Finite Group Databasebeta

← #31817913

Subgroups — #31817913

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×#17984874C211yesmaximal, characteristic, complemented
2C2×SD16×D139C211yesmaximal, characteristic, complemented
3#22581618C211yesmaximal, characteristic, complemented
4#22581596C211yesmaximal, characteristic
5#22581196C211yesmaximal, characteristic, complemented
6#22581191C211yesmaximal, characteristic, complemented
7#22581172C211yesmaximal, characteristic, complemented
8#22581155C211yesmaximal, characteristic, complemented
9#22581114C211yesmaximal, characteristic, complemented
10#22581107C211yesmaximal, characteristic, complemented
11#22581102C211yesmaximal, characteristic, complemented
12#22581090C211yesmaximal, characteristic, complemented
13#22581055C211yesmaximal, characteristic, complemented
14#22581026C211yesmaximal, characteristic, complemented
15#497×C139C211yesmaximal, characteristic, complemented
16C2×#14555786C2211yescharacteristic
17C2×Q8×D139C2211yescharacteristic
18C2×Q8×D139C2211yescharacteristic, complemented
19C2×D4×D139C2211yescharacteristic
20#17984874—24no—
21C2×#14555775C2211yescharacteristic, complemented
22C2×#14555774C2211yescharacteristic
23C2×#14555773C2211yescharacteristic
24C2×#14555772C2211yescharacteristic
25C2×#14555772C2211yescharacteristic
26SD16×D139—24no—
27C2×#14555768C2211yescharacteristic, complemented
28C2×C8×D139C2211yescharacteristic
29#17984841—21no—
30#17984824—21no—
31#17984820—21no—
32#17984820—21no—
33#17984818C2211yescharacteristic
34#17984818C2211yescharacteristic
35#17984818C2211yescharacteristic, complemented
36#17984815—21no—
37#17984813C2211yescharacteristic
38#17984812—21no—
39C4×Dic278C2211yescharacteristic, complemented
40C4×D556C2211yescharacteristic
41C4×D556C2211yescharacteristic
42#17984804—21no—
43C22⋊C4×D139—21no—
44#17984776C2211yescharacteristic, complemented
45C4⋊(C139⋊C8)C2211yescharacteristic, complemented
46#17984767C2211yescharacteristic, complemented
47#17984766C2211yescharacteristic, complemented
48#17984766C2211yescharacteristic, complemented
49#17984766C2211yescharacteristic, complemented
50#17984764C2211yescharacteristic
51#17984763C2211yescharacteristic
52(C2×D139)⋊C8C2211yescharacteristic
53(C2×D139)⋊C8C2211yescharacteristic, complemented
54C2×Q16×C139C2211yescharacteristic, complemented
55C2×Q16×C139C2211yescharacteristic, complemented
56C2×SD16×C139C2211yescharacteristic
57C2×SD16×C139C2211yescharacteristic
58C4⋊Q8×C139C2211yescharacteristic, complemented
59#124×C139C2211yescharacteristic
60#97×C139C2211yescharacteristic, complemented
61C23×D139—21no—
62#14555786—22no—
63#14555786—41no—
64C2×Dic278—21no—
65C2×Dic278—21no—
66C2×Dic278C2311yescharacteristic
67Q8×D139—22no—
68Q8×D139—22no—
69Q8×D139—41no—
70Q8×D139—41no—
71C2×D556—21no—
72C2×D556D411yescharacteristic
73C2×D556C2311yescharacteristic
74C2×C22⋊D139—21no—
75C2×C22⋊D139—21no—
76D4×D139—22no—
77D4×D139—41no—
78C2×C4×D139—21no—
79C2×C4×D139—21no—
80C2×C4×D139—21no—
81C2×C4×D139—21no—
82C2×C4×D139C2311yescharacteristic
83C2×C4×D139C2311yescharacteristic
84#14555775—22no—
85#14555774D414yescomplemented
86#14555773—22no—
87#14555772—22no—
88#14555772—22no—
