Finite Group Databasebeta

← #31818202

Subgroups — #31818202

Order 17792.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#22581758C211yesmaximal, characteristic, complemented
2#22581757C212yesmaximal, complemented
3#22581754C211yesmaximal, characteristic, complemented
4#22581328C211yesmaximal, characteristic, complemented
5#22581224C211yesmaximal, characteristic, complemented
6#22581222C212yesmaximal, complemented
7#22581123C212yesmaximal, complemented
8#22581119C212yesmaximal, complemented
9#22581069C211yesmaximal, characteristic, complemented
10#22581068C211yesmaximal, characteristic, complemented
11#472×C139C211yesmaximal, characteristic, complemented
12#17984888C2212yescomplemented
13#17984887C2212yescomplemented
14#17984870—22no—
15#17984870C2212yescomplemented
16#17984870C2212yescomplemented
17#17984869—22no—
18#17984868C2212yescomplemented
19#17984866—22no—
20#17984866—22no—
21#17984864—21no—
22#17984864C2211yescharacteristic
23#17984863—22no—
24#17984862C2211yescharacteristic
25#17984861—21no—
26C2×D1112—21no—
27C2×D1112C2211yescharacteristic, complemented
28C2×#14555770C2212yescomplemented
29D8×D139—21no—
30D8×D139—22no—
31#17984850C2212yescomplemented
32#17984848C2211yescharacteristic, complemented
33#17984838C2212yescomplemented
34#17984777C2211yescharacteristic, complemented
35#17984769C2211yescharacteristic, complemented
36#17984768C2211yescharacteristic, complemented
37#17984768C2212yescomplemented
38#17984758C2212yescomplemented
39C8×Dic139C2211yescharacteristic, complemented
40AGL(1,ℤ/8)×C139C2212yescomplemented
41C2×D8×C139C2211yescharacteristic, complemented
42#131×C139C2211yescharacteristic, complemented
43#108×C139C2211yescharacteristic, complemented
44#100×C139C2212yescomplemented
45#14555786—22no—
46#14555785—22no—
47#14555785C2311yescharacteristic
48#14555785C2312yes—
49#14555784—22no—
50#14555784—22no—
51C2×D556—22no—
52C2×D556C2312yes—
53C2×C22⋊D139—42no—
54C2×C22⋊D139—42no—
55D4×D139—22no—
56D4×D139—22no—
57D4×D139—22no—
58D4×D139—42no—
59Dic556D411yescharacteristic, complemented
60#14555775D411yescharacteristic, complemented
61#14555774—22no—
62#14555774—22no—
63#14555774D412yescomplemented
64#14555773—22no—
65#14555773D411yescharacteristic, complemented
66#14555772D411yescharacteristic, complemented
67D1112—22no—
68D1112—22no—
69D1112—21no—
70D1112—21no—
71D1112D411yescharacteristic, complemented
72#14555770—22no—
73#14555770—22no—
74#14555770—22no—
75#14555770D411yescharacteristic, complemented
76#14555769C2311yescharacteristic
77#14555768—22no—
78#14555768—21no—
79C2×C139⋊C8C2311yescharacteristic
80C8×D139—22no—
81C8×D139—21no—
82C4×Dic139C2311yescharacteristic
83#41×C139C2311yescharacteristic
84C2×D4×C139C2312yes—
85SD16×C139—22no—
86D8×C139—22no—
87D8×C139—21no—
88D8×C139—21no—
89M16×C139—21no—
90M16×C139C2311yescharacteristic
91M16×C139C2312yes—
92C2×C8×C139—21no—
