Finite Group Databasebeta

← #31796064

Subgroups — #31796064

Order 17536.

322 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 64 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, complemented
2#22578371C211yesmaximal, characteristic, complemented
3#22578246C214yesmaximal, complemented
4C2×#17984059C211yesmaximal, characteristic
5C2×#14555520—21no—
6C2×#14555518—21no—
7#17984145—21no—
8#17984145—21no—
9C22⋊#11764047C2212yes—
10C22⋊#11764047C2212yes—
11C22×C22⋊D137C412yes—
12C22×C4×D137C2211yescharacteristic
13#17984059—21no—
14#17984059—21no—
15#17984059—21no—
16#17984059—21no—
17C2×C4⋊Dic137C411yescharacteristic
18C2×C22⋊Dic137C2212yes—
19C22×D4×C137C411yescharacteristic
20#14555520—42no—
21#14555518—42no—
22C2×#11764047—22no—
23C2×#11764047—22no—
24C23×D137C2×C411yescharacteristic
25C2×C22⋊D137—22no—
26C2×C22⋊D137—22no—
27C2×C22⋊D137—44no—
28C22×Dic137C2×C412yes—
29C22×Dic137C2311yescharacteristic
30C2×C4×D137—42no—
31C2×C4×D137D411yescharacteristic
32C2×C4×D137D411yescharacteristic
33C2×C4×D137D411yescharacteristic
34C2×C4×D137D411yescharacteristic
35C4⋊Dic137—21no—
36C4⋊Dic137—21no—
37C22⋊Dic137—22no—
38C22⋊Dic137—22no—
39C24×C137C2×C412yes—
40C2×D4×C137—21no—
41C2×D4×C137—21no—
42C2×D4×C137—42no—
43C22×C4×C137C2×C411yescharacteristic
44#11764047—42no—
45#11764047—42no—
46C22×D137—22no—
47C22×D137—22no—
48C22×D137—41no—
49C22×D137C22⋊C411yescharacteristic
50C22×D137C22⋊C411yescharacteristic
51C22⋊D137—44no—
52C22⋊D137—44no—
53C2×Dic137—22no—
54C2×Dic137—41no—
55C2×Dic137—42no—
56C2×Dic137C2×D411yescharacteristic
57C2×Dic137C2×D411yescharacteristic
58C4×D137—22no—
59C4×D137—22no—
60C4×D137—41no—
61C4×D137—41no—
62C23×C137—22no—
63C23×C137—24no—
64C23×C137—42no—
65C23×C137—42no—
66C23×C137C22×C411yescharacteristic
67D4×C137—42no—
68D4×C137—42no—
69C2×C4×C137—22no—
70C2×C4×C137C22⋊C411yescharacteristic
71C2×C4×C137C22⋊C411yescharacteristic
72C22×C137—21no—
73C22×C137—21no—
74C22×C137—42no—
75C22×C137—42no—
76C22×C137—42no—
77C22×C137—84no—
78C22×C137C2×C22⋊C411yescharacteristic
79C22×C137#13112yes—
80C2×D137—21no—
81C2×D137—21no—
82C2×D137—21no—
83C2×D137—21no—
84C2×D137—21no—
85C2×D137—44no—
86C2×D137#10012yes—
87C4×C137—21no—
88C4×C137#9912yes—
89Dic137—21no—
90Dic137—21no—
91Dic137—42no—
92C2×C137—22no—
93C2×C137—82no—
94C2×C137#40811yescharacteristic
95C2×C137C2×#9911yescharacteristic
96C2×C137C2×#10011yescharacteristic
97D137—22no—
98D137—41no—
99C137#147111yescharacteristic, complemented, Sylow
100#1471—1371nomaximal, Sylow
101C2×#121—1371no—
102#407—1371no—
103C2×C22⋊C8—1371no—
104#323—1374no—
105C22×D4—1372no—
106C22×D4#1058773911yescharacteristic
107C23×C4—1371no—
108C2×M16—2741no—
109C22×C8—2741no—
110#121—2741no—
111#121—2741no—
112#121—2741no—
113#121—2741no—
114C2×C4⋊C4—1371no—
115C2×C22⋊C4—1372no—
116C22⋊C8—2741no—
117C22⋊C8—2741no—
118C22⋊C8—1372no—
119C22⋊C8—1372no—
120C24—1371no—
121C24C2×#1058773912yes—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—42no—
125C2×D4—2742no—
126C2×D4—2742no—
127C2×D4—5484no—
128C22×C4—5482no—
129C22×C4—1371no—
130C22×C4—1371no—
131C22×C4—1372no—
132C22×C4—1371no—
133C22×C4—1371no—
134C22×C4—1371no—
135C22×C4C2×#1058773911yescharacteristic
136M16—5482no—
137C2×C8—2742no—
138C2×C8—2742no—
139C2×C8—5482no—
140C4⋊C4—2741no—
141C4⋊C4—2741no—
142C22⋊C4—2742no—
143C22⋊C4—2742no—
144C23—22no—
145C23—24no—
146C23—42no—
147C23—42no—
148C23—2742no—
149C23—2742no—
150C23—5481no—
151C23—1371no—
152C23—1371no—
153C23C22×#1058773911yescharacteristic
154D4—42no—
155D4—42no—
156D4—5484no—
157D4—5484no—
158C2×C4—22no—
159C2×C4—2742no—
160C2×C4—2742no—
161C2×C4—2742no—
162C2×C4—5481no—
163C2×C4—5481no—
164C2×C4—5481no—
165C2×C4—5482no—
166C2×C4—1371no—
167C2×C4—1371no—
168C2×C4C22⋊#1058773911yescharacteristic
169C2×C4C22⋊#1058773911yescharacteristic
170C8—5482no—
171C8—5482no—
172C22—21no—
173C22—21no—
174C22—42no—
175C22—42no—
176C22—42no—
177C22—84no—
178C22—2741no—
179C22—2741no—
180C22—2741no—
181C22—2741no—
182C22—2741no—
183C22—5484no—
184C22—1372no—
185C22C2×C22⋊#1058773911yescharacteristic
186C22#1798417512yes—
187C4—21no—
188C4—2741no—
189C4—2741no—
190C4—5482no—
191C4#1798414612yes—
192C2—22no—
193C2—82no—
194C2—2742no—
195C2—5481no—
196C2#2257837711yescharacteristic
197C2C2×#1798414611yescharacteristic
198C2D137⋊#10011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
FittingC22×D4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC2×#17984059C2
cosocleC22×Dic137C23
hypercenterC22×D4#10587739
hypocenterC137#1471
LCSG ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #22578374C2semidirect
2. #22578371C2semidirect
3. #22578246C2semidirect
99. C137#1471semidirect

Maximal subgroups

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

subgroupindex
#225783742
#225783712
#225782462
C2×#179840592
#1471137