Finite Group Databasebeta

← #36352867

Subgroups — #36352867

Order 22144.

228 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 45 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#26012233C211yesmaximal, characteristic
2#26011853C212yesmaximal, complemented
3#26011231C211yesmaximal, characteristic, complemented
4#26011230C212yesmaximal, complemented
5#497×C173C211yesmaximal, characteristic, complemented
6#19149668—21no—
7#19149667C2212yes—
8#19149665—22no—
9#19149649—22no—
10#19149647—21no—
11#19149602—22no—
12#19149599—22no—
13#19149590C2212yescomplemented
14C2×Q16×C173—21no—
15C2×SD16×C173—21no—
16C4⋊Q8×C173C2211yescharacteristic, complemented
17#124×C173C2211yescharacteristic
18#97×C173C2211yescharacteristic, complemented
19C23×D173—21no—
20C2×Dic346D412yes—
21C2×D692—21no—
22C2×D692D412yes—
23C2×C22⋊D173—22no—
24C2×#13063436—22no—
25C22⋊Dic173—21no—
26#14583803—22no—
27#14583803—22no—
28#14583803—22no—
29C4×Dic173—21no—
30C2×Q8×C173—21no—
31C2×Q8×C173D411yescharacteristic
32C2×D4×C173D411yescharacteristic
33Q16×C173—42no—
34SD16×C173—42no—
35C2×C8×C173—21no—
36C4⋊C4×C173—22no—
37C22⋊C4×C173—22no—
38C42×C173C2311yescharacteristic
39C22×D173—22no—
40C22×D173—81no—
41C22×D173—82no—
42Dic346—42no—
43D692—42no—
44D692—81no—
45C22⋊D173—82no—
46C2×Dic173—22no—
47#13063436—42no—
48C23×C173—21no—
49Q8×C173—41no—
50Q8×C173—41no—
51Q8×C173—41no—
52D4×C173—41no—
53C2×C4×C173—21no—
54C2×C4×C173—41no—
55C2×C4×C173C2×D412yes—
56C2×C4×C173C2×D411yescharacteristic
57C8×C173—22no—
58C22×C173—41no—
59C22×C173—42no—
60C22×C173#12011yescharacteristic
61C2×D173—42no—
62C2×D173—42no—
63C2×D173—42no—
64C2×D173—84no—
65C4×C173—21no—
66C4×C173—22no—
67C4×C173—41no—
68C4×C173—81no—
69Dic173—42no—
70C2×C173—81no—
71C2×C173#45311yescharacteristic
72C2×C173#45512yes—
73D173—82no—
74C173#165911yescharacteristic, complemented, Sylow
75#1659—1731nomaximal, Sylow
76H(3,𝔽4)—1731no—
77#497D17311yescharacteristic, complemented
78#489—1732no—
79#329—1731no—
80#329—1732no—
81C2×Q16—21no—
82C2×SD16—21no—
83C4⋊Q8C2×D17311yescharacteristic, complemented
84#124—3462no—
85#124—3461no—
86#124—1732no—
87#124C2×D17311yescharacteristic
88#120—3462no—
89#120—3461no—
90#103—3462no—
91#102—3462no—
92#97—1732no—
93#97C2×D17311yescharacteristic, complemented
94C24—3461no—
95C2×Q8—21no—
96C2×Q8—1732no—
97C2×Q8C22⋊D17311yescharacteristic
98C2×D4—3461no—
99C2×D4—3462no—
100C2×D4—1732no—
101C2×D4C22⋊D17311yescharacteristic
102Q16—42no—
103SD16—42no—
104C2×C8—21no—
105C2×C8—3462no—
106C4⋊C4—22no—
107C22⋊C4—22no—
108C22⋊C4—3462no—
109C22⋊C4—3461no—
110C22⋊C4—3462no—
111C22⋊C4—3462no—
112C42—3461no—
113C42C22×D17311yescharacteristic
114C23—21no—
115C23—3462no—
116C23—13841no—
117C23—13842no—
118Q8—41no—
119Q8—41no—
120Q8—41no—
121Q8—6922no—
122D4—41no—
123D4—13841no—
124D4—13842no—
125D4—6922no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—3462no—
129C2×C4D4×D17312yes—
130C2×C4C2×C22⋊D17311yescharacteristic
131C8—22no—
132C8—6922no—
133C22—41no—
134C22—42no—
135C22—13844no—
136C22—6922no—
137C22—6922no—
138C22—6922no—
139C22#1914964711yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—81no—
144C4—6922no—
145C2—81no—
146C2—13842no—
147C2#2601165811yescharacteristic
148C2#2601166712yes—
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#497×C173C2
socleC22×C173#120
prime radical#26012233C2
hypercenter#497D173
hypocenterC173#1659
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
2. #26011853C2semidirect
3. #26011231C2semidirect
4. #26011230C2semidirect
5. #497×C173C2semidirect
13. #19149590C22semidirect
16. C4⋊Q8×C173C22semidirect
18. #97×C173C22semidirect
74. C173#1659semidirect
77. #497D173semidirect
83. C4⋊Q8C2×D173semidirect
93. #97C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260122332
#260118532
#260112312
#260112302
#497×C1732
#1659173