Finite Group Databasebeta

← C17×#12266 #16870010

Subgroups — C17×#12266 #16870010

Order 4352.

592 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C17×#2670C211yesmaximal, characteristic
2C17×#2029C211yesmaximal, characteristic, complemented
3C17×#1602C211yesmaximal, characteristic, complemented
4C17×#1596C211yesmaximal, characteristic, complemented
5C17×#1351C211yesmaximal, characteristic, complemented
6C17×#1343C211yesmaximal, characteristic, complemented
7C17×#1316C211yesmaximal, characteristic, complemented
8C17×#543C2211yescharacteristic
9C2×C17×#124—21no—
10C2×C17×#120—21no—
11C17×#451—21no—
12C17×#450—21no—
13C17×#394—21no—
14C17×#393C2211yescharacteristic
15C17×#392—21no—
16C17×#392C2211yescharacteristic
17C17×#390—21no—
18C17×#388—21no—
19C17×#379—21no—
20C17×#379C2211yescharacteristic
21C17×#342—21no—
22C17×#342C2211yescharacteristic
23C17×#328—21no—
24C17×#328—21no—
25C17×#323C2211yescharacteristic
26C17×#323C2211yescharacteristic
27C25×C17D411yescharacteristic
28C22×Q8×C17D411yescharacteristic
29C22×D4×C17—21no—
30C22×D4×C17—21no—
31C22×D4×C17—21no—
32C22×D4×C17D411yescharacteristic
33C2×Q16×C17—41no—
34C2×SD16×C17—41no—
35C17×#124—44no—
36C17×C22⋊Q8—21no—
37C17×C22⋊Q8—21no—
38C17×C22⋊Q8—41no—
39C17×C22⋊Q8—41no—
40C17×#120—44no—
41C2×C4⋊C4×C17—21no—
42C2×C22⋊C4×C17—21no—
43C2×C22⋊C4×C17—21no—
44C2×C22⋊C4×C17—21no—
45C2×C22⋊C4×C17—21no—
46C2×C22⋊C4×C17—21no—
47C2×C22⋊C4×C17—21no—
48C2×C22⋊C4×C17—21no—
49C2×C22⋊C4×C17—21no—
50C2×C22⋊C4×C17—21no—
51C2×C22⋊C4×C17—21no—
52C2×C22⋊C4×C17—41no—
53C2×C22⋊C4×C17—41no—
54C2×C22⋊C4×C17D411yescharacteristic
55C2×C22⋊C4×C17C2311yescharacteristic
56C2×C42×C17—21no—
57C17×#103—41no—
58C17×#102—41no—
59C17×C22⋊C8—21no—
60C17×C22⋊C8—21no—
61C17×#95—21no—
62C17×#95D411yescharacteristic
63C17×#95D411yescharacteristic
64C24×C17—21no—
65C24×C17—21no—
66C24×C17—21no—
67C24×C17—21no—
68C24×C17—41no—
69C24×C17—41no—
70C24×C17—42no—
71C24×C17—81no—
72C24×C17C2×D411yescharacteristic
73C2×Q8×C17—21no—
74C2×Q8×C17—21no—
75C2×Q8×C17—42no—
76C2×Q8×C17—41no—
77C2×D4×C17—41no—
78C2×D4×C17—42no—
79C2×D4×C17—42no—
80C2×D4×C17—42no—
81C2×D4×C17—42no—
82C2×D4×C17—41no—
83C2×D4×C17—41no—
84C2×D4×C17—81no—
85C2×D4×C17—81no—
86C2×D4×C17—81no—
87C2×D4×C17—81no—
88C2×D4×C17—81no—
89C22×C4×C17—21no—
90C22×C4×C17—21no—
91C22×C4×C17—21no—
92C22×C4×C17—21no—
93C22×C4×C17—21no—
94C22×C4×C17—21no—
95C22×C4×C17—41no—
96C22×C4×C17C2×D411yescharacteristic
97C22×C4×C17C2×D411yescharacteristic
98Q16×C17—82no—
99SD16×C17—82no—
100C2×C8×C17—41no—
101C2×C8×C17—41no—
102C4⋊C4×C17—21no—
103C4⋊C4×C17—21no—
104C4⋊C4×C17—41no—
105C4⋊C4×C17—41no—
106C4⋊C4×C17—81no—
107C22⋊C4×C17—24no—
108C22⋊C4×C17—41no—
109C22⋊C4×C17—42no—
110C22⋊C4×C17—42no—
111C22⋊C4×C17—42no—
112C22⋊C4×C17—42no—
113C22⋊C4×C17—41no—
114C22⋊C4×C17—42no—
115C22⋊C4×C17—41no—
116C22⋊C4×C17—41no—
117C22⋊C4×C17—41no—
118C22⋊C4×C17—41no—
119C22⋊C4×C17—81no—
120C22⋊C4×C17—81no—
121C22⋊C4×C17—82no—
122C22⋊C4×C17—82no—
123C42×C17—42no—
124#12266C1711yesmaximal, characteristic, direct product factor, Sylow
125C23×C17—21no—
126C23×C17—21no—
127C23×C17—22no—
128C23×C17—41no—
129C23×C17—41no—
130C23×C17—41no—
