Finite Group Databasebeta

← #14539643

Subgroups — #14539643

Order 2176.

422 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763390C211yesmaximal, characteristic, complemented
2#11763375C211yesmaximal, characteristic, complemented
3#11763152C211yesmaximal, characteristic, complemented
4#11763136C211yesmaximal, characteristic, complemented
5#11763127C211yesmaximal, characteristic, complemented
6#11763061C211yesmaximal, characteristic, complemented
7#11763056C211yesmaximal, characteristic, complemented
8#11763046C211yesmaximal, characteristic, complemented
9#11763037C211yesmaximal, characteristic, complemented
10#11762961C211yesmaximal, characteristic, complemented
11#11762956C211yesmaximal, characteristic, complemented
12#11762952C211yesmaximal, characteristic, complemented
13#11762944C211yesmaximal, characteristic, complemented
14#11762774C211yesmaximal, characteristic, complemented
15C17×#480C211yesmaximal, characteristic, complemented
16C2×D4×D17C2211yescharacteristic
17C2×#63252C2211yescharacteristic
18C2×#63252C2211yescharacteristic, complemented
19C17×#121C2211yescharacteristic
20C2×C4⋊C4×C17C2211yescharacteristic, complemented
21#10587626—21no—
22#10587624C2211yescharacteristic
23#10587623—21no—
24C2×#63229C2211yescharacteristic
25C2×#63228C2211yescharacteristic
26#10587616—21no—
27#10587616—21no—
28#10587616C2211yescharacteristic
29C2×D136C2211yescharacteristic
30C2×#63219C2211yescharacteristic
31C2×#63218C2211yescharacteristic, complemented
32C2×C8×D17C2211yescharacteristic, complemented
33#10587583—21no—
34#10587583C2211yescharacteristic
35#10587583C2211yescharacteristic, complemented
36#10587581—21no—
37#10587581C2211yescharacteristic
38#10587581C2211yescharacteristic
39#10587581C2211yescharacteristic, complemented
40#10587578C2211yescharacteristic
41#10587571—21no—
42#10587568—21no—
43C17×#107C2211yescharacteristic, complemented
44C17×#107C2211yescharacteristic, complemented
45C17×#102C2211yescharacteristic
46C17×#102C2211yescharacteristic
47C17×C22⋊C8C2211yescharacteristic, complemented
48C22⋊#3622C2211yescharacteristic, complemented
49#10587492C2211yescharacteristic, complemented
50#10587491C2211yescharacteristic
51#10587491C2211yescharacteristic
52#10587487C2211yescharacteristic, complemented
53#10587481C2211yescharacteristic, complemented
54#10587480C2211yescharacteristic
55#10587480C2211yescharacteristic
56#10587480C2211yescharacteristic, complemented
57#10587479C2211yescharacteristic, complemented
58#10587479C2211yescharacteristic, complemented
59C23×D17—21no—
60C2×D4×C17—21no—
61C2×D4×C17C2311yescharacteristic
62C22×C4×C17—21no—
63C22×C4×C17C2311yescharacteristic
64C2×C22⋊D17—21no—
65C2×C22⋊D17—21no—
66C2×C22⋊D17—21no—
67C2×C22⋊D17—21no—
68C2×C22⋊D17D411yescharacteristic
69C2×C22⋊D17D411yescharacteristic
70D4×D17—41no—
71D4×D17—41no—
72#63252—41no—
73#63252—41no—
74#63252—41no—
75#63252—41no—
76C2×D68—21no—
77C2×D68—21no—
78C2×D68—21no—
79C2×D68D411yescharacteristic
80C2×D68C2311yescharacteristic
81C2×D68C2311yescharacteristic
82C2×C4×D17—21no—
83C2×C4×D17C2311yescharacteristic
84C2×C4×D17C2311yescharacteristic
85C2×Dic34D411yescharacteristic
86C2×Dic34C2311yescharacteristic
87C2×C8×C17C2311yescharacteristic
88C2×C8×C17C2311yescharacteristic
89C4⋊C4×C17—21no—
90C4⋊C4×C17C2311yescharacteristic
91C4⋊C4×C17C2311yescharacteristic
92C4⋊C4×C17C2311yescharacteristic
93C22⋊C4×C17—21no—
94C22⋊Dic17—21no—
95#63229—41no—
96#63228—41no—
