Finite Group Databasebeta

← #17993900

Subgroups — #17993900

Order 4736.

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#14558275C211yesmaximal, characteristic
2#14557895C212yesmaximal, complemented
3#14557273C211yesmaximal, characteristic, complemented
4#14557272C212yesmaximal, complemented
5C37×#497C211yesmaximal, characteristic, complemented
6#11924525—21no—
7#11924524C2212yes—
8#11924522—22no—
9#11924506—22no—
10#11924504—21no—
11#11924459—22no—
12#11924456—22no—
13#11924447C2212yescomplemented
14C2×Q16×C37—21no—
15C2×SD16×C37—21no—
16C4⋊Q8×C37C2211yescharacteristic, complemented
17#124×C37C2211yescharacteristic
18#97×C37C2211yescharacteristic, complemented
19C23×D37—21no—
20C2×Q8×C37—21no—
21C2×Q8×C37D411yescharacteristic
22C2×D4×C37D411yescharacteristic
23C2×C22⋊D37—22no—
24C2×D148—21no—
25C2×D148D412yes—
26C2×Dic74D412yes—
27Q16×C37—42no—
28SD16×C37—42no—
29C2×C8×C37—21no—
30C4⋊C4×C37—22no—
31C22⋊C4×C37—22no—
32C42×C37C2311yescharacteristic
33C22⋊Dic37—21no—
34#10596938—22no—
35#10596938—22no—
36#10596938—22no—
37C4×Dic37—21no—
38C2×#64436—22no—
39C23×C37—21no—
40C22×D37—22no—
41C22×D37—81no—
42C22×D37—82no—
43Q8×C37—41no—
44Q8×C37—41no—
45Q8×C37—41no—
46D4×C37—41no—
47C2×C4×C37—21no—
48C2×C4×C37—41no—
49C2×C4×C37C2×D412yes—
50C2×C4×C37C2×D411yescharacteristic
51C22⋊D37—82no—
52C2×Dic37—22no—
53D148—42no—
54D148—81no—
55Dic74—42no—
56C8×C37—22no—
57#64436—42no—
58C22×C37—41no—
59C22×C37—42no—
60C22×C37#12011yescharacteristic
61C2×D37—42no—
62C2×D37—42no—
63C2×D37—42no—
64C2×D37—84no—
65C4×C37—21no—
66C4×C37—22no—
67C4×C37—41no—
68C4×C37—81no—
69Dic37—42no—
70#1659—371nomaximal, Sylow
71C2×C37—81no—
72C2×C37#45311yescharacteristic
73C2×C37#45512yes—
74D37—82no—
75H(3,𝔽4)—371no—
76#497D3711yescharacteristic, complemented
77#489—372no—
78#329—371no—
79#329—372no—
80C37#165911yescharacteristic, complemented, Sylow
81C2×Q16—21no—
82C2×SD16—21no—
83C4⋊Q8C2×D3711yescharacteristic, complemented
84#124—372no—
85#124—742no—
86#124—741no—
87#124C2×D3711yescharacteristic
88#120—742no—
89#120—741no—
90#103—742no—
91#102—742no—
92#97—372no—
93#97C2×D3711yescharacteristic, complemented
94C24—741no—
95C2×Q8—21no—
96C2×Q8—372no—
97C2×Q8C22⋊D3711yescharacteristic
98C2×D4—372no—
99C2×D4—741no—
100C2×D4—742no—
101C2×D4C22⋊D3711yescharacteristic
102Q16—42no—
103SD16—42no—
104C2×C8—21no—
105C2×C8—742no—
106C4⋊C4—22no—
107C22⋊C4—22no—
108C22⋊C4—742no—
109C22⋊C4—741no—
110C22⋊C4—742no—
111C22⋊C4—742no—
112C42—741no—
113C42C22×D3711yescharacteristic
114C23—21no—
115C23—2961no—
116C23—2962no—
117C23—742no—
118Q8—41no—
119Q8—41no—
120Q8—41no—
121Q8—1482no—
122D4—41no—
123D4—2961no—
124D4—2962no—
125D4—1482no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—742no—
129C2×C4D4×D3712yes—
130C2×C4C2×C22⋊D3711yescharacteristic
131C8—22no—
132C8—1482no—
133C22—41no—
134C22—42no—
135C22—2964no—
136C22—1482no—
137C22—1482no—
138C22—1482no—
139C22#1192450411yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—81no—
144C4—1482no—
145C2—81no—
146C2—2962no—
147C2#1455770011yescharacteristic
148C2#1455770912yes—
subgroupG/H
centerC22#11924504
commutator, cosocleC42×C37C23
FrattiniC42C22×D37
FittingC37×#497C2
socleC22×C37#120
prime radical#14558275C2
hypercenter#497D37
hypocenterC37#1659
LCSG ▷ C42×C37 ▷ C22×C37 ▷ C37
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
2. #14557895C2semidirect
3. #14557273C2semidirect
4. #14557272C2semidirect
5. C37×#497C2semidirect
13. #11924447C22semidirect
16. C4⋊Q8×C37C22semidirect
18. #97×C37C22semidirect
76. #497D37semidirect
80. C37#1659semidirect
83. C4⋊Q8C2×D37semidirect
93. #97C2×D37semidirect

Maximal subgroups

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

subgroupindex
#145582752
#145578952
#145572732
#145572722
C37×#4972
#165937