Finite Group Databasebeta

← #31817923

Subgroups — #31817923

Order 17792.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984861C211yesmaximal, characteristic, complemented
2#22581602C211yesmaximal, characteristic, complemented
3#22581597C211yesmaximal, characteristic, complemented
4#22581344C211yesmaximal, characteristic, complemented
5#22581211C211yesmaximal, characteristic, complemented
6#22581179C211yesmaximal, characteristic, complemented
7#22581166C211yesmaximal, characteristic, complemented
8#22581156C211yesmaximal, characteristic, complemented
9#22581115C211yesmaximal, characteristic, complemented
10#22581112C211yesmaximal, characteristic, complemented
11#22581104C211yesmaximal, characteristic, complemented
12#22581095C211yesmaximal, characteristic, complemented
13#22581065C211yesmaximal, characteristic, complemented
14#22581022C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C139C211yesmaximal, characteristic, complemented
16C2×#14555784C2211yescharacteristic
17C2×#14555784C2211yescharacteristic, complemented
18#17984861—24no—
19C2×Dic556C2211yescharacteristic, complemented
20C2×#14555775C2211yescharacteristic, complemented
21C2×#14555773C2211yescharacteristic
22C2×#14555773C2211yescharacteristic
23C2×#14555773C2211yescharacteristic, complemented
24C2×C8×D139C2211yescharacteristic, complemented
25#17984840C2211yescharacteristic, complemented
26#17984829C2211yescharacteristic
27#17984825—21no—
28#17984823—21no—
29#17984823—21no—
30#17984821C2211yescharacteristic
31#17984821C2211yescharacteristic
32#17984818C2211yescharacteristic, complemented
33#17984814—21no—
34#17984814—21no—
35#17984813C2211yescharacteristic
36#17984811—21no—
37D4×Dic139C2211yescharacteristic
38D4×Dic139C2211yescharacteristic, complemented
39C4×C22⋊D139C2211yescharacteristic
40C2×Dic139⋊C4—21no—
41C2×C22⋊Dic139—21no—
42#17984774C2211yescharacteristic, complemented
43Dic139⋊C8C2211yescharacteristic, complemented
44#17984767C2211yescharacteristic, complemented
45#17984767C2211yescharacteristic, complemented
46#17984766C2211yescharacteristic, complemented
47#17984763C2211yescharacteristic
48#17984763C2211yescharacteristic
49#17984763C2211yescharacteristic, complemented
50C22⋊(C139⋊C8)C2211yescharacteristic
51(C2×D139)⋊C8C2211yescharacteristic, complemented
52#17984752C2211yescharacteristic, complemented
53C2×#41×C139C2211yescharacteristic
54C22×D4×C139C2211yescharacteristic
55AGL(1,ℤ/8)×C139—24no—
56C2×SD16×C139C2211yescharacteristic
57C2×SD16×C139C2211yescharacteristic
58C2×D8×C139C2211yescharacteristic
59C2×D8×C139C2211yescharacteristic
60C2×M16×C139C2211yescharacteristic
61#14555784—22no—
62#14555784—22no—
63#14555784—41no—
64#14555784—41no—
65C2×Dic278C2311yescharacteristic
66C2×Dic278C2311yescharacteristic
67C2×C22⋊D139—21no—
68C2×C22⋊D139—21no—
69C2×C22⋊D139—21no—
70C2×C22⋊D139D411yescharacteristic
71C22×Dic139—21no—
72C22×Dic139—21no—
73C22×Dic139—21no—
74C22×Dic139—21no—
75C2×C4×D139C2311yescharacteristic
76Dic556—22no—
77#14555775—41no—
78#14555773—22no—
79#14555773—22no—
80#14555773—41no—
81C2×C139⋊C8C2311yescharacteristic
82C2×C139⋊C8C2311yescharacteristic
83C8×D139—22no—
84C4⋊Dic139C2311yescharacteristic
85C4⋊Dic139C2311yescharacteristic
86Dic139⋊C4—21no—
87Dic139⋊C4—21no—
88Dic139⋊C4—21no—
89Dic139⋊C4—21no—
90Dic139⋊C4—21no—
91Dic139⋊C4D411yescharacteristic
92C22⋊Dic139—21no—
93C22⋊Dic139—21no—
94C22⋊Dic139—21no—
95C22⋊Dic139—21no—
96C22⋊Dic139—21no—
97C22⋊Dic139—21no—
98C22⋊Dic139—21no—
99C22⋊Dic139D411yescharacteristic
100(C2×D139)⋊C4—21no—
101(C2×D139)⋊C4D411yescharacteristic
102C4×Dic139C2311yescharacteristic
103C24×C139—21no—
104#41×C139—22no—
105#41×C139—41no—
