Finite Group Databasebeta

← #31815655

Subgroups — #31815655

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×#14555786C211yesmaximal, characteristic, complemented
2C2×#17984869C212yesmaximal, complemented
3#121×D139C211yesmaximal, characteristic, complemented
4#22581101C212yesmaximal, complemented
5#22581097C212yesmaximal, complemented
6#22581092C211yesmaximal, characteristic, complemented
7#22581089C211yesmaximal, characteristic, complemented
8#22581086C211yesmaximal, characteristic, complemented
9#22580953C212yesmaximal, complemented
10#22580915C211yesmaximal, characteristic, complemented
11#448×C139C211yesmaximal, characteristic, complemented
12C2×#14555786—28no—
13C2×#14555786—21no—
14C2×#14555786—21no—
15C2×#14555786C2212yes—
16C2×#14555786C2212yescomplemented
17C22×D556—21no—
18C22×D556C2211yescharacteristic, complemented
19C2×D4×D139—21no—
20C2×D4×D139C2211yescharacteristic, complemented
21C22×C4×D139—21no—
22C22×C4×D139C2211yescharacteristic
23#17984869—28no—
24C2×#14555772C2212yescomplemented
25C2×D1112C2212yescomplemented
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#17984766C2212yescomplemented
36#17984764C2212yescomplemented
37#17984762C2212yescomplemented
38C22⋊(C139⋊C8)C2211yescharacteristic, complemented
39(C2×D139)⋊C8C2212yescomplemented
40C22×Q8×C139C2211yescharacteristic, complemented
41C2×SD16×C139C2212yescomplemented
42#121×C139C2211yescharacteristic
43#103×C139C2212yescomplemented
44C22⋊C8×C139C2211yescharacteristic, complemented
45C23×D139—21no—
46C23×D139—21no—
47C23×D139D411yescharacteristic
48#14555786—28no—
49#14555786—28no—
50#14555786—24no—
51#14555786—24no—
52#14555786—24no—
53#14555786D418yescomplemented
54C2×D556—28no—
55C2×D556—22no—
56C2×D556—22no—
57C2×D556—24no—
58C2×D556—21no—
59C2×D556—21no—
60C2×D556C2311yescharacteristic
61C2×D556C2312yes—
62C2×C22⋊D139—21no—
63C2×C22⋊D139—21no—
64C2×C22⋊D139—21no—
65D4×D139—24no—
66D4×D139—44no—
67C22×Dic139D411yescharacteristic
68C2×C4×D139—28no—
69C2×C4×D139—22no—
70C2×C4×D139—22no—
71C2×C4×D139—24no—
72C2×C4×D139—21no—
73C2×C4×D139D411yescharacteristic
74C2×C4×D139D411yescharacteristic
75C2×C4×D139C2311yescharacteristic
76C2×C4×D139C2311yescharacteristic
77#14555772—24no—
78D1112—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—
87C2×Q8×C139—24no—
88C2×Q8×C139—21no—
89C2×Q8×C139C2312yes—
90C2×D4×C139—21no—
91C2×D4×C139C2311yescharacteristic
92C22×C4×C139—21no—
93C22×C4×C139C2311yescharacteristic
94SD16×C139—24no—
95C2×C8×C139C2312yes—
96C4⋊C4×C139C2311yescharacteristic
97C22⋊C4×C139—21no—
98C22×D139—24no—
99C22×D139—22no—
100C22×D139—22no—
101C22×D139—22no—
102C22×D139—21no—
103C22×D139—41no—
104C22×D139—42no—
105C22×D139—42no—
106C22×D139—42no—
107C22×D139—42no—
108C22×D139—42no—
109C22×D139C2×D411yescharacteristic
110C22×D139C2×D411yescharacteristic
111D556—28no—
112D556—24no—
