Finite Group Databasebeta

← C17×#7698 #16865442

Subgroups — C17×#7698 #16865442

Order 4352.

532 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#2854C211yesmaximal, characteristic
2C17×#1797C211yesmaximal, characteristic, complemented
3C17×#1749C211yesmaximal, characteristic, complemented
4C17×#1271C212yesmaximal, complemented
5C17×#1270C212yesmaximal, complemented
6C2×C17×#123—21no—
7C2×C17×#121C411yescharacteristic
8C22×C42×C17C2211yescharacteristic
9C2×C17×C4⋊C8—21no—
10C2×C17×C22⋊C8—21no—
11C2×C17×C22⋊C8—21no—
12C2×C17×#97—21no—
13C17×#390C411yescharacteristic
14C17×#390C2211yescharacteristic
15C17×#389C411yescharacteristic
16C17×#389C2211yescharacteristic
17C17×#388—21no—
18C17×#388—21no—
19C17×#386—21no—
20C17×#380C411yescharacteristic
21C17×#336C2212yes—
22C17×#336C2212yes—
23C22×D4×C17—21no—
24C22×D4×C17C2×C411yescharacteristic
25C23×C4×C17C2×C411yescharacteristic
26C23×C4×C17D411yescharacteristic
27C23×C4×C17D411yescharacteristic
28C22×C8×C17—22no—
29C22×C8×C17—22no—
30C17×#123—44no—
31C17×#121—42no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17C2×C411yescharacteristic
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—24no—
50C17×C22⋊C8—42no—
51C17×#97—42no—
52C17×#95—21no—
53C17×#95—21no—
54C17×#95—21no—
55C24×C17—21no—
56C24×C17C22⋊C411yescharacteristic
57C2×D4×C17—22no—
58C2×D4×C17—41no—
59C2×D4×C17—41no—
60C2×D4×C17—42no—
61C2×D4×C17—42no—
62C2×D4×C17—42no—
63C22×C4×C17—22no—
64C22×C4×C17—22no—
65C22×C4×C17—22no—
66C22×C4×C17—22no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17—21no—
70C22×C4×C17—21no—
71C22×C4×C17—21no—
72C22×C4×C17—41no—
73C22×C4×C17—41no—
74C22×C4×C17—41no—
75C22×C4×C17—41no—
76C22×C4×C17C22⋊C411yescharacteristic
77C22×C4×C17C22⋊C411yescharacteristic
78C22×C4×C17C22⋊C411yescharacteristic
79C22×C4×C17C22×C411yescharacteristic
80C22×C4×C17C2×D411yescharacteristic
81C22×C4×C17C2×D411yescharacteristic
82C2×C8×C17—24no—
83C2×C8×C17—42no—
84C2×C8×C17—42no—
85C2×C8×C17—42no—
86C2×C8×C17—42no—
87C2×C8×C17—42no—
88C4⋊C4×C17—22no—
89C4⋊C4×C17—41no—
90C4⋊C4×C17—41no—
91C4⋊C4×C17—42no—
92C4⋊C4×C17—42no—
93C22⋊C4×C17—42no—
94C22⋊C4×C17—42no—
95C22⋊C4×C17—42no—
96C22⋊C4×C17—42no—
97C22⋊C4×C17—42no—
98C22⋊C4×C17—41no—
99C22⋊C4×C17—41no—
100C42×C17—22no—
101C42×C17—22no—
102C42×C17—41no—
103C42×C17—41no—
104#7698C1711yesmaximal, characteristic, direct product factor, Sylow
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17—21no—
110C23×C17—21no—
111C23×C17—41no—
112C23×C17—41no—
113C23×C17—41no—
114C23×C17—41no—
115C23×C17—41no—
116C23×C17—41no—
117C23×C17—41no—
118C23×C17#10412yes—
119C23×C17C2×C22⋊C411yescharacteristic
120D4×C17—82no—
121D4×C17—84no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
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—44no—
137C2×C4×C17—44no—
138C2×C4×C17—42no—
139C2×C4×C17—42no—
140C2×C4×C17—41no—
141C2×C4×C17—41no—
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#9912yes—
150C2×C4×C17#13112yes—
151C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C17AGL(1,ℤ/8)12yes—
153C2×C4×C17#13711yescharacteristic
154C8×C17—44no—
155C8×C17—82no—
156#2854C2×C1711yescharacteristic
157#1797C2×C1711yescharacteristic, complemented
158#1749C2×C1711yescharacteristic, complemented
159#1271C2×C1712yescomplemented
160#1270C2×C1712yescomplemented
161C22×C17—22no—
162C22×C17—22no—
163C22×C17—22no—
164C22×C17—41no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17—41no—
168C22×C17—81no—
169C22×C17—81no—
170C22×C17—81no—
171C22×C17—81no—
172C22×C17—81no—
173C22×C17—81no—
174C22×C17—81no—
175C22×C17#40811yescharacteristic
176C22×C17C2×#9911yescharacteristic
177C22×C17#41311yescharacteristic
178C22×C17#41711yescharacteristic
179C22×C17#41811yescharacteristic
180C22×C17C2×#10411yescharacteristic
