Finite Group Databasebeta

← #31790432

Subgroups — #31790432

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×#17984094C211yesmaximal, characteristic, complemented
3#22577969C212yesmaximal, complemented
4#22577655C212yesmaximal, complemented
5#22577654C212yesmaximal, complemented
6#22577640C211yesmaximal, characteristic, complemented
7#22577639C211yesmaximal, characteristic, complemented
8#22577636C211yesmaximal, characteristic, complemented
9#22577440C211yesmaximal, characteristic, complemented
10#22577406C212yesmaximal, complemented
11#466×C137C211yesmaximal, characteristic, complemented
12C2×#14555510C2211yescharacteristic
13C2×D4×D137C2212yes—
14C2×D1096C2211yescharacteristic
15C2×#14555495C2212yes—
16D8×D137—24no—
17#17984094—24no—
18C2×#14555494C2211yescharacteristic, complemented
19C2×#14555493C2211yescharacteristic
20C2×#14555493C2211yescharacteristic, 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#17984022C2211yescharacteristic, complemented
32#17984019C2212yescomplemented
33#17984010C2212yescomplemented
34#17984009C2212yescomplemented
35#17984008C2212yescomplemented
36C2×D8×C137C2211yescharacteristic
37C22×C8×C137C2211yescharacteristic, complemented
38#121×C137C2212yescomplemented
39#107×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#14555494—22no—
60#14555493—22no—
61#14555493—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—
76C2×C8×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—21no—
120C22×C137—21no—
121C22×C137—42no—
122C22×C137—42no—
123C22×C137—42no—
124C22×C137C22×D411yescharacteristic
125C2×D137—21no—
126C2×D137—21no—
127C2×D137—21no—
128C2×D137—21no—
129C2×D137—21no—
130C2×D137—42no—
131C2×D137—42no—
132C2×D137—42no—
133C2×D137—88no—
134C2×D137#12112yes—
135C4×C137—21no—
136C4×C137—42no—
137C4×C137C22×D411yescharacteristic
138C4×C137C2×#4111yescharacteristic
139Dic137—21no—
140Dic137—42no—
141Dic137#12112yes—
142C2×C137—41no—
143C2×C137—82no—
144C2×C137C2×#12111yescharacteristic
145C2×C137#57611yescharacteristic
146C2×C137#57611yescharacteristic
147D137—22no—
148D137—41no—
149D137—82no—
150C137#305811yescharacteristic, complemented, Sylow
151#3058—1371nomaximal, Sylow
152C22×D8—1371no—
153C2×#131—1371no—
154#535—1372no—
155#469—1372no—
156#469—1372no—
157#466—1371no—
158#466—1371no—
159#466—1371no—
160#466D13711yescharacteristic, complemented
161#427—1371no—
162#418—1372no—
163C2×#41—1371no—
164C22×D4—1372no—
165C2×D8—2744no—
166C2×D8—1372no—
167C2×D8—1371no—
168C2×D8C2×D13711yescharacteristic
169#131—2744no—
170C2×M16—1371no—
171C2×M16—1371no—
172C2×M16—1371no—
173C22×C8—1371no—
174C22×C8—1371no—
175C22×C8C2×D13711yescharacteristic, complemented
176#127—2742no—
177#124—2742no—
178#121—1372no—
179#121—1372no—
180#121—1372no—
181#121C2×D13712yescomplemented
182#120—2742no—
183#120—2742no—
184#117—1372no—
185#107—1371no—
186#107C2×D13711yescharacteristic, complemented
187#106—1372no—
188#102—1372no—
189#102—1372no—
190#102—1372no—
191#102C2×D13712yescomplemented
192C24—2742no—
193#41—2742no—
194#41—2742no—
