Finite Group Databasebeta

← D17×#2255 #17189817

Subgroups — D17×#2255 #17189817

Order 4352.

518 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D17×#500C211yesmaximal, characteristic
2D17×#499C211yesmaximal, characteristic
3#14531071C211yesmaximal, characteristic, complemented
4#14531070C211yesmaximal, characteristic, complemented
5#14531069C212yesmaximal, complemented
6#14531068C212yesmaximal, complemented
7#14531067C211yesmaximal, characteristic, complemented
8D17×#367C212yesmaximal
9D17×#366C212yesmaximal
10D17×#346C211yesmaximal, characteristic
11C17×#2255C211yesmaximal, characteristic, complemented
12C4⋊Q8×D17—21no—
13#125×D17—21no—
14C2×C16×D17C2212yes—
15#11763214C2211yescharacteristic
16#11763213—21no—
17#11763213C2212yes—
18#11763212—21no—
19#11763211C2211yescharacteristic
20#11763210C2211yescharacteristic
21#11763210C2211yescharacteristic
22#11763209—21no—
23#11763208—21no—
24#107×D17—21no—
25#107×D17C2211yescharacteristic
26#107×D17C2212yes—
27#107×D17C2211yescharacteristic
28#106×D17—21no—
29C4×C8×D17C2211yescharacteristic
30#11762637C2212yes—
31#11762636C2212yes—
32#11762636C2212yes—
33#11762635C2212yes—
34#11762634C2212yes—
35#11762634C2212yes—
36#11762585C2211yescharacteristic
37#11762584C2212yes—
38C17×#500C2211yescharacteristic
39C17×#499C2211yescharacteristic
40C17×#367C2212yes—
41C17×#366C2212yes—
42C17×#346C2211yescharacteristic
43C2×Q8×D17—41no—
44C17×C4⋊Q8—21no—
45C17×#125—21no—
46#10587630—41no—
47C2×C8×D17D411yescharacteristic
48C2×C8×D17C2311yescharacteristic
49C4⋊C4×D17—21no—
50C4⋊C4×D17—21no—
51C4⋊C4×D17—22no—
52C4⋊C4×D17—41no—
53C4⋊C4×D17—41no—
54C4⋊C4×D17—41no—
55#10587576—21no—
56#10587576—21no—
57#10587575—41no—
58#10587575—41no—
59#10587574—22no—
60#10587574—41no—
61C42×D17D411yescharacteristic
62#10587556—21no—
63#10587555—21no—
64C2×C16×C17C2312yes—
65C17×#107—21no—
66C17×#107C2311yescharacteristic
67C17×#107C2311yescharacteristic
68C17×#107C2312yes—
69C17×#106—21no—
70C4×C8×C17C2311yescharacteristic
71#10587489—21no—
72#10587489C2311yescharacteristic
73#10587489C2311yescharacteristic
74#10587489C2312yes—
75#10587488—21no—
76C8×Dic17D411yescharacteristic
77C8×Dic17C2311yescharacteristic
78C2×#63214C2312yes—
79#10587479—21no—
80#10587479—21no—
81#10587478—21no—
82#10587478—21no—
83#10587478—22no—
84#10587478—22no—
85C4×#3622D411yescharacteristic
86C16×D17Q818yescomplemented
87C2×Q8×C17—41no—
88Q8×D17—84no—
89C2×C4×D17—42no—
90C2×C4×D17—41no—
91C2×C4×D17—41no—
92C2×C4×D17D811yescharacteristic
93C2×C4×D17D811yescharacteristic
94C2×C4×D17C2×D411yescharacteristic
95C2×Dic34—41no—
96C2×Dic34—42no—
97C2×C8×C17C2×D411yescharacteristic
98C2×C8×C17C2411yescharacteristic
99C4⋊C4×C17—21no—
100C4⋊C4×C17—21no—
101C4⋊C4×C17—22no—
102C4⋊C4×C17—41no—
103C4⋊C4×C17—41no—
104C4⋊C4×C17—41no—
105C42×C17C2×D411yescharacteristic
106C4⋊Dic17—22no—
107C4⋊Dic17—21no—
108C4⋊Dic17—21no—
