Finite Group Databasebeta

← #31816889

Subgroups — #31816889

Order 17792.

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

normal subgroups only

This group has 596 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×#17984870C211yesmaximal, characteristic, complemented
2C2×#17984862C211yesmaximal, characteristic, complemented
3C22×#14555770C211yesmaximal, characteristic, complemented
4#22581520C211yesmaximal, characteristic, complemented
5#22581517C211yesmaximal, characteristic, complemented
6#22581341C212yesmaximal, complemented
7#22581339C212yesmaximal, complemented
8#22581334C212yesmaximal, complemented
9#22581328C212yesmaximal, complemented
10#22580911C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C139C211yesmaximal, characteristic, complemented
12C2×#14555785C2211yescharacteristic
13C22×D556C2211yescharacteristic
14#17984870—24no—
15#17984862—24no—
16C2×#14555775C2211yescharacteristic, complemented
17C2×#14555774C2212yescomplemented
18C2×#14555773C2211yescharacteristic, complemented
19C2×#14555772C2211yescharacteristic, complemented
20C2×D1112C2212yescomplemented
21C2×#14555770—24no—
22C2×#14555770C2212yes—
23C2×#14555770C2212yes—
24C2×#14555770C2211yescharacteristic, complemented
25C22×C139⋊C8C2211yescharacteristic
26#17984838C2212yescomplemented
27#17984838C2212yescomplemented
28#17984831C2212yescomplemented
29#17984829C2212yescomplemented
30#17984816—22no—
31#17984813—22no—
32#17984811—22no—
33#17984810—22no—
34#17984801C2211yescharacteristic
35#17984752C2212yescomplemented
36C8×Dic139C2212yescomplemented
37C2×#41×C139C2211yescharacteristic
38C22×D4×C139C2211yescharacteristic
39AGL(1,ℤ/8)×C139—24no—
40C2×SD16×C139C2212yescomplemented
41C2×D8×C139C2212yescomplemented
42C2×M16×C139C2211yescharacteristic
43C23×D139—21no—
44#14555785—22no—
45#14555785—22no—
46C2×Dic278C2311yescharacteristic
47C2×D556—22no—
48C2×D556—22no—
49C2×D556—21no—
50C2×D556C2312yes—
51C2×D556C2311yescharacteristic
52C2×C22⋊D139—24no—
53C2×C22⋊D139—24no—
54C2×C22⋊D139—21no—
55C2×C4×D139—21no—
56#14555775—22no—
57#14555774—24no—
58#14555773—22no—
59#14555772—22no—
60D1112—24no—
61#14555770—24no—
62#14555770—24no—
63#14555770—22no—
64C2×C139⋊C8D414yescomplemented
65C2×C139⋊C8C2311yescharacteristic
66C2×C139⋊C8C2311yescharacteristic
67C4⋊Dic139D412yes—
68C22⋊Dic139—24no—
69C22⋊Dic139D412yes—
70(C2×D139)⋊C4—24no—
71C4×Dic139C2312yes—
72C24×C139—21no—
73#41×C139—22no—
74#41×C139—22no—
75C2×Q8×C139C2311yescharacteristic
76C2×D4×C139—21no—
77C2×D4×C139—22no—
78C2×D4×C139—22no—
79C2×D4×C139—21no—
80C2×D4×C139C2312yes—
81C2×D4×C139C2311yescharacteristic
82C22×C4×C139—21no—
83C22×C4×C139C2311yescharacteristic
84SD16×C139—24no—
85D8×C139—24no—
86M16×C139D414yescomplemented
87C2×C8×C139C2312yes—
88C22×D139—22no—
89C22×D139—21no—
90C22×D139—42no—
91C22×D139—82no—
92Dic278—21no—
93Dic278—21no—
94D556—22no—
95D556—21no—
96D556—21no—
97D556—22no—
98D556—42no—
99C22⋊D139—42no—
100C22⋊D139—84no—
101C22⋊D139—84no—
102C2×Dic139—21no—
103C2×Dic139C2×D414yes—
104C4×D139—42no—
105C139⋊C8C2×D412yes—
106C139⋊C8C2×D412yes—
107C23×C139—21no—
108C23×C139—22no—
109C23×C139—42no—
110C23×C139—82no—
111C23×C139C2×D411yescharacteristic
112Q8×C139—21no—
113Q8×C139—21no—
114D4×C139—21no—
115D4×C139—22no—
116D4×C139—22no—
117D4×C139—21no—
118D4×C139—42no—
119D4×C139—42no—
120C2×C4×C139—21no—
121C2×C4×C139—42no—
122C2×C4×C139C2×D411yescharacteristic
123C2×C4×C139C2×D412yes—
124C2×C4×C139C2×D412yes—
125C2×C4×C139C2411yescharacteristic
126C8×C139C2×D414yes—
127C22×C139—22no—
128C22×C139—41no—
129C22×C139—42no—
130C22×C139—44no—
131C22×C139—42no—
132C22×C139—84no—
133C22×C139#12712yes—
134C22×C139C22×D411yescharacteristic
135C2×D139—41no—
136C2×D139—42no—
137C2×D139—42no—
138C2×D139—44no—
139C2×D139—84no—
140C4×C139—41no—
141C4×C139#12712yes—
142C4×C139C22×D411yescharacteristic
143C4×C139C22×D411yescharacteristic
144Dic139—24no—
145Dic139—41no—
146C2×C139—22no—
147C2×C139—81no—
148C2×C139—82no—
149C2×C139C2×#12711yescharacteristic
150C2×C139#57612yes—
151D139—81no—
152D139—82no—
153C139#315511yescharacteristic, complemented, Sylow
