Finite Group Databasebeta

← #31797827

Subgroups — #31797827

Order 17536.

158 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 27 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#22578488C211yesmaximal, characteristic
2#22578488C211yesmaximal, characteristic
3#22578350C211yesmaximal, characteristic
4C22⋊#11764049—21no—
5C22⋊#11764049—21no—
6C23×#10587739C411yescharacteristic
7C2×C4⋊#10587739—21no—
8C2×C22⋊#10587739C2211yescharacteristic
9(C2×#10587739)⋊C4—21no—
10C22⋊C4×D137C411yescharacteristic
11C2×#11764049—41no—
12C2×#11764049—41no—
13C22×#10587739—21no—
14C22×#10587739—41no—
15C22×#10587739—42no—
16C22×#10587739D411yescharacteristic
17C22×#10587739D411yescharacteristic
18C4⋊#10587739—42no—
19C22⋊#10587739—41no—
20C23×D137C2×C411yescharacteristic
21C2×C4×D137—21no—
22C22⋊Dic137C811yescharacteristic, complemented
23#14555487—21no—
24C22⋊C4×C137C811yescharacteristic, complemented
25#11764049—81no—
26#11764049—81no—
27C2×#10587739—22no—
28C2×#10587739—21no—
29C2×#10587739—21no—
30C2×#10587739—41no—
31C2×#10587739—42no—
32C2×#10587739—41no—
33C2×#10587739—41no—
34C2×#10587739—41no—
35C2×#10587739—42no—
36C22×D137—21no—
37C22×D137—21no—
38C22×D137—41no—
39C22×D137—41no—
40C22×D137C22⋊C411yescharacteristic
41C22×D137M1611yescharacteristic
42C2×Dic137—21no—
43C4×D137—41no—
44C4×D137—41no—
45C23×C137C2×C811yescharacteristic
46C2×C4×C137—21no—
47C22×C137—21no—
48C22×C137—41no—
49C22×C137C22⋊C811yescharacteristic
50C2×D137—21no—
51C2×D137—21no—
52C2×D137—21no—
53C2×D137—42no—
54C2×D137—41no—
55C2×D137—41no—
56C2×D137—41no—
57C2×D137#9911yescharacteristic
58C2×D137#10111yescharacteristic
59#10587739—42no—
60#10587739—41no—
61#10587739—41no—
62C4×C137—41no—
63Dic137—41no—
64C2×C137—21no—
65C2×C137—41no—
66C2×C137#32411yescharacteristic
67D137—21no—
68D137—41no—
69D137AΣL(1,𝔽16)11yescharacteristic
70D137#35211yescharacteristic
71C137#132611yescharacteristic, complemented, Sylow
72#1326—1371nomaximal, Sylow
73#385—1371no—
74#323—1371no—
75#323—1371no—
76C23×C4—1371no—
77C2×C4⋊C4—2741no—
78C2×C22⋊C4—1371no—
79C2×C22⋊C4—1371no—
80C22⋊C8—2741no—
81C22⋊C8—2741no—
82#95—2741no—
83C24—1371no—
84C22×C4—2741no—
85C22×C4—2741no—
86C22×C4—5482no—
87C22×C4—5481no—
88C22×C4—1371no—
89C22×C4—1371no—
90C2×C8—5481no—
91C2×C8—5481no—
92C4⋊C4—5482no—
93C22⋊C4—2741no—
94C22⋊C4—5481no—
95C22⋊C4—1371no—
96C22⋊C4#1176404911yescharacteristic, complemented
97C23—2741no—
98C23—2741no—
99C23—5481no—
100C23—5481no—
101C23—1371no—
102C23—1371no—
103C23C2×#1176404911yescharacteristic
104C2×C4—21no—
105C2×C4—2742no—
106C2×C4—2741no—
107C2×C4—2741no—
108C2×C4—2741no—
109C2×C4—5481no—
110C2×C4—5481no—
111C2×C4—5481no—
112C2×C4—5482no—
113C2×C4—5481no—
114C2×C4—5481no—
115C2×C4—5481no—
116C2×C4—5482no—
117C8—10961no—
118C8—10961no—
119C22—21no—
120C22—41no—
121C22—2741no—
122C22—2741no—
123C22—2741no—
124C22—5481no—
125C22—5482no—
126C22—5481no—
127C22—5481no—
128C22—1371no—
129C22—1371no—
130C22C22⋊#1176404911yescharacteristic
131C4—41no—
132C4—5481no—
133C4—5481no—
134C4—5481no—
135C4—5482no—
136C2—21no—
137C2—41no—
138C2—2741no—
139C2—5481no—
140C2—1371no—
141C2—1371no—
142C2#2257848911yescharacteristic
subgroupG/H
centerC2#22578489
commutatorC23×C137C2×C8
FrattiniC23C2×#11764049
FittingC22⋊C4×C137C8
socleC2×C137#324
prime radicalC23×D137C2×C4
cosocleC2×C22⋊#10587739C22
hypercenterC22⋊C4#11764049
hypocenterC137#1326
LCSG ▷ C23×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22 ◁ C23 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
22. C22⋊Dic137C8semidirect
24. C22⋊C4×C137C8semidirect
71. C137#1326semidirect
96. C22⋊C4#11764049semidirect

Maximal subgroups

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

subgroupindex
#225784882
#225783502
#1326137