Finite Group Databasebeta

← #22586491

Subgroups — #22586491

Order 9088.

112 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 25 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#17985794C211yesmaximal, characteristic, complemented
2#17985739C211yesmaximal, characteristic, complemented
3#454×C71C211yesmaximal, characteristic
4#14556043—21no—
5#14556039—21no—
62−1+4×C71C411yescharacteristic, complemented
7AGL(1,ℤ/8)×C71—21no—
8#124×C71C2211yescharacteristic
9#104×C71—21no—
10#101×C71C411yescharacteristic, complemented
11C2×C71⋊C8—41no—
12C22⋊Dic71—41no—
13#41×C71—21no—
14#41×C71—41no—
15#41×C71—41no—
16C2×Q8×C71—41no—
17C2×Q8×C71C2×C411yescharacteristic
18C2×D4×C71D411yescharacteristic
19SD16×C71—41no—
20D8×C71—41no—
21M16×C71—21no—
22C22⋊C4×C71—41no—
23C42×C71D411yescharacteristic
24C23×C71—21no—
25Q8×C71—21no—
26Q8×C71—41no—
27Q8×C71—41no—
28D4×C71—21no—
29D4×C71—41no—
30D4×C71—41no—
31D4×C71—41no—
32C2×C4×C71—21no—
33C2×C4×C71—21no—
34C2×C4×C71—41no—
35C2×C4×C71—42no—
36C2×C4×C71C22⋊C411yescharacteristic
37C2×Dic71—81no—
38C8×C71—41no—
39C71⋊C8—42no—
40C22×C71—41no—
41C22×C71—41no—
42C22×C71—81no—
43C22×C71#9911yescharacteristic
44C4×C71—21no—
45C4×C71—41no—
46C4×C71—41no—
47C4×C71—41no—
48Dic71—161no—
49C2×C71—21no—
50C2×C71—81no—
51C2×C71—81no—
52C2×C71AΣL(1,𝔽16)11yescharacteristic
53#1404—711nomaximal, Sylow
54C71#140411yescharacteristic, complemented, Sylow
55#454D7111yescharacteristic
56#354—711no—
57#329—711no—
582−1+4Dic7111yescharacteristic, complemented
59AGL(1,ℤ/8)—21no—
60#124C2×D7111yescharacteristic
61#104—21no—
62#101Dic7111yescharacteristic, complemented
63#99—1421no—
64#97—1421no—
65#41—21no—
66#41—41no—
67#41—41no—
68C2×Q8—41no—
69C2×Q8C2×Dic7111yescharacteristic
70C2×D4C22⋊D7111yescharacteristic
71SD16—41no—
72D8—41no—
73M16—21no—
74C2×C8—2841no—
75C22⋊C4—41no—
76C22⋊C4—2841no—
77C42C22⋊D7111yescharacteristic
78C23—21no—
79Q8—21no—
80Q8—41no—
81Q8—41no—
82D4—21no—
83D4—41no—
84D4—41no—
85D4—41no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—41no—
89C2×C4—42no—
90C2×C4—5681no—
91C2×C4C22⋊Dic7111yescharacteristic
92C8—41no—
93C8—2842no—
94C22—41no—
95C22—41no—
96C22—81no—
97C22#1455604311yescharacteristic
98C4—21no—
99C4—41no—
100C4—41no—
101C4—41no—
102C4—11361no—
103C2—21no—
104C2—81no—
105C2—81no—
106C2#1798578611yescharacteristic
subgroupG/H
centerC2#17985786
commutatorC2×Q8×C71C2×C4
Frattini#124C2×D71
Fitting#454×C71C2
socleC2×C71AΣL(1,𝔽16)
hypercenter#454D71
hypocenterC71#1404
LCSG ▷ C2×Q8×C71 ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #17985794C2semidirect
2. #17985739C2semidirect
6. 2−1+4×C71C4semidirect
10. #101×C71C4semidirect
54. C71#1404semidirect
58. 2−1+4Dic71semidirect
62. #101Dic71semidirect