Finite Group Databasebeta

← C17×#12622 #16870366

Subgroups — C17×#12622 #16870366

Order 4352.

636 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#2321C211yesmaximal, characteristic
2C2×C17×AΣL(1,𝔽16)C212yesmaximal
3C4×C17×#100C211yesmaximal, characteristic, complemented
4C4×C17×#99C211yesmaximal, characteristic, complemented
5C17×#1390C212yesmaximal, complemented
6C2×C17×#120C2211yescharacteristic
7C2×C4×D4×C17C2211yescharacteristic
8C22×C22⋊C4×C17C411yescharacteristic, complemented
9C2×C17×#100C2212yes—
10C2×C17×#99C2212yes—
11C4×M16×C17—21no—
12C17×#390—21no—
13C17×#388—21no—
14C17×#388—21no—
15C17×#386—21no—
16C17×#386C2211yescharacteristic
17C17×#385C411yescharacteristic, complemented
18C17×#379C411yescharacteristic, complemented
19C4×C22⋊C4×C17—21no—
20C4×C22⋊C4×C17C411yescharacteristic, complemented
21C17×AΣL(1,𝔽16)C418yescomplemented
22C17×#343—21no—
23C17×#342—21no—
24C25×C17C2×C411yescharacteristic
25C22×D4×C17—21no—
26C22×D4×C17C2311yescharacteristic
27C23×C4×C17C2×C411yescharacteristic
28C23×C4×C17C2×C411yescharacteristic
29C23×C4×C17D411yescharacteristic
30C2×M16×C17—22no—
31C17×#120—22no—
32C17×#120C2×C414yes—
33C4×D4×C17—41no—
34C4×D4×C17—41no—
35C2×C4⋊C4×C17—21no—
36C2×C4⋊C4×C17D411yescharacteristic
37C2×C22⋊C4×C17—21no—
38C2×C22⋊C4×C17—22no—
39C2×C22⋊C4×C17—21no—
40C2×C22⋊C4×C17—21no—
41C2×C22⋊C4×C17—21no—
42C2×C22⋊C4×C17—21no—
43C2×C22⋊C4×C17—21no—
44C2×C22⋊C4×C17—41no—
45C2×C22⋊C4×C17—41no—
46C2×C22⋊C4×C17C2×C411yescharacteristic
47C2×C22⋊C4×C17C2×C411yescharacteristic
48C2×C22⋊C4×C17C2×C411yescharacteristic
49C2×C22⋊C4×C17D411yescharacteristic
50C2×C42×C17—21no—
51C2×C42×C17—41no—
52C2×C42×C17D411yescharacteristic
53C17×#100—22no—
54C17×#100C2×C414yes—
55C17×#99—22no—
56C17×#99C2×C414yes—
57C17×#97—41no—
58C4×C8×C17—41no—
59C17×#95—21no—
60C17×#95—21no—
61C17×#95—21no—
62C17×#95—41no—
63C24×C17—21no—
64C24×C17—21no—
65C24×C17—41no—
66C24×C17—41no—
67C24×C17—81no—
68C24×C17—81no—
69C24×C17C4212yescomplemented
70C24×C17C22×C411yescharacteristic
71C24×C17C2×D411yescharacteristic
72C2×D4×C17—24no—
73C2×D4×C17—41no—
74C2×D4×C17—41no—
75C2×D4×C17—81no—
76C2×D4×C17—81no—
77C2×D4×C17C22×C412yes—
78C2×D4×C17#4112yes—
79C22×C4×C17—21no—
80C22×C4×C17—21no—
81C22×C4×C17—21no—
82C22×C4×C17—21no—
83C22×C4×C17—21no—
84C22×C4×C17—21no—
85C22×C4×C17—21no—
86C22×C4×C17—42no—
87C22×C4×C17—41no—
88C22×C4×C17—81no—
89C22×C4×C17—81no—
90C22×C4×C17—81no—
91C22×C4×C17C22⋊C411yescharacteristic
92C22×C4×C17C22⋊C411yescharacteristic
93C22×C4×C17C22⋊C411yescharacteristic
94C22×C4×C17C22⋊C411yescharacteristic
95C22×C4×C17C2×D411yescharacteristic
96M16×C17—24no—
97M16×C17—42no—
98C2×C8×C17—42no—
99C4⋊C4×C17—41no—
100C4⋊C4×C17—81no—
101C22⋊C4×C17—21no—
102C22⋊C4×C17—24no—
103C22⋊C4×C17—24no—
104C22⋊C4×C17—41no—
105C22⋊C4×C17—41no—
106C22⋊C4×C17—42no—
107C22⋊C4×C17—42no—
108C22⋊C4×C17—41no—
109C22⋊C4×C17—41no—
110C22⋊C4×C17—41no—
111C22⋊C4×C17—81no—
112C22⋊C4×C17—81no—
113C22⋊C4×C17—81no—
114C22⋊C4×C17C4212yescomplemented
115C42×C17—21no—
116C42×C17—21no—
117C42×C17—81no—
118C42×C17—82no—
119#12622C1711yesmaximal, characteristic, direct product factor, Sylow
120C23×C17—21no—
121C23×C17—21no—
122C23×C17—21no—
123C23×C17—21no—
124C23×C17—22no—
125C23×C17—41no—
126C23×C17—41no—
127C23×C17—41no—
128C23×C17—41no—
129C23×C17—41no—
130C23×C17—41no—
131C23×C17—41no—
