Finite Group Databasebeta

← #14540613

Subgroups — #14540613

Order 2176.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763455C211yesmaximal, characteristic
2#11763166C211yesmaximal, characteristic, complemented
3#11763139C211yesmaximal, characteristic, complemented
4#11763133C211yesmaximal, characteristic, complemented
5#11763106C211yesmaximal, characteristic, complemented
6#11763048C211yesmaximal, characteristic, complemented
7#11763040C211yesmaximal, characteristic, complemented
8#11763031C211yesmaximal, characteristic, complemented
9#11762969C211yesmaximal, characteristic, complemented
10#11762959C211yesmaximal, characteristic, complemented
11C4×#63230C211yesmaximal, characteristic, complemented
12D4×#3622C211yesmaximal, characteristic, complemented
13#11762803C211yesmaximal, characteristic, complemented
14C2×#10587480C211yesmaximal, characteristic, complemented
15C17×#552C211yesmaximal, characteristic, complemented
16C2×#63252C2211yescharacteristic
17C22×D68C2211yescharacteristic
18C17×#126—21no—
19C17×#125—21no—
20C17×#125C2211yescharacteristic, complemented
21C17×#123—21no—
22C17×C22⋊Q8C2211yescharacteristic
23C17×C22⋊Q8C2211yescharacteristic, complemented
24C4×Q8×C17C2211yescharacteristic, complemented
25C4×D4×C17C2211yescharacteristic
26C17×#117—21no—
27C17×#117C2211yescharacteristic, complemented
28C2×C4⋊C4×C17C2211yescharacteristic
29C2×#63231C2211yescharacteristic, complemented
30C2×#63230C2211yescharacteristic
31C2×#63230C2211yescharacteristic
32#10587623—21no—
33#10587616C2211yescharacteristic
34#10587610C2211yescharacteristic
35C22×#3622C2211yescharacteristic
36C22×#3622C2211yescharacteristic, complemented
37#10587580—21no—
38#10587570—21no—
39C22⋊C4×D17—21no—
40#10587561C2211yescharacteristic
41C4×D68C2211yescharacteristic
42#10587506C2211yescharacteristic, complemented
43C22⋊#3622C2211yescharacteristic
44C22⋊#3622C2211yescharacteristic, complemented
45#10587481C2211yescharacteristic, complemented
46#10587481C2211yescharacteristic, complemented
47#10587481C2211yescharacteristic, complemented
48#10587480C2211yescharacteristic
49#10587480C2211yescharacteristic
50#10587480C2211yescharacteristic
51#10587480C2211yescharacteristic
52#10587480C2211yescharacteristic
53#10587480C2211yescharacteristic
54#10587479C2211yescharacteristic, complemented
55#10587479C2211yescharacteristic, complemented
56#10587478C2211yescharacteristic, complemented
57C4⋊#3622C2211yescharacteristic, complemented
58C4×#3622C2211yescharacteristic, complemented
59C23×D17—21no—
60C2×Q8×C17C2311yescharacteristic
61C2×D4×C17D411yescharacteristic
62C22×C4×C17—21no—
63C22×C4×C17C2311yescharacteristic
64C22×C4×C17C2311yescharacteristic
65C2×C22⋊D17—21no—
66C2×C22⋊D17—21no—
67#63252—22no—
68#63252—41no—
69C2×D68—22no—
70C2×D68—21no—
71C2×D68—41no—
72C2×D68C2311yescharacteristic
73C2×D68C2311yescharacteristic
74C2×C4×D17—21no—
75C2×C4×D17—21no—
76C2×Dic34C2311yescharacteristic
77C4⋊C4×C17—21no—
78C4⋊C4×C17—21no—
79C4⋊C4×C17—21no—
80C4⋊C4×C17—21no—
81C4⋊C4×C17—21no—
82C4⋊C4×C17—21no—
83C4⋊C4×C17D411yescharacteristic
84C4⋊C4×C17C2311yescharacteristic
85C4⋊C4×C17C2311yescharacteristic
86C4⋊C4×C17C2311yescharacteristic
87C22⋊C4×C17—21no—
88C22⋊C4×C17—21no—
89C22⋊C4×C17—21no—
90C22⋊C4×C17D411yescharacteristic
