Finite Group Databasebeta

← #31817951

Subgroups — #31817951

Order 17792.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984870C211yesmaximal, characteristic, complemented
2C2×#17984863C211yesmaximal, characteristic, complemented
3#22581621C211yesmaximal, characteristic, complemented
4#22581594C211yesmaximal, characteristic, complemented
5#22581339C211yesmaximal, characteristic, complemented
6#22581190C211yesmaximal, characteristic, complemented
7#22581178C211yesmaximal, characteristic, complemented
8#22581147C211yesmaximal, characteristic, complemented
9#22581113C211yesmaximal, characteristic, complemented
10#22581101C211yesmaximal, characteristic, complemented
11#22581099C211yesmaximal, characteristic, complemented
12#22581086C211yesmaximal, characteristic, complemented
13#22581052C211yesmaximal, characteristic, complemented
14#22581022C211yesmaximal, characteristic, complemented
15#468×C139C211yesmaximal, characteristic, complemented
16C2×#14555786C2211yescharacteristic
17C2×#14555785C2211yescharacteristic
18C2×#14555784C2211yescharacteristic, complemented
19C22×D556C2211yescharacteristic
20#17984870—24no—
21#17984863—24no—
22C2×#14555775C2211yescharacteristic, complemented
23C2×#14555774C2211yescharacteristic
24C2×#14555774C2211yescharacteristic
25C2×#14555772C2211yescharacteristic, complemented
26C2×D1112C2211yescharacteristic
27C2×D1112C2211yescharacteristic
28C2×#14555770C2211yescharacteristic
29C2×#14555770C2211yescharacteristic
30C2×C8×D139C2211yescharacteristic, complemented
31#17984838—21no—
32#17984838C2211yescharacteristic, complemented
33#17984831C2211yescharacteristic
34#17984827—21no—
35#17984817—21no—
36#17984817C2211yescharacteristic
37#17984817C2211yescharacteristic, complemented
38#17984815—21no—
39#17984814—21no—
40#17984812—21no—
41#17984807C2211yescharacteristic, complemented
42#17984804C2211yescharacteristic
43C4×C22⋊D139C2211yescharacteristic
44(C22⋊D139)⋊C4—21no—
45C22⋊C4×D139—21no—
46Dic139⋊C8C2211yescharacteristic, complemented
47#17984765C2211yescharacteristic, complemented
48#17984762C2211yescharacteristic
49#17984762C2211yescharacteristic
50#17984762C2211yescharacteristic, complemented
51C22⋊(C139⋊C8)C2211yescharacteristic
52(C2×D139)⋊C8C2211yescharacteristic, complemented
53#17984752C2211yescharacteristic, 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×D139—21no—
62#14555786—22no—
63#14555786—41no—
64#14555785—22no—
65#14555785—41no—
66#14555784—22no—
67#14555784—41no—
68C2×Dic278C2311yescharacteristic
69C2×D556—22no—
70C2×D556—21no—
71C2×D556—21no—
72C2×D556—21no—
73C2×D556—21no—
74C2×D556—41no—
75C2×D556C2311yescharacteristic
76C2×D556C2311yescharacteristic
77C2×D556C2311yescharacteristic
78C2×C22⋊D139—21no—
79C2×C22⋊D139—21no—
80C2×C22⋊D139—21no—
81C2×C22⋊D139—21no—
82C2×C22⋊D139D411yescharacteristic
83C22×Dic139—21no—
84C2×C4×D139—21no—
85C2×C4×D139—21no—
86C2×C4×D139—21no—
87C2×C4×D139C2311yescharacteristic
88#14555775—22no—
89#14555774—22no—
90#14555774D414yescomplemented
91#14555772—41no—
92D1112—22no—
93D1112—22no—
94#14555770—22no—
95#14555770—41no—
96C2×C139⋊C8C2311yescharacteristic
97C2×C139⋊C8C2311yescharacteristic
98C8×D139—22no—
99Dic139⋊C4—21no—
100Dic139⋊C4D411yescharacteristic
101C22⋊Dic139D411yescharacteristic
102(C2×D139)⋊C4—21no—
103(C2×D139)⋊C4—21no—
104(C2×D139)⋊C4—21no—
105(C2×D139)⋊C4—21no—
106(C2×D139)⋊C4—21no—
107(C2×D139)⋊C4D411yescharacteristic
108C4×Dic139—21no—
109C4×Dic139C2311yescharacteristic
110C2×Q8×C139C2311yescharacteristic
111C2×D4×C139—21no—
112C2×D4×C139C2311yescharacteristic
113C22×C4×C139C2311yescharacteristic