109C4⋊Dic17—41no—
110C4⋊Dic17—41no—
111C4⋊Dic17—41no—
112#63225—41no—
113#63225—41no—
114#63225—42no—
115#63225—42no—
116#63225—41no—
117#63225—41no—
118#63225—41no—
119#63225—41no—
120C4×Dic17D811yescharacteristic
121C4×Dic17D811yescharacteristic
122C4×Dic17C2×D411yescharacteristic
123C2×#3622C2×D411yescharacteristic
124C2×#3622C2×D411yescharacteristic
125C8×D17Q1614yes—
126C8×D17C2×Q814yes—
127C16×C17C2×Q814yescomplemented
128#63214C2×Q814yescomplemented
129C2×#2255—171nomaximal, Sylow
130C22×D17C2×D811yescharacteristic
131Q8×C17—82no—
132C2×C4×C17—41no—
133C2×C4×C17—41no—
134C2×C4×C17—42no—
135C2×C4×C17C2×D811yescharacteristic
136C2×C4×C17C2×D811yescharacteristic
137C2×C4×C17C22×D411yescharacteristic
138C2×Dic17—42no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17C2×D811yescharacteristic
142C2×Dic17C2×D811yescharacteristic
143C2×Dic17C2×D811yescharacteristic
144C4×D17—22no—
145C4×D17—22no—
146C4×D17—82no—
147C4×D17—84no—
148C4×D17—82no—
149C4×D17C4⋊Q812yes—
150C4×D17C2×Q1612yes—
151Dic34—84no—
152Dic34—82no—
153C8×C17C2×Q1612yes—
154C8×C17C22×Q812yes—
155#3622C4⋊Q812yes—
156#3622C2×Q1612yes—
157C2×#500—171no—
158C2×#499—171no—
159#2255—171no—
160#2255—171no—
161#2255—172no—
162#2255—172no—
163#2255—171no—
164#2255D1711yescharacteristic, direct product factor
165C2×#367—172no—
166C2×#366—172no—
167C2×#346—171no—
168C22×C17C22×D811yescharacteristic
169C2×D17#50012yes—
170C2×D17#50912yes—
171C2×D17#51012yes—
172C4×C17—21no—
173C4×C17—21no—
174C4×C17—82no—
175C4×C17—81no—
176C4×C17—81no—
177C4×C17C2×C4⋊Q811yescharacteristic
178C4×C17C22×Q1611yescharacteristic
179Dic17—21no—
180Dic17—21no—
181Dic17—82no—
182Dic17—81no—
183Dic17—81no—
184Dic17#50012yes—
185C2×C4⋊Q8—341no—
186C2×#125—341no—
187C22×C16—172no—
188#500—172no—
189#500—171no—
190#500—341no—
191#500—341no—
192#500C2×D1711yescharacteristic
193#499—171no—
194#499—171no—
195#499—171no—
196#499—341no—
197#499—341no—
198#499C2×D1711yescharacteristic
199C2×#107—171no—
200C2×#107—171no—
201C2×#107—172no—
202C2×#107—341no—
203C2×#106—341no—
204C2×C4×C8—171no—
205#367—172no—
206#367—172no—
207#367—172no—
208#367C2×D1712yes—
209#366—172no—
210#366—172no—
211#366—172no—
212#366C2×D1712yes—
213#346—172no—
214#346—171no—
215#346C2×D1711yescharacteristic
216C2×C17C2×#50011yescharacteristic
217C2×C17C2×#50911yescharacteristic
218C2×C17C2×#51011yescharacteristic
219D17#225514yesdirect product factor
220C22×Q8—681no—
221C22×C8—171no—
222C22×C8—171no—
223C4⋊Q8—21no—
224C4⋊Q8—342no—
225C4⋊Q8—341no—
226C4⋊Q8—681no—
227C4⋊Q8—681no—
228#125—21no—
229#125—341no—
230#125—341no—
231#125—341no—
232#125—681no—
233#125—681no—
234C2×C4⋊C4—341no—
235C2×C4⋊C4—341no—
236C2×C4⋊C4—342no—
237C2×C4⋊C4—681no—
238C2×C4⋊C4—681no—
239C2×C4⋊C4—681no—
240C2×C42—171no—
241C2×C16—172no—
242C2×C16—178no—
243C2×C16C22×D1712yes—
244#107—21no—
245#107—171no—
246#107—172no—
247#107—171no—
