Finite Group Databasebeta

← #14541019

Subgroups — #14541019

Order 2176.

460 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763523C211yesmaximal, characteristic, complemented
2D4×C22⋊D17C211yesmaximal, characteristic, complemented
3#11763154C211yesmaximal, characteristic, complemented
4#11763134C211yesmaximal, characteristic, complemented
5#11763131C211yesmaximal, characteristic, complemented
6#11763097C211yesmaximal, characteristic, complemented
7#11763060C211yesmaximal, characteristic, complemented
8#11763059C211yesmaximal, characteristic, complemented
9#11763002C211yesmaximal, characteristic, complemented
10#11762941C211yesmaximal, characteristic, complemented
11D8×Dic17C211yesmaximal, characteristic, complemented
12#11762884C211yesmaximal, characteristic, complemented
13#11762814C211yesmaximal, characteristic, complemented
14#11762812C211yesmaximal, characteristic, complemented
15C17×#447C211yesmaximal, characteristic, complemented
16C22×D4×C17C2211yescharacteristic
17C22×C22⋊D17—21no—
18C2×D4×D17C2211yescharacteristic
19C2×D8×C17C2211yescharacteristic
20C2×D8×C17C2211yescharacteristic, complemented
21C17×#121C2211yescharacteristic, complemented
22#10587639—21no—
23#10587626C2211yescharacteristic
24#10587625—21no—
25#10587625—21no—
26#10587624C2211yescharacteristic
27#10587624C2211yescharacteristic, complemented
28#10587623—21no—
29#10587622—21no—
30D4×Dic17C2211yescharacteristic
31D4×Dic17C2211yescharacteristic, complemented
32C2×#63228C2211yescharacteristic, complemented
33#10587616C2211yescharacteristic
34C4×C22⋊D17C2211yescharacteristic
35#10587612C2211yescharacteristic, complemented
36C2×#63223C2211yescharacteristic, complemented
37C2×#63218C2211yescharacteristic, complemented
38#10587578C2211yescharacteristic, complemented
39#10587574C2211yescharacteristic, complemented
40#10587571—21no—
41#10587563—21no—
42C17×#102C2211yescharacteristic
43C17×#102C2211yescharacteristic, complemented
44C17×C22⋊C8C2211yescharacteristic, complemented
45#10587503C2211yescharacteristic
46#10587503C2211yescharacteristic
47#10587503C2211yescharacteristic, complemented
48#10587503C2211yescharacteristic, complemented
49#10587503C2211yescharacteristic, complemented
50#10587492C2211yescharacteristic, complemented
51#10587492C2211yescharacteristic, complemented
52#10587491C2211yescharacteristic, complemented
53#10587489C2211yescharacteristic, complemented
54#10587488C2211yescharacteristic, complemented
55#10587485C2211yescharacteristic, complemented
56C8×Dic17C2211yescharacteristic, complemented
57#10587480C2211yescharacteristic, complemented
58#10587479C2211yescharacteristic, complemented
59C24×C17—21no—
60C23×D17—21no—
61C2×D4×C17—22no—
62C2×D4×C17—21no—
63C2×D4×C17—21no—
64C2×D4×C17—41no—
65C2×D4×C17C2311yescharacteristic
66C2×D4×C17C2311yescharacteristic
67C2×D4×C17C2311yescharacteristic
68C22×C4×C17C2311yescharacteristic
69C2×C22⋊D17—21no—
70C2×C22⋊D17—21no—
71C2×C22⋊D17—21no—
72C2×C22⋊D17—21no—
73C2×C22⋊D17—21no—
74C2×C22⋊D17—21no—
75C2×C22⋊D17—44no—
76C2×C22⋊D17D411yescharacteristic
77C22×Dic17—21no—
78C22×Dic17—21no—
79D4×D17—22no—
80D4×D17—41no—
81C2×D68C2311yescharacteristic
82C2×C4×D17C2311yescharacteristic
83C2×Dic34—21no—
84D8×C17—22no—
85D8×C17—41no—
86C2×C8×C17C2311yescharacteristic
87C2×C8×C17C2311yescharacteristic
88C4⋊C4×C17C2311yescharacteristic
89C22⋊C4×C17—21no—
90C22⋊Dic17—21no—
91C22⋊Dic17—21no—
92C22⋊Dic17—21no—
93C22⋊Dic17—21no—
94C22⋊Dic17—21no—
95C22⋊Dic17D411yescharacteristic
96#63228—41no—
97#63227—21no—
98#63227—21no—
99#63227D411yescharacteristic
100C4⋊Dic17C2311yescharacteristic
