Finite Group Databasebeta

← #31816008

Subgroups — #31816008

Order 17792.

500 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984863C211yesmaximal, characteristic, complemented
2C2×M16×D139C211yesmaximal, characteristic, complemented
3#22581466C211yesmaximal, characteristic, complemented
4#22581463C211yesmaximal, characteristic, complemented
5#22581210C211yesmaximal, characteristic, complemented
6#22581209C211yesmaximal, characteristic, complemented
7#22581205C211yesmaximal, characteristic, complemented
8#22581204C211yesmaximal, characteristic, complemented
9#22581200C211yesmaximal, characteristic, complemented
10#22581199C211yesmaximal, characteristic, complemented
11#22581196C211yesmaximal, characteristic, complemented
12#22580990C211yesmaximal, characteristic, complemented
13#22580948C211yesmaximal, characteristic, complemented
14#22580946C211yesmaximal, characteristic, complemented
15#468×C139C211yesmaximal, characteristic, complemented
16C2×#14555786C2211yescharacteristic
17C2×#14555784C2211yescharacteristic
18C22×C4×D139C2211yescharacteristic
19#17984863—24no—
20C2×#14555775C2211yescharacteristic, complemented
21C2×#14555774C2211yescharacteristic, complemented
22C2×#14555770C2211yescharacteristic, complemented
23C2×#14555769C2211yescharacteristic
24C2×#14555768C2211yescharacteristic
25C2×#14555768C2211yescharacteristic
26M16×D139—24no—
27C2×C8×D139C2211yescharacteristic
28C2×C8×D139C2211yescharacteristic
29#17984827—21no—
30#17984824—21no—
31#17984822C2211yescharacteristic
32#17984821C2211yescharacteristic
33#17984818C2211yescharacteristic
34#17984817C2211yescharacteristic
35#17984816C2211yescharacteristic
36#17984813C2211yescharacteristic
37#17984812—21no—
38#17984810—21no—
39D4×Dic139—21no—
40#17984804—21no—
41(C22⋊D139)⋊C4—21no—
42(C22⋊D139)⋊C4—21no—
43#17984800C2211yescharacteristic
44#17984800C2211yescharacteristic
45#17984776C2211yescharacteristic, complemented
46#17984773C2211yescharacteristic, complemented
47#17984773C2211yescharacteristic, complemented
48#17984767C2211yescharacteristic, complemented
49#17984766C2211yescharacteristic, complemented
50#17984765C2211yescharacteristic, complemented
51#17984764C2211yescharacteristic, complemented
52#17984763C2211yescharacteristic, complemented
53#17984762C2211yescharacteristic, complemented
54C2×SD16×C139C2211yescharacteristic, complemented
55C2×M16×C139C2211yescharacteristic
56C22⋊Q8×C139C2211yescharacteristic
57#121×C139C2211yescharacteristic
58#107×C139C2211yescharacteristic, complemented
59#103×C139C2211yescharacteristic, complemented
60#102×C139C2211yescharacteristic, complemented
61C23×D139D411yescharacteristic
62#14555786—22no—
63#14555786—22no—
64#14555784—22no—
65#14555784—22no—
66C2×Dic278C2311yescharacteristic
67C2×D556—21no—
68C2×D556—21no—
69C2×D556C2311yescharacteristic
70C2×C22⋊D139—21no—
71C2×C22⋊D139—21no—
72C2×C22⋊D139—21no—
73C22×Dic139—21no—
74C22×Dic139D411yescharacteristic
75C2×C4×D139—22no—
76C2×C4×D139—22no—
77C2×C4×D139—21no—
78C2×C4×D139D411yescharacteristic
79C2×C4×D139D411yescharacteristic
80C2×C4×D139C2311yescharacteristic
81C2×C4×D139C2311yescharacteristic
82#14555775—22no—
83#14555774—22no—
84#14555770—22no—
85#14555769—22no—
86#14555768—22no—
87#14555768—22no—
88C2×C139⋊C8C2311yescharacteristic
89C2×C139⋊C8C2311yescharacteristic
90C8×D139—22no—
91C8×D139D414yescomplemented
92C4⋊Dic139—21no—
93C4⋊Dic139C2311yescharacteristic
94C4⋊Dic139C2311yescharacteristic
95Dic139⋊C4—21no—
96Dic139⋊C4—21no—
97C22⋊Dic139—21no—
98C22⋊Dic139—21no—
99(C2×D139)⋊C4—21no—
100(C2×D139)⋊C4—21no—
101(C2×D139)⋊C4—21no—
102(C2×D139)⋊C4—21no—
103(C2×D139)⋊C4—21no—
104(C2×D139)⋊C4—21no—