91C22⋊C4×C17D411yescharacteristic
92C42×C17—21no—
93C42×C17—21no—
94C42×C17C2311yescharacteristic
95C22⋊Dic17—21no—
96#63231—41no—
97#63230—22no—
98#63230—41no—
99#63227—21no—
100#63227—21no—
101#63227—21no—
102#63227—21no—
103C4⋊Dic17C2311yescharacteristic
104#63225—21no—
105C2×#3622—41no—
106C2×#3622—41no—
107C2×#3622C2311yescharacteristic
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110C2×#3622C2311yescharacteristic
111C23×C17C2×D411yescharacteristic
112C23×C17C2×D411yescharacteristic
113C22×D17—21no—
114C22×D17—21no—
115C22×D17—42no—
116C22×D17—42no—
117C22×D17—81no—
118Q8×C17—21no—
119Q8×C17—21no—
120D4×C17—22no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—42no—
130C2×C4×C17C2×D411yescharacteristic
131C2×C4×C17C2×D411yescharacteristic
132C2×C4×C17C2×D411yescharacteristic
133C2×C4×C17C2×D411yescharacteristic
134C2×C4×C17C2411yescharacteristic
135C22⋊D17—42no—
136C22⋊D17—81no—
137C2×Dic17—21no—
138C2×Dic17—21no—
139D68—21no—
140D68—21no—
141D68—21no—
142D68—42no—
143D68#4112yes—
144C4×D17—42no—
145C4×D17—81no—
146Dic34—21no—
147Dic34—21no—
148#3622—21no—
149#3622—21no—
150#3622—21no—
151#3622—21no—
152#3296—171nomaximal, Sylow
153C22×C17—21no—
154C22×C17—21no—
155C22×C17—21no—
156C22×C17—21no—
157C22×C17—21no—
158C22×C17#13512yes—
159C22×C17C22×D411yescharacteristic
160C2×D17—22no—
161C2×D17—41no—
162C2×D17—41no—
163C2×D17—41no—
164C2×D17—41no—
165C2×D17—41no—
166C2×D17—84no—
167C4×C17—21no—
168C4×C17—21no—
169C4×C17—21no—
170C4×C17—21no—
171C4×C17—41no—
172C4×C17—41no—
173C4×C17—41no—
174C4×C17—41no—
175C4×C17C2×#4111yescharacteristic
176C4×C17H(5,𝔽2)11yescharacteristic
177Dic17—41no—
178Dic17—41no—
179#552D1711yescharacteristic, complemented
180#546—171no—
181#488—171no—
182#483—171no—
183#482—171no—
184#478—171no—
185#467—171no—
186#465—171no—
187#464—171no—
188#452—171no—
189#450—171no—
190C4×SD16—171no—
191C8×D4—171no—
192#416—171no—
193C2×#102—171no—
194C2×C17—22no—
195C2×C17—41no—
196C2×C17#54611yescharacteristic
197C2×C17C2×#13511yescharacteristic
198C2×C17#57711yescharacteristic
199D17—42no—
200D17—81no—
201C2×#41—171no—
202C22×D4—171no—
203C2×Q16—171no—
204C2×SD16—171no—
205C2×SD16—171no—
206C22×C8—171no—
207C22×C8—171no—
208#126—21no—
209#125—21no—
210#125C2×D1711yescharacteristic, complemented
211#124—171no—
212#123—21no—
213#123—341no—
214C22⋊Q8—171no—
215C22⋊Q8C2×D1711yescharacteristic
216C22⋊Q8C2×D1711yescharacteristic, complemented
217#121—171no—
218#121—341no—
219#120—341no—
220C4×Q8C2×D1711yescharacteristic, complemented
221C4×D4—171no—
222C4×D4C2×D1711yescharacteristic
223#117—21no—
224#117C2×D1711yescharacteristic, complemented
225C2×C4⋊C4C2×D1711yescharacteristic
226C2×C22⋊C4—341no—
227#107—171no—
228#106—171no—
229#106—171no—
230C4⋊C8—171no—
231#103—171no—
232#103—171no—
233#103—171no—
234#103—171no—
235#102—171no—
236#102—171no—
237#102—171no—
238#102—171no—
239#102—171no—
240#102—171no—
241C22⋊C8—171no—
242C22⋊C8—171no—
243C4×C8—171no—
244C17#329611yescharacteristic, complemented, Sylow
245C24—341no—
246#41—342no—
247#41—681no—
248C2×Q8—171no—
249C2×Q8C22×D1711yescharacteristic
250C2×D4—171no—
251C2×D4—171no—
252C2×D4—342no—
253C2×D4—341no—
254C2×D4—341no—
255C2×D4—341no—
256C2×D4—681no—
257C2×D4C22⋊D1711yescharacteristic
