Finite Group Databasebeta

← C17×#9945 #16867689

Subgroups — C17×#9945 #16867689

Order 4352.

574 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 102 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×#3095C211yesmaximal, characteristic
2C17×#2583C211yesmaximal, characteristic
3C17×#2484C211yesmaximal, characteristic
4C17×#1826C211yesmaximal, characteristic
5C17×#1511C211yesmaximal, characteristic
6C17×#1463C211yesmaximal, characteristic
7C17×#1284C211yesmaximal, characteristic
8C2×C17×C22⋊Q8C2211yescharacteristic
9C2×C17×#117C2211yescharacteristic
10C22×C4⋊C4×C17C2211yescharacteristic
11C17×#485—22no—
12C17×#484—21no—
13C17×#483—21no—
14C17×#428—21no—
15C17×#419—21no—
16C17×#407C2211yescharacteristic
17C17×#407C2211yescharacteristic
18C17×#400—21no—
19C17×#398—21no—
20C17×#398—21no—
21C17×#395—21no—
22C17×#389—22no—
23C17×#387—22no—
24C17×#385—21no—
25C17×#385—21no—
26C17×#385—21no—
27C17×#381—21no—
28C17×#381C2211yescharacteristic
29C4×C22⋊C4×C17C2211yescharacteristic
30C17×#339—22no—
31C17×#328—22no—
32C17×#324—22no—
33C22×Q8×C17D411yescharacteristic
34C23×C4×C17D411yescharacteristic
35C23×C4×C17D411yescharacteristic
36C23×C4×C17C2311yescharacteristic
37C2×M16×C17—21no—
38C2×M16×C17—21no—
39C17×C22⋊Q8—21no—
40C17×C22⋊Q8—21no—
41C17×C22⋊Q8—21no—
42C17×C22⋊Q8—21no—
43C17×#117—21no—
44C17×#117—21no—
45C17×#117—21no—
46C17×#117—21no—
47C2×C4⋊C4×C17—22no—
48C2×C4⋊C4×C17—22no—
49C2×C4⋊C4×C17—21no—
50C2×C4⋊C4×C17—21no—
51C2×C4⋊C4×C17—21no—
52C2×C4⋊C4×C17—21no—
53C2×C4⋊C4×C17—21no—
54C2×C4⋊C4×C17—21no—
55C2×C4⋊C4×C17—21no—
56C2×C4⋊C4×C17—22no—
57C2×C4⋊C4×C17D411yescharacteristic
58C2×C4⋊C4×C17D411yescharacteristic
59C2×C4⋊C4×C17D411yescharacteristic
60C2×C22⋊C4×C17—21no—
61C2×C22⋊C4×C17—21no—
62C2×C22⋊C4×C17—21no—
63C2×C22⋊C4×C17—21no—
64C2×C42×C17—21no—
65C2×C42×C17D412yes—
66C17×#107—22no—
67C17×#106—22no—
68C17×#103—22no—
69C17×#103—22no—
70C17×C22⋊C8—21no—
71C17×C22⋊C8—21no—
72C17×C22⋊C8—21no—
73C17×C22⋊C8D412yes—
74C17×#95—22no—
75C17×#95—22no—
76C17×#95—22no—
77C17×#95—22no—
78C17×#95—22no—
79C17×#95—21no—
80C17×#95—21no—
81C17×#95D412yes—
82C24×C17C2×D411yescharacteristic
83C2×Q8×C17—21no—
84C2×Q8×C17—21no—
85C2×Q8×C17—42no—
86C22×C4×C17—21no—
87C22×C4×C17—21no—
88C22×C4×C17—22no—
89C22×C4×C17—21no—
90C22×C4×C17—21no—
91C22×C4×C17—21no—
92C22×C4×C17—21no—
93C22×C4×C17—22no—
94C22×C4×C17—21no—
95C22×C4×C17—21no—
96C22×C4×C17—21no—
97C22×C4×C17—21no—
98C22×C4×C17—42no—
99C22×C4×C17—42no—
100C22×C4×C17—42no—
101C22×C4×C17C2×D411yescharacteristic
102C22×C4×C17C2×D411yescharacteristic
103C22×C4×C17C2×D411yescharacteristic
104C22×C4×C17C2×D411yescharacteristic
105C22×C4×C17C2×D411yescharacteristic
106C22×C4×C17#4111yescharacteristic
107M16×C17—42no—
108M16×C17—42no—
109C2×C8×C17—22no—
110C2×C8×C17—41no—
111C4⋊C4×C17—21no—
112C4⋊C4×C17—21no—
113C4⋊C4×C17—21no—
114C4⋊C4×C17—21no—
115C4⋊C4×C17—21no—
116C4⋊C4×C17—21no—
117C4⋊C4×C17—44no—
118C4⋊C4×C17—44no—
119C4⋊C4×C17—41no—
120C4⋊C4×C17—41no—
121C4⋊C4×C17—41no—
122C4⋊C4×C17—41no—
123C4⋊C4×C17—41no—
124C4⋊C4×C17—41no—
125C4⋊C4×C17—82no—
126C22⋊C4×C17—24no—
127C22⋊C4×C17—42no—
128C22⋊C4×C17—41no—
129C22⋊C4×C17—41no—
130C22⋊C4×C17—41no—
131C22⋊C4×C17—41no—
132C42×C17—42no—
133C42×C17—41no—
134C42×C17—41no—
135#9945C1711yesmaximal, characteristic, direct product factor, Sylow
136C23×C17—22no—
137C23×C17—22no—
138C23×C17—41no—
139C23×C17#12011yescharacteristic
140C23×C17#12011yescharacteristic
141C23×C17#12111yescharacteristic
142Q8×C17—42no—
143Q8×C17—81no—
144C2×C4×C17—22no—
145C2×C4×C17—22no—
146C2×C4×C17—24no—
147C2×C4×C17—22no—
148C2×C4×C17—22no—
149C2×C4×C17—22no—
150C2×C4×C17—41no—
151C2×C4×C17—41no—
152C2×C4×C17—41no—
153C2×C4×C17—41no—
154C2×C4×C17—42no—
155C2×C4×C17—41no—
156C2×C4×C17—41no—
157C2×C4×C17—41no—
158C2×C4×C17—41no—
159C2×C4×C17—42no—
160C2×C4×C17—42no—
