Finite Group Databasebeta

← C17×#10221 #16867965

Subgroups — C17×#10221 #16867965

Order 4352.

570 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#3227C211yesmaximal, characteristic, complemented
2C17×#3206C211yesmaximal, characteristic, complemented
3C17×#3150C211yesmaximal, characteristic
4C17×#1993C211yesmaximal, characteristic, complemented
5C17×#1517C211yesmaximal, characteristic, complemented
6C17×#1517C211yesmaximal, characteristic, complemented
7C17×#1281C211yesmaximal, characteristic, complemented
8C2×C17×#135—22no—
9C17×#532C2211yescharacteristic
10C17×#532C2211yescharacteristic
11C2×C17×C4⋊Q8C2211yescharacteristic
12C17×#498—21no—
13C17×#497—21no—
14C17×#497—21no—
15C17×#496—21no—
16C17×#496—21no—
17C17×#492—21no—
18C17×#487—21no—
19C17×#486—21no—
20C17×#481—22no—
21C17×#476—22no—
22C17×#461—22no—
23C17×#448—22no—
24C17×#431C2211yescharacteristic
25C17×#431C2211yescharacteristic
26C2×C17×#97C2211yescharacteristic
27C17×#382C2211yescharacteristic
28C17×#340—22no—
29C17×#327—22no—
30C17×#327—22no—
31C2×#41×C17—21no—
32C2×#41×C17—21no—
33C22×Q8×C17—21no—
34C17×#135—44no—
35C17×#135—44no—
36C2×Q16×C17—22no—
37C2×SD16×C17—22no—
38C2×SD16×C17—22no—
39C2×SD16×C17—24no—
40C2×D8×C17—22no—
41C22×C8×C17D412yes—
42C17×C4⋊Q8—21no—
43C17×C4⋊Q8—21no—
44C17×C4⋊Q8—21no—
45C17×C4⋊Q8D411yescharacteristic
46C17×C4⋊Q8D411yescharacteristic
47C17×C4⋊Q8D412yes—
48C17×#127D411yescharacteristic
49C17×#127D411yescharacteristic
50C17×#124—21no—
51C17×#124—21no—
52C17×#121—21no—
53C17×#121—21no—
54C17×#121—21no—
55C17×#121—21no—
56C4×D4×C17—21no—
57C4×D4×C17—21no—
58C4×D4×C17—21no—
59C4×D4×C17—21no—
60C2×C4⋊C4×C17—21no—
61C2×C4⋊C4×C17—21no—
62C2×C4⋊C4×C17—21no—
63C2×C42×C17C2311yescharacteristic
64C17×#106—24no—
65C17×C4⋊C8—21no—
66C17×C4⋊C8—21no—
67C17×#103—24no—
68C17×#102—24no—
69C17×C22⋊C8D412yes—
70C17×C22⋊C8D412yes—
71C17×#97—21no—
72C17×#97—21no—
73C4×C8×C17—21no—
74C4×C8×C17—21no—
75C17×#95—21no—
76C17×#95—21no—
77#41×C17—42no—
78#41×C17—42no—
79#41×C17—42no—
80#41×C17—42no—
81C2×Q8×C17—21no—
82C2×Q8×C17—21no—
83C2×Q8×C17—22no—
84C2×Q8×C17—41no—
85C2×Q8×C17—82no—
86C2×D4×C17—21no—
87C2×D4×C17—21no—
88C2×D4×C17—21no—
89C2×D4×C17—21no—
90C2×D4×C17—21no—
91C2×D4×C17—21no—
92C2×D4×C17—41no—
93C2×D4×C17—41no—
94C22×C4×C17—41no—
95C22×C4×C17—41no—
96C22×C4×C17—41no—
97C22×C4×C17—41no—
98C22×C4×C17C2×D411yescharacteristic
99C22×C4×C17C2×D411yescharacteristic
100C22×C4×C17C2×D411yescharacteristic
101Q16×C17—44no—
102SD16×C17—44no—
103SD16×C17—44no—
104SD16×C17—84no—
105D8×C17—44no—
106C2×C8×C17—22no—
107C2×C8×C17—22no—
108C2×C8×C17—22no—
109C2×C8×C17—22no—
110C2×C8×C17SD1614yescomplemented
111C4⋊C4×C17—22no—
112C4⋊C4×C17—21no—
113C4⋊C4×C17—21no—
114C4⋊C4×C17—22no—
115C4⋊C4×C17—21no—
116C4⋊C4×C17—21no—
117C4⋊C4×C17—41no—
118C4⋊C4×C17—41no—
119C4⋊C4×C17—42no—
120C22⋊C4×C17—41no—
121C22⋊C4×C17—41no—
122C22⋊C4×C17—41no—
123C22⋊C4×C17—41no—
