Finite Group Databasebeta

← #31817926

Subgroups — #31817926

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×#17984869C211yesmaximal, characteristic, complemented
2C2×#17984864C211yesmaximal, characteristic, complemented
3#22581605C211yesmaximal, characteristic, complemented
4#22581600C211yesmaximal, characteristic, complemented
5#22581344C211yesmaximal, characteristic, complemented
6#22581212C211yesmaximal, characteristic, complemented
7#22581167C211yesmaximal, characteristic, complemented
8#22581157C211yesmaximal, characteristic, complemented
9#22581118C211yesmaximal, characteristic, complemented
10#22581113C211yesmaximal, characteristic, complemented
11#22581105C211yesmaximal, characteristic, complemented
12#22581096C211yesmaximal, characteristic, complemented
13#22581064C211yesmaximal, characteristic, complemented
14#22581024C211yesmaximal, characteristic, complemented
15#465×C139C211yesmaximal, characteristic, complemented
16C2×#14555786C2211yescharacteristic
17C2×#14555785C2211yescharacteristic
18C2×#14555785C2211yescharacteristic, complemented
19C2×D4×D139C2211yescharacteristic
20#17984869—24no—
21#17984864—24no—
22C2×Dic556C2211yescharacteristic, complemented
23C2×#14555775C2211yescharacteristic, complemented
24C2×#14555774C2211yescharacteristic
25C2×#14555774C2211yescharacteristic, complemented
26C2×#14555773C2211yescharacteristic
27C2×#14555772C2211yescharacteristic
28C2×D1112C2211yescharacteristic
29C2×#14555770C2211yescharacteristic
30C2×#14555768C2211yescharacteristic
31#17984840C2211yescharacteristic, complemented
32#17984829C2211yescharacteristic
33#17984827—21no—
34#17984824—21no—
35#17984824—21no—
36#17984822C2211yescharacteristic
37#17984822C2211yescharacteristic
38#17984815—21no—
39#17984813—21no—
40#17984810—21no—
41#17984804C2211yescharacteristic
42#17984804C2211yescharacteristic, complemented
43C4×C22⋊D139C2211yescharacteristic
44C4⋊C4×D139—21no—
45C22⋊C4×D139—21no—
46Dic139⋊C8C2211yescharacteristic, complemented
47#17984765C2211yescharacteristic, complemented
48#17984765C2211yescharacteristic, complemented
49#17984762C2211yescharacteristic
50#17984762C2211yescharacteristic, complemented
51C22⋊(C139⋊C8)C2211yescharacteristic, complemented
52(C2×D139)⋊C8C2211yescharacteristic
53#17984752C2211yescharacteristic, complemented
54C2×SD16×C139C2211yescharacteristic
55C22×C8×C139C2211yescharacteristic, complemented
56C22⋊Q8×C139C2211yescharacteristic, complemented
57#121×C139C2211yescharacteristic
58#106×C139C2211yescharacteristic, complemented
59#103×C139C2211yescharacteristic, complemented
60#102×C139C2211yescharacteristic
61C23×D139—21no—
62#14555786—22no—
63#14555786—41no—
64#14555785—22no—
65#14555785—22no—
66#14555785—41no—
67#14555785—41no—
68C2×Dic278C2311yescharacteristic
69C2×Dic278C2311yescharacteristic
70C2×D556—21no—
71C2×D556C2311yescharacteristic
72C2×D556C2311yescharacteristic
73C2×C22⋊D139—21no—
74C2×C22⋊D139—21no—
75C2×C22⋊D139—21no—
76C2×C22⋊D139D411yescharacteristic
77D4×D139—22no—
78D4×D139—41no—
79C2×C4×D139—21no—
80C2×C4×D139—21no—
81C2×C4×D139—21no—
82C2×C4×D139—21no—
83C2×C4×D139—21no—
84C2×C4×D139C2311yescharacteristic
85Dic556—22no—
86#14555775—41no—
87#14555774—22no—
88#14555774—22no—
89#14555773—41no—
90#14555772—22no—
91D1112D414yescomplemented
92#14555770—22no—
93#14555768—22no—
94C2×C139⋊C8C2311yescharacteristic
95C2×C139⋊C8C2311yescharacteristic
96C4⋊Dic139—21no—
97C4⋊Dic139—21no—
98Dic139⋊C4—21no—
99Dic139⋊C4—21no—
100Dic139⋊C4D411yescharacteristic
101C22⋊Dic139—21no—
102C22⋊Dic139—21no—
