Finite Group Databasebeta

← C17×#3321 #14526327

Subgroups — C17×#3321 #14526327

Order 2176.

392 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#555C211yesmaximal, characteristic, complemented
2C17×#548C211yesmaximal, characteristic, complemented
3C17×#489C211yesmaximal, characteristic, complemented
4C17×#485C211yesmaximal, characteristic, complemented
5C17×#482C211yesmaximal, characteristic, complemented
6C17×#475C211yesmaximal, characteristic, complemented
7C17×#469C211yesmaximal, characteristic, complemented
8C17×#465C211yesmaximal, characteristic, complemented
9C17×#463C211yesmaximal, characteristic, complemented
10C17×#449C211yesmaximal, characteristic, complemented
11C17×#448C211yesmaximal, characteristic, complemented
12C17×#440C211yesmaximal, characteristic, complemented
13C17×#435C211yesmaximal, characteristic, complemented
14C17×#419C211yesmaximal, characteristic, complemented
15C17×#416C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C22×Q8×C17C2211yescharacteristic
18C2×SD16×C17C2211yescharacteristic
19C2×SD16×C17C2211yescharacteristic
20C2×D8×C17C2211yescharacteristic
21C2×M16×C17C2211yescharacteristic
22C22×C8×C17C2211yescharacteristic
23C17×C4⋊Q8C2211yescharacteristic
24C17×#124—21no—
25C17×#124—21no—
26C17×#124—21no—
27C17×#124C2211yescharacteristic
28C17×C22⋊Q8—21no—
29C17×C22⋊Q8C2211yescharacteristic
30C17×#121—21no—
31C17×#121—21no—
32C17×#121C2211yescharacteristic
33C17×#121C2211yescharacteristic
34C17×#121C2211yescharacteristic
35C4×Q8×C17C2211yescharacteristic
36C4×D4×C17—21no—
37C4×D4×C17C2211yescharacteristic
38C4×D4×C17C2211yescharacteristic
39C17×#117—21no—
40C17×#117C2211yescharacteristic
41C17×#117C2211yescharacteristic
42C17×#107C2211yescharacteristic
43C17×#106C2211yescharacteristic
44C17×#106C2211yescharacteristic
45C17×C4⋊C8C2211yescharacteristic
46C17×#103C2211yescharacteristic
47C17×#103C2211yescharacteristic
48C17×#103C2211yescharacteristic
49C17×#103C2211yescharacteristic
50C17×#103C2211yescharacteristic
51C17×#103C2211yescharacteristic
52C17×#102C2211yescharacteristic
53C17×#102C2211yescharacteristic
54C17×#102C2211yescharacteristic
55C17×#102C2211yescharacteristic
56C17×C22⋊C8C2211yescharacteristic
57C17×C22⋊C8C2211yescharacteristic
58C17×#97C2211yescharacteristic
59#41×C17—22no—
60#41×C17—41no—
61C2×Q8×C17—21no—
62C2×Q8×C17—21no—
63C2×Q8×C17—22no—
64C2×Q8×C17—41no—
65C2×Q8×C17C2311yescharacteristic
66C2×Q8×C17C2311yescharacteristic
67C2×D4×C17—21no—
68C2×D4×C17—21no—
69C2×D4×C17—21no—
70C2×D4×C17—21no—
71C2×D4×C17D411yescharacteristic
72C2×D4×C17C2311yescharacteristic
73C2×D4×C17C2311yescharacteristic
74C22×C4×C17—21no—
75C22×C4×C17—21no—
76C22×C4×C17—21no—
77C22×C4×C17C2311yescharacteristic
78C22×C4×C17C2311yescharacteristic
79SD16×C17—22no—
80SD16×C17—41no—
81D8×C17—41no—
82M16×C17—41no—
83C2×C8×C17—41no—
84C2×C8×C17C2311yescharacteristic
85C2×C8×C17C2311yescharacteristic
86C2×C8×C17C2311yescharacteristic
87C2×C8×C17C2311yescharacteristic
88C4⋊C4×C17—21no—
89C4⋊C4×C17—21no—
90C4⋊C4×C17—21no—
91C4⋊C4×C17D411yescharacteristic
92C4⋊C4×C17C2311yescharacteristic
93C4⋊C4×C17C2311yescharacteristic
94C4⋊C4×C17C2311yescharacteristic
95C4⋊C4×C17C2311yescharacteristic
96C22⋊C4×C17—21no—
97C22⋊C4×C17—21no—
98C22⋊C4×C17—21no—
99C22⋊C4×C17—21no—
100C22⋊C4×C17—21no—
101C22⋊C4×C17—21no—
102C22⋊C4×C17—21no—
103C22⋊C4×C17—21no—
104C22⋊C4×C17—21no—
105C22⋊C4×C17D411yescharacteristic
106C22⋊C4×C17D411yescharacteristic
107C42×C17—21no—
108C42×C17—21no—
109C42×C17—21no—
110C42×C17C2311yescharacteristic
111C23×C17—21no—
112C23×C17—21no—
113C23×C17C2×D411yescharacteristic
114C23×C17C2×D411yescharacteristic
115Q8×C17—21no—
116Q8×C17—21no—
117Q8×C17—21no—
118Q8×C17—42no—
119Q8×C17—81no—
120Q8×C17#4112yes—
121D4×C17—21no—
122D4×C17—21no—
123D4×C17—21no—
124D4×C17—21no—
125D4×C17—41no—
126D4×C17—42no—
127D4×C17—42no—
128D4×C17—81no—
129D4×C17—81no—
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—21no—
139C2×C4×C17—21no—
140C2×C4×C17—42no—
141C2×C4×C17—42no—
142C2×C4×C17—42no—
143C2×C4×C17C2×D411yescharacteristic
144C2×C4×C17C2×D411yescharacteristic
145C2×C4×C17C2×D411yescharacteristic
146C2×C4×C17C2×D411yescharacteristic
147C2×C4×C17C2411yescharacteristic
148C8×C17—21no—
149C8×C17—21no—
150C8×C17—21no—
151C8×C17—21no—
152#3321C1711yesmaximal, characteristic, direct product factor, Sylow
153C22×C17—21no—
154C22×C17—21no—
155C22×C17—21no—
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—21no—
159C22×C17—41no—
160C22×C17—41no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17—41no—
165C22×C17C22×D411yescharacteristic
166C4×C17—21no—
167C4×C17—21no—
168C4×C17—22no—
169C4×C17—21no—