89#14555768—22no—
90C2×C139⋊C8C2311yescharacteristic
91C2×C139⋊C8C2311yescharacteristic
92C8×D139—22no—
93C4⋊Dic139D411yescharacteristic
94C4⋊Dic139C2311yescharacteristic
95C4⋊Dic139C2311yescharacteristic
96Dic139⋊C4—21no—
97Dic139⋊C4—21no—
98Dic139⋊C4—21no—
99Dic139⋊C4—21no—
100C22⋊Dic139—21no—
101(C2×D139)⋊C4—21no—
102(C2×D139)⋊C4—21no—
103(C2×D139)⋊C4—21no—
104(C2×D139)⋊C4—21no—
105(C2×D139)⋊C4D411yescharacteristic
106(C2×D139)⋊C4D411yescharacteristic
107C4×Dic139—21no—
108C2×Q8×C139C2311yescharacteristic
109C2×Q8×C139C2311yescharacteristic
110C2×Q8×C139C2311yescharacteristic
111C2×D4×C139C2311yescharacteristic
112Q16×C139—22no—
113Q16×C139—41no—
114SD16×C139—22no—
115SD16×C139—41no—
116C2×C8×C139C2311yescharacteristic
117C2×C8×C139C2311yescharacteristic
118C4⋊C4×C139—21no—
119C4⋊C4×C139—21no—
120C22⋊C4×C139—21no—
121C22⋊C4×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—42no—
131Dic278—42no—
132Dic278C2×D412yes—
133D556—22no—
134D556—21no—
135D556—42no—
136D556C2×D412yes—
137C22⋊D139—42no—
138C22⋊D139—81no—
139C2×Dic139—21no—
140C2×Dic139—21no—
141C2×Dic139—21no—
142C2×Dic139C2×D411yescharacteristic
143C2×Dic139C2×D411yescharacteristic
144C4×D139—21no—
145C4×D139—21no—
146C4×D139—21no—
147C4×D139—21no—
148C4×D139—42no—
149C4×D139—42no—
150C4×D139—42no—
151C4×D139—81no—
152C139⋊C8—21no—
153C139⋊C8—21no—
154C23×C139—21no—
155Q8×C139—21no—
156Q8×C139—21no—
157Q8×C139—21no—
158Q8×C139—21no—
159Q8×C139—21no—
160Q8×C139—21no—
161D4×C139—21no—
162D4×C139—21no—
163C2×C4×C139—21no—
164C2×C4×C139—21no—
165C2×C4×C139—21no—
166C2×C4×C139C2×D411yescharacteristic
167C2×C4×C139C2×D411yescharacteristic
168C2×C4×C139C2411yescharacteristic
169C8×C139—21no—
170C8×C139C2×D412yes—
171C22×C139—41no—
172C22×C139—41no—
173C22×C139—41no—
174C22×C139C22×D411yescharacteristic
175C2×D139—21no—
176C2×D139—21no—
177C2×D139—21no—
178C2×D139—22no—
179C2×D139—21no—
180C2×D139—21no—
181C2×D139—41no—
182C2×D139—41no—
183C2×D139—84no—
184C2×D139#13712yescomplemented
185C4×C139—21no—
186C4×C139—21no—
187C4×C139—41no—
188C4×C139—41no—
189C4×C139—41no—
190C4×C139C22×D411yescharacteristic
191C4×C139H(5,𝔽2)11yescharacteristic
192Dic139—21no—
193Dic139—21no—
194Dic139—22no—
195Dic139—41no—
196Dic139—41no—
197C2×C139—81no—
198C2×C139D4211yescharacteristic
199C2×C139C2×#13711yescharacteristic
200C2×C139#57711yescharacteristic
201D139—22no—
202D139—41no—
203D139—42no—
204C139#326611yescharacteristic, complemented, Sylow
205#3266—1391nomaximal, Sylow
206C2×#137—1391no—
207C22×SD16—1391no—
208D4×Q8—1391no—
209#546—1391no—
210#497D13911yescharacteristic, complemented
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×Q8—1391no—
223C22×Q8—1391no—
224C22×D4—1391no—
225#137—2784no—
226C2×Q16—1391no—
227C2×Q16C2×D13911yescharacteristic, complemented
228C2×Q16C2×D13911yescharacteristic, complemented
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⋊Q8C2×D13911yescharacteristic, complemented
240#124C2×D13911yescharacteristic
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×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#103—1391no—
260#103—1391no—
261#102—1391no—
262#102—1391no—
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×Q8C22×D13911yescharacteristic
