Finite Group Databasebeta

← #31817921

Subgroups — #31817921

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×#17984875C211yesmaximal, characteristic, complemented
2C2×SD16×D139C211yesmaximal, characteristic, complemented
3#22581616C211yesmaximal, characteristic, complemented
4#22581600C211yesmaximal, characteristic
5#22581344C211yesmaximal, characteristic, complemented
6#22581195C211yesmaximal, characteristic, complemented
7#22581174C211yesmaximal, characteristic, complemented
8#22581157C211yesmaximal, characteristic, complemented
9#22581113C211yesmaximal, characteristic, complemented
10#22581110C211yesmaximal, characteristic, complemented
11#22581105C211yesmaximal, characteristic, complemented
12#22581091C211yesmaximal, characteristic, complemented
13#22581052C211yesmaximal, characteristic, complemented
14#22581022C211yesmaximal, characteristic, complemented
15#468×C139C211yesmaximal, characteristic, complemented
16C2×#14555785C2211yescharacteristic
17C22×Dic278C2211yescharacteristic, complemented
18C2×Q8×D139C2211yescharacteristic
19C2×D4×D139C2211yescharacteristic
20#17984875—24no—
21C2×Dic556C2211yescharacteristic, complemented
22C2×Dic556C2211yescharacteristic, complemented
23C2×#14555775C2211yescharacteristic, complemented
24C2×#14555774C2211yescharacteristic
25C2×#14555774C2211yescharacteristic
26C2×#14555773C2211yescharacteristic
27C2×#14555773C2211yescharacteristic
28C2×#14555772C2211yescharacteristic
29SD16×D139—24no—
30C2×C8×D139C2211yescharacteristic
31#17984841—21no—
32#17984840C2211yescharacteristic, complemented
33#17984829C2211yescharacteristic
34#17984824—21no—
35#17984822C2211yescharacteristic
36#17984822C2211yescharacteristic
37#17984820—21no—
38#17984819—21no—
39#17984813—21no—
40#17984810—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#17984765C2211yescharacteristic, complemented
49#17984765C2211yescharacteristic, complemented
50#17984762C2211yescharacteristic
51C22⋊(C139⋊C8)C2211yescharacteristic, complemented
52(C2×D139)⋊C8C2211yescharacteristic
53#17984752C2211yescharacteristic, complemented
54C2×SD16×C139C2211yescharacteristic
55C2×M16×C139C2211yescharacteristic, complemented
56C22⋊Q8×C139C2211yescharacteristic, complemented
57#121×C139C2211yescharacteristic
58#107×C139C2211yescharacteristic, complemented
59#103×C139C2211yescharacteristic, complemented
60#102×C139C2211yescharacteristic
61C23×D139—21no—
62#14555785—22no—
63#14555785—41no—
64C2×Dic278—21no—
65C2×Dic278—22no—
66C2×Dic278—21no—
67C2×Dic278—41no—
68C2×Dic278C2311yescharacteristic
69C2×Dic278C2311yescharacteristic
70C2×Dic278C2311yescharacteristic
71Q8×D139—22no—
72Q8×D139—41no—
73C2×D556C2311yescharacteristic
74C2×C22⋊D139—21no—
75C2×C22⋊D139—21no—
76C2×C22⋊D139D411yescharacteristic
77D4×D139—22no—
78D4×D139—41no—
79C22×Dic139—21no—
80C2×C4×D139—21no—
81C2×C4×D139—21no—
82C2×C4×D139—21no—
83C2×C4×D139C2311yescharacteristic
84Dic556—22no—
85Dic556—22no—
86#14555775—41no—
87#14555774—22no—
88#14555774D414yescomplemented
89#14555773—22no—
90#14555773—41no—
91#14555772—22no—
92C2×C139⋊C8C2311yescharacteristic
93C2×C139⋊C8C2311yescharacteristic
