Finite Group Databasebeta

← #16821507

Subgroups — #16821507

Order 3712.

226 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 41 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#14271913C211yesmaximal, characteristic, complemented
2#14271354C212yesmaximal
3#14271340C211yesmaximal, characteristic
4#14270952C212yesmaximal, complemented
5C29×#354C211yesmaximal, characteristic, complemented
6#11745359C2211yescharacteristic
7C29×#124C2211yescharacteristic
8#11745306—21no—
9#11745304—21no—
10#11745297C2212yes—
11#11745248—22no—
12#11745241—22no—
13#11745240—22no—
14#11745232—21no—
15C29×#99C2212yes—
16#11745181C2211yescharacteristic
17#11745154—22no—
18#11745153—21no—
19C2×Q8×C29D411yescharacteristic
20C2×D4×C29C2311yescharacteristic
21C2×C22⋊D29—22no—
22C22×Dic29D412yes—
23#90807—41no—
24D4×D29—41no—
25#90805—21no—
26C2×Dic58—21no—
27C22⋊C4×C29—22no—
28C22⋊C4×C29—22no—
29C42×C29D411yescharacteristic
30C22⋊Dic29—22no—
31C22⋊Dic29D412yes—
32#90785—41no—
33#90784—41no—
34#90781—42no—
35C4⋊Dic29—21no—
36C4⋊Dic29—21no—
37C4⋊Dic29—42no—
38#90779—22no—
39#90779—22no—
40#90779—22no—
41C4×Dic29—21no—
42#90777—21no—
43C23×C29C2×D412yes—
44C22×D29—22no—
45Q8×C29—41no—
46D4×C29—41no—
47C2×C4×C29—21no—
48C2×C4×C29—21no—
49C2×C4×C29—42no—
50C2×C4×C29C2×D411yescharacteristic
51C22⋊D29—42no—
52C22⋊D29—41no—
53C2×Dic29—22no—
54C2×Dic29—22no—
55C2×Dic29—22no—
56C2×Dic29—42no—
57D116—21no—
58C4×D29—41no—
59Dic58—21no—
60Dic58—81no—
61#6597—41no—
62#2201—291nomaximal, Sylow
63C22×C29—22no—
64C22×C29—42no—
65C22×C29#12011yescharacteristic
66C2×D29—41no—
67C2×D29—82no—
68C4×C29—21no—
69C4×C29—41no—
70C4×C29—41no—
71C4×C29—82no—
72Dic29—42no—
73Dic29—41no—
74Dic29—42no—
75#563—291no—
76#458—292no—
77#455—291no—
78#354D2911yescharacteristic, complemented
79#352—292no—
80C2×C29—21no—
81C2×C29—42no—
82C2×C29B(4,𝔽2)11yescharacteristic
83D29—81no—
84H(5,𝔽2)—291no—
85#137—581no—
86C4⋊Q8—581no—
87#126—582no—
88#125—581no—
89#124C2×D2911yescharacteristic
90#123—292no—
91#123—582no—
92C22⋊Q8—582no—
93#104—581no—
94#100—291no—
95#99—582no—
96#99C2×D2912yes—
97C29#220111yescharacteristic, complemented, Sylow
98#41—581no—
99#41—1161no—
100C2×Q8—581no—
101C2×Q8D11611yescharacteristic
102C2×D4—582no—
103C2×D4—1161no—
104C2×D4C22×D2911yescharacteristic
105C22×C4—292no—
106Q16—1161no—
107SD16—1161no—
108M16—581no—
109C4⋊C4—581no—
110C4⋊C4—582no—
111C4⋊C4—582no—
112C4⋊C4—581no—
113C4⋊C4—582no—
114C4⋊C4—1162no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—292no—
118C22⋊C4—582no—
119C22⋊C4—1162no—
120C42—581no—
121C42D11611yescharacteristic
122C23—582no—
123C23D4×D2912yes—
124Q8—41no—
125Q8—581no—
126Q8—2321no—
127D4—41no—
128D4—581no—
129D4—1162no—
130D4—1161no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—42no—
134C2×C4—582no—
135C2×C4—582no—
136C2×C4—582no—
137C2×C4—1161no—
138C2×C4—1162no—
139C2×C4C2×D11611yescharacteristic
140C8—1161no—
141C22—22no—
142C22—42no—
143C22—1161no—
144C22—2322no—
145C22#1174524411yescharacteristic
146C4—21no—
147C4—41no—
148C4—41no—
149C4—82no—
150C4—1162no—
151C4—1161no—
152C4—1162no—
153C2—21no—
154C2—42no—
155C2—2321no—
156C2#1427135111yescharacteristic
subgroupG/H
centerC2#14271351
commutator, cosocleC2×D4×C29C23
FrattiniC2×D4C22×D29
FittingC29×#354C2
socleC2×C29B(4,𝔽2)
prime radical#11745359C22
hypercenter#354D29
hypocenterC29#2201
LCSG ▷ C2×D4×C29 ▷ C22×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
1. #14271913C2semidirect
4. #14270952C2semidirect
5. C29×#354C2semidirect
78. #354D29semidirect
97. C29#2201semidirect

Maximal subgroups

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

subgroupindex
#142719132
#142713542
#142713402
#142709522
C29×#3542
#220129