Finite Group Databasebeta

← #49120570

Subgroups — #49120570

Order 36224.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#31827215C211yesmaximal, characteristic
2#31826871C212yesmaximal, complemented
3#31826206C212yesmaximal, complemented
4#31826205C211yesmaximal, characteristic
5#497×C283C211yesmaximal, characteristic
6#22583608—21no—
7#22583603—22no—
8#22583602C2212yes—
9#22583591—21no—
10#22583587—22no—
11#22583543—22no—
12#22583541—22no—
13#22583521C2212yes—
14C2×Q16×C283—21no—
15C2×SD16×C283—21no—
16C4⋊Q8×C283C2211yescharacteristic
17#124×C283C2211yescharacteristic
18#97×C283C2211yescharacteristic
19C2×Dic566—21no—
20C22×Dic283—21no—
21C2×C283⋊C8—22no—
22C4⋊Dic283D412yes—
23C4⋊Dic283D412yes—
24Dic283⋊C4—22no—
25Dic283⋊C4—22no—
26Dic283⋊C4—22no—
27C22⋊Dic283—22no—
28C22⋊Dic283—21no—
29C4×Dic283—21no—
30C2×Q8×C283—21no—
31C2×Q8×C283D411yescharacteristic
32C2×D4×C283D411yescharacteristic
33Q16×C283—42no—
34SD16×C283—42no—
35C2×C8×C283—21no—
36C4⋊C4×C283—22no—
37C22⋊C4×C283—22no—
38C42×C283C2311yescharacteristic
39Dic566—81no—
40C2×Dic283—22no—
41C2×Dic283—22no—
42C2×Dic283—81no—
43C283⋊C8—24no—
44C23×C283—21no—
45Q8×C283—41no—
46Q8×C283—41no—
47Q8×C283—41no—
48D4×C283—41no—
49C2×C4×C283—21no—
50C2×C4×C283—41no—
51C2×C4×C283C2×D412yes—
52C2×C4×C283C2×D411yescharacteristic
53C8×C283—22no—
54C22×C283—41no—
55C22×C283—42no—
56C22×C283#12011yescharacteristic
57C4×C283—21no—
58C4×C283—22no—
59C4×C283—41no—
60C4×C283—81no—
61Dic283—42no—
62Dic283—42no—
63C2×C283—81no—
64C2×C283#45311yescharacteristic
65C2×C283#45612yes—
66C283#166011yescharacteristic, complemented, Sylow
67#1660—2831nomaximal, Sylow
68#563—2831no—
69#501—2832no—
70#497D28311yescharacteristic
71#330—2832no—
72#329—2831no—
73C2×Q16—21no—
74C2×SD16—21no—
75C4⋊Q8—5661no—
76C4⋊Q8C2×D28311yescharacteristic
77#126—5662no—
78#125—2832no—
79#124C2×D28311yescharacteristic
80#123—5661no—
81C22⋊Q8—5662no—
82#107—5662no—
83#106—5662no—
84#97—2832no—
85#97C2×D28311yescharacteristic
86C2×Q8—21no—
87C2×Q8—5661no—
88C2×Q8C22⋊D28311yescharacteristic
89C2×D4C22⋊D28311yescharacteristic
90C22×C4—5661no—
91Q16—42no—
92SD16—42no—
93C2×C8—21no—
94C2×C8—5662no—
95C4⋊C4—22no—
96C4⋊C4—2832no—
97C4⋊C4—2832no—
98C4⋊C4—5662no—
99C4⋊C4—5662no—
100C4⋊C4—5662no—
101C22⋊C4—22no—
102C22⋊C4—5662no—
103C22⋊C4—5661no—
104C42—5661no—
105C42C22×D28311yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—41no—
109Q8—41no—
110Q8—22641no—
111D4—41no—
112C2×C4—21no—
113C2×C4—41no—
114C2×C4—5662no—
115C2×C4—5662no—
116C2×C4—22641no—
117C2×C4D4×D28312yes—
118C2×C4C2×C22⋊D28311yescharacteristic
119C8—22no—
120C8—5664no—
121C22—41no—
122C22—42no—
123C22#2258357811yescharacteristic
124C4—21no—
125C4—22no—
126C4—41no—
127C4—81no—
128C4—11322no—
129C4—11322no—
130C2—81no—
131C2#3182663211yescharacteristic
132C2#3182664612yes—
subgroupG/H
centerC22#22583578
commutator, cosocleC42×C283C23
FrattiniC42C22×D283
Fitting#497×C283C2
socleC22×C283#120
prime radicalC2×D4×C283D4
hypercenter#497D283
hypocenterC283#1660
LCSG ▷ C42×C283 ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
2. #31826871C2semidirect
3. #31826206C2semidirect
66. C283#1660semidirect

Maximal subgroups

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

subgroupindex
#318272152
#318268712
#318262062
#318262052
#497×C2832
#1660283