Finite Group Databasebeta

← C17×#2737 #14525743

Subgroups — C17×#2737 #14525743

Order 2176.

442 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#397C211yesmaximal, characteristic, complemented
2C17×#397C211yesmaximal, characteristic, complemented
3C17×#395C211yesmaximal, characteristic, complemented
4C17×#394C211yesmaximal, characteristic
5C17×#393C211yesmaximal, characteristic, complemented
6C17×#393C211yesmaximal, characteristic, complemented
7C17×#390C211yesmaximal, characteristic
8C17×#388C211yesmaximal, characteristic, complemented
9C17×#388C211yesmaximal, characteristic, complemented
10C17×#387C211yesmaximal, characteristic, complemented
11C17×#387C211yesmaximal, characteristic, complemented
12C17×#386C211yesmaximal, characteristic
13C17×#383C211yesmaximal, characteristic, complemented
14C17×#381C211yesmaximal, characteristic, complemented
15C17×#380C211yesmaximal, characteristic, complemented
16C22×Q8×C17C2211yescharacteristic
17C22×D4×C17C2211yescharacteristic
18C23×C4×C17C2211yescharacteristic
19C2×C4⋊C4×C17C2211yescharacteristic
20C2×C4⋊C4×C17C2211yescharacteristic
21C2×C4⋊C4×C17C2211yescharacteristic
22C2×C4⋊C4×C17C2211yescharacteristic
23C2×C4⋊C4×C17C2211yescharacteristic
24C2×C22⋊C4×C17C2211yescharacteristic
25C2×C22⋊C4×C17C2211yescharacteristic
26C2×C22⋊C4×C17C2211yescharacteristic
27C2×C22⋊C4×C17C2211yescharacteristic
28C2×C22⋊C4×C17C2211yescharacteristic
29C2×C22⋊C4×C17C2211yescharacteristic
30C2×C22⋊C4×C17C2211yescharacteristic
31C2×C22⋊C4×C17C2211yescharacteristic
32C2×C22⋊C4×C17C2211yescharacteristic
33C2×C22⋊C4×C17C2211yescharacteristic
34C2×C22⋊C4×C17C2211yescharacteristic
35C2×C42×C17C2211yescharacteristic
36C2×C42×C17C2211yescharacteristic
37C17×#95C2211yescharacteristic
38C17×#95C2211yescharacteristic
39C17×#95C2211yescharacteristic
40C17×#95C2211yescharacteristic
41C17×#95C2211yescharacteristic
42C17×#95C2211yescharacteristic
43C17×#95C2211yescharacteristic
44C17×#95C2211yescharacteristic
45C17×#95C2211yescharacteristic
46C17×#95C2211yescharacteristic
47C17×#95C2211yescharacteristic
48C17×#95C2211yescharacteristic
49C17×#95C2211yescharacteristic
50C17×#95C2211yescharacteristic
51C24×C17C2311yescharacteristic
52C24×C17C2311yescharacteristic
53C2×Q8×C17—41no—
54C2×Q8×C17—41no—
55C2×Q8×C17—41no—
56C2×D4×C17—22no—
57C2×D4×C17—22no—
58C2×D4×C17—41no—
59C22×C4×C17—22no—
60C22×C4×C17—22no—
61C22×C4×C17—41no—
62C22×C4×C17C2311yescharacteristic
63C22×C4×C17C2311yescharacteristic
64C22×C4×C17C2311yescharacteristic
65C22×C4×C17C2311yescharacteristic
66C22×C4×C17C2311yescharacteristic
67C22×C4×C17C2311yescharacteristic
68C22×C4×C17C2311yescharacteristic
69C22×C4×C17C2311yescharacteristic
70C22×C4×C17C2311yescharacteristic
71C22×C4×C17C2311yescharacteristic
72C22×C4×C17C2311yescharacteristic
73C22×C4×C17C2311yescharacteristic
74C22×C4×C17C2311yescharacteristic
75C4⋊C4×C17—22no—
76C4⋊C4×C17—22no—
77C4⋊C4×C17—41no—
78C4⋊C4×C17—41no—
79C4⋊C4×C17—41no—
80C22⋊C4×C17—22no—
81C22⋊C4×C17—22no—
82C22⋊C4×C17—41no—
83C22⋊C4×C17—41no—
84C22⋊C4×C17—41no—
85C22⋊C4×C17—41no—
86C22⋊C4×C17—41no—
87C22⋊C4×C17—41no—
88C22⋊C4×C17—41no—
89C22⋊C4×C17—41no—
90C22⋊C4×C17—41no—
91C42×C17—22no—
92C42×C17—41no—
93C23×C17—21no—
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—21no—