131C23×C17—41no—
132C23×C17—41no—
133C23×C17—42no—
134C23×C17—41no—
135C23×C17—41no—
136C23×C17—41no—
137C23×C17—41no—
138C23×C17—42no—
139C23×C17—42no—
140C23×C17—41no—
141C23×C17—41no—
142C23×C17—41no—
143C23×C17—41no—
144C23×C17—41no—
145C23×C17—41no—
146C23×C17—81no—
147C23×C17—82no—
148C23×C17—81no—
149C23×C17—82no—
150C23×C17—84no—
151C23×C17—81no—
152C23×C17#12011yescharacteristic
153C23×C17#13511yescharacteristic
154C23×C17#13711yescharacteristic
155Q8×C17—42no—
156Q8×C17—81no—
157Q8×C17—81no—
158Q8×C17—81no—
159D4×C17—42no—
160D4×C17—82no—
161D4×C17—82no—
162D4×C17—84no—
163D4×C17—81no—
164D4×C17—84no—
165C2×C4×C17—22no—
166C2×C4×C17—22no—
167C2×C4×C17—21no—
168C2×C4×C17—21no—
169C2×C4×C17—41no—
170C2×C4×C17—41no—
171C2×C4×C17—41no—
172C2×C4×C17—41no—
173C2×C4×C17—41no—
174C2×C4×C17—41no—
175C2×C4×C17—41no—
176C2×C4×C17—41no—
177C2×C4×C17—41no—
178C2×C4×C17—41no—
179C2×C4×C17—41no—
180C2×C4×C17—41no—
181C2×C4×C17—41no—
182C2×C4×C17—41no—
183C2×C4×C17—81no—
184C2×C4×C17—81no—
185C2×C4×C17—81no—
186C2×C4×C17—81no—
187C2×C4×C17—81no—
188C8×C17—81no—
189C8×C17—81no—
190#2670C2×C1711yescharacteristic
191#2029C2×C1711yescharacteristic, complemented
192#1602C2×C1711yescharacteristic, complemented
193#1596C2×C1711yescharacteristic, complemented
194#1351C2×C1711yescharacteristic, complemented
195#1343C2×C1711yescharacteristic, complemented
196#1316C2×C1711yescharacteristic, complemented
197C22×C17—21no—
198C22×C17—21no—
199C22×C17—21no—
200C22×C17—21no—
201C22×C17—41no—
202C22×C17—41no—
203C22×C17—42no—
204C22×C17—42no—
205C22×C17—41no—
206C22×C17—41no—
207C22×C17—44no—
208C22×C17—42no—
209C22×C17—81no—
210C22×C17—81no—
211C22×C17—81no—
212C22×C17—81no—
213C22×C17—81no—
214C22×C17—81no—
215C22×C17—81no—
216C22×C17—82no—
217C22×C17—82no—
218C22×C17—82no—
219C22×C17—81no—
220C22×C17—81no—
221C22×C17—81no—
222C22×C17—81no—
223C22×C17—81no—
224C22×C17#45211yescharacteristic
225C22×C17#45511yescharacteristic
226C22×C17B(4,𝔽2)11yescharacteristic
227C4×C17—42no—
228C4×C17—41no—
229C4×C17—42no—
230C4×C17—41no—
231C4×C17—81no—
232C4×C17—81no—
233C4×C17—81no—
234C4×C17—81no—
235C4×C17—161no—
236#543C22×C1711yescharacteristic
237C2×#124—21no—
238C2×#120—21no—
239#451—21no—
240#450—21no—
241#394—21no—
242#393C22×C1711yescharacteristic
243#392—21no—
244#392C22×C1711yescharacteristic
245#390—21no—
246#388—21no—
247#379—21no—
248#379C22×C1711yescharacteristic
249#342—21no—
250#342C22×C1711yescharacteristic
251#328—21no—
252#328—21no—
253#323C22×C1711yescharacteristic
254#323C22×C1711yescharacteristic
255C2×C17—21no—
256C2×C17—21no—
257C2×C17—41no—
258C2×C17—41no—
259C2×C17—82no—
260C2×C17—161no—
261C2×C17#160611yescharacteristic
262C2×C17#219611yescharacteristic
263C2×C17#219911yescharacteristic
264C25D4×C1711yescharacteristic
265C22×Q8D4×C1711yescharacteristic
266C22×D4—21no—
267C22×D4—21no—
268C22×D4—21no—
269C22×D4D4×C1711yescharacteristic
270C2×Q16—41no—
271C2×SD16—41no—
272#124—44no—
273C22⋊Q8—21no—
274C22⋊Q8—21no—
275C22⋊Q8—41no—
276C22⋊Q8—41no—
277#120—44no—
278C2×C4⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—41no—
290C2×C22⋊C4—41no—
291C2×C22⋊C4D4×C1711yescharacteristic
