Finite Group Databasebeta

← #31815710

Subgroups — #31815710

Order 17792.

764 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 149 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×#17984888C211yesmaximal, characteristic, complemented
2C2×#17984870C211yesmaximal, characteristic, complemented
3C2×#17984862C211yesmaximal, characteristic, complemented
4#22581534C211yesmaximal, characteristic, complemented
5#22581118C212yesmaximal, complemented
6#22581105C212yesmaximal, complemented
7#22581101C212yesmaximal, complemented
8#22581088C212yesmaximal, complemented
9C2×#17984762C211yesmaximal, characteristic, complemented
10#22580921C211yesmaximal, characteristic, complemented
11#417×C139C211yesmaximal, characteristic, complemented
12#17984888—24no—
13#17984888—24no—
14C2×#14555786C2212yes—
15C2×#14555785—21no—
16C2×#14555785C2211yescharacteristic
17C2×#14555785C2211yescharacteristic, complemented
18C22×D556—21no—
19C22×D556C2211yescharacteristic
20C2×D4×D139—22no—
21C2×D4×D139C2212yes—
22#17984870—24no—
23#17984862—24no—
24C2×#14555775C2211yescharacteristic, complemented
25C2×#14555774C2212yescomplemented
26C2×#14555773C2211yescharacteristic, complemented
27C2×#14555772C2211yescharacteristic, complemented
28C2×D1112C2212yescomplemented
29C2×#14555770C2211yescharacteristic, complemented
30C22×C139⋊C8C2211yescharacteristic, complemented
31#17984822—21no—
32#17984822C2212yescomplemented
33#17984817—21no—
34#17984817C2212yescomplemented
35#17984816—22no—
36#17984762C2212yescomplemented
37#17984762C2212yescomplemented
38(C2×D139)⋊C8C2212yescomplemented
39(C2×D139)⋊C8C2212yescomplemented
40C2×#41×C139C2211yescharacteristic
41C2×M16×C139C2211yescharacteristic, complemented
42C2×C4⋊C4×C139C2211yescharacteristic, complemented
43#103×C139C2212yescomplemented
44#102×C139C2212yescomplemented
45C23×D139—22no—
46C23×D139—21no—
47#14555786—24no—
48#14555786—24no—
49#14555785—28no—
50#14555785—22no—
51#14555785—22no—
52#14555785—22no—
53#14555785—22no—
54C2×Dic278D411yescharacteristic
55C2×Dic278C2311yescharacteristic
56C2×D556—28no—
57C2×D556—22no—
58C2×D556—22no—
59C2×D556—21no—
60C2×D556—22no—
61C2×D556D411yescharacteristic
62C2×D556D414yescomplemented
63C2×D556C2311yescharacteristic
64C2×D556C2312yes—
65C2×C22⋊D139—22no—
66C2×C22⋊D139—21no—
67C2×C22⋊D139D412yes—
68D4×D139—216no—
69D4×D139—24no—
70D4×D139—24no—
71C2×C4×D139—22no—
72C2×C4×D139—21no—
73C2×C4×D139C2312yes—
74#14555775—22no—
75#14555774—24no—
76#14555773—22no—
77#14555772—22no—
78D1112—24no—
79#14555770—22no—
80C2×C139⋊C8—22no—
81C2×C139⋊C8C2311yescharacteristic
82C2×C139⋊C8C2311yescharacteristic
83C4⋊Dic139—22no—
84(C2×D139)⋊C4—22no—
85(C2×D139)⋊C4—22no—
86#41×C139—22no—
87#41×C139D414yescomplemented
88C2×Q8×C139C2311yescharacteristic
89C2×D4×C139—21no—
90C2×D4×C139C2311yescharacteristic
91C22×C4×C139—21no—
92C22×C4×C139—21no—
93C22×C4×C139C2311yescharacteristic
94M16×C139—22no—
95C2×C8×C139C2312yes—
96C4⋊C4×C139—21no—
97C4⋊C4×C139C2312yes—
98C22×D139—22no—
99C22×D139—28no—
100C22×D139—24no—
101C22×D139—21no—
102C22×D139—82no—
103C22×D139—82no—
104C22×D139—81no—
105C22×D139—81no—
106C22×D139C2×D412yes—