103C22⋊Dic139D411yescharacteristic
104(C2×D139)⋊C4—21no—
105(C2×D139)⋊C4—21no—
106(C2×D139)⋊C4—21no—
107(C2×D139)⋊C4—21no—
108(C2×D139)⋊C4D411yescharacteristic
109C4×Dic139C2311yescharacteristic
110C2×Q8×C139C2311yescharacteristic
111C2×D4×C139—21no—
112C2×D4×C139C2311yescharacteristic
113C22×C4×C139C2311yescharacteristic
114SD16×C139—22no—
115C2×C8×C139—22no—
116C2×C8×C139C2311yescharacteristic
117C2×C8×C139C2311yescharacteristic
118C4⋊C4×C139—21no—
119C4⋊C4×C139C2311yescharacteristic
120C4⋊C4×C139C2311yescharacteristic
121C22⋊C4×C139—21no—
122C22⋊C4×C139—21no—
123C22×D139—21no—
124C22×D139—21no—
125C22×D139—42no—
126C22×D139—42no—
127C22×D139—81no—
128C22×D139C2×D411yescharacteristic
129Dic278—21no—
130Dic278—21no—
131Dic278—21no—
132Dic278—21no—
133D556—21no—
134D556—21no—
135D556—42no—
136D556C2×D412yes—
137D556C2×D412yes—
138C22⋊D139—41no—
139C22⋊D139—42no—
140C22⋊D139—42no—
141C22⋊D139—42no—
142C2×Dic139—21no—
143C2×Dic139—21no—
144C2×Dic139C2×D411yescharacteristic
145C2×Dic139C2×D411yescharacteristic
146C2×Dic139C2×D411yescharacteristic
147C4×D139—21no—
148C4×D139—21no—
149C4×D139—42no—
150C4×D139—42no—
151C4×D139—42no—
152C4×D139—42no—
153C4×D139—81no—
154C139⋊C8—21no—
155C139⋊C8—21no—
156C23×C139—21no—
157C23×C139C2×D411yescharacteristic
158Q8×C139—21no—
159Q8×C139—21no—
160D4×C139—21no—
161D4×C139—21no—
162D4×C139—81no—
163C2×C4×C139—21no—
164C2×C4×C139—21no—
165C2×C4×C139—21no—
166C2×C4×C139—21no—
167C2×C4×C139—21no—
168C2×C4×C139C2×D411yescharacteristic
169C2×C4×C139C2411yescharacteristic
170C8×C139—21no—
171C8×C139C2×D412yes—
172C22×C139—21no—
173C22×C139—21no—
174C22×C139—21no—
175C22×C139—41no—
176C22×C139—41no—
177C22×C139—41no—
178C22×C139C22×D411yescharacteristic
179C2×D139—21no—
180C2×D139—22no—
181C2×D139—22no—
182C2×D139—21no—
183C2×D139—21no—
184C2×D139—41no—
185C2×D139—41no—
186C2×D139—41no—
187C2×D139—41no—
188C2×D139—84no—
189C4×C139—21no—
190C4×C139—41no—
191C4×C139—41no—
192C4×C139—41no—
193C4×C139C22×D411yescharacteristic
194C4×C139H(5,𝔽2)11yescharacteristic
195Dic139—21no—
196Dic139—21no—
197Dic139—21no—
198Dic139—41no—
199Dic139—41no—
200C2×C139—41no—
201C2×C139—81no—
202C2×C139D4211yescharacteristic
203C2×C139#57511yescharacteristic
204C2×C139#57711yescharacteristic
205D139—41no—
206D139—42no—
207D139—42no—
208C139#326711yescharacteristic, complemented, Sylow
209#3267—1391nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1391no—
211C2×#135—1391no—
212#547—1391no—
213#546—1391no—
214#497—1391no—
215#470—1391no—
216#465D13911yescharacteristic, complemented
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×#41—1391no—
228C22×D4—1391no—
229AGL(1,ℤ/8)—2784no—
230#135—2784no—
231C2×Q16—1391no—
232C2×Q16—1391no—
233C2×SD16—1391no—
234C2×SD16—1391no—
235C2×SD16—1391no—
236C2×SD16—1391no—
237C2×SD16C2×D13911yescharacteristic
238C2×D8—1391no—
239C2×D8—1391no—
240C2×M16—1391no—
241C22×C8C2×D13911yescharacteristic, complemented
242C4⋊Q8—1391no—
243#124—1391no—
244#123—2781no—
245#123—2781no—
246#123—2781no—
247C22⋊Q8—1391no—
248C22⋊Q8—1391no—
249C22⋊Q8C2×D13911yescharacteristic, complemented
250#121—2781no—
251#121—2781no—
252#121C2×D13911yescharacteristic
253#120—2781no—
254C4×D4—1391no—
255C4×D4—1391no—
256C4×D4—1391no—
257C2×C4⋊C4—2781no—
258C2×C22⋊C4—2781no—
259#106C2×D13911yescharacteristic, complemented
260C4⋊C8—1391no—
261#103—1391no—
262#103—1391no—
263#103C2×D13911yescharacteristic, complemented
264#102—1391no—
265#102—1391no—
266#102C2×D13911yescharacteristic
267C22⋊C8—1391no—
268C22⋊C8—1391no—
