Finite Group Databasebeta

← C17×#7686 #16865430

Subgroups — C17×#7686 #16865430

Order 4352.

464 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#2860C211yesmaximal, characteristic
2C17×#1767C211yesmaximal, characteristic, complemented
3C4×C17×C4⋊C8C211yesmaximal, characteristic, complemented
4C17×#1271C212yesmaximal, complemented
5C17×#1270C212yesmaximal
6C2×C17×#126—21no—
7C2×C17×C4⋊C8—21no—
8C2×C17×C4⋊C8—21no—
9C2×C17×#97—21no—
10C2×C4×C8×C17—21no—
11C17×#390C411yescharacteristic
12C17×#390C411yescharacteristic
13C17×#390C2211yescharacteristic
14C17×#389C411yescharacteristic
15C17×#389C411yescharacteristic
16C17×#389C2211yescharacteristic
17C17×#388—21no—
18C17×#388—21no—
19C17×#388—21no—
20C43×C17C2211yescharacteristic
21C17×#336C2212yes—
22C17×#336C2212yes—
23C22×D4×C17C2×C411yescharacteristic
24C22×C8×C17—22no—
25C22×C8×C17—22no—
26C17×#126—42no—
27C17×#126—42no—
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17C2×C411yescharacteristic
32C2×C4⋊C4×C17C2×C411yescharacteristic
33C2×C4⋊C4×C17C2×C411yescharacteristic
34C2×C22⋊C4×C17—21no—
35C2×C22⋊C4×C17—21no—
36C2×C22⋊C4×C17—21no—
37C2×C22⋊C4×C17—21no—
38C2×C22⋊C4×C17—21no—
39C2×C22⋊C4×C17—21no—
40C2×C42×C17C2×C411yescharacteristic
41C2×C42×C17C2×C411yescharacteristic
42C2×C42×C17D411yescharacteristic
43C2×C42×C17D411yescharacteristic
44C2×C42×C17D411yescharacteristic
45C2×C42×C17D411yescharacteristic
46C2×C42×C17C2311yescharacteristic
47C17×C4⋊C8—24no—
48C17×C4⋊C8—24no—
49C17×#97—42no—
50C4×C8×C17—42no—
51C17×#95—21no—
52C17×#95—21no—
53C17×#95—21no—
54C24×C17—21no—
55C2×D4×C17—22no—
56C2×D4×C17—41no—
57C2×D4×C17—41no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22×C411yescharacteristic
66C22×C4×C17C2×D411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C2×C8×C17—24no—
69C2×C8×C17—24no—
70C2×C8×C17—42no—
71C2×C8×C17—42no—
72C2×C8×C17—42no—
73C2×C8×C17—42no—
74C4⋊C4×C17—22no—
75C4⋊C4×C17—41no—
76C4⋊C4×C17—41no—
77C4⋊C4×C17—42no—
78C4⋊C4×C17—42no—
79C4⋊C4×C17—41no—
80C4⋊C4×C17—41no—
81C22⋊C4×C17—42no—
82C22⋊C4×C17—42no—
83C22⋊C4×C17—42no—
84C22⋊C4×C17—42no—
85C22⋊C4×C17—41no—
86C22⋊C4×C17—41no—
87C22⋊C4×C17—41no—
88C22⋊C4×C17—41no—
89C42×C17—22no—
90C42×C17—22no—
91C42×C17—22no—
92C42×C17—21no—
93C42×C17—21no—
94C42×C17—41no—
95C42×C17—41no—
96C42×C17D812yes—
97C42×C17SD1612yes—
98#7686C1711yesmaximal, characteristic, direct product factor, Sylow
99C23×C17—41no—
100C23×C17—41no—
101C23×C17—41no—
102C23×C17—41no—
103C23×C17—41no—
104C23×C17—41no—
105C23×C17—41no—
106C23×C17C2×C22⋊C411yescharacteristic
107D4×C17—82no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17#10212yes—
132C2×C4×C17#10212yes—
133C2×C4×C17#10412yes—
134C2×C4×C17#10412yes—
135C2×C4×C17#13112yes—
136C2×C4×C17C2×D811yescharacteristic
137C2×C4×C17C2×SD1611yescharacteristic
138C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C17#13711yescharacteristic
140C8×C17—44no—
141C8×C17—44no—
142#2860C2×C1711yescharacteristic
143#1767C2×C1711yescharacteristic, complemented
144C4×C4⋊C8C2×C1711yescharacteristic, complemented
145#1271C2×C1712yescomplemented
146#1270C2×C1712yes—
147C22×C17—81no—
148C22×C17—81no—
149C22×C17—81no—
150C22×C17—81no—
151C22×C17—81no—
152C22×C17—81no—
153C22×C17—81no—
154C22×C17#40811yescharacteristic
155C22×C17#41011yescharacteristic
156C22×C17#41311yescharacteristic
157C22×C17C2×#10211yescharacteristic
158C22×C17#41711yescharacteristic
159C22×C17C2×#10411yescharacteristic
160C22×C17C2×#10411yescharacteristic
161C4×C17—22no—
162C4×C17—22no—
163C4×C17—22no—
164C4×C17—22no—
165C4×C17—21no—
166C4×C17—21no—
167C4×C17—21no—
168C4×C17—21no—
169C4×C17—22no—
170C4×C17—81no—
171C4×C17—81no—
172C4×C17—81no—
173C2×#126—21no—
174C2×C4⋊C8—21no—
175C2×C4⋊C8—21no—
