Finite Group Databasebeta

← #31784108

Subgroups — #31784108

Order 17536.

1056 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 173 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×#14555510C211yesmaximal, characteristic, complemented
2C2×#17984057C212yesmaximal, complemented
3#22577315C212yesmaximal, complemented
4#22577312C212yesmaximal, complemented
5#22577282C212yesmaximal, complemented
6#22577279C214yesmaximal, complemented
7#22577266C211yesmaximal, characteristic, complemented
8C22×C22⋊C4×C137C211yesmaximal, characteristic, complemented
9C2×#14555510—28no—
10C2×#14555510—24no—
11C22×Dic274C2211yescharacteristic, complemented
12C22×D548C2211yescharacteristic, complemented
13C22×C22⋊D137C2214yes—
14C22×C22⋊D137C2212yes—
15C22×C4×D137C2212yescomplemented
16#17984057—28no—
17C2×C22⋊Dic137C2212yes—
18C2×C22⋊Dic137C2214yes—
19C2×#14555487C2214yescomplemented
20#17983995C2212yescomplemented
21#17983995C2212yescomplemented
22#17983995C2214yescomplemented
23C25×C137C2211yescharacteristic
24C23×C4×C137C2211yescharacteristic
25C23×C4×C137C2211yescharacteristic
26C2×C22⋊C4×C137—24no—
27C2×C22⋊C4×C137C2212yes—
28C2×C22⋊C4×C137C2212yes—
29C23×D137C2312yes—
30#14555510—232no—
31C2×Dic274—24no—
32C2×Dic274—22no—
33C2×D548—24no—
34C2×D548—22no—
35C2×C22⋊D137—28no—
36C2×C22⋊D137—24no—
37C2×C22⋊D137—24no—
38C2×C22⋊D137—28no—
39C2×C22⋊D137—44no—
40C2×C22⋊D137D418yescomplemented
41C22×Dic137C2314yes—
42C22×Dic137C2312yes—
43C2×C4×D137—24no—
44C2×C4×D137—24no—
45C2×C4×D137—24no—
46C22⋊Dic137—28no—
47C22⋊Dic137—24no—
48#14555487—44no—
49C24×C137—22no—
50C24×C137—24no—
51C24×C137—24no—
52C24×C137—42no—
53C24×C137C2312yes—
54C24×C137C2311yescharacteristic
55C22×C4×C137—24no—
56C22×C4×C137—22no—
57C22×C4×C137—42no—
58C22×C4×C137D414yescomplemented
59C22×C4×C137C2312yes—
60C22×C4×C137C2311yescharacteristic
61C22×C4×C137C2311yescharacteristic
62C22⋊C4×C137—24no—
63C22⋊C4×C137—24no—
64C22×D137—22no—
65C22×D137—22no—
66C22×D137—24no—
67C22×D137—24no—
68C22×D137C2×D414yes—
69Dic274—28no—
70D548—28no—
71C22⋊D137—216no—
72C22⋊D137—416no—
73C2×Dic137—22no—
74C2×Dic137—22no—
75C2×Dic137—24no—
76C2×Dic137—24no—
77C2×Dic137—24no—
78C2×Dic137C2×D414yes—
79C4×D137—216no—
80C23×C137—22no—
81C23×C137—22no—
82C23×C137—28no—
83C23×C137—22no—
84C23×C137—24no—
85C23×C137—22no—
86C23×C137—24no—
87C23×C137—416no—
88C23×C137—44no—
89C23×C137—44no—
90C23×C137C2×D414yes—
91C23×C137C2×D412yes—
92C23×C137#4114yescomplemented
93C23×C137C2411yescharacteristic
94C2×C4×C137—28no—
95C2×C4×C137—22no—
96C2×C4×C137—22no—
97C2×C4×C137—22no—
98C2×C4×C137—24no—
99C2×C4×C137—44no—
100C2×C4×C137C2×D412yes—
101C2×C4×C137C2×D412yes—
102C22×C137—22no—
103C22×C137—22no—
104C22×C137—28no—
105C22×C137—24no—
106C22×C137—28no—
107C22×C137—416no—
108C22×C137—42no—
109C22×C137—42no—
110C22×C137—44no—
111C22×C137#12014yes—
112C22×C137C22×D412yes—
113C22×C137C22×D411yescharacteristic
114C22×C137C2×#4112yes—
115C22×C137H(5,𝔽2)12yes—
116C2×D137—24no—
117C2×D137—28no—
118C2×D137—42no—
119C2×D137—42no—
120C2×D137—44no—
121C4×C137—22no—
122C4×C137—22no—
123C4×C137—42no—
124Dic137—24no—
125Dic137—44no—
126C2×C137—24no—
127C2×C137—44no—
128C2×C137C2×#12011yescharacteristic
129C2×C137#52511yescharacteristic
130C2×C137#53511yescharacteristic
131C2×C137#54614yes—
132D137—44no—
133C137#240411yescharacteristic, complemented, Sylow
134#2404—1371nomaximal, Sylow
135C22×#41—1371no—
136C2×#120—1372no—
