Finite Group Databasebeta

← C17×#10464 #16868208

Subgroups — C17×#10464 #16868208

Order 4352.

452 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#3036C211yesmaximal, characteristic
2C17×#1804C211yesmaximal, characteristic, complemented
3C17×#1571C211yesmaximal, characteristic, complemented
4C17×#1540C211yesmaximal, characteristic, complemented
5C17×#1517C211yesmaximal, characteristic, complemented
6C17×#1460C211yesmaximal, characteristic, complemented
7C17×#1286C211yesmaximal, characteristic, complemented
8C2×C17×AGL(1,ℤ/8)—21no—
9C17×#532C2211yescharacteristic
10C2×C4×D4×C17C2211yescharacteristic
11C17×#463—22no—
12C17×#461—21no—
13C17×#459—21no—
14C17×#431C2211yescharacteristic
15C17×#431C2211yescharacteristic
16C17×#423C2211yescharacteristic
17C2×C17×C4⋊C8C2211yescharacteristic
18C17×#417—21no—
19C2×C17×#102—21no—
20C17×#382C2211yescharacteristic
21C17×#340—21no—
22C17×#335—22no—
23C17×#332—21no—
24C17×#331—21no—
25C17×#327—22no—
26C17×#323—22no—
27C2×#41×C17—21no—
28C22×D4×C17D411yescharacteristic
29C23×C4×C17—21no—
30C17×AGL(1,ℤ/8)—42no—
31C17×AGL(1,ℤ/8)—42no—
32C2×SD16×C17—22no—
33C2×D8×C17—22no—
34C2×M16×C17—21no—
35C22×C8×C17—21no—
36C17×C4⋊Q8D411yescharacteristic
37C17×#127D411yescharacteristic
38C17×#124—21no—
39C17×#121—21no—
40C17×#121—21no—
41C4×D4×C17—21no—
42C4×D4×C17—21no—
43C4×D4×C17—21no—
44C4×D4×C17—21no—
45C4×D4×C17—21no—
46C4×D4×C17—21no—
47C2×C4⋊C4×C17—21no—
48C2×C4⋊C4×C17D411yescharacteristic
49C2×C22⋊C4×C17—21no—
50C2×C42×C17C2311yescharacteristic
51C17×C4⋊C8—21no—
52C17×C4⋊C8—21no—
53C17×C4⋊C8—21no—
54C17×C4⋊C8—21no—
55C17×C4⋊C8—21no—
56C17×C4⋊C8D412yes—
57C17×#103—22no—
58C17×#102—22no—
59C17×#102—42no—
60C17×C22⋊C8—21no—
61C17×C22⋊C8D412yes—
62C4×C8×C17—21no—
63C4×C8×C17—21no—
64C17×#95—21no—
65C17×#95—21no—
66C24×C17—21no—
67#41×C17—41no—
68#41×C17—41no—
69#41×C17—41no—
70#41×C17—41no—
71C2×Q8×C17—21no—
72C2×D4×C17—21no—
73C2×D4×C17—21no—
74C2×D4×C17—21no—
75C2×D4×C17—21no—
76C2×D4×C17—21no—
77C2×D4×C17—21no—
78C2×D4×C17—21no—
79C2×D4×C17—21no—
80C2×D4×C17—41no—
81C2×D4×C17D812yes—
82C22×C4×C17—21no—
83C22×C4×C17—41no—
84C22×C4×C17—41no—
85C22×C4×C17—41no—
86C22×C4×C17—41no—
87C22×C4×C17—82no—
88C22×C4×C17C2×D411yescharacteristic
89C22×C4×C17C2×D411yescharacteristic
90C22×C4×C17#4111yescharacteristic
91SD16×C17—44no—
92D8×C17—44no—
93M16×C17—41no—
94M16×C17—41no—
95C2×C8×C17—22no—
96C2×C8×C17—22no—
97C2×C8×C17—41no—
98C2×C8×C17—41no—
99C2×C8×C17—41no—
100C2×C8×C17—41no—
101C4⋊C4×C17—21no—
102C4⋊C4×C17—21no—
103C4⋊C4×C17—21no—
104C4⋊C4×C17—21no—
105C4⋊C4×C17—41no—
106C4⋊C4×C17—41no—
107C22⋊C4×C17—41no—
108C22⋊C4×C17—41no—
109C22⋊C4×C17—41no—
110C22⋊C4×C17—41no—
111C42×C17C2×D411yescharacteristic
112C42×C17C2×D411yescharacteristic
113C42×C17#4111yescharacteristic
114C42×C17#4111yescharacteristic
115#10464C1711yesmaximal, characteristic, direct product factor, Sylow
