Finite Group Databasebeta

← #31795691

Subgroups — #31795691

Order 17536.

200 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 44 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#22578375C211yesmaximal, characteristic
2#22578345C211yesmaximal, characteristic
3#22577365C211yesmaximal, characteristic
4C2×#14555519C412yescomplemented
5C2×C4⋊#10587739—21no—
6C2×C22⋊#10587739C412yes—
7#17984145—21no—
8#17984145—21no—
9(C2×#10587739)⋊C4—21no—
10C22×C4×D137C2211yescharacteristic
11C2×#14555493—21no—
12C22⋊#11764040C411yescharacteristic, complemented
13#17984000—21no—
14C22⋊C8×C137C411yescharacteristic, complemented
15C22×#10587739—22no—
16#14555519—42no—
17C2×#11764047—22no—
18C4⋊#10587739—42no—
19C22⋊#10587739—42no—
20C23×D137C2×C411yescharacteristic
21C22×Dic137C2×C411yescharacteristic
22C2×C4×D137—42no—
23C2×C4×D137D411yescharacteristic
24C2×C4×D137D411yescharacteristic
25C2×C4×D137D411yescharacteristic
26C2×C4×D137Q811yescharacteristic
27#14555493—42no—
28C2×#11764040—21no—
29C22×C4×C137C2×C411yescharacteristic
30C2×C8×C137—21no—
31C2×#10587739—42no—
32C2×#10587739—42no—
33C2×#10587739—42no—
34#11764047—42no—
35C22×D137—21no—
36C22×D137—21no—
37C22×D137—41no—
38C22×D137—41no—
39C22×D137C22⋊C411yescharacteristic
40C22×D137C4⋊C411yescharacteristic
41C2×Dic137—22no—
42C2×Dic137C22⋊C411yescharacteristic
43C2×Dic137C4⋊C411yescharacteristic
44C4×D137—22no—
45C4×D137—22no—
46C4×D137—41no—
47C4×D137—41no—
48#11764040—41no—
49C23×C137C4211yescharacteristic
50C2×C4×C137—41no—
51C2×C4×C137C22⋊C411yescharacteristic
52C2×C4×C137C4⋊C411yescharacteristic
53C8×C137—41no—
54C22×C137—21no—
55C22×C137—21no—
56C22×C137—21no—
57C22×C137#9511yescharacteristic
58C2×D137—21no—
59C2×D137—21no—
60C2×D137—21no—
61C2×D137—21no—
62C2×D137—21no—
63C2×D137—42no—
64C2×D137—42no—
65C2×D137#10011yescharacteristic
66C2×D137#10111yescharacteristic
67#10587739—82no—
68C4×C137—21no—
69C4×C137—21no—
70Dic137—21no—
71Dic137#9912yes—
72C2×C137—41no—
73C2×C137#34211yescharacteristic
74C2×C137#34311yescharacteristic
75C2×C137#34411yescharacteristic
76D137—21no—
77D137—21no—
78D137—41no—
79C137#130511yescharacteristic, complemented, Sylow
80#1305—1371nomaximal, Sylow
81#407—1371no—
82#407—1371no—
83#380—1371no—
84C23×C4—1371no—
85C2×M16—2741no—
86C2×M16—1372no—
87C2×C4⋊C4—2741no—
88C2×C22⋊C4—1372no—
89C22⋊C8—2741no—
90C22⋊C8—2741no—
91C22⋊C8—2741no—
92C22⋊C8—1371no—
93C22⋊C8#1058773911yescharacteristic, complemented
94#95—2741no—
95C24—1371no—
96C22×C4—2742no—
97C22×C4—5482no—
98C22×C4—1371no—
99C22×C4—1371no—
100C22×C4—1371no—
101C22×C4—1371no—
102C22×C4—1371no—
103C22×C4C2×#1058773911yescharacteristic
104M16—5482no—
105M16—5482no—
106C2×C8—21no—
107C2×C8—2742no—
108C2×C8—2741no—
109C4⋊C4—5482no—
110C22⋊C4—5482no—
111C23—2741no—
112C23—2741no—
113C23—5481no—
114C23—5481no—
115C23—1371no—
116C23—1371no—
117C23C4×#1058773911yescharacteristic
118C2×C4—41no—
119C2×C4—2742no—
120C2×C4—2742no—
121C2×C4—2742no—
122C2×C4—5481no—
123C2×C4—5481no—
124C2×C4—5482no—
125C2×C4—5482no—
126C2×C4—5482no—
127C2×C4—1371no—
128C2×C4—1371no—
129C2×C4C22⋊#1058773911yescharacteristic
130C2×C4C4⋊#1058773911yescharacteristic
131C8—41no—
132C8—5481no—
133C8—5482no—
134C22—21no—
135C22—21no—
136C22—21no—
137C22—2741no—
138C22—2741no—
139C22—2741no—
140C22—2741no—
141C22—2741no—
142C22—5482no—
143C22—5482no—
144C22—1371no—
145C22—1371no—
146C22(C2×#10587739)⋊C411yescharacteristic
147C4—21no—
148C4—21no—
149C4—2741no—
150C4—10962no—
151C4—1372no—
152C2—41no—
153C2—2741no—
154C2—2741no—
155C2—5481no—
156C2#2257828011yescharacteristic
157C2#2257828211yescharacteristic
158C2#2257828511yescharacteristic
subgroupG/H
centerC22(C2×#10587739)⋊C4
commutatorC23×C137C42
FrattiniC22×C4C2×#10587739
FittingC22⋊C8×C137C4
socleC22×C137#95
prime radicalC23×D137C2×C4
cosocleC22×C4×D137C22
hypercenterC22⋊C8#10587739
hypocenterC137#1305
LCSG ▷ C23×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
4. C2×#14555519C4semidirect
12. C22⋊#11764040C4semidirect
14. C22⋊C8×C137C4semidirect
79. C137#1305semidirect
93. C22⋊C8#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783752
#225783452
#225773652
#1305137