Finite Group Databasebeta

← #31817935

Subgroups — #31817935

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×#17984863C211yesmaximal, characteristic, complemented
3#22581606C211yesmaximal, characteristic, complemented
4#22581598C211yesmaximal, characteristic, complemented
5#22581345C211yesmaximal, characteristic, complemented
6#22581210C211yesmaximal, characteristic, complemented
7#22581185C211yesmaximal, characteristic, complemented
8#22581169C211yesmaximal, characteristic, complemented
9#22581118C211yesmaximal, characteristic, complemented
10#22581112C211yesmaximal, characteristic, complemented
11#22581102C211yesmaximal, characteristic, complemented
12#22581101C211yesmaximal, characteristic, complemented
13#22581063C211yesmaximal, characteristic, complemented
14#22581023C211yesmaximal, characteristic, complemented
15#461×C139C211yesmaximal, characteristic, complemented
16C2×#14555786C2211yescharacteristic
17C2×#14555786C2211yescharacteristic, complemented
18C2×#14555784C2211yescharacteristic
19C2×D4×D139C2211yescharacteristic
20#17984869—24no—
21#17984863—24no—
22C2×#14555775C2211yescharacteristic, complemented
23C2×#14555775C2211yescharacteristic, complemented
24C2×#14555774C2211yescharacteristic, complemented
25C2×#14555772C2211yescharacteristic
26C2×#14555772C2211yescharacteristic
27C2×D1112C2211yescharacteristic
28C2×#14555770C2211yescharacteristic
29C2×#14555768C2211yescharacteristic
30C2×C8×D139C2211yescharacteristic, complemented
31#17984830C2211yescharacteristic
32#17984827—21no—
33#17984827—21no—
34#17984826—21no—
35#17984822C2211yescharacteristic
36#17984818C2211yescharacteristic
37#17984818C2211yescharacteristic, complemented
38#17984817—21no—
39#17984817C2211yescharacteristic
40#17984816—21no—
41#17984812—21no—
42C4×D556C2211yescharacteristic
43C4×D556C2211yescharacteristic, complemented
44C4⋊C4×D139—21no—
45C2×(C2×D139)⋊C4—21no—
46#17984773C2211yescharacteristic, complemented
47#17984767C2211yescharacteristic, complemented
48#17984764C2211yescharacteristic
49#17984763C2211yescharacteristic
50#17984762C2211yescharacteristic, complemented
51(C2×D139)⋊C8C2211yescharacteristic
52(C2×D139)⋊C8C2211yescharacteristic, complemented
53#17984753C2211yescharacteristic, complemented
54C2×SD16×C139C2211yescharacteristic
55C2×SD16×C139C2211yescharacteristic
56C4⋊Q8×C139C2211yescharacteristic, complemented
57C4×D4×C139C2211yescharacteristic
58C4⋊C8×C139C2211yescharacteristic, complemented
59#103×C139C2211yescharacteristic, complemented
60#103×C139C2211yescharacteristic, complemented
61C23×D139—21no—
62#14555786—22no—
63#14555786—22no—
64#14555786—41no—
65#14555786—41no—
66#14555784—22no—
67#14555784—41no—
68C2×Dic278C2311yescharacteristic
69C2×D556—21no—
70C2×D556—21no—
71C2×D556—21no—
72C2×D556D411yescharacteristic
73C2×D556C2311yescharacteristic
74C2×D556C2311yescharacteristic
75C2×C22⋊D139—21no—
76C2×C22⋊D139—21no—
77D4×D139—22no—
78D4×D139—41no—
79C22×Dic139—21no—
80C2×C4×D139—21no—
81C2×C4×D139—21no—
82C2×C4×D139—21no—
83C2×C4×D139C2311yescharacteristic
84C2×C4×D139C2311yescharacteristic
85#14555775—22no—
86#14555775—41no—
87#14555774—22no—
88#14555772—22no—
89#14555772—41no—
90D1112—22no—
91#14555770D414yescomplemented
92#14555768—22no—
93C2×C139⋊C8C2311yescharacteristic
94C2×C139⋊C8C2311yescharacteristic
95C8×D139—22no—
96C4⋊Dic139—21no—
97C4⋊Dic139D411yescharacteristic
