Finite Group Databasebeta

← C17×#32898 #16890642

Subgroups — C17×#32898 #16890642

Order 4352.

4360 conjugacy classes of subgroups; only the 370 normal subgroups are shown.

normal subgroups only

This group has 4360 conjugacy classes of subgroups — too many to draw. Only its 370 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C17×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C2×C17×B(4,𝔽2)C214yesmaximal, complemented
3C17×#2403C213yesmaximal, complemented
4C17×#1790C214yesmaximal, complemented
5C17×#1781C213yesmaximal, complemented
6C2×C17×H(5,𝔽2)—216no—
7C2×C17×H(5,𝔽2)—26no—
8C2×C17×H(5,𝔽2)C2214yes—
9C22×#41×C17—23no—
10C23×D4×C17—23no—
11C23×D4×C17C2213yes—
12C23×D4×C17C2211yescharacteristic, complemented
13C2×C17×#120—23no—
14C2×C17×#120C2216yescomplemented
15C17×B(4,𝔽2)—216no—
16C2×C17×#99C22112yescomplemented
17C17×#392—26no—
18C17×#386—23no—
19C17×#386C2216yescomplemented
20C17×#379C2213yescomplemented
21C25×C17—21no—
22C25×C17D413yescomplemented
23C25×C17C2311yescharacteristic
24C25×C17C2311yescharacteristic, complemented
25C17×H(5,𝔽2)—248no—
26C17×H(5,𝔽2)—248no—
27C17×H(5,𝔽2)—224no—
28C17×H(5,𝔽2)D4116yescomplemented
29C2×#41×C17—212no—
30C2×#41×C17—248no—
31C2×#41×C17—212no—
32C22×Q8×C17—21no—
33C22×D4×C17—23no—
34C22×D4×C17—248no—
35C22×D4×C17—212no—
36C22×D4×C17—26no—
37C22×D4×C17—212no—
38C22×D4×C17—26no—
39C22×D4×C17—224no—
40C22×D4×C17—23no—
41C22×D4×C17—23no—
42C22×D4×C17—26no—
43C22×D4×C17—44no—
44C22×D4×C17D416yescomplemented
45C22×D4×C17C2316yes—
46C23×C4×C17—23no—
47C23×C4×C17D413yescomplemented
48C23×C4×C17C2311yescharacteristic, complemented
49C17×#120—224no—
50C17×#120—412no—
51C2×C22⋊C4×C17—23no—
52C2×C22⋊C4×C17—26no—
53C2×C22⋊C4×C17—26no—
54C2×C22⋊C4×C17—23no—
55C2×C22⋊C4×C17C2316yescomplemented
56C17×#99—224no—
57C17×#95—26no—
58C24×C17—28no—
59C24×C17—24no—
60C24×C17—23no—
61C24×C17—23no—
62C24×C17—212no—
63C24×C17—24no—
64C24×C17—23no—
65C24×C17—24no—
66C24×C17—42no—
67C24×C17—42no—
68C24×C17—46no—
69C24×C17—81no—
70C24×C17—81no—
71C24×C17—83no—
72C24×C17—82no—
73C24×C17—82no—
74C24×C17C2×D413yes—
75C24×C17C2×D413yes—
76C24×C17C2×D416yes—
77C24×C17C2411yescharacteristic
78#41×C17—296no—
79#41×C17—248no—
80#41×C17—248no—
81C2×Q8×C17—28no—
82C2×Q8×C17—24no—
83C2×D4×C17—224no—
84C2×D4×C17—248no—
85C2×D4×C17—212no—
86C2×D4×C17—296no—
87C2×D4×C17—248no—
88C2×D4×C17—248no—
89C2×D4×C17—212no—
90C2×D4×C17—248no—
91C2×D4×C17—212no—
92C2×D4×C17—212no—
93C2×D4×C17—212no—
94C2×D4×C17—212no—
95C2×D4×C17—212no—
96C2×D4×C17—46no—
97C2×D4×C17—46no—
