Finite Group Databasebeta

← #31790264

Subgroups — #31790264

Order 17536.

584 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 133 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×#17984112C212yesmaximal, complemented
2#127×D137C211yesmaximal, characteristic, complemented
3C4×#14555509C211yesmaximal, characteristic, complemented
4#22577604C211yesmaximal, characteristic, complemented
5#22577602C212yesmaximal, complemented
6#22577600C211yesmaximal, characteristic, complemented
7#22577598C211yesmaximal, characteristic, complemented
8#22577596C212yesmaximal, complemented
9#22577425C211yesmaximal, characteristic, complemented
10#22577406C212yesmaximal, complemented
11#459×C137C211yesmaximal, characteristic, complemented
12C2×#14555509C2211yescharacteristic
13C2×D4×D137—21no—
14C2×D4×D137—21no—
15C2×D4×D137C2212yes—
16#17984112—28no—
17C2×#14555499C2212yescomplemented
18C2×#14555498C2212yescomplemented
19C2×#14555495C2212yescomplemented
20C2×#14555493C2212yes—
21#17984085C2211yescharacteristic
22#17984084—21no—
23#17984084—21no—
24#17984084C2212yes—
25C4×Dic274C2211yescharacteristic, complemented
26#17984053C2211yescharacteristic, complemented
27D4×Dic137C2211yescharacteristic, complemented
28C4×C22⋊D137—21no—
29#17984048—21no—
30#17984046C2211yescharacteristic
31#17984044—21no—
32C2×C4×Dic137—21no—
33C42×D137C2211yescharacteristic
34C4⋊#11764040C2211yescharacteristic, complemented
35#17984017C2212yescomplemented
36#17984010C2212yescomplemented
37#17984009C2212yescomplemented
38#17984008C2212yescomplemented
39C2×D8×C137C2212yescomplemented
40#127×C137C2211yescharacteristic
41C4×D4×C137C2211yescharacteristic, complemented
42C4⋊C8×C137C2211yescharacteristic, complemented
43#102×C137C2212yescomplemented
44C23×D137—22no—
45#14555509—22no—
46#14555509D414yescomplemented
47C2×Dic274C2311yescharacteristic
48C2×D548—21no—
49C2×D548—21no—
50C2×D548C2312yes—
51C2×C22⋊D137—21no—
52C2×C22⋊D137—22no—
53C2×C22⋊D137—22no—
54C2×C22⋊D137—22no—
55D4×D137—24no—
56D4×D137—24no—
57D4×D137—44no—
58D4×D137—44no—
59C22×Dic137—21no—
60C22×Dic137—21no—
61C2×C4×D137D411yescharacteristic
62C2×C4×D137D411yescharacteristic
63C2×C4×D137C2311yescharacteristic
64#14555499—24no—
65#14555498—24no—
66#14555495—24no—
67#14555493—24no—
68C2×#11764040C2312yes—
69C4⋊Dic137C2311yescharacteristic
70#14555489—21no—
71#14555489—21no—
72C22⋊Dic137—21no—
73#14555487—21no—
74C4×Dic137—21no—
75C4×Dic137—21no—
76C4×Dic137—21no—
77C4×Dic137D411yescharacteristic
78C4×Dic137D411yescharacteristic
79C4×Dic137C2311yescharacteristic
80C2×D4×C137—21no—
81C2×D4×C137—21no—
82C2×D4×C137C2312yes—
83C2×D4×C137C2311yescharacteristic
84C22×C4×C137—21no—
85D8×C137—24no—
86C2×C8×C137C2312yes—
87C4⋊C4×C137C2311yescharacteristic
88C22⋊C4×C137—21no—
89C42×C137C2311yescharacteristic
90C22×D137—22no—
91C22×D137—44no—
92C22×D137—44no—
93C22×D137—44no—
94C22×D137C2×D411yescharacteristic
95Dic274—21no—
96Dic274C2×D412yescomplemented
97D548—22no—
98D548—22no—
99D548—42no—
