Finite Group Databasebeta

← #31798010

Subgroups — #31798010

Order 17536.

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

normal subgroups only

This group has 144 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#22578507C211yesmaximal, characteristic, complemented
3#22578504C212yesmaximal, complemented
4#22578498C211yesmaximal, 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#14555522D412yes—
19#14555520—22no—
20#14555520C2×C412yes—
21#14555518—22no—
22#14555518C2311yescharacteristic
23C4×#10587739C2×C411yescharacteristic
24#14555511C2×C412yes—
25#14555497C812yes—
26D1096C811yescharacteristic
27C8×D137C2×C411yescharacteristic
28Q16×C137C811yescharacteristic
29C2×#10587739—21no—
30#11764047—22no—
31#11764047C2×D411yescharacteristic
32#11764047#4111yescharacteristic
33D548—22no—
34D548C2×C812yes—
35C4×D137—22no—
36C4×D137C22×C411yescharacteristic
37#11764040C2×C811yescharacteristic
38Q8×C137C2×C812yes—
39C8×C137C2×C811yescharacteristic
40C2×D137—22no—
41C2×D137#13111yescharacteristic
42#10587739—41no—
43C4×C137—22no—
44C4×C137C22×C811yescharacteristic
45Dic137C4×D411yescharacteristic
46C2×C137C8×D411yescharacteristic
47D137—21no—
48D137—42no—
49C137#217111yescharacteristic, complemented, Sylow
50#2171—1371nomaximal, Sylow
51#504—1372no—
52#443—1371no—
53#364—1371no—
54#350—1372no—
55#346—1371no—
56#135—1371no—
57#131—1372no—
58M32—2742no—
59M32—1372no—
60C2×C16—2742no—
61C2×C16—1371no—
62C2×C16—1371no—
63#108—1371no—
64#104—1372no—
65C4×C8—1371no—
66#41—1372no—
67Q16#1176404911yescharacteristic
68SD16—1372no—
69D8—1371no—
70M16—2742no—
71M16—1372no—
72C2×C8—2742no—
73C2×C8—1371no—
74C2×C8—1371no—
75C42—1371no—
76C16—2742no—
77C16—2741no—
78C16—1372no—
79Q8C2×#1176404912yes—
80D4—2742no—
81D4—1372no—
82C2×C4—2742no—
83C2×C4—2741no—
84C2×C4—1371no—
85C8—2742no—
86C8—1371no—
87C8—1371no—
88C8—1371no—
89C8C2×#1176404911yescharacteristic
90C22—2742no—
91C22—1371no—
92C4—22no—
93C4—5481no—
94C4—1371no—
95C4C22×#1176404911yescharacteristic
96C2—2741no—
97C2—5482no—
98C2D4×#1176404911yescharacteristic
subgroupG/H
centerC2D4×#11764049
commutatorC4×C137C22×C8
FrattiniC4C22×#11764049
FittingQ16×C137C8
socleC2×C137C8×D4
prime radical#17984111C4
cosocle#14555518C23
hypercenterQ16#11764049
hypocenterC137#2171
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. #22578507C2semidirect
3. #22578504C2semidirect
4. #22578498C2semidirect
49. C137#2171semidirect

Maximal subgroups

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

subgroupindex
#225785252
#225785072
#225785042
#225784982
#225784472
#2171137