Finite Group Databasebeta

← #31780113

Subgroups — #31780113

Order 17536.

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

normal subgroups only

This group has 196 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#22578151C211yesmaximal, characteristic
2#22577773C211yesmaximal, characteristic, complemented
3#22577763C212yesmaximal, complemented
4#22577142C212yesmaximal, complemented
5#330×C137C211yesmaximal, characteristic, complemented
6#17984088C2211yescharacteristic
7#17984076C2212yes—
8#17984073—22no—
9#17984066—22no—
10#17984063—22no—
11#17984013—22no—
12#17984011—22no—
13#17984010—22no—
14#17983999C2211yescharacteristic
15#125×C137C2211yescharacteristic
16#97×C137C2212yes—
17C2×Dic274—21no—
18C2×Dic274D412yes—
19C2×D548—22no—
20C2×D548D412yes—
21C2×C4×D137—22no—
22C2×#11764040—21no—
23C4⋊Dic137—22no—
24#14555489—22no—
25#14555487—22no—
26#14555487—22no—
27#14555487—22no—
28C2×C8×C137—22no—
29C4⋊C4×C137—22no—
30C4⋊C4×C137D412yes—
31C42×C137C2311yescharacteristic
32C22×D137—22no—
33Dic274—42no—
34Dic274—42no—
35D548—42no—
36D548—82no—
37C2×Dic137—22no—
38C2×Dic137—41no—
39C4×D137—82no—
40#11764040—41no—
41C2×C4×C137—22no—
42C2×C4×C137C2×D412yes—
43C2×C4×C137C2×D411yescharacteristic
44C8×C137—42no—
45C22×C137#12011yescharacteristic
46C2×D137—42no—
47C2×D137—42no—
48C2×D137—42no—
49C4×C137—21no—
50C4×C137—22no—
51C4×C137—42no—
52Dic137—42no—
53Dic137—81no—
54C2×C137#45411yescharacteristic
55C2×C137#45512yes—
56D137—82no—
57C137#166211yescharacteristic, complemented, Sylow
58#1662—1371nomaximal, Sylow
59#562—1371no—
60#491—1371no—
61#489—1372no—
62#330D13711yescharacteristic, complemented
63#329—1372no—
64C4⋊Q8—1371no—
65#125C2×D13711yescharacteristic
66#124—1372no—
67#123—2742no—
68C22⋊Q8—2742no—
69#121—2742no—
70#103—2742no—
71#103—2742no—
72#102—2742no—
73#97—1371no—
74#97C2×D13712yes—
75C2×Q8—2741no—
76C2×Q8—1372no—
77C2×D4—2742no—
78C2×D4—1372no—
79C22×C4—2742no—
80C2×C8—22no—
81C2×C8—2741no—
82C4⋊C4—22no—
83C4⋊C4—2742no—
84C4⋊C4—2742no—
85C4⋊C4D54812yes—
86C22⋊C4—2742no—
87C22⋊C4—2742no—
88C22⋊C4—2742no—
89C42C22×D13711yescharacteristic
90C23—2742no—
91Q8—5482no—
92Q8—5482no—
93D4—5482no—
94D4—10962no—
95C2×C4—22no—
96C2×C4—2742no—
97C2×C4—5481no—
98C2×C4—10962no—
99C2×C4D4×D13712yes—
100C2×C4C2×D54811yescharacteristic
101C8—42no—
102C8—5481no—
103C22—5482no—
104C22—5482no—
105C22—5482no—
106C22#1798405811yescharacteristic
107C4—21no—
108C4—22no—
109C4—42no—
110C4—5482no—
111C4—10961no—
112C2—10962no—
113C2#2257757811yescharacteristic
114C2#2257758212yes—
subgroupG/H
centerC22#17984058
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
Fitting#330×C137C2
socleC22×C137#120
prime radical#22578151C2
hypercenter#330D137
hypocenterC137#1662
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
2. #22577773C2semidirect
3. #22577763C2semidirect
4. #22577142C2semidirect
5. #330×C137C2semidirect
57. C137#1662semidirect
62. #330D137semidirect

Maximal subgroups

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

subgroupindex
#225781512
#225777732
#225777632
#225771422
#330×C1372
#1662137