Finite Group Databasebeta

← #31796702

Subgroups — #31796702

Order 17536.

272 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 62 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
1C2×#17984148C212yesmaximal
2#22578284C214yesmaximal, complemented
3#22577876C211yesmaximal, characteristic
4C2×#14555519C2214yes—
5#17984148—22no—
6#17984148—22no—
7C22×C22⋊D137C2211yescharacteristic
8C4×C22⋊D137—22no—
9#17984048—22no—
10#17984044C2212yes—
11C2×C22⋊Dic137C411yescharacteristic
12C22⋊C4×D137C412yes—
13C2×C22⋊C4×C137C411yescharacteristic
14#14555519—24no—
15#14555519—24no—
16C2×#11764047—24no—
17C23×D137C2×C411yescharacteristic
18C2×C22⋊D137—24no—
19C2×C22⋊D137D412yes—
20C2×C22⋊D137D412yes—
21C22×Dic137—21no—
22C22×Dic137C2311yescharacteristic
23C2×C4×D137—22no—
24#14555489—22no—
25C22⋊Dic137—21no—
26C22⋊Dic137C2×C412yes—
27#14555487—22no—
28C4×Dic137—22no—
29C24×C137C2×C411yescharacteristic
30C22×C4×C137—21no—
31C22⋊C4×C137—21no—
32C22⋊C4×C137C2×C412yes—
33#11764047—44no—
34C22×D137—22no—
35C22×D137—81no—
36C22×D137C22⋊C412yes—
37C22⋊D137—24no—
38C22⋊D137—24no—
39C2×Dic137—22no—
40C2×Dic137—81no—
41C2×Dic137C2×D412yes—
42C2×Dic137#4114yes—
43C4×D137—82no—
44C23×C137—22no—
45C23×C137—81no—
46C23×C137C22⋊C411yescharacteristic
47C23×C137C22⋊C411yescharacteristic
48C23×C137C22×C411yescharacteristic
49C2×C4×C137—22no—
50C2×C4×C137—81no—
51C22×C137—21no—
52C22×C137—21no—
53C22×C137—41no—
54C22×C137—41no—
55C22×C137—41no—
56C22×C137—82no—
57C22×C137C2×C22⋊C411yescharacteristic
58C22×C137#11712yes—
59C2×D137—22no—
60C2×D137—42no—
61C2×D137—82no—
62C4×C137—42no—
63Dic137—42no—
64Dic137#12314yes—
65C2×C137—21no—
66C2×C137—22no—
67C2×C137—41no—
68C2×C137—41no—
69C2×C137#38111yescharacteristic
70D137—42no—
71C137#182911yescharacteristic, complemented, Sylow
72#1829—1371nomaximal, Sylow
73#518—1371no—
74C2×#100—1372no—
75#344—1374no—
76C22×D4—1371no—
77C2×M16—1374no—
78C4×D4—2742no—
79C4×D4—2742no—
80#117—1372no—
81C2×C22⋊C4—1371no—
82C2×C22⋊C4—1372no—
83C2×C22⋊C4#1058773911yescharacteristic
84#100—2742no—
85#100—2742no—
86C24—1371no—
87C24C2×#1058773911yescharacteristic
88C2×D4—2744no—
89C2×D4—1372no—
90C2×D4—1372no—
91C22×C4—21no—
92C22×C4—2742no—
93C22×C4—2741no—
94C22×C4—1371no—
95M16—2744no—
96M16—2744no—
97C2×C8—2744no—
98C4⋊C4—2742no—
99C22⋊C4—21no—
100C22⋊C4—2742no—
101C22⋊C4—2741no—
102C22⋊C4—1372no—
103C22⋊C4C2×#1058773912yes—
104C42—2742no—
105C23—22no—
106C23—81no—
107C23—2742no—
108C23—10961no—
109C23—1372no—
110C23C22⋊#1058773911yescharacteristic
111C23C22⋊#1058773911yescharacteristic
112C23C22×#1058773911yescharacteristic
113D4—2744no—
114D4—2744no—
115C2×C4—22no—
116C2×C4—81no—
117C2×C4—2742no—
118C2×C4—10962no—
119C2×C4—10961no—
120C2×C4—1372no—
121C2×C4—1374no—
122C8—5484no—
123C22—21no—
124C22—21no—
125C22—41no—
126C22—41no—
127C22—41no—
128C22—82no—
129C22—2742no—
130C22—5482no—
131C22—10962no—
132C22C2×C22⋊#1058773911yescharacteristic
133C22#1798416712yes—
134C4—42no—
135C4—5482no—
136C4—1374no—
137C2—21no—
138C2—22no—
139C2—41no—
140C2—41no—
141C2—5482no—
142C2#2257834611yescharacteristic
subgroupG/H
centerC2#22578346
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
FittingC2×C22⋊C4×C137C4
socleC2×C137#381
prime radicalC22×C22⋊D137C22
cosocleC22×Dic137C23
hypercenterC2×C22⋊C4#10587739
hypocenterC137#1829
LCSG ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #22578284C2semidirect
71. C137#1829semidirect

Maximal subgroups

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

subgroupindex
C2×#179841482
#225782842
#225778762
#1829137