124C42×C17C2×D411yescharacteristic
125C42×C17C2×D411yescharacteristic
126C42×C17C2×D411yescharacteristic
127C42×C17#4111yescharacteristic
128#10221C1711yesmaximal, characteristic, direct product factor, Sylow
129C23×C17—41no—
130C23×C17—41no—
131C23×C17#12711yescharacteristic
132Q8×C17—44no—
133Q8×C17—42no—
134Q8×C17—42no—
135Q8×C17—82no—
136D4×C17—42no—
137D4×C17—42no—
138D4×C17—42no—
139D4×C17—42no—
140D4×C17—81no—
141D4×C17—81no—
142D4×C17—81no—
143D4×C17—81no—
144D4×C17—81no—
145D4×C17—81no—
146C2×C4×C17—41no—
147C2×C4×C17—41no—
148C2×C4×C17—41no—
149C2×C4×C17—41no—
150C2×C4×C17—42no—
151C2×C4×C17—82no—
152C2×C4×C17—82no—
153C2×C4×C17—82no—
154C2×C4×C17—81no—
155C2×C4×C17—81no—
156C2×C4×C17—81no—
157C2×C4×C17#12011yescharacteristic
158C2×C4×C17#12011yescharacteristic
159C2×C4×C17#12011yescharacteristic
160C2×C4×C17#12111yescharacteristic
161C2×C4×C17#12111yescharacteristic
162C2×C4×C17#12111yescharacteristic
163C2×C4×C17C2×SD1612yes—
164C2×C4×C17AGL(1,ℤ/8)12yes—
165C8×C17—24no—
166C8×C17—42no—
167C8×C17—42no—
168#3227C2×C1711yescharacteristic, complemented
169#3206C2×C1711yescharacteristic, complemented
170#3150C2×C1711yescharacteristic
171#1993C2×C1711yescharacteristic, complemented
172#1517C2×C1711yescharacteristic, complemented
173#1517C2×C1711yescharacteristic, complemented
174#1281C2×C1711yescharacteristic, complemented
175C22×C17—21no—
176C22×C17—21no—
177C22×C17—81no—
178C22×C17—81no—
179C22×C17—81no—
180C22×C17—81no—
181C22×C17—81no—
182C22×C17—81no—
183C22×C17#39211yescharacteristic
184C22×C17#49211yescharacteristic
185C22×C17#49611yescharacteristic
186C4×C17—21no—
187C4×C17—21no—
188C4×C17—21no—
189C4×C17—21no—
190C4×C17—82no—
191C4×C17—81no—
192C4×C17—81no—
193C4×C17—161no—
194C4×C17#45012yes—
195C4×C17#46012yes—
196C2×#135—22no—
197#532C22×C1711yescharacteristic
198#532C22×C1711yescharacteristic
199C2×C4⋊Q8C22×C1711yescharacteristic
200#498—21no—
201#497—21no—
202#497—21no—
203#496—21no—
204#496—21no—
205#492—21no—
206#487—21no—
207#486—21no—
208#481—22no—
209#476—22no—
210#461—22no—
211#448—22no—
212#431C22×C1711yescharacteristic
213#431C22×C1711yescharacteristic
214C2×#97C22×C1711yescharacteristic
215#382C22×C1711yescharacteristic
216#340—22no—
217#327—22no—
218#327—22no—
219C2×C17—21no—
220C2×C17—21no—
221C2×C17—161no—
222C2×C17—161no—
223C2×C17#200811yescharacteristic
224C2×C17#200911yescharacteristic
225C2×C17#201411yescharacteristic
226C2×#41—21no—
227C2×#41—21no—
228C22×Q8—21no—
229#135—44no—
230#135—44no—
231C2×Q16—22no—
232C2×SD16—22no—
233C2×SD16—22no—
234C2×SD16—24no—
235C2×D8—22no—
236C22×C8D4×C1712yes—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8D4×C1712yes—
241C4⋊Q8D4×C1711yescharacteristic
242C4⋊Q8D4×C1711yescharacteristic
243#127D4×C1711yescharacteristic
244#127D4×C1711yescharacteristic
245#124—21no—
246#124—21no—
247#121—21no—
248#121—21no—
249#121—21no—