116C23×C17—22no—
117C23×C17—41no—
118C23×C17—41no—
119C23×C17—41no—
120C23×C17—81no—
121C23×C17—81no—
122C23×C17#12111yescharacteristic
123Q8×C17—41no—
124Q8×C17—41no—
125D4×C17—22no—
126D4×C17—22no—
127D4×C17—41no—
128D4×C17—41no—
129D4×C17—41no—
130D4×C17—41no—
131D4×C17—81no—
132D4×C17—81no—
133D4×C17—81no—
134D4×C17—81no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—41no—
138C2×C4×C17—41no—
139C2×C4×C17—82no—
140C2×C4×C17—82no—
141C2×C4×C17—81no—
142C2×C4×C17—81no—
143C2×C4×C17—81no—
144C2×C4×C17—81no—
145C2×C4×C17—81no—
146C2×C4×C17—81no—
147C2×C4×C17#12011yescharacteristic
148C2×C4×C17#12111yescharacteristic
149C2×C4×C17#12111yescharacteristic
150C2×C4×C17#12311yescharacteristic
151C2×C4×C17#12311yescharacteristic
152C2×C4×C17#12411yescharacteristic
153C2×C4×C17C2×D811yescharacteristic
154C2×C4×C17#13511yescharacteristic
155C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C17#13711yescharacteristic
157C8×C17—42no—
158C8×C17—42no—
159C8×C17—81no—
160C8×C17—81no—
161#3036C2×C1711yescharacteristic
162#1804C2×C1711yescharacteristic, complemented
163#1571C2×C1711yescharacteristic, complemented
164#1540C2×C1711yescharacteristic, complemented
165#1517C2×C1711yescharacteristic, complemented
166#1460C2×C1711yescharacteristic, complemented
167#1286C2×C1711yescharacteristic, complemented
168C22×C17—21no—
169C22×C17—21no—
170C22×C17—42no—
171C22×C17—82no—
172C22×C17—82no—
173C22×C17—81no—
174C22×C17—81no—
175C22×C17—81no—
176C22×C17—81no—
177C22×C17—81no—
178C22×C17#39411yescharacteristic
179C22×C17#46611yescharacteristic
180C22×C17#46811yescharacteristic
181C4×C17—21no—
182C4×C17—21no—
183C4×C17—21no—
184C4×C17—21no—
185C4×C17—81no—
186C4×C17—81no—
187C4×C17—161no—
188C4×C17#44711yescharacteristic
189C4×C17#45211yescharacteristic
190C4×C17#48011yescharacteristic
191C4×C17#48211yescharacteristic
192C2×AGL(1,ℤ/8)—21no—
193#532C22×C1711yescharacteristic
194C2×C4×D4C22×C1711yescharacteristic
195#463—22no—
196#461—21no—
197#459—21no—
198#431C22×C1711yescharacteristic
199#431C22×C1711yescharacteristic
200#423C22×C1711yescharacteristic
201C2×C4⋊C8C22×C1711yescharacteristic
202#417—21no—
203C2×#102—21no—
204#382C22×C1711yescharacteristic
205#340—21no—
206#335—22no—
207#332—21no—
208#331—21no—
209#327—22no—
210#323—22no—
211C2×C17—21no—
212C2×C17—21no—
213C2×C17—82no—
214C2×C17—161no—
215C2×C17#203311yescharacteristic
216C2×C17#204811yescharacteristic
217C2×C17#205311yescharacteristic
218C2×#41—21no—
219C22×D4D4×C1711yescharacteristic
220C23×C4—21no—
221AGL(1,ℤ/8)—42no—
222AGL(1,ℤ/8)—42no—
223C2×SD16—22no—
224C2×D8—22no—
225C2×M16—21no—
226C22×C8—21no—
227C4⋊Q8D4×C1711yescharacteristic
228#127D4×C1711yescharacteristic
229#124—21no—
230#121—21no—
231#121—21no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4D4×C1711yescharacteristic
240C2×C22⋊C4—21no—
