Finite Group Databasebeta

← #31817916

Subgroups — #31817916

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×#17984871C211yesmaximal, characteristic, complemented
2C2×SD16×D139C211yesmaximal, characteristic, complemented
3Q8×D556C211yesmaximal, characteristic, complemented
4#22581598C211yesmaximal, characteristic
5#22581347C211yesmaximal, characteristic, complemented
6#22581200C211yesmaximal, characteristic, complemented
7#22581195C211yesmaximal, characteristic, complemented
8#22581175C211yesmaximal, characteristic, complemented
9#22581158C211yesmaximal, characteristic, complemented
10#22581116C211yesmaximal, characteristic, complemented
11#22581110C211yesmaximal, characteristic, complemented
12#22581105C211yesmaximal, characteristic, complemented
13#22581093C211yesmaximal, characteristic, complemented
14#22581026C211yesmaximal, characteristic, complemented
15#440×C139C211yesmaximal, characteristic, complemented
16C2×#14555784C2211yescharacteristic
17C2×Q8×D139C2211yescharacteristic
18C2×Q8×D139C2211yescharacteristic, complemented
19C2×D4×D139C2211yescharacteristic
20#17984871—24no—
21C2×Dic556C2211yescharacteristic, complemented
22C2×Dic556C2211yescharacteristic, complemented
23C2×#14555775C2211yescharacteristic, complemented
24C2×#14555774C2211yescharacteristic
25C2×#14555774C2211yescharacteristic
26C2×#14555773C2211yescharacteristic
27C2×#14555773C2211yescharacteristic
28C2×#14555772C2211yescharacteristic
29SD16×D139—24no—
30C2×#14555768C2211yescharacteristic, complemented
31C2×C8×D139C2211yescharacteristic
32#17984842—21no—
33#17984842C2211yescharacteristic, complemented
34#17984830C2211yescharacteristic
35#17984826—21no—
36#17984822—21no—
37#17984822—21no—
38#17984822C2211yescharacteristic
39#17984822C2211yescharacteristic
40#17984822C2211yescharacteristic, complemented
41#17984817C2211yescharacteristic
42#17984816—21no—
43#17984812—21no—
44C4×D556—21no—
45C4×D556C2211yescharacteristic
46C2×(C2×D139)⋊C4—21no—
47C4⋊(C139⋊C8)C2211yescharacteristic, complemented
48#17984765C2211yescharacteristic, complemented
49#17984765C2211yescharacteristic, complemented
50#17984765C2211yescharacteristic, complemented
51#17984762C2211yescharacteristic
52(C2×D139)⋊C8C2211yescharacteristic
53(C2×D139)⋊C8C2211yescharacteristic, complemented
54C2×SD16×C139C2211yescharacteristic
55C4×Q8×C139C2211yescharacteristic, complemented
56C4×D4×C139C2211yescharacteristic
57#107×C139C2211yescharacteristic, complemented
58#103×C139C2211yescharacteristic, complemented
59#102×C139C2211yescharacteristic
60#97×C139C2211yescharacteristic, complemented
61C23×D139—21no—
62#14555784—22no—
63#14555784—41no—
64C2×Dic278—21no—
65C2×Dic278—21no—
66C2×Dic278C2311yescharacteristic
67C2×Dic278C2311yescharacteristic
68C2×Dic278C2311yescharacteristic
69Q8×D139—22no—
70Q8×D139—22no—
71Q8×D139—41no—
72Q8×D139—41no—
73C2×D556—21no—
74C2×D556D411yescharacteristic
75C2×D556C2311yescharacteristic
76C2×C22⋊D139—21no—
77C2×C22⋊D139—21no—
78D4×D139—22no—
79D4×D139—41no—
80C22×Dic139—21no—
81C2×C4×D139—21no—
82C2×C4×D139—21no—
83C2×C4×D139C2311yescharacteristic
84C2×C4×D139C2311yescharacteristic
85Dic556—22no—
86Dic556—41no—
87#14555775—22no—
88#14555774—22no—
89#14555774—41no—
90#14555773—22no—
91#14555773—22no—
92#14555772—22no—
93#14555768—22no—
94C2×C139⋊C8C2311yescharacteristic
95C2×C139⋊C8C2311yescharacteristic
96C8×D139—22no—
97C4⋊Dic139—21no—
98C4⋊Dic139—21no—
99C4⋊Dic139—21no—
100C4⋊Dic139—21no—
101C4⋊Dic139—21no—
102C4⋊Dic139D411yescharacteristic
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)⋊C4—21no—
109(C2×D139)⋊C4D411yescharacteristic