94C8×D139—22no—
95C4⋊Dic139—21no—
96C4⋊Dic139—21no—
97Dic139⋊C4—21no—
98Dic139⋊C4—21no—
99Dic139⋊C4—21no—
100Dic139⋊C4D411yescharacteristic
101C22⋊Dic139—21no—
102C22⋊Dic139—21no—
103C22⋊Dic139D411yescharacteristic
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—42no—
125C22×D139—42no—
126C22×D139—81no—
127C22×D139C2×D411yescharacteristic
128Dic278—21no—
129Dic278—21no—
130Dic278—21no—
131Dic278—21no—
132Dic278—21no—
133Dic278—42no—
134Dic278—42no—
135Dic278C2×D412yes—
136D556—21no—
137D556C2×D412yes—
138C22⋊D139—22no—
139C22⋊D139—42no—
140C22⋊D139—42no—
141C2×Dic139—21no—
142C2×Dic139—21no—
143C2×Dic139—21no—
144C2×Dic139—42no—
145C2×Dic139C2×D411yescharacteristic
146C2×Dic139C2×D411yescharacteristic
147C2×Dic139C2×D411yescharacteristic
148C4×D139—21no—
149C4×D139—21no—
150C4×D139—42no—
151C4×D139—42no—
152C4×D139—81no—
153C139⋊C8—21no—
154C139⋊C8—21no—
155C23×C139—21no—
156C23×C139C2×D411yescharacteristic
157Q8×C139—21no—
158Q8×C139—21no—
159D4×C139—21no—
160D4×C139—21no—
161D4×C139—81no—
162C2×C4×C139—21no—
163C2×C4×C139—21no—
164C2×C4×C139—21no—
165C2×C4×C139—21no—
166C2×C4×C139—21no—
167C2×C4×C139C2×D411yescharacteristic
168C2×C4×C139C2411yescharacteristic
169C8×C139—21no—
170C8×C139C2×D412yes—
171C22×C139—21no—
172C22×C139—21no—
173C22×C139—21no—
174C22×C139—41no—
175C22×C139—41no—
176C22×C139—41no—
177C22×C139C22×D411yescharacteristic
178C2×D139—21no—
179C2×D139—21no—
180C2×D139—22no—
181C2×D139—41no—
182C2×D139—41no—
183C2×D139—84no—
184C2×D139#13712yescomplemented
185C4×C139—21no—
186C4×C139—41no—
187C4×C139—41no—
188C4×C139—41no—
189C4×C139C22×D411yescharacteristic
190C4×C139H(5,𝔽2)11yescharacteristic
191Dic139—21no—
192Dic139—22no—
193Dic139—21no—
194Dic139—21no—
195Dic139—41no—
196Dic139—41no—
197C2×C139—41no—
198C2×C139—81no—
199C2×C139D4211yescharacteristic
200C2×C139C2×#13711yescharacteristic
201C2×C139#57711yescharacteristic
202D139—22no—
203D139—42no—
204C139#326611yescharacteristic, complemented, Sylow
205#3266—1391nomaximal, Sylow
206C2×#137—1391no—
207C22×SD16—1391no—
208D4×Q8—1391no—
209#546—1391no—
210#497—1391no—
211#468D13911yescharacteristic, complemented
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×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×M16C2×D13911yescharacteristic, complemented
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⋊Q8C2×D13911yescharacteristic, complemented
247#121—2781no—
248#121C2×D13911yescharacteristic
249#120—2781no—
250C4×Q8—1391no—
251C4×D4—2781no—
252C4×D4—1391no—
253C4×D4—1391no—
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#97—1391no—
266C24—2781no—
267#41—2782no—
268#41—5561no—
269C2×Q8—2781no—
270C2×Q8—2782no—
271C2×Q8—2781no—
272C2×Q8—2782no—
273C2×Q8—5561no—
274C2×Q8—5561no—
275C2×Q8—1391no—
276C2×Q8—1391no—
277C2×Q8—1391no—
278C2×Q8C22×D13911yescharacteristic
279C2×D4—21no—
280C2×D4—2782no—
281C2×D4—2781no—
282C2×D4—2781no—
283C2×D4—5561no—
284C2×D4—1391no—
285C2×D4—1391no—
286C2×D4C22×D13911yescharacteristic
287C22×C4—2781no—
288C22×C4—2781no—
289C22×C4—2781no—
290C22×C4—2781no—
291C22×C4—1391no—
292C22×C4C22×D13911yescharacteristic
