Finite Group Databasebeta

← #31795689

Subgroups — #31795689

Order 17536.

200 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 46 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#22578374C211yesmaximal, characteristic
2#22578345C211yesmaximal, characteristic
3#22577366C211yesmaximal, characteristic
4C2×#14555520—21no—
5C2×C4⋊#10587739—21no—
6C2×C22⋊#10587739C412yes—
7#17984145—21no—
8C22⋊#11764047C412yescomplemented
9(C2×#10587739)⋊C4—21no—
10C22×C4×D137C2211yescharacteristic
11C2×#14555494C411yescharacteristic, complemented
12#17984000—21no—
13#17984000—21no—
14C2×M16×C137C411yescharacteristic, complemented
15C22×#10587739—22no—
16#14555520—41no—
17#14555520—41no—
18C2×#11764047—22no—
19C4⋊#10587739—41no—
20C4⋊#10587739—41no—
21C22⋊#10587739—42no—
22C23×D137C2×C411yescharacteristic
23C22×Dic137C2×C411yescharacteristic
24C2×C4×D137—41no—
25C2×C4×D137—41no—
26C2×C4×D137D411yescharacteristic
27C2×C4×D137D411yescharacteristic
28C2×C4×D137D411yescharacteristic
29C2×C4×D137Q811yescharacteristic
30#14555494—41no—
31C2×#11764040—21no—
32C22×C4×C137C2×C411yescharacteristic
33M16×C137—41no—
34C2×C8×C137—21no—
35C2×#10587739—42no—
36C2×#10587739—42no—
37C2×#10587739—42no—
38#11764047—42no—
39C22×D137—21no—
40C22×D137—21no—
41C22×D137—41no—
42C22×D137—41no—
43C22×D137C22⋊C411yescharacteristic
44C22×D137C4⋊C411yescharacteristic
45C2×Dic137—41no—
46C2×Dic137C22⋊C411yescharacteristic
47C2×Dic137C4⋊C411yescharacteristic
48C4×D137—21no—
49C4×D137—21no—
50C4×D137—21no—
51C4×D137—21no—
52C4×D137—41no—
53C4×D137—41no—
54#11764040—41no—
55C23×C137C4211yescharacteristic
56C2×C4×C137—21no—
57C2×C4×C137—21no—
58C2×C4×C137C22⋊C411yescharacteristic
59C2×C4×C137C4⋊C411yescharacteristic
60C8×C137—41no—
61C22×C137—21no—
62C22×C137—21no—
63C22×C137—21no—
64C22×C137#9511yescharacteristic
65C2×D137—21no—
66C2×D137—21no—
67C2×D137—21no—
68C2×D137—21no—
69C2×D137—21no—
70C2×D137—42no—
71C2×D137—42no—
72C2×D137#10011yescharacteristic
73C2×D137#10111yescharacteristic
74#10587739—82no—
75C4×C137—21no—
76C4×C137#9911yescharacteristic
77C4×C137#9911yescharacteristic
78Dic137—21no—
79Dic137—21no—
80C2×C137—41no—
81C2×C137#34211yescharacteristic
82C2×C137#34311yescharacteristic
83C2×C137#34411yescharacteristic
84D137—21no—
85D137—21no—
86D137—41no—
87C137#130511yescharacteristic, complemented, Sylow
88#1305—1371nomaximal, Sylow
89#407—1371no—
90#407—1371no—
91#380—1371no—
92C23×C4—1371no—
93C2×M16—2741no—
94C2×M16—1371no—
95C2×M16#1058773911yescharacteristic, complemented
96C2×C4⋊C4—2741no—
97C2×C22⋊C4—1372no—
98C22⋊C8—2741no—
99C22⋊C8—2741no—
100C22⋊C8—2741no—
101C22⋊C8—1372no—
102#95—2741no—
103C24—1371no—
104C22×C4—2742no—
105C22×C4—5481no—
106C22×C4—5481no—
107C22×C4—1371no—
108C22×C4—1371no—
109C22×C4—1371no—
110C22×C4—1371no—
111C22×C4—1371no—
112C22×C4C2×#1058773911yescharacteristic
113M16—41no—
114M16—5481no—
115M16—5481no—
116M16—5481no—
117C2×C8—21no—
118C2×C8—2742no—
119C2×C8—2741no—
120C4⋊C4—5481no—
121C4⋊C4—5481no—
122C22⋊C4—5482no—
123C23—2741no—
124C23—2741no—
125C23—5481no—
126C23—5481no—
127C23—1371no—
128C23—1371no—
129C23C4×#1058773911yescharacteristic
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—2741no—
133C2×C4—2741no—
134C2×C4—2741no—
135C2×C4—2741no—
136C2×C4—5481no—
137C2×C4—5481no—
138C2×C4—5481no—
139C2×C4—5482no—
140C2×C4—5482no—
141C2×C4—5482no—
142C2×C4—1371no—
143C2×C4—1371no—
144C2×C4C22⋊#1058773911yescharacteristic
145C2×C4C4⋊#1058773911yescharacteristic
146C8—41no—
147C8—5482no—
148C8—5481no—
149C22—21no—
150C22—21no—
151C22—21no—
152C22—2741no—
153C22—2741no—
154C22—2741no—
155C22—2741no—
156C22—2741no—
157C22—5482no—
158C22—5482no—
159C22—1371no—
160C22—1371no—
161C22(C2×#10587739)⋊C411yescharacteristic
162C4—21no—
163C4—2741no—
164C4—2741no—
165C4—10962no—
166C4#1798414611yescharacteristic
167C4#1798414711yescharacteristic
168C2—41no—
169C2—2741no—
170C2—2741no—
171C2—5481no—
172C2#2257828111yescharacteristic
173C2#2257828211yescharacteristic
174C2#2257828611yescharacteristic
subgroupG/H
centerC22(C2×#10587739)⋊C4
commutatorC23×C137C42
FrattiniC22×C4C2×#10587739
FittingC2×M16×C137C4
socleC22×C137#95
prime radicalC23×D137C2×C4
cosocleC22×C4×D137C22
hypercenterC2×M16#10587739
hypocenterC137#1305
LCSG ▷ C23×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
8. C22⋊#11764047C4semidirect
11. C2×#14555494C4semidirect
14. C2×M16×C137C4semidirect
87. C137#1305semidirect
95. C2×M16#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783742
#225783452
#225773662
#1305137