110(C2×D139)⋊C4D411yescharacteristic
111C2×Q8×C139C2311yescharacteristic
112C2×D4×C139C2311yescharacteristic
113C22×C4×C139—21no—
114SD16×C139D414yescomplemented
115C2×C8×C139C2311yescharacteristic
116C2×C8×C139C2311yescharacteristic
117C4⋊C4×C139—21no—
118C4⋊C4×C139C2311yescharacteristic
119C4⋊C4×C139C2311yescharacteristic
120C22⋊C4×C139—21no—
121C42×C139—21no—
122C42×C139C2311yescharacteristic
123C22×D139—21no—
124C22×D139—42no—
125C22×D139—42no—
126C22×D139—81no—
127C22×D139C2×D411yescharacteristic
128C22×D139C2×D411yescharacteristic
129Dic278—21no—
130Dic278—21no—
131Dic278—21no—
132Dic278—21no—
133Dic278—21no—
134Dic278—21no—
135Dic278—42no—
136Dic278—42no—
137D556—22no—
138D556—21no—
139D556—21no—
140D556—81no—
141C22⋊D139—42no—
142C22⋊D139—42no—
143C2×Dic139—21no—
144C2×Dic139—21no—
145C2×Dic139—21no—
146C2×Dic139—42no—
147C2×Dic139C2×D411yescharacteristic
148C2×Dic139C2×D411yescharacteristic
149C4×D139—21no—
150C4×D139—21no—
151C4×D139—21no—
152C4×D139—21no—
153C4×D139—42no—
154C4×D139—42no—
155C139⋊C8—21no—
156C139⋊C8—21no—
157C23×C139—21no—
158Q8×C139—21no—
159Q8×C139C2×D412yes—
160D4×C139—21no—
161D4×C139C2×D412yes—
162C2×C4×C139—21no—
163C2×C4×C139—21no—
164C2×C4×C139—21no—
165C2×C4×C139—81no—
166C2×C4×C139C2×D411yescharacteristic
167C2×C4×C139C2×D411yescharacteristic
168C2×C4×C139C2411yescharacteristic
169C8×C139—21no—
170C8×C139C2×D412yes—
171C22×C139—22no—
172C22×C139—41no—
173C22×C139—41no—
174C22×C139C22×D411yescharacteristic
175C2×D139—21no—
176C2×D139—21no—
177C2×D139—21no—
178C2×D139—21no—
179C2×D139—21no—
180C2×D139—41no—
181C2×D139—41no—
182C2×D139—41no—
183C2×D139—84no—
184C2×D139#13712yescomplemented
185C4×C139—22no—
186C4×C139—21no—
187C4×C139—21no—
188C4×C139—41no—
189C4×C139—41no—
190C4×C139C22×D411yescharacteristic
191C4×C139H(5,𝔽2)11yescharacteristic
192Dic139—21no—
193Dic139—21no—
194Dic139—41no—
195Dic139—41no—
196Dic139—41no—
197C2×C139—42no—
198C2×C139D4211yescharacteristic
199C2×C139C2×#13711yescharacteristic
200C2×C139#57711yescharacteristic
201D139—22no—
202D139—41no—
203D139—81no—
204C139#326611yescharacteristic, complemented, Sylow
205#3266—1391nomaximal, Sylow
206C2×#137—1391no—
207C22×SD16—1391no—
208D4×Q8—1391no—
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#440D13911yescharacteristic, complemented
220#435—1391no—
221C2×#41—1391no—
222C22×Q8—1391no—
223C22×Q8—1391no—
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×SD16C2×D13911yescharacteristic
236C2×M16—1391no—
237C22×C8—1391no—
238C4⋊Q8—2781no—
239C4⋊Q8—1391no—
240#124—1391no—
241#123—2781no—
242C22⋊Q8—2781no—
243C22⋊Q8—2781no—
244C22⋊Q8—1391no—
245C22⋊Q8—1391no—
246C22⋊Q8—1391no—
247#121—2781no—
248#121—1391no—
249#120—2781no—
250C4×Q8C2×D13911yescharacteristic, complemented
251C4×D4—2781no—
252C4×D4—1391no—
253C4×D4C2×D13911yescharacteristic
254C2×C22⋊C4—2781no—
255#107C2×D13911yescharacteristic, complemented
256C4⋊C8—1391no—
257#103—1391no—
258#103—1391no—
259#103—1391no—
260#103C2×D13911yescharacteristic, complemented
261#102—1391no—
262#102C2×D13911yescharacteristic
263C22⋊C8—1391no—
264C22⋊C8—1391no—
265#97C2×D13911yescharacteristic, complemented
266C24—2781no—
267#41—2782no—
268#41—5561no—
269C2×Q8—2781no—
270C2×Q8—2781no—
271C2×Q8—2782no—
272C2×Q8—2782no—
273C2×Q8—5561no—
274C2×Q8—5561no—
275C2×Q8—1391no—
276C2×Q8—1391no—
277C2×Q8—1391no—
278C2×Q8C22×D13911yescharacteristic
279C2×D4—2781no—
280C2×D4—2782no—
281C2×D4—2781no—
282C2×D4—2781no—
283C2×D4—5561no—
284C2×D4—1391no—
285C2×D4—1391no—
286C2×D4C22×D13911yescharacteristic
287C22×C4—21no—
288C22×C4—2781no—
289C22×C4—2781no—
290C22×C4—2781no—
291C22×C4—1391no—
292C22×C4—1391no—
