Finite Group Databasebeta

← #14540313

Subgroups — #14540313

Order 2176.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587600C211yesmaximal, characteristic, complemented
2C2×#10587636C211yesmaximal, characteristic, complemented
3#11763461C211yesmaximal, characteristic, complemented
4#11763447C211yesmaximal, characteristic, complemented
5#11763202C211yesmaximal, characteristic, complemented
6#11763066C211yesmaximal, characteristic, complemented
7#11763040C211yesmaximal, characteristic, complemented
8#11763027C211yesmaximal, characteristic, complemented
9#11763012C211yesmaximal, characteristic, complemented
10#11762970C211yesmaximal, characteristic, complemented
11#11762961C211yesmaximal, characteristic, complemented
12#11762956C211yesmaximal, characteristic, complemented
13#11762918C211yesmaximal, characteristic, complemented
14#11762877C211yesmaximal, characteristic, complemented
15C17×#452C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C2×#63256C2211yescharacteristic
18C2×D4×D17C2211yescharacteristic
19C2×#63252C2211yescharacteristic, complemented
20C2×Q16×C17C2211yescharacteristic, complemented
21C2×SD16×C17C2211yescharacteristic
22C17×C22⋊Q8C2211yescharacteristic, complemented
23#10587636—24no—
24#10587633C2211yescharacteristic
25#10587632C2211yescharacteristic
26#10587630C2211yescharacteristic, complemented
27C2×#63231C2211yescharacteristic, complemented
28C2×#63230C2211yescharacteristic
29C2×#63230C2211yescharacteristic, complemented
30#10587625—21no—
31#10587623—21no—
32D4×Dic17C2211yescharacteristic
33C2×#63229C2211yescharacteristic
34C2×#63228C2211yescharacteristic
35#10587616C2211yescharacteristic
36#10587615—21no—
37C4×C22⋊D17C2211yescharacteristic
38C2×#63227—21no—
39C22×#3622C2211yescharacteristic, complemented
40#10587600—24no—
41C2×D136C2211yescharacteristic
42C2×#63219C2211yescharacteristic
43C2×#63218C2211yescharacteristic
44#10587583C2211yescharacteristic
45#10587580—21no—
46#10587579C2211yescharacteristic, complemented
47C4⋊C4×D17—21no—
48#10587570—21no—
49#10587569—21no—
50C17×#103C2211yescharacteristic, complemented
51C17×#102C2211yescharacteristic
52C17×C22⋊C8C2211yescharacteristic, complemented
53#10587506C2211yescharacteristic, complemented
54#10587492C2211yescharacteristic, complemented
55#10587491C2211yescharacteristic
56#10587487C2211yescharacteristic, complemented
57#10587486C2211yescharacteristic, complemented
58#10587485C2211yescharacteristic, complemented
59#10587480C2211yescharacteristic
60#10587479C2211yescharacteristic, complemented
61C23×D17—21no—
62#41×C17—22no—
63#41×C17—41no—
64C2×Q8×C17C2311yescharacteristic
65C2×Q8×C17C2311yescharacteristic
66C2×D4×C17—21no—
67C2×D4×C17C2311yescharacteristic
68C22×C4×C17—21no—
69C22×C4×C17C2311yescharacteristic
70C2×C22⋊D17—21no—
71C2×C22⋊D17—21no—
72C2×C22⋊D17—21no—
73C2×C22⋊D17D411yescharacteristic
74C22×Dic17—21no—
75#63256—22no—
76#63256—41no—
77D4×D17—22no—
78D4×D17—41no—
79#63252—22no—
80#63252—41no—
81C2×D68—21no—
82C2×D68C2311yescharacteristic
83C2×D68C2311yescharacteristic
84C2×C4×D17—21no—
85C2×C4×D17—21no—
86C2×C4×D17—21no—
87C2×C4×D17C2311yescharacteristic
88C2×Dic34C2311yescharacteristic
89Q16×C17—41no—
90SD16×C17—22no—
91C2×C8×C17C2311yescharacteristic
92C2×C8×C17C2311yescharacteristic
93C4⋊C4×C17—21no—
94C4⋊C4×C17C2311yescharacteristic
95C22⋊C4×C17—21no—
96C22⋊Dic17—21no—
97C22⋊Dic17D411yescharacteristic
98#63231—22no—
99#63230—22no—
100#63230—22no—
101#63229—22no—
102#63228D414yescomplemented
103#63227—21no—
104#63227—21no—
105#63227—21no—
106#63227—21no—
