Finite Group Databasebeta

← #16821513

Subgroups — #16821513

Order 3712.

216 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 216 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#14271345C212yesmaximal, complemented
2#14271335C211yesmaximal, characteristic
3#14270962C212yesmaximal, complemented
4#14270960C211yesmaximal, characteristic, complemented
5C29×#563C211yesmaximal, characteristic, complemented
6#11745362C2211yescharacteristic
7C29×C4⋊Q8C2212yes—
8C29×#126—22no—
9C29×#125—21no—
10C29×#124C2211yescharacteristic
11C29×#123—21no—
12C29×C22⋊Q8—22no—
13#11745304—22no—
14#11745294—21no—
15#11745184C2211yescharacteristic
16#11745184C2212yes—
17#11745153—21no—
18#11745153—22no—
19C2×Q8×C29D412yes—
20C2×Q8×C29C2311yescharacteristic
21C2×D4×C29D411yescharacteristic
22C22×C4×C29—21no—
23#90809—41no—
24Q8×D29—41no—
25#90805—21no—
26#90805—22no—
27C4⋊C4×C29—21no—
28C4⋊C4×C29—22no—
29C4⋊C4×C29—21no—
30C4⋊C4×C29—22no—
31C4⋊C4×C29—22no—
32C22⋊C4×C29—22no—
33C22⋊C4×C29—21no—
34C22⋊C4×C29—22no—
35C42×C29D411yescharacteristic
36C42×C29D412yes—
37#90785—42no—
38#90784—42no—
39#90783—41no—
40#90782—41no—
41#90777—21no—
42#90777—22no—
43C23×C29—21no—
44Q8×C29—41no—
45Q8×C29—42no—
46D4×C29—41no—
47C2×C4×C29—21no—
48C2×C4×C29—22no—
49C2×C4×C29—22no—
50C2×C4×C29—81no—
51C2×C4×C29C2×D412yes—
52C2×C4×C29C2×D411yescharacteristic
53C22⋊D29—41no—
54D116—21no—
55D116—22no—
56C4×D29—41no—
57C4×D29—42no—
58Dic58—21no—
59Dic58—22no—
60#6597—41no—
61#6597—42no—
62#2203—291nomaximal, Sylow
63C22×C29—42no—
64C22×C29—41no—
65C22×C29#12011yescharacteristic
66C2×D29—41no—
67C4×C29—21no—
68C4×C29—22no—
69C4×C29—41no—
70C4×C29—42no—
71C4×C29—42no—
72Dic29—41no—
73#563D2911yescharacteristic, complemented
74#456—292no—
75#454—291no—
76#356—292no—
77#355—291no—
78C2×C29—21no—
79C2×C29—81no—
80C2×C29B(4,𝔽2)11yescharacteristic
81D29—81no—
822−1+4—291no—
83#137—582no—
84AGL(1,ℤ/8)—581no—
85C4⋊Q8C2×D2912yes—
86#126—22no—
87#125—21no—
88#124C2×D2911yescharacteristic
89#123—21no—
90C22⋊Q8—22no—
91#104—581no—
92#104—582no—
93#101—292no—
94#101—291no—
95C29#220311yescharacteristic, complemented, Sylow
96#41—581no—
97#41—582no—
98#41—1161no—
99C2×Q8—1161no—
100C2×Q8C22⋊D2912yes—
101C2×Q8C22×D2911yescharacteristic
102C2×D4C22⋊D2911yescharacteristic
103C22×C4—21no—
104Q16—1162no—
105SD16—1161no—
106SD16—1162no—
107D8—1161no—
108M16—581no—
109M16—582no—
110C4⋊C4—21no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C22⋊C4—22no—
116C22⋊C4—21no—
117C22⋊C4—22no—
118C42C22⋊D2911yescharacteristic
119C42C22⋊D2912yes—
120C23—21no—
121Q8—41no—
122Q8—42no—
123Q8—581no—
124Q8—582no—
125D4—41no—
126D4—581no—
127D4—582no—
128D4—1161no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—81no—
133C2×C4—1161no—
134C2×C4—1162no—
135C2×C4C2×C22⋊D2912yes—
136C2×C4C2×C22⋊D2911yescharacteristic
137C8—1161no—
138C8—1162no—
139C22—42no—
140C22—41no—
141C22—1161no—
142C22#1174531511yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—42no—
147C4—42no—
148C4—1161no—
149C2—21no—
150C2—81no—
151C2—2321no—
152C2#1427135011yescharacteristic
subgroupG/H
centerC2#14271350
commutator, cosocleC2×Q8×C29C23
FrattiniC2×Q8C22×D29
FittingC29×#563C2
socleC2×C29B(4,𝔽2)
prime radical#14271335C2
hypercenter#563D29
hypocenterC29#2203
LCSG ▷ C2×Q8×C29 ▷ C22×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #14271345C2semidirect
3. #14270962C2semidirect
4. #14270960C2semidirect
5. C29×#563C2semidirect
73. #563D29semidirect
95. C29#2203semidirect

Maximal subgroups

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

subgroupindex
#142713452
#142713352
#142709622
#142709602
C29×#5632
#220329