Finite Group Databasebeta

← #38770748

Subgroups — #38770748

Order 25472.

154 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 53 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#27164131C211yesmaximal, characteristic, complemented
3#27163842C211yesmaximal, characteristic, complemented
4#27163830C211yesmaximal, characteristic, complemented
5#27163828C211yesmaximal, characteristic, complemented
6#27163796C211yesmaximal, characteristic, complemented
7#444×C199C211yesmaximal, characteristic, complemented
8C2×D1592C2211yescharacteristic
9C4×D796—21no—
10#20263898C412yescomplemented
11#20263896C412yescomplemented
12#20263895C2211yescharacteristic
13#20263895C2211yescharacteristic
14#20263887C2211yescharacteristic
15#20263877—21no—
16#135×C199C2211yescharacteristic
17#131×C199—21no—
18#108×C199C2211yescharacteristic
19#104×C199—21no—
20#97×C199C2211yescharacteristic
21C2×D796—21no—
22C2×C4×D199—41no—
23D1592—21no—
24D1592C2×C412yes—
25C4⋊Dic199—21no—
26(C2×D199)⋊C4—41no—
27C199⋊C16—21no—
28C199⋊C16C2×C412yescomplemented
29#41×C199—21no—
30Q16×C199C2×C411yescharacteristic, complemented
31SD16×C199—21no—
32D8×C199C2×C411yescharacteristic, complemented
33M16×C199—21no—
34M16×C199—41no—
35C2×C8×C199—41no—
36C2×C8×C199D411yescharacteristic
37C2×C8×C199C2311yescharacteristic
38C42×C199D411yescharacteristic
39C22×D199—41no—
40D796—22no—
41D796—41no—
42C2×Dic199—41no—
43C4×D199—82no—
44Q8×C199—21no—
45D4×C199—21no—
46D4×C199—41no—
47C2×C4×C199—21no—
48C2×C4×C199—41no—
49C2×C4×C199C2×D411yescharacteristic
50C8×C199—21no—
51C8×C199—41no—
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—41no—
61C4×C199C4×D411yescharacteristic
62C4×C199AGL(1,ℤ/8)11yescharacteristic
63Dic199—81no—
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#444D19911yescharacteristic, complemented
72#442—1991no—
73#365—1991no—
74#361—1991no—
75#360—1991no—
76#347—1991no—
77#135C2×D19911yescharacteristic
78C2×D8—1991no—
79#131—21no—
80C4×D4—3981no—
81SD32—1992no—
82D16—1992no—
83M32—1991no—
84M32—1991no—
85#108C2×D19911yescharacteristic
86#106—1991no—
87#104—21no—
88#102—3981no—
89#97C2×D19911yescharacteristic
90#41—21no—
91C2×D4—3981no—
92C22×C4—7961no—
93Q16C4×D19911yescharacteristic, complemented
94SD16—21no—
95D8—3981no—
96D8—1992no—
97D8C4×D19911yescharacteristic, complemented
98M16—21no—
99M16—41no—
100C2×C8—41no—
101C2×C8C22⋊D19911yescharacteristic
102C2×C8C22×D19911yescharacteristic
103C4⋊C4—3981no—
104C22⋊C4—7961no—
105C42C22⋊D19911yescharacteristic
106C16—3981no—
107C16—1992no—
108C23—7961no—
109Q8—21no—
110D4—21no—
111D4—41no—
112D4—7961no—
113D4—3982no—
114C2×C4—21no—
115C2×C4—41no—
116C2×C4—7961no—
117C2×C4—15922no—
118C2×C4C2×C22⋊D19911yescharacteristic
119C8—21no—
120C8—41no—
121C8C2×C4×D19911yescharacteristic
122C8#1569188611yescharacteristic
123C22—41no—
124C22—7962no—
125C22—15921no—
126C22—15921no—
127C22#2026398011yescharacteristic
128C4—41no—
129C4—41no—
130C4—15921no—
131C4C4×C22⋊D19911yescharacteristic
132C4#2026398111yescharacteristic
133C2—21no—
134C2—81no—
135C2—15922no—
136C2#2716412811yescharacteristic
subgroupG/H
centerC2#27164128
commutatorC8×C199C22×C4
FrattiniC2×C8C22×D199
Fitting#444×C199C2
socleC2×C199#442
prime radical#27164453C2
cosocleC2×C8×C199C23
hypercenter#444D199
hypocenterC199AGL(1,ℤ/16)
LCSG ▷ C8×C199 ▷ C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #444

Complements

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

NH ≅ G/Ntype
2. #27164131C2semidirect
3. #27163842C2semidirect
4. #27163830C2semidirect
5. #27163828C2semidirect
6. #27163796C2semidirect
7. #444×C199C2semidirect
10. #20263898C4semidirect
11. #20263896C4semidirect
28. C199⋊C16C2×C4semidirect
30. Q16×C199C2×C4semidirect
32. D8×C199C2×C4semidirect
68. C199AGL(1,ℤ/16)semidirect
71. #444D199semidirect
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
#271641312
#271638422
#271638302
#271638282
#271637962
#444×C1992
AGL(1,ℤ/16)199