Finite Group Databasebeta

← #31817917

Subgroups — #31817917

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
2C2×SD16×D139C211yesmaximal, characteristic, complemented
3Q8×C22⋊D139C211yesmaximal, characteristic, complemented
4#22581600C211yesmaximal, characteristic
5#22581344C211yesmaximal, characteristic, complemented
6#22581197C211yesmaximal, characteristic, complemented
7#22581193C211yesmaximal, characteristic, complemented
8#22581173C211yesmaximal, characteristic, complemented
9#22581157C211yesmaximal, characteristic, complemented
10#22581117C211yesmaximal, characteristic, complemented
11#22581110C211yesmaximal, characteristic, complemented
12#22581102C211yesmaximal, characteristic, complemented
13#22581056C211yesmaximal, characteristic, complemented
14#22581027C211yesmaximal, characteristic, complemented
15#448×C139C211yesmaximal, characteristic, complemented
16C2×#14555785C2211yescharacteristic
17C2×Q8×D139C2211yescharacteristic
18C2×D4×D139C2211yescharacteristic
19#17984873—24no—
20C2×Dic556C2211yescharacteristic, complemented
21C2×#14555775C2211yescharacteristic, complemented
22C2×#14555775C2211yescharacteristic, complemented
23C2×#14555774C2211yescharacteristic
24C2×#14555773C2211yescharacteristic
25C2×#14555772C2211yescharacteristic
26C2×#14555772C2211yescharacteristic
27SD16×D139—24no—
28C2×#14555769C2211yescharacteristic, complemented
29C2×C8×D139C2211yescharacteristic
30#17984840—21no—
31#17984840C2211yescharacteristic, complemented
32#17984829C2211yescharacteristic
33#17984824—21no—
34#17984822C2211yescharacteristic
35#17984821—21no—
36#17984821C2211yescharacteristic, complemented
37#17984818—21no—
38#17984818C2211yescharacteristic
39#17984813—21no—
40#17984810—21no—
41Q8×Dic139C2211yescharacteristic, complemented
42#17984804C2211yescharacteristic
43C4×C22⋊D139—21no—
44C4×C22⋊D139C2211yescharacteristic
45C22⋊C4×D139—21no—
46#17984775C2211yescharacteristic, complemented
47Dic139⋊C8C2211yescharacteristic, complemented
48#17984767C2211yescharacteristic, complemented
49#17984765C2211yescharacteristic, complemented
50#17984764C2211yescharacteristic
51#17984763C2211yescharacteristic
52(C2×D139)⋊C8C2211yescharacteristic
53#17984752C2211yescharacteristic, complemented
54C22×Q8×C139C2211yescharacteristic, complemented
55C2×SD16×C139C2211yescharacteristic
56C2×SD16×C139C2211yescharacteristic
57#121×C139C2211yescharacteristic
58#103×C139C2211yescharacteristic, complemented
59#103×C139C2211yescharacteristic, complemented
60C22⋊C8×C139C2211yescharacteristic, complemented
61C23×D139—21no—
62#14555785—22no—
63#14555785—41no—
64C2×Dic278—21no—
65C2×Dic278C2311yescharacteristic
66C2×Dic278C2311yescharacteristic
67Q8×D139—22no—
68Q8×D139—41no—
69C2×D556C2311yescharacteristic
70C2×C22⋊D139—21no—
71C2×C22⋊D139—21no—
72C2×C22⋊D139D411yescharacteristic
73D4×D139—22no—
74D4×D139—41no—
75C2×C4×D139—21no—
76C2×C4×D139—21no—
77C2×C4×D139—21no—
78C2×C4×D139C2311yescharacteristic
79Dic556—41no—
80#14555775—22no—
81#14555775—22no—
82#14555774—22no—
83#14555773—22no—
84#14555772—22no—
