Finite Group Databasebeta

← C17×#7730 #16865474

Subgroups — C17×#7730 #16865474

Order 4352.

520 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#2850C211yesmaximal, characteristic
2C17×#1817C211yesmaximal, characteristic, complemented
3C17×#1797C211yesmaximal, characteristic, complemented
4C17×#1272C212yesmaximal, complemented
5C17×#1270C212yesmaximal, complemented
6C2×C17×#124C411yescharacteristic
7C2×C17×C22⋊Q8C411yescharacteristic
8C2×C17×#121C411yescharacteristic
9C22×C42×C17C2211yescharacteristic
10C2×C17×C4⋊C8—21no—
11C2×C17×C22⋊C8—21no—
12C2×C17×C22⋊C8—21no—
13C17×#391C2211yescharacteristic
14C17×#390C2211yescharacteristic
15C17×#388—21no—
16C17×#388—21no—
17C17×#386—21no—
18C17×#386—21no—
19C17×#382C411yescharacteristic
20C17×#336—21no—
21C17×#336C2212yes—
22C17×#336C2212yes—
23C22×Q8×C17C2×C411yescharacteristic
24C22×D4×C17—21no—
25C22×D4×C17C2×C411yescharacteristic
26C23×C4×C17C2×C411yescharacteristic
27C23×C4×C17D411yescharacteristic
28C23×C4×C17D411yescharacteristic
29C22×C8×C17—22no—
30C22×C8×C17—22no—
31C17×#124—42no—
32C17×C22⋊Q8—42no—
33C17×#121—42no—
34C2×C4⋊C4×C17—21no—
35C2×C4⋊C4×C17C2×C411yescharacteristic
36C2×C4⋊C4×C17C2×C411yescharacteristic
37C2×C22⋊C4×C17—21no—
38C2×C22⋊C4×C17—21no—
39C2×C22⋊C4×C17—21no—
40C2×C22⋊C4×C17—21no—
41C2×C22⋊C4×C17—21no—
42C2×C22⋊C4×C17—21no—
43C2×C42×C17—42no—
44C2×C42×C17C2×C411yescharacteristic
45C2×C42×C17D411yescharacteristic
46C2×C42×C17D411yescharacteristic
47C2×C42×C17C2311yescharacteristic
48C17×C4⋊C8—42no—
49C17×C22⋊C8—42no—
50C17×C22⋊C8—42no—
51C17×#95—21no—
52C17×#95—21no—
53C17×#95—21no—
54C17×#95—21no—
55C24×C17—21no—
56C24×C17C22⋊C411yescharacteristic
57C2×Q8×C17—22no—
58C2×Q8×C17—41no—
59C2×Q8×C17—41no—
60C2×D4×C17—22no—
61C2×D4×C17—41no—
62C2×D4×C17—41no—
63C2×D4×C17—42no—
64C2×D4×C17—42no—
65C2×D4×C17—42no—
66C22×C4×C17—22no—
67C22×C4×C17—22no—
68C22×C4×C17—22no—
69C22×C4×C17—22no—
70C22×C4×C17—21no—
71C22×C4×C17—21no—
72C22×C4×C17—21no—
73C22×C4×C17—21no—
74C22×C4×C17—21no—
75C22×C4×C17—41no—
76C22×C4×C17—41no—
77C22×C4×C17—41no—
78C22×C4×C17—41no—
79C22×C4×C17C22⋊C411yescharacteristic
80C22×C4×C17C22⋊C411yescharacteristic
81C22×C4×C17C22⋊C411yescharacteristic
82C22×C4×C17C22×C411yescharacteristic
83C22×C4×C17C2×D411yescharacteristic
84C22×C4×C17C2×D411yescharacteristic
85C2×C8×C17—42no—
86C2×C8×C17—42no—
87C2×C8×C17—42no—
88C2×C8×C17—42no—
89C2×C8×C17—42no—
90C2×C8×C17—42no—
91C4⋊C4×C17—41no—
92C4⋊C4×C17—41no—
93C4⋊C4×C17—42no—
94C22⋊C4×C17—42no—
95C22⋊C4×C17—42no—
96C22⋊C4×C17—42no—
97C22⋊C4×C17—42no—
98C22⋊C4×C17—42no—
99C22⋊C4×C17—41no—
100C22⋊C4×C17—41no—
101C42×C17—22no—
102C42×C17—22no—
103C42×C17—41no—
104C42×C17—41no—
105#7730C1711yesmaximal, characteristic, direct product factor, Sylow
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17—21no—
110C23×C17—21no—
111C23×C17—21no—
112C23×C17—41no—
113C23×C17—41no—
114C23×C17—41no—
115C23×C17—41no—
116C23×C17—41no—
117C23×C17—41no—
118C23×C17—41no—
119C23×C17#10012yes—
120C23×C17C2×C22⋊C411yescharacteristic
121Q8×C17—82no—
122D4×C17—82no—
123D4×C17—84no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17—21no—
136C2×C4×C17—21no—
137C2×C4×C17—21no—
138C2×C4×C17—44no—
139C2×C4×C17—44no—
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#9912yes—
152C2×C4×C17#13112yes—
153C2×C4×C17#13512yes—
154C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C17#13711yescharacteristic
156C8×C17—82no—
157C8×C17—82no—
158#2850C2×C1711yescharacteristic
159#1817C2×C1711yescharacteristic, complemented
160#1797C2×C1711yescharacteristic, complemented
161#1272C2×C1712yescomplemented
162#1270C2×C1712yescomplemented
163C22×C17—22no—
164C22×C17—22no—
165C22×C17—22no—
166C22×C17—41no—
167C22×C17—41no—
168C22×C17—41no—
169C22×C17—41no—
170C22×C17—81no—
171C22×C17—81no—
172C22×C17—81no—
173C22×C17—81no—
174C22×C17—81no—
175C22×C17—81no—
176C22×C17—81no—
177C22×C17#40811yescharacteristic
178C22×C17C2×#9911yescharacteristic
179C22×C17C2×#10011yescharacteristic
180C22×C17#41611yescharacteristic
