Finite Group Databasebeta

← #36351713

Subgroups — #36351713

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#26011663C211yesmaximal, characteristic
2#26011231C211yesmaximal, characteristic, complemented
3#354×C173C211yesmaximal, characteristic, complemented
4#19149727C411yescharacteristic, complemented
5#19149705—21no—
6#19149607—21no—
7#19149598C411yescharacteristic, complemented
8#19149590—21no—
9#124×C173C2211yescharacteristic
10#99×C173—21no—
11#14583827—41no—
12#14583826—21no—
13#14583826—41no—
14Q8×D173—41no—
15#14583814—41no—
16#14583811—41no—
17#14583810—21no—
18C2×#13063436—41no—
19C2×Q8×C173C2×C411yescharacteristic
20C2×D4×C173D411yescharacteristic
21C22⋊C4×C173—41no—
22C22⋊C4×C173—41no—
23C42×C173D411yescharacteristic
24Dic346—21no—
25Dic346—41no—
26D692—21no—
27D692—41no—
28C22⋊D173—41no—
29C4×D173—41no—
30C4×D173—42no—
31#13063436—41no—
32#13063436—42no—
33C23×C173—21no—
34Q8×C173—41no—
35D4×C173—41no—
36C2×C4×C173—21no—
37C2×C4×C173—21no—
38C2×C4×C173—81no—
39C2×C4×C173C22⋊C411yescharacteristic
40C22×C173—41no—
41C22×C173—81no—
42C22×C173#9911yescharacteristic
43C2×D173—41no—
44C4×C173—21no—
45C4×C173—41no—
46C4×C173—41no—
47C4×C173—161no—
48Dic173—41no—
49C2×C173—21no—
50C2×C173—81no—
51C2×C173AΣL(1,𝔽16)11yescharacteristic
52D173—81no—
53C173#140411yescharacteristic, complemented, Sylow
54#1404—1731nomaximal, Sylow
55#454—1731no—
56#354D17311yescharacteristic, complemented
57#329—1731no—
582−1+4—1731no—
59AGL(1,ℤ/8)—3461no—
60#124C2×D17311yescharacteristic
61#104—3461no—
62#101—1731no—
63#99—21no—
64#97—3461no—
65#41—3461no—
66#41—6921no—
67#41—6921no—
68C2×Q8—6921no—
69C2×Q8C4×D17311yescharacteristic
70C2×D4C22⋊D17311yescharacteristic
71SD16—6921no—
72D8—6921no—
73M16—3461no—
74C2×C8—6921no—
75C22⋊C4—41no—
76C22⋊C4—41no—
77C42D69211yescharacteristic
78C23—21no—
79Q8—41no—
80Q8—3461no—
81Q8—6921no—
82D4—41no—
83D4—3461no—
84D4—6921no—
85D4—6921no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—81no—
89C2×C4—6921no—
90C2×C4—6922no—
91C2×C4#1458380311yescharacteristic
92C8—6921no—
93C8—6922no—
94C22—41no—
95C22—81no—
96C22—6921no—
97C22#1914959311yescharacteristic
98C4—21no—
99C4—41no—
100C4—41no—
101C4—161no—
102C4—6921no—
103C2—21no—
104C2—81no—
105C2—13841no—
106C2#2601127711yescharacteristic
subgroupG/H
centerC2#26011277
commutatorC2×Q8×C173C2×C4
Frattini#124C2×D173
Fitting#354×C173C2
socleC2×C173AΣL(1,𝔽16)
prime radical#26011663C2
cosocle#124×C173C22
hypercenter#354D173
hypocenterC173#1404
LCSG ▷ C2×Q8×C173 ▷ C2×C4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #354

Complements

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

NH ≅ G/Ntype
2. #26011231C2semidirect
3. #354×C173C2semidirect
4. #19149727C4semidirect
7. #19149598C4semidirect
53. C173#1404semidirect
56. #354D173semidirect

Maximal subgroups

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

subgroupindex
#260116632
#260112312
#354×C1732
#1404173