106C2×Q8×C139C2311yescharacteristic
107C2×D4×C139—22no—
108C2×D4×C139—21no—
109C2×D4×C139—21no—
110C2×D4×C139—41no—
111C2×D4×C139C2311yescharacteristic
112C2×D4×C139C2311yescharacteristic
113C2×D4×C139C2311yescharacteristic
114C22×C4×C139—21no—
115C22×C4×C139C2311yescharacteristic
116SD16×C139—22no—
117SD16×C139—22no—
118D8×C139—22no—
119D8×C139D414yescomplemented
120M16×C139—22no—
121C2×C8×C139C2311yescharacteristic
122C2×C8×C139C2311yescharacteristic
123C22×D139C2×D411yescharacteristic
124Dic278—21no—
125Dic278—21no—
126Dic278—21no—
127Dic278—21no—
128C22⋊D139—41no—
129C22⋊D139—42no—
130C22⋊D139—42no—
131C22⋊D139—81no—
132C2×Dic139—21no—
133C2×Dic139—21no—
134C2×Dic139—21no—
135C2×Dic139—21no—
136C2×Dic139—42no—
137C2×Dic139—42no—
138C2×Dic139—42no—
139C2×Dic139—81no—
140C2×Dic139C2×D411yescharacteristic
141C2×Dic139C2×D411yescharacteristic
142C2×Dic139C2×D411yescharacteristic
143C4×D139—21no—
144C4×D139—21no—
145C139⋊C8—21no—
146C139⋊C8—21no—
147C23×C139—21no—
148C23×C139—21no—
149C23×C139—21no—
150C23×C139—42no—
151C23×C139—42no—
152C23×C139—81no—
153C23×C139C2×D411yescharacteristic
154Q8×C139—21no—
155Q8×C139—21no—
156D4×C139—21no—
157D4×C139—21no—
158D4×C139—21no—
159D4×C139—21no—
160D4×C139—42no—
161D4×C139—42no—
162D4×C139C2×D412yes—
163D4×C139C2×D412yes—
164C2×C4×C139—21no—
165C2×C4×C139—21no—
166C2×C4×C139—21no—
167C2×C4×C139—42no—
168C2×C4×C139C2×D411yescharacteristic
169C2×C4×C139C2411yescharacteristic
170C8×C139—21no—
171C8×C139C2×D412yes—
172C22×C139—21no—
173C22×C139—22no—
174C22×C139—22no—
175C22×C139—21no—
176C22×C139—21no—
177C22×C139—41no—
178C22×C139—41no—
179C22×C139—41no—
180C22×C139—41no—
181C22×C139—41no—
182C22×C139—41no—
183C22×C139—41no—
184C22×C139—84no—
185C22×C139C22×D411yescharacteristic
186C2×D139—21no—
187C2×D139—21no—
188C2×D139—21no—
189C4×C139—21no—
190C4×C139—41no—
191C4×C139C22×D411yescharacteristic
192C4×C139H(5,𝔽2)11yescharacteristic
193Dic139—21no—
194Dic139—21no—
195Dic139—21no—
196Dic139—41no—
197Dic139—41no—
198Dic139—41no—
199Dic139—41no—
200C2×C139—41no—
201C2×C139—42no—
202C2×C139—42no—
203C2×C139—81no—
204C2×C139D4211yescharacteristic
205C2×C139#57511yescharacteristic
206C2×C139#57711yescharacteristic
207D139—41no—
208C139#326711yescharacteristic, complemented, Sylow
209#3267—1391nomaximal, Sylow
210C2×AGL(1,ℤ/8)D13911yescharacteristic, complemented
211C2×#135—1391no—
212#547—1391no—
213#546—1391no—
214#497—1391no—
215#470—1391no—
216#465—1391no—
217#463—1391no—
218#461—1391no—
219#452—1391no—
220#452—1391no—
221#450—1391no—
222#449—1391no—
223#442—1391no—
224#435—1391no—
225C2×#41—1391no—
226C2×#41—1391no—
227C2×#41C2×D13911yescharacteristic
228C22×D4C2×D13911yescharacteristic
229AGL(1,ℤ/8)—24no—
230#135—2784no—
231C2×Q16—1391no—
232C2×Q16—1391no—
233C2×SD16—1391no—
234C2×SD16—1391no—
235C2×SD16—1391no—
236C2×SD16C2×D13911yescharacteristic
237C2×SD16C2×D13911yescharacteristic
238C2×D8C2×D13911yescharacteristic
239C2×D8C2×D13911yescharacteristic
240C2×M16C2×D13911yescharacteristic
241C22×C8—1391no—
242C4⋊Q8—1391no—
243#124—1391no—
244#123—2781no—
245#123—2781no—
246#123—2781no—
247C22⋊Q8—1391no—
248C22⋊Q8—1391no—
249C22⋊Q8—1391no—
250#121—2781no—
251#121—2781no—
252#121—1391no—
253#120—2781no—
254C4×D4—1391no—
255C4×D4—1391no—
256C4×D4—1391no—
257C2×C4⋊C4—2781no—
258C2×C22⋊C4—2781no—
259#106—1391no—
260C4⋊C8—1391no—
261#103—1391no—
262#103—1391no—
263#103—1391no—
264#102—1391no—
265#102—1391no—
266#102—1391no—
267C22⋊C8—1391no—
268C22⋊C8—1391no—
269#97—1391no—
270C24—21no—
271#41—22no—
272#41—41no—
273#41—2782no—
274#41—2782no—
275#41—5561no—
276#41—5561no—
277C2×Q8—1391no—
278C2×Q8—1391no—
279C2×Q8C22×D13911yescharacteristic
