Finite Group Databasebeta

← C17×#9166 #16866910

Subgroups — C17×#9166 #16866910

Order 4352.

436 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#3113C211yesmaximal, characteristic
2C17×#2984C211yesmaximal, characteristic
3C17×#1763C211yesmaximal, characteristic
4C17×#1531C211yesmaximal, characteristic
5C17×#1503C211yesmaximal, characteristic
6C17×#1302C211yesmaximal, characteristic
7C17×#1285C211yesmaximal, characteristic
8C17×#533C2211yescharacteristic
9C2×C4×Q8×C17C2211yescharacteristic
10C17×#483—21no—
11C17×#475C411yescharacteristic, complemented
12C17×#475C411yescharacteristic, complemented
13C17×#462C411yescharacteristic, complemented
14C17×#462C411yescharacteristic, complemented
15C17×#451—21no—
16C17×#446—21no—
17C17×#446—22no—
18C8×Q8×C17—21no—
19C17×#441—21no—
20C17×#441—21no—
21C17×#432—21no—
22C17×#432C2211yescharacteristic
23C17×#432C2211yescharacteristic
24C17×#431—21no—
25C17×#431C2211yescharacteristic
26C2×C17×#103—21no—
27C2×C17×#97C2211yescharacteristic
28C4×C4⋊C4×C17C2211yescharacteristic
29C17×#340—21no—
30C17×#330—22no—
31C17×#328—21no—
32C17×#328—21no—
33C22×Q8×C17D411yescharacteristic
34C2×Q16×C17—21no—
35C2×Q16×C17—21no—
36C22×C8×C17—21no—
37C17×C4⋊Q8C2×C411yescharacteristic
38C17×C4⋊Q8C2×C411yescharacteristic
39C17×#125—21no—
40C17×C22⋊Q8—21no—
41C17×C22⋊Q8—21no—
42C4×Q8×C17—21no—
43C4×Q8×C17—21no—
44C4×Q8×C17—21no—
45C4×Q8×C17—21no—
46C4×Q8×C17C2×C411yescharacteristic
47C4×Q8×C17C2×C411yescharacteristic
48C4×Q8×C17D411yescharacteristic
49C4×Q8×C17D411yescharacteristic
50C17×#117—21no—
51C2×C4⋊C4×C17—21no—
52C2×C4⋊C4×C17—21no—
53C2×C4⋊C4×C17D411yescharacteristic
54C2×C42×C17—21no—
55C2×C42×C17—21no—
56C2×C42×C17C2311yescharacteristic
57C17×#106—21no—
58C17×#106—21no—
59C17×C4⋊C8—22no—
60C17×C4⋊C8—22no—
61C17×C4⋊C8—21no—
62C17×C4⋊C8—21no—
63C17×C4⋊C8C2×C411yescharacteristic
64C17×C4⋊C8C2×C411yescharacteristic
65C17×#103—24no—
66C17×#103—21no—
67C17×#103—21no—
68C17×#103—21no—
69C17×#103—21no—
70C17×C22⋊C8—21no—
71C17×C22⋊C8—21no—
72C17×C22⋊C8D411yescharacteristic
73C17×C22⋊C8D411yescharacteristic
74C17×#97—22no—
75C17×#97—21no—
76C17×#97—21no—
77C17×#97—21no—
78C17×#97D412yes—
79C4×C8×C17—22no—
80C4×C8×C17—21no—
81C17×#95—21no—
82C2×Q8×C17—21no—
83C2×Q8×C17—22no—
84C2×Q8×C17—21no—
85C2×Q8×C17—21no—
86C2×Q8×C17—41no—
87C2×Q8×C17C22⋊C411yescharacteristic
88C2×Q8×C17C22⋊C411yescharacteristic
89C22×C4×C17—21no—
90C22×C4×C17—21no—
91C22×C4×C17—41no—
92C22×C4×C17C2×D411yescharacteristic
93C22×C4×C17C2×D411yescharacteristic
94C22×C4×C17#4111yescharacteristic
95Q16×C17—42no—
96Q16×C17—42no—
97C2×C8×C17—22no—
98C2×C8×C17—21no—
99C2×C8×C17—21no—
100C2×C8×C17—21no—
101C2×C8×C17—21no—
102C2×C8×C17—22no—
103C2×C8×C17—41no—
104C2×C8×C17—41no—
105C4⋊C4×C17—21no—
106C4⋊C4×C17—21no—
107C4⋊C4×C17—21no—
108C4⋊C4×C17—21no—
109C4⋊C4×C17—21no—
110C4⋊C4×C17—22no—
111C4⋊C4×C17—21no—
112C4⋊C4×C17—21no—
113C4⋊C4×C17—21no—
114C4⋊C4×C17—21no—
115C4⋊C4×C17—41no—
116C4⋊C4×C17—41no—
117C4⋊C4×C17—41no—
118C4⋊C4×C17C22⋊C411yescharacteristic