98C2×D4×C17—46no—
99C2×D4×C17—43no—
100C2×D4×C17—41no—
101C2×D4×C17—412no—
102C2×D4×C17—412no—
103C2×D4×C17—86no—
104C2×D4×C17C2×D4124yes—
105C22×C4×C17—224no—
106C22×C4×C17—212no—
107C22×C4×C17—23no—
108C22×C4×C17—23no—
109C22×C4×C17—212no—
110C22×C4×C17—23no—
111C22×C4×C17—26no—
112C22×C4×C17—42no—
113C22×C4×C17C2×D416yes—
114C22⋊C4×C17—212no—
115C22⋊C4×C17—412no—
116C22⋊C4×C17—46no—
117C22⋊C4×C17—412no—
118C22⋊C4×C17—83no—
119#32898C1711yesmaximal, characteristic, direct product factor, Sylow
120C23×C17—212no—
121C23×C17—212no—
122C23×C17—216no—
123C23×C17—26no—
124C23×C17—26no—
125C23×C17—26no—
126C23×C17—26no—
127C23×C17—26no—
128C23×C17—26no—
129C23×C17—41no—
130C23×C17—43no—
131C23×C17—41no—
132C23×C17—43no—
133C23×C17—43no—
134C23×C17—46no—
135C23×C17—46no—
136C23×C17—46no—
137C23×C17—43no—
138C23×C17—46no—
139C23×C17—46no—
140C23×C17—46no—
141C23×C17—41no—
142C23×C17—83no—
143C23×C17—83no—
144C23×C17—88no—
145C23×C17—88no—
146C23×C17—824no—
147C23×C17—816no—
148C23×C17—84no—
149C23×C17—84no—
150C23×C17#12013yes—
151C23×C17#12011yescharacteristic
152C23×C17#120112yescomplemented
153C23×C17C22×D413yes—
154C23×C17C22×D414yes—
155Q8×C17—216no—
156D4×C17—248no—
157D4×C17—248no—
158D4×C17—248no—
159D4×C17—412no—
160D4×C17—44no—
161D4×C17—824no—
162C2×C4×C17—212no—
163C2×C4×C17—212no—
164C2×C4×C17—248no—
165C2×C4×C17—26no—
166C2×C4×C17—26no—
167C2×C4×C17—26no—
168C2×C4×C17—43no—
169C2×C4×C17—41no—
170C2×C4×C17—46no—
171C2×C4×C17—46no—
172C2×C4×C17—46no—
173C2×C4×C17#120112yescomplemented
174C22×H(5,𝔽2)C2×C1711yescharacteristic, complemented
175C2×B(4,𝔽2)C2×C1714yescomplemented
176#2403C2×C1713yescomplemented
177#1790C2×C1714yescomplemented
178#1781C2×C1713yescomplemented
179C22×C17—21no—
180C22×C17—23no—
181C22×C17—23no—
182C22×C17—21no—
183C22×C17—23no—
184C22×C17—212no—
185C22×C17—23no—
186C22×C17—23no—
187C22×C17—44no—
188C22×C17—412no—
189C22×C17—412no—
190C22×C17—424no—
191C22×C17—412no—
192C22×C17—46no—
193C22×C17—46no—
194C22×C17—42no—
195C22×C17—86no—
196C22×C17—86no—
197C22×C17—86no—
198C22×C17—812no—
199C22×C17—816no—
200C22×C17—81no—
201C22×C17—81no—
202C22×C17B(4,𝔽2)12yescomplemented
203C22×C17C2×#12016yes—
204C22×C17C2×#12011yescharacteristic
205C4×C17—212no—
206C4×C17—42no—
207C4×C17—86no—
208C2×H(5,𝔽2)—216no—
209C2×H(5,𝔽2)—26no—
210C2×H(5,𝔽2)C22×C1714yes—
211C22×#41—23no—
212C23×D4—23no—
213C23×D4C22×C1713yes—
214C23×D4C22×C1711yescharacteristic, complemented
215C2×#120—23no—
216C2×#120C22×C1716yescomplemented
217B(4,𝔽2)—216no—
218C2×#99C22×C17112yescomplemented