195C2×Q8—1371no—
196C2×D4—22no—
197C2×D4—2742no—
198C2×D4—2742no—
199C2×D4—2742no—
200C2×D4—2744no—
201C2×D4—2744no—
202C2×D4—2742no—
203C2×D4—1371no—
204C2×D4—1371no—
205C2×D4—1371no—
206C2×D4—1372no—
207C2×D4C22×D13712yes—
208C22×C4—1371no—
209C22×C4—1371no—
210C22×C4C22×D13711yescharacteristic
211D8—22no—
212D8—2744no—
213D8—2742no—
214M16—2742no—
215M16—2742no—
216M16—2742no—
217C2×C8—22no—
218C2×C8—2742no—
219C2×C8—1371no—
220C2×C8—1371no—
221C2×C8—1374no—
222C2×C8C22×D13711yescharacteristic
223C2×C8C22×D13711yescharacteristic
224C4⋊C4—1372no—
225C4⋊C4C22×D13712yes—
226C22⋊C4—22no—
227C22⋊C4—2742no—
228C22⋊C4—2742no—
229C22⋊C4—2742no—
230C22⋊C4—2742no—
231C42—2742no—
232C23—22no—
233C23—2742no—
234C23—5484no—
235C23—10962no—
236C23—10962no—
237C23—1371no—
238C23—1371no—
239C23D4×D13711yescharacteristic
240Q8—2742no—
241D4—22no—
242D4—22no—
243D4—82no—
244D4—2742no—
245D4—2742no—
246D4—2742no—
247D4—2742no—
248D4—2742no—
249D4—5484no—
250D4—10962no—
251D4—10962no—
252D4—10962no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—2741no—
257C2×C4—2741no—
258C2×C4—2742no—
259C2×C4—1371no—
260C2×C4—1371no—
261C2×C4—1372no—
262C2×C4—1372no—
263C2×C4D4×D13711yescharacteristic
264C2×C4C23×D13711yescharacteristic
265C8—21no—
266C8—2741no—
267C8—1372no—
268C8D4×D13712yes—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—2741no—
276C22—2741no—
277C22—2741no—
278C22—2741no—
279C22—2741no—
280C22—5482no—
281C22—5482no—
282C22—5482no—
283C22—10968no—
284C22—1372no—
285C22C2×D4×D13711yescharacteristic
286C4—21no—
287C4—42no—
288C4—2741no—
289C4—5482no—
290C4—1372no—
291C4C2×D4×D13711yescharacteristic
292C4#41×D13711yescharacteristic
293C2—41no—
294C2—82no—
295C2—2742no—
296C2—5481no—
297C2—10962no—
298C2#121×D13711yescharacteristic
299C2#2257820411yescharacteristic
300C2#2257820511yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#466×C137C2
socleC22×C137C22×D4
hypercenter#466D137
hypocenterC137#3058
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×D8×D137C2semidirect
2. C2×#17984094C2semidirect
3. #22577969C2semidirect
4. #22577655C2semidirect
5. #22577654C2semidirect
6. #22577640C2semidirect
7. #22577639C2semidirect
8. #22577636C2semidirect
9. #22577440C2semidirect
10. #22577406C2semidirect
11. #466×C137C2semidirect
18. C2×#14555494C22semidirect
20. C2×#14555493C22semidirect
22. C2×C8×D137C22semidirect
25. #17984063C22semidirect
26. #17984062C22semidirect
27. #17984059C22semidirect
31. #17984022C22semidirect
32. #17984019C22semidirect
33. #17984010C22semidirect
34. #17984009C22semidirect
35. #17984008C22semidirect
37. C22×C8×C137C22semidirect
38. #121×C137C22semidirect
39. #107×C137C22semidirect
40. #102×C137C22semidirect
65. C8×D137D4semidirect
150. C137#3058semidirect
160. #466D137semidirect
175. C22×C8C2×D137semidirect
181. #121C2×D137semidirect
186. #107C2×D137semidirect
191. #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×#179840942
#225779692
#225776552
#225776542
#225776402
#225776392
#225776362
#225774402
#225774062
#466×C1372
#3058137