105C4×Dic139—21no—
106C4×Dic139—21no—
107C2×Q8×C139C2311yescharacteristic
108C2×D4×C139—21no—
109C2×D4×C139C2311yescharacteristic
110C22×C4×C139C2311yescharacteristic
111SD16×C139—22no—
112M16×C139—22no—
113C2×C8×C139C2311yescharacteristic
114C2×C8×C139C2311yescharacteristic
115C4⋊C4×C139—21no—
116C4⋊C4×C139C2311yescharacteristic
117C4⋊C4×C139C2311yescharacteristic
118C22⋊C4×C139—21no—
119C22⋊C4×C139—21no—
120C22×D139—22no—
121C22×D139—21no—
122C22×D139—41no—
123C22×D139—41no—
124C22×D139C2×D411yescharacteristic
125C22×D139C2×D411yescharacteristic
126Dic278—21no—
127Dic278—21no—
128D556—21no—
129D556—21no—
130D556—42no—
131D556—42no—
132C22⋊D139—42no—
133C22⋊D139—42no—
134C22⋊D139—42no—
135C2×Dic139—22no—
136C2×Dic139—21no—
137C2×Dic139—21no—
138C2×Dic139—21no—
139C2×Dic139—42no—
140C2×Dic139C2×D411yescharacteristic
141C2×Dic139C2×D411yescharacteristic
142C4×D139—22no—
143C4×D139—21no—
144C4×D139—21no—
145C4×D139—22no—
146C4×D139—42no—
147C4×D139C2×D412yes—
148C4×D139#4112yes—
149C139⋊C8—21no—
150C139⋊C8C2×D412yes—
151C23×C139—21no—
152C23×C139C2×D411yescharacteristic
153Q8×C139—21no—
154Q8×C139—21no—
155D4×C139—21no—
156D4×C139—21no—
157D4×C139—42no—
158C2×C4×C139—21no—
159C2×C4×C139—21no—
160C2×C4×C139—21no—
161C2×C4×C139—21no—
162C2×C4×C139—21no—
163C2×C4×C139C2×D411yescharacteristic
164C2×C4×C139C2411yescharacteristic
165C8×C139—21no—
166C8×C139C2×D412yes—
167C22×C139—21no—
168C22×C139—21no—
169C22×C139—21no—
170C22×C139—41no—
171C22×C139—41no—
172C22×C139—41no—
173C22×C139C22×D411yescharacteristic
174C2×D139—21no—
175C2×D139—21no—
176C2×D139—21no—
177C2×D139—21no—
178C2×D139—21no—
179C2×D139—44no—
180C2×D139—41no—
181C2×D139—41no—
182C2×D139—41no—
183C2×D139#12112yes—
184C4×C139—21no—
185C4×C139—41no—
186C4×C139—41no—
187C4×C139—41no—
188C4×C139C22×D411yescharacteristic
189C4×C139C2×#4111yescharacteristic
190Dic139—21no—
191Dic139—41no—
192Dic139—41no—
193Dic139—41no—
194Dic139#12112yes—
195C2×C139—41no—
196C2×C139—81no—
197C2×C139C2×#12111yescharacteristic
198C2×C139#57511yescharacteristic
199C2×C139#57511yescharacteristic
200D139—22no—
201D139—41no—
202D139—81no—
203C139#305411yescharacteristic, complemented, Sylow
204#3054—1391nomaximal, Sylow
205C2×#135—1391no—
206C22×M16—1391no—
207#525—1391no—
208#525—1391no—
209#470—1391no—
210#470—1391no—
211#469—1391no—
212#469—1391no—
213#468—1391no—
214#468—1391no—
215#468—1391no—
216#468D13911yescharacteristic, complemented
217#427—1391no—
218#416—1391no—
219#416—1391no—
220C2×#41—1391no—
221C2×#41—1391no—
222C23×C4—1391no—
223#135—2784no—
224C2×Q16—1391no—
225C2×SD16—1391no—
226C2×SD16C2×D13911yescharacteristic, complemented
227C2×D8—1391no—
228C2×M16—2784no—
229C2×M16—1391no—
230C2×M16—1391no—
231C2×M16—1391no—
232C2×M16C2×D13911yescharacteristic
233C22×C8—1391no—
234C22×C8—1391no—
235#123—2781no—
236#123—2781no—
237C22⋊Q8—1391no—
238C22⋊Q8—1391no—
239C22⋊Q8—1391no—
240C22⋊Q8C2×D13911yescharacteristic
241#121—1391no—
242#121—1391no—
243#121—1391no—
244#121C2×D13911yescharacteristic
245#120—2781no—
246#120—2781no—
247C4×D4—2781no—
248C4×D4—2781no—
249C4×D4—2781no—
250C4×D4—2781no—
251#117—1391no—
252#117—1391no—
253#107—1391no—
254#107C2×D13911yescharacteristic, complemented
255#106—1391no—
256#106—1391no—
257#103—1391no—
258#103—1391no—
259#103—1391no—
260#103C2×D13911yescharacteristic, complemented
261#102—1391no—
262#102—1391no—
263#102—1391no—
264#102C2×D13911yescharacteristic, complemented
265C24—1391no—
266#41—2782no—
267#41—2782no—
268#41—2782no—
269#41—2782no—
270C2×Q8—1391no—
271C2×Q8C22×D13911yescharacteristic
272C2×D4—21no—
273C2×D4—2781no—
274C2×D4—2781no—
275C2×D4—2781no—