85#14555772D414yescomplemented
86#14555769—22no—
87C2×C139⋊C8C2311yescharacteristic
88C2×C139⋊C8C2311yescharacteristic
89C8×D139—22no—
90C4⋊Dic139—21no—
91C4⋊Dic139—21no—
92C4⋊Dic139C2311yescharacteristic
93Dic139⋊C4—21no—
94Dic139⋊C4—21no—
95Dic139⋊C4—21no—
96Dic139⋊C4—21no—
97Dic139⋊C4D411yescharacteristic
98C22⋊Dic139—21no—
99C22⋊Dic139—21no—
100C22⋊Dic139—21no—
101C22⋊Dic139D411yescharacteristic
102(C2×D139)⋊C4—21no—
103(C2×D139)⋊C4—21no—
104(C2×D139)⋊C4—21no—
105(C2×D139)⋊C4D411yescharacteristic
106C4×Dic139—21no—
107C4×Dic139C2311yescharacteristic
108C2×Q8×C139—21no—
109C2×Q8×C139—22no—
110C2×Q8×C139—41no—
111C2×Q8×C139C2311yescharacteristic
112C2×Q8×C139C2311yescharacteristic
113C2×D4×C139—21no—
114C2×D4×C139C2311yescharacteristic
115C22×C4×C139—21no—
116C22×C4×C139C2311yescharacteristic
117SD16×C139—22no—
118SD16×C139—41no—
119C2×C8×C139C2311yescharacteristic
120C2×C8×C139C2311yescharacteristic
121C4⋊C4×C139C2311yescharacteristic
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—42no—
133D556—21no—
134D556C2×D412yes—
135C22⋊D139—22no—
136C22⋊D139—42no—
137C22⋊D139—42no—
138C2×Dic139—21no—
139C2×Dic139—21no—
140C2×Dic139—21no—
141C2×Dic139C2×D411yescharacteristic
142C2×Dic139C2×D411yescharacteristic
143C2×Dic139C2×D411yescharacteristic
144C4×D139—21no—
145C4×D139—21no—
146C4×D139—42no—
147C4×D139—42no—
148C4×D139—81no—
149C139⋊C8—21no—
150C139⋊C8C2×D412yes—
151C23×C139—21no—
152C23×C139C2×D411yescharacteristic
153Q8×C139—21no—
154Q8×C139—21no—
155Q8×C139—21no—
156Q8×C139—42no—
157Q8×C139C2×D412yes—
158D4×C139—21no—
159D4×C139—21no—
160D4×C139—81no—
161C2×C4×C139—21no—
162C2×C4×C139—21no—
163C2×C4×C139—21no—
164C2×C4×C139—21no—
165C2×C4×C139—21no—
166C2×C4×C139—42no—
167C2×C4×C139C2×D411yescharacteristic
168C2×C4×C139C2411yescharacteristic
169C8×C139—21no—
170C8×C139—21no—
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—22no—
187C4×C139—41no—
188C4×C139—41no—
189C4×C139C22×D411yescharacteristic
190C4×C139H(5,𝔽2)11yescharacteristic
191Dic139—21no—
192Dic139—21no—
193Dic139—21no—
194Dic139—41no—
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#468—1391no—
212#467—1391no—
213#464—1391no—
214#461—1391no—
215#452—1391no—
216#451—1391no—
217#450—1391no—
218#448D13911yescharacteristic, complemented
219#440—1391no—
220#435—1391no—
221C2×#41—1391no—
222C22×Q8—1391no—
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×SD16C2×D13911yescharacteristic
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#121C2×D13911yescharacteristic
249#120—2781no—
250C4×Q8—1391no—
251C4×D4—2781no—
252C4×D4—1391no—
253C4×D4—1391no—
254C2×C22⋊C4—2781no—
255#107—1391no—
256C4⋊C8—1391no—
257#103—1391no—
258#103—1391no—
259#103C2×D13911yescharacteristic, complemented
260#103C2×D13911yescharacteristic, complemented
261#102—1391no—
262#102—1391no—
263C22⋊C8—1391no—
264C22⋊C8C2×D13911yescharacteristic, complemented
265#97—1391no—
266C24—2781no—
267#41—2782no—
268#41—5561no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—2781no—
273C2×Q8—2782no—
274C2×Q8—5561no—
275C2×Q8—1391no—