293Q16—2782no—
294Q16—2782no—
295Q16—5561no—
296SD16—2782no—
297SD16—2782no—
298SD16—2782no—
299SD16—2782no—
300SD16—5561no—
301SD16D55614yescomplemented
302M16—2782no—
303C2×C8—2782no—
304C2×C8—1391no—
305C2×C8—1391no—
306C2×C8C22×D13911yescharacteristic
307C2×C8C22×D13911yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—2781no—
310C4⋊C4—2781no—
311C4⋊C4—2781no—
312C4⋊C4—2781no—
313C4⋊C4—2781no—
314C4⋊C4—1391no—
315C4⋊C4C22×D13911yescharacteristic
316C4⋊C4C22×D13911yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—2781no—
319C22⋊C4—2781no—
320C22⋊C4—2781no—
321C22⋊C4—2781no—
322C22⋊C4—2781no—
323C22⋊C4—2781no—
324C22⋊C4—1391no—
325C22⋊C4—1391no—
326C42—21no—
327C42C22×D13911yescharacteristic
328C23—21no—
329C23—2781no—
330C23—5562no—
331C23—5562no—
332C23—11121no—
333C23—1391no—
334C23—1391no—
335Q8—21no—
336Q8—2781no—
337Q8—2781no—
338Q8—2781no—
339Q8—2781no—
340Q8—2781no—
341Q8—2781no—
342Q8—5562no—
343Q8—5562no—
344Q8C2×D55612yes—
345D4—21no—
346D4—2782no—
347D4—2781no—
348D4—2781no—
349D4—5562no—
350D4—5562no—
351D4—11121no—
352D4C2×D55612yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—81no—
357C2×C4—2781no—
358C2×C4—2781no—
359C2×C4—2781no—
360C2×C4—2781no—
361C2×C4—2781no—
362C2×C4—2781no—
363C2×C4—2781no—
364C2×C4—5562no—
365C2×C4—5562no—
366C2×C4—5562no—
367C2×C4—1391no—
368C2×C4—1391no—
369C2×C4D4×D13911yescharacteristic
370C2×C4D4×D13911yescharacteristic
371C2×C4C23×D13911yescharacteristic
372C8—21no—
373C8—2781no—
374C8—2781no—
375C8C2×D55612yes—
376C22—22no—
377C22—41no—
378C22—41no—
379C22—2781no—
380C22—2781no—
381C22—2781no—
382C22—2781no—
383C22—2781no—
384C22—5561no—
385C22—5561no—
386C22—5561no—
387C22—11124no—
388C22—1392no—
389C22C2×D4×D13911yescharacteristic
390C4—21no—
391C4—22no—
392C4—21no—
393C4—41no—
394C4—41no—
395C4—2781no—
396C4—2781no—
397C4—5561no—
398C4—5561no—
399C4—5561no—
400C4C22×D55611yescharacteristic
401C4#1798488811yescharacteristic
402C2—42no—
403C2—2782no—
404C2—5561no—
405C2—11121no—
406C2D4×D55611yescharacteristic
407C2#137×D13911yescharacteristic
408C2#2258176311yescharacteristic
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#440×C139C2
socleC22×C139C22×D4
prime radical#22581598C2
hypercenter#440D139
hypocenterC139#3266
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. C2×#17984871C2semidirect
2. C2×SD16×D139C2semidirect
3. Q8×D556C2semidirect
5. #22581347C2semidirect
6. #22581200C2semidirect
7. #22581195C2semidirect
8. #22581175C2semidirect
9. #22581158C2semidirect
10. #22581116C2semidirect
11. #22581110C2semidirect
12. #22581105C2semidirect
13. #22581093C2semidirect
14. #22581026C2semidirect
15. #440×C139C2semidirect
18. C2×Q8×D139C22semidirect
21. C2×Dic556C22semidirect
22. C2×Dic556C22semidirect
23. C2×#14555775C22semidirect
30. C2×#14555768C22semidirect
33. #17984842C22semidirect
40. #17984822C22semidirect
47. C4⋊(C139⋊C8)C22semidirect
48. #17984765C22semidirect
49. #17984765C22semidirect
50. #17984765C22semidirect
53. (C2×D139)⋊C8C22semidirect
55. C4×Q8×C139C22semidirect
57. #107×C139C22semidirect
58. #103×C139C22semidirect
60. #97×C139C22semidirect
114. SD16×C139D4semidirect
184. C2×D139#137semidirect
204. C139#3266semidirect
219. #440D139semidirect
250. C4×Q8C2×D139semidirect
255. #107C2×D139semidirect
260. #103C2×D139semidirect
265. #97C2×D139semidirect
301. SD16D556semidirect

Maximal subgroups

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

subgroupindex
C2×#179848712
C2×SD16×D1392
Q8×D5562
#225815982
#225813472
#225812002
#225811952
#225811752
#225811582
#225811162
#225811102
#225811052
#225810932
#225810262
#440×C1392
#3266139