Finite Group Databasebeta

← #31795773

Subgroups — #31795773

Order 17536.

124 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 30 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#22578283C211yesmaximal, characteristic
2#22578249C211yesmaximal, characteristic
3#22577334C211yesmaximal, characteristic
4C2×#14555519—21no—
5C22⋊#11764047—21no—
6C2×#14555489—21no—
7C2×#14555489C2211yescharacteristic
8#17983995—21no—
9#17983995C411yescharacteristic
10C2×C4⋊C4×C137C411yescharacteristic
11#14555519—22no—
12#14555519—41no—
13C2×#11764047—41no—
14C2×#11764047—41no—
15C22×Dic137—21no—
16C22×Dic137D411yescharacteristic
17C22×Dic137D411yescharacteristic
18#14555489—41no—
19#14555489—41no—
20#14555489—41no—
21C22×C4×C137—21no—
22C22×C4×C137C2×C411yescharacteristic
23C4⋊C4×C137—41no—
24#11764047—42no—
25#11764047—81no—
26C2×Dic137—21no—
27C2×Dic137—21no—
28C2×Dic137—22no—
29C2×Dic137—21no—
30C2×Dic137—41no—
31C2×Dic137—41no—
32C2×Dic137—41no—
33C2×Dic137D811yescharacteristic
34C2×Dic137SD1611yescharacteristic
35C23×C137C22⋊C411yescharacteristic
36C2×C4×C137—21no—
37C2×C4×C137—41no—
38C2×C4×C137—41no—
39C2×C4×C137—41no—
40C2×C4×C137—41no—
41C22×C137—21no—
42C22×C137—21no—
43C22×C137#9911yescharacteristic
44C22×C137#10211yescharacteristic
45C22×C137#10411yescharacteristic
46C4×C137—41no—
47C4×C137—81no—
48Dic137—21no—
49Dic137—21no—
50Dic137—42no—
51Dic137—41no—
52C2×C137—21no—
53C2×C137—21no—
54C2×C137#32711yescharacteristic
55C2×C137#35211yescharacteristic
56C2×C137#35611yescharacteristic
57C137#134611yescharacteristic, complemented, Sylow
58#1346—1371nomaximal, Sylow
59#398—1371no—
60#343—1371no—
61#324—1371no—
62C2×M16—2741no—
63C2×C4⋊C4—2741no—
64C2×C4⋊C4—1371no—
65C2×C4⋊C4#1058773911yescharacteristic
66C22⋊C8—2741no—
67#95—2741no—
68#95—1371no—
69C22×C4—21no—
70C22×C4—2741no—
71C22×C4—1371no—
72C22×C4—1371no—
73C22×C4C2×#1058773911yescharacteristic
74M16—2742no—
75M16—5481no—
76C2×C8—5481no—
77C2×C8—5481no—
78C4⋊C4—41no—
79C4⋊C4—5481no—
80C4⋊C4—5481no—
81C4⋊C4—5481no—
82C23C22⋊#1058773911yescharacteristic
83C2×C4—21no—
84C2×C4—41no—
85C2×C4—41no—
86C2×C4—41no—
87C2×C4—41no—
88C2×C4—2741no—
89C2×C4—2741no—
90C2×C4—2742no—
91C2×C4—2741no—
92C2×C4—5481no—
93C2×C4—5481no—
94C2×C4—5481no—
95C2×C4—1371no—
96C2×C4—1371no—
97C8—5482no—
98C8—10961no—
99C22—21no—
100C22—21no—
101C22#1798414611yescharacteristic
102C22#1798415011yescharacteristic
103C22#1798415211yescharacteristic
104C4—41no—
105C4—81no—
106C4—2741no—
107C4—2741no—
108C4—5482no—
109C4—5481no—
110C2—21no—
111C2—21no—
112C2#2257825411yescharacteristic
113C2#2257830511yescharacteristic
114C2#2257831311yescharacteristic
subgroupG/H
centerC22#17984146
commutatorC22×C4×C137C2×C4
FrattiniC22×C4C2×#10587739
FittingC2×C4⋊C4×C137C4
socleC22×C137#99
prime radicalC23×C137C22⋊C4
cosocleC2×#14555489C22
hypercenterC2×C4⋊C4#10587739
hypocenterC137#1346
LCSG ▷ C22×C4×C137 ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
57. C137#1346semidirect

Maximal subgroups

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

subgroupindex
#225782832
#225782492
#225773342
#1346137