Finite Group Databasebeta

← #31817915

Subgroups — #31817915

Order 17792.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984873C211yesmaximal, characteristic, complemented
2C22×#14555772C211yesmaximal, characteristic, complemented
3#22581599C211yesmaximal, characteristic
4#22581345C211yesmaximal, characteristic, complemented
5#22581201C211yesmaximal, characteristic, complemented
6#22581192C211yesmaximal, characteristic, complemented
7#22581175C211yesmaximal, characteristic, complemented
8#22581155C211yesmaximal, characteristic, complemented
9#22581113C211yesmaximal, characteristic, complemented
10#22581109C211yesmaximal, characteristic, complemented
11#22581103C211yesmaximal, characteristic, complemented
12#22581092C211yesmaximal, characteristic, complemented
13#22581051C211yesmaximal, characteristic, complemented
14#22581025C211yesmaximal, characteristic, complemented
15D4×Q8×C139C211yesmaximal, characteristic, complemented
16C2×#14555785C2211yescharacteristic
17C22×D556C2211yescharacteristic
18#17984873—24no—
19C2×#14555775C2211yescharacteristic, complemented
20C2×#14555775C2211yescharacteristic, complemented
21C2×#14555775C2211yescharacteristic, complemented
22C2×#14555772—24no—
23C2×#14555772C2211yescharacteristic
24C2×#14555772C2211yescharacteristic
25C2×#14555772C2211yescharacteristic
26C2×#14555772C2211yescharacteristic
27C2×#14555772C2211yescharacteristic
28C2×#14555772C2211yescharacteristic
29C2×#14555769C2211yescharacteristic, complemented
30C22×C139⋊C8C2211yescharacteristic
31#17984830C2211yescharacteristic
32#17984827—21no—
33#17984821C2211yescharacteristic
34#17984816C2211yescharacteristic
35#17984815—21no—
36#17984810—21no—
37C4×D556C2211yescharacteristic
38C22⋊C4×D139—21no—
39#17984775C2211yescharacteristic, complemented
40C4⋊(C139⋊C8)C2211yescharacteristic, complemented
41#17984766C2211yescharacteristic, complemented
42#17984766C2211yescharacteristic, complemented
43#17984766C2211yescharacteristic, complemented
44#17984765C2211yescharacteristic, complemented
45#17984762C2211yescharacteristic
46#17984762C2211yescharacteristic
47C22⋊(C139⋊C8)C2211yescharacteristic
48C22⋊(C139⋊C8)C2211yescharacteristic, complemented
49#17984753C2211yescharacteristic, complemented
50C22×Q8×C139C2211yescharacteristic
51C22×Q8×C139C2211yescharacteristic, complemented
52C4⋊Q8×C139—21no—
53C4⋊Q8×C139C2211yescharacteristic, complemented
54C22⋊Q8×C139—21no—
55C22⋊Q8×C139—21no—
56C22⋊Q8×C139C2211yescharacteristic
57C22⋊Q8×C139C2211yescharacteristic, complemented
58C4×Q8×C139C2211yescharacteristic, complemented
59C4×D4×C139—21no—
60C4×D4×C139C2211yescharacteristic
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—22no—
76#14555775—41no—
77#14555772—22no—
78#14555772—22no—
79#14555772—22no—
80#14555772—22no—
81#14555772—41no—
82#14555772D414yescomplemented
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—
96C2×Q8×C139—21no—
97C2×Q8×C139—21no—
98C2×Q8×C139—22no—
99C2×Q8×C139—22no—
100C2×Q8×C139—41no—
101C2×Q8×C139—41no—
102C2×Q8×C139C2311yescharacteristic
103C2×Q8×C139C2311yescharacteristic
104C2×Q8×C139C2311yescharacteristic
105C2×D4×C139D411yescharacteristic
106C22×C4×C139—21no—
107C22×C4×C139—21no—
108C22×C4×C139C2311yescharacteristic
109C22×C4×C139C2311yescharacteristic
110C4⋊C4×C139—21no—