154#3155—1391nomaximal, Sylow
155C2×AGL(1,ℤ/8)—1391no—
156C2×AGL(1,ℤ/8)D13911yescharacteristic, complemented
157C2×#135—1391no—
158C22×D8—1391no—
159#535—1391no—
160#535—1391no—
161#496—1392no—
162#496—1392no—
163#495—1392no—
164#493—1392no—
165#405—1391no—
166C2×#41—1391no—
167C2×#41C2×D13911yescharacteristic
168C22×D4—1391no—
169C22×D4C2×D13911yescharacteristic
170AGL(1,ℤ/8)—24no—
171AGL(1,ℤ/8)—2784no—
172#135—2784no—
173C2×Q16—1391no—
174C2×SD16—1391no—
175C2×SD16—1391no—
176C2×SD16—1392no—
177C2×SD16C2×D13912yescomplemented
178C2×D8—2784no—
179C2×D8—1392no—
180C2×D8—1392no—
181C2×D8—1391no—
182C2×D8—1392no—
183C2×D8C2×D13912yescomplemented
184C2×M16C2×D13911yescharacteristic
185C22×C8—1391no—
186#127—1392no—
187#127—1392no—
188#124—1392no—
189#124—1392no—
190#121—2782no—
191#121—2782no—
192#120—2782no—
193#120—2782no—
194#117—1391no—
195#97—1392no—
196C4×C8—1392no—
197C24—21no—
198C24—2781no—
199#41—22no—
200#41—22no—
201#41—2782no—
202#41—2782no—
203C2×Q8—1391no—
204C2×Q8C22×D13911yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—2781no—
210C2×D4—2784no—
211C2×D4—2784no—
212C2×D4—2782no—
213C2×D4—2781no—
214C2×D4—2782no—
215C2×D4—1392no—
216C2×D4—1391no—
217C2×D4C22×D13912yes—
218C2×D4C22×D13911yescharacteristic
219C22×C4—21no—
220C22×C4—2781no—
221C22×C4C22×D13911yescharacteristic
222Q16—2782no—
223SD16—24no—
224SD16—2782no—
225SD16—2782no—
226SD16—2784no—
227D8—24no—
228D8—2784no—
229D8—2784no—
230D8—2782no—
231D8—2784no—
232M16C22⋊D13914yescomplemented
233C2×C8—1394no—
234C2×C8—1391no—
235C2×C8—1391no—
236C2×C8C22×D13912yes—
237C4⋊C4—1392no—
238C22⋊C4—2784no—
239C22⋊C4—2784no—
240C22⋊C4—1392no—
241C42—1392no—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—82no—
246C23—2782no—
247C23—2781no—
248C23—5562no—
249C23—11122no—
250C23D4×D13911yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—2781no—
254Q8—2781no—
255D4—22no—
256D4—21no—
257D4—21no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—2782no—
262D4—2781no—
263D4—2781no—
264D4—2782no—
265D4—5562no—
266D4—5562no—
267D4—11124no—
268D4—11124no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—2781no—
272C2×C4—5562no—
273C2×C4—1394no—
274C2×C4D4×D13911yescharacteristic
275C2×C4D4×D13912yes—
276C2×C4C2×C22⋊D13912yes—
277C2×C4C23×D13911yescharacteristic
278C8—1392no—
279C8—1392no—
280C8C2×C22⋊D13914yes—
281C22—22no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—44no—
286C22—84no—
287C22—5561no—
288C22—5562no—
289C22—5562no—
290C22—5564no—
291C22—11124no—
292C22#1798483812yes—
293C22C2×D4×D13911yescharacteristic
294C4—41no—
295C4—2784no—
296C4—5561no—
297C4#1798483812yes—
298C4C2×D4×D13911yescharacteristic
299C4C22×C22⋊D13911yescharacteristic
300C2—22no—
301C2—82no—
302C2—81no—
303C2—11122no—
304C2—11121no—
305C2C2×#1798483811yescharacteristic
306C2#2258175712yes—
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
FittingC2×AGL(1,ℤ/8)×C139C2
socleC22×C139C22×D4
hypercenterC2×AGL(1,ℤ/8)D139
hypocenterC139#3155
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#17984870C2semidirect
2. C2×#17984862C2semidirect
3. C22×#14555770C2semidirect
4. #22581520C2semidirect
5. #22581517C2semidirect
6. #22581341C2semidirect
7. #22581339C2semidirect
8. #22581334C2semidirect
9. #22581328C2semidirect
10. #22580911C2semidirect
11. C2×AGL(1,ℤ/8)×C139C2semidirect
16. C2×#14555775C22semidirect
17. C2×#14555774C22semidirect
18. C2×#14555773C22semidirect
19. C2×#14555772C22semidirect
20. C2×D1112C22semidirect
24. C2×#14555770C22semidirect
26. #17984838C22semidirect
27. #17984838C22semidirect
28. #17984831C22semidirect
29. #17984829C22semidirect
35. #17984752C22semidirect
36. C8×Dic139C22semidirect
40. C2×SD16×C139C22semidirect
41. C2×D8×C139C22semidirect
64. C2×C139⋊C8D4semidirect
86. M16×C139D4semidirect
153. C139#3155semidirect
156. C2×AGL(1,ℤ/8)D139semidirect
177. C2×SD16C2×D139semidirect
183. C2×D8C2×D139semidirect
232. M16C22⋊D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848702
C2×#179848622
C22×#145557702
#225815202
#225815172
#225813412
#225813392
#225813342
#225813282
#225809112
C2×AGL(1,ℤ/8)×C1392
#3155139