Finite Group Databasebeta

← #14539854

Subgroups — #14539854

Order 2176.

396 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763419C211yesmaximal, characteristic, complemented
2#11763373C211yesmaximal, characteristic
3#11763202C211yesmaximal, characteristic, complemented
4#11763180C211yesmaximal, characteristic, complemented
5#11763172C211yesmaximal, characteristic, complemented
6#11763165C211yesmaximal, characteristic, complemented
7#11763034C211yesmaximal, characteristic, complemented
8#11763033C211yesmaximal, characteristic, complemented
9#11763008C211yesmaximal, characteristic, complemented
10#11763007C211yesmaximal, characteristic, complemented
11#11762971C211yesmaximal, characteristic, complemented
12#11762962C211yesmaximal, characteristic, complemented
13#11762959C211yesmaximal, characteristic, complemented
14#11762867C211yesmaximal, characteristic, complemented
15C17×#452C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C22×D68C2211yescharacteristic
18C2×Q16×C17C2211yescharacteristic, complemented
19C2×SD16×C17C2211yescharacteristic, complemented
20C17×C22⋊Q8C2211yescharacteristic
21#10587633C2211yescharacteristic
22#10587633C2211yescharacteristic
23Q8×Dic17C2211yescharacteristic, complemented
24#10587630C2211yescharacteristic, complemented
25C2×#63231C2211yescharacteristic, complemented
26C2×#63230C2211yescharacteristic
27C2×#63230C2211yescharacteristic
28#10587626C2211yescharacteristic
29#10587626C2211yescharacteristic
30#10587623—21no—
31#10587623—21no—
32C2×#63229C2211yescharacteristic, complemented
33#10587616—21no—
34#10587616—21no—
35#10587612C2211yescharacteristic
36#10587610C2211yescharacteristic
37C2×#63219C2211yescharacteristic
38C2×#63219C2211yescharacteristic
39#10587576—21no—
40#10587576C2211yescharacteristic, complemented
41#10587575—21no—
42#10587573C2211yescharacteristic, complemented
43#10587565—21no—
44#10587565—21no—
45C17×#103C2211yescharacteristic, complemented
46C17×#102C2211yescharacteristic, complemented
47C17×C22⋊C8C2211yescharacteristic
48#10587506C2211yescharacteristic, complemented
49#10587503C2211yescharacteristic, complemented
50C22⋊#3622C2211yescharacteristic
51#10587492C2211yescharacteristic
52#10587492C2211yescharacteristic
53#10587486C2211yescharacteristic, complemented
54#10587485C2211yescharacteristic, complemented
55#10587485C2211yescharacteristic, complemented
56C8×Dic17C2211yescharacteristic, complemented
57#10587480C2211yescharacteristic
58#10587480C2211yescharacteristic
59C23×D17—21no—
60#41×C17—41no—
61#41×C17—41no—
62C2×Q8×C17C2311yescharacteristic
63C2×Q8×C17C2311yescharacteristic
64C2×D4×C17—21no—
65C2×D4×C17C2311yescharacteristic
66C22×C4×C17—21no—
67C22×C4×C17C2311yescharacteristic
68C2×C22⋊D17—21no—
69C2×C22⋊D17—21no—
70C2×C22⋊D17—21no—
71C2×C22⋊D17—21no—
72C2×D68—21no—
73C2×D68—41no—
74C2×D68—41no—
75C2×D68C2311yescharacteristic
76C2×D68C2311yescharacteristic
77C2×Dic34C2311yescharacteristic
78Q16×C17—41no—
79SD16×C17—41no—
80C2×C8×C17C2311yescharacteristic
81C2×C8×C17C2311yescharacteristic
82C4⋊C4×C17—21no—
83C4⋊C4×C17C2311yescharacteristic
84C22⋊C4×C17—21no—
85C22⋊Dic17—21no—
86C22⋊Dic17D411yescharacteristic
87C22⋊Dic17D411yescharacteristic
88#63231—41no—
89#63230—41no—
90#63230—41no—
91#63229—41no—
92#63227—21no—
93#63227—21no—
94#63227—21no—
95#63227—21no—
