Finite Group Databasebeta

← #31782461

Subgroups — #31782461

Order 17536.

270 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 73 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#22577377C212yesmaximal, complemented
2#22577196C214yesmaximal, complemented
3#535×C137C211yesmaximal, characteristic
4#17984047—22no—
5#17984003—22no—
6#17984003C2214yescomplemented
7C2×#41×C137C2211yescharacteristic
8C22×D4×C137C411yescharacteristic, complemented
9#127×C137C2212yes—
10#124×C137C412yescomplemented
11#121×C137—22no—
12#120×C137—22no—
13#117×C137C411yescharacteristic, complemented
14C4⋊Dic137—42no—
15C22⋊Dic137—24no—
16C4×Dic137—42no—
17C24×C137—21no—
18#41×C137—42no—
19C2×Q8×C137C2×C411yescharacteristic
20C2×D4×C137—24no—
21C2×D4×C137—21no—
22C2×D4×C137—42no—
23C2×D4×C137C2×C412yes—
24C2×D4×C137D412yes—
25C2×D4×C137C2311yescharacteristic
26C22×C4×C137C2×C411yescharacteristic
27C22×C4×C137D412yes—
28C4⋊C4×C137—21no—
29C22⋊C4×C137—21no—
30C22⋊C4×C137—24no—
31C42×C137C2×C412yes—
32C2×Dic137—44no—
33C23×C137—22no—
34C23×C137—42no—
35C23×C137—82no—
36C23×C137C22⋊C411yescharacteristic
37C23×C137C2×D412yes—
38Q8×C137—41no—
39D4×C137—42no—
40D4×C137—41no—
41D4×C137—42no—
42D4×C137—42no—
43D4×C137—84no—
44C2×C4×C137—22no—
45C2×C4×C137—22no—
46C2×C4×C137—22no—
47C2×C4×C137—22no—
48C2×C4×C137C22⋊C412yes—
49C2×C4×C137C22⋊C411yescharacteristic
50C2×C4×C137C22×C411yescharacteristic
51C22×C137—21no—
52C22×C137—22no—
53C22×C137—42no—
54C22×C137—41no—
55C22×C137—44no—
56C22×C137—42no—
57C22×C137—84no—
58C22×C137C2×C22⋊C411yescharacteristic
59C4×C137—22no—
60C4×C137—21no—
61C4×C137—42no—
62C4×C137#9912yes—
63Dic137—84no—
64C2×C137—21no—
65C2×C137—42no—
66C2×C137—41no—
67C2×C137—82no—
68C2×C137C2×#9911yescharacteristic
69C137#212811yescharacteristic, complemented, Sylow
70#2128—1371nomaximal, Sylow
71#535D13711yescharacteristic
72#410—1372no—
73#353—1374no—
74C2×#41C2×D13711yescharacteristic
75C22×D4Dic13711yescharacteristic, complemented
76#127C2×D13712yes—
77#124Dic13712yescomplemented
78#121—22no—
79#120—22no—
80#117—2742no—
81#117Dic13711yescharacteristic, complemented
82#99—2742no—
83#99—1374no—
84C24—21no—
85#41—42no—
86C2×Q8C2×Dic13711yescharacteristic
87C2×D4—24no—
88C2×D4—21no—
89C2×D4—42no—
90C2×D4C2×Dic13712yes—
91C2×D4C22⋊D13712yes—
92C2×D4C22×D13711yescharacteristic
93C22×C4C2×Dic13711yescharacteristic
94C22×C4C22⋊D13712yes—
95C4⋊C4—21no—
96C4⋊C4—5482no—
97C22⋊C4—21no—
98C22⋊C4—24no—
99C22⋊C4—2744no—
100C42—5482no—
101C42C2×Dic13712yes—
102C23—22no—
103C23—42no—
104C23—82no—
105C23C22⋊Dic13711yescharacteristic
106C23C2×C22⋊D13712yes—
107Q8—41no—
108D4—42no—
109D4—41no—
110D4—42no—
111D4—42no—
112D4—84no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—5484no—
118C2×C4C22⋊Dic13712yes—
119C2×C4C22⋊Dic13711yescharacteristic
120C2×C4C22×Dic13711yescharacteristic
121C22—22no—
122C22—21no—
123C22—42no—
124C22—41no—
125C22—44no—
126C22—42no—
127C22—84no—
128C22C2×C22⋊Dic13711yescharacteristic
129C4—22no—
130C4—21no—
131C4—42no—
132C4—10964no—
133C4#1798400312yes—
134C2—21no—
135C2—42no—
136C2—41no—
137C2—82no—
138C2C2×#1798400311yescharacteristic
subgroupG/H
centerC2C2×#17984003
commutatorC2×C4×C137C22×C4
FrattiniC2×D4C22×D137
Fitting, prime radical#535×C137C2
socleC2×C137C2×#99
cosocleC2×D4×C137C23
hypercenter#535D137
hypocenterC137#2128
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #22577377C2semidirect
2. #22577196C2semidirect
6. #17984003C22semidirect
8. C22×D4×C137C4semidirect
10. #124×C137C4semidirect
13. #117×C137C4semidirect
69. C137#2128semidirect
75. C22×D4Dic137semidirect
77. #124Dic137semidirect
81. #117Dic137semidirect

Maximal subgroups

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

subgroupindex
#225773772
#225771962
#535×C1372
#2128137