Finite Group Databasebeta

← #31798008

Subgroups — #31798008

Order 17536.

146 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 53 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#22578525C212yesmaximal, complemented
2#22578508C211yesmaximal, characteristic, complemented
3#22578504C212yesmaximal, complemented
4#10587739⋊C16C211yesmaximal, characteristic, complemented
5#22578447C211yesmaximal, characteristic
6#17984185—22no—
7#17984185C2212yes—
8#17984183—22no—
9#17984183C2211yescharacteristic
10#17984183C2211yescharacteristic
11#17984176C2212yes—
12#17984159C411yescharacteristic
13#17984153C412yes—
14C8×#10587739C2211yescharacteristic
15#17984111C411yescharacteristic
16#14555522—22no—
17#14555522—21no—
18#14555522D411yescharacteristic
19#14555522D411yescharacteristic
20#14555520—22no—
21#14555520C2×C412yes—
22#14555518—22no—
23#14555518C2311yescharacteristic
24C4×#10587739C2×C411yescharacteristic
25#14555511C2×C412yes—
26#14555497C812yes—
27D1096C811yescharacteristic
28C8×D137C2×C411yescharacteristic
29Q16×C137C811yescharacteristic
30C2×#10587739—21no—
31#11764047—22no—
32#11764047C2×D411yescharacteristic
33#11764047#4111yescharacteristic
34D548—22no—
35D548C2×C812yes—
36C4×D137—22no—
37C4×D137C22×C411yescharacteristic
38#11764040C2×C811yescharacteristic
39Q8×C137C2×C812yes—
40C8×C137C2×C811yescharacteristic
41C2×D137—22no—
42C2×D137#13111yescharacteristic
43#10587739—22no—
44C4×C137—22no—
45C4×C137C22×C811yescharacteristic
46Dic137C4×D411yescharacteristic
47C2×C137C8×D411yescharacteristic
48D137—21no—
49D137—42no—
50C137#217011yescharacteristic, complemented, Sylow
51#2170—1371nomaximal, Sylow
52#504—1372no—
53#443—1371no—
54#364—1371no—
55#350—1372no—
56C4×C16—1371no—
57#135—1371no—
58#131—1372no—
59M32—2742no—
60M32—1372no—
61C2×C16—2742no—
62C2×C16—1371no—
63C2×C16—1371no—
64#108—1371no—
65#104—1372no—
66C4×C8—1371no—
67#41—1372no—
68Q16#1176404911yescharacteristic
69SD16—1372no—
70D8—1371no—
71M16—2742no—
72M16—1372no—
73C2×C8—2742no—
74C2×C8—1371no—
75C2×C8—1371no—
76C42—1371no—
77C16—2742no—
78C16—2741no—
79C16—1371no—
80C16—1371no—
81Q8C2×#1176404912yes—
82D4—2742no—
83D4—1372no—
84C2×C4—2742no—
85C2×C4—2741no—
86C2×C4—1371no—
87C8—2742no—
88C8—1371no—
89C8—1371no—
90C8—1371no—
91C8C2×#1176404911yescharacteristic
92C22—2742no—
93C22—1371no—
94C4—22no—
95C4—2742no—
96C4—1371no—
97C4C22×#1176404911yescharacteristic
98C2—2741no—
99C2—5482no—
100C2D4×#1176404911yescharacteristic
subgroupG/H
centerC2D4×#11764049
commutatorC4×C137C22×C8
FrattiniC4C22×#11764049
FittingQ16×C137C8
socleC2×C137C8×D4
prime radical#17984111C4
cosocle#14555518C23
hypercenterQ16#11764049
hypocenterC137#2170
LCSG ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C4 ◁ Q16

Complements

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

NH ≅ G/Ntype
1. #22578525C2semidirect
2. #22578508C2semidirect
3. #22578504C2semidirect
4. #10587739⋊C16C2semidirect
50. C137#2170semidirect

Maximal subgroups

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

subgroupindex
#225785252
#225785082
#225785042
#10587739⋊C162
#225784472
#2170137