Finite Group Databasebeta

← #49120572

Subgroups — #49120572

Order 36224.

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#31826875C212yesmaximal, complemented
2#31826860C211yesmaximal, characteristic
3#31826207C212yesmaximal, complemented
4#31826205C211yesmaximal, characteristic
5#563×C283C211yesmaximal, characteristic
6C2×#17985382—21no—
7C2×#17985380—21no—
8#22583610C2211yescharacteristic
9#22583543—22no—
10#22583541—22no—
11#22583521C2212yes—
12#22583521C2211yescharacteristic
13C4⋊Q8×C283—21no—
14#126×C283—22no—
15#125×C283C2212yes—
16#124×C283C2211yescharacteristic
17#123×C283—21no—
18C22⋊Q8×C283—22no—
19C2×Dic566—21no—
20#17985382—42no—
21#17985380—42no—
22C2×C283⋊C8—21no—
23C2×C283⋊C8—22no—
24C4⋊Dic283—22no—
25C2×Q8×C283—21no—
26C2×Q8×C283D411yescharacteristic
27C2×D4×C283D411yescharacteristic
28C22×C4×C283—21no—
29C4⋊C4×C283—22no—
30C4⋊C4×C283—22no—
31C4⋊C4×C283—22no—
32C4⋊C4×C283D412yes—
33C4⋊C4×C283D412yes—
34C22⋊C4×C283—22no—
35C22⋊C4×C283—21no—
36C22⋊C4×C283—22no—
37C42×C283—21no—
38C42×C283C2311yescharacteristic
39Dic566—41no—
40Dic566—41no—
41C2×Dic283—41no—
42C283⋊C8—24no—
43C283⋊C8—22no—
44C23×C283—21no—
45Q8×C283—41no—
46Q8×C283—81no—
47D4×C283—41no—
48C2×C4×C283—22no—
49C2×C4×C283—21no—
50C2×C4×C283—22no—
51C2×C4×C283—81no—
52C2×C4×C283C2×D412yes—
53C2×C4×C283C2×D411yescharacteristic
54C22×C283—41no—
55C22×C283—42no—
56C22×C283#12011yescharacteristic
57C4×C283—21no—
58C4×C283—22no—
59C4×C283—42no—
60C4×C283—41no—
61C4×C283—42no—
62Dic283—81no—
63C2×C283—81no—
64C2×C283#45311yescharacteristic
65C2×C283#45612yes—
66C283#166011yescharacteristic, complemented, Sylow
67#1660—2831nomaximal, Sylow
68#563D28311yescharacteristic
69#501—2832no—
70#497—2831no—
71#330—2832no—
72#329—2831no—
73C2×Q16—5661no—
74C2×SD16—5661no—
75C4⋊Q8—21no—
76C4⋊Q8—2831no—
77#126—22no—
78#125C2×D28312yes—
79#124C2×D28311yescharacteristic
80#123—21no—
81C22⋊Q8—22no—
82#107—5662no—
83#106—5662no—
84#97—2832no—
85#97—2831no—
86C2×Q8—21no—
87C2×Q8—5661no—
88C2×Q8C22⋊D28311yescharacteristic
89C2×D4C22⋊D28311yescharacteristic
90C22×C4—21no—
91Q16—11322no—
92SD16—11322no—
93C2×C8—5662no—
94C2×C8—5661no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—5662no—
99C4⋊C4C22⋊D28312yes—
100C4⋊C4C22⋊D28312yes—
101C22⋊C4—22no—
102C22⋊C4—21no—
103C22⋊C4—22no—
104C42—21no—
105C42C22×D28311yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—81no—
109Q8—11321no—
110Q8—11321no—
111D4—41no—
112C2×C4—22no—
113C2×C4—21no—
114C2×C4—22no—
115C2×C4—81no—
116C2×C4—11321no—
117C2×C4C2×C22⋊D28312yes—
118C2×C4C2×C22⋊D28311yescharacteristic
119C8—5662no—
120C8—5664no—
121C22—41no—
122C22—42no—
123C22#2258357911yescharacteristic
124C4—21no—
125C4—22no—
126C4—42no—
127C4—41no—
128C4—42no—
129C4—22641no—
130C2—81no—
131C2#3182663311yescharacteristic
132C2#3182664812yes—
subgroupG/H
centerC22#22583579
commutator, cosocleC42×C283C23
FrattiniC42C22×D283
Fitting#563×C283C2
socleC22×C283#120
prime radicalC2×D4×C283D4
hypercenter#563D283
hypocenterC283#1660
LCSG ▷ C42×C283 ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #31826875C2semidirect
3. #31826207C2semidirect
66. C283#1660semidirect

Maximal subgroups

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

subgroupindex
#318268752
#318268602
#318262072
#318262052
#563×C2832
#1660283