Finite Group Databasebeta

← #31790435

Subgroups — #31790435

Order 17536.

524 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 127 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×D8×D137C211yesmaximal, characteristic, complemented
2C2×#17984096C211yesmaximal, characteristic, complemented
3#22577969C212yesmaximal, complemented
4#22577657C211yesmaximal, characteristic, complemented
5#22577656C211yesmaximal, characteristic, complemented
6#22577653C211yesmaximal, characteristic, complemented
7#22577638C212yesmaximal, complemented
8#22577637C212yesmaximal, complemented
9#22577438C211yesmaximal, characteristic, complemented
10#22577406C212yesmaximal, complemented
11#469×C137C211yesmaximal, characteristic, complemented
12C2×#14555510C2211yescharacteristic
13C2×D4×D137C2212yes—
14C2×D1096C2211yescharacteristic
15C2×#14555495C2212yes—
16D8×D137—24no—
17#17984096—24no—
18C2×#14555493C2211yescharacteristic
19C2×#14555493C2211yescharacteristic, complemented
20C22×#11764040C2211yescharacteristic, complemented
21C2×C8×D137C2211yescharacteristic
22C2×C8×D137C2211yescharacteristic, complemented
23#17984084—22no—
24#17984074—22no—
25#17984063C2212yescomplemented
26#17984062C2212yescomplemented
27#17984059C2212yescomplemented
28#17984058—22no—
29#17984056—22no—
30#17984046C2212yes—
31#17984021C2212yescomplemented
32#17984020C2211yescharacteristic, complemented
33#17984010C2212yescomplemented
34#17984009C2212yescomplemented
35#17984008C2212yescomplemented
36C2×D8×C137C2211yescharacteristic
37C2×M16×C137C2211yescharacteristic, complemented
38#121×C137C2212yescomplemented
39#106×C137C2211yescharacteristic, complemented
40#102×C137C2212yescomplemented
41C23×D137—22no—
42#14555510—22no—
43#14555510—22no—
44C2×Dic274D411yescharacteristic
45C2×D548—22no—
46C2×D548D411yescharacteristic
47C2×D548C2312yes—
48C2×C22⋊D137—22no—
49C2×C22⋊D137—22no—
50C2×C22⋊D137—22no—
51C2×C22⋊D137D411yescharacteristic
52C2×C22⋊D137D411yescharacteristic
53D4×D137—24no—
54D4×D137—24no—
55C2×C4×D137C2311yescharacteristic
56C2×C4×D137C2311yescharacteristic
57D1096—22no—
58#14555495—24no—
59#14555493—22no—
60#14555493—22no—
61C2×#11764040—22no—
62C2×#11764040C2311yescharacteristic
63C2×#11764040C2311yescharacteristic
64C8×D137—22no—
65C8×D137D414yescomplemented
66C4⋊Dic137C2312yes—
67C22⋊Dic137—22no—
68#14555487—22no—
69#14555487—22no—
70#14555487—22no—
71C4×Dic137—22no—
72C2×D4×C137—22no—
73C2×D4×C137C2312yes—
74C22×C4×C137C2311yescharacteristic
75D8×C137—22no—
76M16×C137—22no—
77C2×C8×C137C2311yescharacteristic
78C2×C8×C137C2311yescharacteristic
79C4⋊C4×C137C2312yes—
80C22⋊C4×C137—22no—
81C22×D137—22no—
82C22×D137—44no—
83C22×D137—82no—
84C22×D137—82no—
85C22×D137C2×D411yescharacteristic
86C22×D137C2×D411yescharacteristic
87Dic274—22no—
88D548—22no—
89D548—22no—
90D548—22no—
91D548—82no—
92C22⋊D137—22no—
93C22⋊D137—22no—
94C22⋊D137—44no—
95C22⋊D137—82no—
96C22⋊D137—82no—
97C2×Dic137—22no—
98C2×Dic137C2×D411yescharacteristic
99C2×Dic137C2×D411yescharacteristic
100C4×D137—21no—
101C4×D137—21no—
102C4×D137C2×D412yes—
103C4×D137#4112yes—
104#11764040—21no—
105#11764040C2×D412yes—
106C23×C137—22no—
107C23×C137C2×D411yescharacteristic
108D4×C137—22no—
109D4×C137—22no—
110D4×C137—82no—
111C2×C4×C137—21no—
112C2×C4×C137—21no—
113C2×C4×C137—22no—
114C2×C4×C137C2×D411yescharacteristic
115C2×C4×C137C2411yescharacteristic
116C8×C137—21no—
117C8×C137C2×D412yes—
118C22×C137—21no—
119C22×C137—22no—
120C22×C137—42no—
121C22×C137—42no—
122C22×C137—42no—
123C22×C137C22×D411yescharacteristic
124C2×D137—22no—
125C2×D137—21no—
126C2×D137—22no—
127C2×D137—42no—
128C2×D137—42no—
129C2×D137—42no—
130C2×D137—88no—
131C2×D137#12112yes—
132C4×C137—21no—
133C4×C137—42no—
134C4×C137C22×D411yescharacteristic
135C4×C137C2×#4111yescharacteristic
136Dic137—21no—
137Dic137—42no—
138Dic137#12112yes—
139C2×C137—41no—
140C2×C137—82no—
141C2×C137C2×#12111yescharacteristic
142C2×C137#57612yes—
143D137—22no—
144D137—41no—
145D137—82no—
