Finite Group Databasebeta

← #37508292

Subgroups — #37508292

Order 22912.

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

normal subgroups only

This group has 230 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#26039187C211yesmaximal, characteristic, complemented
2#26038826C212yesmaximal, complemented
3#26038181C211yesmaximal, characteristic, complemented
4#26038178C212yesmaximal, complemented
5#490×C179C211yesmaximal, characteristic, complemented
6C2×#15668247—22no—
7C2×#15668244—22no—
8#19153101C2212yes—
9#19153098—22no—
10#19153091—22no—
11#19153088—22no—
12#19153024C2212yes—
13#127×C179C2211yescharacteristic
14#125×C179C2211yescharacteristic
15#102×C179—22no—
16#97×C179C2211yescharacteristic
17C2×Dic358D412yes—
18C2×D716—22no—
19C2×D716D412yes—
20C2×C4×D179—22no—
21#15668247—44no—
22#15668244—44no—
23C2×C179⋊C8—22no—
24Dic179⋊C4—22no—
25(C2×D179)⋊C4—22no—
26(C2×D179)⋊C4—22no—
27(C2×D179)⋊C4—22no—
28C2×D4×C179—21no—
29C2×D4×C179—22no—
30C2×C8×C179—21no—
31C4⋊C4×C179—22no—
32C4⋊C4×C179D412yes—
33C42×C179C2311yescharacteristic
34C22×D179—22no—
35Dic358—42no—
36D716—42no—
37D716—82no—
38C2×Dic179—22no—
39C4×D179—82no—
40C179⋊C8—24no—
41C23×C179—41no—
42D4×C179—42no—
43D4×C179—42no—
44D4×C179—42no—
45C2×C4×C179—22no—
46C2×C4×C179C2×D412yes—
47C2×C4×C179C2×D411yescharacteristic
48C8×C179—41no—
49C22×C179—82no—
50C22×C179—81no—
51C22×C179#12011yescharacteristic
52C2×D179—42no—
53C2×D179—42no—
54C2×D179—42no—
55C4×C179—22no—
56C4×C179—21no—
57C4×C179—42no—
58Dic179—42no—
59C2×C179—161no—
60C2×C179#45312yes—
61C2×C179#45411yescharacteristic
62D179—82no—
63C179#165611yescharacteristic, complemented, Sylow
64#1656—1791nomaximal, Sylow
65#562—1791no—
66#496—1792no—
67#490D17911yescharacteristic, complemented
68#330—1791no—
69#329—1792no—
70C2×SD16—3582no—
71C2×D8—3582no—
72#127C2×D17911yescharacteristic
73#125C2×D17911yescharacteristic
74#124—1792no—
75#123—3582no—
76C22⋊Q8—3582no—
77#121—3582no—
78#102—22no—
79#97—1792no—
80#97C2×D17911yescharacteristic
81C2×Q8—1792no—
82C2×D4—22no—
83C2×D4—21no—
84C2×D4—3582no—
85C2×D4—1792no—
86C22×C4—3582no—
87SD16—7164no—
88D8—7164no—
89C2×C8—21no—
90C2×C8—3582no—
91C4⋊C4—22no—
92C4⋊C4—3582no—
93C4⋊C4C22⋊D17912yes—
94C22⋊C4—3582no—
95C22⋊C4—3582no—
96C22⋊C4—3582no—
97C42C22×D17911yescharacteristic
98C23—41no—
99C23—3582no—
100Q8—7162no—
101D4—42no—
102D4—42no—
103D4—42no—
104D4—14322no—
105D4—7162no—
106C2×C4—22no—
107C2×C4—3582no—
108C2×C4—14322no—
109C2×C4D4×D17912yes—
110C2×C4C2×C22⋊D17911yescharacteristic
111C8—41no—
112C8—3584no—
113C22—82no—
114C22—81no—
115C22—7162no—
116C22—7162no—
117C22—7162no—
118C22#1915308111yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—7162no—
123C2—161no—
124C2—14322no—
125C2#2603860512yes—
126C2#2603861111yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#490×C179C2
socleC22×C179#120
hypercenter#490D179
hypocenterC179#1656
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #26039187C2semidirect
2. #26038826C2semidirect
3. #26038181C2semidirect
4. #26038178C2semidirect
5. #490×C179C2semidirect
63. C179#1656semidirect
67. #490D179semidirect

Maximal subgroups

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

subgroupindex
#260391872
#260388262
#260381812
#260381782
#490×C1792
#1656179