Finite Group Databasebeta

← #14539853

Subgroups — #14539853

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#11763417C211yesmaximal, characteristic, complemented
2#11763377C211yesmaximal, characteristic
3#11763202C211yesmaximal, characteristic, complemented
4#11763180C211yesmaximal, characteristic, complemented
5#11763172C211yesmaximal, characteristic, complemented
6#11763165C211yesmaximal, characteristic, complemented
7#11763028C211yesmaximal, characteristic, complemented
8#11763010C211yesmaximal, characteristic, complemented
9#11763009C211yesmaximal, characteristic, complemented
10#11762974C211yesmaximal, characteristic, complemented
11#11762970C211yesmaximal, characteristic, complemented
12#11762961C211yesmaximal, characteristic, complemented
13#11762958C211yesmaximal, characteristic, complemented
14#11762869C211yesmaximal, characteristic, complemented
15C17×#464C211yesmaximal, characteristic, complemented
16C2×#63256C2211yescharacteristic
17C2×D4×D17C2211yescharacteristic
18C2×Q16×C17C2211yescharacteristic, complemented
19C2×SD16×C17C2211yescharacteristic
20C17×#124C2211yescharacteristic
21C4×Q8×C17C2211yescharacteristic, complemented
22#10587633C2211yescharacteristic
23#10587632C2211yescharacteristic
24#10587630C2211yescharacteristic, complemented
25C2×#63231C2211yescharacteristic, complemented
26C2×#63230C2211yescharacteristic
27C2×#63230C2211yescharacteristic, complemented
28#10587626C2211yescharacteristic
29C2×#63229C2211yescharacteristic
30C2×#63219C2211yescharacteristic
31C2×#63219C2211yescharacteristic, complemented
32#10587584—21no—
33#10587583C2211yescharacteristic
34#10587576C2211yescharacteristic, complemented
35#10587575—21no—
36#10587570—21no—
37#10587570—21no—
38#10587568—21no—
39#10587568—21no—
40#10587562—21no—
41#10587560C2211yescharacteristic
42#10587558C2211yescharacteristic
43#10587556—21no—
44C4×Dic34C2211yescharacteristic, complemented
45C17×C4⋊C8C2211yescharacteristic, complemented
46C17×#103C2211yescharacteristic, complemented
47C17×#102C2211yescharacteristic
48#10587506C2211yescharacteristic, complemented
49#10587503C2211yescharacteristic
50#10587492C2211yescharacteristic
51#10587492C2211yescharacteristic, complemented
52#10587491C2211yescharacteristic
53#10587491C2211yescharacteristic
54#10587486C2211yescharacteristic, complemented
55C8×Dic17C2211yescharacteristic, complemented
56#10587480C2211yescharacteristic
57#10587480C2211yescharacteristic, complemented
58C4⋊#3622C2211yescharacteristic, complemented
59C23×D17—21no—
60C2×Q8×C17C2311yescharacteristic
61C2×Q8×C17C2311yescharacteristic
62C2×D4×C17C2311yescharacteristic
63C2×C22⋊D17—21no—
64C2×C22⋊D17—21no—
65C2×C22⋊D17—21no—
66#63256—41no—
67#63256—41no—
68D4×D17—41no—
69D4×D17—41no—
70C2×D68—21no—
71C2×D68—21no—
72C2×D68—21no—
73C2×D68C2311yescharacteristic
74C2×D68C2311yescharacteristic
75C2×C4×D17—21no—
76C2×C4×D17C2311yescharacteristic
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⋊C4×C17—21no—
86C42×C17—21no—
87C42×C17C2311yescharacteristic
88#63231—41no—
89#63230—41no—
90#63230—41no—
91#63229—41no—
92#63227—21no—
93#63227—21no—
94#63227—21no—
95#63227—21no—
96#63227D411yescharacteristic
97#63227D411yescharacteristic
98C4⋊Dic17—21no—
99C4⋊Dic17—21no—
100C4⋊Dic17C2311yescharacteristic
101#63225—21no—
102#63225—21no—
103#63225—21no—
104#63225—21no—
105#63225—21no—
106#63225D411yescharacteristic
107#63225D411yescharacteristic
108C4×Dic17—21no—
109C4×Dic17C2311yescharacteristic
110C2×#3622C2311yescharacteristic
111C2×#3622C2311yescharacteristic
112#63219—41no—
113#63219—41no—
114C23×C17—21no—
115C22×D17—21no—
116C22×D17—21no—
117C22×D17—42no—
118C22×D17—81no—
119C22×D17—81no—
120C22×D17C2×D411yescharacteristic
121Q8×C17—21no—
122Q8×C17—21no—
123Q8×C17—21no—
124Q8×C17—21no—
125D4×C17—21no—
126D4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17C2×D411yescharacteristic
131C2×C4×C17C2×D411yescharacteristic
132C2×C4×C17C2411yescharacteristic
133C22⋊D17—42no—
134C22⋊D17—81no—
135C22⋊D17—81no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17C2×D411yescharacteristic
139C2×Dic17C2×D411yescharacteristic
140C2×Dic17C2×D411yescharacteristic
141D68—21no—
142D68—21no—
143D68—21no—
144D68—21no—
145D68—42no—
146D68—81no—
147D68—81no—
148C4×D17—21no—
149C4×D17—21no—
150C4×D17—42no—
151Dic34—21no—
152Dic34—21no—
153C8×C17—21no—
154C8×C17—21no—
155#3622—21no—
156#3622—21no—