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×C139D411yescharacteristic
120C22⋊C4×C139D411yescharacteristic
121C42×C139—21no—
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×C139C2×D411yescharacteristic
146C23×C139C2×D411yescharacteristic
147Q8×C139—21no—
148Q8×C139—21no—
149Q8×C139—21no—
150Q8×C139—21no—
151Q8×C139—21no—
152Q8×C139—42no—
153Q8×C139—42no—
154Q8×C139C2×D412yes—
155D4×C139—22no—
156C2×C4×C139—21no—
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×C139C2×D411yescharacteristic
167C2×C4×C139C2×D411yescharacteristic
168C2×C4×C139C2×D411yescharacteristic
169C2×C4×C139C2×D411yescharacteristic
170C2×C4×C139C2411yescharacteristic
171C22×C139—21no—
172C22×C139—21no—
173C22×C139—21no—
174C22×C139—21no—
175C22×C139—21no—
176C22×C139#13712yescomplemented
177C22×C139C22×D411yescharacteristic
178C2×D139—22no—
179C2×D139—41no—
180C2×D139—41no—
181C2×D139—41no—
182C2×D139—41no—
183C2×D139—41no—
184C2×D139—84no—
185C4×C139—21no—
186C4×C139—21no—
187C4×C139—22no—
188C4×C139—21no—
189C4×C139—21no—
190C4×C139—41no—
191C4×C139—41no—
192C4×C139—41no—
193C4×C139C22×D411yescharacteristic
194C4×C139H(5,𝔽2)11yescharacteristic
195Dic139—41no—
196Dic139—41no—
197C2×C139—22no—
198C2×C139—41no—
199C2×C139D4211yescharacteristic
200C2×C139C2×#13711yescharacteristic
201C2×C139#57711yescharacteristic
202D139—42no—
203D139—81no—
204C139#326611yescharacteristic, complemented, Sylow
205#3266—1391nomaximal, Sylow
206C2×#137—1391no—
207C22×SD16—1391no—
208D4×Q8D13911yescharacteristic, complemented
209#546—1391no—
210#497—1391no—
211#468—1391no—
212#467—1391no—
213#464—1391no—
214#461—1391no—
215#452—1391no—
216#451—1391no—
217#450—1391no—
218#448—1391no—
219#440—1391no—
220#435—1391no—
221C2×#41—1391no—
222C22×Q8C2×D13911yescharacteristic
223C22×Q8C2×D13911yescharacteristic, complemented
224C22×D4—1391no—
225#137—2784no—
226C2×Q16—1391no—
227C2×Q16—1391no—
228C2×Q16—1391no—
229C2×SD16—2784no—
230C2×SD16—1391no—
231C2×SD16—1391no—
232C2×SD16—1391no—
233C2×SD16—1391no—
234C2×SD16—1391no—
235C2×SD16—1391no—
236C2×M16—1391no—
237C22×C8—1391no—
238C4⋊Q8—21no—
239C4⋊Q8C2×D13911yescharacteristic, complemented
240#124—1391no—
241#123—2781no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—1391no—
245C22⋊Q8C2×D13911yescharacteristic
246C22⋊Q8C2×D13911yescharacteristic, complemented
247#121—2781no—
248#121—1391no—
249#120—2781no—
250C4×Q8C2×D13911yescharacteristic, complemented
251C4×D4—21no—
252C4×D4—1391no—
253C4×D4C2×D13911yescharacteristic
254C2×C22⋊C4—2781no—
255#107—1391no—
256C4⋊C8—1391no—
257#103—1391no—
258#103—1391no—
259#103—1391no—
260#103—1391no—
261#102—1391no—
262#102—1391no—
263C22⋊C8—1391no—
264C22⋊C8—1391no—
265#97—1391no—
266C24—2781no—
267#41—2782no—
268#41—5561no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—1391no—
276C2×Q8C22×D13911yescharacteristic
277C2×Q8C22×D13911yescharacteristic
278C2×Q8C22×D13911yescharacteristic
279C2×D4—2782no—
280C2×D4—2781no—
281C2×D4—2781no—
282C2×D4—2781no—
283C2×D4—5561no—
284C2×D4—1391no—
285C2×D4—1391no—
286C2×D4C22⋊D13911yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—2781no—
290C22×C4—2781no—
291C22×C4C22×D13911yescharacteristic
292C22×C4C22×D13911yescharacteristic
293Q16—2782no—
294Q16—2782no—
295Q16—5561no—
296SD16—2782no—
297SD16—2782no—
298SD16—2782no—
299SD16—2782no—
300SD16—5561no—
301SD16—1394no—
302M16—2782no—
