Finite Group Databasebeta

← #31816866

Subgroups — #31816866

Order 17792.

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

normal subgroups only

This group has 572 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×#17984875C211yesmaximal, characteristic, complemented
2C2×#17984872C211yesmaximal, characteristic, complemented
3C2×#17984868C211yesmaximal, characteristic, complemented
4#22581499C211yesmaximal, characteristic, complemented
5C2×#17984829C211yesmaximal, characteristic, complemented
6#22581344C212yesmaximal, complemented
7#22581341C212yesmaximal, complemented
8#22581336C212yesmaximal, complemented
9#22581334C212yesmaximal, complemented
10M16×Dic139C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C139C211yesmaximal, characteristic, complemented
12C2×#14555785C2211yescharacteristic
13C22×Dic278C2211yescharacteristic
14#17984875—24no—
15#17984872—24no—
16#17984868—24no—
17C2×Dic556C2212yescomplemented
18C2×#14555775C2212yescomplemented
19C2×#14555774C2212yescomplemented
20C2×#14555773C2212yes—
21C2×#14555773C2212yescomplemented
22C2×#14555770C2212yes—
23C2×#14555769C2211yescharacteristic
24#17984840C2211yescharacteristic, complemented
25#17984838C2211yescharacteristic, complemented
26#17984831C2211yescharacteristic, complemented
27#17984829—22no—
28#17984829C2212yes—
29#17984829C2212yes—
30#17984829C2211yescharacteristic, complemented
31#17984814—22no—
32C4×C22⋊D139—22no—
33C2×C22⋊Dic139—22no—
34C2×C4×Dic139C2211yescharacteristic
35#17984752C2212yescomplemented
36C8×Dic139C2212yescomplemented
37C2×#41×C139C2211yescharacteristic
38C22×D4×C139C2211yescharacteristic
39AGL(1,ℤ/8)×C139—24no—
40C2×SD16×C139C2212yes—
41C2×D8×C139C2212yes—
42C2×M16×C139C2211yescharacteristic
43#14555785—22no—
44#14555785—22no—
45C2×Dic278—21no—
46C2×Dic278—22no—
47C2×Dic278—22no—
48C2×Dic278C2312yes—
49C2×Dic278C2311yescharacteristic
50C2×D556C2311yescharacteristic
51C2×C22⋊D139—22no—
52C2×C22⋊D139—21no—
53C22×Dic139—21no—
54C22×Dic139D412yes—
55C2×C4×D139—21no—
56Dic556—24no—
57#14555775—24no—
58#14555774—24no—
59#14555773—24no—
60#14555773—24no—
61#14555770—24no—
62#14555769D414yescomplemented
63C2×C139⋊C8C2312yes—
64Dic139⋊C4—22no—
65C22⋊Dic139—24no—
66C22⋊Dic139—24no—
67C22⋊Dic139—22no—
68(C2×D139)⋊C4—22no—
69C4×Dic139D412yes—
70C4×Dic139C2311yescharacteristic
71C4×Dic139C2311yescharacteristic
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—21no—
89Dic278—21no—
90Dic278—22no—
91Dic278—22no—
92Dic278—21no—
93Dic278—42no—
94D556—21no—
95D556—21no—
96C22⋊D139—44no—
97C22⋊D139—42no—
98C2×Dic139—22no—
99C2×Dic139—21no—
100C2×Dic139—42no—
101C2×Dic139—42no—
102C2×Dic139C2×D412yes—
103C2×Dic139C2×D412yes—
104C4×D139—42no—
105C139⋊C8C2×D414yes—
106C23×C139—21no—
107C23×C139—22no—
108C23×C139—44no—
109C23×C139—42no—
110C23×C139C2×D411yescharacteristic
111Q8×C139—21no—
112Q8×C139—21no—
113D4×C139—21no—
114D4×C139—22no—
115D4×C139—22no—
116D4×C139—21no—
117D4×C139—42no—
118D4×C139—42no—
119C2×C4×C139—21no—
120C2×C4×C139—42no—
121C2×C4×C139C2×D411yescharacteristic
122C2×C4×C139C2×D412yes—
123C2×C4×C139C2×D412yes—
124C2×C4×C139C2411yescharacteristic
125C8×C139C2×D414yes—
126C22×C139—22no—
127C22×C139—41no—
128C22×C139—42no—
129C22×C139—44no—
130C22×C139—42no—
131C22×C139—84no—
132C22×C139#12712yes—
133C22×C139C22×D411yescharacteristic
134C2×D139—41no—
135C2×D139—42no—
136C4×C139—41no—
137C4×C139#12712yes—
138C4×C139C22×D411yescharacteristic
139C4×C139C22×D411yescharacteristic
140Dic139—22no—
141Dic139—22no—
142Dic139—41no—
143Dic139—42no—
144C2×C139—22no—
145C2×C139—81no—
146C2×C139—82no—
147C2×C139C2×#12711yescharacteristic
148C2×C139#57512yes—
149D139—81no—
150C139#315311yescharacteristic, complemented, Sylow
151#3153—1391nomaximal, Sylow