276C2×D4—2781no—
277C2×D4—2781no—
278C2×D4—1391no—
279C2×D4C22×D13911yescharacteristic
280C22×C4—2782no—
281C22×C4—2782no—
282C22×C4—2781no—
283C22×C4—2781no—
284C22×C4—1391no—
285C22×C4—1391no—
286C22×C4—1391no—
287C22×C4—1391no—
288C22×C4—1391no—
289C22×C4C22×D13911yescharacteristic
290Q16—2782no—
291SD16—22no—
292SD16—2782no—
293D8—2782no—
294M16—22no—
295M16—2782no—
296M16—2782no—
297M16—2782no—
298C2×C8—2782no—
299C2×C8—1391no—
300C2×C8—1391no—
301C2×C8—1394no—
302C2×C8C22×D13911yescharacteristic
303C2×C8C22×D13911yescharacteristic
304C4⋊C4—21no—
305C4⋊C4—2781no—
306C4⋊C4—2781no—
307C4⋊C4—2781no—
308C4⋊C4—1391no—
309C4⋊C4—1391no—
310C4⋊C4C22×D13911yescharacteristic
311C4⋊C4C22×D13911yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—2781no—
315C22⋊C4—2781no—
316C22⋊C4—2781no—
317C22⋊C4—2781no—
318C22⋊C4—2781no—
319C22⋊C4—2781no—
320C22⋊C4—2781no—
321C22⋊C4—2781no—
322C42—2781no—
323C42—2781no—
324C23—21no—
325C23—2782no—
326C23—2781no—
327C23—5561no—
328C23—5561no—
329C23—1391no—
330C23—1391no—
331C23D4×D13911yescharacteristic
332Q8—21no—
333Q8—21no—
334Q8—2781no—
335Q8—2781no—
336D4—21no—
337D4—21no—
338D4—42no—
339D4—2781no—
340D4—2781no—
341D4—5562no—
342D4—5562no—
343D4—5562no—
344D4—5562no—
345D4—5562no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—2782no—
352C2×C4—2782no—
353C2×C4—2782no—
354C2×C4—2781no—
355C2×C4—2781no—
356C2×C4—2781no—
357C2×C4—2781no—
358C2×C4—2781no—
359C2×C4—5562no—
360C2×C4—5562no—
361C2×C4—1391no—
362C2×C4—1391no—
363C2×C4—1392no—
364C2×C4—1392no—
365C2×C4D4×D13911yescharacteristic
366C2×C4C23×D13911yescharacteristic
367C8—21no—
368C8—2781no—
369C8—1392no—
370C8D4×D13912yes—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—2781no—
378C22—2781no—
379C22—2781no—
380C22—2781no—
381C22—2781no—
382C22—5564no—
383C22—5561no—
384C22—5561no—
385C22—5561no—
386C22—1392no—
387C22C2×D4×D13911yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—2781no—
393C4—5561no—
394C4—5561no—
395C4—5561no—
396C4—1392no—
397C4C2×D4×D13911yescharacteristic
398C4#41×D13911yescharacteristic
399C2—41no—
400C2—81no—
401C2—2782no—
402C2—5561no—
403C2—11121no—
404C2#121×D13911yescharacteristic
405C2#2258174911yescharacteristic
406C2#2258175211yescharacteristic
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#468×C139C2
socleC22×C139C22×D4
hypercenter#468D139
hypocenterC139#3054
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#17984863C2semidirect
2. C2×M16×D139C2semidirect
3. #22581466C2semidirect
4. #22581463C2semidirect
5. #22581210C2semidirect
6. #22581209C2semidirect
7. #22581205C2semidirect
8. #22581204C2semidirect
9. #22581200C2semidirect
10. #22581199C2semidirect
11. #22581196C2semidirect
12. #22580990C2semidirect
13. #22580948C2semidirect
14. #22580946C2semidirect
15. #468×C139C2semidirect
20. C2×#14555775C22semidirect
21. C2×#14555774C22semidirect
22. C2×#14555770C22semidirect
45. #17984776C22semidirect
46. #17984773C22semidirect
47. #17984773C22semidirect
48. #17984767C22semidirect
49. #17984766C22semidirect
50. #17984765C22semidirect
51. #17984764C22semidirect
52. #17984763C22semidirect
53. #17984762C22semidirect
54. C2×SD16×C139C22semidirect
58. #107×C139C22semidirect
59. #103×C139C22semidirect
60. #102×C139C22semidirect
91. C8×D139D4semidirect
203. C139#3054semidirect
216. #468D139semidirect
226. C2×SD16C2×D139semidirect
254. #107C2×D139semidirect
260. #103C2×D139semidirect
264. #102C2×D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848632
C2×M16×D1392
#225814662
#225814632
#225812102
#225812092
#225812052
#225812042
#225812002
#225811992
#225811962
#225809902
#225809482
#225809462
#468×C1392
#3054139