Finite Group Databasebeta

← #14539658

Subgroups — #14539658

Order 2176.

410 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763385C211yesmaximal, characteristic, complemented
2#11763375C211yesmaximal, characteristic
3#11763153C211yesmaximal, characteristic, complemented
4#11763152C211yesmaximal, characteristic, complemented
5#11763133C211yesmaximal, characteristic, complemented
6#11763056C211yesmaximal, characteristic, complemented
7#11763054C211yesmaximal, characteristic, complemented
8#11763039C211yesmaximal, characteristic, complemented
9#11763037C211yesmaximal, characteristic, complemented
10#11762961C211yesmaximal, characteristic, complemented
11#11762961C211yesmaximal, characteristic, complemented
12#11762950C211yesmaximal, characteristic, complemented
13#11762947C211yesmaximal, characteristic, complemented
14#11762774C211yesmaximal, characteristic, complemented
15C17×#482C211yesmaximal, characteristic, complemented
16C2×D4×D17C2211yescharacteristic
17C2×#63252C2211yescharacteristic
18C22×Dic34C2211yescharacteristic, complemented
19C17×#121C2211yescharacteristic
20C17×#117C2211yescharacteristic, complemented
21#10587626—21no—
22#10587624C2211yescharacteristic
23#10587623—21no—
24C2×#63229C2211yescharacteristic
25C2×#63229C2211yescharacteristic
26#10587616C2211yescharacteristic
27C2×#63219C2211yescharacteristic
28C2×#63219C2211yescharacteristic
29C2×#63218C2211yescharacteristic, complemented
30C2×C8×D17C2211yescharacteristic, complemented
31#10587583C2211yescharacteristic
32#10587583C2211yescharacteristic
33#10587583C2211yescharacteristic, complemented
34#10587583C2211yescharacteristic, complemented
35#10587581C2211yescharacteristic
36#10587578C2211yescharacteristic
37#10587572—21no—
38#10587572—21no—
39#10587571—21no—
40#10587568—21no—
41#10587561—21no—
42#10587555—21no—
43C17×#107C2211yescharacteristic, complemented
44C17×#106C2211yescharacteristic, complemented
45C17×#102C2211yescharacteristic
46C17×#102C2211yescharacteristic
47C17×C22⋊C8C2211yescharacteristic, complemented
48C22⋊#3622C2211yescharacteristic, complemented
49#10587491C2211yescharacteristic
50#10587491C2211yescharacteristic
51#10587487C2211yescharacteristic, complemented
52#10587487C2211yescharacteristic, complemented
53#10587481C2211yescharacteristic, complemented
54#10587481C2211yescharacteristic, complemented
55#10587480C2211yescharacteristic
56#10587480C2211yescharacteristic
57#10587479C2211yescharacteristic, complemented
58#10587478C2211yescharacteristic, complemented
59C23×D17—21no—
60C2×D4×C17—21no—
61C2×D4×C17C2311yescharacteristic
62C22×C4×C17C2311yescharacteristic
63C2×C22⋊D17—21no—
64C2×C22⋊D17—21no—
65C2×C22⋊D17—21no—
66C2×C22⋊D17D411yescharacteristic
67C2×C22⋊D17D411yescharacteristic
68C22×Dic17—21no—
69D4×D17—41no—
70D4×D17—41no—
71#63252—41no—
72#63252—41no—
73C2×D68—21no—
74C2×D68D411yescharacteristic
75C2×D68C2311yescharacteristic
76C2×C4×D17C2311yescharacteristic
77C2×C4×D17C2311yescharacteristic
78C2×Dic34—21no—
79C2×Dic34—41no—
80C2×Dic34—41no—
81C2×Dic34D411yescharacteristic
82C2×Dic34C2311yescharacteristic
83C2×Dic34C2311yescharacteristic
84C2×C8×C17C2311yescharacteristic
85C2×C8×C17C2311yescharacteristic
86C4⋊C4×C17C2311yescharacteristic
87C4⋊C4×C17C2311yescharacteristic
88C4⋊C4×C17C2311yescharacteristic
89C22⋊C4×C17—21no—
90C22⋊C4×C17—21no—
91C42×C17—21no—
92C22⋊Dic17—21no—
93#63229—41no—
