Finite Group Databasebeta

← C17×#10200 #16867944

Subgroups — C17×#10200 #16867944

Order 4352.

550 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#3245C211yesmaximal, characteristic
2C17×#3227C211yesmaximal, characteristic, complemented
3C17×#3141C211yesmaximal, characteristic, complemented
4C17×#1981C211yesmaximal, characteristic, complemented
5C17×#1980C211yesmaximal, characteristic, complemented
6C17×#1502C211yesmaximal, characteristic, complemented
7C17×#1283C211yesmaximal, characteristic, complemented
8C17×#532C2211yescharacteristic
9C17×#532C2211yescharacteristic
10C2×C17×C4⋊Q8C2211yescharacteristic
11C17×#498—21no—
12C17×#489—22no—
13C17×#487—21no—
14C17×#486—21no—
15C17×#486—21no—
16C17×#481—22no—
17C17×#476—22no—
18C17×#474—21no—
19C17×#474—21no—
20C17×#466—22no—
21C17×#459—21no—
22C17×#459—21no—
23C17×#431C2211yescharacteristic
24C2×C17×C4⋊C8C2211yescharacteristic
25C2×C17×#104—22no—
26C17×#417—22no—
27C2×C17×#103—22no—
28C4×M16×C17C2211yescharacteristic
29C4×C4⋊C4×C17C2211yescharacteristic
30C17×#327—22no—
31C2×#41×C17—21no—
32C2×#41×C17—21no—
33C22×Q8×C17—21no—
34C2×D8×C17—22no—
35C2×M16×C17D412yes—
36C22×C8×C17D412yes—
37C17×C4⋊Q8—21no—
38C17×C4⋊Q8—21no—
39C17×C4⋊Q8—21no—
40C17×C4⋊Q8D411yescharacteristic
41C17×C4⋊Q8D412yes—
42C17×C4⋊Q8D411yescharacteristic
43C17×#127D411yescharacteristic
44C17×#127D411yescharacteristic
45C17×#124—21no—
46C17×#124—21no—
47C17×#121—21no—
48C17×#121—21no—
49C17×#121—21no—
50C17×#121—21no—
51C4×D4×C17—21no—
52C4×D4×C17—21no—
53C4×D4×C17—21no—
54C4×D4×C17—21no—
55C2×C4⋊C4×C17—21no—
56C2×C4⋊C4×C17—21no—
57C2×C4⋊C4×C17—21no—
58C2×C42×C17—21no—
59C2×C42×C17C2311yescharacteristic
60C17×#107—22no—
61C17×#106—24no—
62C17×C4⋊C8—21no—
63C17×C4⋊C8—21no—
64C17×C4⋊C8—21no—
65C17×#104—44no—
66C17×#103—24no—
67C17×#103—44no—
68C17×#102—24no—
69C17×#102—22no—
70C17×#102—22no—
71C17×#102—24no—
72C17×C22⋊C8D412yes—
73C17×#97—21no—
74C4×C8×C17—21no—
75C4×C8×C17—21no—
76C17×#95—21no—
77#41×C17—42no—
78#41×C17—42no—
79#41×C17—42no—
80#41×C17—42no—
81C2×Q8×C17—21no—
82C2×Q8×C17—21no—
83C2×Q8×C17—22no—
84C2×Q8×C17—42no—
85C2×Q8×C17—42no—
86C2×Q8×C17—41no—
87C2×D4×C17—21no—
88C2×D4×C17—21no—
89C2×D4×C17—21no—
90C2×D4×C17—21no—
91C2×D4×C17—21no—
92C2×D4×C17—21no—
93C2×D4×C17—41no—
94C2×D4×C17—41no—
95C22×C4×C17—41no—
96C22×C4×C17—41no—
97C22×C4×C17—41no—
98C22×C4×C17—41no—
99C22×C4×C17C2×D411yescharacteristic
100C22×C4×C17C2×D411yescharacteristic
101C22×C4×C17C2×D411yescharacteristic
102D8×C17—82no—
103M16×C17—22no—
104M16×C17—22no—
105C2×C8×C17—22no—
106C2×C8×C17—22no—
107C2×C8×C17—22no—
108C2×C8×C17—42no—
109C4⋊C4×C17—22no—
110C4⋊C4×C17—22no—
111C4⋊C4×C17—22no—
112C4⋊C4×C17—21no—
113C4⋊C4×C17—21no—
114C4⋊C4×C17—21no—
115C4⋊C4×C17—21no—
116C4⋊C4×C17—41no—
117C4⋊C4×C17—41no—
118C4⋊C4×C17—41no—
119C22⋊C4×C17—41no—
120C22⋊C4×C17—41no—
121C22⋊C4×C17—41no—
122C22⋊C4×C17—41no—
123C42×C17—42no—
124C42×C17C2×D411yescharacteristic
125C42×C17C2×D411yescharacteristic