100D548—42no—
101C22⋊D137—24no—
102C22⋊D137—44no—
103C22⋊D137—44no—
104C22⋊D137—44no—
105C2×Dic137—21no—
106C2×Dic137—24no—
107C2×Dic137—21no—
108C2×Dic137—42no—
109C2×Dic137C2×D411yescharacteristic
110C2×Dic137C2×D411yescharacteristic
111C2×Dic137C2×D411yescharacteristic
112C4×D137—22no—
113C4×D137—22no—
114C4×D137C2×D412yes—
115C4×D137#4112yes—
116#11764040—22no—
117C23×C137—21no—
118C23×C137—22no—
119D4×C137—22no—
120D4×C137—21no—
121D4×C137—22no—
122D4×C137—42no—
123D4×C137—42no—
124D4×C137C2×D412yescomplemented
125C2×C4×C137—21no—
126C2×C4×C137—42no—
127C2×C4×C137C2×D411yescharacteristic
128C2×C4×C137C2×D411yescharacteristic
129C2×C4×C137C2411yescharacteristic
130C8×C137—22no—
131C22×C137—22no—
132C22×C137—41no—
133C22×C137—41no—
134C22×C137—42no—
135C22×C137—42no—
136C22×C137—42no—
137C22×C137C22×D411yescharacteristic
138C2×D137—21no—
139C2×D137—21no—
140C2×D137—42no—
141C2×D137—42no—
142C2×D137—42no—
143C2×D137—88no—
144C2×D137#12112yes—
145C4×C137—21no—
146C4×C137—41no—
147C4×C137AGL(1,ℤ/8)12yescomplemented
148C4×C137C22×D411yescharacteristic
149C4×C137C2×#4111yescharacteristic
150Dic137—22no—
151Dic137—21no—
152Dic137—41no—
153Dic137#12112yes—
154Dic137AGL(1,ℤ/8)12yescomplemented
155C2×C137—42no—
156C2×C137—82no—
157C2×C137C2×#12111yescharacteristic
158C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
159C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
160D137—22no—
161D137—82no—
162C137#303811yescharacteristic, complemented, Sylow
163#3038—1371nomaximal, Sylow
164C2×AGL(1,ℤ/8)—1372no—
165C2×#127—1371no—
166C4×#41—1371no—
167#460—1371no—
168#460—1371no—
169#460—1372no—
170#459—1372no—
171#459—1371no—
172#459D13711yescharacteristic, complemented
173#423—1371no—
174#418—1372no—
175C2×#41—1371no—
176C22×D4—2741no—
177C22×D4—2741no—
178C22×D4—1372no—
179AGL(1,ℤ/8)—2748no—
180C2×SD16—1372no—
181C2×SD16—1372no—
182C2×D8—1372no—
183C2×D8C2×D13712yescomplemented
184C2×M16—1372no—
185#127—2741no—
186#127—2741no—
187#127—1372no—
188#127—1371no—
189#127C2×D13711yescharacteristic
190C4×Q8—1371no—
191C4×Q8—1371no—
192C4×D4—2741no—
193C4×D4—2741no—
194C4×D4—1371no—
195C4×D4C2×D13711yescharacteristic, complemented
196#117—2741no—
197#117—1371no—
198C2×C42—2741no—
199C2×C42—1371no—
200C4⋊C8—1372no—
201C4⋊C8—1371no—
202C4⋊C8C2×D13711yescharacteristic, complemented
203#102—1372no—
204#102—1372no—
205#102—1372no—
206#102C2×D13712yescomplemented
207C24—2742no—
208#41—2742no—
209#41—1374no—
210C2×Q8—1371no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—2741no—
214C2×D4—2744no—
215C2×D4—2744no—
216C2×D4—2742no—
217C2×D4—2742no—
218C2×D4—2742no—
219C2×D4—2741no—
220C2×D4—2741no—
221C2×D4—5484no—
222C2×D4—5484no—
223C2×D4—1372no—
224C2×D4C22×D13712yes—
225C2×D4C22×D13711yescharacteristic
226C22×C4—21no—
227C22×C4—2741no—
228C22×C4—2741no—
229C22×C4—1371no—
230C22×C4—1371no—
231C22×C4—1371no—
232SD16—2744no—
233SD16—2744no—
234D8—24no—
235D8—2744no—
236M16—2744no—
237C2×C8—1372no—
238C2×C8C22×D13712yes—
