Finite Group Databasebeta

← #31815651

Subgroups — #31815651

Order 17792.

806 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14555784C211yesmaximal, characteristic, complemented
2C2×#17984866C212yesmaximal, complemented
3#121×D139C211yesmaximal, characteristic, complemented
4#22581099C212yesmaximal, complemented
5#22581095C212yesmaximal, complemented
6#22581094C211yesmaximal, characteristic, complemented
7#22581091C211yesmaximal, characteristic, complemented
8#22581087C211yesmaximal, characteristic, complemented
9#22580951C212yesmaximal, complemented
10#22580915C211yesmaximal, characteristic, complemented
11#447×C139C211yesmaximal, characteristic, complemented
12C2×#14555784—28no—
13C2×#14555784—21no—
14C2×#14555784—21no—
15C2×#14555784C2212yes—
16C2×#14555784C2212yescomplemented
17C22×Dic278C2211yescharacteristic, complemented
18C22×C22⋊D139—21no—
19C2×D4×D139—21no—
20C2×D4×D139C2211yescharacteristic, complemented
21C23×Dic139—21no—
22C22×C4×D139C2211yescharacteristic
23#17984866—28no—
24C2×#14555774C2212yescomplemented
25C2×#14555773C2212yescomplemented
26C2×#14555770C2212yescomplemented
27C2×#14555768C2212yescomplemented
28#17984817C2211yescharacteristic, complemented
29#17984816C2211yescharacteristic
30#17984815—21no—
31#17984814—21no—
32#17984813C2211yescharacteristic, complemented
33C4⋊C4×D139C2211yescharacteristic, complemented
34C22⋊C4×D139—21no—
35#17984767C2212yescomplemented
36#17984765C2212yescomplemented
37#17984763C2212yescomplemented
38C22⋊(C139⋊C8)C2211yescharacteristic, complemented
39(C2×D139)⋊C8C2212yescomplemented
40C22×D4×C139C2211yescharacteristic, complemented
41C2×D8×C139C2212yescomplemented
42#121×C139C2211yescharacteristic
43#102×C139C2212yescomplemented
44C22⋊C8×C139C2211yescharacteristic, complemented
45C23×D139—21no—
46C23×D139D411yescharacteristic
47#14555784—28no—
48#14555784—28no—
49#14555784—24no—
50#14555784—24no—
51#14555784—24no—
52#14555784D418yescomplemented
53C2×Dic278—24no—
54C2×Dic278—21no—
55C2×Dic278C2312yes—
56C2×D556—21no—
57C2×D556C2311yescharacteristic
58C2×C22⋊D139—28no—
59C2×C22⋊D139—22no—
60C2×C22⋊D139—22no—
61C2×C22⋊D139—21no—
62C2×C22⋊D139—21no—
63C2×C22⋊D139—21no—
64D4×D139—24no—
65D4×D139—44no—
66C22×Dic139—28no—
67C22×Dic139—22no—
68C22×Dic139—22no—
69C22×Dic139—21no—
70C22×Dic139D411yescharacteristic
71C2×C4×D139—24no—
72C2×C4×D139—21no—
73C2×C4×D139D411yescharacteristic
74C2×C4×D139D411yescharacteristic
75C2×C4×D139C2311yescharacteristic
76C2×C4×D139C2311yescharacteristic
77#14555774—24no—
78#14555773—24no—
79#14555770—24no—
80#14555768—24no—
81C2×C139⋊C8C2312yes—
82C4⋊Dic139C2311yescharacteristic
83Dic139⋊C4—21no—
84C22⋊Dic139—21no—
85(C2×D139)⋊C4—21no—
86(C2×D139)⋊C4—21no—
87C24×C139—21no—
88C2×D4×C139—24no—
89C2×D4×C139—21no—
90C2×D4×C139—21no—
91C2×D4×C139C2311yescharacteristic
92C2×D4×C139C2312yes—
93C22×C4×C139C2311yescharacteristic
94D8×C139—24no—
95C2×C8×C139C2312yes—
96C4⋊C4×C139C2311yescharacteristic
97C22⋊C4×C139—21no—
98C22×D139—22no—
99C22×D139—22no—
100C22×D139—21no—
101C22×D139—41no—
102C22×D139—42no—
103C22×D139—42no—
104C22×D139—42no—
105C22×D139C2×D411yescharacteristic
106C22×D139C2×D411yescharacteristic
107Dic278—24no—
108Dic278—22no—
109Dic278C2×D414yescomplemented
110D556—22no—
111D556—42no—
