Finite Group Databasebeta

← #31797268

Subgroups — #31797268

Order 17536.

816 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 210 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
1C22×C22⋊#10587739C212yesmaximal
2#22578350C218yesmaximal, complemented
3#22578345C214yesmaximal, complemented
4C2×C22⋊C4×D137C211yesmaximal, characteristic
5C23×#10587739C2214yes—
6C2×C4⋊#10587739C2218yescomplemented
7C2×C22⋊#10587739—28no—
8C2×C22⋊#10587739C2214yes—
9C2×C22⋊#10587739C2214yes—
10(C2×#10587739)⋊C4C2218yescomplemented
11C24×D137C2211yescharacteristic
12C22×C4×D137C2212yes—
13C2×C22⋊Dic137C411yescharacteristic, complemented
14C2×#14555487C412yescomplemented
15C22⋊C4×D137C414yescomplemented
16C22⋊C4×D137C2214yes—
17C2×C22⋊C4×C137C411yescharacteristic, complemented
18C22×#10587739—28no—
19C22×#10587739—48no—
20C22×#10587739C2318yes—
21C4⋊#10587739—48no—
22C22⋊#10587739—28no—
23C22⋊#10587739—28no—
24C23×D137—44no—
25C23×D137C2×C412yes—
26C23×D137C2×C411yescharacteristic
27C23×D137D414yes—
28C23×D137Q814yes—
29C23×D137C2312yes—
30C23×D137C2311yescharacteristic
31C22×Dic137C2×C412yescomplemented
32C2×C4×D137—44no—
33C2×C4×D137C2×C414yescomplemented
34C2×C4×D137C2314yes—
35C22⋊Dic137C2×C414yescomplemented
36#14555487C2×C418yescomplemented
37C24×C137C2×C411yescharacteristic
38C22×C4×C137C2×C412yescomplemented
39C22⋊C4×C137C2×C414yescomplemented
40C2×#10587739—28no—
41C2×#10587739—216no—
42C2×#10587739—416no—
43C22×D137—24no—
44C22×D137—28no—
45C22×D137—48no—
46C22×D137—48no—
47C22×D137—44no—
48C22×D137—42no—
49C22×D137C4⋊C418yescomplemented
50C22×D137C4⋊C414yescomplemented
51C22×D137C22×C412yes—
52C22×D137C22×C411yescharacteristic
53C22×D137C2×D412yes—
54C22×D137C2×D414yes—
55C22×D137C2×Q814yes—
56C22×D137C2×Q812yes—
57C22×D137C2411yescharacteristic
58C2×Dic137—42no—
59C2×Dic137C22×C414yes—
60C4×D137—28no—
61C4×D137—44no—
62C23×C137—42no—
63C23×C137C4⋊C414yescomplemented
64C23×C137C22×C412yes—
65C23×C137C22×C411yescharacteristic
66C2×C4×C137—42no—
67C2×C4×C137C22×C414yes—
68C22×C137—22no—
69C22×C137—24no—
70C22×C137—44no—
71C22×C137C2×C4⋊C414yes—
72C22×C137C2×C4⋊C412yes—
73C22×C137C23×C411yescharacteristic
74C2×D137—22no—
75C2×D137—24no—
76C2×D137—24no—
77C2×D137—28no—
78C2×D137—24no—
79C2×D137—28no—
80C2×D137—48no—
81C2×D137—44no—
82C2×D137C2×C4⋊C414yes—
83C2×D137C2×C4⋊C412yes—
84C2×D137C22×D411yescharacteristic
85C2×D137C22×Q811yescharacteristic
86C2×D137H(5,𝔽2)14yes—
87#10587739—48no—
88C4×C137—24no—
89Dic137—24no—
90C2×C137—22no—
91C2×C137—24no—
92C2×C137C22×C4⋊C411yescharacteristic
93C2×C137#51812yes—
94D137—22no—
95D137—24no—
96D137#53412yes—
97D137#54312yes—
98C137#231511yescharacteristic, complemented, Sylow
99#2315—1371nomaximal, Sylow
100C22×C22⋊C4—1372no—
101C22×C22⋊C4—1371no—
102#385—1378no—
