Finite Group Databasebeta

← #16818798

Subgroups — #16818798

Order 3712.

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#14271558C211yesmaximal, characteristic, complemented
2#14271522C212yesmaximal, complemented
3#14270903C212yesmaximal, complemented
4#14270903C211yesmaximal, characteristic, complemented
5C29×H(3,𝔽4)C211yesmaximal, characteristic
6C29×#124—21no—
7C29×#124—22no—
8C29×#124C2212yes—
9C29×#124C2211yescharacteristic
10C29×#120—21no—
11C29×#120—22no—
12C2×#90785—21no—
13C2×#90783—21no—
14#11745231C2211yescharacteristic, complemented
15#11745183—22no—
16#11745180—22no—
17#11745151C2212yescomplemented
18#11745151C2211yescharacteristic, complemented
19C24×C29—21no—
20C2×Q8×C29D411yescharacteristic
21C2×Q8×C29D412yes—
22C2×D4×C29—21no—
23C2×D4×C29—22no—
24C2×D4×C29D411yescharacteristic
25C2×D4×C29D412yes—
26C2×Dic58—21no—
27C22⋊C4×C29—22no—
28C22⋊C4×C29—21no—
29C22⋊C4×C29—22no—
30C22⋊C4×C29—22no—
31C22⋊C4×C29—22no—
32C42×C29—21no—
33C42×C29C2311yescharacteristic
34#90785—42no—
35#90783—42no—
36C4⋊Dic29—22no—
37C2×#6597—21no—
38C2×#6597—22no—
39C23×C29—21no—
40C23×C29—22no—
41C23×C29—81no—
42C23×C29—82no—
43Q8×C29—41no—
44Q8×C29—42no—
45D4×C29—41no—
46D4×C29—42no—
47D4×C29—81no—
48D4×C29—82no—
49C2×C4×C29—21no—
50C2×C4×C29—22no—
51C2×C4×C29C2×D412yes—
52C2×C4×C29C2×D411yescharacteristic
53C2×Dic29—41no—
54Dic58—41no—
55Dic58—41no—
56#6597—22no—
57#6597—42no—
58#1659—291nomaximal, Sylow
59C22×C29—41no—
60C22×C29—42no—
61C22×C29—42no—
62C22×C29—42no—
63C22×C29—42no—
64C22×C29—84no—
65C22×C29#12011yescharacteristic
66C4×C29—21no—
67C4×C29—22no—
68C4×C29—41no—
69C4×C29—42no—
70Dic29—81no—
71H(3,𝔽4)D2911yescharacteristic
72#497—291no—
73#489—292no—
74#329—291no—
75#329—292no—
76C2×C29—81no—
77C2×C29—82no—
78C2×C29#45311yescharacteristic
79C2×C29#45512yes—
80C2×Q16—581no—
81C2×SD16—581no—
82C4⋊Q8—291no—
83#124—22no—
84#124—21no—
85#124C2×D2912yes—
86#124C2×D2911yescharacteristic
87#120—22no—
88#120—21no—
89#103—582no—
90#102—582no—
91#97—292no—
92#97—291no—
93C29#165911yescharacteristic, complemented, Sylow
94C24—21no—
95C2×Q8—581no—
96C2×Q8C22⋊D2912yes—
97C2×Q8C22⋊D2911yescharacteristic
98C2×D4—21no—
99C2×D4—22no—
100C2×D4C22⋊D2912yes—
101C2×D4C22⋊D2911yescharacteristic
102Q16—1162no—
103SD16—1162no—
104C2×C8—581no—
105C2×C8—582no—
106C4⋊C4—582no—
107C22⋊C4—22no—
108C22⋊C4—21no—
109C22⋊C4—22no—
110C22⋊C4—22no—
111C22⋊C4—22no—
112C42—21no—
113C42C22×D2911yescharacteristic
114C23—21no—
115C23—22no—
116C23—81no—
117C23—82no—
118Q8—41no—
119Q8—42no—
120Q8—1161no—
121Q8—1161no—
122D4—41no—
123D4—42no—
124D4—81no—
125D4—82no—
126C2×C4—21no—
127C2×C4—22no—
128C2×C4—1161no—
129C2×C4C2×C22⋊D2912yes—
130C2×C4C2×C22⋊D2911yescharacteristic
131C8—582no—
132C8—1162no—
133C22—41no—
134C22—42no—
135C22—42no—
136C22—42no—
137C22—42no—
138C22—84no—
139C22#1174531511yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—42no—
144C4—2321no—
145C2—81no—
146C2—82no—
147C2#1427133111yescharacteristic
148C2#1427134112yes—
subgroupG/H
centerC22#11745315
commutator, cosocleC42×C29C23
FrattiniC42C22×D29
Fitting, prime radicalC29×H(3,𝔽4)C2
socleC22×C29#120
hypercenterH(3,𝔽4)D29
hypocenterC29#1659
LCSG ▷ C42×C29 ▷ C22×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #14271558C2semidirect
2. #14271522C2semidirect
3. #14270903C2semidirect
4. #14270903C2semidirect
14. #11745231C22semidirect
17. #11745151C22semidirect
18. #11745151C22semidirect
93. C29#1659semidirect

Maximal subgroups

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

subgroupindex
#142715582
#142715222
#142709032
C29×H(3,𝔽4)2
#165929