152C2×#137—1391no—
153C2×#137—1391no—
154C2×AGL(1,ℤ/8)—1391no—
155C2×AGL(1,ℤ/8)D13911yescharacteristic, complemented
156#532—1391no—
157C2×#124—1391no—
158#497—1392no—
159#496—1392no—
160#495—1392no—
161#495—1392no—
162C4×M16—1391no—
163C2×#41—1391no—
164C2×#41C2×D13911yescharacteristic
165C22×Q8—1391no—
166C22×D4C2×D13911yescharacteristic
167#137—2784no—
168#137—2784no—
169AGL(1,ℤ/8)—24no—
170AGL(1,ℤ/8)—2784no—
171C2×Q16—1392no—
172C2×Q16—1392no—
173C2×SD16—1392no—
174C2×SD16—1392no—
175C2×SD16—1392no—
176C2×SD16C2×D13912yes—
177C2×D8—1392no—
178C2×D8C2×D13912yes—
179C2×M16—1391no—
180C2×M16C2×D13911yescharacteristic
181C4⋊Q8—1391no—
182#127—1391no—
183#124—2782no—
184#124—1391no—
185#124—1392no—
186#124—1392no—
187#124—1391no—
188#121—2782no—
189C4×D4—2782no—
190C2×C22⋊C4—2782no—
191C2×C42—1391no—
192#97—1392no—
193C4×C8—1392no—
194C24—21no—
195#41—22no—
196#41—22no—
197#41—2782no—
198#41—2782no—
199C2×Q8—2781no—
200C2×Q8—2782no—
201C2×Q8—2782no—
202C2×Q8—1392no—
203C2×Q8—1391no—
204C2×Q8C22×D13911yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—2781no—
210C2×D4—2782no—
211C2×D4—1391no—
212C2×D4C22×D13912yes—
213C2×D4C22×D13911yescharacteristic
214C22×C4—21no—
215C22×C4—2781no—
216C22×C4—2781no—
217C22×C4—1392no—
218C22×C4C22×D13911yescharacteristic
219Q16—2784no—
220Q16—2784no—
221SD16—24no—
222SD16—2784no—
223SD16—2784no—
224SD16—2784no—
225D8—24no—
226D8—2784no—
227M16—1394no—
228M16C22⋊D13914yescomplemented
229C2×C8—1392no—
230C2×C8C22×D13912yes—
231C4⋊C4—2782no—
232C22⋊C4—2782no—
233C22⋊C4—2784no—
234C22⋊C4—2784no—
235C22⋊C4—2782no—
236C42—1392no—
237C42—1391no—
238C42—1391no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—42no—
243C23—2781no—
244C23D4×D13911yescharacteristic
245Q8—21no—
246Q8—21no—
247Q8—2781no—
248Q8—2782no—
249Q8—2782no—
250Q8—2781no—
251Q8—5562no—
252D4—22no—
253D4—21no—
254D4—21no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—2781no—
259D4—2781no—
260D4—5564no—
261D4—5562no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—2782no—
265C2×C4—2781no—
266C2×C4—5562no—
267C2×C4—5562no—
268C2×C4—5562no—
269C2×C4—1392no—
270C2×C4—1392no—
271C2×C4D4×D13911yescharacteristic
272C2×C4D4×D13912yes—
273C2×C4C2×C22⋊D13912yes—
274C2×C4C23×D13911yescharacteristic
275C8—1394no—
276C8C2×C22⋊D13914yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—44no—
282C22—84no—
283C22—5561no—
284C22—5562no—
285C22#1798483812yes—
286C22C2×D4×D13911yescharacteristic
287C4—41no—
288C4—2782no—
289C4—2782no—
290C4—5561no—
291C4—5562no—
292C4#1798483812yes—
293C4C2×D4×D13911yescharacteristic
294C4C22×C22⋊D13911yescharacteristic
295C2—22no—
296C2—82no—
297C2—81no—
298C2—11121no—
299C2C2×#1798483811yescharacteristic
300C2#2258174912yes—
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#3153
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×#17984875C2semidirect
2. C2×#17984872C2semidirect
3. C2×#17984868C2semidirect
4. #22581499C2semidirect
5. C2×#17984829C2semidirect
6. #22581344C2semidirect
7. #22581341C2semidirect
8. #22581336C2semidirect
9. #22581334C2semidirect
10. M16×Dic139C2semidirect
11. C2×AGL(1,ℤ/8)×C139C2semidirect
17. C2×Dic556C22semidirect
18. C2×#14555775C22semidirect
19. C2×#14555774C22semidirect
21. C2×#14555773C22semidirect
24. #17984840C22semidirect
25. #17984838C22semidirect
26. #17984831C22semidirect
30. #17984829C22semidirect
35. #17984752C22semidirect
36. C8×Dic139C22semidirect
62. #14555769D4semidirect
86. M16×C139D4semidirect
150. C139#3153semidirect
155. C2×AGL(1,ℤ/8)D139semidirect
228. M16C22⋊D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848752
C2×#179848722
C2×#179848682
#225814992
C2×#179848292
#225813442
#225813412
#225813362
#225813342
M16×Dic1392
C2×AGL(1,ℤ/8)×C1392
#3153139