101C4⋊Dic17C2311yescharacteristic
102C4⋊Dic17C2311yescharacteristic
103#63225—21no—
104#63225D411yescharacteristic
105C4×Dic17—21no—
106C4×Dic17—21no—
107C4×Dic17C2311yescharacteristic
108#63223—41no—
109C2×#3622C2311yescharacteristic
110C2×#3622C2311yescharacteristic
111#63218—41no—
112C23×C17—21no—
113C23×C17—21no—
114C23×C17—21no—
115C23×C17—42no—
116C23×C17—42no—
117C23×C17—81no—
118C23×C17C2×D411yescharacteristic
119C22×D17—21no—
120C22×D17—42no—
121C22×D17—42no—
122C22×D17—81no—
123C22×D17C2×D411yescharacteristic
124D4×C17—21no—
125D4×C17—21no—
126D4×C17—21no—
127D4×C17—21no—
128D4×C17—21no—
129D4×C17—42no—
130D4×C17—81no—
131D4×C17#4112yes—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17C2×D411yescharacteristic
136C2×C4×C17C2411yescharacteristic
137C22⋊D17—22no—
138C22⋊D17—42no—
139C22⋊D17—42no—
140C22⋊D17—42no—
141C22⋊D17—42no—
142C22⋊D17—81no—
143C22⋊D17—81no—
144C2×Dic17—21no—
145C2×Dic17—21no—
146C2×Dic17—21no—
147C2×Dic17—42no—
148C2×Dic17—42no—
149C2×Dic17C2×D411yescharacteristic
150C2×Dic17C2×D411yescharacteristic
151C2×Dic17C2×D411yescharacteristic
152D68—21no—
153D68—21no—
154C4×D17—21no—
155C4×D17—21no—
156Dic34—81no—
157C8×C17—21no—
158C8×C17—21no—
159#3622—21no—
160#3622—21no—
161#3325—171nomaximal, Sylow
162C22×C17—22no—
163C22×C17—21no—
164C22×C17—21no—
165C22×C17—21no—
166C22×C17—41no—
167C22×C17—41no—
168C22×C17—41no—
169C22×C17—41no—
170C22×C17—41no—
171C22×C17—41no—
172C22×C17—41no—
173C22×C17—41no—
174C22×C17—84no—
175C22×C17C22×D411yescharacteristic
176C2×D17—21no—
177C2×D17—21no—
178C2×D17—21no—
179C2×D17—41no—
180C2×D17—41no—
181C2×D17—41no—
182C2×D17—84no—
183C4×C17—21no—
184C4×C17—41no—
185C4×C17C2×#4111yescharacteristic
186C4×C17H(5,𝔽2)11yescharacteristic
187Dic17—21no—
188Dic17—21no—
189Dic17—41no—
190Dic17—41no—
191Dic17—41no—
192Dic17AGL(1,ℤ/8)12yes—
193#555—171no—
194D42—171no—
195#486—171no—
196#482—171no—
197#482—171no—
198#476—171no—
199#469—171no—
200#469—171no—
201#459—171no—
202#447—171no—
203#447D1711yescharacteristic, complemented
204C4×D8—171no—
205#436—171no—
206#418—171no—
207#418—171no—
208C2×C17—42no—
209C2×C17—41no—
210C2×C17—81no—
211C2×C17—81no—
212C2×C17#54611yescharacteristic
213C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C17#57611yescharacteristic
215D17—41no—
216D17—81no—
217C22×D4—171no—
218C22×D4—341no—
219C22×D4C2×D1711yescharacteristic
220C2×D8—171no—
221C2×D8C2×D1711yescharacteristic
222C2×D8C2×D1711yescharacteristic, complemented
223C2×M16—171no—
224C2×M16—171no—
225C4⋊Q8—171no—
226#127—171no—
227#124—341no—
228#124—341no—
229#121—171no—
230#121—171no—
231#121—171no—
232#121—341no—
233#121—341no—
234#121C2×D1711yescharacteristic, complemented
235#120—341no—
236#120—341no—
237C4×D4—171no—
238C4×D4—171no—
239C4×D4—171no—
240#117—171no—
241#117—171no—
242#117—341no—
243#107—171no—
244#106—171no—
245#106—171no—
246C4⋊C8—171no—
247#102—171no—
248#102—171no—
249#102—171no—
250#102—171no—
251#102—171no—
252#102—171no—
253#102—171no—
254#102—171no—
255#102C2×D1711yescharacteristic
256#102C2×D1711yescharacteristic, complemented
257C22⋊C8—171no—
258C22⋊C8C2×D1711yescharacteristic, complemented
259C4×C8—171no—
260C17#332511yescharacteristic, complemented, Sylow
261C24—21no—
262C24—341no—
263C2×Q8—341no—
264C2×D4—22no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—41no—
268C2×D4—171no—
269C2×D4—171no—
270C2×D4—341no—
271C2×D4—341no—
272C2×D4—341no—
273C2×D4—342no—
274C2×D4—341no—
275C2×D4—341no—
276C2×D4—341no—
277C2×D4—681no—
278C2×D4—684no—
279C2×D4C22×D1711yescharacteristic
280C2×D4C22×D1711yescharacteristic