93C2×C8×C139C2311yescharacteristic
94C22×D139—22no—
95C22×D139—42no—
96C22×D139—42no—
97Dic278C2×D412yes—
98Dic278C2×D411yescharacteristic
99D556—22no—
100D556—22no—
101D556—42no—
102D556C2×D412yes—
103D556C2×D411yescharacteristic
104C22⋊D139—22no—
105C22⋊D139—21no—
106C22⋊D139—42no—
107C22⋊D139—42no—
108C22⋊D139—42no—
109C22⋊D139—84no—
110C2×Dic139—21no—
111C2×Dic139—42no—
112C4×D139—21no—
113C4×D139—22no—
114C4×D139—42no—
115C139⋊C8—21no—
116C139⋊C8C2×D411yescharacteristic
117C139⋊C8C2×D411yescharacteristic
118C23×C139—22no—
119Q8×C139C2×D411yescharacteristic
120D4×C139—21no—
121D4×C139—22no—
122D4×C139—22no—
123D4×C139C2×D411yescharacteristic
124C2×C4×C139—21no—
125C2×C4×C139C2411yescharacteristic
126C8×C139—21no—
127C8×C139—22no—
128C8×C139C2×D411yescharacteristic
129C8×C139C2×D411yescharacteristic
130C22×C139—21no—
131C22×C139—42no—
132C22×C139—82no—
133C22×C139H(5,𝔽2)11yescharacteristic
134C2×D139—21no—
135C2×D139—22no—
136C2×D139—42no—
137C2×D139—82no—
138C2×D139—162no—
139C2×D139—162no—
140C4×C139—21no—
141C4×C139C22×D411yescharacteristic
142C4×C139C22×D411yescharacteristic
143Dic139—22no—
144Dic139—22no—
145Dic139—21no—
146C2×C139—21no—
147C2×C139—41no—
148C2×C139—82no—
149C2×C139D4211yescharacteristic
150D139—41no—
151D139—42no—
152D139—82no—
153C139#329211yescharacteristic, complemented, Sylow
154#3292—1391nomaximal, Sylow
155#576—1392no—
156#576—1391no—
157#576—1391no—
158#493—1391no—
159#472—1391no—
160#472—1392no—
161#472D13911yescharacteristic, complemented
162#453—1392no—
163#453—1392no—
164#443—1391no—
165#443—1391no—
166H(5,𝔽2)—1392no—
167H(5,𝔽2)—1392no—
168AGL(1,ℤ/8)—2782no—
169AGL(1,ℤ/8)—2782no—
170AGL(1,ℤ/8)—2782no—
171AGL(1,ℤ/8)—2782no—
172AGL(1,ℤ/8)—1392no—
173AGL(1,ℤ/8)—1392no—
174AGL(1,ℤ/8)—1392no—
175AGL(1,ℤ/8)C2×D13912yescomplemented
176#135—2781no—
177#135—2781no—
178#135—2782no—
179#135—1391no—
180#135—1391no—
181C2×D8—2781no—
182C2×D8—2782no—
183C2×D8—2781no—
184C2×D8—1392no—
185C2×D8—1391no—
186C2×D8C2×D13911yescharacteristic, complemented
187#131—1392no—
188#131—1391no—
189#131C2×D13911yescharacteristic, complemented
190#127—1392no—
191#108—1391no—
192#108C2×D13911yescharacteristic, complemented
193#104—1391no—
194#104—1392no—
195#104—1391no—
196#100—1392no—
197#100C2×D13912yescomplemented
198C4×C8—1391no—
199#41—2782no—
200#41—2782no—
201#41—2782no—
202#41—2782no—
203#41—1392no—
204#41—1391no—
205#41C22×D13911yescharacteristic
206C2×D4—2782no—
207C2×D4—2782no—
208C2×D4—2782no—
209C2×D4—2782no—
210C2×D4—5562no—
211C2×D4—5562no—
212C2×D4—5562no—
213C2×D4—1392no—
214C2×D4C22×D13912yes—
215Q16—1391no—
216Q16—1391no—
217SD16—22no—
218SD16—2782no—
219SD16—2782no—
220SD16—2782no—
221SD16—1391no—
222SD16—1391no—
223SD16—1392no—
224D8—22no—
225D8—21no—
226D8—21no—
227D8—2782no—
228D8—2782no—
229D8—2782no—
230D8—2782no—
231D8—2782no—
232D8—2781no—
233D8—2781no—
234D8—1391no—
235D8—1391no—
236M16—21no—