157#3222—171nomaximal, Sylow
158C22×C17—41no—
159C22×C17—41no—
160C22×C17—41no—
161C22×C17C22×D411yescharacteristic
162C2×D17—21no—
163C2×D17—21no—
164C2×D17—21no—
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—21no—
174C4×C17—41no—
175C4×C17—41no—
176C4×C17—41no—
177C4×C17H(5,𝔽2)11yescharacteristic
178C4×C17H(5,𝔽2)11yescharacteristic
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#464D1711yescharacteristic, complemented
192#460—171no—
193#460—171no—
194#452—171no—
195#452—171no—
196#450—171no—
197#450—171no—
198#433—171no—
199C2×C17—81no—
200C2×C17#53511yescharacteristic
201C2×C17#57711yescharacteristic
202C2×C17#57711yescharacteristic
203D17—41no—
204D17—81no—
205D17—81no—
206C2×#41—171no—
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
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#124C2×D1711yescharacteristic
226C22⋊Q8—171no—
227C22⋊Q8—171no—
228#121—341no—
229#121—341no—
230#120—341no—
231#120—341no—
232C4×Q8—171no—
233C4×Q8C2×D1711yescharacteristic, complemented
234#117—171no—
235C4⋊C8—171no—
236C4⋊C8C2×D1711yescharacteristic, complemented
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
245C22⋊C8—171no—
246C22⋊C8—171no—
247#97—171no—
248C4×C8—171no—
249C17#322211yescharacteristic, complemented, Sylow
250C24—341no—
251#41—681no—
252#41—681no—
253C2×Q8—171no—
254C2×Q8C22×D1711yescharacteristic
255C2×Q8C22×D1711yescharacteristic
256C2×D4—171no—
257C2×D4—171no—
258C2×D4—341no—
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—171no—
268C22×C4—341no—
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—21no—
295C22⋊C4—171no—
296C22⋊C4—171no—
297C22⋊C4—341no—
298C22⋊C4—341no—
299C22⋊C4—341no—
300C22⋊C4—341no—
301C42—21no—
302C42—171no—
303C42—341no—
304C42C22×D1711yescharacteristic
305C23—21no—
306C23—171no—
307C23—341no—
308C23—341no—
309C23—682no—
310C23—1361no—
311C23—1361no—
312Q8—21no—
313Q8—21no—
314Q8—21no—
315Q8—21no—
316Q8—341no—
317Q8—341no—
318D4—21no—
319D4—21no—
320D4—341no—
321D4—341no—
322D4—341no—
323D4—341no—
324D4—682no—
325D4—682no—
326D4—1361no—
327D4—1361no—
328D4—1361no—
329D4—1361no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—171no—
334C2×C4—171no—
335C2×C4—171no—
336C2×C4—341no—
337C2×C4—341no—
338C2×C4—341no—
339C2×C4—341no—
340C2×C4—682no—
341C2×C4D4×D1711yescharacteristic
342C2×C4D4×D1711yescharacteristic
343C2×C4C23×D1711yescharacteristic
344C8—21no—
345C8—21no—
346C8—341no—
347C8—341no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—341no—
352C22—341no—
353C22—341no—
354C22—681no—
355C22—681no—
356C22—681no—
357C22—681no—
358C22—681no—
359C22—681no—
360C22—1364no—
361C22C2×D4×D1711yescharacteristic
362C4—21no—
363C4—21no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—341no—
368C4—341no—
369C4—341no—
370C4—681no—
371C4—681no—
372C4#1058769111yescharacteristic
373C4#1058769611yescharacteristic
374C2—81no—
375C2—681no—
376C2—1361no—
377C2—1361no—
378C2#1176337511yescharacteristic
379C2#1176361611yescharacteristic
380C2#1176361811yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#464C2
socleC22×C17C22×D4
prime radical#11763377C2
hypercenter#464D17
hypocenterC17#3222
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. #11763417C2semidirect
3. #11763202C2semidirect
4. #11763180C2semidirect
5. #11763172C2semidirect
6. #11763165C2semidirect
7. #11763028C2semidirect
8. #11763010C2semidirect
9. #11763009C2semidirect
10. #11762974C2semidirect
11. #11762970C2semidirect
12. #11762961C2semidirect
13. #11762958C2semidirect
14. #11762869C2semidirect
15. C17×#464C2semidirect
18. C2×Q16×C17C22semidirect
21. C4×Q8×C17C22semidirect
24. #10587630C22semidirect
25. C2×#63231C22semidirect
27. C2×#63230C22semidirect
31. C2×#63219C22semidirect
34. #10587576C22semidirect
44. C4×Dic34C22semidirect
45. C17×C4⋊C8C22semidirect
46. C17×#103C22semidirect
48. #10587506C22semidirect
51. #10587492C22semidirect
54. #10587486C22semidirect
55. C8×Dic17C22semidirect
57. #10587480C22semidirect
58. C4⋊#3622C22semidirect
191. #464D17semidirect
209. C2×Q16C2×D17semidirect
233. C4×Q8C2×D17semidirect
236. C4⋊C8C2×D17semidirect
238. #103C2×D17semidirect
249. C17#3222semidirect

Maximal subgroups

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

subgroupindex
#117634172
#117633772
#117632022
#117631802
#117631722
#117631652
#117630282
#117630102
#117630092
#117629742
#117629702
#117629612
#117629582
#117628692
C17×#4642
#322217