107Dic278—22no—
108Dic278—21no—
109Dic278—21no—
110D556—28no—
111D556—22no—
112D556—24no—
113D556—22no—
114D556—21no—
115D556—21no—
116D556—82no—
117D556C2×D414yes—
118C22⋊D139—28no—
119C22⋊D139—24no—
120C22⋊D139—42no—
121C2×Dic139—21no—
122C2×Dic139C2×D412yes—
123C4×D139—28no—
124C4×D139—22no—
125C4×D139—22no—
126C4×D139—42no—
127C139⋊C8—21no—
128C139⋊C8—21no—
129C23×C139—21no—
130C23×C139C2×D411yescharacteristic
131Q8×C139—21no—
132Q8×C139C2×D412yes—
133D4×C139—24no—
134D4×C139—21no—
135D4×C139C2×D412yes—
136C2×C4×C139—21no—
137C2×C4×C139—24no—
138C2×C4×C139—22no—
139C2×C4×C139—81no—
140C2×C4×C139C2×D411yescharacteristic
141C2×C4×C139C2×D412yes—
142C2×C4×C139C2×D412yes—
143C2×C4×C139C2411yescharacteristic
144C8×C139—22no—
145C22×C139—21no—
146C22×C139—21no—
147C22×C139—22no—
148C22×C139—41no—
149C22×C139—41no—
150C22×C139#12012yes—
151C22×C139C22×D411yescharacteristic
152C2×D139—22no—
153C2×D139—24no—
154C2×D139—22no—
155C2×D139—22no—
156C2×D139—42no—
157C2×D139—42no—
158C2×D139—41no—
159C2×D139—41no—
160C2×D139—41no—
161C2×D139—84no—
162C2×D139—88no—
163C4×C139—22no—
164C4×C139—42no—
165C4×C139#12012yes—
166C4×C139C22×D411yescharacteristic
167C4×C139C22×D411yescharacteristic
168Dic139—22no—
169Dic139—41no—
170C2×C139—22no—
171C2×C139—42no—
172C2×C139C2×#12011yescharacteristic
173C2×C139#57611yescharacteristic
174C2×C139#57711yescharacteristic
175D139—42no—
176D139—44no—
177D139—81no—
178C139#301211yescharacteristic, complemented, Sylow
179#3012—1391nomaximal, Sylow
180C2×H(5,𝔽2)—1391no—
181C2×AGL(1,ℤ/8)—1391no—
182C2×#135—1391no—
183#537—1391no—
184#452—1392no—
185#450—1392no—
186#449—1392no—
187#447—1392no—
188#417D13911yescharacteristic, complemented
189C2×#102—1391no—
190#408—1391no—
191H(5,𝔽2)—2784no—
192H(5,𝔽2)—2784no—
193C2×#41—2781no—
194C2×#41—1392no—
195C2×#41—1391no—
196C2×#41—1391no—
197C2×#41C2×D13911yescharacteristic
198C22×D4—2781no—
199C22×D4—2782no—
200C22×D4—1392no—
201C22×D4—1391no—
202AGL(1,ℤ/8)—2784no—
203#135—2784no—
204C2×Q16—1391no—
205C2×SD16—1391no—
206C2×SD16—1391no—
207C2×SD16—1392no—
208C2×D8—1391no—
209C2×D8—1392no—
210C2×M16C2×D13911yescharacteristic, complemented
211C22×C8—1391no—
212C22⋊Q8—2781no—
213C22⋊Q8—1392no—
214#121—2781no—
215#121—2782no—
216#121—1392no—
217C2×C4⋊C4C2×D13911yescharacteristic, complemented
218#103C2×D13912yescomplemented
219#102—1392no—
220#102—1392no—
221#102C2×D13912yescomplemented
222C22⋊C8—1392no—
223C22⋊C8—1392no—
224C24—2782no—
225C24—2781no—
226#41—22no—
227#41—2788no—
228#41—2784no—
229#41—2784no—
230#41—2782no—
231#41—2782no—
232#41—2782no—
233#41—2782no—
234#41D55614yescomplemented
235C2×Q8—1391no—
236C2×Q8—1391no—
237C2×Q8C22×D13911yescharacteristic
238C2×D4—21no—
239C2×D4—2788no—
240C2×D4—2782no—
241C2×D4—2782no—
242C2×D4—27816no—
243C2×D4—2784no—
244C2×D4—2784no—
245C2×D4—2781no—
246C2×D4—2782no—
247C2×D4—2781no—
248C2×D4—2782no—
249C2×D4—1392no—
250C2×D4—1391no—
251C2×D4—1394no—
252C2×D4—1391no—
253C2×D4—1392no—
254C2×D4C22×D13911yescharacteristic
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—2782no—
258C22×C4—2781no—
259C22×C4—1392no—
260C22×C4C22×D13911yescharacteristic