276C2×Q8—1391no—
277C2×Q8C22×D13911yescharacteristic
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—21no—
288C22×C4—2781no—
289C22×C4—2781no—
290C22×C4—2781no—
291C22×C4—1391no—
292C22×C4C22×D13911yescharacteristic
293Q16—2782no—
294Q16—2782no—
295Q16—5561no—
296SD16—22no—
297SD16—41no—
298SD16—2782no—
299SD16—2782no—
300SD16—2782no—
301SD16—1394no—
302M16—2782no—
303C2×C8—2782no—
304C2×C8—1391no—
305C2×C8—1391no—
306C2×C8C22×D13911yescharacteristic
307C2×C8C22×D13911yescharacteristic
308C4⋊C4—2781no—
309C4⋊C4—2781no—
310C4⋊C4—2781no—
311C4⋊C4—2781no—
312C4⋊C4—2781no—
313C4⋊C4—2781no—
314C4⋊C4—1391no—
315C4⋊C4—1391no—
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—2781no—
327C42—1391no—
328C23—21no—
329C23—2781no—
330C23—5562no—
331C23—5562no—
332C23—11121no—
333C23—1391no—
334C23D4×D13911yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—42no—
339Q8—2781no—
340Q8—2781no—
341Q8—2781no—
342Q8—2781no—
343Q8—5562no—
344Q8D4×D13912yes—
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—42no—
359C2×C4—2781no—
360C2×C4—2781no—
361C2×C4—2781no—
362C2×C4—2781no—
363C2×C4—2781no—
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—21no—
374C8—2781no—
375C8—1392no—
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—22no—
392C4—41no—
393C4—41no—
394C4—2781no—
395C4—2781no—
396C4—2781no—
397C4—5561no—
398C4—5561no—
399C4—5561no—
400C4C2×D4×D13911yescharacteristic
401C4#1798488711yescharacteristic
402C2—41no—
403C2—81no—
404C2—2782no—
405C2—5562no—
406C2#2258158511yescharacteristic
407C2#137×D13911yescharacteristic
408C2#2258175911yescharacteristic
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#448×C139C2
socleC22×C139C22×D4
prime radical#22581600C2
hypercenter#448D139
hypocenterC139#3266
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#17984873C2semidirect
2. C2×SD16×D139C2semidirect
3. Q8×C22⋊D139C2semidirect
5. #22581344C2semidirect
6. #22581197C2semidirect
7. #22581193C2semidirect
8. #22581173C2semidirect
9. #22581157C2semidirect
10. #22581117C2semidirect
11. #22581110C2semidirect
12. #22581102C2semidirect
13. #22581056C2semidirect
14. #22581027C2semidirect
15. #448×C139C2semidirect
20. C2×Dic556C22semidirect
21. C2×#14555775C22semidirect
22. C2×#14555775C22semidirect
28. C2×#14555769C22semidirect
31. #17984840C22semidirect
36. #17984821C22semidirect
41. Q8×Dic139C22semidirect
46. #17984775C22semidirect
47. Dic139⋊C8C22semidirect
48. #17984767C22semidirect
49. #17984765C22semidirect
53. #17984752C22semidirect
54. C22×Q8×C139C22semidirect
58. #103×C139C22semidirect
59. #103×C139C22semidirect
60. C22⋊C8×C139C22semidirect
85. #14555772D4semidirect
184. C2×D139#137semidirect
204. C139#3266semidirect
218. #448D139semidirect
223. C22×Q8C2×D139semidirect
259. #103C2×D139semidirect
260. #103C2×D139semidirect
264. C22⋊C8C2×D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848732
C2×SD16×D1392
Q8×C22⋊D1392
#225816002
#225813442
#225811972
#225811932
#225811732
#225811572
#225811172
#225811102
#225811022
#225810562
#225810272
#448×C1392
#3266139