Finite Group Databasebeta

← #36369842

Subgroups — #36369842

Order 22144.

270 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 54 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#26012470C212yesmaximal, complemented
2#26012395C214yesmaximal, complemented
3#26012058C211yesmaximal, characteristic
4#19149758—22no—
5#19149738C2214yescomplemented
6#19149737—22no—
7C2×#14583826C2211yescharacteristic
8#19149675C2212yes—
9#19149666C412yescomplemented
10#19149650—22no—
11#19149648—22no—
12#19149638C411yescharacteristic, complemented
13C22×D4×C173C411yescharacteristic, complemented
14C4⋊#10621924—42no—
15C22⋊#10621924—24no—
16C4×#10621924—42no—
17#14583826—42no—
18C2×Dic346C2×C411yescharacteristic
19C2×D692C2311yescharacteristic
20C2×C22⋊D173—24no—
21C2×C22⋊D173D412yes—
22C2×C4×D173D412yes—
23C4⋊Dic173—21no—
24C22⋊Dic173—24no—
25C22⋊Dic173—21no—
26C4×Dic173C2×C412yes—
27C24×C173—21no—
28C2×D4×C173—21no—
29C2×D4×C173—42no—
30C2×D4×C173C2×C412yes—
31C22×C4×C173C2×C411yescharacteristic
32C2×#10621924—44no—
33C22×D173C2×D412yes—
34Dic346—41no—
35D692—41no—
36C22⋊D173—42no—
37C22⋊D173—84no—
38C2×Dic173—22no—
39C2×Dic173C22⋊C412yes—
40C4×D173—22no—
41C4×D173—22no—
42C23×C173—22no—
43C23×C173—42no—
44C23×C173—82no—
45C23×C173C22⋊C411yescharacteristic
46D4×C173—42no—
47D4×C173—42no—
48C2×C4×C173—22no—
49C2×C4×C173C22⋊C411yescharacteristic
50C2×C4×C173C22×C411yescharacteristic
51C22×C173—21no—
52C22×C173—41no—
53C22×C173—44no—
54C22×C173—42no—
55C22×C173—84no—
56C22×C173C2×C22⋊C411yescharacteristic
57C2×D173—22no—
58C2×D173—42no—
59#10621924—84no—
60C4×C173—21no—
61C4×C173#9912yes—
62Dic173—22no—
63Dic173—42no—
64C2×C173—21no—
65C2×C173—41no—
66C2×C173—82no—
67C2×C173C2×#9911yescharacteristic
68D173—42no—
69C173#212811yescharacteristic, complemented, Sylow
70#2128—1731nomaximal, Sylow
71#535—1731no—
72#410—1732no—
73#353—1734no—
74C2×#41—1731no—
75C22×D4#1062192411yescharacteristic, complemented
76#127—1732no—
77#124—1732no—
78#121—3462no—
79#120—3462no—
80#117—3462no—
81#117—1731no—
82#99—3462no—
83#99—1734no—
84C24—21no—
85#41—6922no—
86C2×Q8—1731no—
87C2×D4—21no—
88C2×D4—42no—
89C2×D4—3464no—
90C2×D4—1732no—
91C2×D4—1731no—
92C2×D4C2×#1062192412yes—
93C22×C4—1732no—
94C22×C4C2×#1062192411yescharacteristic
95C4⋊C4—3461no—
96C4⋊C4—6922no—
97C22⋊C4—3461no—
98C22⋊C4—3464no—
99C22⋊C4—3464no—
100C42—1732no—
101C42—6922no—
102C23—22no—
103C23—42no—
104C23—82no—
105C23—1732no—
106C23C22⋊#1062192411yescharacteristic
107Q8—6921no—
108D4—42no—
109D4—42no—
110D4—13844no—
111D4—6921no—
112D4—6922no—
113C2×C4—22no—
114C2×C4—3462no—
115C2×C4—3462no—
116C2×C4—3462no—
117C2×C4—1732no—
118C2×C4—6924no—
119C2×C4C22⋊#1062192411yescharacteristic
120C2×C4C22×#1062192411yescharacteristic
121C22—21no—
122C22—41no—
123C22—44no—
124C22—42no—
125C22—84no—
126C22—3462no—
127C22—6922no—
128C22C2×C22⋊#1062192411yescharacteristic
129C4—21no—
130C4—3462no—
131C4—13844no—
132C4—6922no—
133C4#1914973712yes—
134C2—21no—
135C2—41no—
136C2—82no—
137C2—6922no—
138C2C2×#1914973711yescharacteristic
subgroupG/H
centerC2C2×#19149737
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC22×D4×C173C4
socleC2×C173C2×#99
prime radical#26012058C2
cosocleC2×D692C23
hypercenterC22×D4#10621924
hypocenterC173#2128
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #26012470C2semidirect
2. #26012395C2semidirect
5. #19149738C22semidirect
9. #19149666C4semidirect
12. #19149638C4semidirect
13. C22×D4×C173C4semidirect
69. C173#2128semidirect
75. C22×D4#10621924semidirect

Maximal subgroups

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

subgroupindex
#260124702
#260123952
#260120582
#2128173