98C4⋊Dic139C2311yescharacteristic
99Dic139⋊C4—21no—
100Dic139⋊C4—21no—
101(C2×D139)⋊C4—21no—
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)⋊C4—21no—
108(C2×D139)⋊C4D411yescharacteristic
109(C2×D139)⋊C4D411yescharacteristic
110C2×Q8×C139C2311yescharacteristic
111C2×Q8×C139C2311yescharacteristic
112C2×D4×C139C2311yescharacteristic
113C22×C4×C139—21no—
114SD16×C139—22no—
115SD16×C139—22no—
116C2×C8×C139C2311yescharacteristic
117C2×C8×C139C2311yescharacteristic
118C4⋊C4×C139—21no—
119C4⋊C4×C139—21no—
120C4⋊C4×C139C2311yescharacteristic
121C22⋊C4×C139—21no—
122C42×C139C2311yescharacteristic
123C22×D139—21no—
124C22×D139—21no—
125C22×D139—42no—
126C22×D139—42no—
127C22×D139—81no—
128C22×D139C2×D411yescharacteristic
129C22×D139C2×D411yescharacteristic
130Dic278—21no—
131Dic278—21no—
132D556—21no—
133D556—21no—
134D556—21no—
135D556—41no—
136D556—42no—
137D556—42no—
138D556—81no—
139D556C2×D412yes—
140C22⋊D139—42no—
141C22⋊D139—42no—
142C2×Dic139—21no—
143C2×Dic139—21no—
144C2×Dic139—42no—
145C2×Dic139C2×D411yescharacteristic
146C2×Dic139C2×D411yescharacteristic
147C4×D139—21no—
148C4×D139—21no—
149C4×D139—21no—
150C4×D139—21no—
151C4×D139—42no—
152C4×D139—42no—
153C4×D139—42no—
154C139⋊C8—21no—
155C139⋊C8C2×D412yes—
156C23×C139—21no—
157Q8×C139—21no—
158Q8×C139—21no—
159Q8×C139—21no—
160Q8×C139—21no—
161D4×C139—21no—
162D4×C139C2×D412yes—
163C2×C4×C139—21no—
164C2×C4×C139—21no—
165C2×C4×C139—21no—
166C2×C4×C139—81no—
167C2×C4×C139C2×D411yescharacteristic
168C2×C4×C139C2×D411yescharacteristic
169C2×C4×C139C2411yescharacteristic
170C8×C139—21no—
171C8×C139—21no—
172C22×C139—22no—
173C22×C139—41no—
174C22×C139—41no—
175C22×C139C22×D411yescharacteristic
176C2×D139—21no—
177C2×D139—21no—
178C2×D139—22no—
179C2×D139—21no—
180C2×D139—21no—
181C2×D139—21no—
182C2×D139—21no—
183C2×D139—41no—
184C2×D139—41no—
185C2×D139—41no—
186C2×D139—41no—
187C2×D139—41no—
188C2×D139—84no—
189C4×C139—21no—
190C4×C139—21no—
191C4×C139—41no—
192C4×C139—41no—
193C4×C139—41no—
194C4×C139C22×D411yescharacteristic
195C4×C139H(5,𝔽2)11yescharacteristic
196Dic139—21no—
197Dic139—21no—
198Dic139—41no—
199Dic139—41no—
200C2×C139—42no—
201C2×C139D4211yescharacteristic
202C2×C139#57511yescharacteristic
203C2×C139#57711yescharacteristic
204D139—41no—
205D139—41no—
206D139—42no—
207D139—81no—
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#465—1391no—
217#463—1391no—
218#461D13911yescharacteristic, complemented
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×SD16C2×D13911yescharacteristic
237C2×SD16C2×D13911yescharacteristic
238C2×D8—1391no—
239C2×D8—1391no—
240C2×M16—1391no—
241C22×C8—1391no—
242C4⋊Q8C2×D13911yescharacteristic, complemented
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×D4C2×D13911yescharacteristic
257C2×C4⋊C4—2781no—
258C2×C22⋊C4—2781no—
259#106—1391no—
260C4⋊C8C2×D13911yescharacteristic, complemented
261#103—1391no—
262#103C2×D13911yescharacteristic, complemented
263#103C2×D13911yescharacteristic, complemented
264#102—1391no—
265#102—1391no—
266#102—1391no—
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×Q8C22×D13911yescharacteristic
279C2×Q8C22×D13911yescharacteristic
280C2×D4—2781no—
281C2×D4—2781no—
282C2×D4—2781no—
283C2×D4—2782no—
284C2×D4—2781no—
285C2×D4—2781no—
286C2×D4—5561no—
287C2×D4—1391no—
288C2×D4—1391no—
289C2×D4—1391no—