119C4⋊C4×C17C22⋊C411yescharacteristic
120C22⋊C4×C17—41no—
121C42×C17—21no—
122C42×C17—21no—
123C42×C17—21no—
124C42×C17—21no—
125C42×C17—41no—
126C42×C17—81no—
127C42×C17C22×C411yescharacteristic
128C42×C17C2×D411yescharacteristic
129C42×C17C2×D411yescharacteristic
130C42×C17#4111yescharacteristic
131#9166C1711yesmaximal, characteristic, direct product factor, Sylow
132C23×C17#12111yescharacteristic
133Q8×C17—21no—
134Q8×C17—21no—
135Q8×C17—21no—
136Q8×C17—42no—
137Q8×C17—41no—
138Q8×C17—41no—
139Q8×C17—81no—
140Q8×C17#10412yes—
141C2×C4×C17—21no—
142C2×C4×C17—21no—
143C2×C4×C17—21no—
144C2×C4×C17—21no—
145C2×C4×C17—21no—
146C2×C4×C17—21no—
147C2×C4×C17—21no—
148C2×C4×C17—21no—
149C2×C4×C17—41no—
150C2×C4×C17—42no—
151C2×C4×C17—41no—
152C2×C4×C17—41no—
153C2×C4×C17—42no—
154C2×C4×C17—41no—
155C2×C4×C17—41no—
156C2×C4×C17—81no—
157C2×C4×C17—81no—
158C2×C4×C17C2×C22⋊C411yescharacteristic
159C2×C4×C17C4×D411yescharacteristic
160C2×C4×C17C4×D411yescharacteristic
161C2×C4×C17#12011yescharacteristic
162C2×C4×C17#12111yescharacteristic
163C2×C4×C17#12311yescharacteristic
164C8×C17—41no—
165C8×C17—41no—
166C8×C17—42no—
167C8×C17—42no—
168C8×C17—41no—
169C8×C17—41no—
170#3113C2×C1711yescharacteristic
171#2984C2×C1711yescharacteristic
172#1763C2×C1711yescharacteristic
173#1531C2×C1711yescharacteristic
174#1503C2×C1711yescharacteristic
175#1302C2×C1711yescharacteristic
176#1285C2×C1711yescharacteristic
177C22×C17—21no—
178C22×C17—21no—
179C22×C17#38611yescharacteristic
180C22×C17#47111yescharacteristic
181C22×C17#47311yescharacteristic
182C4×C17—21no—
183C4×C17—22no—
184C4×C17—21no—
185C4×C17—21no—
186C4×C17—21no—
187C4×C17—21no—
188C4×C17—41no—
189C4×C17—41no—
190C4×C17—41no—
191C4×C17—82no—
192C4×C17—81no—
193C4×C17—81no—
194C4×C17C2×#10411yescharacteristic
195C4×C17#42111yescharacteristic
196#533C22×C1711yescharacteristic
197C2×C4×Q8C22×C1711yescharacteristic
198#483—21no—
199#475C4×C1711yescharacteristic, complemented
200#475C4×C1711yescharacteristic, complemented
201#462C4×C1711yescharacteristic, complemented
202#462C4×C1711yescharacteristic, complemented
203#451—21no—
204#446—21no—
205#446—22no—
206C8×Q8—21no—
207#441—21no—
208#441—21no—
209#432—21no—
210#432C22×C1711yescharacteristic
211#432C22×C1711yescharacteristic
212#431—21no—
213#431C22×C1711yescharacteristic
214C2×#103—21no—
215C2×#97C22×C1711yescharacteristic
216C4×C4⋊C4C22×C1711yescharacteristic
217#340—21no—
218#330—22no—
219#328—21no—
220#328—21no—
221C2×C17—21no—
222C2×C17—21no—
223C2×C17#187111yescharacteristic
224C2×C17#190511yescharacteristic
225C2×C17#190811yescharacteristic
226C22×Q8D4×C1711yescharacteristic
227C2×Q16—21no—
228C2×Q16—21no—
229C22×C8—21no—
230C4⋊Q8C2×C4×C1711yescharacteristic
231C4⋊Q8C2×C4×C1711yescharacteristic
232#125—21no—
233C22⋊Q8—21no—
234C22⋊Q8—21no—
235C4×Q8—21no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×Q8C2×C4×C1711yescharacteristic
240C4×Q8C2×C4×C1711yescharacteristic
241C4×Q8D4×C1711yescharacteristic
242C4×Q8D4×C1711yescharacteristic
