Finite Group Databasebeta

← C17×#3257 #14526263

Subgroups — C17×#3257 #14526263

Order 2176.

358 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#542C211yesmaximal, characteristic, complemented
2C17×#540C211yesmaximal, characteristic, complemented
3C17×#479C211yesmaximal, characteristic, complemented
4C17×#478C211yesmaximal, characteristic, complemented
5C17×#476C211yesmaximal, characteristic, complemented
6C17×#474C211yesmaximal, characteristic, complemented
7C17×#469C211yesmaximal, characteristic, complemented
8C17×#468C211yesmaximal, characteristic, complemented
9C17×#466C211yesmaximal, characteristic, complemented
10C17×#465C211yesmaximal, characteristic, complemented
11C17×#463C211yesmaximal, characteristic, complemented
12C17×#463C211yesmaximal, characteristic, complemented
13C17×#460C211yesmaximal, characteristic, complemented
14C17×#459C211yesmaximal, characteristic, complemented
15C17×#424C211yesmaximal, characteristic, complemented
16C2×SD16×C17C2211yescharacteristic
17C2×SD16×C17C2211yescharacteristic
18C2×D8×C17C2211yescharacteristic
19C2×D8×C17C2211yescharacteristic
20C2×M16×C17C2211yescharacteristic
21C22×C8×C17C2211yescharacteristic
22C17×C4⋊Q8C2211yescharacteristic
23C17×#127C2211yescharacteristic
24C17×#126—21no—
25C17×#125C2211yescharacteristic
26C17×#124—21no—
27C17×#124C2211yescharacteristic
28C17×#123—21no—
29C17×#123—21no—
30C17×#123—21no—
31C17×C22⋊Q8—21no—
32C17×C22⋊Q8C2211yescharacteristic
33C17×#121—21no—
34C17×#121—21no—
35C17×#121C2211yescharacteristic
36C17×#121C2211yescharacteristic
37C17×#121C2211yescharacteristic
38C4×Q8×C17C2211yescharacteristic
39C4×D4×C17C2211yescharacteristic
40C4×D4×C17C2211yescharacteristic
41C4×D4×C17C2211yescharacteristic
42C17×#117C2211yescharacteristic
43C17×#107C2211yescharacteristic
44C17×#107C2211yescharacteristic
45C17×#106C2211yescharacteristic
46C17×#106C2211yescharacteristic
47C17×C4⋊C8C2211yescharacteristic
48C17×C4⋊C8C2211yescharacteristic
49C17×C4⋊C8C2211yescharacteristic
50C17×C4⋊C8C2211yescharacteristic
51C17×#103C2211yescharacteristic
52C17×#103C2211yescharacteristic
53C17×#102C2211yescharacteristic
54C17×#102C2211yescharacteristic
55C17×#102C2211yescharacteristic
56C17×#102C2211yescharacteristic
57C17×#102C2211yescharacteristic
58C17×#102C2211yescharacteristic
59C2×Q8×C17—21no—
60C2×Q8×C17C2311yescharacteristic
61C2×D4×C17—21no—
62C2×D4×C17—21no—
63C2×D4×C17—21no—
64C2×D4×C17—21no—
65C2×D4×C17—21no—
66C2×D4×C17C2311yescharacteristic
67C2×D4×C17C2311yescharacteristic
68C2×D4×C17C2311yescharacteristic
69C22×C4×C17—21no—
70C22×C4×C17—21no—
71C22×C4×C17—21no—
72C22×C4×C17C2311yescharacteristic
73SD16×C17—41no—
74SD16×C17—41no—
75D8×C17—41no—
76D8×C17—41no—
77M16×C17—41no—
78C2×C8×C17—41no—
79C2×C8×C17C2311yescharacteristic
80C2×C8×C17C2311yescharacteristic
81C2×C8×C17C2311yescharacteristic
82C2×C8×C17C2311yescharacteristic
83C4⋊C4×C17—21no—
84C4⋊C4×C17—21no—
85C4⋊C4×C17—21no—
86C4⋊C4×C17—21no—
87C4⋊C4×C17—21no—
88C4⋊C4×C17D411yescharacteristic
89C4⋊C4×C17D411yescharacteristic
90C4⋊C4×C17C2311yescharacteristic
91C4⋊C4×C17C2311yescharacteristic
92C4⋊C4×C17C2311yescharacteristic
93C4⋊C4×C17C2311yescharacteristic
94C22⋊C4×C17—21no—
95C22⋊C4×C17—21no—
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×C17D411yescharacteristic
104C22⋊C4×C17D411yescharacteristic