290C2×D4C22×D13911yescharacteristic
291C22×C4—21no—
292C22×C4—2781no—
293C22×C4—2781no—
294C22×C4—2781no—
295C22×C4—2781no—
296C22×C4—1391no—
297C22×C4—1391no—
298Q16—2782no—
299Q16—5561no—
300SD16—22no—
301SD16—22no—
302SD16—2782no—
303SD16—2782no—
304SD16—5561no—
305D8—2782no—
306D8—1394no—
307M16—2782no—
308C2×C8—2782no—
309C2×C8—1391no—
310C2×C8—1391no—
311C2×C8C22×D13911yescharacteristic
312C2×C8C22×D13911yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—2781no—
316C4⋊C4—2781no—
317C4⋊C4—2781no—
318C4⋊C4—1391no—
319C4⋊C4—1391no—
320C4⋊C4C22×D13911yescharacteristic
321C22⋊C4—21no—
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—
331C42C22×D13911yescharacteristic
332C23—21no—
333C23—2781no—
334C23—2781no—
335C23—5562no—
336C23—5562no—
337C23—11121no—
338C23—1391no—
339C23—1391no—
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—2781no—
345Q8—2781no—
346D4—21no—
347D4—2781no—
348D4—2781no—
349D4—2781no—
350D4—5561no—
351D4—5562no—
352D4—5562no—
353D4—5562no—
354D4—5562no—
355D4—11121no—
356D4—1392no—
357D4D4×D13912yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—81no—
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—5562no—
372C2×C4—1391no—
373C2×C4—1391no—
374C2×C4D4×D13911yescharacteristic
375C2×C4D4×D13911yescharacteristic
376C2×C4C23×D13911yescharacteristic
377C8—21no—
378C8—21no—
379C8—2781no—
380C8—1392no—
381C22—22no—
382C22—41no—
383C22—41no—
384C22—2781no—
385C22—2781no—
386C22—2782no—
387C22—2781no—
388C22—2781no—
389C22—2781no—
390C22—2781no—
391C22—5561no—
392C22—5561no—
393C22—5561no—
394C22—5561no—
395C22—5561no—
396C22—11124no—
397C22C2×D4×D13911yescharacteristic
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—2781no—
404C4—2781no—
405C4—5561no—
406C4—5561no—
407C4C2×D4×D13911yescharacteristic
408C4#1798488811yescharacteristic
409C2—42no—
410C2—5561no—
411C2—5561no—
412C2—5562no—
413C2—11121no—
414C2#2258158711yescharacteristic
415C2#2258175211yescharacteristic
416C2#2258175911yescharacteristic
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#461×C139C2
socleC22×C139C22×D4
hypercenter#461D139
hypocenterC139#3267
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#17984869C2semidirect
2. C2×#17984863C2semidirect
3. #22581606C2semidirect
4. #22581598C2semidirect
5. #22581345C2semidirect
6. #22581210C2semidirect
7. #22581185C2semidirect
8. #22581169C2semidirect
9. #22581118C2semidirect
10. #22581112C2semidirect
11. #22581102C2semidirect
12. #22581101C2semidirect
13. #22581063C2semidirect
14. #22581023C2semidirect
15. #461×C139C2semidirect
17. C2×#14555786C22semidirect
22. C2×#14555775C22semidirect
23. C2×#14555775C22semidirect
24. C2×#14555774C22semidirect
30. C2×C8×D139C22semidirect
37. #17984818C22semidirect
43. C4×D556C22semidirect
46. #17984773C22semidirect
47. #17984767C22semidirect
50. #17984762C22semidirect
52. (C2×D139)⋊C8C22semidirect
53. #17984753C22semidirect
56. C4⋊Q8×C139C22semidirect
58. C4⋊C8×C139C22semidirect
59. #103×C139C22semidirect
60. #103×C139C22semidirect
91. #14555770D4semidirect
208. C139#3267semidirect
218. #461D139semidirect
242. C4⋊Q8C2×D139semidirect
260. C4⋊C8C2×D139semidirect
262. #103C2×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×#179848632
#225816062
#225815982
#225813452
#225812102
#225811852
#225811692
#225811182
#225811122
#225811022
#225811012
#225810632
#225810232
#461×C1392
#3267139