241C2×C42C23×C1711yescharacteristic
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8D4×C1712yes—
248#103—22no—
249#102—22no—
250#102—42no—
251C22⋊C8—21no—
252C22⋊C8D4×C1712yes—
253C4×C8—21no—
254C4×C8—21no—
255#95—21no—
256#95—21no—
257C17#1046411yescharacteristic, direct product factor, Sylow
258C24—21no—
259#41—41no—
260#41—41no—
261#41—41no—
262#41—41no—
263C2×Q8—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—41no—
273C2×D4D8×C1712yes—
274C22×C4—21no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—82no—
280C22×C4C2×D4×C1711yescharacteristic
281C22×C4C2×D4×C1711yescharacteristic
282C22×C4#41×C1711yescharacteristic
283SD16—44no—
284D8—44no—
285M16—41no—
286M16—41no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C42C2×D4×C1711yescharacteristic
304C42C2×D4×C1711yescharacteristic
305C42#41×C1711yescharacteristic
306C42#41×C1711yescharacteristic
307C23—22no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—81no—
312C23—81no—
313C23C17×#12111yescharacteristic
314Q8—41no—
315Q8—41no—
316D4—22no—
317D4—22no—
318D4—41no—
319D4—41no—
320D4—41no—
321D4—41no—
322D4—81no—
323D4—81no—
324D4—81no—
325D4—81no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4C17×#12011yescharacteristic
339C2×C4C17×#12111yescharacteristic
340C2×C4C17×#12111yescharacteristic
341C2×C4C17×#12311yescharacteristic
342C2×C4C17×#12311yescharacteristic
343C2×C4C17×#12411yescharacteristic
344C2×C4C2×D8×C1711yescharacteristic
345C2×C4C17×#13511yescharacteristic
346C2×C4C17×AGL(1,ℤ/8)11yescharacteristic
347C2×C4C17×#13711yescharacteristic
348C8—42no—
349C8—42no—
350C8—81no—
351C8—81no—
352C22—21no—
353C22—21no—
354C22—42no—
355C22—82no—
356C22—82no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22C17×#39411yescharacteristic
363C22C17×#46611yescharacteristic
364C22C17×#46811yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—81no—
370C4—81no—
371C4—161no—
372C4C17×#44711yescharacteristic
373C4C17×#45211yescharacteristic
374C4C17×#48011yescharacteristic
375C4C17×#48211yescharacteristic
376C2—21no—
377C2—21no—
378C2—82no—
379C2—161no—
380C2C17×#203311yescharacteristic
381C2C17×#204811yescharacteristic
382C2C17×#205311yescharacteristic
subgroupG/H
center, socleC22×C17#394
commutator, Frattini, cosocleC2×C42C23×C17
prime radicalC17×#3036C2
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
2. C17×#1804C2semidirect
3. C17×#1571C2semidirect
4. C17×#1540C2semidirect
5. C17×#1517C2semidirect
6. C17×#1460C2semidirect
7. C17×#1286C2semidirect
115. #10464C17direct (N × H)
162. #1804C2×C17semidirect
163. #1571C2×C17semidirect
164. #1540C2×C17semidirect
165. #1517C2×C17semidirect
166. #1460C2×C17semidirect
167. #1286C2×C17semidirect
257. C17#10464direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#30362
C17×#18042
C17×#15712
C17×#15402
C17×#15172
C17×#14602
C17×#12862
#1046417