Finite Group Databasebeta

← #36352870

Subgroups — #36352870

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#26011886C211yesmaximal, characteristic, complemented
3#26011851C212yesmaximal, complemented
4#26011230C212yesmaximal, complemented
5#329×C173C211yesmaximal, characteristic, complemented
6C2×#14583816—21no—
7C2×#14583814—21no—
8#19149679C2211yescharacteristic, complemented
9#19149668—21no—
10#19149667C2212yes—
11#19149665—22no—
12#19149649—22no—
13#19149647—21no—
14#19149604—22no—
15#19149601—22no—
16#19149590C2211yescharacteristic, complemented
17#124×C173C2211yescharacteristic
18#97×C173C2212yescomplemented
19C23×D173—21no—
20C2×Dic346—21no—
21C2×Dic346D412yes—
22C2×D692—21no—
23C2×D692D412yes—
24C2×C22⋊D173—22no—
25#14583816—42no—
26#14583814—42no—
27C2×#13063436—21no—
28C4⋊Dic173—22no—
29C22⋊Dic173—21no—
30#14583803—22no—
31#14583803—22no—
32#14583803—22no—
33C4×Dic173—21no—
34C2×Q8×C173D411yescharacteristic
35C2×D4×C173D411yescharacteristic
36C2×C8×C173—22no—
37C22⋊C4×C173—22no—
38C42×C173C2311yescharacteristic
39C22×D173—22no—
40C22×D173—81no—
41C22×D173—82no—
42Dic346—42no—
43Dic346—41no—
44Dic346—41no—
45D692—42no—
46D692—81no—
47C22⋊D173—82no—
48C2×Dic173—22no—
49C2×Dic173—41no—
50#13063436—22no—
51C23×C173—21no—
52Q8×C173—41no—
53D4×C173—41no—
54C2×C4×C173—21no—
55C2×C4×C173C2×D412yes—
56C2×C4×C173C2×D411yescharacteristic
57C8×C173—42no—
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—
68Dic173—42no—
69Dic173—81no—
70C2×C173—81no—
71C2×C173#45311yescharacteristic
72C2×C173#45512yes—
73D173—82no—
74C173#165911yescharacteristic, complemented, Sylow
75#1659—1731nomaximal, Sylow
76H(3,𝔽4)—1731no—
77#497—1731no—
78#489—1732no—
79#329—1732no—
80#329D17311yescharacteristic, complemented
81C2×Q16—3461no—
82C2×SD16—3461no—
83C4⋊Q8—1731no—
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—1731no—
93#97C2×D17312yescomplemented
94C24—3461no—
95C2×Q8—3461no—
96C2×Q8—1732no—
97C2×Q8D69211yescharacteristic
98C2×D4—3461no—
99C2×D4—3462no—
100C2×D4—1732no—
101C2×D4D69211yescharacteristic
102Q16—6922no—
103SD16—6922no—
104C2×C8—22no—
105C2×C8—3461no—
106C4⋊C4—3462no—
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—6921no—
120Q8—6921no—
121Q8—6922no—
122D4—41no—
123D4—13841no—
124D4—13842no—
125D4—6922no—
126C2×C4—21no—
127C2×C4—3462no—
128C2×C4—6921no—
129C2×C4D4×D17312yes—
130C2×C4C2×D69211yescharacteristic
131C8—42no—
132C8—3462no—
133C22—41no—
134C22—42no—
135C22—13844no—
136C22—6922no—
137C22—6922no—
138C22—6922no—
139C22#1914964911yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—13841no—
144C4—6922no—
145C2—81no—
146C2—13842no—
147C2#2601166111yescharacteristic
148C2#2601167012yes—
subgroupG/H
centerC22#19149649
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#329×C173C2
socleC22×C173#120
prime radical#26012233C2
hypercenter#329D173
hypocenterC173#1659
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #26011886C2semidirect
3. #26011851C2semidirect
4. #26011230C2semidirect
5. #329×C173C2semidirect
8. #19149679C22semidirect
16. #19149590C22semidirect
18. #97×C173C22semidirect
74. C173#1659semidirect
80. #329D173semidirect
93. #97C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260122332
#260118862
#260118512
#260112302
#329×C1732
#1659173