239C4⋊C4—2741no—
240C4⋊C4—2741no—
241C4⋊C4—1371no—
242C4⋊C4C22×D13711yescharacteristic
243C22⋊C4—21no—
244C22⋊C4—2741no—
245C22⋊C4—2741no—
246C42—2741no—
247C42—2741no—
248C42—2741no—
249C42—1371no—
250C42—1371no—
251C42—1371no—
252C42C22×D13711yescharacteristic
253C23—21no—
254C23—22no—
255C23—2742no—
256C23—5484no—
257C23—5484no—
258C23—5484no—
259C23—1371no—
260Q8—2741no—
261Q8—1372no—
262D4—22no—
263D4—21no—
264D4—22no—
265D4—42no—
266D4—42no—
267D4—2744no—
268D4—2742no—
269D4—2742no—
270D4—5482no—
271D4—5484no—
272D4—5484no—
273D4—5484no—
274D4—5482no—
275D4D4×D13712yescomplemented
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—2742no—
279C2×C4—2741no—
280C2×C4—2744no—
281C2×C4—2742no—
282C2×C4—2741no—
283C2×C4—5482no—
284C2×C4—1371no—
285C2×C4—1371no—
286C2×C4—1371no—
287C2×C4—1372no—
288C2×C4—1372no—
289C2×C4D4×D13711yescharacteristic
290C2×C4D4×D13711yescharacteristic
291C2×C4C23×D13711yescharacteristic
292C8—22no—
293C8—2742no—
294C22—22no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—2741no—
301C22—2741no—
302C22—5482no—
303C22—5482no—
304C22—5482no—
305C22—10968no—
306C22—1372no—
307C22C2×D4×D13711yescharacteristic
308C4—21no—
309C4—41no—
310C4—2741no—
311C4—2742no—
312C4—5481no—
313C4—1372no—
314C4—1372no—
315C4#1798411212yescomplemented
316C4C2×D4×D13711yescharacteristic
317C4#41×D13711yescharacteristic
318C2—42no—
319C2—82no—
320C2—2742no—
321C2—10962no—
322C2#121×D13711yescharacteristic
323C2AGL(1,ℤ/8)×D13711yescharacteristic
324C2C2×#1798411211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#459×C137C2
socleC22×C137C22×D4
hypercenter#459D137
hypocenterC137#3038
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#17984112C2semidirect
2. #127×D137C2semidirect
3. C4×#14555509C2semidirect
4. #22577604C2semidirect
5. #22577602C2semidirect
6. #22577600C2semidirect
7. #22577598C2semidirect
8. #22577596C2semidirect
9. #22577425C2semidirect
10. #22577406C2semidirect
11. #459×C137C2semidirect
17. C2×#14555499C22semidirect
18. C2×#14555498C22semidirect
19. C2×#14555495C22semidirect
25. C4×Dic274C22semidirect
26. #17984053C22semidirect
27. D4×Dic137C22semidirect
34. C4⋊#11764040C22semidirect
35. #17984017C22semidirect
36. #17984010C22semidirect
37. #17984009C22semidirect
38. #17984008C22semidirect
39. C2×D8×C137C22semidirect
41. C4×D4×C137C22semidirect
42. C4⋊C8×C137C22semidirect
43. #102×C137C22semidirect
46. #14555509D4semidirect
96. Dic274C2×D4semidirect
124. D4×C137C2×D4semidirect
147. C4×C137AGL(1,ℤ/8)semidirect
154. Dic137AGL(1,ℤ/8)semidirect
162. C137#3038semidirect
172. #459D137semidirect
183. C2×D8C2×D137semidirect
195. C4×D4C2×D137semidirect
202. C4⋊C8C2×D137semidirect
206. #102C2×D137semidirect
275. D4D4×D137semidirect
315. C4#17984112semidirect

Maximal subgroups

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

subgroupindex
C2×#179841122
#127×D1372
C4×#145555092
#225776042
#225776022
#225776002
#225775982
#225775962
#225774252
#225774062
#459×C1372
#3038137