94#63229—41no—
95#63227—21no—
96#63227—21no—
97#63227—21no—
98#63227—21no—
99#63227—21no—
100#63227—21no—
101#63227—21no—
102C4⋊Dic17—21no—
103C4⋊Dic17—21no—
104C4⋊Dic17—21no—
105C4⋊Dic17—21no—
106C4⋊Dic17C2311yescharacteristic
107C4×Dic17—21no—
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110#63219—41no—
111#63219—41no—
112#63218—41no—
113C8×D17—41no—
114C23×C17—21no—
115C23×C17C2×D411yescharacteristic
116C22×D17—21no—
117C22×D17—42no—
118C22×D17—81no—
119C22×D17—81no—
120C22×D17C2×D411yescharacteristic
121C22×D17C2×D411yescharacteristic
122D4×C17—21no—
123D4×C17—21no—
124D4×C17—81no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17C2×D411yescharacteristic
131C2×C4×C17C2411yescharacteristic
132C22⋊D17—41no—
133C22⋊D17—41no—
134C22⋊D17—42no—
135C22⋊D17—81no—
136C22⋊D17—81no—
137C2×Dic17—21no—
138C2×Dic17—21no—
139C2×Dic17—21no—
140C2×Dic17—42no—
141C2×Dic17C2×D411yescharacteristic
142C2×Dic17C2×D411yescharacteristic
143D68—21no—
144D68—21no—
145D68—41no—
146D68—81no—
147C4×D17—21no—
148C4×D17—21no—
149C4×D17—21no—
150C4×D17—21no—
151Dic34—21no—
152Dic34—21no—
153Dic34—21no—
154Dic34—21no—
155Dic34—41no—
156Dic34—42no—
157C8×C17—21no—
158C8×C17—21no—
159#3622—21no—
160#3622—21no—
161#3200—171nomaximal, Sylow
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—21no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17—41no—
168C22×C17C22×D411yescharacteristic
169C2×D17—21no—
170C2×D17—21no—
171C2×D17—21no—
172C2×D17—21no—
173C2×D17—21no—
174C2×D17—21no—
175C2×D17—41no—
176C2×D17—41no—
177C2×D17—41no—
178C2×D17—84no—
179C4×C17—21no—
180C4×C17—41no—
181C4×C17—41no—
182C4×C17—41no—
183C4×C17H(5,𝔽2)11yescharacteristic
184C4×C17H(5,𝔽2)11yescharacteristic
185Dic17—21no—
186Dic17—21no—
187Dic17—41no—
188Dic17—41no—
189Dic17—41no—
190#536—171no—
191#535—171no—
192#485—171no—
193#485—171no—
194#482—171no—
195#482D1711yescharacteristic, complemented
196#468—171no—
197#468—171no—
198#465—171no—
199#465—171no—
200#450—171no—
201#450—171no—
202#448—171no—
203#448—171no—
204#408—171no—
205C2×C17—41no—
206C2×C17—81no—
207C2×C17#53411yescharacteristic
208C2×C17#57711yescharacteristic
209C2×C17#57711yescharacteristic
210D17—41no—
211D17—41no—
212D17—81no—
213C2×#41—171no—
214C22×Q8—171no—
215C22×D4—171no—
216C2×SD16—171no—
217C2×SD16—171no—
218C2×SD16—171no—
219C2×SD16—171no—
220C2×M16—171no—
221C22×C8—171no—
222C4⋊Q8—341no—
223#127—341no—
224#124—341no—
225#124—341no—
226#123—341no—
227#123—341no—
228C22⋊Q8—171no—
229C22⋊Q8—171no—
230C22⋊Q8—171no—
231C22⋊Q8—171no—
232#121—171no—
233#121—171no—
234#121—171no—
235#121C2×D1711yescharacteristic
236#120—341no—
237#120—341no—
238#117—171no—
239#117C2×D1711yescharacteristic, complemented
240#107—171no—
241#107C2×D1711yescharacteristic, complemented
242#106—171no—
243#106C2×D1711yescharacteristic, complemented
244#103—171no—
245#103—171no—
246#103—171no—
247#103—171no—
248#102—171no—
249#102—171no—
250#102C2×D1711yescharacteristic
251#102C2×D1711yescharacteristic
252C22⋊C8—171no—
253C22⋊C8—171no—
254C22⋊C8—171no—
255C22⋊C8C2×D1711yescharacteristic, complemented
256C17#320011yescharacteristic, complemented, Sylow
257C24—341no—
258#41—681no—
259#41—681no—
260C2×Q8—171no—
261C2×Q8—171no—
262C2×Q8—171no—
263C2×Q8—341no—
264C2×Q8—681no—
265C2×Q8—681no—
266C2×D4—21no—
267C2×D4—171no—
268C2×D4—171no—
269C2×D4—171no—
270C2×D4—171no—
271C2×D4—341no—
272C2×D4—341no—