219#392—26no—
220#386—23no—
221#386C22×C1716yescomplemented
222#379C22×C1713yescomplemented
223C2×C17—26no—
224C2×C17—22no—
225C2×C17—43no—
226C2×C17—412no—
227C2×C17—82no—
228C2×C17#284611yescharacteristic
229C2×C17C2×B(4,𝔽2)11yescharacteristic
230C2×C17#302611yescharacteristic
231C25—21no—
232C25D4×C1713yescomplemented
233C25C23×C1711yescharacteristic
234C25C23×C1711yescharacteristic, complemented
235H(5,𝔽2)—248no—
236H(5,𝔽2)—248no—
237H(5,𝔽2)—224no—
238H(5,𝔽2)D4×C17116yescomplemented
239C2×#41—212no—
240C2×#41—248no—
241C2×#41—212no—
242C22×Q8—21no—
243C22×D4—23no—
244C22×D4—248no—
245C22×D4—212no—
246C22×D4—26no—
247C22×D4—212no—
248C22×D4—26no—
249C22×D4—224no—
250C22×D4—23no—
251C22×D4—23no—
252C22×D4—26no—
253C22×D4—44no—
254C22×D4D4×C1716yescomplemented
255C22×D4C23×C1716yes—
256C23×C4—23no—
257C23×C4D4×C1713yescomplemented
258C23×C4C23×C1711yescharacteristic, complemented
259#120—224no—
260#120—412no—
261C2×C22⋊C4—23no—
262C2×C22⋊C4—26no—
263C2×C22⋊C4—26no—
264C2×C22⋊C4—23no—
265C2×C22⋊C4C23×C1716yescomplemented
266#99—224no—
267#95—26no—
268C17#3289811yescharacteristic, direct product factor, Sylow
269C24—28no—
270C24—24no—
271C24—23no—
272C24—24no—
273C24—23no—
274C24—212no—
275C24—23no—
276C24—24no—
277C24—42no—
278C24—42no—
279C24—46no—
280C24—81no—
281C24—81no—
282C24—83no—
283C24—82no—
284C24—82no—
285C24C2×D4×C1713yes—
286C24C2×D4×C1713yes—
287C24C2×D4×C1716yes—
288C24C24×C1711yescharacteristic
289#41—296no—
290#41—248no—
291#41—248no—
292C2×Q8—28no—
293C2×Q8—24no—
294C2×D4—224no—
295C2×D4—248no—
296C2×D4—212no—
297C2×D4—296no—
298C2×D4—248no—
299C2×D4—248no—
300C2×D4—212no—
301C2×D4—212no—
302C2×D4—212no—
303C2×D4—248no—
304C2×D4—212no—
305C2×D4—212no—
306C2×D4—212no—
307C2×D4—46no—
308C2×D4—46no—
309C2×D4—46no—
310C2×D4—43no—
311C2×D4—41no—
312C2×D4—412no—
313C2×D4—412no—
314C2×D4—86no—
315C2×D4C2×D4×C17124yes—
316C22×C4—224no—
317C22×C4—212no—
318C22×C4—23no—
319C22×C4—23no—
320C22×C4—212no—
321C22×C4—23no—
322C22×C4—26no—
323C22×C4—42no—
324C22×C4C2×D4×C1716yes—
325C22⋊C4—212no—
326C22⋊C4—412no—
327C22⋊C4—46no—
328C22⋊C4—412no—
329C22⋊C4—83no—
330C23—212no—
331C23—212no—
332C23—216no—
333C23—26no—
334C23—26no—
335C23—26no—
336C23—26no—
337C23—26no—
338C23—26no—
339C23—41no—
340C23—43no—
341C23—41no—
342C23—43no—
343C23—43no—
344C23—46no—
345C23—46no—
346C23—46no—
347C23—43no—
348C23—46no—
349C23—46no—
350C23—46no—
351C23—41no—
352C23—83no—
353C23—83no—
354C23—88no—
355C23—88no—
356C23—824no—
357C23—816no—
358C23—84no—
359C23—84no—
