Finite Group Databasebeta

← C17×#11566 #16869310

Subgroups — C17×#11566 #16869310

Order 4352.

494 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#2544C211yesmaximal, characteristic
2C17×#1465C211yesmaximal, characteristic, complemented
3C17×#1459C211yesmaximal, characteristic, complemented
4C17×#1319C212yesmaximal, complemented
5C17×#1318C212yesmaximal, complemented
6C2×C17×#127C411yescharacteristic
7C2×C17×#124C411yescharacteristic
8C2×C17×#124C411yescharacteristic
9C2×C4×D4×C17C2211yescharacteristic
10C17×#407—21no—
11C2×C17×C22⋊C8—21no—
12C17×#394—21no—
13C17×#394—21no—
14C17×#391C2211yescharacteristic
15C17×#390C2211yescharacteristic
16C17×#386—21no—
17C17×#386—21no—
18C4×C4⋊C4×C17C411yescharacteristic
19C17×#324C2212yes—
20C17×#324C2212yes—
21C17×#323—21no—
22C17×#323—21no—
23C22×Q8×C17C2×C411yescharacteristic
24C22×D4×C17—21no—
25C22×D4×C17—21no—
26C22×D4×C17C2×C411yescharacteristic
27C22×D4×C17C2×C411yescharacteristic
28C23×C4×C17D411yescharacteristic
29C23×C4×C17D411yescharacteristic
30C2×M16×C17—41no—
31C22×C8×C17—41no—
32C17×#127—42no—
33C17×#124—41no—
34C17×#124—41no—
35C17×#124—42no—
36C4×D4×C17—41no—
37C4×D4×C17—41no—
38C2×C4⋊C4×C17—21no—
39C2×C4⋊C4×C17C2311yescharacteristic
40C2×C22⋊C4×C17—21no—
41C2×C22⋊C4×C17—21no—
42C2×C22⋊C4×C17—21no—
43C2×C22⋊C4×C17—21no—
44C2×C22⋊C4×C17—21no—
45C2×C22⋊C4×C17—21no—
46C2×C22⋊C4×C17D411yescharacteristic
47C2×C22⋊C4×C17D411yescharacteristic
48C2×C42×C17C2×C411yescharacteristic
49C2×C42×C17C2×C411yescharacteristic
50C2×C42×C17C2×C411yescharacteristic
51C17×C22⋊C8—22no—
52C17×C22⋊C8—22no—
53C17×C22⋊C8—41no—
54C17×C22⋊C8—41no—
55C17×#95—21no—
56C17×#95—21no—
57C17×#95—21no—
58C24×C17—21no—
59C24×C17C22⋊C411yescharacteristic
60C24×C17C22⋊C411yescharacteristic
61C2×Q8×C17—22no—
62C2×Q8×C17—41no—
63C2×Q8×C17—41no—
64C2×D4×C17—22no—
65C2×D4×C17—22no—
66C2×D4×C17—22no—
67C2×D4×C17—41no—
68C2×D4×C17—41no—
69C2×D4×C17—41no—
70C2×D4×C17—42no—
71C2×D4×C17—42no—
72C2×D4×C17—42no—
73C2×D4×C17—42no—
74C2×D4×C17—81no—
75C2×D4×C17—81no—
76C22×C4×C17—21no—
77C22×C4×C17—21no—
78C22×C4×C17—21no—
79C22×C4×C17—21no—
80C22×C4×C17—21no—
81C22×C4×C17—21no—
82C22×C4×C17—21no—
83C22×C4×C17—81no—
84C22×C4×C17—81no—
85C22×C4×C17C22⋊C411yescharacteristic
86C22×C4×C17C22⋊C411yescharacteristic
87C22×C4×C17C22×C411yescharacteristic
88C22×C4×C17C2×D411yescharacteristic
89C22×C4×C17C2×D411yescharacteristic
90M16×C17—82no—
91C2×C8×C17—42no—
92C2×C8×C17—42no—
93C2×C8×C17—82no—
94C4⋊C4×C17—41no—
95C4⋊C4×C17—42no—
96C22⋊C4×C17—41no—
97C22⋊C4×C17—41no—
98C22⋊C4×C17—42no—
99C22⋊C4×C17—42no—
100C22⋊C4×C17—42no—
101C22⋊C4×C17—41no—
102C22⋊C4×C17—41no—
103C22⋊C4×C17—81no—
104C22⋊C4×C17—81no—
105C42×C17—21no—
106C42×C17—21no—
107C42×C17—41no—
108C42×C17—41no—
109#11566C1711yesmaximal, characteristic, direct product factor, Sylow
110C23×C17—21no—
111C23×C17—21no—
112C23×C17—21no—
113C23×C17—21no—
114C23×C17—21no—
115C23×C17—41no—
116C23×C17—41no—
117C23×C17—41no—
118C23×C17—41no—
119C23×C17—41no—
120C23×C17—41no—
121C23×C17—41no—
122C23×C17—41no—
123C23×C17—41no—
124C23×C17—41no—
125C23×C17—41no—
126C23×C17#10012yes—
127C23×C17C2×C22⋊C411yescharacteristic
128Q8×C17—82no—
129D4×C17—44no—
130D4×C17—82no—
131D4×C17—84no—
132D4×C17—84no—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17—22no—
136C2×C4×C17—21no—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
139C2×C4×C17—41no—
140C2×C4×C17—42no—
141C2×C4×C17—42no—
142C2×C4×C17—41no—
143C2×C4×C17—41no—
144C2×C4×C17—41no—
145C2×C4×C17—41no—
146C2×C4×C17—41no—
147C2×C4×C17—41no—
148C2×C4×C17—41no—
149C2×C4×C17—41no—
150C2×C4×C17—41no—
151C2×C4×C17—41no—
152C2×C4×C17—41no—
153C2×C4×C17—81no—
154C2×C4×C17—81no—
155C2×C4×C17#9911yescharacteristic
156C2×C4×C17#9911yescharacteristic
157C2×C4×C17#13112yes—
158C8×C17—82no—
159C8×C17—82no—
160#2544C2×C1711yescharacteristic
161#1465C2×C1711yescharacteristic, complemented
162#1459C2×C1711yescharacteristic, complemented
163#1319C2×C1712yescomplemented
164#1318C2×C1712yescomplemented
165C22×C17—21no—
