Finite Group Databasebeta

← #31796184

Subgroups — #31796184

Order 17536.

346 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 60 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, complemented
2C4×C22⋊#10587739C211yesmaximal, characteristic, complemented
3#22578255C212yesmaximal, complemented
4#22578254C212yesmaximal, complemented
5#121×D137C211yesmaximal, characteristic
6C2×#14555519—21no—
7C2×C22⋊#10587739—21no—
8C2×C4×#10587739—21no—
9#17984145—21no—
10#17984145C2212yescomplemented
11(C2×#10587739)⋊C4C2212yescomplemented
12C2×D4×D137—21no—
13C2×D4×D137C411yescharacteristic, complemented
14C22×C4×D137C2211yescharacteristic
15#17984063C2211yescharacteristic
16#17984062C411yescharacteristic, complemented
17#17984061—21no—
18#17984060—21no—
19#17984059C2211yescharacteristic
20C4⋊C4×D137C411yescharacteristic, complemented
21C22⋊C4×D137—21no—
22#121×C137C411yescharacteristic, complemented
23C22×#10587739—22no—
24#14555519—42no—
25C2×#11764047—22no—
26C22⋊#10587739—24no—
27C4×#10587739—42no—
28C23×D137—21no—
29C23×D137D411yescharacteristic
30C2×D548—21no—
31C2×D548C2×C411yescharacteristic
32C2×C22⋊D137—21no—
33C2×C22⋊D137—21no—
34C2×C22⋊D137—21no—
35D4×D137—42no—
36D4×D137—44no—
37C22×Dic137D411yescharacteristic
38C2×C4×D137—21no—
39C2×C4×D137—42no—
40C2×C4×D137C2×C411yescharacteristic
41C2×C4×D137D411yescharacteristic
42C2×C4×D137D411yescharacteristic
43C2×C4×D137C2311yescharacteristic
44C4⋊Dic137C2×C411yescharacteristic
45#14555489—21no—
46C22⋊Dic137—21no—
47#14555487—21no—
48#14555487—21no—
49C2×D4×C137—21no—
50C2×D4×C137C2×C411yescharacteristic
51C22×C4×C137C2×C411yescharacteristic
52C4⋊C4×C137C2×C411yescharacteristic
53C22⋊C4×C137—21no—
54C2×#10587739—24no—
55C2×#10587739—42no—
56C2×#10587739—42no—
57#11764047—42no—
58C22×D137—22no—
59C22×D137—22no—
60C22×D137—21no—
61C22×D137—41no—
62C22×D137—42no—
63C22×D137—42no—
64C22×D137—42no—
65C22×D137C22⋊C411yescharacteristic
66C22×D137C2×D411yescharacteristic
67D548—41no—
68D548—42no—
69C22⋊D137—42no—
70C22⋊D137—42no—
71C22⋊D137—42no—
72C2×Dic137—22no—
73C2×Dic137—21no—
74C2×Dic137C22⋊C411yescharacteristic
75C2×Dic137C2×D411yescharacteristic
76C4×D137—22no—
77C4×D137—22no—
78C4×D137—41no—
79C4×D137—41no—
80C4×D137—42no—
81C23×C137—21no—
82C23×C137C22⋊C411yescharacteristic
83D4×C137—41no—
84D4×C137—42no—
85C2×C4×C137—21no—
86C2×C4×C137—41no—
87C2×C4×C137C22⋊C411yescharacteristic
88C2×C4×C137C22×C411yescharacteristic
89C22×C137—21no—
90C22×C137—21no—
91C22×C137—21no—
92C22×C137—41no—
93C22×C137—41no—
94C22×C137—41no—
95C22×C137C2×C22⋊C411yescharacteristic
96C2×D137—21no—
97C2×D137—21no—
98C2×D137—21no—
99C2×D137—21no—
100C2×D137—44no—
101C2×D137—41no—
102C2×D137—41no—
103C2×D137—41no—
104C2×D137—84no—
105C2×D137#10412yes—
106C2×D137AGL(1,ℤ/8)12yes—
107#10587739—44no—
108C4×C137—21no—
109C4×C137—21no—
110C4×C137—41no—
111Dic137—21no—
112Dic137—41no—
113Dic137#9912yes—
114C2×C137—41no—
115C2×C137—81no—
116C2×C137C2×#9911yescharacteristic
117C2×C137#41811yescharacteristic
