Finite Group Databasebeta

← #22586495

Subgroups — #22586495

Order 9088.

106 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 106 conjugacy classes of subgroups — too many to draw. Only its 23 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#17986181C211yesmaximal, characteristic
2#17985798C211yesmaximal, characteristic, complemented
3#333×C71C211yesmaximal, characteristic, complemented
4#14556176C411yescharacteristic
5#14556159—21no—
6#14556056—21no—
7#14556047—21no—
8#14556047C411yescharacteristic
9C4⋊Q8×C71C2211yescharacteristic
10C4⋊C8×C71—21no—
11#11765866—41no—
12#11765865—21no—
13#11765865—41no—
14Q8×D71—41no—
15#11765855—41no—
16#11765852—41no—
17#11765849—21no—
18#11765849—41no—
19C2×Q8×C71C2×C411yescharacteristic
20C2×Q8×C71D411yescharacteristic
21C2×C8×C71—41no—
22C4⋊C4×C71—41no—
23C42×C71D411yescharacteristic
24Q8×C71—41no—
25Q8×C71—41no—
26C2×C4×C71—21no—
27C2×C4×C71—21no—
28C2×C4×C71—21no—
29C2×C4×C71C22⋊C411yescharacteristic
30C22⋊D71—41no—
31D284—21no—
32D284—41no—
33C4×D71—41no—
34C4×D71—42no—
35Dic142—21no—
36Dic142—41no—
37C8×C71—81no—
38C71⋊C8—41no—
39C71⋊C8—81no—
40C22×C71#9911yescharacteristic
41C2×D71—41no—
42C4×C71—21no—
43C4×C71—21no—
44C4×C71—21no—
45C4×C71—41no—
46C4×C71—41no—
47Dic71—41no—
48C2×C71—21no—
49C2×C71AΣL(1,𝔽16)11yescharacteristic
50D71—81no—
51#1405—711nomaximal, Sylow
52C71#140511yescharacteristic, complemented, Sylow
53#456—711no—
54#356—711no—
55#333D7111yescharacteristic, complemented
562−1+4—711no—
57#137—1421no—
58C4⋊Q8C2×D7111yescharacteristic
59C4⋊C8—21no—
60#104—1421no—
61#101—711no—
62#101—1421no—
63#41—2841no—
64#41—2841no—
65#41—1421no—
66C2×Q8—2841no—
67C2×Q8C4×D7111yescharacteristic
68C2×Q8D28411yescharacteristic
69Q16—2841no—
70SD16—2841no—
71M16—2841no—
72M16—1421no—
73C2×C8—41no—
74C4⋊C4—41no—
75C42C22⋊D7111yescharacteristic
76Q8—41no—
77Q8—41no—
78Q8—2841no—
79Q8—1421no—
80D4—2841no—
81D4—2841no—
82D4—1421no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—2841no—
87C2×C4—2842no—
88C2×C4(C2×D71)⋊C411yescharacteristic
89C8—81no—
90C8—2841no—
91C8—5681no—
92C22—2841no—
93C22#1455604211yescharacteristic
94C4—21no—
95C4—21no—
96C4—21no—
97C4—41no—
98C4—41no—
99C4—2841no—
100C2—21no—
101C2—5681no—
102C2#1798578711yescharacteristic
subgroupG/H
centerC2#17985787
commutatorC2×Q8×C71C2×C4
FrattiniC4⋊Q8C2×D71
Fitting#333×C71C2
socleC2×C71AΣL(1,𝔽16)
hypercenter#333D71
hypocenterC71#1405
LCSG ▷ C2×Q8×C71 ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #17985798C2semidirect
3. #333×C71C2semidirect
52. C71#1405semidirect
55. #333D71semidirect