Finite Group Databasebeta

← #31817930

Subgroups — #31817930

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×#17984867C211yesmaximal, characteristic, complemented
2C2×#17984862C211yesmaximal, characteristic, complemented
3#22581599C211yesmaximal, characteristic, complemented
4#22581345C211yesmaximal, characteristic, complemented
5#22581209C211yesmaximal, characteristic, complemented
6#22581184C211yesmaximal, characteristic, complemented
7#22581168C211yesmaximal, characteristic, complemented
8#22581155C211yesmaximal, characteristic, complemented
9#22581115C211yesmaximal, characteristic, complemented
10#22581113C211yesmaximal, characteristic, complemented
11#22581103C211yesmaximal, characteristic, complemented
12#22581098C211yesmaximal, characteristic, complemented
13#22581063C211yesmaximal, characteristic, complemented
14#22581025C211yesmaximal, characteristic, complemented
15#547×C139C211yesmaximal, characteristic, complemented
16C2×#14555785C2211yescharacteristic
17C22×D556C2211yescharacteristic
18#17984867—24no—
19#17984862—24no—
20C2×#14555775C2211yescharacteristic, complemented
21C2×#14555775C2211yescharacteristic, complemented
22C2×#14555773C2211yescharacteristic, complemented
23C2×#14555772C2211yescharacteristic
24C2×#14555772C2211yescharacteristic
25C2×#14555772C2211yescharacteristic
26C2×#14555772C2211yescharacteristic
27C2×#14555770C2211yescharacteristic
28C2×#14555770C2211yescharacteristic
29C2×#14555769C2211yescharacteristic
30C22×C139⋊C8C2211yescharacteristic, complemented
31#17984830C2211yescharacteristic
32#17984827—21no—
33#17984821C2211yescharacteristic
34#17984816C2211yescharacteristic
35#17984815—21no—
36#17984810—21no—
37C4×D556C2211yescharacteristic
38C22⋊C4×D139—21no—
39#17984773C2211yescharacteristic, complemented
40C4⋊(C139⋊C8)C2211yescharacteristic, complemented
41#17984766C2211yescharacteristic, complemented
42#17984766C2211yescharacteristic, complemented
43#17984765C2211yescharacteristic, complemented
44#17984763C2211yescharacteristic, complemented
45#17984762C2211yescharacteristic
46#17984762C2211yescharacteristic
47C22⋊(C139⋊C8)C2211yescharacteristic
48C22⋊(C139⋊C8)C2211yescharacteristic, complemented
49#17984753C2211yescharacteristic, complemented
50C2×#41×C139C2211yescharacteristic
51C2×#41×C139C2211yescharacteristic, complemented
52C4⋊Q8×C139C2211yescharacteristic, complemented
53#123×C139—21no—
54#123×C139—21no—
55C22⋊Q8×C139C2211yescharacteristic
56C22⋊Q8×C139C2211yescharacteristic, complemented
57#121×C139—21no—
58C4×D4×C139C2211yescharacteristic
59C4×D4×C139C2211yescharacteristic, complemented
60C2×C4⋊C4×C139—21no—
61C23×D139—21no—
62#14555785—22no—
63#14555785—41no—
64C2×Dic278C2311yescharacteristic
65C2×D556—22no—
66C2×D556—21no—
67C2×D556—41no—
68C2×D556C2311yescharacteristic
69C2×D556C2311yescharacteristic
70C2×C22⋊D139—21no—
71C2×C22⋊D139—21no—
72C2×C4×D139—21no—
73C2×C4×D139—21no—
74#14555775—22no—
75#14555775—41no—
76#14555773—22no—
77#14555772—22no—
78#14555772—22no—
79#14555772—22no—
80#14555772—41no—
81#14555770—22no—
82#14555770D414yescomplemented
83#14555769—22no—
84C2×C139⋊C8—22no—
85C2×C139⋊C8C2311yescharacteristic
86C2×C139⋊C8C2311yescharacteristic
87C2×C139⋊C8C2311yescharacteristic
88C2×C139⋊C8C2311yescharacteristic
89C4⋊Dic139C2311yescharacteristic
90Dic139⋊C4—21no—
91C22⋊Dic139—21no—
92(C2×D139)⋊C4—21no—
93(C2×D139)⋊C4—21no—
94(C2×D139)⋊C4—21no—
95(C2×D139)⋊C4—21no—
96#41×C139—22no—
97#41×C139—22no—
98#41×C139—41no—
99#41×C139—41no—
100C2×Q8×C139C2311yescharacteristic