166C22×C17—21no—
167C22×C17—22no—
168C22×C17—42no—
169C22×C17—42no—
170C22×C17—41no—
171C22×C17—41no—
172C22×C17—41no—
173C22×C17—41no—
174C22×C17—41no—
175C22×C17—81no—
176C22×C17—81no—
177C22×C17—81no—
178C22×C17—81no—
179C22×C17—81no—
180C22×C17—81no—
181C22×C17—81no—
182C22×C17—81no—
183C22×C17—81no—
184C22×C17#40811yescharacteristic
185C22×C17C2×#9911yescharacteristic
186C22×C17C2×#10011yescharacteristic
187C4×C17—21no—
188C4×C17—21no—
189C4×C17—22no—
190C4×C17—42no—
191C4×C17—41no—
192C4×C17—41no—
193C4×C17—41no—
194C4×C17—81no—
195C4×C17—81no—
196C2×#127C4×C1711yescharacteristic
197C2×#124C4×C1711yescharacteristic
198C2×#124C4×C1711yescharacteristic
199C2×C4×D4C22×C1711yescharacteristic
200#407—21no—
201C2×C22⋊C8—21no—
202#394—21no—
203#394—21no—
204#391C22×C1711yescharacteristic
205#390C22×C1711yescharacteristic
206#386—21no—
207#386—21no—
208C4×C4⋊C4C4×C1711yescharacteristic
209#324C22×C1712yes—
210#324C22×C1712yes—
211#323—21no—
212#323—21no—
213C2×C17—21no—
214C2×C17—21no—
215C2×C17—42no—
216C2×C17—81no—
217C2×C17—161no—
218C2×C17#147111yescharacteristic
219C2×C17#212711yescharacteristic
220C2×C17#213211yescharacteristic
221C22×Q8C2×C4×C1711yescharacteristic
222C22×D4—21no—
223C22×D4—21no—
224C22×D4C2×C4×C1711yescharacteristic
225C22×D4C2×C4×C1711yescharacteristic
226C23×C4D4×C1711yescharacteristic
227C23×C4D4×C1711yescharacteristic
228C2×M16—41no—
229C22×C8—41no—
230#127—42no—
231#124—41no—
232#124—41no—
233#124—42no—
234C4×D4—41no—
235C4×D4—41no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4C23×C1711yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4D4×C1711yescharacteristic
245C2×C22⋊C4D4×C1711yescharacteristic
246C2×C42C2×C4×C1711yescharacteristic
247C2×C42C2×C4×C1711yescharacteristic
248C2×C42C2×C4×C1711yescharacteristic
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—41no—
252C22⋊C8—41no—
253#95—21no—
254#95—21no—
255#95—21no—
256C17#1156611yescharacteristic, direct product factor, Sylow
257C24—21no—
258C24C22⋊C4×C1711yescharacteristic
259C24C22⋊C4×C1711yescharacteristic
260C2×Q8—22no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—81no—
274C2×D4—81no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—81no—
283C22×C4—81no—
284C22×C4C22⋊C4×C1711yescharacteristic
285C22×C4C22⋊C4×C1711yescharacteristic
286C22×C4C22×C4×C1711yescharacteristic
287C22×C4C2×D4×C1711yescharacteristic
288C22×C4C2×D4×C1711yescharacteristic
289M16—82no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—82no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—81no—
303C22⋊C4—81no—
304C42—21no—
305C42—21no—
306C42—41no—
307C42—41no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23C17×#10012yes—
325C23C2×C22⋊C4×C1711yescharacteristic
326Q8—82no—
327D4—44no—
328D4—82no—
329D4—84no—
330D4—84no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4C17×#9911yescharacteristic
354C2×C4C17×#9911yescharacteristic
355C2×C4C17×#13112yes—
356C8—82no—
357C8—82no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—42no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22C17×#40811yescharacteristic
378C22C2×C17×#9911yescharacteristic
379C22C2×C17×#10011yescharacteristic
380C4—22no—
381C4—21no—
382C4—21no—
383C4—42no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—81no—
388C4—81no—
389C2—21no—
390C2—21no—
391C2—42no—
392C2—81no—
393C2—161no—
394C2C17×#147111yescharacteristic
395C2C17×#212711yescharacteristic
396C2C17×#213211yescharacteristic
subgroupG/H
center, socleC22×C17C2×#99
commutatorC22×C4C22×C4×C17
Frattini, cosocleC2×C4⋊C4C23×C17
prime radicalC2×C17×#127C4
LCSG ▷ C22×C4 ▷ 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
2. C17×#1465C2semidirect
3. C17×#1459C2semidirect
4. C17×#1319C2semidirect
5. C17×#1318C2semidirect
109. #11566C17direct (N × H)
161. #1465C2×C17semidirect
162. #1459C2×C17semidirect
163. #1319C2×C17semidirect
164. #1318C2×C17semidirect
256. C17#11566direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#25442
C17×#14652
C17×#14592
C17×#13192
C17×#13182
#1156617