170C4×C17—21no—
171C4×C17—41no—
172C4×C17—41no—
173C4×C17—41no—
174C4×C17—41no—
175C4×C17—41no—
176C4×C17C2×#4111yescharacteristic
177C4×C17H(5,𝔽2)11yescharacteristic
178#555C2×C1711yescharacteristic, complemented
179#548C2×C1711yescharacteristic, complemented
180#489C2×C1711yescharacteristic, complemented
181#485C2×C1711yescharacteristic, complemented
182#482C2×C1711yescharacteristic, complemented
183#475C2×C1711yescharacteristic, complemented
184#469C2×C1711yescharacteristic, complemented
185#465C2×C1711yescharacteristic, complemented
186#463C2×C1711yescharacteristic, complemented
187#449C2×C1711yescharacteristic, complemented
188#448C2×C1711yescharacteristic, complemented
189#440C2×C1711yescharacteristic, complemented
190#435C2×C1711yescharacteristic, complemented
191#419C2×C1711yescharacteristic, complemented
192#416C2×C1711yescharacteristic, complemented
193C2×C17—41no—
194C2×C17—41no—
195C2×C17—81no—
196C2×C17—81no—
197C2×C17#54611yescharacteristic
198C2×C17#57511yescharacteristic
199C2×C17#57711yescharacteristic
200C2×#41C22×C1711yescharacteristic
201C22×Q8C22×C1711yescharacteristic
202C2×SD16C22×C1711yescharacteristic
203C2×SD16C22×C1711yescharacteristic
204C2×D8C22×C1711yescharacteristic
205C2×M16C22×C1711yescharacteristic
206C22×C8C22×C1711yescharacteristic
207C4⋊Q8C22×C1711yescharacteristic
208#124—21no—
209#124—21no—
210#124—21no—
211#124C22×C1711yescharacteristic
212C22⋊Q8—21no—
213C22⋊Q8C22×C1711yescharacteristic
214#121—21no—
215#121—21no—
216#121C22×C1711yescharacteristic
217#121C22×C1711yescharacteristic
218#121C22×C1711yescharacteristic
219C4×Q8C22×C1711yescharacteristic
220C4×D4—21no—
221C4×D4C22×C1711yescharacteristic
222C4×D4C22×C1711yescharacteristic
223#117—21no—
224#117C22×C1711yescharacteristic
225#117C22×C1711yescharacteristic
226#107C22×C1711yescharacteristic
227#106C22×C1711yescharacteristic
228#106C22×C1711yescharacteristic
229C4⋊C8C22×C1711yescharacteristic
230#103C22×C1711yescharacteristic
231#103C22×C1711yescharacteristic
232#103C22×C1711yescharacteristic
233#103C22×C1711yescharacteristic
234#103C22×C1711yescharacteristic
235#103C22×C1711yescharacteristic
236#102C22×C1711yescharacteristic
237#102C22×C1711yescharacteristic
238#102C22×C1711yescharacteristic
239#102C22×C1711yescharacteristic
240C22⋊C8C22×C1711yescharacteristic
241C22⋊C8C22×C1711yescharacteristic
242#97C22×C1711yescharacteristic
243C17#332111yescharacteristic, direct product factor, Sylow
244#41—22no—
245#41—41no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—22no—
249C2×Q8—41no—
250C2×Q8C23×C1711yescharacteristic
251C2×Q8C23×C1711yescharacteristic
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4D4×C1711yescharacteristic
257C2×D4C23×C1711yescharacteristic
258C2×D4C23×C1711yescharacteristic
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4C23×C1711yescharacteristic
263C22×C4C23×C1711yescharacteristic
264SD16—22no—
265SD16—41no—
266D8—41no—
267M16—41no—
268C2×C8—41no—
269C2×C8C23×C1711yescharacteristic
270C2×C8C23×C1711yescharacteristic
271C2×C8C23×C1711yescharacteristic
272C2×C8C23×C1711yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4D4×C1711yescharacteristic
277C4⋊C4C23×C1711yescharacteristic
278C4⋊C4C23×C1711yescharacteristic
279C4⋊C4C23×C1711yescharacteristic
280C4⋊C4C23×C1711yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4D4×C1711yescharacteristic
291C22⋊C4D4×C1711yescharacteristic
292C42—21no—
293C42—21no—
294C42—21no—
295C42C23×C1711yescharacteristic
296C23—21no—
297C23—21no—
298C23C2×D4×C1711yescharacteristic
299C23C2×D4×C1711yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—42no—
304Q8—81no—
305Q8#41×C1712yes—
306D4—21no—
307D4—21no—
308D4—21no—
309D4—21no—
310D4—41no—
311D4—42no—
312D4—42no—
313D4—81no—
314D4—81no—
315C2×C4—21no—
316C2×C4—21no—
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—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4C2×D4×C1711yescharacteristic
329C2×C4C2×D4×C1711yescharacteristic
330C2×C4C2×D4×C1711yescharacteristic
331C2×C4C2×D4×C1711yescharacteristic
332C2×C4C24×C1711yescharacteristic
333C8—21no—
334C8—21no—
335C8—21no—
336C8—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22C22×D4×C1711yescharacteristic
350C4—21no—
351C4—21no—
352C4—22no—
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4C2×#41×C1711yescharacteristic
361C4C17×H(5,𝔽2)11yescharacteristic
362C2—41no—
363C2—41no—
364C2—81no—
365C2—81no—
366C2C17×#54611yescharacteristic
367C2C17×#57511yescharacteristic
368C2C17×#57711yescharacteristic
subgroupG/H
center, socleC22×C17C22×D4
commutator, Frattini, cosocleC2×C4C24×C17
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C17 ◁ C4×D4×C17 ◁ G