237M16—2782no—
238M16—2781no—
239M16—1391no—
240M16C22×D13912yes—
241M16C22×D13911yescharacteristic
242C2×C8—21no—
243C2×C8—2782no—
244C2×C8—2781no—
245C2×C8—1391no—
246C2×C8C22×D13911yescharacteristic
247C42—1391no—
248C23—22no—
249C23—2782no—
250C23—5562no—
251C23—5562no—
252Q8—1391no—
253Q8—1392no—
254Q8D4×D13911yescharacteristic
255D4—21no—
256D4—22no—
257D4—22no—
258D4—2782no—
259D4—2781no—
260D4—2782no—
261D4—2782no—
262D4—5562no—
263D4—5562no—
264D4—5562no—
265D4—5562no—
266D4—11124no—
267D4—1391no—
268D4—1392no—
269D4D4×D13911yescharacteristic
270C2×C4—21no—
271C2×C4—2781no—
272C2×C4—2782no—
273C2×C4—2781no—
274C2×C4—5562no—
275C2×C4—5562no—
276C2×C4C23×D13911yescharacteristic
277C8—21no—
278C8—22no—
279C8—2781no—
280C8—1391no—
281C8—1391no—
282C8D4×D13911yescharacteristic
283C8D4×D13911yescharacteristic
284C22—21no—
285C22—42no—
286C22—82no—
287C22—2782no—
288C22—2781no—
289C22—5562no—
290C22—11122no—
291C22—22242no—
292C22—22242no—
293C22#1798488811yescharacteristic
294C4—21no—
295C4—2782no—
296C4—2782no—
297C4—2781no—
298C4C2×D4×D13911yescharacteristic
299C4C2×D4×D13911yescharacteristic
300C2—21no—
301C2—41no—
302C2—82no—
303C2—5562no—
304C2—5561no—
305C2—11122no—
306C2#2258158711yescharacteristic
subgroupG/H
centerC2#22581587
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#472×C139C2
socleC2×C139D42
hypercenter#472D139
hypocenterC139#3292
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #22581758C2semidirect
2. #22581757C2semidirect
3. #22581754C2semidirect
4. #22581328C2semidirect
5. #22581224C2semidirect
6. #22581222C2semidirect
7. #22581123C2semidirect
8. #22581119C2semidirect
9. #22581069C2semidirect
10. #22581068C2semidirect
11. #472×C139C2semidirect
12. #17984888C22semidirect
13. #17984887C22semidirect
15. #17984870C22semidirect
16. #17984870C22semidirect
18. #17984868C22semidirect
27. C2×D1112C22semidirect
28. C2×#14555770C22semidirect
31. #17984850C22semidirect
32. #17984848C22semidirect
33. #17984838C22semidirect
34. #17984777C22semidirect
35. #17984769C22semidirect
36. #17984768C22semidirect
37. #17984768C22semidirect
38. #17984758C22semidirect
39. C8×Dic139C22semidirect
40. AGL(1,ℤ/8)×C139C22semidirect
41. C2×D8×C139C22semidirect
42. #131×C139C22semidirect
43. #108×C139C22semidirect
44. #100×C139C22semidirect
59. Dic556D4semidirect
60. #14555775D4semidirect
63. #14555774D4semidirect
65. #14555773D4semidirect
66. #14555772D4semidirect
71. D1112D4semidirect
75. #14555770D4semidirect
153. C139#3292semidirect
161. #472D139semidirect
175. AGL(1,ℤ/8)C2×D139semidirect
186. C2×D8C2×D139semidirect
189. #131C2×D139semidirect
192. #108C2×D139semidirect
197. #100C2×D139semidirect

Maximal subgroups

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

subgroupindex
#225817582
#225817572
#225817542
#225813282
#225812242
#225812222
#225811232
#225811192
#225810692
#225810682
#472×C1392
#3292139