Finite Group Databasebeta

← #36352855

Subgroups — #36352855

Order 22144.

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#26012239C211yesmaximal, characteristic, complemented
2#26011878C212yesmaximal, complemented
3#26011233C211yesmaximal, characteristic, complemented
4#26011230C212yesmaximal, complemented
5#490×C173C211yesmaximal, characteristic, complemented
6C2×#14583814—22no—
7C2×#14583811—22no—
8#19149667C2212yes—
9#19149664—22no—
10#19149657—22no—
11#19149654—22no—
12#19149590C2212yes—
13#127×C173C2211yescharacteristic
14#125×C173C2211yescharacteristic
15#102×C173—22no—
16#97×C173C2211yescharacteristic
17C2×Dic346D412yes—
18C2×D692—22no—
19C2×D692D412yes—
20C2×C4×D173—22no—
21#14583814—44no—
22#14583811—44no—
23C2×#13063436—22no—
24#14583805—22no—
25#14583803—22no—
26#14583803—22no—
27#14583803—22no—
28C2×D4×C173—22no—
29C2×D4×C173—21no—
30C2×C8×C173—21no—
31C4⋊C4×C173—22no—
32C4⋊C4×C173D412yes—
33C42×C173C2311yescharacteristic
34C22×D173—22no—
35Dic346—42no—
36D692—42no—
37D692—82no—
38C2×Dic173—22no—
39C4×D173—82no—
40#13063436—24no—
41C23×C173—41no—
42D4×C173—42no—
43D4×C173—42no—
44D4×C173—42no—
45C2×C4×C173—22no—
46C2×C4×C173C2×D412yes—
47C2×C4×C173C2×D411yescharacteristic
48C8×C173—41no—
49C22×C173—82no—
50C22×C173—81no—
51C22×C173#12011yescharacteristic
52C2×D173—42no—
53C2×D173—42no—
54C2×D173—42no—
55C4×C173—22no—
56C4×C173—21no—
57C4×C173—42no—
58Dic173—42no—
59C2×C173—161no—
60C2×C173#45312yes—
61C2×C173#45411yescharacteristic
62D173—82no—
63C173#165611yescharacteristic, complemented, Sylow
64#1656—1731nomaximal, Sylow
65#562—1731no—
66#496—1732no—
67#490D17311yescharacteristic, complemented
68#330—1731no—
69#329—1732no—
70C2×SD16—3462no—
71C2×D8—3462no—
72#127C2×D17311yescharacteristic
73#125C2×D17311yescharacteristic
74#124—1732no—
75#123—3462no—
76C22⋊Q8—3462no—
77#121—3462no—
78#102—22no—
79#97—1732no—
80#97C2×D17311yescharacteristic
81C2×Q8—1732no—
82C2×D4—22no—
83C2×D4—21no—
84C2×D4—3462no—
85C2×D4—1732no—
86C22×C4—3462no—
87SD16—6924no—
88D8—6924no—
89C2×C8—21no—
90C2×C8—3462no—
91C4⋊C4—22no—
92C4⋊C4—3462no—
93C4⋊C4C22⋊D17312yes—
94C22⋊C4—3462no—
95C22⋊C4—3462no—
96C22⋊C4—3462no—
97C42C22×D17311yescharacteristic
98C23—41no—
99C23—3462no—
100Q8—6922no—
101D4—42no—
102D4—42no—
103D4—42no—
104D4—13842no—
105D4—6922no—
106C2×C4—22no—
107C2×C4—3462no—
108C2×C4—13842no—
109C2×C4D4×D17312yes—
110C2×C4C2×C22⋊D17311yescharacteristic
111C8—41no—
112C8—3464no—
113C22—82no—
114C22—81no—
115C22—6922no—
116C22—6922no—
117C22—6922no—
118C22#1914964711yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—6922no—
123C2—161no—
124C2—13842no—
125C2#2601165712yes—
126C2#2601166311yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#490×C173C2
socleC22×C173#120
hypercenter#490D173
hypocenterC173#1656
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #26012239C2semidirect
2. #26011878C2semidirect
3. #26011233C2semidirect
4. #26011230C2semidirect
5. #490×C173C2semidirect
63. C173#1656semidirect
67. #490D173semidirect

Maximal subgroups

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

subgroupindex
#260122392
#260118782
#260112332
#260112302
#490×C1732
#1656173