280C2×D4—22no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C2×D4—2781no—
285C2×D4—2781no—
286C2×D4—2781no—
287C2×D4—1391no—
288C2×D4C22×D13911yescharacteristic
289C2×D4C22×D13911yescharacteristic
290C2×D4C22×D13911yescharacteristic
291C22×C4—21no—
292C22×C4—2781no—
293C22×C4—2781no—
294C22×C4—2781no—
295C22×C4—2781no—
296C22×C4—1391no—
297C22×C4C22×D13911yescharacteristic
298Q16—2782no—
299Q16—5561no—
300SD16—22no—
301SD16—22no—
302SD16—2782no—
303SD16—2782no—
304SD16—5561no—
305D8—22no—
306D8C22⋊D13914yescomplemented
307M16—22no—
308C2×C8—2782no—
309C2×C8—1391no—
310C2×C8—1391no—
311C2×C8C22×D13911yescharacteristic
312C2×C8C22×D13911yescharacteristic
313C4⋊C4—2781no—
314C4⋊C4—2781no—
315C4⋊C4—2781no—
316C4⋊C4—2781no—
317C4⋊C4—2781no—
318C4⋊C4—1391no—
319C4⋊C4—1391no—
320C4⋊C4—1391no—
321C22⋊C4—2781no—
322C22⋊C4—2781no—
323C22⋊C4—2781no—
324C22⋊C4—2781no—
325C22⋊C4—2781no—
326C22⋊C4—2781no—
327C22⋊C4—2781no—
328C22⋊C4—2781no—
329C22⋊C4—1391no—
330C22⋊C4—1391no—
331C42—1391no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—42no—
336C23—42no—
337C23—81no—
338C23—1391no—
339C23D4×D13911yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—2781no—
343Q8—2781no—
344Q8—2781no—
345Q8—2781no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—42no—
351D4—42no—
352D4—5561no—
353D4—5562no—
354D4—5562no—
355D4—11121no—
356D4C2×C22⋊D13912yes—
357D4C2×C22⋊D13912yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—2781no—
363C2×C4—2781no—
364C2×C4—2781no—
365C2×C4—2781no—
366C2×C4—2781no—
367C2×C4—2781no—
368C2×C4—5562no—
369C2×C4—5562no—
370C2×C4—5562no—
371C2×C4—11121no—
372C2×C4—1391no—
373C2×C4—1391no—
374C2×C4—1391no—
375C2×C4D4×D13911yescharacteristic
376C2×C4C23×D13911yescharacteristic
377C8—21no—
378C8—2781no—
379C8—2781no—
380C8C2×C22⋊D13912yes—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—22no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—84no—
394C22—2781no—
395C22—2781no—
396C22—2781no—
397C22C2×D4×D13911yescharacteristic
398C4—21no—
399C4—41no—
400C4—2781no—
401C4—2781no—
402C4—2781no—
403C4—5561no—
404C4—5561no—
405C4—5561no—
406C4—5561no—
407C4C22×C22⋊D13911yescharacteristic
408C4#1798488711yescharacteristic
409C2—41no—
410C2—42no—
411C2—42no—
412C2—81no—
413C2—5561no—
414C2D4×C22⋊D13911yescharacteristic
415C2#2258174911yescharacteristic
416C2#2258176311yescharacteristic
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#3267
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×#17984861C2semidirect
2. #22581602C2semidirect
3. #22581597C2semidirect
4. #22581344C2semidirect
5. #22581211C2semidirect
6. #22581179C2semidirect
7. #22581166C2semidirect
8. #22581156C2semidirect
9. #22581115C2semidirect
10. #22581112C2semidirect
11. #22581104C2semidirect
12. #22581095C2semidirect
13. #22581065C2semidirect
14. #22581022C2semidirect
15. C2×AGL(1,ℤ/8)×C139C2semidirect
17. C2×#14555784C22semidirect
19. C2×Dic556C22semidirect
20. C2×#14555775C22semidirect
23. C2×#14555773C22semidirect
24. C2×C8×D139C22semidirect
25. #17984840C22semidirect
32. #17984818C22semidirect
38. D4×Dic139C22semidirect
42. #17984774C22semidirect
43. Dic139⋊C8C22semidirect
44. #17984767C22semidirect
45. #17984767C22semidirect
46. #17984766C22semidirect
49. #17984763C22semidirect
51. (C2×D139)⋊C8C22semidirect
52. #17984752C22semidirect
119. D8×C139D4semidirect
208. C139#3267semidirect
210. C2×AGL(1,ℤ/8)D139semidirect
306. D8C22⋊D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848612
#225816022
#225815972
#225813442
#225812112
#225811792
#225811662
#225811562
#225811152
#225811122
#225811042
#225810952
#225810652
#225810222
C2×AGL(1,ℤ/8)×C1392
#3267139