Finite Group Databasebeta

← #36368682

Subgroups — #36368682

Order 22144.

122 conjugacy classes of subgroups; only the 20 normal subgroups are shown.

normal subgroups only

This group has 122 conjugacy classes of subgroups — too many to draw. Only its 20 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#26012395C211yesmaximal, characteristic, complemented
2#26012349C211yesmaximal, characteristic, complemented
3#26011659C211yesmaximal, characteristic
4#19149746—21no—
5#19149738—21no—
6#19149704—21no—
7#19149675C2211yescharacteristic
8#19149606—21no—
9#19149596C411yescharacteristic, complemented
10H(5,𝔽2)×C173C411yescharacteristic, complemented
11C2×#13063443—41no—
12C22⋊#10621924—41no—
13C2×D692D411yescharacteristic
14C2×C22⋊D173—41no—
15#14583813—41no—
16#14583811—41no—
17#14583810—21no—
18C4×Dic173D411yescharacteristic
19#41×C173—21no—
20#41×C173—41no—
21C2×D4×C173—41no—
22C2×D4×C173—41no—
23C2×D4×C173C2×C411yescharacteristic
24C2×#10621924—81no—
25#13063443—81no—
26C22×D173—21no—
27D692—41no—
28C22⋊D173—81no—
29C22⋊D173—81no—
30C2×Dic173—21no—
31#13063436—41no—
32C23×C173—21no—
33C23×C173—41no—
34Q8×C173—21no—
35D4×C173—21no—
36D4×C173—41no—
37D4×C173—42no—
38D4×C173—41no—
39C2×C4×C173—41no—
40C2×C4×C173—41no—
41C2×C4×C173C22⋊C411yescharacteristic
42C22×C173—41no—
43C22×C173—41no—
44C22×C173—81no—
45C22×C173—161no—
46C22×C173#9911yescharacteristic
47C2×D173—41no—
48C2×D173—81no—
49#10621924—161no—
50C4×C173—21no—
51C4×C173—41no—
52Dic173—21no—
53Dic173—21no—
54C2×C173—21no—
55C2×C173—81no—
56C2×C173—81no—
57C2×C173AΣL(1,𝔽16)11yescharacteristic
58D173—81no—
59C173#140211yescharacteristic, complemented, Sylow
60#1402—1731nomaximal, Sylow
61#453—1731no—
62#353—1731no—
63#331—1731no—
64H(5,𝔽2)#1062192411yescharacteristic, complemented
65AGL(1,ℤ/8)—3461no—
66#127—1731no—
67C4⋊C8—3461no—
68#104—3461no—
69#100—1731no—
70#99—3461no—
71#41—21no—
72#41—41no—
73C2×D4—41no—
74C2×D4—41no—
75C2×D4—1731no—
76C2×D4—6921no—
77C2×D4C2×#1062192411yescharacteristic
78SD16—6921no—
79D8—6921no—
80M16—3461no—
81C2×C8—6921no—
82C22⋊C4—6921no—
83C42—1731no—
84C23—21no—
85C23—41no—
86C23—3461no—
87Q8—21no—
88D4—21no—
89D4—41no—
90D4—42no—
91D4—41no—
92D4—13841no—
93D4—13841no—
94D4—6921no—
95C2×C4—41no—
96C2×C4—41no—
97C2×C4—3461no—
98C2×C4—13841no—
99C2×C4C22⋊#1062192411yescharacteristic
100C8—13841no—
101C8—6921no—
102C22—41no—
103C22—41no—
104C22—81no—
105C22—161no—
106C22—13841no—
107C22—6921no—
108C22#1914973711yescharacteristic
109C4—21no—
110C4—41no—
111C4—3461no—
112C4—3461no—
113C4—27681no—
114C2—21no—
115C2—81no—
116C2—81no—
117C2—13841no—
118C2#2601239111yescharacteristic
subgroupG/H
centerC2#26012391
commutatorC2×D4×C173C2×C4
FrattiniC2×D4C2×#10621924
FittingH(5,𝔽2)×C173C4
socleC2×C173AΣL(1,𝔽16)
prime radical#26011659C2
cosocle#19149675C22
hypercenterH(5,𝔽2)#10621924
hypocenterC173#1402
LCSG ▷ C2×D4×C173 ▷ C2×C4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C2×D4 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #26012395C2semidirect
2. #26012349C2semidirect
9. #19149596C4semidirect
10. H(5,𝔽2)×C173C4semidirect
59. C173#1402semidirect
64. H(5,𝔽2)#10621924semidirect

Maximal subgroups

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

subgroupindex
#260123952
#260123492
#260116592
#1402173