Finite Group Databasebeta

← #31816852

Subgroups — #31816852

Order 17792.

656 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 153 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×#17984868C211yesmaximal, characteristic, complemented
3C2×#17984867C211yesmaximal, characteristic, complemented
4#22581499C211yesmaximal, characteristic, complemented
5C2×#17984838C211yesmaximal, characteristic, complemented
6#22581341C212yesmaximal, complemented
7#22581339C212yesmaximal, complemented
8#22581328C212yesmaximal, complemented
9#22581324C212yesmaximal, complemented
10M16×Dic139C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C139C211yesmaximal, characteristic, complemented
12C2×#14555785C2211yescharacteristic
13C22×D556C2211yescharacteristic
14C22×C22⋊D139—22no—
15#17984870—24no—
16#17984868—24no—
17#17984867—24no—
18C2×#14555774C2212yescomplemented
19C2×#14555773C2212yescomplemented
20C2×#14555772C2212yescomplemented
21C2×D1112C2212yescomplemented
22C2×#14555770C2212yescomplemented
23C2×#14555770C2212yescomplemented
24C2×#14555769C2211yescharacteristic
25#17984840C2211yescharacteristic, complemented
26#17984838—22no—
27#17984838C2212yes—
28#17984838C2212yes—
29#17984838C2211yescharacteristic, complemented
30#17984831C2211yescharacteristic, complemented
31#17984829C2211yescharacteristic, complemented
32#17984814—22no—
33C4×C22⋊D139—22no—
34C2×C4×Dic139C2211yescharacteristic
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×D556C2311yescharacteristic
51C2×D556C2312yes—
52C2×C22⋊D139—22no—
53C2×C22⋊D139—24no—
54C2×C22⋊D139—24no—
55C2×C22⋊D139—21no—
56C2×C22⋊D139—48no—
57C22×Dic139D412yes—
58C2×C4×D139—21no—
59#14555774—24no—
60#14555773—24no—
61#14555772—24no—
62D1112—24no—
63#14555770—24no—
64#14555770—24no—
65#14555769D414yescomplemented
66C2×C139⋊C8C2312yes—
67Dic139⋊C4—22no—
68C22⋊Dic139—22no—
69(C2×D139)⋊C4—22no—
70C4×Dic139D412yes—
71C4×Dic139C2311yescharacteristic
72C4×Dic139C2311yescharacteristic
73C24×C139—21no—
74#41×C139—22no—
75#41×C139—22no—
76C2×Q8×C139C2311yescharacteristic
77C2×D4×C139—21no—
78C2×D4×C139—22no—
79C2×D4×C139—22no—
80C2×D4×C139—21no—
81C2×D4×C139C2311yescharacteristic
82C2×D4×C139C2312yes—
83C22×C4×C139—21no—
84C22×C4×C139C2311yescharacteristic
85SD16×C139—24no—
86D8×C139—24no—
87M16×C139D414yescomplemented
88C2×C8×C139C2312yes—
89C22×D139—21no—
90C22×D139—22no—
91C22×D139—44no—
92C22×D139—42no—
93Dic278—21no—
94Dic278—21no—
95D556—21no—
96D556—22no—
97D556—21no—
98D556—22no—
99D556—42no—
100C22⋊D139—48no—
101C22⋊D139—42no—
102C22⋊D139—44no—
103C22⋊D139—48no—
104C2×Dic139—24no—
105C2×Dic139—21no—
106C2×Dic139C2×D412yes—
107C2×Dic139C2×D412yes—
108C4×D139—42no—
109C139⋊C8C2×D414yescomplemented
110C23×C139—21no—
111C23×C139—22no—
112C23×C139—44no—
113C23×C139—42no—
114C23×C139C2×D411yescharacteristic
115Q8×C139—21no—
116Q8×C139—21no—
117D4×C139—21no—
118D4×C139—22no—
119D4×C139—22no—
120D4×C139—21no—
121D4×C139—42no—
122D4×C139—42no—
123C2×C4×C139—21no—
124C2×C4×C139—42no—
125C2×C4×C139C2×D411yescharacteristic
126C2×C4×C139C2×D412yes—
127C2×C4×C139C2×D412yes—
128C2×C4×C139C2411yescharacteristic
129C8×C139C2×D414yescomplemented
130C22×C139—22no—
131C22×C139—41no—
132C22×C139—42no—
133C22×C139—44no—
134C22×C139—42no—
135C22×C139—84no—
136C22×C139#12712yes—
137C22×C139C22×D411yescharacteristic
138C2×D139—41no—
139C2×D139—42no—
140C2×D139—44no—
141C2×D139—42no—
142C2×D139—84no—
143C4×C139—41no—
144C4×C139#12712yes—
145C4×C139C22×D411yescharacteristic
146C4×C139C22×D411yescharacteristic
147Dic139—22no—
148Dic139—41no—
149Dic139AGL(1,ℤ/8)14yescomplemented
150C2×C139—22no—
151C2×C139—81no—
152C2×C139—82no—
153C2×C139C2×#12711yescharacteristic
154C2×C139C2×AGL(1,ℤ/8)12yes—
155D139—81no—
156D139—82no—
157C139#315111yescharacteristic, complemented, Sylow
158#3151—1391nomaximal, Sylow
159C2×AGL(1,ℤ/8)—1391no—
160C2×AGL(1,ℤ/8)—1391no—
161C2×AGL(1,ℤ/8)—1391no—
162C2×AGL(1,ℤ/8)D13911yescharacteristic, complemented
163#532—1391no—
