Finite Group Databasebeta

← #37508117

Subgroups — #37508117

Order 22912.

314 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 57 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#26039074C211yesmaximal, characteristic
2#26038811C211yesmaximal, characteristic, complemented
3#26038730C212yesmaximal, complemented
4#26038166C212yesmaximal, complemented
5#331×C179C211yesmaximal, characteristic, complemented
6C2×D4×D179—22no—
7C2×#15668247—22no—
8#19153113C2211yescharacteristic, complemented
9#19153084—22no—
10#19153081—22no—
11C4×D716C2212yes—
12#19153045—22no—
13#19153035—22no—
14C4×C179⋊C8C2211yescharacteristic, complemented
15#127×C179C2211yescharacteristic
16C4⋊C8×C179C2212yescomplemented
17C23×D179—22no—
18C2×Dic358—21no—
19C2×D716D412yes—
20C2×C22⋊D179—24no—
21D4×D179—44no—
22D4×D179—44no—
23C2×C4×D179—22no—
24#15668247—44no—
25C2×C179⋊C8—21no—
26C4⋊Dic179—22no—
27C4⋊Dic179D412yes—
28C22⋊Dic179—22no—
29(C2×D179)⋊C4—22no—
30C2×D4×C179—22no—
31C2×D4×C179D412yes—
32C2×C8×C179—22no—
33C42×C179C2311yescharacteristic
34C22×D179—22no—
35C22×D179—48no—
36C22×D179—82no—
37Dic358—42no—
38D716—22no—
39D716SD1614yescomplemented
40C22⋊D179—48no—
41C2×Dic179—22no—
42C2×Dic179—41no—
43C4×D179—42no—
44C4×D179—42no—
45C179⋊C8—22no—
46C23×C179—22no—
47D4×C179—42no—
48D4×C179—42no—
49D4×C179—42no—
50C2×C4×C179C2×D412yes—
51C2×C4×C179C2×D411yescharacteristic
52C8×C179—42no—
53C22×C179—44no—
54C22×C179—42no—
55C22×C179#12011yescharacteristic
56C2×D179—24no—
57C2×D179—42no—
58C2×D179—42no—
59C2×D179—88no—
60C4×C179—21no—
61C4×C179C2×SD1612yes—
62C4×C179AGL(1,ℤ/8)12yes—
63Dic179—42no—
64Dic179—81no—
65C2×C179—82no—
66C2×C179#45012yes—
67C2×C179#45311yescharacteristic
68D179—44no—
69C179#162911yescharacteristic, complemented, Sylow
70#1629—1791nomaximal, Sylow
71D42—1791no—
72#492—1791no—
73#476—1792no—
74#331D17911yescharacteristic, complemented
75D4⋊C8—1792no—
76C22×D4—3582no—
77C2×SD16—3582no—
78C4⋊Q8—1791no—
79#127C2×D17911yescharacteristic
80#121—3582no—
81#120—3582no—
82C4×D4—1792no—
83#106—3582no—
84C4⋊C8C2×D17912yescomplemented
85#102—3582no—
86C4×C8—1791no—
87C24—3582no—
88C2×Q8—3581no—
89C2×D4—22no—
90C2×D4—3584no—
91C2×D4—1792no—
92C2×D4—7164no—
93C2×D4—7164no—
94C2×D4D71612yes—
95C22×C4—3582no—
96SD16—7164no—
97C2×C8—22no—
98C2×C8—3581no—
99C4⋊C4—3582no—
100C4⋊C4—1792no—
101C22⋊C4—3582no—
102C22⋊C4—3582no—
103C42C22×D17911yescharacteristic
104C23—22no—
105C23—3582no—
106C23—14322no—
107C23—7168no—
108Q8—7162no—
109D4—42no—
110D4—42no—
111D4—42no—
112D4—3582no—
113D4—1794no—
114D4—7168no—
115C2×C4—3582no—
116C2×C4—7161no—
117C2×C4—7162no—
118C2×C4—7162no—
119C2×C4D4×D17912yes—
120C2×C4C2×D71611yescharacteristic
121C8—42no—
122C8—3582no—
123C22—44no—
124C22—42no—
125C22—3584no—
126C22—14328no—
127C22—7162no—
128C22—7162no—
129C22#1915308311yescharacteristic
130C4—21no—
131C4—14321no—
132C4—7162no—
133C4SD16×D17912yes—
134C4#1915313712yes—
135C2—82no—
136C2—7164no—
137C2#2603859112yes—
138C2#2603860911yescharacteristic
subgroupG/H
centerC22#19153083
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#331×C179C2
socleC22×C179#120
prime radical#26039074C2
hypercenter#331D179
hypocenterC179#1629
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #26038811C2semidirect
3. #26038730C2semidirect
4. #26038166C2semidirect
5. #331×C179C2semidirect
8. #19153113C22semidirect
14. C4×C179⋊C8C22semidirect
16. C4⋊C8×C179C22semidirect
39. D716SD16semidirect
69. C179#1629semidirect
74. #331D179semidirect
84. C4⋊C8C2×D179semidirect

Maximal subgroups

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

subgroupindex
#260390742
#260388112
#260387302
#260381662
#331×C1792
#1629179