103#380—1374no—
104C25—1371no—
105C23×C4—1374no—
106C23×C4—1372no—
107C2×C4⋊C4—1378no—
108C2×C22⋊C4—2748no—
109C2×C22⋊C4—1374no—
110C2×C22⋊C4—1374no—
111C2×C22⋊C4—1371no—
112C2×C22⋊C4—1372no—
113C2×C22⋊C4—1374no—
114C2×C22⋊C4—1374no—
115C2×C22⋊C4#1058773911yescharacteristic, complemented
116#95—1378no—
117C24—5484no—
118C24—1371no—
119C24—1372no—
120C24—1374no—
121C24—1374no—
122C24—1371no—
123C24—1372no—
124C24C2×#1058773911yescharacteristic
125C22×C4—2748no—
126C22×C4—5484no—
127C22×C4—5488no—
128C22×C4—1378no—
129C22×C4—1374no—
130C22×C4—1372no—
131C22×C4—1374no—
132C22×C4C2×#1058773912yescomplemented
133C4⋊C4—5488no—
134C22⋊C4—2748no—
135C22⋊C4—2748no—
136C22⋊C4—1374no—
137C22⋊C4—1378no—
138C22⋊C4C2×#1058773914yescomplemented
139C23—42no—
140C23—2744no—
141C23—2748no—
142C23—5482no—
143C23—5484no—
144C23—5488no—
145C23—5488no—
146C23—1372no—
147C23—1374no—
148C23—1372no—
149C23—1374no—
150C23—1372no—
151C23—1371no—
152C23—1378no—
153C23—1374no—
154C23—1371no—
155C23C4⋊#1058773914yescomplemented
156C23C22×#1058773912yes—
157C23C22×#1058773911yescharacteristic
158C2×C4—42no—
159C2×C4—2748no—
160C2×C4—2748no—
161C2×C4—27416no—
162C2×C4—5484no—
163C2×C4—5482no—
164C2×C4—54816no—
165C2×C4—1374no—
166C2×C4C22×#1058773914yes—
167C22—22no—
168C22—24no—
169C22—44no—
170C22—2748no—
171C22—2744no—
172C22—2744no—
173C22—2748no—
174C22—2742no—
175C22—2744no—
176C22—5484no—
177C22—5488no—
178C22—1372no—
179C22—1374no—
180C22—1371no—
181C22—1371no—
182C22—1374no—
183C22C2×C4⋊#1058773912yes—
184C22C2×C4⋊#1058773914yes—
185C22C23×#1058773911yescharacteristic
186C4—24no—
187C4—2744no—
188C4—5488no—
189C2—22no—
190C2—24no—
191C2—2744no—
192C2—2742no—
193C2—1372no—
194C2—1372no—
195C2C22×C4⋊#1058773911yescharacteristic
196C2#2257847012yes—
subgroupG/H
center, FrattiniC22C23×#10587739
commutator, socleC22×C137C23×C4
FittingC2×C22⋊C4×C137C4
prime radicalC24×D137C22
cosocleC22×D137C24
hypercenterC2×C22⋊C4#10587739
hypocenterC137#2315
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #22578350C2semidirect
3. #22578345C2semidirect
6. C2×C4⋊#10587739C22semidirect
10. (C2×#10587739)⋊C4C22semidirect
13. C2×C22⋊Dic137C4semidirect
14. C2×#14555487C4semidirect
15. C22⋊C4×D137C4semidirect
17. C2×C22⋊C4×C137C4semidirect
31. C22×Dic137C2×C4semidirect
33. C2×C4×D137C2×C4semidirect
35. C22⋊Dic137C2×C4semidirect
36. #14555487C2×C4semidirect
38. C22×C4×C137C2×C4semidirect
39. C22⋊C4×C137C2×C4semidirect
49. C22×D137C4⋊C4semidirect
50. C22×D137C4⋊C4semidirect
63. C23×C137C4⋊C4semidirect
98. C137#2315semidirect
115. C2×C22⋊C4#10587739semidirect
132. C22×C4C2×#10587739semidirect
138. C22⋊C4C2×#10587739semidirect
155. C23C4⋊#10587739semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊#105877392
#225783502
#225783452
C2×C22⋊C4×D1372
#2315137