132C23×C17—41no—
133C23×C17—42no—
134C23×C17—42no—
135C23×C17—41no—
136C23×C17—41no—
137C23×C17—82no—
138C23×C17—82no—
139C23×C17—82no—
140C23×C17—82no—
141C23×C17—84no—
142C23×C17C2×C4211yescharacteristic
143C23×C17C2×C22⋊C411yescharacteristic
144C23×C17#11711yescharacteristic
145C23×C17C4×D412yes—
146D4×C17—42no—
147D4×C17—42no—
148D4×C17—84no—
149C2×C4×C17—21no—
150C2×C4×C17—21no—
151C2×C4×C17—22no—
152C2×C4×C17—21no—
153C2×C4×C17—21no—
154C2×C4×C17—21no—
155C2×C4×C17—41no—
156C2×C4×C17—42no—
157C2×C4×C17—41no—
158C2×C4×C17—41no—
159C2×C4×C17—41no—
160C2×C4×C17—41no—
161C2×C4×C17—41no—
162C2×C4×C17—41no—
163C2×C4×C17—44no—
164C2×C4×C17—41no—
165C2×C4×C17—41no—
166C2×C4×C17—41no—
167C2×C4×C17—41no—
168C2×C4×C17—41no—
169C2×C4×C17—81no—
170C2×C4×C17—81no—
171C2×C4×C17—81no—
172C2×C4×C17—82no—
173C2×C4×C17—82no—
174C2×C4×C17—82no—
175C2×C4×C17#9911yescharacteristic
176C2×C4×C17#9911yescharacteristic
177C2×C4×C17C4×D412yes—
178C8×C17—44no—
179#2321C2×C1711yescharacteristic
180C2×AΣL(1,𝔽16)C2×C1712yes—
181C4×#100C2×C1711yescharacteristic, complemented
182C4×#99C2×C1711yescharacteristic, complemented
183#1390C2×C1712yescomplemented
184C22×C17—21no—
185C22×C17—21no—
186C22×C17—22no—
187C22×C17—21no—
188C22×C17—21no—
189C22×C17—42no—
190C22×C17—41no—
191C22×C17—41no—
192C22×C17—42no—
193C22×C17—42no—
194C22×C17—42no—
195C22×C17—41no—
196C22×C17—41no—
197C22×C17—41no—
198C22×C17—41no—
199C22×C17—42no—
200C22×C17—81no—
201C22×C17—81no—
202C22×C17—81no—
203C22×C17—84no—
204C22×C17—84no—
205C22×C17—81no—
206C22×C17—81no—
207C22×C17C4×C22⋊C411yescharacteristic
208C22×C17C2×#9911yescharacteristic
209C22×C17#41011yescharacteristic
210C4×C17—21no—
211C4×C17—21no—
212C4×C17—22no—
213C4×C17—42no—
214C4×C17—41no—
215C4×C17—41no—
216C4×C17—41no—
217C4×C17—81no—
218C4×C17—84no—
219C4×C17—81no—
220C2×#120C22×C1711yescharacteristic
221C2×C4×D4C22×C1711yescharacteristic
222C22×C22⋊C4C4×C1711yescharacteristic, complemented
223C2×#100C22×C1712yes—
224C2×#99C22×C1712yes—
225C4×M16—21no—
226#390—21no—
227#388—21no—
228#388—21no—
229#386—21no—
230#386C22×C1711yescharacteristic
231#385C4×C1711yescharacteristic, complemented
232#379C4×C1711yescharacteristic, complemented
233C4×C22⋊C4—21no—
234C4×C22⋊C4C4×C1711yescharacteristic, complemented
235AΣL(1,𝔽16)C4×C1718yescomplemented
236#343—21no—
237#342—21no—
238C2×C17—21no—
239C2×C17—21no—
240C2×C17—42no—
241C2×C17—41no—
242C2×C17—41no—
243C2×C17—82no—
244C2×C17C4×#9911yescharacteristic
245C2×C17#212111yescharacteristic
246C2×C17#212211yescharacteristic
247C25C2×C4×C1711yescharacteristic
248C22×D4—21no—
249C22×D4C23×C1711yescharacteristic
250C23×C4C2×C4×C1711yescharacteristic
251C23×C4C2×C4×C1711yescharacteristic
252C23×C4D4×C1711yescharacteristic
253C2×M16—22no—
254#120—22no—
255#120C2×C4×C1714yes—
256C4×D4—41no—
257C4×D4—41no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4D4×C1711yescharacteristic
260C2×C22⋊C4—21no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—41no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4C2×C4×C1711yescharacteristic
270C2×C22⋊C4C2×C4×C1711yescharacteristic
271C2×C22⋊C4C2×C4×C1711yescharacteristic
272C2×C22⋊C4D4×C1711yescharacteristic
273C2×C42—21no—
274C2×C42—41no—
275C2×C42D4×C1711yescharacteristic
276#100—22no—
277#100C2×C4×C1714yes—
278#99—22no—
279#99C2×C4×C1714yes—
280#97—41no—
