Finite Group Databasebeta

← #38768161

Subgroups — #38768161

Order 25472.

196 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 49 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#27164428C212yesmaximal, complemented
2#27164413C211yesmaximal, characteristic
3#27163760C212yesmaximal, complemented
4#27163758C211yesmaximal, characteristic
5#563×C199C211yesmaximal, characteristic
6C2×#15691876—21no—
7C2×#15691874—21no—
8#20263955C2211yescharacteristic
9#20263888—22no—
10#20263886—22no—
11#20263866C2212yes—
12#20263866C2211yescharacteristic
13C4⋊Q8×C199—21no—
14#126×C199—22no—
15#125×C199C2212yes—
16#124×C199C2211yescharacteristic
17#123×C199—21no—
18C22⋊Q8×C199—22no—
19C2×Dic398—21no—
20#15691876—42no—
21#15691874—42no—
22C2×C199⋊C8—21no—
23C2×C199⋊C8—22no—
24C4⋊Dic199—22no—
25C2×Q8×C199—21no—
26C2×Q8×C199D411yescharacteristic
27C2×D4×C199D411yescharacteristic
28C22×C4×C199—21no—
29C4⋊C4×C199—22no—
30C4⋊C4×C199—22no—
31C4⋊C4×C199—22no—
32C4⋊C4×C199D412yes—
33C4⋊C4×C199D412yes—
34C22⋊C4×C199—22no—
35C22⋊C4×C199—21no—
36C22⋊C4×C199—22no—
37C42×C199—21no—
38C42×C199C2311yescharacteristic
39Dic398—41no—
40Dic398—41no—
41C2×Dic199—41no—
42C199⋊C8—24no—
43C199⋊C8—22no—
44C23×C199—21no—
45Q8×C199—41no—
46Q8×C199—81no—
47D4×C199—41no—
48C2×C4×C199—22no—
49C2×C4×C199—21no—
50C2×C4×C199—22no—
51C2×C4×C199—81no—
52C2×C4×C199C2×D412yes—
53C2×C4×C199C2×D411yescharacteristic
54C22×C199—41no—
55C22×C199—42no—
56C22×C199#12011yescharacteristic
57C4×C199—21no—
58C4×C199—22no—
59C4×C199—42no—
60C4×C199—41no—
61C4×C199—42no—
62Dic199—81no—
63C2×C199—81no—
64C2×C199#45311yescharacteristic
65C2×C199#45612yes—
66C199#166011yescharacteristic, complemented, Sylow
67#1660—1991nomaximal, Sylow
68#563D19911yescharacteristic
69#501—1992no—
70#497—1991no—
71#330—1992no—
72#329—1991no—
73C2×Q16—3981no—
74C2×SD16—3981no—
75C4⋊Q8—21no—
76C4⋊Q8—1991no—
77#126—22no—
78#125C2×D19912yes—
79#124C2×D19911yescharacteristic
80#123—21no—
81C22⋊Q8—22no—
82#107—3982no—
83#106—3982no—
84#97—1992no—
85#97—1991no—
86C2×Q8—21no—
87C2×Q8—3981no—
88C2×Q8C22⋊D19911yescharacteristic
89C2×D4C22⋊D19911yescharacteristic
90C22×C4—21no—
91Q16—7962no—
92SD16—7962no—
93C2×C8—3982no—
94C2×C8—3981no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—3982no—
99C4⋊C4C22⋊D19912yes—
100C4⋊C4C22⋊D19912yes—
101C22⋊C4—22no—
102C22⋊C4—21no—
103C22⋊C4—22no—
104C42—21no—
105C42C22×D19911yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—81no—
109Q8—7961no—
110Q8—7961no—
111D4—41no—
112C2×C4—22no—
113C2×C4—21no—
114C2×C4—22no—
115C2×C4—81no—
116C2×C4—7961no—
117C2×C4C2×C22⋊D19912yes—
118C2×C4C2×C22⋊D19911yescharacteristic
119C8—3982no—
120C8—3984no—
121C22—41no—
122C22—42no—
123C22#2026392411yescharacteristic
124C4—21no—
125C4—22no—
126C4—42no—
127C4—41no—
128C4—42no—
129C4—15921no—
130C2—81no—
131C2#2716418611yescharacteristic
132C2#2716420112yes—
subgroupG/H
centerC22#20263924
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#563×C199C2
socleC22×C199#120
prime radicalC2×D4×C199D4
hypercenter#563D199
hypocenterC199#1660
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #27164428C2semidirect
3. #27163760C2semidirect
66. C199#1660semidirect

Maximal subgroups

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

subgroupindex
#271644282
#271644132
#271637602
#271637582
#563×C1992
#1660199