Finite Group Databasebeta

← C17×#9173 #16866917

Subgroups — C17×#9173 #16866917

Order 4352.

562 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#3038C211yesmaximal, characteristic
2C17×#2906C211yesmaximal, characteristic, complemented
3C17×#1545C211yesmaximal, characteristic, complemented
4C2×C17×#331C211yesmaximal, characteristic, complemented
5C17×#1477C211yesmaximal, characteristic, complemented
6C17×#1301C211yesmaximal, characteristic, complemented
7C17×#1292C211yesmaximal, characteristic, complemented
8C2×C17×AGL(1,ℤ/8)—21no—
9C2×C17×#131—21no—
10C2×C17×#127C2211yescharacteristic
11C4×#41×C17C2211yescharacteristic
12C17×#460—21no—
13C17×#460C412yes—
14C17×#459—21no—
15C17×#459C412yes—
16C17×#431C2211yescharacteristic
17C17×#424—21no—
18C17×#424—22no—
19C17×#423—21no—
20C17×#423C2211yescharacteristic
21C17×#423C2211yescharacteristic
22C2×C17×C4⋊C8—21no—
23C2×C17×C4⋊C8C2211yescharacteristic
24C2×C17×C4⋊C8C2211yescharacteristic
25C17×#418—21no—
26C17×#331—21no—
27C17×#331—21no—
28C17×#331—22no—
29C17×Q8⋊C8—21no—
30C17×D4⋊C8—21no—
31C2×#41×C17D411yescharacteristic
32C22×D4×C17—21no—
33C22×D4×C17—21no—
34C22×D4×C17—21no—
35C17×AGL(1,ℤ/8)—42no—
36C17×AGL(1,ℤ/8)—42no—
37C2×SD16×C17—22no—
38C2×D8×C17—22no—
39C17×#131—42no—
40C17×#131—42no—
41C2×M16×C17—22no—
42C2×M16×C17—21no—
43C2×M16×C17—21no—
44C22×C8×C17—22no—
45C22×C8×C17—21no—
46C22×C8×C17—21no—
47C17×#127—21no—
48C17×#127—21no—
49C17×#127—21no—
50C17×#127C2×C412yes—
51C4×Q8×C17C2×C411yescharacteristic
52C4×Q8×C17D411yescharacteristic
53C4×D4×C17—21no—
54C4×D4×C17—21no—
55C4×D4×C17C2×C411yescharacteristic
56C4×D4×C17D411yescharacteristic
57C17×#117—21no—
58C17×#117D411yescharacteristic
59C2×C42×C17—21no—
60C2×C42×C17C2311yescharacteristic
61C17×C4⋊C8—22no—
62C17×C4⋊C8—22no—
63C17×C4⋊C8—21no—
64C17×C4⋊C8—21no—
65C17×C4⋊C8—21no—
66C17×C4⋊C8—21no—
67C17×C4⋊C8—21no—
68C17×C4⋊C8—21no—
69C17×C4⋊C8C2×C412yes—
70C17×C4⋊C8D411yescharacteristic
71C17×C4⋊C8D411yescharacteristic
72C17×C4⋊C8D412yes—
73C17×#102—22no—
74C17×#102—22no—
75C17×C22⋊C8—21no—
76C4×C8×C17—21no—
77C24×C17—41no—
78#41×C17—21no—
79#41×C17—21no—
80#41×C17D812yes—
81#41×C17SD1612yes—
82C2×Q8×C17C22⋊C411yescharacteristic
83C2×D4×C17—21no—
84C2×D4×C17—22no—
85C2×D4×C17—22no—
86C2×D4×C17—22no—
87C2×D4×C17—41no—
88C2×D4×C17—41no—
89C2×D4×C17—41no—
90C2×D4×C17—41no—
91C2×D4×C17—41no—
92C2×D4×C17—41no—
93C2×D4×C17—41no—
94C2×D4×C17—81no—
95C2×D4×C17—81no—
96C2×D4×C17—82no—
97C2×D4×C17C22⋊C411yescharacteristic
98C22×C4×C17—21no—
99C22×C4×C17—21no—
100C22×C4×C17—21no—
101C22×C4×C17C2×D411yescharacteristic
102C22×C4×C17#4111yescharacteristic
103C22×C4×C17#4111yescharacteristic
104SD16×C17—44no—
105D8×C17—44no—
106M16×C17—44no—
107M16×C17—41no—
108M16×C17—41no—
109M16×C17—41no—
110M16×C17—41no—
111C2×C8×C17—22no—
112C2×C8×C17—21no—
113C2×C8×C17—22no—
114C2×C8×C17—21no—
115C2×C8×C17—44no—
116C2×C8×C17—41no—
117C2×C8×C17—41no—
118C2×C8×C17—41no—
119C2×C8×C17—41no—
120C2×C8×C17—81no—
121C4⋊C4×C17—21no—
122C4⋊C4×C17—21no—
123C4⋊C4×C17C22⋊C411yescharacteristic
124C4⋊C4×C17C22⋊C411yescharacteristic
125C22⋊C4×C17—21no—
126C22⋊C4×C17—21no—
127C22⋊C4×C17—21no—
128C42×C17—21no—
129C42×C17—21no—
130C42×C17—21no—
131C42×C17C22×C411yescharacteristic
132C42×C17C2×D411yescharacteristic
133C42×C17C2×D411yescharacteristic
134C42×C17C2×D411yescharacteristic