114SD16×C139—22no—
115M16×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—21no—
126C22×D139—42no—
127C22×D139—42no—
128C22×D139—81no—
129C22×D139C2×D411yescharacteristic
130Dic278—21no—
131Dic278C2×D412yes—
132D556—21no—
133D556—21no—
134D556—21no—
135D556—21no—
136D556—21no—
137D556—42no—
138D556—42no—
139D556—81no—
140D556—81no—
141D556C2×D412yes—
142C22⋊D139—41no—
143C22⋊D139—42no—
144C22⋊D139—42no—
145C22⋊D139—42no—
146C22⋊D139—81no—
147C2×Dic139—21no—
148C2×Dic139—42no—
149C2×Dic139C2×D411yescharacteristic
150C2×Dic139C2×D411yescharacteristic
151C2×Dic139C2×D411yescharacteristic
152C4×D139—21no—
153C4×D139—21no—
154C4×D139—42no—
155C4×D139—42no—
156C4×D139—81no—
157C139⋊C8—21no—
158C139⋊C8—21no—
159C23×C139—21no—
160C23×C139C2×D411yescharacteristic
161Q8×C139—21no—
162Q8×C139—21no—
163D4×C139—21no—
164D4×C139—21no—
165D4×C139—81no—
166C2×C4×C139—21no—
167C2×C4×C139—21no—
168C2×C4×C139—21no—
169C2×C4×C139—21no—
170C2×C4×C139—21no—
171C2×C4×C139C2×D411yescharacteristic
172C2×C4×C139C2411yescharacteristic
173C8×C139—21no—
174C8×C139C2×D412yes—
175C22×C139—21no—
176C22×C139—21no—
177C22×C139—21no—
178C22×C139—41no—
179C22×C139—41no—
180C22×C139—41no—
181C22×C139C22×D411yescharacteristic
182C2×D139—21no—
183C2×D139—22no—
184C2×D139—21no—
185C2×D139—21no—
186C2×D139—41no—
187C2×D139—41no—
188C2×D139—41no—
189C2×D139—41no—
190C2×D139—41no—
191C2×D139—41no—
192C2×D139—41no—
193C2×D139—41no—
194C2×D139—84no—
195C4×C139—21no—
196C4×C139—41no—
197C4×C139—41no—
198C4×C139—41no—
199C4×C139C22×D411yescharacteristic
200C4×C139H(5,𝔽2)11yescharacteristic
201Dic139—21no—
202Dic139—22no—
203Dic139—21no—
204Dic139—21no—
205C2×C139—41no—
206C2×C139—81no—
207C2×C139D4211yescharacteristic
208C2×C139#57511yescharacteristic
209C2×C139#57611yescharacteristic
210D139—41no—
211D139—42no—
212D139—81no—
213D139—81no—
214C139#326811yescharacteristic, complemented, Sylow
215#3268—1391nomaximal, Sylow
216C2×AGL(1,ℤ/8)—1391no—
217C2×#135—1391no—
218#550—1391no—
219#546—1391no—
220#496—1391no—
221#468D13911yescharacteristic, complemented
222#466—1391no—
223#464—1391no—
224#459—1391no—
225#452—1391no—
226#449—1391no—
227#449—1391no—
228#447—1391no—
229#440—1391no—
230#435—1391no—
231C2×#41—1391no—
232C2×#41—1391no—
233C2×#41—1391no—
234C22×D4—1391no—
235AGL(1,ℤ/8)—2784no—
236#135—2784no—
237C2×Q16—1391no—
238C2×SD16—1391no—
239C2×SD16—1391no—
240C2×SD16—1391no—
241C2×SD16C2×D13911yescharacteristic, complemented
242C2×D8—1391no—
243C2×D8—1391no—
244C2×D8—1391no—
245C2×D8—1391no—
246C2×M16C2×D13911yescharacteristic
247C22×C8—1391no—
248#127—2781no—
249#127—1391no—
250#124—1391no—
251#123—2781no—
252C22⋊Q8C2×D13911yescharacteristic
253#121—2781no—
254#121—2781no—
255#121—2781no—
256#121—1391no—
257#121—1391no—
258#121C2×D13911yescharacteristic
259#120—2781no—
260C4×Q8—1391no—
261C4×D4—2781no—
262C4×D4—1391no—
263C4×D4—1391no—
264C2×C22⋊C4—2781no—
265#107C2×D13911yescharacteristic, complemented
266C4⋊C8—1391no—
267#103—1391no—
268#103C2×D13911yescharacteristic, complemented
269#102—1391no—
270#102—1391no—
271#102—1391no—
272#102C2×D13911yescharacteristic, complemented
273C22⋊C8—1391no—
274C22⋊C8—1391no—
275#97—1391no—
276C24—2781no—
277#41—2782no—
278#41—2782no—
279#41—2782no—
280#41—5561no—
281#41—5561no—
282#41—5561no—
283C2×Q8—1391no—
284C2×Q8C22×D13911yescharacteristic
285C2×D4—21no—
286C2×D4—2782no—
287C2×D4—2781no—
288C2×D4—2781no—
289C2×D4—2781no—
290C2×D4—2781no—
291C2×D4—2781no—
292C2×D4—2781no—
293C2×D4—2781no—
294C2×D4—2781no—
295C2×D4—5561no—
296C2×D4—1391no—
297C2×D4—1391no—
298C2×D4—1391no—
299C2×D4—1391no—
300C2×D4C22×D13911yescharacteristic