105C42×C17—21no—
106C42×C17C2311yescharacteristic
107C42×C17C2311yescharacteristic
108C23×C17—21no—
109C23×C17—21no—
110C23×C17—21no—
111C23×C17C2×D411yescharacteristic
112Q8×C17—21no—
113Q8×C17—21no—
114Q8×C17—81no—
115D4×C17—21no—
116D4×C17—21no—
117D4×C17—21no—
118D4×C17—21no—
119D4×C17—21no—
120D4×C17—21no—
121D4×C17—81no—
122D4×C17—81no—
123D4×C17—81no—
124D4×C17—81no—
125D4×C17—81no—
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—81no—
134C2×C4×C17—81no—
135C2×C4×C17—81no—
136C2×C4×C17C2×D411yescharacteristic
137C2×C4×C17C2×D411yescharacteristic
138C2×C4×C17C2×D411yescharacteristic
139C2×C4×C17C2×D411yescharacteristic
140C2×C4×C17C2×D411yescharacteristic
141C2×C4×C17C2411yescharacteristic
142C8×C17—21no—
143C8×C17—21no—
144C8×C17—21no—
145C8×C17—21no—
146#3257C1711yesmaximal, characteristic, direct product factor, Sylow
147C22×C17—21no—
148C22×C17—21no—
149C22×C17—21no—
150C22×C17—41no—
151C22×C17—41no—
152C22×C17—41no—
153C22×C17—41no—
154C22×C17—41no—
155C22×C17—41no—
156C22×C17—41no—
157C22×C17—41no—
158C22×C17—41no—
159C22×C17C22×D411yescharacteristic
160C4×C17—21no—
161C4×C17—21no—
162C4×C17—21no—
163C4×C17—21no—
164C4×C17—21no—
165C4×C17—41no—
166C4×C17—41no—
167C4×C17—41no—
168C4×C17—41no—
169C4×C17—41no—
170C4×C17H(5,𝔽2)11yescharacteristic
171C4×C172−1+411yescharacteristic
172#542C2×C1711yescharacteristic, complemented
173#540C2×C1711yescharacteristic, complemented
174#479C2×C1711yescharacteristic, complemented
175#478C2×C1711yescharacteristic, complemented
176#476C2×C1711yescharacteristic, complemented
177#474C2×C1711yescharacteristic, complemented
178#469C2×C1711yescharacteristic, complemented
179#468C2×C1711yescharacteristic, complemented
180#466C2×C1711yescharacteristic, complemented
181#465C2×C1711yescharacteristic, complemented
182#463C2×C1711yescharacteristic, complemented
183#463C2×C1711yescharacteristic, complemented
184#460C2×C1711yescharacteristic, complemented
185#459C2×C1711yescharacteristic, complemented
186#424C2×C1711yescharacteristic, complemented
187C2×C17—41no—
188C2×C17—81no—
189C2×C17—81no—
190C2×C17—81no—
191C2×C17#53711yescharacteristic
192C2×C17#57611yescharacteristic
193C2×C17#57711yescharacteristic
194C2×SD16C22×C1711yescharacteristic
195C2×SD16C22×C1711yescharacteristic
196C2×D8C22×C1711yescharacteristic
197C2×D8C22×C1711yescharacteristic
198C2×M16C22×C1711yescharacteristic
199C22×C8C22×C1711yescharacteristic
200C4⋊Q8C22×C1711yescharacteristic
201#127C22×C1711yescharacteristic
202#126—21no—
203#125C22×C1711yescharacteristic
204#124—21no—
205#124C22×C1711yescharacteristic
206#123—21no—
207#123—21no—
208#123—21no—
209C22⋊Q8—21no—
210C22⋊Q8C22×C1711yescharacteristic
211#121—21no—
212#121—21no—
213#121C22×C1711yescharacteristic
214#121C22×C1711yescharacteristic
215#121C22×C1711yescharacteristic
216C4×Q8C22×C1711yescharacteristic
217C4×D4C22×C1711yescharacteristic
218C4×D4C22×C1711yescharacteristic
219C4×D4C22×C1711yescharacteristic
220#117C22×C1711yescharacteristic
221#107C22×C1711yescharacteristic
222#107C22×C1711yescharacteristic
223#106C22×C1711yescharacteristic
224#106C22×C1711yescharacteristic
225C4⋊C8C22×C1711yescharacteristic
226C4⋊C8C22×C1711yescharacteristic
227C4⋊C8C22×C1711yescharacteristic
228C4⋊C8C22×C1711yescharacteristic
229#103C22×C1711yescharacteristic
