Finite Group Databasebeta

← #31795783

Subgroups — #31795783

Order 17536.

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

normal subgroups only

This group has 162 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#22578281C211yesmaximal, characteristic
2#22578247C211yesmaximal, characteristic
3#22577316C211yesmaximal, characteristic
4C2×C22⋊#10587739—21no—
5C22⋊#11764047—21no—
6C22×Dic274C411yescharacteristic, complemented
7C2×#14555487—21no—
8C2×#14555487C2211yescharacteristic
9#17983995—21no—
10#95×C137C411yescharacteristic, complemented
11C22×#10587739—41no—
12C2×#11764047—41no—
13C22⋊#10587739—22no—
14C22⋊#10587739—41no—
15C23×D137D411yescharacteristic
16C2×Dic274—41no—
17C2×Dic274—42no—
18C22×Dic137—21no—
19C22×Dic137D411yescharacteristic
20#14555487—41no—
21#14555487—42no—
22C22×C4×C137—21no—
23C22×C4×C137C2×C411yescharacteristic
24C2×#10587739—42no—
25C2×#10587739—81no—
26C2×#10587739—81no—
27#11764047—81no—
28C22×D137—21no—
29C22×D137—21no—
30C22×D137—41no—
31C22×D137—41no—
32C22×D137SD1611yescharacteristic
33C22×D137Q1611yescharacteristic
34Dic274—42no—
35Dic274—42no—
36C2×Dic137—22no—
37C2×Dic137—21no—
38C2×Dic137—41no—
39C2×Dic137—41no—
40C2×Dic137—41no—
41C23×C137C22⋊C411yescharacteristic
42C2×C4×C137—21no—
43C2×C4×C137—41no—
44C2×C4×C137—41no—
45C2×C4×C137—41no—
46C2×C4×C137—41no—
47C22×C137—21no—
48C22×C137—21no—
49C22×C137#9911yescharacteristic
50C22×C137#10311yescharacteristic
51C22×C137#10411yescharacteristic
52C2×D137—21no—
53C2×D137—21no—
54C2×D137—41no—
55C2×D137—41no—
56C2×D137—41no—
57C2×D137—41no—
58C2×D137—81no—
59#10587739—82no—
60C4×C137—41no—
61C4×C137—81no—
62Dic137—42no—
63Dic137—41no—
64C2×C137—21no—
65C2×C137—21no—
66C2×C137#32811yescharacteristic
67C2×C137AΣL(1,𝔽16)11yescharacteristic
68C2×C137#35411yescharacteristic
69D137—41no—
70D137—41no—
71C137#135111yescharacteristic, complemented, Sylow
72#1351—1371nomaximal, Sylow
73#393—1371no—
74#342—1371no—
75#323—1371no—
76C22×Q8—1371no—
77C2×C22⋊C4—2741no—
78C2×C22⋊C4—2741no—
79C2×C22⋊C4—1371no—
80C22⋊C8—2741no—
81#95—2741no—
82#95#1058773911yescharacteristic, complemented
83C24—1371no—
84C2×Q8—5481no—
85C2×Q8—5482no—
86C22×C4—21no—
87C22×C4—2741no—
88C22×C4—5481no—
89C22×C4—1371no—
90C22×C4C2×#1058773911yescharacteristic
91C2×C8—5481no—
92C22⋊C4—2742no—
93C22⋊C4—5481no—
94C22⋊C4—5482no—
95C22⋊C4—5481no—
96C23—2741no—
97C23—2741no—
98C23—5481no—
99C23—5481no—
100C23—1371no—
101C23—1371no—
102C23C22⋊#1058773911yescharacteristic
103Q8—5482no—
104Q8—5482no—
105C2×C4—21no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—2742no—
111C2×C4—2741no—
112C2×C4—5481no—
113C2×C4—5481no—
114C2×C4—5481no—
115C2×C4—5482no—
116C2×C4—10961no—
117C2×C4—10961no—
118C8—10961no—
119C22—21no—
120C22—21no—
121C22—2741no—
122C22—2741no—
123C22—5481no—
124C22—5481no—
125C22—5481no—
126C22—5481no—
127C22—10961no—
128C22#1798414711yescharacteristic
129C22#1798415111yescharacteristic
130C22#1798415211yescharacteristic
131C4—41no—
132C4—81no—
133C4—5482no—
134C4—5481no—
135C4—10962no—
136C2—21no—
137C2—21no—
138C2—5481no—
139C2—5481no—
140C2#2257825611yescharacteristic
141C2#2257830411yescharacteristic
142C2#2257831011yescharacteristic
subgroupG/H
centerC22#17984147
commutatorC22×C4×C137C2×C4
FrattiniC22×C4C2×#10587739
Fitting#95×C137C4
socleC22×C137#99
prime radicalC23×D137D4
cosocleC2×#14555487C22
hypercenter#95#10587739
hypocenterC137#1351
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
6. C22×Dic274C4semidirect
10. #95×C137C4semidirect
71. C137#1351semidirect
82. #95#10587739semidirect

Maximal subgroups

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

subgroupindex
#225782812
#225782472
#225773162
#1351137