146C137#305811yescharacteristic, complemented, Sylow
147#3058—1371nomaximal, Sylow
148C22×D8—1371no—
149C2×#131—1371no—
150#535—1372no—
151#469—1371no—
152#469—1371no—
153#469—1371no—
154#469D13711yescharacteristic, complemented
155#466—1372no—
156#466—1372no—
157#427—1371no—
158#418—1372no—
159C2×#41—1371no—
160C22×D4—1372no—
161C2×D8—2744no—
162C2×D8—1372no—
163C2×D8—1371no—
164C2×D8C2×D13711yescharacteristic
165#131—2744no—
166C2×M16—1371no—
167C2×M16—1371no—
168C2×M16C2×D13711yescharacteristic, complemented
169C22×C8—1371no—
170C22×C8—1371no—
171C22×C8—1371no—
172#127—2742no—
173#124—2742no—
174#121—1372no—
175#121—1372no—
176#121—1372no—
177#121C2×D13712yescomplemented
178#120—2742no—
179#120—2742no—
180#117—1372no—
181#107—1372no—
182#106—1371no—
183#106C2×D13711yescharacteristic, complemented
184#102—1372no—
185#102—1372no—
186#102—1372no—
187#102C2×D13712yescomplemented
188C24—2742no—
189#41—2742no—
190#41—2742no—
191C2×Q8—1371no—
192C2×D4—22no—
193C2×D4—2742no—
194C2×D4—2742no—
195C2×D4—2742no—
196C2×D4—2744no—
197C2×D4—2744no—
198C2×D4—2742no—
199C2×D4—1371no—
200C2×D4—1371no—
201C2×D4—1371no—
202C2×D4—1372no—
203C2×D4C22×D13712yes—
204C22×C4—1371no—
205C22×C4—1371no—
206C22×C4C22×D13711yescharacteristic
207D8—22no—
208D8—2744no—
209D8—2742no—
210M16—22no—
211M16—2742no—
212M16—2742no—
213C2×C8—2742no—
214C2×C8—2742no—
215C2×C8—1371no—
216C2×C8—1371no—
217C2×C8—1374no—
218C2×C8C22×D13711yescharacteristic
219C2×C8C22×D13711yescharacteristic
220C4⋊C4—1372no—
221C4⋊C4C22×D13712yes—
222C22⋊C4—22no—
223C22⋊C4—2742no—
224C22⋊C4—2742no—
225C22⋊C4—2742no—
226C22⋊C4—2742no—
227C42—2742no—
228C23—22no—
229C23—2742no—
230C23—5484no—
231C23—10962no—
232C23—10962no—
233C23—1371no—
234C23—1371no—
235C23D4×D13711yescharacteristic
236Q8—2742no—
237D4—22no—
238D4—22no—
239D4—82no—
240D4—2742no—
241D4—2742no—
242D4—2742no—
243D4—2742no—
244D4—2742no—
245D4—5484no—
246D4—10962no—
247D4—10962no—
248D4—10962no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—2741no—
253C2×C4—2741no—
254C2×C4—2742no—
255C2×C4—1371no—
256C2×C4—1371no—
257C2×C4—1372no—
258C2×C4—1372no—
259C2×C4D4×D13711yescharacteristic
260C2×C4C23×D13711yescharacteristic
261C8—21no—
262C8—2741no—
263C8—1372no—
264C8D4×D13712yes—
265C22—22no—
266C22—21no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—2742no—
271C22—2742no—
272C22—2741no—
273C22—5482no—
274C22—5482no—
275C22—5482no—
276C22—10968no—
277C22—1372no—
278C22C2×D4×D13711yescharacteristic
279C4—21no—
280C4—42no—
281C4—2741no—
282C4—5482no—
283C4—1372no—
284C4C2×D4×D13711yescharacteristic
285C4#41×D13711yescharacteristic
286C2—41no—
287C2—82no—
288C2—2742no—
289C2—5481no—
290C2—10962no—
291C2#121×D13711yescharacteristic
292C2#2257820712yes—
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#469×C137C2
socleC22×C137C22×D4
hypercenter#469D137
hypocenterC137#3058
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×D8×D137C2semidirect
2. C2×#17984096C2semidirect
3. #22577969C2semidirect
4. #22577657C2semidirect
5. #22577656C2semidirect
6. #22577653C2semidirect
7. #22577638C2semidirect
8. #22577637C2semidirect
9. #22577438C2semidirect
10. #22577406C2semidirect
11. #469×C137C2semidirect
19. C2×#14555493C22semidirect
20. C22×#11764040C22semidirect
22. C2×C8×D137C22semidirect
25. #17984063C22semidirect
26. #17984062C22semidirect
27. #17984059C22semidirect
31. #17984021C22semidirect
32. #17984020C22semidirect
33. #17984010C22semidirect
34. #17984009C22semidirect
35. #17984008C22semidirect
37. C2×M16×C137C22semidirect
38. #121×C137C22semidirect
39. #106×C137C22semidirect
40. #102×C137C22semidirect
65. C8×D137D4semidirect
146. C137#3058semidirect
154. #469D137semidirect
168. C2×M16C2×D137semidirect
177. #121C2×D137semidirect
183. #106C2×D137semidirect
187. #102C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1372
C2×#179840962
#225779692
#225776572
#225776562
#225776532
#225776382
#225776372
#225774382
#225774062
#469×C1372
#3058137