107#63227—21no—
108#63227—21no—
109#63227D411yescharacteristic
110C4⋊Dic17—21no—
111C4⋊Dic17C2311yescharacteristic
112#63225—21no—
113#63225—21no—
114#63225—21no—
115#63225D411yescharacteristic
116C4×Dic17C2311yescharacteristic
117C2×#3622—22no—
118C2×#3622C2311yescharacteristic
119C2×#3622C2311yescharacteristic
120D136—22no—
121#63219—41no—
122#63218—22no—
123C23×C17—21no—
124C23×C17C2×D411yescharacteristic
125C22×D17—21no—
126C22×D17—21no—
127C22×D17—42no—
128C22×D17—42no—
129C22×D17—81no—
130C22×D17C2×D411yescharacteristic
131Q8×C17—21no—
132Q8×C17—21no—
133Q8×C17—21no—
134Q8×C17—21no—
135D4×C17—21no—
136D4×C17—42no—
137D4×C17C2×D412yes—
138C2×C4×C17—21no—
139C2×C4×C17—21no—
140C2×C4×C17—21no—
141C2×C4×C17—21no—
142C2×C4×C17—21no—
143C2×C4×C17—42no—
144C2×C4×C17C2×D411yescharacteristic
145C2×C4×C17C2411yescharacteristic
146C22⋊D17—41no—
147C22⋊D17—42no—
148C22⋊D17—42no—
149C22⋊D17—81no—
150C2×Dic17—21no—
151C2×Dic17—21no—
152C2×Dic17—81no—
153C2×Dic17C2×D411yescharacteristic
154C2×Dic17C2×D411yescharacteristic
155C2×Dic17C2×D411yescharacteristic
156D68—21no—
157D68—21no—
158D68—21no—
159D68—42no—
160D68C2×D412yes—
161C4×D17—21no—
162C4×D17—21no—
163C4×D17—42no—
164C4×D17—42no—
165C4×D17—42no—
166Dic34—21no—
167Dic34—21no—
168C8×C17—21no—
169C8×C17—21no—
170#3622—21no—
171#3622C2×D412yes—
172#3267—171nomaximal, Sylow
173C22×C17—21no—
174C22×C17—22no—
175C22×C17—21no—
176C22×C17—21no—
177C22×C17—41no—
178C22×C17—41no—
179C22×C17C22×D411yescharacteristic
180C2×D17—21no—
181C2×D17—22no—
182C2×D17—21no—
183C2×D17—21no—
184C2×D17—41no—
185C2×D17—41no—
186C2×D17—41no—
187C2×D17—41no—
188C2×D17—41no—
189C2×D17—84no—
190C4×C17—21no—
191C4×C17—41no—
192C4×C17—41no—
193C4×C17—41no—
194C4×C17C22×D411yescharacteristic
195C4×C17H(5,𝔽2)11yescharacteristic
196Dic17—21no—
197Dic17—21no—
198Dic17—21no—
199Dic17—41no—
200Dic17—41no—
201C2×AGL(1,ℤ/8)—171no—
202C2×#135—171no—
203#547—171no—
204#546—171no—
205#497—171no—
206#470—171no—
207#465—171no—
208#463—171no—
209#461—171no—
210#452—171no—
211#452D1711yescharacteristic, complemented
212#450—171no—
213#449—171no—
214#442—171no—
215#435—171no—
216C2×C17—42no—
217C2×C17—41no—
218C2×C17D4211yescharacteristic
219C2×C17#57511yescharacteristic
220C2×C17#57711yescharacteristic
221D17—41no—
222D17—42no—
223D17—81no—
224C2×#41—171no—
225C2×#41—171no—
226C2×#41C2×D1711yescharacteristic
227C22×D4—171no—
228AGL(1,ℤ/8)—344no—
229#135—344no—
230C2×Q16—171no—
231C2×Q16C2×D1711yescharacteristic, complemented
232C2×SD16—171no—
233C2×SD16—171no—
234C2×SD16—171no—
235C2×SD16—171no—
236C2×SD16C2×D1711yescharacteristic
237C2×D8—171no—
238C2×D8—171no—
239C2×M16—171no—
240C22×C8—171no—
241C4⋊Q8—171no—
242#124—171no—
243#123—341no—
244#123—341no—
245#123—341no—
246C22⋊Q8—171no—
247C22⋊Q8—171no—
248C22⋊Q8C2×D1711yescharacteristic, complemented
249#121—171no—
250#121—341no—
251#121—341no—
252#120—341no—
253C4×D4—171no—
254C4×D4—171no—
255C4×D4—171no—
256C2×C4⋊C4—341no—
257C2×C22⋊C4—341no—
258#106—171no—
259C4⋊C8—171no—
260#103—171no—
261#103—171no—
262#103C2×D1711yescharacteristic, complemented
263#102—171no—
264#102—171no—
265#102C2×D1711yescharacteristic
266C22⋊C8—171no—
267C22⋊C8C2×D1711yescharacteristic, complemented
268#97—171no—
269C17#326711yescharacteristic, complemented, Sylow
270C24—341no—
271#41—22no—
272#41—41no—
273#41—342no—
274#41—342no—
275#41—681no—
276#41—681no—
277C2×Q8—171no—
278C2×Q8C22×D1711yescharacteristic
279C2×Q8C22×D1711yescharacteristic
280C2×D4—21no—
281C2×D4—171no—
282C2×D4—171no—
283C2×D4—171no—
284C2×D4—341no—
