Finite Group Databasebeta

← #16818803

Subgroups — #16818803

Order 3712.

196 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 196 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#14271573C212yesmaximal, complemented
2#14271558C211yesmaximal, characteristic
3#14270905C212yesmaximal, complemented
4#14270903C211yesmaximal, characteristic
5C29×#563C211yesmaximal, characteristic
6C29×C4⋊Q8—21no—
7C29×#126—22no—
8C29×#125C2212yes—
9C29×#124C2211yescharacteristic
10C29×#123—21no—
11C29×C22⋊Q8—22no—
12C2×#90785—21no—
13C2×#90783—21no—
14#11745231C2211yescharacteristic
15#11745156—22no—
16#11745155—22no—
17#11745151C2211yescharacteristic
18#11745151C2212yes—
19C2×Q8×C29—21no—
20C2×Q8×C29D411yescharacteristic
21C2×D4×C29D411yescharacteristic
22C22×C4×C29—21no—
23C2×Dic58—21no—
24C4⋊C4×C29—22no—
25C4⋊C4×C29—22no—
26C4⋊C4×C29—22no—
27C4⋊C4×C29D412yes—
28C4⋊C4×C29D412yes—
29C22⋊C4×C29—22no—
30C22⋊C4×C29—21no—
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—
40Q8×C29—41no—
41Q8×C29—81no—
42D4×C29—41no—
43C2×C4×C29—22no—
44C2×C4×C29—21no—
45C2×C4×C29—22no—
46C2×C4×C29—81no—
47C2×C4×C29C2×D412yes—
48C2×C4×C29C2×D411yescharacteristic
49C2×Dic29—41no—
50Dic58—41no—
51Dic58—41no—
52#6597—24no—
53#6597—22no—
54#1660—291nomaximal, Sylow
55C22×C29—41no—
56C22×C29—42no—
57C22×C29#12011yescharacteristic
58C4×C29—21no—
59C4×C29—22no—
60C4×C29—42no—
61C4×C29—41no—
62C4×C29—42no—
63Dic29—81no—
64#563D2911yescharacteristic
65#501—292no—
66#497—291no—
67#330—292no—
68#329—291no—
69C2×C29—81no—
70C2×C29#45311yescharacteristic
71C2×C29#45612yes—
72C2×Q16—581no—
73C2×SD16—581no—
74C4⋊Q8—21no—
75C4⋊Q8—291no—
76#126—22no—
77#125C2×D2912yes—
78#124C2×D2911yescharacteristic
79#123—21no—
80C22⋊Q8—22no—
81#107—582no—
82#106—582no—
83#97—292no—
84#97—291no—
85C29#166011yescharacteristic, complemented, Sylow
86C2×Q8—21no—
87C2×Q8—581no—
88C2×Q8C22⋊D2911yescharacteristic
89C2×D4C22⋊D2911yescharacteristic
90C22×C4—21no—
91Q16—1162no—
92SD16—1162no—
93C2×C8—582no—
94C2×C8—581no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—582no—
99C4⋊C4C22⋊D2912yes—
100C4⋊C4C22⋊D2912yes—
101C22⋊C4—22no—
102C22⋊C4—21no—
103C22⋊C4—22no—
104C42—21no—
105C42C22×D2911yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—81no—
109Q8—1161no—
110Q8—1161no—
111D4—41no—
112C2×C4—22no—
113C2×C4—21no—
114C2×C4—22no—
115C2×C4—81no—
116C2×C4—1161no—
117C2×C4C2×C22⋊D2912yes—
118C2×C4C2×C22⋊D2911yescharacteristic
119C8—582no—
120C8—584no—
121C22—41no—
122C22—42no—
123C22#1174531511yescharacteristic
124C4—21no—
125C4—22no—
126C4—42no—
127C4—41no—
128C4—42no—
129C4—2321no—
130C2—81no—
131C2#1427133111yescharacteristic
132C2#1427134612yes—
subgroupG/H
centerC22#11745315
commutator, cosocleC42×C29C23
FrattiniC42C22×D29
FittingC29×#563C2
socleC22×C29#120
prime radicalC2×D4×C29D4
hypercenter#563D29
hypocenterC29#1660
LCSG ▷ C42×C29 ▷ C22×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #14271573C2semidirect
3. #14270905C2semidirect
85. C29#1660semidirect

Maximal subgroups

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

subgroupindex
#142715732
#142715582
#142709052
#142709032
C29×#5632
#166029