281C2×D4C22×D1711yescharacteristic
282C22×C4—171no—
283C22×C4—341no—
284C22×C4—341no—
285C22×C4C22×D1711yescharacteristic
286D8—22no—
287D8—41no—
288D8—681no—
289M16—681no—
290M16—681no—
291C2×C8—171no—
292C2×C8—171no—
293C2×C8C22×D1711yescharacteristic
294C2×C8C22×D1711yescharacteristic
295C4⋊C4—171no—
296C4⋊C4—171no—
297C4⋊C4—171no—
298C4⋊C4—171no—
299C4⋊C4—341no—
300C4⋊C4C22×D1711yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—171no—
303C22⋊C4—171no—
304C22⋊C4—341no—
305C22⋊C4—341no—
306C22⋊C4—341no—
307C22⋊C4—341no—
308C22⋊C4—341no—
309C22⋊C4—341no—
310C22⋊C4—341no—
311C42—171no—
312C42—341no—
313C42—341no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—42no—
318C23—42no—
319C23—81no—
320C23—171no—
321C23—341no—
322C23—682no—
323C23—682no—
324C23—1361no—
325C23D4×D1711yescharacteristic
326Q8—1361no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—21no—
332D4—42no—
333D4—81no—
334D4—342no—
335D4—341no—
336D4—341no—
337D4—682no—
338D4—682no—
339D4—682no—
340D4—682no—
341D4—1361no—
342D4—1361no—
343D4#6325412yes—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—171no—
348C2×C4—171no—
349C2×C4—171no—
350C2×C4—341no—
351C2×C4—341no—
352C2×C4—341no—
353C2×C4—341no—
354C2×C4—341no—
355C2×C4—682no—
356C2×C4—682no—
357C2×C4D4×D1711yescharacteristic
358C2×C4C23×D1711yescharacteristic
359C8—21no—
360C8—21no—
361C8—341no—
362C8—341no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—22no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—84no—
376C22—341no—
377C22—341no—
378C22—341no—
379C22—681no—
380C22—681no—
381C22—681no—
382C22—1364no—
383C22C2×D4×D1711yescharacteristic
384C4—21no—
385C4—41no—
386C4—172no—
387C4—341no—
388C4—341no—
389C4—681no—
390C4—681no—
391C4—681no—
392C4C2×#6325411yescharacteristic
393C4#1058769111yescharacteristic
394C2—41no—
395C2—42no—
396C2—81no—
397C2—81no—
398C2—681no—
399C2—1361no—
400C2#1176344611yescharacteristic
401C2AGL(1,ℤ/8)×D1711yescharacteristic
402C2#1176361011yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#447C2
socleC22×C17C22×D4
hypercenter#447D17
hypocenterC17#3325
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #11763523C2semidirect
2. D4×C22⋊D17C2semidirect
3. #11763154C2semidirect
4. #11763134C2semidirect
5. #11763131C2semidirect
6. #11763097C2semidirect
7. #11763060C2semidirect
8. #11763059C2semidirect
9. #11763002C2semidirect
10. #11762941C2semidirect
11. D8×Dic17C2semidirect
12. #11762884C2semidirect
13. #11762814C2semidirect
14. #11762812C2semidirect
15. C17×#447C2semidirect
20. C2×D8×C17C22semidirect
21. C17×#121C22semidirect
27. #10587624C22semidirect
31. D4×Dic17C22semidirect
32. C2×#63228C22semidirect
35. #10587612C22semidirect
36. C2×#63223C22semidirect
37. C2×#63218C22semidirect
38. #10587578C22semidirect
39. #10587574C22semidirect
43. C17×#102C22semidirect
44. C17×C22⋊C8C22semidirect
47. #10587503C22semidirect
48. #10587503C22semidirect
49. #10587503C22semidirect
50. #10587492C22semidirect
51. #10587492C22semidirect
52. #10587491C22semidirect
53. #10587489C22semidirect
54. #10587488C22semidirect
55. #10587485C22semidirect
56. C8×Dic17C22semidirect
57. #10587480C22semidirect
58. #10587479C22semidirect
203. #447D17semidirect
222. C2×D8C2×D17semidirect
234. #121C2×D17semidirect
256. #102C2×D17semidirect
258. C22⋊C8C2×D17semidirect
260. C17#3325semidirect

Maximal subgroups

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

subgroupindex
#117635232
D4×C22⋊D172
#117631542
#117631342
#117631312
#117630972
#117630602
#117630592
#117630022
#117629412
D8×Dic172
#117628842
#117628142
#117628122
C17×#4472
#332517