137C22×C22⋊C4D13711yescharacteristic, complemented
138#393—1372no—
139#392—1372no—
140#386—1372no—
141#386—1374no—
142#381—1371no—
143C25C2×D13711yescharacteristic
144C2×#41—2748no—
145C2×#41—2744no—
146C22×Q8—1371no—
147C22×D4—1374no—
148C22×D4—1372no—
149C22×D4—1371no—
150C23×C4—1372no—
151C23×C4C2×D13711yescharacteristic
152C23×C4C2×D13711yescharacteristic
153#120—2748no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—1372no—
156C2×C22⋊C4—1374no—
157C2×C22⋊C4—1374no—
158C2×C22⋊C4C2×D13712yes—
159C2×C22⋊C4C2×D13712yes—
160#95—1372no—
161#95—1372no—
162#95—1374no—
163C24—22no—
164C24—24no—
165C24—24no—
166C24—42no—
167C24—1372no—
168C24C22×D13712yes—
169C24C22×D13711yescharacteristic
170#41—27432no—
171C2×Q8—2744no—
172C2×Q8—2742no—
173C2×D4—2744no—
174C2×D4—2744no—
175C2×D4—2744no—
176C2×D4—2748no—
177C2×D4—2748no—
178C2×D4—2742no—
179C2×D4—5484no—
180C2×D4—1378no—
181C22×C4—24no—
182C22×C4—22no—
183C22×C4—42no—
184C22×C4—2744no—
185C22×C4—2744no—
186C22×C4—2744no—
187C22×C4—1374no—
188C22×C4—1372no—
189C22×C4C22⋊D13714yescomplemented
190C22×C4C22×D13712yes—
191C22×C4C22×D13711yescharacteristic
192C22×C4C22×D13711yescharacteristic
193C22⋊C4—24no—
194C22⋊C4—24no—
195C22⋊C4—2744no—
196C22⋊C4—2748no—
197C22⋊C4—5484no—
198C23—22no—
199C23—22no—
200C23—28no—
201C23—24no—
202C23—22no—
203C23—22no—
204C23—24no—
205C23—416no—
206C23—44no—
207C23—44no—
208C23—2742no—
209C23—2742no—
210C23—2744no—
211C23—2744no—
212C23—1374no—
213C23D4×D13714yes—
214C23C2×C22⋊D13712yes—
215C23#1455551014yescomplemented
216C23C23×D13711yescharacteristic
217Q8—2748no—
218D4—2748no—
219D4—27416no—
220D4—54816no—
221C2×C4—28no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—24no—
226C2×C4—44no—
227C2×C4—27416no—
228C2×C4—2742no—
229C2×C4—2742no—
230C2×C4—2744no—
231C2×C4—2744no—
232C2×C4—2744no—
233C2×C4—1374no—
234C2×C4C2×C22⋊D13712yes—
235C2×C4C2×C22⋊D13712yes—
236C22—22no—
237C22—28no—
238C22—22no—
239C22—28no—
240C22—24no—
241C22—416no—
242C22—42no—
243C22—44no—
244C22—42no—
245C22—2748no—
246C22—2744no—
247C22—5482no—
248C22—5484no—
249C22—5482no—
250C22#1798405614yes—
251C22C2×D4×D13712yes—
252C22C22×C22⋊D13711yescharacteristic
253C22C2×#1455551012yes—
254C22#1798413312yes—
255C4—22no—
256C4—22no—
257C4—42no—
258C4—2744no—
259C4—5484no—
260C2—24no—
261C2—44no—
262C2—5484no—
263C2C2×#1798405611yescharacteristic
264C2#2257791411yescharacteristic
265C2#2257797011yescharacteristic
266C2#2257804314yes—
subgroupG/H
center, FrattiniC23C23×D137
commutator, socle, cosocleC23×C137C24
FittingC22×C22⋊C4×C137C2
hypercenterC22×C22⋊C4D137
hypocenterC137#2404
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#14555510C2semidirect
2. C2×#17984057C2semidirect
3. #22577315C2semidirect
4. #22577312C2semidirect
5. #22577282C2semidirect
6. #22577279C2semidirect
7. #22577266C2semidirect
8. C22×C22⋊C4×C137C2semidirect
11. C22×Dic274C22semidirect
12. C22×D548C22semidirect
15. C22×C4×D137C22semidirect
19. C2×#14555487C22semidirect
20. #17983995C22semidirect
21. #17983995C22semidirect
22. #17983995C22semidirect
40. C2×C22⋊D137D4semidirect
58. C22×C4×C137D4semidirect
92. C23×C137#41semidirect
133. C137#2404semidirect
137. C22×C22⋊C4D137semidirect
189. C22×C4C22⋊D137semidirect
215. C23#14555510semidirect

Maximal subgroups

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

subgroupindex
C22×#145555102
C2×#179840572
#225773152
#225773122
#225772822
#225772792
#225772662
C22×C22⋊C4×C1372
#2404137