Finite Group Databasebeta

← #22590535

Subgroups — #22590535

Order 9088.

140 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 53 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
1C2×#14556062C211yesmaximal, characteristic
2#17985826C212yesmaximal, complemented
3#17985803C212yesmaximal, complemented
4C22⋊(C71⋊C16)C211yesmaximal, characteristic
5#467×C71C211yesmaximal, characteristic
6C2×C4⋊Dic71—21no—
7C2×C71⋊C16C2212yes—
8#14556062—21no—
9#14556062C2212yes—
10C2×Q16×C71C2211yescharacteristic
11C22×C8×C71C2211yescharacteristic
12C22⋊Q8×C71—21no—
13#107×C71C2211yescharacteristic
14#103×C71—21no—
15C22×Dic71—41no—
16C4⋊Dic71—22no—
17C4⋊Dic71—41no—
18C71⋊C16—22no—
19C2×Q8×C71—21no—
20C22×C4×C71D411yescharacteristic
21Q16×C71—22no—
22C2×C8×C71—22no—
23C2×C8×C71D411yescharacteristic
24C2×C8×C71C2311yescharacteristic
25C4⋊C4×C71—21no—
26C4⋊C4×C71—41no—
27C22⋊C4×C71—41no—
28C23×C71D811yescharacteristic
29Q8×C71—41no—
30Q8×C71—41no—
31C2×C4×C71—21no—
32C2×C4×C71—21no—
33C2×C4×C71—41no—
34C2×C4×C71—41no—
35C2×C4×C71D811yescharacteristic
36C2×C4×C71C2×D411yescharacteristic
37C2×Dic71—42no—
38C2×Dic71—82no—
39C8×C71—21no—
40C8×C71#4112yes—
41C22×C71—21no—
42C22×C71—21no—
43C22×C71SD3212yes—
44C22×C71C2×D811yescharacteristic
45C4×C71—21no—
46C4×C71—81no—
47C4×C71—81no—
48C4×C71#12311yescharacteristic
49C4×C71#13711yescharacteristic
50Dic71—82no—
51C2×C71—22no—
52C2×C71#48011yescharacteristic
53C2×C71C2×SD3211yescharacteristic
54C2×C71#51011yescharacteristic
55#2235—711nomaximal, Sylow
56C71#223511yescharacteristic, complemented, Sylow
57#467D7111yescharacteristic
58C2×#107—711no—
59#367—712no—
60#358—712no—
61C22⋊C16—711no—
62C2×Q16C2×D7111yescharacteristic
63C22×C8C2×D7111yescharacteristic
64C22⋊Q8—21no—
65C2×C4⋊C4—1421no—
66C2×C16—712no—
67#107—712no—
68#107—1421no—
69#107C2×D7111yescharacteristic
70#103—21no—
71C2×Q8—21no—
72C22×C4—2841no—
73C22×C4C22⋊D7111yescharacteristic
74Q16—22no—
75C2×C8—22no—
76C2×C8C22⋊D7111yescharacteristic
77C2×C8C22×D7111yescharacteristic
78C4⋊C4—21no—
79C4⋊C4—41no—
80C4⋊C4—2841no—
81C4⋊C4—1422no—
82C22⋊C4—41no—
83C16—1422no—
84C23#1176585011yescharacteristic
85Q8—41no—
86Q8—41no—
87C2×C4—21no—
88C2×C4—21no—
89C2×C4—41no—
90C2×C4—41no—
91C2×C4—2842no—
92C2×C4—5682no—
93C2×C4#1176585011yescharacteristic
94C2×C4C2×C22⋊D7111yescharacteristic
95C8—21no—
96C8#1176586412yes—
97C22—21no—
98C22—21no—
99C22#1455607112yes—
100C22C2×#1176585011yescharacteristic
101C4—21no—
102C4—81no—
103C4—81no—
104C4—5682no—
105C4#1455610911yescharacteristic
106C4#1455615811yescharacteristic
107C2—22no—
108C2#1798631011yescharacteristic
109C2C2×#1455607111yescharacteristic
110C2#1798643911yescharacteristic
subgroupG/H
centerC22C2×#11765850
commutatorC2×C8×C71C23
FrattiniC2×C8C22×D71
Fitting#467×C71C2
socleC22×C71C2×D8
hypercenter#467D71
hypocenterC71#2235
LCSG ▷ C2×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #467

Complements

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

NH ≅ G/Ntype
2. #17985826C2semidirect
3. #17985803C2semidirect
56. C71#2235semidirect