277C2×Q8C22×D13911yescharacteristic
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—2781no—
288C22×C4—2781no—
289C22×C4—2781no—
290C22×C4—2781no—
291C22×C4—1391no—
292C22×C4—1391no—
293Q16—22no—
294Q16—41no—
295Q16—2782no—
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—21no—
309C4⋊C4—21no—
310C4⋊C4—2781no—
311C4⋊C4—2781no—
312C4⋊C4—2781no—
313C4⋊C4—2781no—
314C4⋊C4—1391no—
315C4⋊C4—1391no—
316C4⋊C4—1391no—
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—
327C42C22×D13911yescharacteristic
328C23—21no—
329C23—2781no—
330C23—5562no—
331C23—5562no—
332C23—11121no—
333C23—1391no—
334C23—1391no—
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—21no—
341Q8—2781no—
342Q8—5562no—
343Q8—5562no—
344Q8—1392no—
345D4—21no—
346D4—21no—
347D4—2782no—
348D4—2781no—
349D4—5562no—
350D4—5562no—
351D4—11121no—
352D4—1392no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—2781no—
357C2×C4—2781no—
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×C4D4×D13911yescharacteristic
370C2×C4D4×D13911yescharacteristic
371C2×C4C23×D13911yescharacteristic
372C8—21no—
373C8—2781no—
374C8—2781no—
375C8D4×D13912yes—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—2781no—
380C22—2781no—
381C22—2781no—
382C22—2781no—
383C22—2782no—
384C22—2781no—
385C22—5561no—
386C22—5561no—
387C22—11124no—
388C22—1392no—
389C22C2×D4×D13911yescharacteristic
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—2781no—
396C4—2781no—
397C4—2782no—
398C4—5561no—
399C4—5561no—
400C4C2×D4×D13911yescharacteristic
401C4#1798488711yescharacteristic
402C2—81no—
403C2—2782no—
404C2—5561no—
405C2—5562no—
406C2#2258158811yescharacteristic
407C2#137×D13911yescharacteristic
408C2#2258176211yescharacteristic
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#497×C139C2
socleC22×C139C22×D4
prime radical#22581596C2
hypercenter#497D139
hypocenterC139#3266
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#17984874C2semidirect
2. C2×SD16×D139C2semidirect
3. #22581618C2semidirect
5. #22581196C2semidirect
6. #22581191C2semidirect
7. #22581172C2semidirect
8. #22581155C2semidirect
9. #22581114C2semidirect
10. #22581107C2semidirect
11. #22581102C2semidirect
12. #22581090C2semidirect
13. #22581055C2semidirect
14. #22581026C2semidirect
15. #497×C139C2semidirect
18. C2×Q8×D139C22semidirect
21. C2×#14555775C22semidirect
27. C2×#14555768C22semidirect
35. #17984818C22semidirect
39. C4×Dic278C22semidirect
44. #17984776C22semidirect
45. C4⋊(C139⋊C8)C22semidirect
46. #17984767C22semidirect
47. #17984766C22semidirect
48. #17984766C22semidirect
49. #17984766C22semidirect
53. (C2×D139)⋊C8C22semidirect
54. C2×Q16×C139C22semidirect
55. C2×Q16×C139C22semidirect
58. C4⋊Q8×C139C22semidirect
60. #97×C139C22semidirect
85. #14555774D4semidirect
184. C2×D139#137semidirect
204. C139#3266semidirect
210. #497D139semidirect
227. C2×Q16C2×D139semidirect
228. C2×Q16C2×D139semidirect
239. C4⋊Q8C2×D139semidirect
265. #97C2×D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848742
C2×SD16×D1392
#225816182
#225815962
#225811962
#225811912
#225811722
#225811552
#225811142
#225811072
#225811022
#225810902
#225810552
#225810262
#497×C1392
#3266139