97#63227—21no—
98#63227—21no—
99#63227—21no—
100#63227—21no—
101#63227—21no—
102#63227—21no—
103#63227—21no—
104C4⋊Dic17—21no—
105C4⋊Dic17—21no—
106C4⋊Dic17C2311yescharacteristic
107C4×Dic17—21no—
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110D136—41no—
111#63219—41no—
112#63218—41no—
113C8×D17—41no—
114C23×C17—21no—
115C23×C17C2×D411yescharacteristic
116C22×D17—21no—
117C22×D17—21no—
118C22×D17—42no—
119C22×D17—81no—
120C22×D17—81no—
121C22×D17C2×D411yescharacteristic
122C22×D17C2×D411yescharacteristic
123D4×C17—21no—
124D4×C17—21no—
125D4×C17—81no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—81no—
132C2×C4×C17C2×D411yescharacteristic
133C2×C4×C17C2411yescharacteristic
134C22⋊D17—41no—
135C22⋊D17—41no—
136C22⋊D17—42no—
137C22⋊D17—42no—
138C22⋊D17—81no—
139C22⋊D17—81no—
140C2×Dic17—21no—
141C2×Dic17—21no—
142C2×Dic17C2×D411yescharacteristic
143C2×Dic17C2×D411yescharacteristic
144D68—21no—
145D68—21no—
146D68—21no—
147D68—21no—
148D68—41no—
149D68—81no—
150D68—81no—
151D68—81no—
152C4×D17—21no—
153C4×D17—21no—
154C4×D17—21no—
155C4×D17—21no—
156C4×D17—42no—
157Dic34—21no—
158Dic34—21no—
159Dic34—41no—
160C8×C17—21no—
161C8×C17—21no—
162#3622—21no—
163#3622—21no—
164#3198—171nomaximal, Sylow
165C22×C17—21no—
166C22×C17—21no—
167C22×C17—21no—
168C22×C17—41no—
169C22×C17—41no—
170C22×C17—41no—
171C22×C17C22×D411yescharacteristic
172C2×D17—21no—
173C2×D17—21no—
174C2×D17—21no—
175C2×D17—21no—
176C2×D17—21no—
177C2×D17—21no—
178C2×D17—41no—
179C2×D17—41no—
180C2×D17—41no—
181C2×D17—41no—
182C2×D17—41no—
183C2×D17—41no—
184C2×D17—84no—
185C4×C17—21no—
186C4×C17—41no—
187C4×C17—41no—
188C4×C17—41no—
189C4×C17H(5,𝔽2)11yescharacteristic
190C4×C17H(5,𝔽2)11yescharacteristic
191Dic17—21no—
192Dic17—21no—
193Dic17—41no—
194Dic17—41no—
195#537—171no—
196#535—171no—
197#485—171no—
198#482—171no—
199#481—171no—
200#480D1711yescharacteristic, complemented
201#469—171no—
202#468—171no—
203#466—171no—
204#465—171no—
205#450—171no—
206#449—171no—
207#449—171no—
208#447—171no—
209#408—171no—
210C2×C17—41no—
211C2×C17—81no—
212C2×C17#53411yescharacteristic
213C2×C17#57611yescharacteristic
214C2×C17#57711yescharacteristic
215D17—41no—
216D17—41no—
217D17—81no—
218D17—81no—
219C2×#41—171no—
220C2×#41—171no—
221C22×D4—171no—
222C2×SD16—171no—
223C2×SD16—171no—
224C2×D8—171no—
225C2×D8—171no—
226C2×M16—171no—
227C22×C8—171no—
228#127—341no—
229#124—341no—
230C22⋊Q8—171no—
231C22⋊Q8—171no—
232C22⋊Q8—341no—
233#121—171no—
234#121—171no—
235#121—171no—
236#121—171no—
237#121—171no—
238#121—341no—
239#121—341no—
240#121—341no—
241#121C2×D1711yescharacteristic
242#120—341no—
243#120—341no—
244#117—171no—
245C2×C4⋊C4C2×D1711yescharacteristic, complemented
246#107C2×D1711yescharacteristic, complemented
247#107C2×D1711yescharacteristic, complemented
248#106—171no—
249#106—171no—
250#103—171no—
251#103—171no—
252#102—171no—
253#102—171no—
254#102—171no—
255#102—171no—
256#102C2×D1711yescharacteristic
257#102C2×D1711yescharacteristic
258C22⋊C8—171no—
259C22⋊C8—171no—
260C22⋊C8—171no—
261C22⋊C8C2×D1711yescharacteristic, complemented
262C17#319811yescharacteristic, complemented, Sylow
263C24—341no—
264#41—681no—
265#41—681no—
266#41—681no—
267#41—681no—
268C2×Q8—171no—
269C2×Q8—171no—
270C2×D4—21no—
271C2×D4—171no—
272C2×D4—171no—
273C2×D4—171no—
274C2×D4—171no—
275C2×D4—171no—
276C2×D4—341no—
277C2×D4—341no—
278C2×D4—341no—
279C2×D4—341no—
280C2×D4—341no—
281C2×D4—341no—