285C2×D4—341no—
286C2×D4—342no—
287C2×D4—341no—
288C2×D4—341no—
289C2×D4—681no—
290C2×D4C22×D1711yescharacteristic
291C22×C4—21no—
292C22×C4—171no—
293C22×C4—341no—
294C22×C4—341no—
295C22×C4—341no—
296C22×C4—341no—
297C22×C4C22×D1711yescharacteristic
298Q16—41no—
299Q16—342no—
300SD16—22no—
301SD16—342no—
302SD16—342no—
303SD16—342no—
304SD16—681no—
305D8—174no—
306D8—342no—
307M16—342no—
308C2×C8—171no—
309C2×C8—171no—
310C2×C8—342no—
311C2×C8C22×D1711yescharacteristic
312C2×C8C22×D1711yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—171no—
315C4⋊C4—171no—
316C4⋊C4—341no—
317C4⋊C4—341no—
318C4⋊C4—341no—
319C4⋊C4—341no—
320C4⋊C4C22×D1711yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—171no—
323C22⋊C4—171no—
324C22⋊C4—341no—
325C22⋊C4—341no—
326C22⋊C4—341no—
327C22⋊C4—341no—
328C22⋊C4—341no—
329C22⋊C4—341no—
330C22⋊C4—341no—
331C42—171no—
332C23—21no—
333C23—171no—
334C23—341no—
335C23—341no—
336C23—682no—
337C23—682no—
338C23—1361no—
339C23D4×D1711yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—341no—
345Q8—341no—
346D4—21no—
347D4—42no—
348D4—172no—
349D4—341no—
350D4—341no—
351D4—341no—
352D4—682no—
353D4—681no—
354D4—682no—
355D4—682no—
356D4—1361no—
357D4D4×D1712yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—42no—
364C2×C4—171no—
365C2×C4—171no—
366C2×C4—171no—
367C2×C4—341no—
368C2×C4—341no—
369C2×C4—341no—
370C2×C4—341no—
371C2×C4—682no—
372C2×C4—682no—
373C2×C4—682no—
374C2×C4—1361no—
375C2×C4D4×D1711yescharacteristic
376C2×C4C23×D1711yescharacteristic
377C8—21no—
378C8—21no—
379C8—172no—
380C8—341no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—341no—
388C22—341no—
389C22—341no—
390C22—342no—
391C22—681no—
392C22—681no—
393C22—681no—
394C22—681no—
395C22—681no—
396C22—1364no—
397C22C2×D4×D1711yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—341no—
403C4—341no—
404C4—341no—
405C4—681no—
406C4—681no—
407C4C2×D4×D1711yescharacteristic
408C4#1058769111yescharacteristic
409C2—41no—
410C2—42no—
411C2—681no—
412C2—682no—
413C2—1361no—
414C2#1176344111yescharacteristic
415C2#1176360611yescharacteristic
416C2#1176361811yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#452C2
socleC22×C17C22×D4
hypercenter#452D17
hypocenterC17#3267
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#10587600C2semidirect
2. C2×#10587636C2semidirect
3. #11763461C2semidirect
4. #11763447C2semidirect
5. #11763202C2semidirect
6. #11763066C2semidirect
7. #11763040C2semidirect
8. #11763027C2semidirect
9. #11763012C2semidirect
10. #11762970C2semidirect
11. #11762961C2semidirect
12. #11762956C2semidirect
13. #11762918C2semidirect
14. #11762877C2semidirect
15. C17×#452C2semidirect
19. C2×#63252C22semidirect
20. C2×Q16×C17C22semidirect
22. C17×C22⋊Q8C22semidirect
26. #10587630C22semidirect
27. C2×#63231C22semidirect
29. C2×#63230C22semidirect
39. C22×#3622C22semidirect
46. #10587579C22semidirect
50. C17×#103C22semidirect
52. C17×C22⋊C8C22semidirect
53. #10587506C22semidirect
54. #10587492C22semidirect
56. #10587487C22semidirect
57. #10587486C22semidirect
58. #10587485C22semidirect
60. #10587479C22semidirect
102. #63228D4semidirect
211. #452D17semidirect
231. C2×Q16C2×D17semidirect
248. C22⋊Q8C2×D17semidirect
262. #103C2×D17semidirect
267. C22⋊C8C2×D17semidirect
269. C17#3267semidirect

Maximal subgroups

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

subgroupindex
C2×#105876002
C2×#105876362
#117634612
#117634472
#117632022
#117630662
#117630402
#117630272
#117630122
#117629702
#117629612
#117629562
#117629182
#117628772
C17×#4522
#326717