261Q16—2782no—
262SD16—2784no—
263SD16—2782no—
264SD16—2782no—
265D8—2784no—
266D8—2782no—
267M16—22no—
268C2×C8—2782no—
269C2×C8—1391no—
270C2×C8—1391no—
271C2×C8C22×D13912yes—
272C4⋊C4—21no—
273C4⋊C4—2782no—
274C4⋊C4C22×D13912yes—
275C22⋊C4—2782no—
276C22⋊C4—2782no—
277C23—21no—
278C23—2788no—
279C23—2782no—
280C23—2784no—
281C23—2781no—
282C23—11121no—
283C23—11121no—
284C23—11122no—
285C23—11122no—
286C23—1392no—
287C23D4×D13911yescharacteristic
288Q8—21no—
289Q8—2782no—
290Q8—2781no—
291Q8—2781no—
292Q8C2×D55612yes—
293D4—24no—
294D4—21no—
295D4—2788no—
296D4—2788no—
297D4—2784no—
298D4—2782no—
299D4—2784no—
300D4—2781no—
301D4—2782no—
302D4—2781no—
303D4—5562no—
304D4—11122no—
305D4—1394no—
306D4C2×D55612yes—
307C2×C4—21no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—81no—
311C2×C4—2788no—
312C2×C4—2782no—
313C2×C4—2782no—
314C2×C4—2781no—
315C2×C4—5562no—
316C2×C4—1392no—
317C2×C4D4×D13911yescharacteristic
318C2×C4D4×D13912yes—
319C2×C4C2×D55612yes—
320C2×C4C23×D13911yescharacteristic
321C8—22no—
322C8—2781no—
323C8—2781no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—41no—
328C22—41no—
329C22—2782no—
330C22—2782no—
331C22—2784no—
332C22—2782no—
333C22—5562no—
334C22—5562no—
335C22—5561no—
336C22—5561no—
337C22—5561no—
338C22—11124no—
339C22—11128no—
340C22#1798481212yes—
341C22C2×D4×D13911yescharacteristic
342C4—22no—
343C4—42no—
344C4—2782no—
345C4—5561no—
346C4#1798481212yes—
347C4C2×D4×D13911yescharacteristic
348C4C22×D55611yescharacteristic
349C2—22no—
350C2—42no—
351C2—5562no—
352C2—5564no—
353C2—11121no—
354C2C2×#1798481211yescharacteristic
355C2#2258175711yescharacteristic
356C2#2258176211yescharacteristic
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#417×C139C2
socleC22×C139C22×D4
hypercenter#417D139
hypocenterC139#3012
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#17984888C2semidirect
2. C2×#17984870C2semidirect
3. C2×#17984862C2semidirect
4. #22581534C2semidirect
5. #22581118C2semidirect
6. #22581105C2semidirect
7. #22581101C2semidirect
8. #22581088C2semidirect
9. C2×#17984762C2semidirect
10. #22580921C2semidirect
11. #417×C139C2semidirect
17. C2×#14555785C22semidirect
24. C2×#14555775C22semidirect
25. C2×#14555774C22semidirect
26. C2×#14555773C22semidirect
27. C2×#14555772C22semidirect
28. C2×D1112C22semidirect
29. C2×#14555770C22semidirect
30. C22×C139⋊C8C22semidirect
32. #17984822C22semidirect
34. #17984817C22semidirect
36. #17984762C22semidirect
37. #17984762C22semidirect
38. (C2×D139)⋊C8C22semidirect
39. (C2×D139)⋊C8C22semidirect
41. C2×M16×C139C22semidirect
42. C2×C4⋊C4×C139C22semidirect
43. #103×C139C22semidirect
44. #102×C139C22semidirect
62. C2×D556D4semidirect
87. #41×C139D4semidirect
178. C139#3012semidirect
188. #417D139semidirect
210. C2×M16C2×D139semidirect
217. C2×C4⋊C4C2×D139semidirect
218. #103C2×D139semidirect
221. #102C2×D139semidirect
234. #41D556semidirect

Maximal subgroups

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

subgroupindex
C2×#179848882
C2×#179848702
C2×#179848622
#225815342
#225811182
#225811052
#225811012
#225810882
C2×#179847622
#225809212
#417×C1392
#3012139