176C2×#97—21no—
177C2×C4×C8—21no—
178#390C4×C1711yescharacteristic
179#390C4×C1711yescharacteristic
180#390C22×C1711yescharacteristic
181#389C4×C1711yescharacteristic
182#389C4×C1711yescharacteristic
183#389C22×C1711yescharacteristic
184#388—21no—
185#388—21no—
186#388—21no—
187C43C22×C1711yescharacteristic
188#336C22×C1712yes—
189#336C22×C1712yes—
190C2×C17—161no—
191C2×C17#147311yescharacteristic
192C2×C17#148311yescharacteristic
193C2×C17#149211yescharacteristic
194C2×C17#150211yescharacteristic
195C2×C17#150911yescharacteristic
196C2×C17#152411yescharacteristic
197C2×C17#154711yescharacteristic
198C22×D4C2×C4×C1711yescharacteristic
199C22×C8—22no—
200C22×C8—22no—
201#126—42no—
202#126—42no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4C2×C4×C1711yescharacteristic
207C2×C4⋊C4C2×C4×C1711yescharacteristic
208C2×C4⋊C4C2×C4×C1711yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C42C2×C4×C1711yescharacteristic
216C2×C42C2×C4×C1711yescharacteristic
217C2×C42D4×C1711yescharacteristic
218C2×C42D4×C1711yescharacteristic
219C2×C42D4×C1711yescharacteristic
220C2×C42D4×C1711yescharacteristic
221C2×C42C23×C1711yescharacteristic
222C4⋊C8—24no—
223C4⋊C8—24no—
224#97—42no—
225C4×C8—42no—
226#95—21no—
227#95—21no—
228#95—21no—
229C17#768611yescharacteristic, direct product factor, Sylow
230C24—21no—
231C2×D4—22no—
232C2×D4—41no—
233C2×D4—41no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4C22⋊C4×C1711yescharacteristic
238C22×C4C22⋊C4×C1711yescharacteristic
239C22×C4C22⋊C4×C1711yescharacteristic
240C22×C4C22⋊C4×C1711yescharacteristic
241C22×C4C22×C4×C1711yescharacteristic
242C22×C4C2×D4×C1711yescharacteristic
243C22×C4C2×D4×C1711yescharacteristic
244C2×C8—24no—
245C2×C8—24no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C4⋊C4—22no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C42—22no—
266C42—21no—
267C42—21no—
268C42—22no—
269C42—22no—
270C42—41no—
271C42—41no—
272C42D8×C1712yes—
273C42SD16×C1712yes—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23C2×C22⋊C4×C1711yescharacteristic
282D4—82no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4C17×#10212yes—
307C2×C4C17×#10212yes—
308C2×C4C17×#10412yes—
309C2×C4C17×#10412yes—
310C2×C4C17×#13112yes—
311C2×C4C2×D8×C1711yescharacteristic
312C2×C4C2×SD16×C1711yescharacteristic
313C2×C4C17×AGL(1,ℤ/8)11yescharacteristic
314C2×C4C17×#13711yescharacteristic
315C8—44no—
316C8—44no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22C17×#40811yescharacteristic
325C22C17×#41011yescharacteristic
326C22C17×#41311yescharacteristic
327C22C2×C17×#10211yescharacteristic
328C22C17×#41711yescharacteristic
329C22C2×C17×#10411yescharacteristic
330C22C2×C17×#10411yescharacteristic
331C4—22no—
332C4—22no—
333C4—22no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—22no—
339C4—22no—
340C4—81no—
341C4—81no—
342C4—81no—
343C2—161no—
344C2C17×#147311yescharacteristic
345C2C17×#148311yescharacteristic
346C2C17×#149211yescharacteristic
347C2C17×#150211yescharacteristic
348C2C17×#150911yescharacteristic
349C2C17×#152411yescharacteristic
350C2C17×#154711yescharacteristic
subgroupG/H
center, socleC23×C17C2×C22⋊C4
commutatorC22×C4C22×C4×C17
Frattini, cosocleC2×C42C23×C17
prime radicalC22×D4×C17C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C17 ◁ C43×C17 ◁ G

Complements

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

NH ≅ G/Ntype
2. C17×#1767C2semidirect
3. C4×C17×C4⋊C8C2semidirect
4. C17×#1271C2semidirect
98. #7686C17direct (N × H)
143. #1767C2×C17semidirect
144. C4×C4⋊C8C2×C17semidirect
145. #1271C2×C17semidirect
229. C17#7686direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#28602
C17×#17672
C4×C17×C4⋊C82
C17×#12712
C17×#12702
#768617