Finite Group Databasebeta

← #16818799

Subgroups — #16818799

Order 3712.

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#14271905C211yesmaximal, characteristic
2#14271558C211yesmaximal, characteristic, complemented
3#14271523C212yesmaximal, complemented
4#14270902C212yesmaximal, complemented
5C29×#329C211yesmaximal, characteristic, complemented
6C29×#124C2211yescharacteristic
7#11745309—21no—
8C2×#90785—21no—
9#11745299—21no—
10C2×#90783—21no—
11#11745247—22no—
12#11745244—22no—
13#11745237C2212yes—
14#11745231C2211yescharacteristic, complemented
15C29×#97C2212yescomplemented
16#11745169—22no—
17#11745164—22no—
18#11745151C2211yescharacteristic, complemented
19C23×D29—21no—
20C2×Q8×C29D411yescharacteristic
21C2×D4×C29D411yescharacteristic
22C2×C22⋊D29—22no—
23C2×D116—21no—
24C2×D116D412yes—
25C2×Dic58—21no—
26C2×Dic58D412yes—
27C2×C8×C29—22no—
28C22⋊C4×C29—22no—
29C42×C29C2311yescharacteristic
30C22⋊Dic29—21no—
31#90785—42no—
32#90783—42no—
33#90781—22no—
34#90781—22no—
35#90781—22no—
36C4⋊Dic29—22no—
37C4×Dic29—21no—
38C2×#6597—21no—
39C23×C29—21no—
40C22×D29—22no—
41C22×D29—81no—
42C22×D29—82no—
43Q8×C29—41no—
44D4×C29—41no—
45C2×C4×C29—21no—
46C2×C4×C29C2×D411yescharacteristic
47C2×C4×C29C2×D412yes—
48C22⋊D29—82no—
49C2×Dic29—22no—
50C2×Dic29—41no—
51D116—42no—
52D116—81no—
53Dic58—42no—
54Dic58—41no—
55Dic58—41no—
56C8×C29—42no—
57#6597—22no—
58#1659—291nomaximal, Sylow
59C22×C29—41no—
60C22×C29—42no—
61C22×C29#12011yescharacteristic
62C2×D29—42no—
63C2×D29—42no—
64C2×D29—42no—
65C2×D29—84no—
66C4×C29—21no—
67C4×C29—22no—
68C4×C29—41no—
69Dic29—42no—
70Dic29—81no—
71H(3,𝔽4)—291no—
72#497—291no—
73#489—292no—
74#329—292no—
75#329D2911yescharacteristic, complemented
76C2×C29—81no—
77C2×C29#45311yescharacteristic
78C2×C29#45512yes—
79D29—82no—
80C2×Q16—581no—
81C2×SD16—581no—
82C4⋊Q8—291no—
83#124—292no—
84#124—582no—
85#124—581no—
86#124C2×D2911yescharacteristic
87#120—582no—
88#120—581no—
89#103—582no—
90#102—582no—
91#97—291no—
92#97C2×D2912yescomplemented
93C29#165911yescharacteristic, complemented, Sylow
94C24—581no—
95C2×Q8—292no—
96C2×Q8—581no—
97C2×Q8D11611yescharacteristic
98C2×D4—292no—
99C2×D4—581no—
100C2×D4—582no—
101C2×D4D11611yescharacteristic
102Q16—1162no—
103SD16—1162no—
104C2×C8—22no—
105C2×C8—581no—
106C4⋊C4—582no—
107C22⋊C4—22no—
108C22⋊C4—582no—
109C22⋊C4—581no—
110C22⋊C4—582no—
111C22⋊C4—582no—
112C42—581no—
113C42C22×D2911yescharacteristic
114C23—21no—
115C23—582no—
116C23—2321no—
117C23—2322no—
118Q8—41no—
119Q8—1162no—
120Q8—1161no—
121Q8—1161no—
122D4—41no—
123D4—1162no—
124D4—2321no—
125D4—2322no—
126C2×C4—21no—
127C2×C4—582no—
128C2×C4—1161no—
129C2×C4C2×D11611yescharacteristic
130C2×C4D4×D2912yes—
131C8—42no—
132C8—582no—
133C22—41no—
134C22—42no—
135C22—1162no—
136C22—1162no—
137C22—1162no—
138C22—2324no—
139C22#1174524411yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—1162no—
144C4—2321no—
145C2—81no—
146C2—2322no—
147C2#1427133311yescharacteristic
148C2#1427134212yes—
subgroupG/H
centerC22#11745244
commutator, cosocleC42×C29C23
FrattiniC42C22×D29
FittingC29×#329C2
socleC22×C29#120
prime radical#14271905C2
hypercenter#329D29
hypocenterC29#1659
LCSG ▷ C42×C29 ▷ C22×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #14271558C2semidirect
3. #14271523C2semidirect
4. #14270902C2semidirect
5. C29×#329C2semidirect
14. #11745231C22semidirect
15. C29×#97C22semidirect
18. #11745151C22semidirect
75. #329D29semidirect
92. #97C2×D29semidirect
93. C29#1659semidirect

Maximal subgroups

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

subgroupindex
#142719052
#142715582
#142715232
#142709022
C29×#3292
#165929