112C22⋊D139—28no—
113C22⋊D139—28no—
114C22⋊D139—42no—
115C22⋊D139—42no—
116C22⋊D139—42no—
117C2×Dic139—24no—
118C2×Dic139—28no—
119C2×Dic139—22no—
120C2×Dic139—22no—
121C2×Dic139—28no—
122C2×Dic139—21no—
123C2×Dic139C2×D411yescharacteristic
124C2×Dic139C2×D411yescharacteristic
125C4×D139—22no—
126C4×D139—22no—
127C4×D139—22no—
128C4×D139—42no—
129C4×D139C2×D414yes—
130C139⋊C8—22no—
131C23×C139—24no—
132C23×C139—22no—
133C23×C139—21no—
134C23×C139—42no—
135C23×C139—42no—
136C23×C139C2×D411yescharacteristic
137D4×C139—24no—
138D4×C139—22no—
139D4×C139—22no—
140D4×C139—42no—
141D4×C139C2×D414yescomplemented
142C2×C4×C139—22no—
143C2×C4×C139—21no—
144C2×C4×C139C2×D411yescharacteristic
145C2×C4×C139C2411yescharacteristic
146C8×C139—22no—
147C22×C139—21no—
148C22×C139—21no—
149C22×C139—24no—
150C22×C139—48no—
151C22×C139—42no—
152C22×C139—42no—
153C22×C139—41no—
154C22×C139—41no—
155C22×C139—41no—
156C22×C139AGL(1,ℤ/8)12yescomplemented
157C22×C139C22×D411yescharacteristic
158C2×D139—21no—
159C2×D139—21no—
160C2×D139—21no—
161C2×D139—21no—
162C2×D139—21no—
163C2×D139—44no—
164C2×D139—41no—
165C2×D139—41no—
166C2×D139—41no—
167C2×D139—84no—
168C2×D139#12012yes—
169C4×C139—21no—
170C4×C139—41no—
171C4×C139C22×D412yes—
172Dic139—21no—
173Dic139—24no—
174Dic139—41no—
175Dic139#12012yes—
176C2×C139—22no—
177C2×C139—44no—
178C2×C139—81no—
179C2×C139C2×#12011yescharacteristic
180C2×C139C2×AGL(1,ℤ/8)11yescharacteristic
181C2×C139#57511yescharacteristic
182D139—22no—
183D139—41no—
184D139—81no—
185C139#300611yescharacteristic, complemented, Sylow
186#3006—1391nomaximal, Sylow
187C22×#41—1391no—
188C2×AGL(1,ℤ/8)—1392no—
189C2×#121—1391no—
190#449—1392no—
191#449—1392no—
192#448—1391no—
193#448—1391no—
194#447—1391no—
195#447D13911yescharacteristic, complemented
196#417—1392no—
197#407—1391no—
198C2×#41—2788no—
199C2×#41—2781no—
200C2×#41—2781no—
201C2×#41—1392no—
202C2×#41—1392no—
203C22×Q8—1391no—
204C22×D4—2781no—
205C22×D4—2781no—
206C22×D4—1391no—
207C22×D4C2×D13911yescharacteristic, complemented
208C23×C4—2781no—
209C23×C4—1391no—
210AGL(1,ℤ/8)—2788no—
211C2×SD16—1392no—
212C2×SD16—1392no—
213C2×D8—1392no—
214C2×D8C2×D13912yescomplemented
215C2×M16—1392no—
216#121—2781no—
217#121—2781no—
218#121—1391no—
219#121—1391no—
220#121—1391no—
221#121C2×D13911yescharacteristic
222C2×C4⋊C4—1391no—
223C2×C22⋊C4—2781no—
224#103—1392no—
225#103—1392no—
226#102—1392no—
227#102C2×D13912yescomplemented
228C22⋊C8—1391no—
229C22⋊C8—1392no—
230C22⋊C8C2×D13911yescharacteristic, complemented
231C24—21no—
232C24—2781no—
233C24—1391no—
234#41—2788no—
235#41—2788no—
236#41—2784no—
237#41—2784no—
238#41—2784no—
239#41—1398no—
240C2×Q8—2784no—
241C2×Q8—2781no—
242C2×Q8—1392no—
243C2×D4—24no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—2781no—
247C2×D4—2788no—
248C2×D4—2782no—
249C2×D4—2782no—
250C2×D4—2781no—
251C2×D4—2781no—
252C2×D4—2784no—
253C2×D4—2781no—
254C2×D4—5564no—
255C2×D4—1391no—
256C2×D4C22×D13911yescharacteristic
257C2×D4C22×D13912yes—
258C22×C4—2784no—
259C22×C4—2781no—
260C22×C4—2788no—
261C22×C4—2781no—
262C22×C4—2782no—
263C22×C4—2782no—
264C22×C4—1391no—
265C22×C4—1391no—
266C22×C4—1391no—
267C22×C4—1391no—
268C22×C4—1391no—
269C22×C4C22×D13911yescharacteristic
270SD16—2784no—
271SD16—2784no—
