Finite Group Databasebeta

← #38768528

Subgroups — #38768528

Order 25472.

176 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 65 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#27164421C211yesmaximal, characteristic
2#27164416C211yesmaximal, characteristic
3#27164397C211yesmaximal, characteristic
4#27164370C211yesmaximal, characteristic
5#27164368C211yesmaximal, characteristic
6#27163739C211yesmaximal, characteristic
7#500×C199C211yesmaximal, characteristic
8C2×#15691876—22no—
9#20263955C2211yescharacteristic
10#20263955C2211yescharacteristic
11#20263888—22no—
12#20263887—22no—
13#20263879—22no—
14#20263878—22no—
15C4×C199⋊C8C2211yescharacteristic
16C4×C199⋊C8C2211yescharacteristic
17C4⋊Q8×C199C2211yescharacteristic
18C4⋊Q8×C199C2211yescharacteristic
19#107×C199—22no—
20C4×C8×C199C2211yescharacteristic
21C2×Dic398—21no—
22C2×Dic398—21no—
23#15691876—44no—
24C2×C199⋊C8D412yes—
25C2×C199⋊C8D412yes—
26C4⋊Dic199—21no—
27C4⋊Dic199—21no—
28C4⋊Dic199—21no—
29C4⋊Dic199—21no—
30C2×Q8×C199—21no—
31C2×Q8×C199—21no—
32C2×C8×C199D412yes—
33C4⋊C4×C199—21no—
34C4⋊C4×C199—21no—
35C4⋊C4×C199—21no—
36C4⋊C4×C199—21no—
37C42×C199C2311yescharacteristic
38Dic398—42no—
39Dic398—42no—
40C2×Dic199—41no—
41C2×Dic199—41no—
42C199⋊C8—22no—
43C199⋊C8—22no—
44Q8×C199—42no—
45Q8×C199—42no—
46C2×C4×C199—41no—
47C2×C4×C199—41no—
48C2×C4×C199C2×D411yescharacteristic
49C2×C4×C199C2×D411yescharacteristic
50C2×C4×C199C2×D411yescharacteristic
51C8×C199Q1614yescomplemented
52C22×C199#12711yescharacteristic
53C4×C199—81no—
54C4×C199—81no—
55C4×C199C2×Q1612yes—
56C4×C199AGL(1,ℤ/8)12yes—
57C4×C199#13712yes—
58Dic199—81no—
59Dic199—81no—
60C2×C199#49411yescharacteristic
61C2×C199#49611yescharacteristic
62C2×C199#49711yescharacteristic
63C199#172311yescharacteristic, complemented, Sylow
64#1723—1991nomaximal, Sylow
65#500—1991no—
66#500D19911yescharacteristic
67#498—1991no—
68#494—1991no—
69#487—1991no—
70#487—1991no—
71#322—1991no—
72C2×Q16—3982no—
73C4⋊Q8—1991no—
74C4⋊Q8—1991no—
75C4⋊Q8C2×D19911yescharacteristic
76C4⋊Q8C2×D19911yescharacteristic
77#107—22no—
78#107—3982no—
79#106—3982no—
80#103—3982no—
81#103—3982no—
82C4×C8—1991no—
83C4×C8—1991no—
84C4×C8C2×D19911yescharacteristic
85C2×Q8—21no—
86C2×Q8—21no—
87C2×Q8—3981no—
88C2×Q8—3981no—
89Q16—7964no—
90C2×C8—1992no—
91C2×C8—1992no—
92C2×C8C22⋊D19912yes—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—21no—
96C4⋊C4—21no—
97C4⋊C4—3981no—
98C4⋊C4—3981no—
99C4⋊C4—3981no—
100C4⋊C4—3981no—
101C42C22×D19911yescharacteristic
102Q8—42no—
103Q8—42no—
104Q8—7962no—
105Q8—7962no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—7961no—
109C2×C4—7961no—
110C2×C4D4×D19911yescharacteristic
111C2×C4D4×D19911yescharacteristic
112C2×C4C2×C22⋊D19911yescharacteristic
113C8—3982no—
114C8—3982no—
115C8#1569187614yescomplemented
116C22#2026395111yescharacteristic
117C4—81no—
118C4—81no—
119C4—15921no—
120C4—15921no—
121C4C2×#1569187612yes—
122C4#2026397912yes—
123C4#2026398712yes—
124C2#2716439711yescharacteristic
125C2#2716440611yescharacteristic
126C2#2716441111yescharacteristic
subgroupG/H
centerC22#20263951
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#500×C199C2
socle, prime radicalC22×C199#127
hypercenter#500D199
hypocenterC199#1723
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
51. C8×C199Q16semidirect
63. C199#1723semidirect
115. C8#15691876semidirect

Maximal subgroups

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

subgroupindex
#271644212
#271644162
#271643972
#271643702
#271643682
#271637392
#500×C1992
#1723199