269#97—1391no—
270C24—2781no—
271#41—2782no—
272#41—2782no—
273#41—2782no—
274#41—5561no—
275#41—5561no—
276#41—5561no—
277C2×Q8—1391no—
278C2×Q8—1391no—
279C2×Q8C22×D13911yescharacteristic
280C2×D4—21no—
281C2×D4—2781no—
282C2×D4—2782no—
283C2×D4—2781no—
284C2×D4—2781no—
285C2×D4—2781no—
286C2×D4—5561no—
287C2×D4—1391no—
288C2×D4—1391no—
289C2×D4—1391no—
290C2×D4C22×D13911yescharacteristic
291C22×C4—2781no—
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—2782no—
302SD16—2782no—
303SD16—2782no—
304SD16—5561no—
305D8—2782no—
306D8—1394no—
307M16—2782no—
308C2×C8—22no—
309C2×C8—1391no—
310C2×C8—1391no—
311C2×C8C22×D13911yescharacteristic
312C2×C8C22×D13911yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—2781no—
315C4⋊C4—2781no—
316C4⋊C4—2781no—
317C4⋊C4—2781no—
318C4⋊C4—1391no—
319C4⋊C4C22×D13911yescharacteristic
320C4⋊C4C22×D13911yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
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—2781no—
334C23—2781no—
335C23—5562no—
336C23—5562no—
337C23—11121no—
338C23—1391no—
339C23D4×D13911yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—2781no—
343Q8—2781no—
344Q8—2781no—
345Q8—2781no—
346D4—21no—
347D4—21no—
348D4—81no—
349D4—2781no—
350D4—2781no—
351D4—5562no—
352D4—5561no—
353D4—5562no—
354D4—5562no—
355D4—5562no—
356D4—1392no—
357D4—1392no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—2781no—
364C2×C4—2781no—
365C2×C4—2781no—
366C2×C4—2781no—
367C2×C4—5562no—
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—
380C8D4×D13912yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—2781no—
388C22—2782no—
389C22—2782no—
390C22—2781no—
391C22—2781no—
392C22—5561no—
393C22—5561no—
394C22—5561no—
395C22—5561no—
396C22—11124no—
397C22C2×D4×D13911yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—2781no—
403C4—2781no—
404C4—2781no—
405C4—5561no—
406C4—5561no—
407C4C2×D4×D13911yescharacteristic
408C4#1798488811yescharacteristic
409C2—41no—
410C2—81no—
411C2—5561no—
412C2—5562no—
413C2—5562no—
414C2#2258158711yescharacteristic
415C2#2258175011yescharacteristic
416C2#2258175911yescharacteristic
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#465×C139C2
socleC22×C139C22×D4
hypercenter#465D139
hypocenterC139#3267
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#17984869C2semidirect
2. C2×#17984864C2semidirect
3. #22581605C2semidirect
4. #22581600C2semidirect
5. #22581344C2semidirect
6. #22581212C2semidirect
7. #22581167C2semidirect
8. #22581157C2semidirect
9. #22581118C2semidirect
10. #22581113C2semidirect
11. #22581105C2semidirect
12. #22581096C2semidirect
13. #22581064C2semidirect
14. #22581024C2semidirect
15. #465×C139C2semidirect
18. C2×#14555785C22semidirect
22. C2×Dic556C22semidirect
23. C2×#14555775C22semidirect
25. C2×#14555774C22semidirect
31. #17984840C22semidirect
42. #17984804C22semidirect
46. Dic139⋊C8C22semidirect
47. #17984765C22semidirect
48. #17984765C22semidirect
50. #17984762C22semidirect
51. C22⋊(C139⋊C8)C22semidirect
53. #17984752C22semidirect
55. C22×C8×C139C22semidirect
56. C22⋊Q8×C139C22semidirect
58. #106×C139C22semidirect
59. #103×C139C22semidirect
91. D1112D4semidirect
208. C139#3267semidirect
216. #465D139semidirect
241. C22×C8C2×D139semidirect
249. C22⋊Q8C2×D139semidirect
259. #106C2×D139semidirect
263. #103C2×D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848692
C2×#179848642
#225816052
#225816002
#225813442
#225812122
#225811672
#225811572
#225811182
#225811132
#225811052
#225810962
#225810642
#225810242
#465×C1392
#3267139