311yescharacteristic
32C22×D199—22no—
33Dic398—42no—
34Dic398—42no—
35D796—42no—
36D796—82no—
37C2×Dic199—22no—
38C2×Dic199—41no—
39C4×D199—82no—
40C199⋊C8—41no—
41C2×C4×C199—22no—
42C2×C4×C199C2×D412yes—
43C2×C4×C199C2×D411yescharacteristic
44C8×C199—42no—
45C22×C199#12011yescharacteristic
46C2×D199—42no—
47C2×D199—42no—
48C2×D199—42no—
49C4×C199—21no—
50C4×C199—22no—
51C4×C199—42no—
52Dic199—42no—
53Dic199—81no—
54C2×C199#45411yescharacteristic
55C2×C199#45512yes—
56D199—82no—
57C199#166211yescharacteristic, complemented, Sylow
58#1662—1991nomaximal, Sylow
59#562—1991no—
60#491—1991no—
61#489—1992no—
62#330D19911yescharacteristic, complemented
63#329—1992no—
64C4⋊Q8—1991no—
65#125C2×D19911yescharacteristic
66#124—1992no—
67#123—3982no—
68C22⋊Q8—3982no—
69#121—3982no—
70#103—3982no—
71#103—3982no—
72#102—3982no—
73#97—1991no—
74#97C2×D19912yes—
75C2×Q8—3981no—
76C2×Q8—1992no—
77C2×D4—3982no—
78C2×D4—1992no—
79C22×C4—3982no—
80C2×C8—22no—
81C2×C8—3981no—
82C4⋊C4—22no—
83C4⋊C4—3982no—
84C4⋊C4—3982no—
85C4⋊C4D79612yes—
86C22⋊C4—3982no—
87C22⋊C4—3982no—
88C22⋊C4—3982no—
89C42C22×D19911yescharacteristic
90C23—3982no—
91Q8—7962no—
92Q8—7962no—
93D4—7962no—
94D4—15922no—
95C2×C4—22no—
96C2×C4—7961no—
97C2×C4—15922no—
98C2×C4—3982no—
99C2×C4D4×D19912yes—
100C2×C4C2×D79611yescharacteristic
101C8—42no—
102C8—7961no—
103C22—7962no—
104C22—7962no—
105C22—7962no—
106C22#2026392511yescharacteristic
107C4—21no—
108C4—22no—
109C4—42no—
110C4—7962no—
111C4—15921no—
112C2—15922no—
113C2#2716419311yescharacteristic
114C2#2716419712yes—
subgroupG/H
centerC22#20263925
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#330×C199C2
socleC22×C199#120
prime radical#27164766C2
hypercenter#330D199
hypocenterC199#1662
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
2. #27164388C2semidirect
3. #27164378C2semidirect
4. #27163757C2semidirect
5. #330×C199C2semidirect
57. C199#1662semidirect
62. #330D199semidirect

Maximal subgroups

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

subgroupindex
#271647662
#271643882
#271643782
#271637572
#330×C1992
#1662199