Finite Group Databasebeta

← #36351709

Subgroups — #36351709

Order 22144.

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

normal subgroups only

This group has 122 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#26011658C211yesmaximal, characteristic
2#26011284C211yesmaximal, characteristic, complemented
3#331×C173C211yesmaximal, characteristic, complemented
4#19149724C411yescharacteristic, complemented
5#19149705—21no—
6#19149607—21no—
7#19149596C411yescharacteristic, complemented
8#19149594—21no—
9#127×C173C2211yescharacteristic
10C4⋊C8×C173—21no—
11#14583826—21no—
12#14583825—41no—
13C2×C22⋊D173—41no—
14D4×D173—41no—
15#14583814—41no—
16#14583811—41no—
17#14583810—21no—
18C22⋊Dic173—41no—
19C2×D4×C173—41no—
20C2×D4×C173C2×C411yescharacteristic
21C2×D4×C173D411yescharacteristic
22C2×C8×C173—41no—
23C42×C173D411yescharacteristic
24C22×D173—41no—
25Dic346—21no—
26D692—21no—
27C22⋊D173—42no—
28C22⋊D173—41no—
29C2×Dic173—41no—
30C2×Dic173—81no—
31C4×D173—41no—
32#13063436—41no—
33C23×C173—21no—
34C23×C173—21no—
35D4×C173—41no—
36D4×C173—41no—
37D4×C173—81no—
38D4×C173—81no—
39C2×C4×C173—21no—
40C2×C4×C173C22⋊C411yescharacteristic
41C8×C173—81no—
42C22×C173—41no—
43C22×C173—41no—
44C22×C173—81no—
45C22×C173—81no—
46C22×C173#9911yescharacteristic
47C2×D173—41no—
48C2×D173—161no—
49C4×C173—21no—
50C4×C173—21no—
51C4×C173—21no—
52Dic173—41no—
53Dic173—161no—
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#331D17311yescharacteristic, complemented
64H(5,𝔽2)—1731no—
65AGL(1,ℤ/8)—3461no—
66#127C2×D17311yescharacteristic
67C4⋊C8—21no—
68#104—3461no—
69#100—1731no—
70#99—3461no—
71#41—3461no—
72#41—6921no—
73C2×D4—41no—
74C2×D4—6921no—
75C2×D4—6921no—
76C2×D4C4×D17311yescharacteristic
77C2×D4D69211yescharacteristic
78SD16—6921no—
79D8—6921no—
80M16—3461no—
81C2×C8—41no—
82C22⋊C4—6921no—
83C42C22⋊D17311yescharacteristic
84C23—21no—
85C23—21no—
86C23—6921no—
87Q8—3461no—
88D4—41no—
89D4—41no—
90D4—81no—
91D4—81no—
92D4—3461no—
93D4—6922no—
94D4—6921no—
95C2×C4—21no—
96C2×C4—13841no—
97C2×C4—6921no—
98C2×C4—6921no—
99C2×C4#1458380311yescharacteristic
100C8—81no—
101C8—6921no—
102C22—41no—
103C22—41no—
104C22—81no—
105C22—81no—
106C22—6921no—
107C22—27681no—
108C22#1914959311yescharacteristic
109C4—21no—
110C4—21no—
111C4—21no—
112C4—6921no—
113C4—27681no—
114C2—21no—
115C2—81no—
116C2—81no—
117C2—13841no—
118C2#2601127911yescharacteristic
subgroupG/H
centerC2#26011279
commutatorC2×D4×C173C2×C4
Frattini#127C2×D173
Fitting#331×C173C2
socleC2×C173AΣL(1,𝔽16)
prime radical#26011658C2
cosocle#127×C173C22
hypercenter#331D173
hypocenterC173#1402
LCSG ▷ C2×D4×C173 ▷ C2×C4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #26011284C2semidirect
3. #331×C173C2semidirect
4. #19149724C4semidirect
7. #19149596C4semidirect
59. C173#1402semidirect
63. #331D173semidirect

Maximal subgroups

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

subgroupindex
#260116582
#260112842
#331×C1732
#1402173