250#121—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C42C23×C1711yescharacteristic
259#106—24no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262#103—24no—
263#102—24no—
264C22⋊C8D4×C1712yes—
265C22⋊C8D4×C1712yes—
266#97—21no—
267#97—21no—
268C4×C8—21no—
269C4×C8—21no—
270#95—21no—
271#95—21no—
272C17#1022111yescharacteristic, direct product factor, Sylow
273#41—42no—
274#41—42no—
275#41—42no—
276#41—42no—
277C2×Q8—22no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—41no—
281C2×Q8—82no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4C2×D4×C1711yescharacteristic
295C22×C4C2×D4×C1711yescharacteristic
296C22×C4C2×D4×C1711yescharacteristic
297Q16—44no—
298SD16—44no—
299SD16—44no—
300SD16—84no—
301D8—44no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8SD16×C1714yescomplemented
307C4⋊C4—22no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C42C2×D4×C1711yescharacteristic
321C42C2×D4×C1711yescharacteristic
322C42C2×D4×C1711yescharacteristic
323C42#41×C1711yescharacteristic
324C23—41no—
325C23—41no—
326C23C17×#12711yescharacteristic
327Q8—44no—
328Q8—42no—
329Q8—42no—
330Q8—82no—
331D4—42no—
332D4—42no—
333D4—42no—
334D4—42no—
335D4—81no—
336D4—81no—
337D4—81no—
338D4—81no—
339D4—81no—
340D4—81no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4C17×#12011yescharacteristic
353C2×C4C17×#12011yescharacteristic
354C2×C4C17×#12011yescharacteristic
355C2×C4C17×#12111yescharacteristic
356C2×C4C17×#12111yescharacteristic
357C2×C4C17×#12111yescharacteristic
358C2×C4C2×SD16×C1712yes—
359C2×C4C17×AGL(1,ℤ/8)12yes—
360C8—24no—
361C8—42no—
362C8—42no—
363C22—21no—
364C22—21no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22C17×#39211yescharacteristic
372C22C17×#49211yescharacteristic
373C22C17×#49611yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—82no—
379C4—81no—
380C4—81no—
381C4—161no—
382C4C17×#45012yes—
383C4C17×#46012yes—
384C2—21no—
385C2—21no—
386C2—161no—
387C2—161no—
388C2C17×#200811yescharacteristic
389C2C17×#200911yescharacteristic
390C2C17×#201411yescharacteristic
subgroupG/H
center, socleC22×C17#392
commutator, Frattini, cosocleC2×C42C23×C17
prime radicalC17×#3150C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C17 ◁ C2×C42×C17 ◁ G

Complements

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

NH ≅ G/Ntype
1. C17×#3227C2semidirect
2. C17×#3206C2semidirect
4. C17×#1993C2semidirect
5. C17×#1517C2semidirect
6. C17×#1517C2semidirect
7. C17×#1281C2semidirect
110. C2×C8×C17SD16semidirect
128. #10221C17direct (N × H)
168. #3227C2×C17semidirect
169. #3206C2×C17semidirect
171. #1993C2×C17semidirect
172. #1517C2×C17semidirect
173. #1517C2×C17semidirect
174. #1281C2×C17semidirect
272. C17#10221direct (N × H)
306. C2×C8SD16×C17semidirect

Maximal subgroups

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

subgroupindex
C17×#32272
C17×#32062
C17×#31502
C17×#19932
C17×#15172
C17×#12812
#1022117