164C2×#127—1391no—
165#496—1392no—
166#496—1392no—
167#493—1392no—
168#492—1392no—
169C4×M16—1391no—
170C2×#41—1391no—
171C2×#41C2×D13911yescharacteristic
172C22×D4—2782no—
173C22×D4—1391no—
174C22×D4C2×D13911yescharacteristic
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—2784no—
177AGL(1,ℤ/8)—2784no—
178AGL(1,ℤ/8)—2784no—
179C2×SD16—1392no—
180C2×SD16—1392no—
181C2×SD16—1392no—
182C2×SD16C2×D13912yescomplemented
183C2×D8—1392no—
184C2×D8—1392no—
185C2×D8—1392no—
186C2×D8C2×D13912yescomplemented
187C2×M16—1391no—
188C2×M16C2×D13911yescharacteristic
189C4⋊Q8—1391no—
190#127—2782no—
191#127—1392no—
192#127—1392no—
193#127—1391no—
194#124—1391no—
195#124—1391no—
196#121—2782no—
197C4×D4—2782no—
198C2×C42—1391no—
199#97—1392no—
200C4×C8—1392no—
201C24—21no—
202C24—2781no—
203#41—22no—
204#41—22no—
205#41—2782no—
206#41—2782no—
207C2×Q8—1391no—
208C2×Q8C22×D13911yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—2781no—
214C2×D4—2782no—
215C2×D4—2784no—
216C2×D4—2784no—
217C2×D4—2782no—
218C2×D4—2781no—
219C2×D4—2782no—
220C2×D4—5568no—
221C2×D4—1391no—
222C2×D4—1392no—
223C2×D4C22×D13911yescharacteristic
224C2×D4C22×D13912yes—
225C22×C4—21no—
226C22×C4—2781no—
227C22×C4—1392no—
228C22×C4C22×D13911yescharacteristic
229SD16—24no—
230SD16—2784no—
231SD16—2784no—
232SD16—2784no—
233D8—24no—
234D8—2784no—
235D8—2784no—
236D8—2784no—
237M16—1394no—
238M16C22⋊D13914yescomplemented
239C2×C8—1392no—
240C2×C8C22×D13912yes—
241C4⋊C4—2782no—
242C22⋊C4—2782no—
243C22⋊C4—2782no—
244C42—1392no—
245C42—1391no—
246C42—1391no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—2781no—
252C23—2782no—
253C23—5564no—
254C23—5562no—
255C23D4×D13911yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—2781no—
259Q8—2781no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—2781no—
267D4—2782no—
268D4—2781no—
269D4—2782no—
270D4—5568no—
271D4—5562no—
272D4—5568no—
273D4—5564no—
274D4—5562no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—2784no—
278C2×C4—2781no—
279C2×C4—5562no—
280C2×C4—1392no—
281C2×C4—1392no—
282C2×C4D4×D13911yescharacteristic
283C2×C4D4×D13912yes—
284C2×C4C2×C22⋊D13912yes—
285C2×C4C23×D13911yescharacteristic
286C8—1394no—
287C8C2×C22⋊D13914yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—5564no—
295C22—5562no—
296C22—5561no—
297C22—5562no—
298C22—11124no—
299C22#1798483812yes—
300C22C2×D4×D13911yescharacteristic
301C4—41no—
302C4—2782no—
303C4—5561no—
304C4—1394no—
305C4#1798483812yes—
306C4C2×D4×D13911yescharacteristic
307C4C22×C22⋊D13911yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—11122no—
312C2—11121no—
313C2C2×#1798483811yescharacteristic
314C2AGL(1,ℤ/8)×D13912yes—
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#3151
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×#17984868C2semidirect
3. C2×#17984867C2semidirect
4. #22581499C2semidirect
5. C2×#17984838C2semidirect
6. #22581341C2semidirect
7. #22581339C2semidirect
8. #22581328C2semidirect
9. #22581324C2semidirect
10. M16×Dic139C2semidirect
11. C2×AGL(1,ℤ/8)×C139C2semidirect
18. C2×#14555774C22semidirect
19. C2×#14555773C22semidirect
20. C2×#14555772C22semidirect
21. C2×D1112C22semidirect
22. C2×#14555770C22semidirect
23. C2×#14555770C22semidirect
25. #17984840C22semidirect
29. #17984838C22semidirect
30. #17984831C22semidirect
31. #17984829C22semidirect
35. #17984752C22semidirect
36. C8×Dic139C22semidirect
40. C2×SD16×C139C22semidirect
41. C2×D8×C139C22semidirect
65. #14555769D4semidirect
87. M16×C139D4semidirect
109. C139⋊C8C2×D4semidirect
129. C8×C139C2×D4semidirect
149. Dic139AGL(1,ℤ/8)semidirect
157. C139#3151semidirect
162. C2×AGL(1,ℤ/8)D139semidirect
182. C2×SD16C2×D139semidirect
186. C2×D8C2×D139semidirect
238. M16C22⋊D139semidirect
287. C8C2×C22⋊D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848702
C2×#179848682
C2×#179848672
#225814992
C2×#179848382
#225813412
#225813392
#225813282
#225813242
M16×Dic1392
C2×AGL(1,ℤ/8)×C1392
#3151139