Complements

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

NH ≅ G/Ntype
1. C17×#555C2semidirect
2. C17×#548C2semidirect
3. C17×#489C2semidirect
4. C17×#485C2semidirect
5. C17×#482C2semidirect
6. C17×#475C2semidirect
7. C17×#469C2semidirect
8. C17×#465C2semidirect
9. C17×#463C2semidirect
10. C17×#449C2semidirect
11. C17×#448C2semidirect
12. C17×#440C2semidirect
13. C17×#435C2semidirect
14. C17×#419C2semidirect
15. C17×#416C2semidirect
152. #3321C17direct (N × H)
178. #555C2×C17semidirect
179. #548C2×C17semidirect
180. #489C2×C17semidirect
181. #485C2×C17semidirect
182. #482C2×C17semidirect
183. #475C2×C17semidirect
184. #469C2×C17semidirect
185. #465C2×C17semidirect
186. #463C2×C17semidirect
187. #449C2×C17semidirect
188. #448C2×C17semidirect
189. #440C2×C17semidirect
190. #435C2×C17semidirect
191. #419C2×C17semidirect
192. #416C2×C17semidirect
243. C17#3321direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#5552
C17×#5482
C17×#4892
C17×#4852
C17×#4822
C17×#4752
C17×#4692
C17×#4652
C17×#4632
C17×#4492
C17×#4482
C17×#4402
C17×#4352
C17×#4192
C17×#4162
#332117