Finite Group Databasebeta

← #36368689

Subgroups — #36368689

Order 22144.

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

normal subgroups only

This group has 106 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#26012401C211yesmaximal, characteristic, complemented
2#26012351C211yesmaximal, characteristic, complemented
3#26011676C211yesmaximal, characteristic
4#19149746—21no—
5#19149740—21no—
6#19149727C411yescharacteristic
7#19149712—21no—
8#19149677C2211yescharacteristic
9#19149605—21no—
10#101×C173C411yescharacteristic
11#14583835—41no—
12C2×#13063443—41no—
13#14583827—41no—
14#14583826—21no—
15#14583826—41no—
16C2×Dic346D411yescharacteristic
17Q8×D173—41no—
18Dic692—41no—
19#14583815—41no—
20#14583805—41no—
21C4×Dic173D411yescharacteristic
22C2×Q8×C173C2×C411yescharacteristic
23M16×C173—21no—
24#13063443—81no—
25#13063443—81no—
26Dic346—21no—
27Dic346—41no—
28Dic346—41no—
29D692—21no—
30D692—41no—
31C22⋊D173—41no—
32C2×Dic173—21no—
33C2×Dic173—21no—
34C4×D173—41no—
35C4×D173—42no—
36Q8×C173—41no—
37C2×C4×C173—21no—
38C2×C4×C173C22⋊C411yescharacteristic
39C8×C173—41no—
40C22×C173#9911yescharacteristic
41C2×D173—41no—
42C4×C173—21no—
43C4×C173—41no—
44Dic173—21no—
45Dic173—21no—
46Dic173—41no—
47Dic173—41no—
48C2×C173—21no—
49C2×C173AΣL(1,𝔽16)11yescharacteristic
50D173—81no—
51C173#140511yescharacteristic, complemented, Sylow
52#1405—1731nomaximal, Sylow
53#456—1731no—
54#356—1731no—
55#333—1731no—
562−1+4—1731no—
57#137—3461no—
58C4⋊Q8—1731no—
59C4⋊C8—3461no—
60#104—3461no—
61#101—3461no—
62#101#1062192411yescharacteristic
63#41—3461no—
64#41—6921no—
65#41—6921no—
66C2×Q8—1731no—
67C2×Q8—6921no—
68C2×Q8C2×#1062192411yescharacteristic
69Q16—6921no—
70SD16—6921no—
71M16—21no—
72M16—6921no—
73C2×C8—6921no—
74C4⋊C4—6921no—
75C42—1731no—
76Q8—41no—
77Q8—3461no—
78Q8—6921no—
79Q8—6921no—
80D4—3461no—
81D4—6921no—
82D4—6921no—
83C2×C4—21no—
84C2×C4—3461no—
85C2×C4—3461no—
86C2×C4—6921no—
87C2×C4—6922no—
88C2×C4C22⋊#1062192411yescharacteristic
89C8—41no—
90C8—13841no—
91C8—13841no—
92C22—6921no—
93C22#1914973811yescharacteristic
94C4—21no—
95C4—41no—
96C4—3461no—
97C4—3461no—
98C4—6921no—
99C4—6921no—
100C2—21no—
101C2—13841no—
102C2#2601239211yescharacteristic
subgroupG/H
centerC2#26012392
commutatorC2×Q8×C173C2×C4
FrattiniC2×Q8C2×#10621924
Fitting#101×C173C4
socleC2×C173AΣL(1,𝔽16)
prime radical#19149727C4
cosocle#19149677C22
hypercenter#101#10621924
hypocenterC173#1405
LCSG ▷ C2×Q8×C173 ▷ C2×C4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C2×Q8 ◁ #101

Complements

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

NH ≅ G/Ntype
1. #26012401C2semidirect
2. #26012351C2semidirect
51. C173#1405semidirect

Maximal subgroups

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

subgroupindex
#260124012
#260123512
#260116762
#1405173