Finite Group Databasebeta

← #36352869

Subgroups — #36352869

Order 22144.

228 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 49 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#26011886C211yesmaximal, characteristic, complemented
2#26011850C212yesmaximal, complemented
3#26011231C211yesmaximal, characteristic, complemented
4#26011231C212yesmaximal, complemented
5H(3,𝔽4)×C173C211yesmaximal, characteristic
6C2×#14583816—21no—
7C2×#14583814—21no—
8#19149679C2211yescharacteristic, complemented
9#19149603—22no—
10#19149600—22no—
11#19149590C2212yescomplemented
12#19149590C2211yescharacteristic, complemented
13#124×C173—22no—
14#124×C173—21no—
15#124×C173C2212yes—
16#124×C173C2211yescharacteristic
17#120×C173—22no—
18#120×C173—21no—
19C2×Dic346—21no—
20#14583816—42no—
21#14583814—42no—
22C2×#13063436—21no—
23C2×#13063436—22no—
24C4⋊Dic173—22no—
25C24×C173—21no—
26C2×Q8×C173D412yes—
27C2×Q8×C173D411yescharacteristic
28C2×D4×C173—21no—
29C2×D4×C173—22no—
30C2×D4×C173D412yes—
31C2×D4×C173D411yescharacteristic
32C22⋊C4×C173—22no—
33C22⋊C4×C173—21no—
34C22⋊C4×C173—22no—
35C22⋊C4×C173—22no—
36C22⋊C4×C173—22no—
37C42×C173—21no—
38C42×C173C2311yescharacteristic
39Dic346—41no—
40Dic346—41no—
41C2×Dic173—41no—
42#13063436—22no—
43#13063436—42no—
44C23×C173—21no—
45C23×C173—22no—
46C23×C173—81no—
47C23×C173—82no—
48Q8×C173—41no—
49Q8×C173—42no—
50D4×C173—41no—
51D4×C173—42no—
52D4×C173—81no—
53D4×C173—82no—
54C2×C4×C173—21no—
55C2×C4×C173—22no—
56C2×C4×C173C2×D411yescharacteristic
57C2×C4×C173C2×D412yes—
58C22×C173—41no—
59C22×C173—42no—
60C22×C173—42no—
61C22×C173—42no—
62C22×C173—42no—
63C22×C173—84no—
64C22×C173#12011yescharacteristic
65C4×C173—21no—
66C4×C173—22no—
67C4×C173—41no—
68C4×C173—42no—
69Dic173—81no—
70C2×C173—81no—
71C2×C173—82no—
72C2×C173#45311yescharacteristic
73C2×C173#45512yes—
74C173#165911yescharacteristic, complemented, Sylow
75#1659—1731nomaximal, Sylow
76H(3,𝔽4)D17311yescharacteristic
77#497—1731no—
78#489—1732no—
79#329—1731no—
80#329—1732no—
81C2×Q16—3461no—
82C2×SD16—3461no—
83C4⋊Q8—1731no—
84#124—22no—
85#124—21no—
86#124C2×D17312yes—
87#124C2×D17311yescharacteristic
88#120—22no—
89#120—21no—
90#103—3462no—
91#102—3462no—
92#97—1732no—
93#97—1731no—
94C24—21no—
95C2×Q8—3461no—
96C2×Q8C22⋊D17312yes—
97C2×Q8C22⋊D17311yescharacteristic
98C2×D4—21no—
99C2×D4—22no—
100C2×D4C22⋊D17312yes—
101C2×D4C22⋊D17311yescharacteristic
102Q16—6922no—
103SD16—6922no—
104C2×C8—3461no—
105C2×C8—3462no—
106C4⋊C4—3462no—
107C22⋊C4—22no—
108C22⋊C4—21no—
109C22⋊C4—22no—
110C22⋊C4—22no—
111C22⋊C4—22no—
112C42—21no—
113C42C22×D17311yescharacteristic
114C23—21no—
115C23—22no—
116C23—81no—
117C23—82no—
118Q8—41no—
119Q8—42no—
120Q8—6921no—
121Q8—6921no—
122D4—41no—
123D4—42no—
124D4—81no—
125D4—82no—
126C2×C4—21no—
127C2×C4—22no—
128C2×C4—6921no—
129C2×C4C2×C22⋊D17312yes—
130C2×C4C2×C22⋊D17311yescharacteristic
131C8—3462no—
132C8—6922no—
133C22—41no—
134C22—42no—
135C22—42no—
136C22—42no—
137C22—42no—
138C22—84no—
139C22#1914964811yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—42no—
144C4—13841no—
145C2—81no—
146C2—82no—
147C2#2601165911yescharacteristic
148C2#2601166912yes—
subgroupG/H
centerC22#19149648
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting, prime radicalH(3,𝔽4)×C173C2
socleC22×C173#120
hypercenterH(3,𝔽4)D173
hypocenterC173#1659
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #26011886C2semidirect
2. #26011850C2semidirect
3. #26011231C2semidirect
4. #26011231C2semidirect
8. #19149679C22semidirect
11. #19149590C22semidirect
12. #19149590C22semidirect
74. C173#1659semidirect

Maximal subgroups

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

subgroupindex
#260118862
#260118502
#260112312
H(3,𝔽4)×C1732
#1659173