258C22×C4—21no—
259C22×C4—341no—
260C22×C4—341no—
261C22×C4C22×D1711yescharacteristic
262C22×C4C22×D1711yescharacteristic
263Q16—681no—
264SD16—342no—
265SD16—681no—
266C2×C8—171no—
267C2×C8—171no—
268C2×C8—171no—
269C2×C8—171no—
270C2×C8—681no—
271C2×C8—681no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—171no—
279C4⋊C4—341no—
280C4⋊C4C22⋊D1711yescharacteristic
281C4⋊C4C22×D1711yescharacteristic
282C4⋊C4C22×D1711yescharacteristic
283C4⋊C4C22×D1711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—341no—
288C22⋊C4—341no—
289C22⋊C4—341no—
290C22⋊C4—341no—
291C22⋊C4—341no—
292C22⋊C4C22⋊D1711yescharacteristic
293C22⋊C4C22⋊D1711yescharacteristic
294C42—21no—
295C42—21no—
296C42C22×D1711yescharacteristic
297C23—341no—
298C23—341no—
299C23—682no—
300C23—682no—
301C23—1361no—
302C23C2×C22⋊D1711yescharacteristic
303C23C2×C22⋊D1711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—341no—
307Q8—341no—
308D4—22no—
309D4—172no—
310D4—341no—
311D4—341no—
312D4—341no—
313D4—682no—
314D4—682no—
315D4—1361no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—341no—
326C2×C4—341no—
327C2×C4—682no—
328C2×C4—1361no—
329C2×C4C2×C22⋊D1711yescharacteristic
330C2×C4C2×C22⋊D1711yescharacteristic
331C2×C4C2×C22⋊D1711yescharacteristic
332C2×C4C2×C22⋊D1711yescharacteristic
333C2×C4C23×D1711yescharacteristic
334C8—341no—
335C8—341no—
336C8—341no—
337C8—341no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—342no—
344C22—681no—
345C22—681no—
346C22—681no—
347C22—681no—
348C22—681no—
349C22—1364no—
350C22#1058763612yes—
351C22C22×C22⋊D1711yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—681no—
361C4—681no—
362C4#41×D1711yescharacteristic
363C4#1058769611yescharacteristic
364C2—22no—
365C2—41no—
366C2—682no—
367C2—1361no—
368C2#1176344711yescharacteristic
369C2C2×#1058763611yescharacteristic
370C2#1176361711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#552C2
socleC22×C17C22×D4
prime radical#11763455C2
hypercenter#552D17
hypocenterC17#3296
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #11763166C2semidirect
3. #11763139C2semidirect
4. #11763133C2semidirect
5. #11763106C2semidirect
6. #11763048C2semidirect
7. #11763040C2semidirect
8. #11763031C2semidirect
9. #11762969C2semidirect
10. #11762959C2semidirect
11. C4×#63230C2semidirect
12. D4×#3622C2semidirect
13. #11762803C2semidirect
14. C2×#10587480C2semidirect
15. C17×#552C2semidirect
20. C17×#125C22semidirect
23. C17×C22⋊Q8C22semidirect
24. C4×Q8×C17C22semidirect
27. C17×#117C22semidirect
29. C2×#63231C22semidirect
36. C22×#3622C22semidirect
42. #10587506C22semidirect
44. C22⋊#3622C22semidirect
45. #10587481C22semidirect
46. #10587481C22semidirect
47. #10587481C22semidirect
54. #10587479C22semidirect
55. #10587479C22semidirect
56. #10587478C22semidirect
57. C4⋊#3622C22semidirect
58. C4×#3622C22semidirect
179. #552D17semidirect
210. #125C2×D17semidirect
216. C22⋊Q8C2×D17semidirect
220. C4×Q8C2×D17semidirect
224. #117C2×D17semidirect
244. C17#3296semidirect

Maximal subgroups

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

subgroupindex
#117634552
#117631662
#117631392
#117631332
#117631062
#117630482
#117630402
#117630312
#117629692
#117629592
C4×#632302
D4×#36222
#117628032
C2×#105874802
C17×#5522
#329617