282C2×D4—341no—
283C2×D4—681no—
284C2×D4—681no—
285C2×D4C22×D1711yescharacteristic
286C22×C4—21no—
287C22×C4—171no—
288C22×C4—171no—
289C22×C4—341no—
290C22×C4C22×D1711yescharacteristic
291SD16—681no—
292SD16—681no—
293D8—681no—
294D8—681no—
295M16—681no—
296C2×C8—171no—
297C2×C8—171no—
298C2×C8—681no—
299C2×C8C22×D1711yescharacteristic
300C2×C8C22×D1711yescharacteristic
301C4⋊C4—21no—
302C4⋊C4—171no—
303C4⋊C4—341no—
304C4⋊C4—341no—
305C4⋊C4C22×D1711yescharacteristic
306C4⋊C4C22×D1711yescharacteristic
307C4⋊C4C22×D1711yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—341no—
310C22⋊C4—341no—
311C22⋊C4—341no—
312C22⋊C4—341no—
313C22⋊C4—341no—
314C22⋊C4—341no—
315C22⋊C4—341no—
316C22⋊C4—341no—
317C42—341no—
318C23—21no—
319C23—171no—
320C23—171no—
321C23—341no—
322C23—341no—
323C23—682no—
324C23—1361no—
325C23—1361no—
326C23D4×D1711yescharacteristic
327Q8—341no—
328Q8—341no—
329Q8—681no—
330D4—21no—
331D4—21no—
332D4—81no—
333D4—341no—
334D4—341no—
335D4—341no—
336D4—341no—
337D4—681no—
338D4—681no—
339D4—681no—
340D4—682no—
341D4—682no—
342D4—1361no—
343D4—1361no—
344D4—1361no—
345D4—1361no—
346D4—1361no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—81no—
353C2×C4—171no—
354C2×C4—171no—
355C2×C4—341no—
356C2×C4—341no—
357C2×C4—341no—
358C2×C4—341no—
359C2×C4—341no—
360C2×C4—341no—
361C2×C4—682no—
362C2×C4D4×D1711yescharacteristic
363C2×C4C23×D1711yescharacteristic
364C8—21no—
365C8—21no—
366C8—341no—
367C8—341no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—341no—
375C22—341no—
376C22—341no—
377C22—341no—
378C22—341no—
379C22—341no—
380C22—681no—
381C22—681no—
382C22—681no—
383C22—681no—
384C22—681no—
385C22—681no—
386C22—1364no—
387C22C2×D4×D1711yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—341no—
393C4—341no—
394C4—681no—
395C4—681no—
396C4#1058769611yescharacteristic
397C4#1058769611yescharacteristic
398C2—41no—
399C2—81no—
400C2—681no—
401C2—681no—
402C2—1361no—
403C2—1361no—
404C2#1176337011yescharacteristic
405C2#1176361011yescharacteristic
406C2#1176361811yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#480C2
socleC22×C17C22×D4
hypercenter#480D17
hypocenterC17#3198
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #11763390C2semidirect
2. #11763375C2semidirect
3. #11763152C2semidirect
4. #11763136C2semidirect
5. #11763127C2semidirect
6. #11763061C2semidirect
7. #11763056C2semidirect
8. #11763046C2semidirect
9. #11763037C2semidirect
10. #11762961C2semidirect
11. #11762956C2semidirect
12. #11762952C2semidirect
13. #11762944C2semidirect
14. #11762774C2semidirect
15. C17×#480C2semidirect
18. C2×#63252C22semidirect
20. C2×C4⋊C4×C17C22semidirect
31. C2×#63218C22semidirect
32. C2×C8×D17C22semidirect
35. #10587583C22semidirect
39. #10587581C22semidirect
43. C17×#107C22semidirect
44. C17×#107C22semidirect
47. C17×C22⋊C8C22semidirect
48. C22⋊#3622C22semidirect
49. #10587492C22semidirect
52. #10587487C22semidirect
53. #10587481C22semidirect
56. #10587480C22semidirect
57. #10587479C22semidirect
58. #10587479C22semidirect
200. #480D17semidirect
245. C2×C4⋊C4C2×D17semidirect
246. #107C2×D17semidirect
247. #107C2×D17semidirect
261. C22⋊C8C2×D17semidirect
262. C17#3198semidirect

Maximal subgroups

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

subgroupindex
#117633902
#117633752
#117631522
#117631362
#117631272
#117630612
#117630562
#117630462
#117630372
#117629612
#117629562
#117629522
#117629442
#117627742
C17×#4802
#319817