Finite Group Databasebeta

← #31795772

Subgroups — #31795772

Order 17536.

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

normal subgroups only

This group has 168 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, complemented
2#22578249C211yesmaximal, characteristic, complemented
3#22577321C211yesmaximal, characteristic
4C2×#14555519—21no—
5C22⋊#11764047—21no—
6C22×D548C411yescharacteristic
7C2×#14555489C2211yescharacteristic
8C2×#14555487—21no—
9C2×#14555487—21no—
10#95×C137C411yescharacteristic
11#14555519—22no—
12#14555519—41no—
13C2×#11764047—41no—
14C2×#11764047—41no—
15C23×D137—21no—
16C2×D548—41no—
17C2×D548—42no—
18C22×Dic137D411yescharacteristic
19C22×Dic137D411yescharacteristic
20#14555489—41no—
21#14555487—42no—
22#14555487—41no—
23#14555487—41no—
24C22×C4×C137—21no—
25C22×C4×C137C2×C411yescharacteristic
26#11764047—42no—
27#11764047—81no—
28C22×D137—22no—
29C22×D137—41no—
30C22×D137—41no—
31C22×D137—41no—
32C22×D137—41no—
33C22×D137—41no—
34C22×D137—41no—
35D548—42no—
36D548—42no—
37C2×Dic137—21no—
38C2×Dic137—21no—
39C2×Dic137—21no—
40C2×Dic137—41no—
41C2×Dic137D811yescharacteristic
42C2×Dic137SD1611yescharacteristic
43C23×C137C22⋊C411yescharacteristic
44C2×C4×C137—21no—
45C2×C4×C137—41no—
46C2×C4×C137—41no—
47C2×C4×C137—41no—
48C2×C4×C137—41no—
49C22×C137—21no—
50C22×C137—21no—
51C22×C137#9911yescharacteristic
52C22×C137#10211yescharacteristic
53C22×C137#10411yescharacteristic
54C2×D137—42no—
55C2×D137—42no—
56C2×D137—42no—
57C2×D137—81no—
58C2×D137—81no—
59C2×D137—81no—
60C2×D137—81no—
61C4×C137—41no—
62C4×C137—81no—
63Dic137—21no—
64Dic137—21no—
65Dic137—41no—
66C2×C137—21no—
67C2×C137—21no—
68C2×C137#32711yescharacteristic
69C2×C137AΣL(1,𝔽16)11yescharacteristic
70C2×C137#35511yescharacteristic
71D137—82no—
72C137#134511yescharacteristic, complemented, Sylow
73#1345—1371nomaximal, Sylow
74#394—1371no—
75#343—1371no—
76#324—1371no—
77C22×D4—1371no—
78C2×M16—2741no—
79C2×C4⋊C4—1371no—
80C2×C22⋊C4—2741no—
81C2×C22⋊C4—2741no—
82C22⋊C8—2741no—
83#95#1058773911yescharacteristic
84C24—2741no—
85C2×D4—5481no—
86C2×D4—5482no—
87C22×C4—21no—
88C22×C4—1371no—
89C22×C4—1371no—
90C22×C4C2×#1058773911yescharacteristic
91M16—2742no—
92M16—5481no—
93C2×C8—5481no—
94C2×C8—5481no—
95C4⋊C4—5481no—
96C22⋊C4—5482no—
97C22⋊C4—5481no—
98C22⋊C4—5481no—
99C23—2742no—
100C23—5481no—
101C23—5481no—
102C23—5481no—
103C23—5481no—
104C23—5481no—
105C23—5481no—
106C23C22⋊#1058773911yescharacteristic
107D4—5482no—
108D4—5482no—
109C2×C4—21no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—2741no—
115C2×C4—2741no—
116C2×C4—2741no—
117C2×C4—5481no—
118C2×C4—1371no—
119C2×C4—1371no—
120C8—5482no—
121C8—10961no—
122C22—21no—
123C22—21no—
124C22—5482no—
125C22—5482no—
126C22—5482no—
127C22—10961no—
128C22—10961no—
129C22—10961no—
130C22—10961no—
131C22#1798414711yescharacteristic
132C22#1798415011yescharacteristic
133C22#1798415311yescharacteristic
134C4—41no—
135C4—81no—
136C4—2741no—
137C4—2741no—
138C4—5481no—
139C2—21no—
140C2—21no—
141C2—10962no—
142C2#2257825511yescharacteristic
143C2#2257830411yescharacteristic
144C2#2257831211yescharacteristic
subgroupG/H
centerC22#17984147
commutatorC22×C4×C137C2×C4
FrattiniC22×C4C2×#10587739
Fitting#95×C137C4
socleC22×C137#99
prime radicalC22×D548C4
cosocleC2×#14555489C22
hypercenter#95#10587739
hypocenterC137#1345
LCSG ▷ C22×C4×C137 ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #95

Complements

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

NH ≅ G/Ntype
1. #22578283C2semidirect
2. #22578249C2semidirect
72. C137#1345semidirect

Maximal subgroups

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

subgroupindex
#225782832
#225782492
#225773212
#1345137