Finite Group Databasebeta

← #31784104

Subgroups — #31784104

Order 17536.

1056 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 185 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
1C2×#17984056C212yesmaximal, complemented
2C22×#14555487C211yesmaximal, characteristic, complemented
3#22577314C212yesmaximal, complemented
4#22577311C212yesmaximal, complemented
5#22577283C212yesmaximal, complemented
6#22577282C214yesmaximal, complemented
7#22577266C211yesmaximal, characteristic, complemented
8C22×#41×C137C211yesmaximal, characteristic, complemented
9C24×D137C2211yescharacteristic
10C22×C22⋊D137C2214yes—
11C23×Dic137C2211yescharacteristic
12#17984056—28no—
13C2×C22⋊Dic137C2212yes—
14C2×C22⋊Dic137C2214yescomplemented
15C2×#14555487—24no—
16C2×#14555487C2212yes—
17C2×#14555487C2214yes—
18C2×#14555487C2212yes—
19#17983995C2212yescomplemented
20#17983995C2212yescomplemented
21#17983995C2214yescomplemented
22C2×#41×C137—28no—
23C2×#41×C137—24no—
24C22×Q8×C137C2211yescharacteristic, complemented
25C22×D4×C137C2212yes—
26C22×D4×C137C2211yescharacteristic, complemented
27C23×C4×C137C2211yescharacteristic
28C23×C4×C137C2212yescomplemented
29C23×D137—24no—
30C23×D137—22no—
31C23×D137—24no—
32C23×D137—42no—
33C23×D137C2312yes—
34C2×C22⋊D137—28no—
35C2×C22⋊D137—28no—
36C2×C22⋊D137—44no—
37C22×Dic137—22no—
38C22×Dic137—42no—
39C22×Dic137C2312yes—
40C22×Dic137C2314yes—
41C22⋊Dic137—24no—
42C22⋊Dic137—44no—
43#14555487—24no—
44#14555487—24no—
45#14555487—28no—
46C24×C137C2312yes—
47C24×C137C2311yescharacteristic
48#41×C137—232no—
49C2×Q8×C137—24no—
50C2×Q8×C137—22no—
51C2×D4×C137—24no—
52C2×D4×C137—24no—
53C2×D4×C137—24no—
54C2×D4×C137—22no—
55C2×D4×C137D418yescomplemented
56C22×C4×C137—24no—
57C22×C4×C137—24no—
58C22×C4×C137—24no—
59C22×C4×C137—24no—
60C22×C4×C137D414yescomplemented
61C22×C4×C137C2311yescharacteristic
62C22×C4×C137C2311yescharacteristic
63C22×C4×C137C2312yes—
64C22×D137—28no—
65C22×D137—22no—
66C22×D137—24no—
67C22×D137—22no—
68C22×D137—24no—
69C22×D137—416no—
70C22×D137—44no—
71C22×D137—44no—
72C22×D137#4114yescomplemented
73C22⋊D137—416no—
74C2×Dic137—22no—
75C2×Dic137—24no—
76C2×Dic137—24no—
77C2×Dic137—24no—
78C2×Dic137—24no—
79C2×Dic137—44no—
80C23×C137—22no—
81C23×C137—22no—
82C23×C137—22no—
83C23×C137—22no—
84C23×C137—24no—
85C23×C137—24no—
86C23×C137C2×D414yes—
87C23×C137C2×D414yes—
88C23×C137C2×D412yes—
89C23×C137C2411yescharacteristic
90Q8×C137—28no—
91D4×C137—28no—
92D4×C137—216no—
93C2×C4×C137—216no—
94C2×C4×C137—28no—
95C2×C4×C137—22no—
96C2×C4×C137—22no—
97C2×C4×C137—22no—
98C2×C4×C137—22no—
99C2×C4×C137C2×D412yes—
100C2×C4×C137C2×D414yes—
101C2×C4×C137C2×D412yes—
102C22×C137—22no—
103C22×C137—22no—
104C22×C137—28no—
105C22×C137—24no—
106C22×C137—28no—
107C22×C137—42no—
108C22×C137—42no—
109C22×C137—44no—
110C22×C137#12014yes—
111C22×C137C22×D411yescharacteristic
112C22×C137C22×D412yes—
113C22×C137C2×#4112yes—
114C22×C137H(5,𝔽2)12yes—
115C2×D137—24no—
116C2×D137—28no—
117C2×D137—416no—
118C2×D137—42no—
119C2×D137—42no—
120C2×D137—44no—
121C4×C137—22no—
122C4×C137—24no—
123C4×C137—22no—
124Dic137—42no—
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×#41D13711yescharacteristic, complemented
136C2×#120—1372no—
137C22×C22⋊C4—1371no—
138#393—1372no—
139#392—1372no—
140#386—1372no—
141#386—1374no—
142#381—1371no—