135#9173C1711yesmaximal, characteristic, direct product factor, Sylow
136C23×C17—21no—
137C23×C17—42no—
138C23×C17—82no—
139C23×C17—82no—
140C23×C17—81no—
141C23×C17—81no—
142C23×C17#12311yescharacteristic
143Q8×C17—21no—
144Q8×C17#10212yes—
145D4×C17—22no—
146D4×C17—22no—
147D4×C17—21no—
148D4×C17—42no—
149D4×C17—42no—
150D4×C17—82no—
151D4×C17—82no—
152D4×C17—82no—
153D4×C17—81no—
154D4×C17—81no—
155D4×C17—81no—
156D4×C17—81no—
157D4×C17#10212yes—
158C2×C4×C17—21no—
159C2×C4×C17—22no—
160C2×C4×C17—22no—
161C2×C4×C17—21no—
162C2×C4×C17—21no—
163C2×C4×C17—21no—
164C2×C4×C17—21no—
165C2×C4×C17—41no—
166C2×C4×C17—41no—
167C2×C4×C17—41no—
168C2×C4×C17—81no—
169C2×C4×C17C2×C22⋊C411yescharacteristic
170C2×C4×C17C4×D411yescharacteristic
171C2×C4×C17C4×D411yescharacteristic
172C2×C4×C17#12011yescharacteristic
173C2×C4×C17#12111yescharacteristic
174C2×C4×C17#12111yescharacteristic
175C2×C4×C17C2×D811yescharacteristic
176C2×C4×C17C2×SD1611yescharacteristic
177C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
178C2×C4×C17#13711yescharacteristic
179C8×C17—42no—
180C8×C17—41no—
181C8×C17—42no—
182C8×C17—41no—
183C8×C17—81no—
184C8×C17—81no—
185#3038C2×C1711yescharacteristic
186#2906C2×C1711yescharacteristic, complemented
187#1545C2×C1711yescharacteristic, complemented
188C2×#331C2×C1711yescharacteristic, complemented
189#1477C2×C1711yescharacteristic, complemented
190#1301C2×C1711yescharacteristic, complemented
191#1292C2×C1711yescharacteristic, complemented
192C22×C17—21no—
193C22×C17—21no—
194C22×C17—22no—
195C22×C17—41no—
196C22×C17—41no—
197C22×C17—82no—
198C22×C17—82no—
199C22×C17—82no—
200C22×C17—162no—
201C22×C17—162no—
202C22×C17#38611yescharacteristic
203C22×C17#48011yescharacteristic
204C22×C17#48111yescharacteristic
205C4×C17—22no—
206C4×C17—21no—
207C4×C17—21no—
208C4×C17—21no—
209C4×C17—41no—
210C4×C17—41no—
211C4×C17C2×#10211yescharacteristic
212C4×C17#41711yescharacteristic
213C4×C17#44711yescharacteristic
214C4×C17#44811yescharacteristic
215C4×C17#47111yescharacteristic
216C4×C17#47211yescharacteristic
217C2×AGL(1,ℤ/8)—21no—
218C2×#131—21no—
219C2×#127C22×C1711yescharacteristic
220C4×#41C22×C1711yescharacteristic
221#460—21no—
222#460C4×C1712yes—
223#459—21no—
224#459C4×C1712yes—
225#431C22×C1711yescharacteristic
226#424—21no—
227#424—22no—
228#423—21no—
229#423C22×C1711yescharacteristic
230#423C22×C1711yescharacteristic
231C2×C4⋊C8—21no—
232C2×C4⋊C8C22×C1711yescharacteristic
233C2×C4⋊C8C22×C1711yescharacteristic
234#418—21no—
235#331—21no—
236#331—21no—
237#331—22no—
238Q8⋊C8—21no—
239D4⋊C8—21no—
240C2×C17—21no—
241C2×C17—21no—
242C2×C17—42no—
243C2×C17—162no—
244C2×C17#187411yescharacteristic
245C2×C17#190311yescharacteristic
246C2×C17#190711yescharacteristic
247C2×#41D4×C1711yescharacteristic
248C22×D4—21no—
249C22×D4—21no—
250C22×D4—21no—
251AGL(1,ℤ/8)—42no—
252AGL(1,ℤ/8)—42no—
253C2×SD16—22no—
254C2×D8—22no—
255#131—42no—
256#131—42no—
257C2×M16—22no—
258C2×M16—21no—
259C2×M16—21no—
260C22×C8—22no—
261C22×C8—21no—
262C22×C8—21no—
263#127—21no—
264#127—21no—
265#127—21no—
266#127C2×C4×C1712yes—
267C4×Q8C2×C4×C1711yescharacteristic
268C4×Q8D4×C1711yescharacteristic
269C4×D4—21no—
270C4×D4—21no—
271C4×D4C2×C4×C1711yescharacteristic
272C4×D4D4×C1711yescharacteristic
273#117—21no—
274#117D4×C1711yescharacteristic
275C2×C42—21no—
276C2×C42C23×C1711yescharacteristic
277C4⋊C8—22no—