97C23×C17—21no—
98C23×C17—21no—
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17#4112yes—
107C23×C17C2411yescharacteristic
108Q8×C17—44no—
109D4×C17—44no—
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—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—21no—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
139C2×C4×C17—21no—
140C2×C4×C17—21no—
141C2×C4×C17—21no—
142C2×C4×C17—21no—
143C2×C4×C17—21no—
144C2×C4×C17—21no—
145C2×C4×C17—21no—
146C2×C4×C17—21no—
147C2×C4×C17—44no—
148C2×C4×C17#4112yes—
149C2×C4×C17#4112yes—
150#2737C1711yesmaximal, characteristic, direct product factor, Sylow
151C22×C17—22no—
152C22×C17—22no—
153C22×C17—22no—
154C22×C17—41no—
155C22×C17—41no—
156C22×C17—41no—
157C22×C17—41no—
158C22×C17—41no—
159C22×C17—41no—
160C22×C17—41no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17—41no—
165C22×C17C2×#4111yescharacteristic
166C22×C17C2×#4111yescharacteristic
167C22×C17C2×#4111yescharacteristic
168C22×C17H(5,𝔽2)11yescharacteristic
169C22×C17H(5,𝔽2)11yescharacteristic
170C22×C17H(5,𝔽2)11yescharacteristic
171C22×C172−1+411yescharacteristic
172C4×C17—22no—
173C4×C17—22no—
174C4×C17—41no—
175C4×C17—41no—
176C4×C17—41no—
177C4×C17—41no—
178C4×C17—41no—
179C4×C17—41no—
180C4×C17—41no—
181C4×C17—41no—
182C4×C17—41no—
183C4×C17—41no—
184C4×C17—41no—
185#397C2×C1711yescharacteristic, complemented
186#397C2×C1711yescharacteristic, complemented
187#395C2×C1711yescharacteristic, complemented
188#394C2×C1711yescharacteristic
189#393C2×C1711yescharacteristic, complemented
190#393C2×C1711yescharacteristic, complemented
191#390C2×C1711yescharacteristic
192#388C2×C1711yescharacteristic, complemented
193#388C2×C1711yescharacteristic, complemented
194#387C2×C1711yescharacteristic, complemented
195#387C2×C1711yescharacteristic, complemented
196#386C2×C1711yescharacteristic
197#383C2×C1711yescharacteristic, complemented
198#381C2×C1711yescharacteristic, complemented
199#380C2×C1711yescharacteristic, complemented
200C2×C17—42no—
201C2×C17—81no—
202C2×C17#53911yescharacteristic
203C2×C17#54211yescharacteristic
204C2×C17#54211yescharacteristic
205C2×C17#55111yescharacteristic
206C2×C17#55111yescharacteristic
207C2×C17#55511yescharacteristic
208C2×C17#56011yescharacteristic
209C22×Q8C22×C1711yescharacteristic
210C22×D4C22×C1711yescharacteristic
211C23×C4C22×C1711yescharacteristic
212C2×C4⋊C4C22×C1711yescharacteristic
213C2×C4⋊C4C22×C1711yescharacteristic
214C2×C4⋊C4C22×C1711yescharacteristic
215C2×C4⋊C4C22×C1711yescharacteristic
216C2×C4⋊C4C22×C1711yescharacteristic
217C2×C22⋊C4C22×C1711yescharacteristic
218C2×C22⋊C4C22×C1711yescharacteristic
219C2×C22⋊C4C22×C1711yescharacteristic
220C2×C22⋊C4C22×C1711yescharacteristic
221C2×C22⋊C4C22×C1711yescharacteristic
222C2×C22⋊C4C22×C1711yescharacteristic
223C2×C22⋊C4C22×C1711yescharacteristic
224C2×C22⋊C4C22×C1711yescharacteristic
225C2×C22⋊C4C22×C1711yescharacteristic
226C2×C22⋊C4C22×C1711yescharacteristic
227C2×C22⋊C4C22×C1711yescharacteristic
228C2×C42C22×C1711yescharacteristic
229C2×C42C22×C1711yescharacteristic
230#95C22×C1711yescharacteristic
231#95C22×C1711yescharacteristic
232#95C22×C1711yescharacteristic
233#95C22×C1711yescharacteristic
234#95C22×C1711yescharacteristic
235#95C22×C1711yescharacteristic