113D556—28no—
114D556—22no—
115D556—22no—
116D556—42no—
117D556C2×D414yescomplemented
118C22⋊D139—42no—
119C22⋊D139—42no—
120C22⋊D139—42no—
121C2×Dic139—22no—
122C2×Dic139—21no—
123C2×Dic139C2×D411yescharacteristic
124C2×Dic139C2×D411yescharacteristic
125C4×D139—28no—
126C4×D139—22no—
127C4×D139—22no—
128C4×D139—28no—
129C4×D139—22no—
130C4×D139—42no—
131C4×D139C2×D414yes—
132C139⋊C8—22no—
133C23×C139—21no—
134C23×C139C2×D411yescharacteristic
135Q8×C139—24no—
136Q8×C139—22no—
137Q8×C139C2×D414yescomplemented
138D4×C139—22no—
139D4×C139—42no—
140C2×C4×C139—24no—
141C2×C4×C139—22no—
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—41no—
150C22×C139—41no—
151C22×C139—41no—
152C22×C139AGL(1,ℤ/8)12yescomplemented
153C22×C139C22×D411yescharacteristic
154C2×D139—21no—
155C2×D139—21no—
156C2×D139—21no—
157C2×D139—24no—
158C2×D139—21no—
159C2×D139—21no—
160C2×D139—44no—
161C2×D139—48no—
162C2×D139—42no—
163C2×D139—42no—
164C2×D139—41no—
165C2×D139—41no—
166C2×D139—41no—
167C2×D139—84no—
168C2×D139#12012yes—
169C4×C139—24no—
170C4×C139—21no—
171C4×C139—41no—
172C4×C139C22×D412yes—
173Dic139—21no—
174Dic139—41no—
175Dic139#12012yes—
176C2×C139—22no—
177C2×C139—81no—
178C2×C139C2×#12011yescharacteristic
179C2×C139C2×AGL(1,ℤ/8)11yescharacteristic
180C2×C139#57511yescharacteristic
181D139—22no—
182D139—41no—
183D139—44no—
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#448D13911yescharacteristic, complemented
194#447—1391no—
195#447—1391no—
196#417—1392no—
197#407—1391no—
198C2×#41—2788no—
199C2×#41—2781no—
200C2×#41—2781no—
201C2×#41—1392no—
202C2×#41—1392no—
203C22×Q8C2×D13911yescharacteristic, complemented
204C22×D4—2781no—
205C22×D4—2781no—
206C22×D4—1391no—
207C22×D4—1391no—
208C23×C4—2781no—
209C23×C4—1391no—
210AGL(1,ℤ/8)—2788no—
211C2×SD16—1392no—
212C2×SD16C2×D13912yescomplemented
213C2×D8—1392no—
214C2×D8—1392no—
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#103C2×D13912yescomplemented
226#102—1392no—
227#102—1392no—
228C22⋊C8—1391no—
229C22⋊C8—1392no—
230C22⋊C8C2×D13911yescharacteristic, complemented
231C24—2781no—
232C24—2781no—
233C24—1391no—
234#41—2788no—
235#41—2788no—
236#41—2784no—
237#41—2784no—
238#41—2784no—
239#41—1398no—
240C2×Q8—24no—
241C2×Q8—21no—
242C2×Q8C22×D13912yes—
243C2×D4—21no—
244C2×D4—2788no—
245C2×D4—2782no—
246C2×D4—2782no—
247C2×D4—2784no—
248C2×D4—2781no—
249C2×D4—2781no—
250C2×D4—2781no—
251C2×D4—2784no—
252C2×D4—2781no—
253C2×D4—2781no—
254C2×D4—5564no—
255C2×D4—1391no—
256C2×D4—1392no—
257C2×D4C22×D13911yescharacteristic
258C22×C4—21no—
259C22×C4—2788no—
260C22×C4—2782no—
261C22×C4—2782no—
262C22×C4—2784no—
263C22×C4—2781no—
264C22×C4—1391no—
265C22×C4—1391no—
266C22×C4—1391no—
267C22×C4—1391no—
268C22×C4—1391no—
269C22×C4C22×D13911yescharacteristic
270SD16—24no—
271SD16—2784no—
272D8—2784no—