272D8—24no—
273D8—2784no—
274M16—2784no—
275C2×C8—1392no—
276C2×C8C22×D13912yes—
277C4⋊C4—2781no—
278C4⋊C4—1391no—
279C4⋊C4C22×D13911yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—2781no—
282C22⋊C4—2781no—
283C22⋊C4—2781no—
284C23—24no—
285C23—22no—
286C23—21no—
287C23—42no—
288C23—42no—
289C23—2782no—
290C23—2782no—
291C23—2781no—
292C23—5561no—
293C23—5562no—
294C23—5562no—
295C23—5562no—
296C23—1391no—
297C23—1391no—
298C23D4×D13911yescharacteristic
299Q8—2784no—
300Q8—2782no—
301Q8—1394no—
302D4—24no—
303D4—22no—
304D4—22no—
305D4—42no—
306D4—2788no—
307D4—2788no—
308D4—2782no—
309D4—5562no—
310D4—5562no—
311D4—5562no—
312D4—5562no—
313D4D4×D13914yescomplemented
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—2782no—
317C2×C4—2782no—
318C2×C4—2784no—
319C2×C4—2788no—
320C2×C4—2782no—
321C2×C4—2782no—
322C2×C4—2788no—
323C2×C4—2782no—
324C2×C4—2781no—
325C2×C4—5562no—
326C2×C4—1391no—
327C2×C4—1391no—
328C2×C4—1394no—
329C2×C4D4×D13911yescharacteristic
330C2×C4C23×D13911yescharacteristic
331C8—22no—
332C8—2782no—
333C22—21no—
334C22—21no—
335C22—24no—
336C22—48no—
337C22—42no—
338C22—42no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—2781no—
343C22—2781no—
344C22—2781no—
345C22—2781no—
346C22—2781no—
347C22—5564no—
348C22—5561no—
349C22—5561no—
350C22—5561no—
351C22—11124no—
352C22—1392no—
353C22#1798486612yescomplemented
354C22C2×D4×D13911yescharacteristic
355C4—21no—
356C4—41no—
357C4—2781no—
358C4—2784no—
359C4—5561no—
360C4—1392no—
361C4C2×D4×D13912yes—
362C2—22no—
363C2—44no—
364C2—81no—
365C2—2782no—
366C2—5561no—
367C2—11121no—
368C2#120×D13911yescharacteristic
369C2C2×#1798486611yescharacteristic
370C2#2258174911yescharacteristic
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#447×C139C2
socleC22×C139C22×D4
hypercenter#447D139
hypocenterC139#3006
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C22×#14555784C2semidirect
2. C2×#17984866C2semidirect
3. #121×D139C2semidirect
4. #22581099C2semidirect
5. #22581095C2semidirect
6. #22581094C2semidirect
7. #22581091C2semidirect
8. #22581087C2semidirect
9. #22580951C2semidirect
10. #22580915C2semidirect
11. #447×C139C2semidirect
16. C2×#14555784C22semidirect
17. C22×Dic278C22semidirect
20. C2×D4×D139C22semidirect
24. C2×#14555774C22semidirect
25. C2×#14555773C22semidirect
26. C2×#14555770C22semidirect
27. C2×#14555768C22semidirect
28. #17984817C22semidirect
32. #17984813C22semidirect
33. C4⋊C4×D139C22semidirect
35. #17984767C22semidirect
36. #17984765C22semidirect
37. #17984763C22semidirect
38. C22⋊(C139⋊C8)C22semidirect
39. (C2×D139)⋊C8C22semidirect
40. C22×D4×C139C22semidirect
41. C2×D8×C139C22semidirect
43. #102×C139C22semidirect
44. C22⋊C8×C139C22semidirect
52. #14555784D4semidirect
109. Dic278C2×D4semidirect
141. D4×C139C2×D4semidirect
156. C22×C139AGL(1,ℤ/8)semidirect
185. C139#3006semidirect
195. #447D139semidirect
207. C22×D4C2×D139semidirect
214. C2×D8C2×D139semidirect
227. #102C2×D139semidirect
230. C22⋊C8C2×D139semidirect
313. D4D4×D139semidirect
353. C22#17984866semidirect

Maximal subgroups

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

subgroupindex
C22×#145557842
C2×#179848662
#121×D1392
#225810992
#225810952
#225810942
#225810912
#225810872
#225809512
#225809152
#447×C1392
#3006139