Finite Group Databasebeta

← #37508312

Subgroups — #37508312

Order 22912.

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#26039189C211yesmaximal, characteristic
2#26038848C212yesmaximal, complemented
3#26038834C211yesmaximal, characteristic
4#26038180C212yesmaximal, complemented
5#329×C179C211yesmaximal, characteristic
6C2×#15668249—21no—
7C2×#15668247—21no—
8#19153113C2211yescharacteristic
9#19153111—21no—
10#19153106—22no—
11#19153105C2212yes—
12#19153094—21no—
13#19153090—22no—
14#19153047—22no—
15#19153045—22no—
16#19153024C2211yescharacteristic
17#124×C179C2211yescharacteristic
18#97×C179C2212yes—
19C2×Dic358—21no—
20C2×Dic358—21no—
21C22×Dic179—21no—
22#15668249—42no—
23#15668247—42no—
24C2×C179⋊C8—21no—
25C4⋊Dic179—22no—
26C4⋊Dic179D412yes—
27C4⋊Dic179D412yes—
28Dic179⋊C4—22no—
29Dic179⋊C4—22no—
30Dic179⋊C4—22no—
31C22⋊Dic179—22no—
32C22⋊Dic179—21no—
33C4×Dic179—21no—
34C2×Q8×C179D411yescharacteristic
35C2×D4×C179D411yescharacteristic
36C2×C8×C179—22no—
37C22⋊C4×C179—22no—
38C42×C179C2311yescharacteristic
39Dic358—41no—
40Dic358—41no—
41Dic358—81no—
42C2×Dic179—22no—
43C2×Dic179—22no—
44C2×Dic179—41no—
45C2×Dic179—81no—
46C179⋊C8—22no—
47C23×C179—21no—
48Q8×C179—41no—
49D4×C179—41no—
50C2×C4×C179—21no—
51C2×C4×C179C2×D412yes—
52C2×C4×C179C2×D411yescharacteristic
53C8×C179—24no—
54C22×C179—41no—
55C22×C179—42no—
56C22×C179#12011yescharacteristic
57C4×C179—21no—
58C4×C179—22no—
59C4×C179—41no—
60Dic179—42no—
61Dic179—42no—
62Dic179—81no—
63C2×C179—81no—
64C2×C179#45311yescharacteristic
65C2×C179#45612yes—
66C179#166011yescharacteristic, complemented, Sylow
67#1660—1791nomaximal, Sylow
68#563—1791no—
69#501—1792no—
70#497—1791no—
71#330—1792no—
72#329D17911yescharacteristic
73C2×Q16—3581no—
74C2×SD16—3581no—
75C4⋊Q8—3581no—
76C4⋊Q8—1791no—
77#126—3582no—
78#125—1792no—
79#124C2×D17911yescharacteristic
80#123—3581no—
81C22⋊Q8—3582no—
82#107—3582no—
83#106—3582no—
84#97—1791no—
85#97C2×D17912yes—
86C2×Q8—3581no—
87C2×Q8—3581no—
88C2×Q8D71611yescharacteristic
89C2×D4D71611yescharacteristic
90C22×C4—3581no—
91Q16—7162no—
92SD16—7162no—
93C2×C8—22no—
94C2×C8—3581no—
95C4⋊C4—3582no—
96C4⋊C4—3582no—
97C4⋊C4—3582no—
98C4⋊C4—3582no—
99C4⋊C4—1792no—
100C4⋊C4—1792no—
101C22⋊C4—22no—
102C22⋊C4—3582no—
103C22⋊C4—3581no—
104C42—3581no—
105C42C22×D17911yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—14321no—
109Q8—7161no—
110Q8—7161no—
111D4—41no—
112C2×C4—21no—
113C2×C4—3582no—
114C2×C4—3582no—
115C2×C4—14321no—
116C2×C4—7161no—
117C2×C4D4×D17912yes—
118C2×C4C2×D71611yescharacteristic
119C8—24no—
120C8—3582no—
121C22—41no—
122C22—42no—
123C22#1915308311yescharacteristic
124C4—21no—
125C4—22no—
126C4—41no—
127C4—14321no—
128C4—7162no—
129C4—7162no—
130C2—81no—
131C2#2603860911yescharacteristic
132C2#2603862312yes—
subgroupG/H
centerC22#19153083
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#329×C179C2
socleC22×C179#120
prime radicalC2×D4×C179D4
hypercenter#329D179
hypocenterC179#1660
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #26038848C2semidirect
4. #26038180C2semidirect
66. C179#1660semidirect

Maximal subgroups

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

subgroupindex
#260391892
#260388482
#260388342
#260381802
#329×C1792
#1660179