118C2×C137C2×#10411yescharacteristic
119D137—22no—
120D137—22no—
121D137—81no—
122C137#150711yescharacteristic, complemented, Sylow
123#1507—1371nomaximal, Sylow
124C2×#121—1371no—
125#407—1371no—
126C4×C22⋊C4—1371no—
127#327—1372no—
128#327—1372no—
129C22×D4—2741no—
130C22×D4—1371no—
131C23×C4—1371no—
132C2×M16—2741no—
133#121—2741no—
134#121—2741no—
135#121—1371no—
136#121—1371no—
137#121—1371no—
138#121#1058773911yescharacteristic, complemented
139C2×C4⋊C4—1371no—
140C2×C22⋊C4—2741no—
141C2×C22⋊C4—2741no—
142C2×C42—2741no—
143C22⋊C8—2741no—
144C22⋊C8—1372no—
145#95—1372no—
146C24—2741no—
147C24—1371no—
148C2×D4—21no—
149C2×D4—2741no—
150C2×D4—2741no—
151C2×D4—2741no—
152C2×D4—2741no—
153C2×D4—5482no—
154C2×D4—5484no—
155C2×D4—1371no—
156C2×D4C2×#1058773911yescharacteristic
157C22×C4—2741no—
158C22×C4—2742no—
159C22×C4—5482no—
160C22×C4—1371no—
161C22×C4—1371no—
162C22×C4—1371no—
163C22×C4—1371no—
164C22×C4—1371no—
165C22×C4C2×#1058773911yescharacteristic
166M16—5482no—
167C2×C8—2742no—
168C4⋊C4—2741no—
169C4⋊C4—1371no—
170C4⋊C4C2×#1058773911yescharacteristic
171C22⋊C4—21no—
172C22⋊C4—2741no—
173C22⋊C4—2744no—
174C22⋊C4—2741no—
175C22⋊C4—2741no—
176C42—5482no—
177C23—21no—
178C23—2742no—
179C23—2742no—
180C23—2741no—
181C23—5481no—
182C23—5482no—
183C23—5482no—
184C23—5482no—
185C23—1371no—
186C23—1371no—
187C23C22⋊#1058773911yescharacteristic
188D4—41no—
189D4—42no—
190D4—5481no—
191D4—5482no—
192D4—5482no—
193D4—5482no—
194D4—5482no—
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—2742no—
198C2×C4—2742no—
199C2×C4—2742no—
200C2×C4—2744no—
201C2×C4—2741no—
202C2×C4—5481no—
203C2×C4—5481no—
204C2×C4—5482no—
205C2×C4—5482no—
206C2×C4—5482no—
207C2×C4—1371no—
208C2×C4—1371no—
209C2×C4C22⋊#1058773911yescharacteristic
210C2×C4C22×#1058773911yescharacteristic
211C8—5482no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—2741no—
219C22—2741no—
220C22—2741no—
221C22—2741no—
222C22—5484no—
223C22—5481no—
224C22—5481no—
225C22—5481no—
226C22—10964no—
227C22—1372no—
228C22—1372no—
229C22C2×C22⋊#1058773911yescharacteristic
230C4—21no—
231C4—21no—
232C4—41no—
233C4—2741no—
234C4—5484no—
235C4—5481no—
236C4—1372no—
237C2—41no—
238C2—81no—
239C2—2742no—
240C2—2742no—
241C2—10961no—
242C2D137⋊#9911yescharacteristic
243C2#2257839511yescharacteristic
244C2D137⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#121×C137C4
socleC22×C137C2×C22⋊C4
prime radical#121×D137C2
cosocleC2×C4×D137C23
hypercenter#121#10587739
hypocenterC137#1507
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #22578375C2semidirect
2. C4×C22⋊#10587739C2semidirect
3. #22578255C2semidirect
4. #22578254C2semidirect
10. #17984145C22semidirect
11. (C2×#10587739)⋊C4C22semidirect
13. C2×D4×D137C4semidirect
16. #17984062C4semidirect
20. C4⋊C4×D137C4semidirect
22. #121×C137C4semidirect
122. C137#1507semidirect
138. #121#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783752
C4×C22⋊#105877392
#225782552
#225782542
#121×D1372
#1507137