273C2×D4—341no—
274C2×D4—341no—
275C2×D4—681no—
276C2×D4—681no—
277C2×D4C22×D1711yescharacteristic
278C22×C4—171no—
279C22×C4—171no—
280C22×C4—341no—
281C22×C4C22×D1711yescharacteristic
282SD16—681no—
283SD16—681no—
284SD16—681no—
285SD16—681no—
286M16—681no—
287C2×C8—171no—
288C2×C8—171no—
289C2×C8—681no—
290C2×C8C22×D1711yescharacteristic
291C2×C8C22×D1711yescharacteristic
292C4⋊C4—171no—
293C4⋊C4—341no—
294C4⋊C4—341no—
295C4⋊C4—341no—
296C4⋊C4—341no—
297C4⋊C4C22×D1711yescharacteristic
298C4⋊C4C22×D1711yescharacteristic
299C4⋊C4C22×D1711yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—341no—
303C22⋊C4—341no—
304C22⋊C4—341no—
305C22⋊C4—341no—
306C22⋊C4—341no—
307C22⋊C4—341no—
308C22⋊C4—341no—
309C22⋊C4—341no—
310C42—21no—
311C42—341no—
312C23—21no—
313C23—171no—
314C23—171no—
315C23—341no—
316C23—682no—
317C23—1361no—
318C23—1361no—
319C23D4×D1711yescharacteristic
320Q8—341no—
321Q8—341no—
322Q8—341no—
323Q8—341no—
324Q8—681no—
325Q8—682no—
326D4—21no—
327D4—21no—
328D4—81no—
329D4—341no—
330D4—341no—
331D4—681no—
332D4—681no—
333D4—681no—
334D4—682no—
335D4—1361no—
336D4—1361no—
337D4—1361no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—171no—
344C2×C4—171no—
345C2×C4—341no—
346C2×C4—341no—
347C2×C4—341no—
348C2×C4—341no—
349C2×C4—341no—
350C2×C4—341no—
351C2×C4—341no—
352C2×C4—682no—
353C2×C4D4×D1711yescharacteristic
354C2×C4C23×D1711yescharacteristic
355C8—21no—
356C8—21no—
357C8—341no—
358C8—341no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—341no—
366C22—341no—
367C22—341no—
368C22—341no—
369C22—341no—
370C22—341no—
371C22—681no—
372C22—681no—
373C22—681no—
374C22—1364no—
375C22C2×D4×D1711yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—341no—
381C4—341no—
382C4—681no—
383C4—681no—
384C4—681no—
385C4#1058769611yescharacteristic
386C4#1058769611yescharacteristic
387C2—41no—
388C2—81no—
389C2—681no—
390C2—681no—
391C2—1361no—
392C2#1176337011yescharacteristic
393C2#1176361611yescharacteristic
394C2#1176361911yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#482C2
socleC22×C17C22×D4
prime radical#11763375C2
hypercenter#482D17
hypocenterC17#3200
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #11763385C2semidirect
3. #11763153C2semidirect
4. #11763152C2semidirect
5. #11763133C2semidirect
6. #11763056C2semidirect
7. #11763054C2semidirect
8. #11763039C2semidirect
9. #11763037C2semidirect
10. #11762961C2semidirect
11. #11762961C2semidirect
12. #11762950C2semidirect
13. #11762947C2semidirect
14. #11762774C2semidirect
15. C17×#482C2semidirect
18. C22×Dic34C22semidirect
20. C17×#117C22semidirect
29. C2×#63218C22semidirect
30. C2×C8×D17C22semidirect
33. #10587583C22semidirect
34. #10587583C22semidirect
43. C17×#107C22semidirect
44. C17×#106C22semidirect
47. C17×C22⋊C8C22semidirect
48. C22⋊#3622C22semidirect
51. #10587487C22semidirect
52. #10587487C22semidirect
53. #10587481C22semidirect
54. #10587481C22semidirect
57. #10587479C22semidirect
58. #10587478C22semidirect
195. #482D17semidirect
239. #117C2×D17semidirect
241. #107C2×D17semidirect
243. #106C2×D17semidirect
255. C22⋊C8C2×D17semidirect
256. C17#3200semidirect

Maximal subgroups

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

subgroupindex
#117633852
#117633752
#117631532
#117631522
#117631332
#117630562
#117630542
#117630392
#117630372
#117629612
#117629502
#117629472
#117627742
C17×#4822
#320017