126C42×C17C2×D411yescharacteristic
127C42×C17#4111yescharacteristic
128#10200C1711yesmaximal, characteristic, direct product factor, Sylow
129C23×C17—41no—
130C23×C17—41no—
131C23×C17#12711yescharacteristic
132Q8×C17—44no—
133Q8×C17—42no—
134Q8×C17—42no—
135Q8×C17—81no—
136Q8×C17—81no—
137D4×C17—42no—
138D4×C17—42no—
139D4×C17—41no—
140D4×C17—41no—
141D4×C17—41no—
142D4×C17—41no—
143D4×C17—42no—
144D4×C17—81no—
145D4×C17—81no—
146D4×C17—81no—
147D4×C17—81no—
148D4×C17—81no—
149C2×C4×C17—21no—
150C2×C4×C17—21no—
151C2×C4×C17—41no—
152C2×C4×C17—42no—
153C2×C4×C17—41no—
154C2×C4×C17—41no—
155C2×C4×C17—42no—
156C2×C4×C17—81no—
157C2×C4×C17—82no—
158C2×C4×C17—81no—
159C2×C4×C17—82no—
160C2×C4×C17—82no—
161C2×C4×C17#12011yescharacteristic
162C2×C4×C17#12011yescharacteristic
163C2×C4×C17#12011yescharacteristic
164C2×C4×C17#12111yescharacteristic
165C2×C4×C17#12111yescharacteristic
166C2×C4×C17#12111yescharacteristic
167C2×C4×C17#13512yes—
168C2×C4×C17AGL(1,ℤ/8)12yes—
169C8×C17—42no—
170C8×C17—42no—
171C8×C17—42no—
172#3245C2×C1711yescharacteristic
173#3227C2×C1711yescharacteristic, complemented
174#3141C2×C1711yescharacteristic, complemented
175#1981C2×C1711yescharacteristic, complemented
176#1980C2×C1711yescharacteristic, complemented
177#1502C2×C1711yescharacteristic, complemented
178#1283C2×C1711yescharacteristic, complemented
179C22×C17—21no—
180C22×C17—21no—
181C22×C17—81no—
182C22×C17—81no—
183C22×C17—81no—
184C22×C17—81no—
185C22×C17—81no—
186C22×C17—81no—
187C22×C17#39211yescharacteristic
188C22×C17#49511yescharacteristic
189C22×C17#49611yescharacteristic
190C4×C17—21no—
191C4×C17—21no—
192C4×C17—21no—
193C4×C17—82no—
194C4×C17—82no—
195C4×C17—81no—
196C4×C17—81no—
197C4×C17#44912yes—
198C4×C17#45311yescharacteristic
199C4×C17#45411yescharacteristic
200C4×C17#46312yes—
201#532C22×C1711yescharacteristic
202#532C22×C1711yescharacteristic
203C2×C4⋊Q8C22×C1711yescharacteristic
204#498—21no—
205#489—22no—
206#487—21no—
207#486—21no—
208#486—21no—
209#481—22no—
210#476—22no—
211#474—21no—
212#474—21no—
213#466—22no—
214#459—21no—
215#459—21no—
216#431C22×C1711yescharacteristic
217C2×C4⋊C8C22×C1711yescharacteristic
218C2×#104—22no—
219#417—22no—
220C2×#103—22no—
221C4×M16C22×C1711yescharacteristic
222C4×C4⋊C4C22×C1711yescharacteristic
223#327—22no—
224C2×C17—21no—
225C2×C17—21no—
226C2×C17—161no—
227C2×C17—161no—
228C2×C17#200611yescharacteristic
229C2×C17#200911yescharacteristic
230C2×C17#201211yescharacteristic
231C2×#41—21no—
232C2×#41—21no—
233C22×Q8—21no—
234C2×D8—22no—
235C2×M16D4×C1712yes—
236C22×C8D4×C1712yes—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8D4×C1712yes—
241C4⋊Q8D4×C1711yescharacteristic
242C4⋊Q8D4×C1711yescharacteristic
243#127D4×C1711yescharacteristic
244#127D4×C1711yescharacteristic
245#124—21no—
246#124—21no—
247#121—21no—
248#121—21no—
249#121—21no—
250#121—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C42—21no—
259C2×C42C23×C1711yescharacteristic