278C4⋊C8—22no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8C2×C4×C1712yes—
286C4⋊C8D4×C1711yescharacteristic
287C4⋊C8D4×C1711yescharacteristic
288C4⋊C8D4×C1712yes—
289#102—22no—
290#102—22no—
291C22⋊C8—21no—
292C4×C8—21no—
293C17#917311yescharacteristic, direct product factor, Sylow
294C24—41no—
295#41—21no—
296#41—21no—
297#41D8×C1712yes—
298#41SD16×C1712yes—
299C2×Q8C22⋊C4×C1711yescharacteristic
300C2×D4—21no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—81no—
312C2×D4—81no—
313C2×D4—82no—
314C2×D4C22⋊C4×C1711yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4C2×D4×C1711yescharacteristic
319C22×C4#41×C1711yescharacteristic
320C22×C4#41×C1711yescharacteristic
321SD16—44no—
322D8—44no—
323M16—44no—
324M16—41no—
325M16—41no—
326M16—41no—
327M16—41no—
328C2×C8—22no—
329C2×C8—21no—
330C2×C8—22no—
331C2×C8—21no—
332C2×C8—44no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—81no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4C22⋊C4×C1711yescharacteristic
341C4⋊C4C22⋊C4×C1711yescharacteristic
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42C22×C4×C1711yescharacteristic
349C42C2×D4×C1711yescharacteristic
350C42C2×D4×C1711yescharacteristic
351C42C2×D4×C1711yescharacteristic
352C23—21no—
353C23—42no—
354C23—82no—
355C23—82no—
356C23—81no—
357C23—81no—
358C23C17×#12311yescharacteristic
359Q8—21no—
360Q8C17×#10212yes—
361D4—22no—
362D4—22no—
363D4—21no—
364D4—42no—
365D4—42no—
366D4—82no—
367D4—82no—
368D4—82no—
369D4—81no—
370D4—81no—
371D4—81no—
372D4—81no—
373D4C17×#10212yes—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—81no—
385C2×C4C2×C22⋊C4×C1711yescharacteristic
386C2×C4C4×D4×C1711yescharacteristic
387C2×C4C4×D4×C1711yescharacteristic
388C2×C4C17×#12011yescharacteristic
389C2×C4C17×#12111yescharacteristic
390C2×C4C17×#12111yescharacteristic
391C2×C4C2×D8×C1711yescharacteristic
392C2×C4C2×SD16×C1711yescharacteristic
393C2×C4C17×AGL(1,ℤ/8)11yescharacteristic
394C2×C4C17×#13711yescharacteristic
395C8—42no—
396C8—41no—
397C8—42no—
398C8—41no—
399C8—81no—
400C8—81no—
401C22—21no—
402C22—21no—
403C22—22no—
404C22—41no—
405C22—41no—
406C22—82no—
407C22—82no—
408C22—82no—
409C22—162no—
410C22—162no—
411C22C17×#38611yescharacteristic
412C22C17×#48011yescharacteristic
413C22C17×#48111yescharacteristic
414C4—22no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—41no—
419C4—41no—
420C4C2×C17×#10211yescharacteristic
421C4C17×#41711yescharacteristic
422C4C17×#44711yescharacteristic
423C4C17×#44811yescharacteristic
424C4C17×#47111yescharacteristic
425C4C17×#47211yescharacteristic
426C2—21no—
427C2—21no—
428C2—42no—
429C2—162no—
430C2C17×#187411yescharacteristic
431C2C17×#190311yescharacteristic
432C2C17×#190711yescharacteristic
subgroupG/H
center, socleC22×C17#386
commutatorC42C22×C4×C17
Frattini, cosocleC2×C42C23×C17
prime radicalC17×#3038C2
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
2. C17×#2906C2semidirect
3. C17×#1545C2semidirect
4. C2×C17×#331C2semidirect
5. C17×#1477C2semidirect
6. C17×#1301C2semidirect
7. C17×#1292C2semidirect
135. #9173C17direct (N × H)
186. #2906C2×C17semidirect
187. #1545C2×C17semidirect
188. C2×#331C2×C17semidirect
189. #1477C2×C17semidirect
190. #1301C2×C17semidirect
191. #1292C2×C17semidirect
293. C17#9173direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#30382
C17×#29062
C17×#15452
C2×C17×#3312
C17×#14772
C17×#13012
C17×#12922
#917317