281C4×C8—41no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—41no—
286C17#1262211yescharacteristic, direct product factor, Sylow
287C24—21no—
288C24—21no—
289C24—41no—
290C24—41no—
291C24—81no—
292C24—81no—
293C24C42×C1712yescomplemented
294C24C22×C4×C1711yescharacteristic
295C24C2×D4×C1711yescharacteristic
296C2×D4—24no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—81no—
300C2×D4—81no—
301C2×D4C22×C4×C1712yes—
302C2×D4#41×C1712yes—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—81no—
313C22×C4—81no—
314C22×C4—81no—
315C22×C4C22⋊C4×C1711yescharacteristic
316C22×C4C22⋊C4×C1711yescharacteristic
317C22×C4C22⋊C4×C1711yescharacteristic
318C22×C4C22⋊C4×C1711yescharacteristic
319C22×C4C2×D4×C1711yescharacteristic
320M16—24no—
321M16—42no—
322C2×C8—42no—
323C4⋊C4—41no—
324C4⋊C4—81no—
325C22⋊C4—21no—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—81no—
336C22⋊C4—81no—
337C22⋊C4—81no—
338C22⋊C4C42×C1712yescomplemented
339C42—21no—
340C42—21no—
341C42—81no—
342C42—82no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—22no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23—42no—
358C23—41no—
359C23—41no—
360C23—82no—
361C23—82no—
362C23—82no—
363C23—82no—
364C23—84no—
365C23C2×C42×C1711yescharacteristic
366C23C2×C22⋊C4×C1711yescharacteristic
367C23C17×#11711yescharacteristic
368C23C4×D4×C1712yes—
369D4—42no—
370D4—42no—
371D4—84no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—44no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4C17×#9911yescharacteristic
399C2×C4C17×#9911yescharacteristic
400C2×C4C4×D4×C1712yes—
401C8—44no—
402C22—21no—
403C22—21no—
404C22—22no—
405C22—21no—
406C22—21no—
407C22—42no—
408C22—41no—
409C22—41no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—42no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—84no—
422C22—84no—
423C22—81no—
424C22—81no—
425C22C4×C22⋊C4×C1711yescharacteristic
426C22C2×C17×#9911yescharacteristic
427C22C17×#41011yescharacteristic
428C4—22no—
429C4—21no—
430C4—21no—
431C4—42no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—81no—
436C4—84no—
437C4—81no—
438C2—21no—
439C2—21no—
440C2—42no—
441C2—41no—
442C2—41no—
443C2—82no—
444C2C4×C17×#9911yescharacteristic
445C2C17×#212111yescharacteristic
446C2C17×#212211yescharacteristic
subgroupG/H
center, socleC22×C17C2×#99
commutatorC23C2×C42×C17
Frattini, cosocleC22×D4C23×C17
prime radicalC2×C17×#120C22
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C17 ◁ C22×C4×C17 ◁ C2×C4×D4×C17 ◁ G

Complements

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

NH ≅ G/Ntype
3. C4×C17×#100C2semidirect
4. C4×C17×#99C2semidirect
5. C17×#1390C2semidirect
8. C22×C22⋊C4×C17C4semidirect
17. C17×#385C4semidirect
18. C17×#379C4semidirect
20. C4×C22⋊C4×C17C4semidirect
21. C17×AΣL(1,𝔽16)C4semidirect
69. C24×C17C42semidirect
114. C22⋊C4×C17C42semidirect
119. #12622C17direct (N × H)
181. C4×#100C2×C17semidirect
182. C4×#99C2×C17semidirect
183. #1390C2×C17semidirect
222. C22×C22⋊C4C4×C17semidirect
231. #385C4×C17semidirect
232. #379C4×C17semidirect
234. C4×C22⋊C4C4×C17semidirect
235. AΣL(1,𝔽16)C4×C17semidirect
286. C17#12622direct (N × H)
293. C24C42×C17semidirect
338. C22⋊C4C42×C17semidirect

Maximal subgroups

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

subgroupindex
C17×#23212
C2×C17×AΣL(1,𝔽16)2
C4×C17×#1002
C4×C17×#992
C17×#13902
#1262217