230#103C22×C1711yescharacteristic
231#102C22×C1711yescharacteristic
232#102C22×C1711yescharacteristic
233#102C22×C1711yescharacteristic
234#102C22×C1711yescharacteristic
235#102C22×C1711yescharacteristic
236#102C22×C1711yescharacteristic
237C17#325711yescharacteristic, direct product factor, Sylow
238C2×Q8—21no—
239C2×Q8C23×C1711yescharacteristic
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4C23×C1711yescharacteristic
246C2×D4C23×C1711yescharacteristic
247C2×D4C23×C1711yescharacteristic
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4C23×C1711yescharacteristic
252SD16—41no—
253SD16—41no—
254D8—41no—
255D8—41no—
256M16—41no—
257C2×C8—41no—
258C2×C8C23×C1711yescharacteristic
259C2×C8C23×C1711yescharacteristic
260C2×C8C23×C1711yescharacteristic
261C2×C8C23×C1711yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4D4×C1711yescharacteristic
268C4⋊C4D4×C1711yescharacteristic
269C4⋊C4C23×C1711yescharacteristic
270C4⋊C4C23×C1711yescharacteristic
271C4⋊C4C23×C1711yescharacteristic
272C4⋊C4C23×C1711yescharacteristic
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4D4×C1711yescharacteristic
283C22⋊C4D4×C1711yescharacteristic
284C42—21no—
285C42C23×C1711yescharacteristic
286C42C23×C1711yescharacteristic
287C23—21no—
288C23—21no—
289C23—21no—
290C23C2×D4×C1711yescharacteristic
291Q8—21no—
292Q8—21no—
293Q8—81no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—21no—
300D4—81no—
301D4—81no—
302D4—81no—
303D4—81no—
304D4—81no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4C2×D4×C1711yescharacteristic
316C2×C4C2×D4×C1711yescharacteristic
317C2×C4C2×D4×C1711yescharacteristic
318C2×C4C2×D4×C1711yescharacteristic
319C2×C4C2×D4×C1711yescharacteristic
320C2×C4C24×C1711yescharacteristic
321C8—21no—
322C8—21no—
323C8—21no—
324C8—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22C22×D4×C1711yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4C17×H(5,𝔽2)11yescharacteristic
349C4C17×2−1+411yescharacteristic
350C2—41no—
351C2—81no—
352C2—81no—
353C2—81no—
354C2C17×#53711yescharacteristic
355C2C17×#57611yescharacteristic
356C2C17×#57711yescharacteristic
subgroupG/H
center, socleC22×C17C22×D4
commutator, Frattini, cosocleC2×C4C24×C17
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C17 ◁ C17×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C17×#542C2semidirect
2. C17×#540C2semidirect
3. C17×#479C2semidirect
4. C17×#478C2semidirect
5. C17×#476C2semidirect
6. C17×#474C2semidirect
7. C17×#469C2semidirect
8. C17×#468C2semidirect
9. C17×#466C2semidirect
10. C17×#465C2semidirect
11. C17×#463C2semidirect
12. C17×#463C2semidirect
13. C17×#460C2semidirect
14. C17×#459C2semidirect
15. C17×#424C2semidirect
146. #3257C17direct (N × H)
172. #542C2×C17semidirect
173. #540C2×C17semidirect
174. #479C2×C17semidirect
175. #478C2×C17semidirect
176. #476C2×C17semidirect
177. #474C2×C17semidirect
178. #469C2×C17semidirect
179. #468C2×C17semidirect
180. #466C2×C17semidirect
181. #465C2×C17semidirect
182. #463C2×C17semidirect
183. #463C2×C17semidirect
184. #460C2×C17semidirect
185. #459C2×C17semidirect
186. #424C2×C17semidirect
237. C17#3257direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#5422
C17×#5402
C17×#4792
C17×#4782
C17×#4762
C17×#4742
C17×#4692
C17×#4682
C17×#4662
C17×#4652
C17×#4632
C17×#4602
C17×#4592
C17×#4242
#325717