236#95C22×C1711yescharacteristic
237#95C22×C1711yescharacteristic
238#95C22×C1711yescharacteristic
239#95C22×C1711yescharacteristic
240#95C22×C1711yescharacteristic
241#95C22×C1711yescharacteristic
242#95C22×C1711yescharacteristic
243#95C22×C1711yescharacteristic
244C17#273711yescharacteristic, direct product factor, Sylow
245C24C23×C1711yescharacteristic
246C24C23×C1711yescharacteristic
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—41no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—41no—
256C22×C4C23×C1711yescharacteristic
257C22×C4C23×C1711yescharacteristic
258C22×C4C23×C1711yescharacteristic
259C22×C4C23×C1711yescharacteristic
260C22×C4C23×C1711yescharacteristic
261C22×C4C23×C1711yescharacteristic
262C22×C4C23×C1711yescharacteristic
263C22×C4C23×C1711yescharacteristic
264C22×C4C23×C1711yescharacteristic
265C22×C4C23×C1711yescharacteristic
266C22×C4C23×C1711yescharacteristic
267C22×C4C23×C1711yescharacteristic
268C22×C4C23×C1711yescharacteristic
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C22⋊C4—22no—
275C22⋊C4—22no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C42—22no—
286C42—41no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23#41×C1712yes—
301C23C24×C1711yescharacteristic
302Q8—44no—
303D4—44no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
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—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—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—44no—
342C2×C4#41×C1712yes—
343C2×C4#41×C1712yes—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22C2×#41×C1711yescharacteristic
359C22C2×#41×C1711yescharacteristic
360C22C2×#41×C1711yescharacteristic
361C22C17×H(5,𝔽2)11yescharacteristic
362C22C17×H(5,𝔽2)11yescharacteristic
363C22C17×H(5,𝔽2)11yescharacteristic
364C22C17×2−1+411yescharacteristic
365C4—22no—
366C4—22no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—41no—
378C2—42no—
379C2—81no—
380C2C17×#53911yescharacteristic
381C2C17×#54211yescharacteristic
382C2C17×#54211yescharacteristic
383C2C17×#55111yescharacteristic
384C2C17×#55111yescharacteristic
385C2C17×#55511yescharacteristic
386C2C17×#56011yescharacteristic
subgroupG/H
center, socleC23×C17C24
commutator, Frattini, cosocleC23C24×C17
prime radicalC22×D4×C17C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C17 ◁ G

Complements

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

NH ≅ G/Ntype
1. C17×#397C2semidirect
2. C17×#397C2semidirect
3. C17×#395C2semidirect
5. C17×#393C2semidirect
6. C17×#393C2semidirect
8. C17×#388C2semidirect
9. C17×#388C2semidirect
10. C17×#387C2semidirect
11. C17×#387C2semidirect
13. C17×#383C2semidirect
14. C17×#381C2semidirect
15. C17×#380C2semidirect
150. #2737C17direct (N × H)
185. #397C2×C17semidirect
186. #397C2×C17semidirect
187. #395C2×C17semidirect
189. #393C2×C17semidirect
190. #393C2×C17semidirect
192. #388C2×C17semidirect
193. #388C2×C17semidirect
194. #387C2×C17semidirect
195. #387C2×C17semidirect
197. #383C2×C17semidirect
198. #381C2×C17semidirect
199. #380C2×C17semidirect
244. C17#2737direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#3972
C17×#3952
C17×#3942
C17×#3932
C17×#3902
C17×#3882
C17×#3872
C17×#3862
C17×#3832
C17×#3812
C17×#3802
#273717