181C22×C17#41711yescharacteristic
182C22×C17#42111yescharacteristic
183C22×C17#42111yescharacteristic
184C4×C17—22no—
185C4×C17—21no—
186C4×C17—21no—
187C4×C17—22no—
188C4×C17—22no—
189C4×C17—41no—
190C4×C17—41no—
191C4×C17—81no—
192C4×C17—81no—
193C4×C17—81no—
194C2×#124C4×C1711yescharacteristic
195C2×C22⋊Q8C4×C1711yescharacteristic
196C2×#121C4×C1711yescharacteristic
197C22×C42C22×C1711yescharacteristic
198C2×C4⋊C8—21no—
199C2×C22⋊C8—21no—
200C2×C22⋊C8—21no—
201#391C22×C1711yescharacteristic
202#390C22×C1711yescharacteristic
203#388—21no—
204#388—21no—
205#386—21no—
206#386—21no—
207#382C4×C1711yescharacteristic
208#336—21no—
209#336C22×C1712yes—
210#336C22×C1712yes—
211C2×C17—42no—
212C2×C17—161no—
213C2×C17#147111yescharacteristic
214C2×C17#148211yescharacteristic
215C2×C17#149511yescharacteristic
216C2×C17#151311yescharacteristic
217C2×C17#151611yescharacteristic
218C2×C17#153311yescharacteristic
219C2×C17#155111yescharacteristic
220C22×Q8C2×C4×C1711yescharacteristic
221C22×D4—21no—
222C22×D4C2×C4×C1711yescharacteristic
223C23×C4C2×C4×C1711yescharacteristic
224C23×C4D4×C1711yescharacteristic
225C23×C4D4×C1711yescharacteristic
226C22×C8—22no—
227C22×C8—22no—
228#124—42no—
229C22⋊Q8—42no—
230#121—42no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4C2×C4×C1711yescharacteristic
233C2×C4⋊C4C2×C4×C1711yescharacteristic
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C42—42no—
241C2×C42C2×C4×C1711yescharacteristic
242C2×C42D4×C1711yescharacteristic
243C2×C42D4×C1711yescharacteristic
244C2×C42C23×C1711yescharacteristic
245C4⋊C8—42no—
246C22⋊C8—42no—
247C22⋊C8—42no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252C17#773011yescharacteristic, direct product factor, Sylow
253C24—21no—
254C24C22⋊C4×C1711yescharacteristic
255C2×Q8—22no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×D4—22no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4C22⋊C4×C1711yescharacteristic
278C22×C4C22⋊C4×C1711yescharacteristic
279C22×C4C22⋊C4×C1711yescharacteristic
280C22×C4C22×C4×C1711yescharacteristic
281C22×C4C2×D4×C1711yescharacteristic
282C22×C4C2×D4×C1711yescharacteristic
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—42no—
299C42—22no—
300C42—22no—
301C42—41no—
302C42—41no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23C17×#10012yes—
317C23C2×C22⋊C4×C1711yescharacteristic
318Q8—82no—
319D4—82no—
320D4—84no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
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×C4C17×#9912yes—
349C2×C4C17×#13112yes—
350C2×C4C17×#13512yes—
351C2×C4C17×AGL(1,ℤ/8)11yescharacteristic
352C2×C4C17×#13711yescharacteristic
353C8—82no—
354C8—82no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22C17×#40811yescharacteristic
370C22C2×C17×#9911yescharacteristic
371C22C2×C17×#10011yescharacteristic
372C22C17×#41611yescharacteristic
373C22C17×#41711yescharacteristic
374C22C17×#42111yescharacteristic
375C22C17×#42111yescharacteristic
376C4—22no—
377C4—21no—
378C4—21no—
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—81no—
384C4—81no—
385C4—81no—
386C2—42no—
387C2—161no—
388C2C17×#147111yescharacteristic
389C2C17×#148211yescharacteristic
390C2C17×#149511yescharacteristic
391C2C17×#151311yescharacteristic
392C2C17×#151611yescharacteristic
393C2C17×#153311yescharacteristic
394C2C17×#155111yescharacteristic
subgroupG/H
center, socleC23×C17C2×C22⋊C4
commutatorC22×C4C22×C4×C17
Frattini, cosocleC2×C42C23×C17
prime radicalC2×C17×#121C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C17 ◁ C22×C42×C17 ◁ G

Complements

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

NH ≅ G/Ntype
2. C17×#1817C2semidirect
3. C17×#1797C2semidirect
4. C17×#1272C2semidirect
5. C17×#1270C2semidirect
105. #7730C17direct (N × H)
159. #1817C2×C17semidirect
160. #1797C2×C17semidirect
161. #1272C2×C17semidirect
162. #1270C2×C17semidirect
252. C17#7730direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#28502
C17×#18172
C17×#17972
C17×#12722
C17×#12702
#773017