96C4⋊Dic17—21no—
97C4⋊Dic17—21no—
98C4⋊Dic17D411yescharacteristic
99C4⋊Dic17D411yescharacteristic
100C4⋊Dic17C2311yescharacteristic
101#63225—21no—
102#63225—21no—
103#63225—21no—
104#63225—21no—
105#63225—21no—
106C4×Dic17—21no—
107C4×Dic17—21no—
108C4×Dic17C2311yescharacteristic
109C4×Dic17C2311yescharacteristic
110C2×#3622C2311yescharacteristic
111C2×#3622C2311yescharacteristic
112#63219—41no—
113#63219—41no—
114C23×C17—21no—
115C23×C17C2×D411yescharacteristic
116C22×D17—21no—
117C22×D17—21no—
118C22×D17—42no—
119C22×D17—81no—
120C22×D17—81no—
121Q8×C17—21no—
122Q8×C17—21no—
123Q8×C17—21no—
124Q8×C17—21no—
125D4×C17—21no—
126D4×C17—21no—
127D4×C17—42no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—42no—
134C2×C4×C17C2×D411yescharacteristic
135C2×C4×C17C2411yescharacteristic
136C22⋊D17—81no—
137C22⋊D17—81no—
138C22⋊D17—81no—
139C22⋊D17—81no—
140C2×Dic17—21no—
141C2×Dic17—21no—
142C2×Dic17C2×D411yescharacteristic
143C2×Dic17C2×D411yescharacteristic
144C2×Dic17C2×D411yescharacteristic
145C2×Dic17C2×D411yescharacteristic
146D68—21no—
147D68—21no—
148D68—21no—
149D68—21no—
150D68—42no—
151Dic34—21no—
152Dic34—21no—
153C8×C17—21no—
154C8×C17—21no—
155#3622—21no—
156#3622—21no—
157#3222—171nomaximal, Sylow
158C22×C17—21no—
159C22×C17—21no—
160C22×C17—21no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17C22×D411yescharacteristic
165C2×D17—41no—
166C2×D17—41no—
167C2×D17—41no—
168C2×D17—41no—
169C2×D17—41no—
170C2×D17—41no—
171C2×D17—84no—
172C4×C17—21no—
173C4×C17—41no—
174C4×C17—41no—
175C4×C17—41no—
176C4×C17H(5,𝔽2)11yescharacteristic
177C4×C17H(5,𝔽2)11yescharacteristic
178Dic17—21no—
179Dic17—21no—
180Dic17—21no—
181Dic17—21no—
182Dic17—41no—
183Dic17—41no—
184#541—171no—
185#535—171no—
186#497—171no—
187#492—171no—
188#490—171no—
189#488—171no—
190#464—171no—
191#464—171no—
192#460—171no—
193#460—171no—
194#452—171no—
195#452D1711yescharacteristic, complemented
196#450—171no—
197#450—171no—
198#433—171no—
199C2×C17—41no—
200C2×C17—81no—
201C2×C17#53511yescharacteristic
202C2×C17#57711yescharacteristic
203C2×C17#57711yescharacteristic
204D17—81no—
205D17—81no—
206C2×#41C2×D1711yescharacteristic
207C22×D4—171no—
208C2×Q16—171no—
209C2×Q16C2×D1711yescharacteristic, complemented
210C2×SD16—171no—
211C2×SD16—171no—
212C2×SD16—171no—
213C2×SD16—171no—
214C2×SD16—171no—
215C2×SD16C2×D1711yescharacteristic, complemented
216C4⋊Q8—171no—
217#127—171no—
218#127—171no—
219#126—341no—
220#126—341no—
221#125—171no—
222#125—341no—
223#125—341no—
224#124—171no—
225#124—171no—
226C22⋊Q8—171no—
227C22⋊Q8C2×D1711yescharacteristic
228#121—341no—
229#121—341no—
230#120—341no—
231#120—341no—
232C4×Q8—171no—
233C4×Q8—171no—
234#117—171no—
235C4⋊C8—171no—
236C4⋊C8—171no—
237#103—171no—
238#103C2×D1711yescharacteristic, complemented
239#102—171no—
240#102—171no—
241#102—171no—
242#102—171no—
243#102—171no—
244#102C2×D1711yescharacteristic, complemented
245C22⋊C8—171no—
246C22⋊C8C2×D1711yescharacteristic
247#97—171no—
248C4×C8—171no—
249C17#322211yescharacteristic, complemented, Sylow
250C24—341no—
251#41—41no—
252#41—41no—
253C2×Q8—171no—
254C2×Q8C22×D1711yescharacteristic
255C2×Q8C22×D1711yescharacteristic
256C2×D4—21no—
257C2×D4—171no—
258C2×D4—171no—
259C2×D4—341no—
260C2×D4—341no—
261C2×D4—341no—
262C2×D4—341no—
263C2×D4—341no—