248#107—341no—
249#107—341no—
250#107—342no—
251#107—342no—
252#107—341no—
253#107C22×D1711yescharacteristic
254#107C22×D1712yes—
255#107C22×D1711yescharacteristic
256#106—21no—
257#106—341no—
258#106—341no—
259#106—341no—
260C4×C8—171no—
261C4×C8—171no—
262C4×C8—171no—
263C4×C8C22×D1711yescharacteristic
264C17C2×#225511yescharacteristic, complemented, Sylow
265C2×Q8—41no—
266C2×Q8—682no—
267C2×Q8—681no—
268C2×Q8—1364no—
269C22×C4—171no—
270C22×C4—171no—
271C22×C4—171no—
272C22×C4—681no—
273C22×C4—681no—
274C22×C4—682no—
275C2×C8—171no—
276C2×C8—171no—
277C2×C8—174no—
278C2×C8—174no—
279C2×C8D4×D1711yescharacteristic
280C2×C8C23×D1711yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—22no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—341no—
288C4⋊C4—341no—
289C4⋊C4—342no—
290C4⋊C4—681no—
291C4⋊C4—681no—
292C4⋊C4—682no—
293C4⋊C4—681no—
294C4⋊C4—681no—
295C4⋊C4—681no—
296C4⋊C4—681no—
297C4⋊C4—682no—
298C4⋊C4—681no—
299C4⋊C4—681no—
300C4⋊C4—681no—
301C42—171no—
302C42—171no—
303C42—171no—
304C42D4×D1711yescharacteristic
305C16—174no—
306C16Q8×D1714yescomplemented
307C23—171no—
308Q8—82no—
309Q8—1362no—
310Q8—1364no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—171no—
315C2×C4—172no—
316C2×C4—172no—
317C2×C4—171no—
318C2×C4—171no—
319C2×C4—342no—
320C2×C4—342no—
321C2×C4—681no—
322C2×C4—681no—
323C2×C4—682no—
324C2×C4—1364no—
325C2×C4—1362no—
326C2×C4—1362no—
327C2×C4D8×D1711yescharacteristic
328C2×C4D8×D1711yescharacteristic
329C2×C4C2×D4×D1711yescharacteristic
330C8—172no—
331C8—172no—
332C8Q16×D1712yes—
333C8C2×Q8×D1712yes—
334C22—172no—
335C22—172no—
336C22—172no—
337C22C2×D8×D1711yescharacteristic
338C4—21no—
339C4—21no—
340C4—82no—
341C4—81no—
342C4—81no—
343C4—172no—
344C4—341no—
345C4—341no—
346C4—1362no—
347C4—1361no—
348C4—1361no—
349C4C4⋊Q8×D1711yescharacteristic
350C4C2×Q16×D1711yescharacteristic
351C2—174no—
352C2D17×#50011yescharacteristic
353C2D17×#50911yescharacteristic
354C2D17×#51011yescharacteristic
subgroupG/H
centerC22C2×D8×D17
commutator, cosocleC2×C8×C17C24
FrattiniC2×C8C23×D17
FittingC17×#2255C2
socleC22×C17C22×D8
prime radicalC22×D17C2×D8
hypercenter#2255D17
hypocenterC17C2×#2255
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2255

Complements

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

NH ≅ G/Ntype
3. #14531071C2semidirect
4. #14531070C2semidirect
5. #14531069C2semidirect
6. #14531068C2semidirect
7. #14531067C2semidirect
11. C17×#2255C2semidirect
86. C16×D17Q8semidirect
127. C16×C17C2×Q8semidirect
128. #63214C2×Q8semidirect
164. #2255D17direct (N × H)
219. D17#2255direct (N × H)
264. C17C2×#2255semidirect
306. C16Q8×D17semidirect

Maximal subgroups

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

subgroupindex
D17×#5002
D17×#4992
#145310712
#145310702
#145310692
#145310682
#145310672
D17×#3672
D17×#3662
D17×#3462
C17×#22552
C2×#225517