260#107—22no—
261#106—24no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265#104—44no—
266#103—24no—
267#103—44no—
268#102—24no—
269#102—22no—
270#102—22no—
271#102—24no—
272C22⋊C8D4×C1712yes—
273#97—21no—
274C4×C8—21no—
275C4×C8—21no—
276#95—21no—
277C17#1020011yescharacteristic, direct product factor, Sylow
278#41—42no—
279#41—42no—
280#41—42no—
281#41—42no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—22no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—41no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—41no—
295C2×D4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4C2×D4×C1711yescharacteristic
301C22×C4C2×D4×C1711yescharacteristic
302C22×C4C2×D4×C1711yescharacteristic
303D8—82no—
304M16—22no—
305M16—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—42no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C42—42no—
325C42C2×D4×C1711yescharacteristic
326C42C2×D4×C1711yescharacteristic
327C42C2×D4×C1711yescharacteristic
328C42#41×C1711yescharacteristic
329C23—41no—
330C23—41no—
331C23C17×#12711yescharacteristic
332Q8—44no—
333Q8—42no—
334Q8—42no—
335Q8—81no—
336Q8—81no—
337D4—42no—
338D4—42no—
339D4—41no—
340D4—41no—
341D4—41no—
342D4—41no—
343D4—42no—
344D4—81no—
345D4—81no—
346D4—81no—
347D4—81no—
348D4—81no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—82no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4C17×#12011yescharacteristic
362C2×C4C17×#12011yescharacteristic
363C2×C4C17×#12011yescharacteristic
364C2×C4C17×#12111yescharacteristic
365C2×C4C17×#12111yescharacteristic
366C2×C4C17×#12111yescharacteristic
367C2×C4C17×#13512yes—
368C2×C4C17×AGL(1,ℤ/8)12yes—
369C8—42no—
370C8—42no—
371C8—42no—
372C22—21no—
373C22—21no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22C17×#39211yescharacteristic
381C22C17×#49511yescharacteristic
382C22C17×#49611yescharacteristic
383C4—21no—
384C4—21no—
385C4—21no—
386C4—82no—
387C4—82no—
388C4—81no—
389C4—81no—
390C4C17×#44912yes—
391C4C17×#45311yescharacteristic
392C4C17×#45411yescharacteristic
393C4C17×#46312yes—
394C2—21no—
395C2—21no—
396C2—161no—
397C2—161no—
398C2C17×#200611yescharacteristic
399C2C17×#200911yescharacteristic
400C2C17×#201211yescharacteristic
subgroupG/H
center, socleC22×C17#392
commutator, Frattini, cosocleC2×C42C23×C17
prime radicalC17×#3245C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C17 ◁ C2×C42×C17 ◁ G

Complements

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

NH ≅ G/Ntype
2. C17×#3227C2semidirect
3. C17×#3141C2semidirect
4. C17×#1981C2semidirect
5. C17×#1980C2semidirect
6. C17×#1502C2semidirect
7. C17×#1283C2semidirect
128. #10200C17direct (N × H)
173. #3227C2×C17semidirect
174. #3141C2×C17semidirect
175. #1981C2×C17semidirect
176. #1980C2×C17semidirect
177. #1502C2×C17semidirect
178. #1283C2×C17semidirect
277. C17#10200direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#32452
C17×#32272
C17×#31412
C17×#19812
C17×#19802
C17×#15022
C17×#12832
#1020017