Finite Group Databasebeta

← #38767933

Subgroups — #38767933

Order 25472.

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

normal subgroups only

This group has 240 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#27164392C211yesmaximal, characteristic, complemented
2#27164213C212yesmaximal, complemented
3#27163767C211yesmaximal, characteristic, complemented
4D4⋊(C199⋊C8)C212yesmaximal, complemented
5#555×C199C211yesmaximal, characteristic, complemented
6C2×#15691873—22no—
7C2×#15691871—22no—
8#20263952C2211yescharacteristic
9C4⋊(C199⋊C8)C2212yes—
10#20263875—22no—
11C4×C199⋊C8C2211yescharacteristic
12C4⋊Q8×C199C2211yescharacteristic
13#124×C199—22no—
14#121×C199—22no—
15C4×D4×C199C2212yes—
16#117×C199—22no—
17C2×D796—22no—
18C2×D796—21no—
19#15691873—44no—
20#15691871—44no—
21C2×C199⋊C8—21no—
22C2×C199⋊C8—22no—
23C2×Q8×C199D412yes—
24C2×D4×C199—22no—
25C2×D4×C199D412yes—
26C22×C4×C199—22no—
27C4⋊C4×C199—22no—
28C4⋊C4×C199D412yes—
29C22⋊C4×C199—24no—
30C22⋊C4×C199—22no—
31C42×C199—22no—
32C42×C199C2311yescharacteristic
33C22×D199—41no—
34D796—42no—
35D796—42no—
36D796—42no—
37C199⋊C8—22no—
38C199⋊C8—42no—
39C23×C199—22no—
40Q8×C199—42no—
41D4×C199—22no—
42D4×C199—22no—
43D4×C199—82no—
44C2×C4×C199—22no—
45C2×C4×C199—22no—
46C2×C4×C199—42no—
47C2×C4×C199—42no—
48C2×C4×C199C2×D411yescharacteristic
49C2×C4×C199C2×D412yes—
50C22×C199—42no—
51C22×C199—42no—
52C22×C199—42no—
53C22×C199#12011yescharacteristic
54C2×D199—82no—
55C2×D199—81no—
56C4×C199—21no—
57C4×C199—42no—
58C4×C199—42no—
59C4×C199#13512yes—
60C4×C199AGL(1,ℤ/8)12yes—
61C2×C199—82no—
62C2×C199#44912yes—
63C2×C199#45311yescharacteristic
64D199—161no—
65C199#162411yescharacteristic, complemented, Sylow
66#1624—1991nomaximal, Sylow
67#555D19911yescharacteristic, complemented
68#492—1991no—
69#459—1992no—
70#333—1991no—
71D4⋊C8—1992no—
72C2×SD16—3982no—
73C2×D8—3982no—
74C4⋊Q8C2×D19911yescharacteristic
75#127—1991no—
76#124—22no—
77#121—22no—
78C4×D4C2×D19912yes—
79#117—22no—
80C4⋊C8—1992no—
81#102—3982no—
82C4×C8—1991no—
83C2×Q8C22⋊D19912yes—
84C2×D4—22no—
85C2×D4—3982no—
86C2×D4—3981no—
87C2×D4C22⋊D19912yes—
88C22×C4—22no—
89SD16—7964no—
90D8—7964no—
91C2×C8—3982no—
92C2×C8—3981no—
93C4⋊C4—22no—
94C4⋊C4C22⋊D19912yes—
95C22⋊C4—24no—
96C22⋊C4—22no—
97C42—22no—
98C42C22×D19911yescharacteristic
99C23—22no—
100C23—7961no—
101Q8—42no—
102D4—22no—
103D4—22no—
104D4—82no—
105D4—7962no—
106D4—7962no—
107D4—7962no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4C2×C22⋊D19911yescharacteristic
113C2×C4C2×C22⋊D19912yes—
114C8—7962no—
115C8—3982no—
116C22—42no—
117C22—42no—
118C22—42no—
119C22—15922no—
120C22—15921no—
121C22#2026392411yescharacteristic
122C4—21no—
123C4—42no—
124C4—42no—
125C4#2026397512yes—
126C4#2026398012yes—
127C2—82no—
128C2—31841no—
129C2#2716416312yes—
130C2#2716418611yescharacteristic
subgroupG/H
centerC22#20263924
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#555×C199C2
socleC22×C199#120
hypercenter#555D199
hypocenterC199#1624
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #27164392C2semidirect
2. #27164213C2semidirect
3. #27163767C2semidirect
4. D4⋊(C199⋊C8)C2semidirect
5. #555×C199C2semidirect
65. C199#1624semidirect
67. #555D199semidirect

Maximal subgroups

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

subgroupindex
#271643922
#271642132
#271637672
D4⋊(C199⋊C8)2
#555×C1992
#1624199