Finite Group Databasebeta

← #36351712

Subgroups — #36351712

Order 22144.

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

normal subgroups only

This group has 112 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#26011668C211yesmaximal, characteristic
2#26011288C211yesmaximal, characteristic, complemented
3#329×C173C211yesmaximal, characteristic, complemented
4#19149724C411yescharacteristic
5#19149710—21no—
6#19149607—21no—
7#19149598—21no—
8#19149596C411yescharacteristic
9#124×C173C2211yescharacteristic
10#97×C173—21no—
11#14583826—21no—
12#14583825—41no—
13C2×C22⋊D173—41no—
14D4×D173—41no—
15#14583816—41no—
16#14583813—41no—
17#14583810—21no—
18#14583810—41no—
19C2×Q8×C173D411yescharacteristic
20C2×D4×C173C2×C411yescharacteristic
21C2×C8×C173—41no—
22C22⋊C4×C173—41no—
23C42×C173D411yescharacteristic
24C22×D173—41no—
25Dic346—21no—
26D692—21no—
27C22⋊D173—42no—
28C22⋊D173—41no—
29C2×Dic173—41no—
30C4×D173—41no—
31#13063436—41no—
32#13063436—81no—
33C23×C173—21no—
34Q8×C173—41no—
35D4×C173—41no—
36C2×C4×C173—21no—
37C2×C4×C173—21no—
38C2×C4×C173C22⋊C411yescharacteristic
39C8×C173—42no—
40C22×C173—41no—
41C22×C173—81no—
42C22×C173#9911yescharacteristic
43C2×D173—41no—
44C2×D173—161no—
45C4×C173—21no—
46C4×C173—41no—
47C4×C173—41no—
48Dic173—41no—
49C2×C173—21no—
50C2×C173—81no—
51C2×C173AΣL(1,𝔽16)11yescharacteristic
52D173—81no—
53C173#140311yescharacteristic, complemented, Sylow
54#1403—1731nomaximal, Sylow
55#455—1731no—
56#355—1731no—
57#329D17311yescharacteristic, complemented
58H(5,𝔽2)—1731no—
59#137—3461no—
60#124C2×D17311yescharacteristic
61#104—3461no—
62#101—3461no—
63#100—1731no—
64#97—21no—
65#41—3461no—
66#41—6921no—
67C2×Q8D69211yescharacteristic
68C2×D4—6921no—
69C2×D4—6921no—
70C2×D4C4×D17311yescharacteristic
71Q16—6921no—
72SD16—6921no—
73M16—3461no—
74M16—6921no—
75C2×C8—41no—
76C22⋊C4—41no—
77C42C22⋊D17311yescharacteristic
78C23—21no—
79C23—6921no—
80Q8—41no—
81Q8—3461no—
82D4—41no—
83D4—3461no—
84D4—6922no—
85D4—6921no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—6921no—
89C2×C4—6921no—
90C2×C4#1458380311yescharacteristic
91C8—42no—
92C8—13841no—
93C8—6921no—
94C22—41no—
95C22—81no—
96C22—6921no—
97C22—27681no—
98C22#1914959311yescharacteristic
99C4—21no—
100C4—41no—
101C4—41no—
102C4—6921no—
103C2—21no—
104C2—81no—
105C2—13841no—
106C2#2601127911yescharacteristic
subgroupG/H
centerC2#26011279
commutatorC2×D4×C173C2×C4
Frattini#124C2×D173
Fitting#329×C173C2
socleC2×C173AΣL(1,𝔽16)
prime radical#19149724C4
cosocle#124×C173C22
hypercenter#329D173
hypocenterC173#1403
LCSG ▷ C2×D4×C173 ▷ C2×C4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #26011288C2semidirect
3. #329×C173C2semidirect
53. C173#1403semidirect
57. #329D173semidirect

Maximal subgroups

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

subgroupindex
#260116682
#260112882
#329×C1732
#1403173