301C22×C4—2781no—
302C22×C4—2781no—
303C22×C4—2781no—
304C22×C4—2781no—
305C22×C4—1391no—
306C22×C4C22×D13911yescharacteristic
307Q16—2782no—
308SD16—22no—
309SD16—2782no—
310SD16—5561no—
311SD16—1394no—
312D8—2782no—
313D8—2782no—
314D8—2782no—
315D8—5561no—
316M16—22no—
317C2×C8—2782no—
318C2×C8—1391no—
319C2×C8—1391no—
320C2×C8C22×D13911yescharacteristic
321C2×C8C22×D13911yescharacteristic
322C4⋊C4—21no—
323C4⋊C4—2781no—
324C4⋊C4—1391no—
325C4⋊C4C22×D13911yescharacteristic
326C4⋊C4C22×D13911yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—2781no—
330C22⋊C4—2781no—
331C22⋊C4—2781no—
332C22⋊C4—2781no—
333C22⋊C4—2781no—
334C22⋊C4—1391no—
335C22⋊C4—1391no—
336C42—2781no—
337C42—1391no—
338C23—21no—
339C23—2781no—
340C23—2781no—
341C23—2781no—
342C23—5562no—
343C23—5562no—
344C23—11121no—
345C23—1391no—
346C23D4×D13911yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—2781no—
350Q8—1392no—
351D4—21no—
352D4—21no—
353D4—81no—
354D4—2781no—
355D4—2781no—
356D4—2781no—
357D4—2781no—
358D4—2781no—
359D4—5562no—
360D4—5562no—
361D4—5561no—
362D4—5562no—
363D4—5562no—
364D4—5562no—
365D4—11121no—
366D4—11121no—
367D4—11121no—
368D4—1392no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—2781no—
375C2×C4—2781no—
376C2×C4—2781no—
377C2×C4—5562no—
378C2×C4—5562no—
379C2×C4—5562no—
380C2×C4—11121no—
381C2×C4—1391no—
382C2×C4—1391no—
383C2×C4—1391no—
384C2×C4D4×D13911yescharacteristic
385C2×C4C23×D13911yescharacteristic
386C8—21no—
387C8—2781no—
388C8—2781no—
389C8D4×D13912yes—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—2781no—
397C22—2782no—
398C22—2781no—
399C22—2781no—
400C22—5561no—
401C22—5561no—
402C22—5561no—
403C22—5561no—
404C22—5561no—
405C22—5561no—
406C22—5561no—
407C22—5561no—
408C22—11124no—
409C22C2×D4×D13911yescharacteristic
410C4—21no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—2781no—
415C4—2782no—
416C4—2781no—
417C4—2781no—
418C4C2×D4×D13911yescharacteristic
419C4#1798488811yescharacteristic
420C2—41no—
421C2—81no—
422C2—5561no—
423C2—5562no—
424C2—11121no—
425C2—11121no—
426C2#2258158711yescharacteristic
427C2#2258175211yescharacteristic
428C2#2258175711yescharacteristic
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#468×C139C2
socleC22×C139C22×D4
hypercenter#468D139
hypocenterC139#3268
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×#17984870C2semidirect
2. C2×#17984863C2semidirect
3. #22581621C2semidirect
4. #22581594C2semidirect
5. #22581339C2semidirect
6. #22581190C2semidirect
7. #22581178C2semidirect
8. #22581147C2semidirect
9. #22581113C2semidirect
10. #22581101C2semidirect
11. #22581099C2semidirect
12. #22581086C2semidirect
13. #22581052C2semidirect
14. #22581022C2semidirect
15. #468×C139C2semidirect
18. C2×#14555784C22semidirect
22. C2×#14555775C22semidirect
25. C2×#14555772C22semidirect
30. C2×C8×D139C22semidirect
32. #17984838C22semidirect
37. #17984817C22semidirect
41. #17984807C22semidirect
46. Dic139⋊C8C22semidirect
47. #17984765C22semidirect
50. #17984762C22semidirect
52. (C2×D139)⋊C8C22semidirect
53. #17984752C22semidirect
54. C2×SD16×C139C22semidirect
58. #107×C139C22semidirect
59. #103×C139C22semidirect
60. #102×C139C22semidirect
90. #14555774D4semidirect
214. C139#3268semidirect
221. #468D139semidirect
241. C2×SD16C2×D139semidirect
265. #107C2×D139semidirect
268. #103C2×D139semidirect
272. #102C2×D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848702
C2×#179848632
#225816212
#225815942
#225813392
#225811902
#225811782
#225811472
#225811132
#225811012
#225810992
#225810862
#225810522
#225810222
#468×C1392
#3268139