101C2×Q8×C139C2311yescharacteristic
102C2×D4×C139—21no—
103C2×D4×C139—21no—
104C2×D4×C139D411yescharacteristic
105C2×D4×C139C2311yescharacteristic
106C22×C4×C139—21no—
107C22×C4×C139—21no—
108C22×C4×C139—21no—
109C22×C4×C139C2311yescharacteristic
110C22×C4×C139C2311yescharacteristic
111C4⋊C4×C139—21no—
112C4⋊C4×C139—21no—
113C4⋊C4×C139—21no—
114C4⋊C4×C139—21no—
115C4⋊C4×C139D411yescharacteristic
116C4⋊C4×C139C2311yescharacteristic
117C22⋊C4×C139—21no—
118C22⋊C4×C139—21no—
119C22⋊C4×C139—21no—
120C22⋊C4×C139D411yescharacteristic
121C22⋊C4×C139D411yescharacteristic
122C42×C139C2311yescharacteristic
123C22×D139—21no—
124C22×D139—21no—
125C22×D139—42no—
126C22×D139—42no—
127C22×D139—81no—
128Dic278—21no—
129Dic278—21no—
130D556—21no—
131D556—21no—
132D556—21no—
133D556—42no—
134D556C2×D412yes—
135C22⋊D139—42no—
136C22⋊D139—81no—
137C2×Dic139—21no—
138C2×Dic139—21no—
139C4×D139—42no—
140C4×D139—81no—
141C139⋊C8—21no—
142C139⋊C8—21no—
143C139⋊C8—21no—
144C139⋊C8C2×D412yes—
145C23×C139—21no—
146C23×C139C2×D411yescharacteristic
147C23×C139C2×D411yescharacteristic
148Q8×C139—21no—
149Q8×C139—21no—
150Q8×C139—21no—
151Q8×C139—21no—
152D4×C139—21no—
153D4×C139—41no—
154D4×C139—42no—
155D4×C139—42no—
156D4×C139C2×D412yes—
157C2×C4×C139—21no—
158C2×C4×C139—21no—
159C2×C4×C139—21no—
160C2×C4×C139—21no—
161C2×C4×C139—21no—
162C2×C4×C139—21no—
163C2×C4×C139—21no—
164C2×C4×C139—42no—
165C2×C4×C139—42no—
166C2×C4×C139—42no—
167C2×C4×C139C2×D411yescharacteristic
168C2×C4×C139C2×D411yescharacteristic
169C2×C4×C139C2×D411yescharacteristic
170C2×C4×C139C2×D411yescharacteristic
171C2×C4×C139C2411yescharacteristic
172C22×C139—21no—
173C22×C139—21no—
174C22×C139—22no—
175C22×C139—21no—
176C22×C139—21no—
177C22×C139—21no—
178C22×C139—21no—
179C22×C139—41no—
180C22×C139—41no—
181C22×C139C22×D411yescharacteristic
182C2×D139—22no—
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—21no—
192C4×C139—21no—
193C4×C139—41no—
194C4×C139—41no—
195C4×C139—41no—
196C4×C139C22×D411yescharacteristic
197C4×C139H(5,𝔽2)11yescharacteristic
198Dic139—41no—
199Dic139—41no—
200C2×C139—41no—
201C2×C139—41no—
202C2×C139—42no—
203C2×C139D4211yescharacteristic
204C2×C139#57511yescharacteristic
205C2×C139#57711yescharacteristic
206D139—42no—
207D139—81no—
208C139#326711yescharacteristic, complemented, Sylow
209#3267—1391nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1391no—
211C2×#135—1391no—
212#547D13911yescharacteristic, complemented
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×#41C2×D13911yescharacteristic
227C2×#41C2×D13911yescharacteristic, complemented
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×SD16—1391no—
238C2×D8—1391no—
239C2×D8—1391no—
240C2×M16—1391no—
241C22×C8—1391no—
242C4⋊Q8C2×D13911yescharacteristic, complemented
243#124—1391no—
244#123—21no—
245#123—21no—
246#123—2781no—
247C22⋊Q8—1391no—
248C22⋊Q8C2×D13911yescharacteristic
249C22⋊Q8C2×D13911yescharacteristic, complemented
250#121—21no—
251#121—2781no—
252#121—1391no—
253#120—2781no—
254C4×D4—1391no—
255C4×D4C2×D13911yescharacteristic
256C4×D4C2×D13911yescharacteristic, complemented
257C2×C4⋊C4—21no—
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—2781no—
271#41—22no—
272#41—22no—
273#41—41no—
274#41—41no—
275#41—2782no—
276#41—5561no—
277C2×Q8—1391no—
278C2×Q8C22×D13911yescharacteristic
279C2×Q8C22×D13911yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—2782no—
283C2×D4—2781no—
284C2×D4—2781no—
285C2×D4—2781no—
286C2×D4—5561no—
287C2×D4—1391no—
288C2×D4—1391no—
