Finite Group Databasebeta

← #37508321

Subgroups — #37508321

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#26039187C211yesmaximal, characteristic
2#26038801C212yesmaximal, complemented
3#26038181C211yesmaximal, characteristic, complemented
4#26038178C212yesmaximal, complemented
5#491×C179C211yesmaximal, characteristic, complemented
6#19153101C2212yes—
7#19153098—22no—
8#19153091—22no—
9#19153088—22no—
10#19153036—22no—
11#19153033—22no—
12#19153024C2212yes—
13C4⋊Q8×C179C2211yescharacteristic
14#125×C179C2211yescharacteristic
15#103×C179—22no—
16#97×C179C2211yescharacteristic
17C2×Dic358D412yes—
18C2×D716—22no—
19C2×D716D412yes—
20C2×C4×D179—22no—
21C2×C179⋊C8—22no—
22Dic179⋊C4—22no—
23(C2×D179)⋊C4—22no—
24(C2×D179)⋊C4—22no—
25(C2×D179)⋊C4—22no—
26C2×Q8×C179—21no—
27C2×C8×C179—21no—
28C4⋊C4×C179—22no—
29C4⋊C4×C179—22no—
30C4⋊C4×C179D412yes—
31C42×C179C2311yescharacteristic
32C22×D179—22no—
33Dic358—42no—
34D716—42no—
35D716—82no—
36C2×Dic179—22no—
37C4×D179—82no—
38C179⋊C8—42no—
39Q8×C179—42no—
40C2×C4×C179—22no—
41C2×C4×C179—41no—
42C2×C4×C179C2×D412yes—
43C2×C4×C179C2×D411yescharacteristic
44C8×C179—41no—
45C22×C179#12011yescharacteristic
46C2×D179—42no—
47C2×D179—42no—
48C2×D179—42no—
49C4×C179—21no—
50C4×C179—22no—
51C4×C179—42no—
52C4×C179—81no—
53Dic179—42no—
54C2×C179#45411yescharacteristic
55C2×C179#45512yes—
56D179—82no—
57C179#166211yescharacteristic, complemented, Sylow
58#1662—1791nomaximal, Sylow
59#562—1791no—
60#491D17911yescharacteristic, complemented
61#489—1792no—
62#330—1791no—
63#329—1792no—
64C4⋊Q8C2×D17911yescharacteristic
65#125C2×D17911yescharacteristic
66#124—1792no—
67#123—3582no—
68C22⋊Q8—3582no—
69#121—3582no—
70#103—22no—
71#103—3582no—
72#102—3582no—
73#97—1792no—
74#97C2×D17911yescharacteristic
75C2×Q8—21no—
76C2×Q8—1792no—
77C2×D4—3582no—
78C2×D4—1792no—
79C22×C4—3582no—
80C2×C8—21no—
81C2×C8—3582no—
82C4⋊C4—22no—
83C4⋊C4—22no—
84C4⋊C4—3582no—
85C4⋊C4C22⋊D17912yes—
86C22⋊C4—3582no—
87C22⋊C4—3582no—
88C22⋊C4—3582no—
89C42C22×D17911yescharacteristic
90C23—3582no—
91Q8—42no—
92Q8—7162no—
93D4—14322no—
94D4—7162no—
95C2×C4—22no—
96C2×C4—41no—
97C2×C4—3582no—
98C2×C4—14322no—
99C2×C4D4×D17912yes—
100C2×C4C2×C22⋊D17911yescharacteristic
101C8—41no—
102C8—7162no—
103C22—7162no—
104C22—7162no—
105C22—7162no—
106C22#1915308111yescharacteristic
107C4—21no—
108C4—22no—
109C4—42no—
110C4—81no—
111C4—7162no—
112C2—14322no—
113C2#2603861111yescharacteristic
114C2#2603861512yes—
subgroupG/H
centerC22#19153081
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#491×C179C2
socleC22×C179#120
prime radical#26039187C2
hypercenter#491D179
hypocenterC179#1662
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
2. #26038801C2semidirect
3. #26038181C2semidirect
4. #26038178C2semidirect
5. #491×C179C2semidirect
57. C179#1662semidirect
60. #491D179semidirect

Maximal subgroups

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

subgroupindex
#260391872
#260388012
#260381812
#260381782
#491×C1792
#1662179