293Q16—2782no—
294Q16—2782no—
295Q16—5561no—
296SD16—22no—
297SD16—2782no—
298SD16—2782no—
299SD16—2782no—
300SD16—5561no—
301SD16—1394no—
302M16—22no—
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—21no—
319C22⋊C4—2781no—
320C22⋊C4—2781no—
321C22⋊C4—2781no—
322C22⋊C4—2781no—
323C22⋊C4—2781no—
324C22⋊C4—1391no—
325C22⋊C4—1391no—
326C42—2781no—
327C42—1391no—
328C23—21no—
329C23—2781no—
330C23—5562no—
331C23—5562no—
332C23—11121no—
333C23—1391no—
334C23D4×D13911yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—2781no—
338Q8—2781no—
339Q8—2781no—
340Q8—2781no—
341Q8—2781no—
342Q8—5562no—
343Q8—5562no—
344Q8—1392no—
345D4—21no—
346D4—21no—
347D4—81no—
348D4—2782no—
349D4—2781no—
350D4—5562no—
351D4—5562no—
352D4—1392no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—2781no—
359C2×C4—2781no—
360C2×C4—2781no—
361C2×C4—2781no—
362C2×C4—2781no—
363C2×C4—5562no—
364C2×C4—5562no—
365C2×C4—5562no—
366C2×C4—11121no—
367C2×C4—1391no—
368C2×C4—1391no—
369C2×C4—1391no—
370C2×C4D4×D13911yescharacteristic
371C2×C4C23×D13911yescharacteristic
372C8—21no—
373C8—2781no—
374C8—2781no—
375C8D4×D13912yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—2781no—
383C22—2781no—
384C22—2782no—
385C22—5561no—
386C22—5561no—
387C22—11124no—
388C22—1392no—
389C22C2×D4×D13911yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—2781no—
395C4—2781no—
396C4—2782no—
397C4—2781no—
398C4—5561no—
399C4—5561no—
400C4C2×D4×D13911yescharacteristic
401C4#1798488811yescharacteristic
402C2—41no—
403C2—81no—
404C2—2782no—
405C2—5562no—
406C2#2258158711yescharacteristic
407C2#137×D13911yescharacteristic
408C2#2258176311yescharacteristic
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#468×C139C2
socleC22×C139C22×D4
prime radical#22581600C2
hypercenter#468D139
hypocenterC139#3266
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×#17984875C2semidirect
2. C2×SD16×D139C2semidirect
3. #22581616C2semidirect
5. #22581344C2semidirect
6. #22581195C2semidirect
7. #22581174C2semidirect
8. #22581157C2semidirect
9. #22581113C2semidirect
10. #22581110C2semidirect
11. #22581105C2semidirect
12. #22581091C2semidirect
13. #22581052C2semidirect
14. #22581022C2semidirect
15. #468×C139C2semidirect
17. C22×Dic278C22semidirect
21. C2×Dic556C22semidirect
22. C2×Dic556C22semidirect
23. C2×#14555775C22semidirect
32. #17984840C22semidirect
41. #17984807C22semidirect
46. Dic139⋊C8C22semidirect
47. #17984765C22semidirect
48. #17984765C22semidirect
49. #17984765C22semidirect
51. C22⋊(C139⋊C8)C22semidirect
53. #17984752C22semidirect
55. C2×M16×C139C22semidirect
56. C22⋊Q8×C139C22semidirect
58. #107×C139C22semidirect
59. #103×C139C22semidirect
88. #14555774D4semidirect
184. C2×D139#137semidirect
204. C139#3266semidirect
211. #468D139semidirect
236. C2×M16C2×D139semidirect
246. C22⋊Q8C2×D139semidirect
255. #107C2×D139semidirect
260. #103C2×D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848752
C2×SD16×D1392
#225816162
#225816002
#225813442
#225811952
#225811742
#225811572
#225811132
#225811102
#225811052
#225810912
#225810522
#225810222
#468×C1392
#3266139