303C2×C8—2782no—
304C2×C8—1391no—
305C2×C8—1391no—
306C2×C8—1391no—
307C2×C8—1391no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—2781no—
314C4⋊C4—1391no—
315C4⋊C4C22⋊D13911yescharacteristic
316C4⋊C4C22×D13911yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—2781no—
320C22⋊C4—2781no—
321C22⋊C4—2781no—
322C22⋊C4—2781no—
323C22⋊C4—2781no—
324C22⋊C4C22⋊D13911yescharacteristic
325C22⋊C4C22⋊D13911yescharacteristic
326C42—21no—
327C42C22×D13911yescharacteristic
328C23—2781no—
329C23—2781no—
330C23—5562no—
331C23—5562no—
332C23—11121no—
333C23C2×C22⋊D13911yescharacteristic
334C23C2×C22⋊D13911yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—42no—
342Q8—2781no—
343Q8—2781no—
344Q8D4×D13912yes—
345D4—22no—
346D4—2781no—
347D4—2781no—
348D4—2781no—
349D4—5562no—
350D4—5562no—
351D4—11121no—
352D4—1392no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—2781no—
364C2×C4—2781no—
365C2×C4—5562no—
366C2×C4—11121no—
367C2×C4C2×C22⋊D13911yescharacteristic
368C2×C4C2×C22⋊D13911yescharacteristic
369C2×C4C2×C22⋊D13911yescharacteristic
370C2×C4C2×C22⋊D13911yescharacteristic
371C2×C4C23×D13911yescharacteristic
372C8—2781no—
373C8—2781no—
374C8—2781no—
375C8—1392no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—2782no—
382C22—5561no—
383C22—5561no—
384C22—5561no—
385C22—5561no—
386C22—5561no—
387C22—11124no—
388C22#1798487312yescomplemented
389C22C22×C22⋊D13911yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—22no—
394C4—21no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—5561no—
399C4—5561no—
400C4C2×D4×D13911yescharacteristic
401C4#1798488711yescharacteristic
402C2—22no—
403C2—41no—
404C2—5562no—
405C2—11121no—
406C2D4×C22⋊D13911yescharacteristic
407C2C2×#1798487311yescharacteristic
408C2#2258176111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
FittingD4×Q8×C139C2
socleC22×C139C22×D4
prime radical#22581599C2
hypercenterD4×Q8D139
hypocenterC139#3266
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#17984873C2semidirect
2. C22×#14555772C2semidirect
4. #22581345C2semidirect
5. #22581201C2semidirect
6. #22581192C2semidirect
7. #22581175C2semidirect
8. #22581155C2semidirect
9. #22581113C2semidirect
10. #22581109C2semidirect
11. #22581103C2semidirect
12. #22581092C2semidirect
13. #22581051C2semidirect
14. #22581025C2semidirect
15. D4×Q8×C139C2semidirect
19. C2×#14555775C22semidirect
20. C2×#14555775C22semidirect
21. C2×#14555775C22semidirect
29. C2×#14555769C22semidirect
39. #17984775C22semidirect
40. C4⋊(C139⋊C8)C22semidirect
41. #17984766C22semidirect
42. #17984766C22semidirect
43. #17984766C22semidirect
44. #17984765C22semidirect
48. C22⋊(C139⋊C8)C22semidirect
49. #17984753C22semidirect
51. C22×Q8×C139C22semidirect
53. C4⋊Q8×C139C22semidirect
57. C22⋊Q8×C139C22semidirect
58. C4×Q8×C139C22semidirect
82. #14555772D4semidirect
176. C22×C139#137semidirect
204. C139#3266semidirect
208. D4×Q8D139semidirect
223. C22×Q8C2×D139semidirect
239. C4⋊Q8C2×D139semidirect
246. C22⋊Q8C2×D139semidirect
250. C4×Q8C2×D139semidirect
388. C22#17984873semidirect

Maximal subgroups

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

subgroupindex
C2×#179848732
C22×#145557722
#225815992
#225813452
#225812012
#225811922
#225811752
#225811552
#225811132
#225811092
#225811032
#225810922
#225810512
#225810252
D4×Q8×C1392
#3266139