243#117—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4D4×C1711yescharacteristic
247C2×C42—21no—
248C2×C42—21no—
249C2×C42C23×C1711yescharacteristic
250#106—21no—
251#106—21no—
252C4⋊C8—22no—
253C4⋊C8—22no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8C2×C4×C1711yescharacteristic
257C4⋊C8C2×C4×C1711yescharacteristic
258#103—24no—
259#103—21no—
260#103—21no—
261#103—21no—
262#103—21no—
263C22⋊C8—21no—
264C22⋊C8—21no—
265C22⋊C8D4×C1711yescharacteristic
266C22⋊C8D4×C1711yescharacteristic
267#97—22no—
268#97—21no—
269#97—21no—
270#97—21no—
271#97D4×C1712yes—
272C4×C8—22no—
273C4×C8—21no—
274#95—21no—
275C17#916611yescharacteristic, direct product factor, Sylow
276C2×Q8—21no—
277C2×Q8—22no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—41no—
281C2×Q8C22⋊C4×C1711yescharacteristic
282C2×Q8C22⋊C4×C1711yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—41no—
286C22×C4C2×D4×C1711yescharacteristic
287C22×C4C2×D4×C1711yescharacteristic
288C22×C4#41×C1711yescharacteristic
289Q16—42no—
290Q16—42no—
291C2×C8—22no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—22no—
297C2×C8—41no—
298C2×C8—41no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4C22⋊C4×C1711yescharacteristic
313C4⋊C4C22⋊C4×C1711yescharacteristic
314C22⋊C4—41no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—41no—
320C42—81no—
321C42C22×C4×C1711yescharacteristic
322C42C2×D4×C1711yescharacteristic
323C42C2×D4×C1711yescharacteristic
324C42#41×C1711yescharacteristic
325C23C17×#12111yescharacteristic
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—42no—
330Q8—41no—
331Q8—41no—
332Q8—81no—
333Q8C17×#10412yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4C2×C22⋊C4×C1711yescharacteristic
352C2×C4C4×D4×C1711yescharacteristic
353C2×C4C4×D4×C1711yescharacteristic
354C2×C4C17×#12011yescharacteristic
355C2×C4C17×#12111yescharacteristic
356C2×C4C17×#12311yescharacteristic
357C8—42no—
358C8—41no—
359C8—41no—
360C8—42no—
361C8—41no—
362C8—41no—
363C22—21no—
364C22—21no—
365C22C17×#38611yescharacteristic
366C22C17×#47111yescharacteristic
367C22C17×#47311yescharacteristic
368C4—21no—
369C4—22no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—82no—
378C4—81no—
379C4—81no—
380C4C2×C17×#10411yescharacteristic
381C4C17×#42111yescharacteristic
382C2—21no—
383C2—21no—
384C2C17×#187111yescharacteristic
385C2C17×#190511yescharacteristic
386C2C17×#190811yescharacteristic
subgroupG/H
center, socleC22×C17#386
commutatorC42C22×C4×C17
Frattini, cosocleC2×C42C23×C17
prime radicalC23×C17#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C17 ◁ C2×C42×C17 ◁ G

Complements

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

NH ≅ G/Ntype
11. C17×#475C4semidirect
12. C17×#475C4semidirect
13. C17×#462C4semidirect
14. C17×#462C4semidirect
131. #9166C17direct (N × H)
199. #475C4×C17semidirect
200. #475C4×C17semidirect
201. #462C4×C17semidirect
202. #462C4×C17semidirect
275. C17#9166direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#31132
C17×#29842
C17×#17632
C17×#15312
C17×#15032
C17×#13022
C17×#12852
#916617