292C2×C22⋊C4C23×C1711yescharacteristic
293C2×C42—21no—
294#103—41no—
295#102—41no—
296C22⋊C8—21no—
297C22⋊C8—21no—
298#95—21no—
299#95D4×C1711yescharacteristic
300#95D4×C1711yescharacteristic
301C17#1226611yescharacteristic, direct product factor, Sylow
302C24—21no—
303C24—21no—
304C24—21no—
305C24—21no—
306C24—41no—
307C24—42no—
308C24—41no—
309C24—81no—
310C24C2×D4×C1711yescharacteristic
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—42no—
314C2×Q8—41no—
315C2×D4—41no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—81no—
323C2×D4—81no—
324C2×D4—81no—
325C2×D4—81no—
326C2×D4—81no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—41no—
334C22×C4C2×D4×C1711yescharacteristic
335C22×C4C2×D4×C1711yescharacteristic
336Q16—82no—
337SD16—82no—
338C2×C8—41no—
339C2×C8—41no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—81no—
345C22⋊C4—24no—
346C22⋊C4—41no—
347C22⋊C4—42no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—41no—
352C22⋊C4—42no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—81no—
358C22⋊C4—81no—
359C22⋊C4—82no—
360C22⋊C4—82no—
361C42—42no—
362C23—21no—
363C23—21no—
364C23—22no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—42no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—42no—
376C23—42no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—81no—
384C23—82no—
385C23—81no—
386C23—82no—
387C23—84no—
388C23—81no—
389C23C17×#12011yescharacteristic
390C23C17×#13511yescharacteristic
391C23C17×#13711yescharacteristic
392Q8—42no—
393Q8—81no—
394Q8—81no—
395Q8—81no—
396D4—42no—
397D4—82no—
398D4—82no—
399D4—84no—
400D4—81no—
401D4—84no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C8—81no—
426C8—81no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—41no—
432C22—41no—
433C22—42no—
434C22—42no—
435C22—41no—
436C22—41no—
437C22—44no—
438C22—42no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22—82no—
447C22—82no—
448C22—82no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22C17×#45211yescharacteristic
455C22C17×#45511yescharacteristic
456C22C17×B(4,𝔽2)11yescharacteristic
457C4—42no—
458C4—41no—
459C4—42no—
460C4—41no—
461C4—81no—
462C4—81no—
463C4—81no—
464C4—81no—
465C4—161no—
466C2—21no—
467C2—21no—
468C2—41no—
469C2—41no—
470C2—82no—
471C2—161no—
472C2C17×#160611yescharacteristic
473C2C17×#219611yescharacteristic
474C2C17×#219911yescharacteristic
subgroupG/H
center, socleC22×C17B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C17
prime radicalC17×#2670C2
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C17 ◁ C23×C17 ◁ C2×C22⋊C4×C17 ◁ G

Complements

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

NH ≅ G/Ntype
2. C17×#2029C2semidirect
3. C17×#1602C2semidirect
4. C17×#1596C2semidirect
5. C17×#1351C2semidirect
6. C17×#1343C2semidirect
7. C17×#1316C2semidirect
124. #12266C17direct (N × H)
191. #2029C2×C17semidirect
192. #1602C2×C17semidirect
193. #1596C2×C17semidirect
194. #1351C2×C17semidirect
195. #1343C2×C17semidirect
196. #1316C2×C17semidirect
301. C17#12266direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#26702
C17×#20292
C17×#16022
C17×#15962
C17×#13512
C17×#13432
C17×#13162
#1226617