Finite Group Databasebeta

← #31782856

Subgroups — #31782856

Order 17536.

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

normal subgroups only

This group has 154 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#22577843C211yesmaximal, characteristic, complemented
2#22577786C211yesmaximal, characteristic, complemented
3#22577520C211yesmaximal, characteristic, complemented
4#22577223C211yesmaximal, characteristic, complemented
5#22577220C211yesmaximal, characteristic, complemented
6#22577216C211yesmaximal, characteristic, complemented
7#509×C137C211yesmaximal, characteristic
8C2×Dic548C2211yescharacteristic
9C2×#14555498—21no—
10#17984095—21no—
11#17984075—21no—
12#17984034—21no—
13#17984032—21no—
14#17984028C2211yescharacteristic
15#17984024C2211yescharacteristic
16#17984015—21no—
17C8×Dic137C2211yescharacteristic
18#135×C137C2211yescharacteristic
19C2×D8×C137C2211yescharacteristic
20SD32×C137—21no—
21D16×C137—21no—
22M32×C137C2211yescharacteristic
23C2×Dic274—21no—
24Dic548—21no—
25Dic548—21no—
26#14555498—41no—
27#14555498—41no—
28#14555494—21no—
29#14555494—41no—
30C2×#11764040—41no—
31C2×#11764040D411yescharacteristic
32C22⋊Dic137—42no—
33C4×Dic137D411yescharacteristic
34#14555485—21no—
35#41×C137—21no—
36C2×D4×C137—21no—
37Q16×C137D411yescharacteristic
38SD16×C137—21no—
39D8×C137—21no—
40D8×C137—21no—
41D8×C137D411yescharacteristic
42C2×C8×C137C2311yescharacteristic
43C16×C137—21no—
44Dic274—41no—
45Dic274—41no—
46C2×Dic137—21no—
47C2×Dic137—41no—
48#11764040—41no—
49#11764040D811yescharacteristic
50#11764040D811yescharacteristic
51C23×C137—41no—
52Q8×C137—21no—
53D4×C137—21no—
54D4×C137—41no—
55D4×C137—41no—
56D4×C137—41no—
57C2×C4×C137—41no—
58C2×C4×C137C2×D411yescharacteristic
59C8×C137C2×D411yescharacteristic
60C8×C137#4111yescharacteristic
61C22×C137—41no—
62C22×C137—82no—
63C22×C137—81no—
64C22×C137AGL(1,ℤ/8)11yescharacteristic
65C4×C137—41no—
66C4×C137#12111yescharacteristic
67C4×C137C2×D811yescharacteristic
68Dic137—41no—
69Dic137—81no—
70C2×C137—21no—
71C2×C137—81no—
72C2×C137—161no—
73C2×C137#45911yescharacteristic
74C137#221211yescharacteristic, complemented, Sylow
75#2212—1371nomaximal, Sylow
76#510—1371no—
77#509D13711yescharacteristic
78#495—1371no—
79#443—1371no—
80#364—1371no—
81#362—1371no—
82#361—1371no—
83#135C2×D13711yescharacteristic
84C2×Q16—1371no—
85C2×SD16—2741no—
86C2×D8C2×D13711yescharacteristic
87#131—2741no—
88#124—2741no—
89Q32—2741no—
90SD32—21no—
91SD32—2741no—
92D16—21no—
93M32—1371no—
94M32C2×D13711yescharacteristic
95#108—1371no—
96#104—2741no—
97C4×C8—1371no—
98#41—21no—
99C2×Q8—2741no—
100C2×D4—21no—
101Q16—2741no—
102Q16—2741no—
103Q16C22⋊D13711yescharacteristic
104SD16—21no—
105SD16—5481no—
106SD16—5481no—
107D8—21no—
108D8—21no—
109D8C22⋊D13711yescharacteristic
110M16—2741no—
111M16—5481no—
112C2×C8—5481no—
113C2×C8—1371no—
114C2×C8C22×D13711yescharacteristic
115C22⋊C4—5482no—
116C42—1371no—
117C16—21no—
118C16—2741no—
119C23—41no—
120Q8—21no—
121Q8—5481no—
122Q8—5481no—
123D4—21no—
124D4—41no—
125D4—41no—
126D4—41no—
127C2×C4—41no—
128C2×C4—2741no—
129C2×C4—5481no—
130C2×C4D4×D13711yescharacteristic
131C8—5481no—
132C8—1371no—
133C8—1371no—
134C8C2×C22⋊D13711yescharacteristic
135C8#1455550911yescharacteristic
136C22—41no—
137C22—82no—
138C22—81no—
139C22#1798411211yescharacteristic
140C4—41no—
141C4—5481no—
142C4—10961no—
143C4#1798406011yescharacteristic
144C4D8×D13711yescharacteristic
145C2—21no—
146C2—81no—
147C2—161no—
148C2#2257759611yescharacteristic
subgroupG/H
centerC2#22577596
commutator, cosocleC2×C8×C137C23
FrattiniC2×C8C22×D137
Fitting, prime radical#509×C137C2
socleC2×C137#459
hypercenter#509D137
hypocenterC137#2212
LCSG ▷ C2×C8×C137 ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #22577843C2semidirect
2. #22577786C2semidirect
3. #22577520C2semidirect
4. #22577223C2semidirect
5. #22577220C2semidirect
6. #22577216C2semidirect
74. C137#2212semidirect

Maximal subgroups

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

subgroupindex
#225778432
#225777862
#225775202
#225772232
#225772202
#225772162
#509×C1372
#2212137