Finite Group Databasebeta

← #16821523

Subgroups — #16821523

Order 3712.

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

normal subgroups only

This group has 224 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#14271893C211yesmaximal, characteristic, complemented
2#14271348C212yesmaximal
3#14271345C211yesmaximal, characteristic
4#14270961C212yesmaximal, complemented
5C29×#356C211yesmaximal, characteristic, complemented
6#11745362C2211yescharacteristic
7C29×C4⋊Q8C2211yescharacteristic
8#11745306—21no—
9#11745306C2212yes—
10#11745304—21no—
11#11745271—22no—
12#11745250—22no—
13#11745249—22no—
14C4×Dic58—21no—
15C29×#101C2212yes—
16#11745184C2211yescharacteristic
17#11745173—22no—
18#11745153—21no—
19C2×Q8×C29D411yescharacteristic
20C2×Q8×C29C2311yescharacteristic
21#90809—41no—
22Q8×D29—41no—
23#90805—21no—
24#90805—22no—
25C2×Dic58—21no—
26C2×Dic58—21no—
27C2×Dic58D412yes—
28M16×C29—22no—
29C4⋊C4×C29—22no—
30C42×C29D411yescharacteristic
31#90785—41no—
32#90784—41no—
33C4⋊Dic29—21no—
34#90779—22no—
35#90779—22no—
36#90779—22no—
37C4×Dic29—21no—
38C4×Dic29—22no—
39C4×Dic29D412yes—
40#90777—21no—
41Dic116—42no—
42#90773—42no—
43Q8×C29—41no—
44Q8×C29—41no—
45C2×C4×C29—21no—
46C2×C4×C29—21no—
47C2×C4×C29C2×D411yescharacteristic
48C2×C4×C29C2×D412yes—
49C22⋊D29—41no—
50C2×Dic29—22no—
51C2×Dic29—22no—
52D116—22no—
53D116—21no—
54C4×D29—41no—
55C4×D29—42no—
56Dic58—24no—
57Dic58—22no—
58Dic58—21no—
59Dic58—42no—
60Dic58—81no—
61C8×C29—42no—
62#6597—41no—
63#2205—291nomaximal, Sylow
64C22×C29#12011yescharacteristic
65C2×D29—41no—
66C4×C29—22no—
67C4×C29—21no—
68C4×C29—22no—
69C4×C29—41no—
70Dic29—24no—
71Dic29—41no—
72Dic29—42no—
73Q82—291no—
74#456—291no—
75#456—292no—
76#356—292no—
77#356D2911yescharacteristic, complemented
78C2×C29—21no—
79C2×C29B(4,𝔽2)11yescharacteristic
80D29—81no—
812−1+4—291no—
82#137—582no—
83#137—581no—
84C4⋊Q8—292no—
85C4⋊Q8—582no—
86C4⋊Q8—581no—
87C4⋊Q8C2×D2911yescharacteristic
88C4×Q8—582no—
89C4×Q8—581no—
90#104—582no—
91#104—581no—
92#101—291no—
93#101C2×D2912yes—
94C29#220511yescharacteristic, complemented, Sylow
95#41—581no—
96#41—582no—
97#41—1161no—
98C2×Q8—292no—
99C2×Q8—581no—
100C2×Q8—581no—
101C2×Q8—1161no—
102C2×Q8D11611yescharacteristic
103C2×Q8C22×D2911yescharacteristic
104Q16—1162no—
105Q16—1161no—
106SD16—1162no—
107SD16—1161no—
108M16—22no—
109M16—581no—
110C4⋊C4—22no—
111C4⋊C4—582no—
112C4⋊C4—581no—
113C4⋊C4—582no—
114C4⋊C4—582no—
115C42—292no—
116C42—582no—
117C42—581no—
118C42D11611yescharacteristic
119Q8—41no—
120Q8—41no—
121Q8—584no—
122Q8—582no—
123Q8—581no—
124Q8—1162no—
125Q8—2321no—
126D4—582no—
127D4—581no—
128D4—1161no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—582no—
132C2×C4—582no—
133C2×C4—1162no—
134C2×C4—1161no—
135C2×C4C2×D11611yescharacteristic
136C2×C4D4×D2912yes—
137C8—42no—
138C8—1161no—
139C22—1161no—
140C22#1174524411yescharacteristic
141C4—22no—
142C4—22no—
143C4—21no—
144C4—41no—
145C4—584no—
146C4—1162no—
147C4—1161no—
148C2—21no—
149C2—2321no—
150C2#1427135111yescharacteristic
subgroupG/H
centerC2#14271351
commutator, cosocleC2×Q8×C29C23
FrattiniC2×Q8C22×D29
FittingC29×#356C2
socleC2×C29B(4,𝔽2)
prime radical#11745362C22
hypercenter#356D29
hypocenterC29#2205
LCSG ▷ C2×Q8×C29 ▷ C22×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
1. #14271893C2semidirect
4. #14270961C2semidirect
5. C29×#356C2semidirect
77. #356D29semidirect
94. C29#2205semidirect

Maximal subgroups

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

subgroupindex
#142718932
#142713482
#142713452
#142709612
C29×#3562
#220529