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—21no—
285C23—2784no—
286C23—2782no—
287C23—2782no—
288C23—2782no—
289C23—2781no—
290C23—5561no—
291C23—5562no—
292C23—5562no—
293C23—5562no—
294C23—5562no—
295C23—5562no—
296C23—1391no—
297C23—1391no—
298C23D4×D13911yescharacteristic
299Q8—24no—
300Q8—22no—
301Q8D4×D13914yescomplemented
302D4—22no—
303D4—42no—
304D4—2788no—
305D4—2784no—
306D4—2788no—
307D4—2782no—
308D4—2782no—
309D4—5562no—
310D4—5562no—
311D4—5562no—
312D4—5562no—
313D4—1394no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—2788no—
319C2×C4—2782no—
320C2×C4—2788no—
321C2×C4—2782no—
322C2×C4—2782no—
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—41no—
336C22—41no—
337C22—41no—
338C22—2781no—
339C22—2781no—
340C22—2781no—
341C22—2784no—
342C22—2781no—
343C22—2781no—
344C22—5564no—
345C22—5568no—
346C22—5562no—
347C22—5562no—
348C22—5561no—
349C22—5561no—
350C22—5561no—
351C22—11124no—
352C22—1392no—
353C22#1798486912yescomplemented
354C22C2×D4×D13911yescharacteristic
355C4—24no—
356C4—21no—
357C4—41no—
358C4—2781no—
359C4—5561no—
360C4—1392no—
361C4C2×D4×D13912yes—
362C2—22no—
363C2—81no—
364C2—2782no—
365C2—5561no—
366C2—5564no—
367C2—11121no—
368C2#120×D13911yescharacteristic
369C2C2×#1798486911yescharacteristic
370C2#2258175211yescharacteristic
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#448×C139C2
socleC22×C139C22×D4
hypercenter#448D139
hypocenterC139#3006
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×#14555786C2semidirect
2. C2×#17984869C2semidirect
3. #121×D139C2semidirect
4. #22581101C2semidirect
5. #22581097C2semidirect
6. #22581092C2semidirect
7. #22581089C2semidirect
8. #22581086C2semidirect
9. #22580953C2semidirect
10. #22580915C2semidirect
11. #448×C139C2semidirect
16. C2×#14555786C22semidirect
18. C22×D556C22semidirect
20. C2×D4×D139C22semidirect
24. C2×#14555772C22semidirect
25. C2×D1112C22semidirect
26. C2×#14555770C22semidirect
27. C2×#14555768C22semidirect
28. #17984817C22semidirect
32. #17984813C22semidirect
33. C4⋊C4×D139C22semidirect
35. #17984766C22semidirect
36. #17984764C22semidirect
37. #17984762C22semidirect
38. C22⋊(C139⋊C8)C22semidirect
39. (C2×D139)⋊C8C22semidirect
40. C22×Q8×C139C22semidirect
41. C2×SD16×C139C22semidirect
43. #103×C139C22semidirect
44. C22⋊C8×C139C22semidirect
53. #14555786D4semidirect
117. D556C2×D4semidirect
137. Q8×C139C2×D4semidirect
152. C22×C139AGL(1,ℤ/8)semidirect
185. C139#3006semidirect
193. #448D139semidirect
203. C22×Q8C2×D139semidirect
212. C2×SD16C2×D139semidirect
225. #103C2×D139semidirect
230. C22⋊C8C2×D139semidirect
301. Q8D4×D139semidirect
353. C22#17984869semidirect

Maximal subgroups

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

subgroupindex
C22×#145557862
C2×#179848692
#121×D1392
#225811012
#225810972
#225810922
#225810892
#225810862
#225809532
#225809152
#448×C1392
#3006139