264C2×D4—681no—
265C2×D4—681no—
266C2×D4C22×D1711yescharacteristic
267C22×C4—21no—
268C22×C4C22×D1711yescharacteristic
269Q16—41no—
270Q16—681no—
271SD16—41no—
272SD16—681no—
273SD16—681no—
274SD16—681no—
275SD16—681no—
276SD16—681no—
277C2×C8—171no—
278C2×C8—171no—
279C2×C8C22×D1711yescharacteristic
280C2×C8C22×D1711yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—171no—
283C4⋊C4—171no—
284C4⋊C4—171no—
285C4⋊C4—341no—
286C4⋊C4—341no—
287C4⋊C4—341no—
288C4⋊C4—341no—
289C4⋊C4—341no—
290C4⋊C4—341no—
291C4⋊C4—341no—
292C4⋊C4C22×D1711yescharacteristic
293C22⋊C4—21no—
294C22⋊C4—171no—
295C22⋊C4—171no—
296C22⋊C4—341no—
297C22⋊C4—341no—
298C22⋊C4—341no—
299C22⋊C4—341no—
300C22⋊C4—341no—
301C42—171no—
302C42—171no—
303C42—341no—
304C42—341no—
305C23—21no—
306C23—341no—
307C23—341no—
308C23—682no—
309C23—1361no—
310C23—1361no—
311C23D4×D1711yescharacteristic
312Q8—21no—
313Q8—21no—
314Q8—21no—
315Q8—21no—
316Q8—341no—
317Q8—341no—
318D4—21no—
319D4—21no—
320D4—42no—
321D4—341no—
322D4—341no—
323D4—341no—
324D4—341no—
325D4—682no—
326D4—1361no—
327D4—1361no—
328D4—1361no—
329D4—1361no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—42no—
336C2×C4—171no—
337C2×C4—171no—
338C2×C4—171no—
339C2×C4—171no—
340C2×C4—341no—
341C2×C4—341no—
342C2×C4D4×D1711yescharacteristic
343C2×C4C23×D1711yescharacteristic
344C8—21no—
345C8—21no—
346C8—341no—
347C8—341no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—681no—
355C22—681no—
356C22—681no—
357C22—681no—
358C22—681no—
359C22—681no—
360C22—1364no—
361C22C2×D4×D1711yescharacteristic
362C4—21no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—341no—
367C4—341no—
368C4—341no—
369C4—341no—
370C4—681no—
371C4—681no—
372C4#1058769111yescharacteristic
373C4#1058769111yescharacteristic
374C2—41no—
375C2—81no—
376C2—1361no—
377C2—1361no—
378C2#1176337411yescharacteristic
379C2#1176361611yescharacteristic
380C2#1176361811yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#452C2
socleC22×C17C22×D4
prime radical#11763373C2
hypercenter#452D17
hypocenterC17#3222
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. #11763419C2semidirect
3. #11763202C2semidirect
4. #11763180C2semidirect
5. #11763172C2semidirect
6. #11763165C2semidirect
7. #11763034C2semidirect
8. #11763033C2semidirect
9. #11763008C2semidirect
10. #11763007C2semidirect
11. #11762971C2semidirect
12. #11762962C2semidirect
13. #11762959C2semidirect
14. #11762867C2semidirect
15. C17×#452C2semidirect
18. C2×Q16×C17C22semidirect
19. C2×SD16×C17C22semidirect
23. Q8×Dic17C22semidirect
24. #10587630C22semidirect
25. C2×#63231C22semidirect
32. C2×#63229C22semidirect
40. #10587576C22semidirect
42. #10587573C22semidirect
45. C17×#103C22semidirect
46. C17×#102C22semidirect
48. #10587506C22semidirect
49. #10587503C22semidirect
53. #10587486C22semidirect
54. #10587485C22semidirect
55. #10587485C22semidirect
56. C8×Dic17C22semidirect
195. #452D17semidirect
209. C2×Q16C2×D17semidirect
215. C2×SD16C2×D17semidirect
238. #103C2×D17semidirect
244. #102C2×D17semidirect
249. C17#3222semidirect

Maximal subgroups

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

subgroupindex
#117634192
#117633732
#117632022
#117631802
#117631722
#117631652
#117630342
#117630332
#117630082
#117630072
#117629712
#117629622
#117629592
#117628672
C17×#4522
#322217