181C22×C17#42111yescharacteristic
182C4×C17—22no—
183C4×C17—22no—
184C4×C17—22no—
185C4×C17—21no—
186C4×C17—21no—
187C4×C17—41no—
188C4×C17—41no—
189C4×C17—81no—
190C4×C17—81no—
191C4×C17—81no—
192C2×#123—21no—
193C2×#121C4×C1711yescharacteristic
194C22×C42C22×C1711yescharacteristic
195C2×C4⋊C8—21no—
196C2×C22⋊C8—21no—
197C2×C22⋊C8—21no—
198C2×#97—21no—
199#390C4×C1711yescharacteristic
200#390C22×C1711yescharacteristic
201#389C4×C1711yescharacteristic
202#389C22×C1711yescharacteristic
203#388—21no—
204#388—21no—
205#386—21no—
206#380C4×C1711yescharacteristic
207#336C22×C1712yes—
208#336C22×C1712yes—
209C2×C17—42no—
210C2×C17—161no—
211C2×C17#146811yescharacteristic
212C2×C17#148311yescharacteristic
213C2×C17#149611yescharacteristic
214C2×C17#150711yescharacteristic
215C2×C17#151311yescharacteristic
216C2×C17#153211yescharacteristic
217C2×C17#155711yescharacteristic
218C22×D4—21no—
219C22×D4C2×C4×C1711yescharacteristic
220C23×C4C2×C4×C1711yescharacteristic
221C23×C4D4×C1711yescharacteristic
222C23×C4D4×C1711yescharacteristic
223C22×C8—22no—
224C22×C8—22no—
225#123—44no—
226#121—42no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4C2×C4×C1711yescharacteristic
230C2×C4⋊C4C2×C4×C1711yescharacteristic
231C2×C4⋊C4C2×C4×C1711yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C42—42no—
239C2×C42C2×C4×C1711yescharacteristic
240C2×C42D4×C1711yescharacteristic
241C2×C42D4×C1711yescharacteristic
242C2×C42C23×C1711yescharacteristic
243C4⋊C8—42no—
244C22⋊C8—24no—
245C22⋊C8—42no—
246#97—42no—
247#95—21no—
248#95—21no—
249#95—21no—
250C17#769811yescharacteristic, direct product factor, Sylow
251C24—21no—
252C24C22⋊C4×C1711yescharacteristic
253C2×D4—22no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4C22⋊C4×C1711yescharacteristic
273C22×C4C22⋊C4×C1711yescharacteristic
274C22×C4C22⋊C4×C1711yescharacteristic
275C22×C4C22×C4×C1711yescharacteristic
276C22×C4C2×D4×C1711yescharacteristic
277C22×C4C2×D4×C1711yescharacteristic
278C2×C8—24no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C4⋊C4—22no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—41no—
293C22⋊C4—42no—
294C22⋊C4—41no—
295C22⋊C4—42no—
296C42—22no—
297C42—22no—
298C42—41no—
299C42—41no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23C17×#10412yes—
314C23C2×C22⋊C4×C1711yescharacteristic
315D4—82no—
316D4—84no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
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—44no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4C17×#9912yes—
345C2×C4C17×#13112yes—
346C2×C4C17×AGL(1,ℤ/8)11yescharacteristic
347C2×C4C17×AGL(1,ℤ/8)12yes—
348C2×C4C17×#13711yescharacteristic
349C8—44no—
350C8—82no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22C17×#40811yescharacteristic
366C22C2×C17×#9911yescharacteristic
367C22C17×#41311yescharacteristic
368C22C17×#41711yescharacteristic
369C22C17×#41811yescharacteristic
370C22C2×C17×#10411yescharacteristic
371C22C17×#42111yescharacteristic
372C4—22no—
373C4—22no—
374C4—22no—
375C4—21no—
376C4—21no—
377C4—41no—
378C4—41no—
379C4—81no—
380C4—81no—
381C4—81no—
382C2—42no—
383C2—161no—
384C2C17×#146811yescharacteristic
385C2C17×#148311yescharacteristic
386C2C17×#149611yescharacteristic
387C2C17×#150711yescharacteristic
388C2C17×#151311yescharacteristic
389C2C17×#153211yescharacteristic
390C2C17×#155711yescharacteristic
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×#1797C2semidirect
3. C17×#1749C2semidirect
4. C17×#1271C2semidirect
5. C17×#1270C2semidirect
104. #7698C17direct (N × H)
157. #1797C2×C17semidirect
158. #1749C2×C17semidirect
159. #1271C2×C17semidirect
160. #1270C2×C17semidirect
250. C17#7698direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#28542
C17×#17972
C17×#17492
C17×#12712
C17×#12702
#769817