360C23C17×#12013yes—
361C23C17×#12011yescharacteristic
362C23C17×#120112yescomplemented
363C23C22×D4×C1713yes—
364C23C22×D4×C1714yes—
365Q8—216no—
366D4—248no—
367D4—248no—
368D4—248no—
369D4—412no—
370D4—44no—
371D4—824no—
372C2×C4—212no—
373C2×C4—212no—
374C2×C4—248no—
375C2×C4—26no—
376C2×C4—26no—
377C2×C4—26no—
378C2×C4—43no—
379C2×C4—41no—
380C2×C4—46no—
381C2×C4—46no—
382C2×C4—46no—
383C2×C4C17×#120112yescomplemented
384C22—23no—
385C22—23no—
386C22—21no—
387C22—21no—
388C22—23no—
389C22—23no—
390C22—23no—
391C22—212no—
392C22—44no—
393C22—412no—
394C22—412no—
395C22—424no—
396C22—412no—
397C22—46no—
398C22—46no—
399C22—42no—
400C22—86no—
401C22—86no—
402C22—86no—
403C22—812no—
404C22—816no—
405C22—81no—
406C22—81no—
407C22C17×B(4,𝔽2)12yescomplemented
408C22C2×C17×#12016yes—
409C22C2×C17×#12011yescharacteristic
410C4—212no—
411C4—42no—
412C4—86no—
413C2—26no—
414C2—22no—
415C2—43no—
416C2—412no—
417C2—82no—
418C2C17×#284611yescharacteristic
419C2C2×C17×B(4,𝔽2)11yescharacteristic
420C2C17×#302611yescharacteristic
subgroupG/H
center, socleC22×C17C2×#120
commutator, Frattini, cosocleC24C24×C17
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C17 ◁ C25×C17 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C17×H(5,𝔽2)C2semidirect
2. C2×C17×B(4,𝔽2)C2semidirect
3. C17×#2403C2semidirect
4. C17×#1790C2semidirect
5. C17×#1781C2semidirect
12. C23×D4×C17C22semidirect
14. C2×C17×#120C22semidirect
16. C2×C17×#99C22semidirect
19. C17×#386C22semidirect
20. C17×#379C22semidirect
22. C25×C17D4semidirect
24. C25×C17C23semidirect
28. C17×H(5,𝔽2)D4semidirect
44. C22×D4×C17D4semidirect
47. C23×C4×C17D4semidirect
48. C23×C4×C17C23semidirect
55. C2×C22⋊C4×C17C23semidirect
119. #32898C17direct (N × H)
152. C23×C17#120semidirect
173. C2×C4×C17#120semidirect
174. C22×H(5,𝔽2)C2×C17semidirect
175. C2×B(4,𝔽2)C2×C17semidirect
176. #2403C2×C17semidirect
177. #1790C2×C17semidirect
178. #1781C2×C17semidirect
202. C22×C17B(4,𝔽2)semidirect
214. C23×D4C22×C17semidirect
216. C2×#120C22×C17semidirect
218. C2×#99C22×C17semidirect
221. #386C22×C17semidirect
222. #379C22×C17semidirect
232. C25D4×C17semidirect
234. C25C23×C17semidirect
238. H(5,𝔽2)D4×C17semidirect
254. C22×D4D4×C17semidirect
257. C23×C4D4×C17semidirect
258. C23×C4C23×C17semidirect
265. C2×C22⋊C4C23×C17semidirect
268. C17#32898direct (N × H)
362. C23C17×#120semidirect
383. C2×C4C17×#120semidirect
407. C22C17×B(4,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
C22×C17×H(5,𝔽2)2
C2×C17×B(4,𝔽2)2
C17×#24032
C17×#17902
C17×#17812
#3289817