289C2×D4C22⋊D13911yescharacteristic
290C2×D4C22×D13911yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—2781no—
295C22×C4—2781no—
296C22×C4C22×D13911yescharacteristic
297C22×C4C22×D13911yescharacteristic
298Q16—2782no—
299Q16—5561no—
300SD16—2782no—
301SD16—2782no—
302SD16—2782no—
303SD16—2782no—
304SD16—5561no—
305D8—2782no—
306D8—1394no—
307M16—2782no—
308C2×C8—2782no—
309C2×C8—1391no—
310C2×C8—1391no—
311C2×C8—1391no—
312C2×C8—1391no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—2781no—
318C4⋊C4—1391no—
319C4⋊C4C22⋊D13911yescharacteristic
320C4⋊C4C22×D13911yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—2781no—
325C22⋊C4—2781no—
326C22⋊C4—2781no—
327C22⋊C4—2781no—
328C22⋊C4—2781no—
329C22⋊C4C22⋊D13911yescharacteristic
330C22⋊C4C22⋊D13911yescharacteristic
331C42C22×D13911yescharacteristic
332C23—21no—
333C23—2781no—
334C23—2781no—
335C23—5562no—
336C23—5562no—
337C23—11121no—
338C23C2×C22⋊D13911yescharacteristic
339C23C2×C22⋊D13911yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—2781no—
345Q8—2781no—
346D4—21no—
347D4—41no—
348D4—42no—
349D4—42no—
350D4—2781no—
351D4—2781no—
352D4—2781no—
353D4—5562no—
354D4—5562no—
355D4—11121no—
356D4—1392no—
357D4D4×D13912yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—2781no—
369C2×C4—2781no—
370C2×C4—5562no—
371C2×C4—11121no—
372C2×C4C2×C22⋊D13911yescharacteristic
373C2×C4C2×C22⋊D13911yescharacteristic
374C2×C4C2×C22⋊D13911yescharacteristic
375C2×C4C2×C22⋊D13911yescharacteristic
376C2×C4C23×D13911yescharacteristic
377C8—2781no—
378C8—2781no—
379C8—2781no—
380C8—1392no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—22no—
386C22—21no—
387C22—21no—
388C22—41no—
389C22—41no—
390C22—2782no—
391C22—5561no—
392C22—5561no—
393C22—5561no—
394C22—5561no—
395C22—5561no—
396C22—11124no—
397C22C22×C22⋊D13911yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—5561no—
406C4—5561no—
407C4C2×D4×D13911yescharacteristic
408C4#1798488711yescharacteristic
409C2—41no—
410C2—41no—
411C2—42no—
412C2—5562no—
413C2—11121no—
414C2D4×C22⋊D13911yescharacteristic
415C2#2258175111yescharacteristic
416C2#2258176111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#547×C139C2
socleC22×C139C22×D4
hypercenter#547D139
hypocenterC139#3267
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#17984867C2semidirect
2. C2×#17984862C2semidirect
3. #22581599C2semidirect
4. #22581345C2semidirect
5. #22581209C2semidirect
6. #22581184C2semidirect
7. #22581168C2semidirect
8. #22581155C2semidirect
9. #22581115C2semidirect
10. #22581113C2semidirect
11. #22581103C2semidirect
12. #22581098C2semidirect
13. #22581063C2semidirect
14. #22581025C2semidirect
15. #547×C139C2semidirect
20. C2×#14555775C22semidirect
21. C2×#14555775C22semidirect
22. C2×#14555773C22semidirect
30. C22×C139⋊C8C22semidirect
39. #17984773C22semidirect
40. C4⋊(C139⋊C8)C22semidirect
41. #17984766C22semidirect
42. #17984766C22semidirect
43. #17984765C22semidirect
44. #17984763C22semidirect
48. C22⋊(C139⋊C8)C22semidirect
49. #17984753C22semidirect
51. C2×#41×C139C22semidirect
52. C4⋊Q8×C139C22semidirect
56. C22⋊Q8×C139C22semidirect
59. C4×D4×C139C22semidirect
82. #14555770D4semidirect
208. C139#3267semidirect
212. #547D139semidirect
227. C2×#41C2×D139semidirect
242. C4⋊Q8C2×D139semidirect
249. C22⋊Q8C2×D139semidirect
256. C4×D4C2×D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848672
C2×#179848622
#225815992
#225813452
#225812092
#225811842
#225811682
#225811552
#225811152
#225811132
#225811032
#225810982
#225810632
#225810252
#547×C1392
#3267139