143C25—1371no—
144C2×#41—28no—
145C2×#41—24no—
146C22×Q8C2×D13711yescharacteristic, complemented
147C22×D4—1374no—
148C22×D4C2×D13712yes—
149C22×D4C2×D13711yescharacteristic, complemented
150C23×C4—1371no—
151C23×C4C2×D13711yescharacteristic
152C23×C4C2×D13712yescomplemented
153#120—2748no—
154C2×C22⋊C4—2744no—
155C2×C22⋊C4—1372no—
156C2×C22⋊C4—1372no—
157C2×C22⋊C4—1372no—
158C2×C22⋊C4—1374no—
159C2×C22⋊C4—1374no—
160#95—1372no—
161#95—1372no—
162#95—1374no—
163C24—2744no—
164C24—2742no—
165C24—2744no—
166C24—5482no—
167C24—1372no—
168C24C22×D13712yes—
169C24C22×D13711yescharacteristic
170#41—232no—
171C2×Q8—24no—
172C2×Q8—22no—
173C2×D4—24no—
174C2×D4—24no—
175C2×D4—24no—
176C2×D4—22no—
177C2×D4—2748no—
178C2×D4—2748no—
179C2×D4—5484no—
180C2×D4C22⋊D13718yescomplemented
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—2742no—
186C22×C4—5482no—
187C22×C4—1374no—
188C22×C4—1372no—
189C22×C4C22⋊D13714yescomplemented
190C22×C4C22×D13711yescharacteristic
191C22×C4C22×D13711yescharacteristic
192C22×C4C22×D13712yes—
193C22⋊C4—2744no—
194C22⋊C4—2744no—
195C22⋊C4—2744no—
196C22⋊C4—2748no—
197C22⋊C4—5484no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—24no—
203C23—24no—
204C23—2748no—
205C23—2742no—
206C23—2744no—
207C23—2742no—
208C23—2744no—
209C23—54816no—
210C23—5484no—
211C23—5484no—
212C23—1374no—
213C23C2×C22⋊D13714yes—
214C23C2×C22⋊D13714yes—
215C23C2×C22⋊D13712yes—
216C23C23×D13711yescharacteristic
217Q8—28no—
218D4—28no—
219D4—216no—
220D4—54816no—
221C2×C4—216no—
222C2×C4—28no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—2742no—
228C2×C4—2744no—
229C2×C4—2744no—
230C2×C4—2744no—
231C2×C4—2744no—
232C2×C4—5484no—
233C2×C4C2×C22⋊D13712yes—
234C2×C4C2×C22⋊D13714yes—
235C2×C4C2×C22⋊D13712yes—
236C22—22no—
237C22—28no—
238C22—22no—
239C22—28no—
240C22—24no—
241C22—42no—
242C22—44no—
243C22—42no—
244C22—2748no—
245C22—2744no—
246C22—54816no—
247C22—5482no—
248C22—5484no—
249C22—5482no—
250C22#1798405714yes—
251C22C22×C22⋊D13711yescharacteristic
252C22C22×C22⋊D13712yes—
253C22#41×D13712yes—
254C22#1798413412yes—
255C4—22no—
256C4—24no—
257C4—22no—
258C4—5482no—
259C4—5484no—
260C2—24no—
261C2—44no—
262C2—5484no—
263C2C2×#1798405711yescharacteristic
264C2#2257791011yescharacteristic
265C2#2257796711yescharacteristic
266C2#2257804114yes—
subgroupG/H
center, FrattiniC23C23×D137
commutator, socle, cosocleC23×C137C24
FittingC22×#41×C137C2
hypercenterC22×#41D137
hypocenterC137#2404
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#17984056C2semidirect
2. C22×#14555487C2semidirect
3. #22577314C2semidirect
4. #22577311C2semidirect
5. #22577283C2semidirect
6. #22577282C2semidirect
7. #22577266C2semidirect
8. C22×#41×C137C2semidirect
14. C2×C22⋊Dic137C22semidirect
19. #17983995C22semidirect
20. #17983995C22semidirect
21. #17983995C22semidirect
24. C22×Q8×C137C22semidirect
26. C22×D4×C137C22semidirect
28. C23×C4×C137C22semidirect
55. C2×D4×C137D4semidirect
60. C22×C4×C137D4semidirect
72. C22×D137#41semidirect
133. C137#2404semidirect
135. C22×#41D137semidirect
146. C22×Q8C2×D137semidirect
149. C22×D4C2×D137semidirect
152. C23×C4C2×D137semidirect
180. C2×D4C22⋊D137semidirect
189. C22×C4C22⋊D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840562
C22×#145554872
#225773142
#225773112
#225772832
#225772822
#225772662
C22×#41×C1372
#2404137