Finite Group Databasebeta

← #38770745

Subgroups — #38770745

Order 25472.

154 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 51 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#27164453C211yesmaximal, characteristic
2#27164140C211yesmaximal, characteristic, complemented
3#27164129C211yesmaximal, characteristic, complemented
4#27163840C211yesmaximal, characteristic, complemented
5#27163832C211yesmaximal, characteristic, complemented
6#27163795C211yesmaximal, characteristic, complemented
7#360×C199C211yesmaximal, characteristic, complemented
8C2×D1592C2211yescharacteristic
9#20263962—21no—
10#20263917—21no—
11#20263898C412yescomplemented
12#20263896C412yescomplemented
13#20263895C2211yescharacteristic
14#20263891C2211yescharacteristic
15#20263882—21no—
16#20263875—21no—
17#20263865C2211yescharacteristic
18#135×C199C2211yescharacteristic
19M32×C199C2211yescharacteristic
20#106×C199C2211yescharacteristic
21C2×D796—21no—
22C2×C4×D199—41no—
23D1592—21no—
24D1592C2×C412yes—
25#15691870—21no—
26#15691870—41no—
27C2×C199⋊C8—41no—
28C2×C199⋊C8D411yescharacteristic
29(C2×D199)⋊C4—41no—
30C4×Dic199D411yescharacteristic
31C199⋊C16C2×C412yescomplemented
32#41×C199—21no—
33Q16×C199C2×C411yescharacteristic, complemented
34SD16×C199—21no—
35D8×C199C2×C411yescharacteristic, complemented
36C2×C8×C199C2311yescharacteristic
37C4⋊C4×C199—21no—
38C16×C199—21no—
39C22×D199—41no—
40D796—22no—
41D796—41no—
42C2×Dic199—21no—
43C4×D199—82no—
44C199⋊C8—21no—
45C199⋊C8—41no—
46Q8×C199—21no—
47D4×C199—21no—
48D4×C199—41no—
49C2×C4×C199—41no—
50C2×C4×C199—41no—
51C2×C4×C199C2×D411yescharacteristic
52C8×C199C22×C411yescharacteristic
53C8×C199#4111yescharacteristic
54C22×C199—41no—
55C22×C199AGL(1,ℤ/8)11yescharacteristic
56C2×D199—42no—
57C2×D199—81no—
58C2×D199—81no—
59C4×C199—41no—
60C4×C199—81no—
61C4×C199C4×D411yescharacteristic
62C4×C199AGL(1,ℤ/8)11yescharacteristic
63Dic199—41no—
64C2×C199—21no—
65C2×C199—81no—
66C2×C199#44211yescharacteristic
67D199—82no—
68C199AGL(1,ℤ/16)11yescharacteristic, complemented, Sylow
69AGL(1,ℤ/16)—1991nomaximal, Sylow
70#509—1991no—
71#444—1991no—
72#442—1991no—
73#365—1991no—
74#361—1991no—
75#360D19911yescharacteristic, complemented
76#347—1991no—
77#135C2×D19911yescharacteristic
78C2×D8—1991no—
79#131—3981no—
80C4×D4—3981no—
81SD32—1992no—
82D16—1992no—
83M32—1991no—
84M32C2×D19911yescharacteristic
85#108—1991no—
86#106C2×D19911yescharacteristic
87#104—3981no—
88#102—3981no—
89#97—1991no—
90#41—21no—
91C2×D4—3981no—
92C22×C4—7961no—
93Q16C4×D19911yescharacteristic, complemented
94SD16—21no—
95D8—3981no—
96D8—1992no—
97D8C4×D19911yescharacteristic, complemented
98M16—7961no—
99M16—3981no—
100C2×C8—7961no—
101C2×C8—1991no—
102C2×C8C22×D19911yescharacteristic
103C4⋊C4—21no—
104C22⋊C4—7961no—
105C42—1991no—
106C16—21no—
107C16—1992no—
108C23—7961no—
109Q8—21no—
110D4—21no—
111D4—41no—
112D4—7961no—
113D4—3982no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—15922no—
117C2×C4—3981no—
118C2×C4D4×D19911yescharacteristic
119C8—7961no—
120C8—3981no—
121C8C2×C4×D19911yescharacteristic
122C8#1569188511yescharacteristic
123C22—41no—
124C22—7962no—
125C22—15921no—
126C22—15921no—
127C22#2026397911yescharacteristic
128C4—41no—
129C4—81no—
130C4—7961no—
131C4(C22⋊D199)⋊C411yescharacteristic
132C4#2026398211yescharacteristic
133C2—21no—
134C2—81no—
135C2—15922no—
136C2#2716412711yescharacteristic
subgroupG/H
centerC2#27164127
commutatorC8×C199C22×C4
FrattiniC2×C8C22×D199
Fitting#360×C199C2
socleC2×C199#442
prime radical#27164453C2
cosocleC2×C8×C199C23
hypercenter#360D199
hypocenterC199AGL(1,ℤ/16)
LCSG ▷ C8×C199 ▷ C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #360

Complements

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

NH ≅ G/Ntype
2. #27164140C2semidirect
3. #27164129C2semidirect
4. #27163840C2semidirect
5. #27163832C2semidirect
6. #27163795C2semidirect
7. #360×C199C2semidirect
11. #20263898C4semidirect
12. #20263896C4semidirect
31. C199⋊C16C2×C4semidirect
33. Q16×C199C2×C4semidirect
35. D8×C199C2×C4semidirect
68. C199AGL(1,ℤ/16)semidirect
75. #360D199semidirect
93. Q16C4×D199semidirect
97. D8C4×D199semidirect

Maximal subgroups

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

subgroupindex
#271644532
#271641402
#271641292
#271638402
#271638322
#271637952
#360×C1992
AGL(1,ℤ/16)199