161C2×C4×C17—42no—
162C2×C4×C17—42no—
163C2×C4×C17—42no—
164C2×C4×C17—42no—
165C2×C4×C17—41no—
166C2×C4×C17—41no—
167C2×C4×C17—41no—
168C2×C4×C17—41no—
169C2×C4×C17—41no—
170C2×C4×C17—42no—
171C2×C4×C17—42no—
172C2×C4×C17—42no—
173C2×C4×C17—42no—
174C2×C4×C17—81no—
175C2×C4×C17#12011yescharacteristic
176C2×C4×C17#12111yescharacteristic
177C2×C4×C17#12111yescharacteristic
178C2×C4×C17#12711yescharacteristic
179C8×C17—42no—
180C8×C17—81no—
181#3095C2×C1711yescharacteristic
182#2583C2×C1711yescharacteristic
183#2484C2×C1711yescharacteristic
184#1826C2×C1711yescharacteristic
185#1511C2×C1711yescharacteristic
186#1463C2×C1711yescharacteristic
187#1284C2×C1711yescharacteristic
188C22×C17—21no—
189C22×C17—21no—
190C22×C17—21no—
191C22×C17—21no—
192C22×C17—21no—
193C22×C17—21no—
194C22×C17—21no—
195C22×C17—44no—
196C22×C17#39211yescharacteristic
197C22×C17#45612yes—
198C22×C17#45812yes—
199C4×C17—21no—
200C4×C17—21no—
201C4×C17—21no—
202C4×C17—21no—
203C4×C17—44no—
204C4×C17—42no—
205C4×C17—82no—
206C4×C17—81no—
207C4×C17—81no—
208C4×C17—81no—
209C4×C17—81no—
210C4×C17—81no—
211C2×C22⋊Q8C22×C1711yescharacteristic
212C2×#117C22×C1711yescharacteristic
213C22×C4⋊C4C22×C1711yescharacteristic
214#485—22no—
215#484—21no—
216#483—21no—
217#428—21no—
218#419—21no—
219#407C22×C1711yescharacteristic
220#407C22×C1711yescharacteristic
221#400—21no—
222#398—21no—
223#398—21no—
224#395—21no—
225#389—22no—
226#387—22no—
227#385—21no—
228#385—21no—
229#385—21no—
230#381—21no—
231#381C22×C1711yescharacteristic
232C4×C22⋊C4C22×C1711yescharacteristic
233#339—22no—
234#328—22no—
235#324—22no—
236C2×C17—22no—
237C2×C17—22no—
238C2×C17—41no—
239C2×C17#200511yescharacteristic
240C2×C17#201011yescharacteristic
241C2×C17#202211yescharacteristic
242C22×Q8D4×C1711yescharacteristic
243C23×C4D4×C1711yescharacteristic
244C23×C4D4×C1711yescharacteristic
245C23×C4C23×C1711yescharacteristic
246C2×M16—21no—
247C2×M16—21no—
248C22⋊Q8—21no—
249C22⋊Q8—21no—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252#117—21no—
253#117—21no—
254#117—21no—
255#117—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4D4×C1711yescharacteristic
267C2×C4⋊C4D4×C1711yescharacteristic
268C2×C4⋊C4D4×C1711yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C42—21no—
274C2×C42D4×C1712yes—
275#107—22no—
276#106—22no—
277#103—22no—
278#103—22no—
279C22⋊C8—21no—
280C22⋊C8—21no—
281C22⋊C8—21no—
282C22⋊C8D4×C1712yes—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—21no—
289#95—21no—
290#95D4×C1712yes—
291C17#994511yescharacteristic, direct product factor, Sylow
292C24C2×D4×C1711yescharacteristic
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—42no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4C2×D4×C1711yescharacteristic
312C22×C4C2×D4×C1711yescharacteristic
313C22×C4C2×D4×C1711yescharacteristic
314C22×C4C2×D4×C1711yescharacteristic
315C22×C4C2×D4×C1711yescharacteristic
316C22×C4#41×C1711yescharacteristic
317M16—42no—
318M16—42no—
319C2×C8—22no—
320C2×C8—41no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—82no—
336C22⋊C4—24no—
337C22⋊C4—42no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C42—42no—
343C42—41no—
344C42—41no—
345C23—22no—
346C23—22no—
347C23—41no—
348C23C17×#12011yescharacteristic
349C23C17×#12011yescharacteristic
350C23C17×#12111yescharacteristic
351Q8—42no—
352Q8—81no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—81no—
384C2×C4C17×#12011yescharacteristic
385C2×C4C17×#12111yescharacteristic
386C2×C4C17×#12111yescharacteristic
387C2×C4C17×#12711yescharacteristic
388C8—42no—
389C8—81no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—44no—
398C22C17×#39211yescharacteristic
399C22C17×#45612yes—
400C22C17×#45812yes—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—44no—
406C4—42no—
407C4—82no—
408C4—81no—
409C4—81no—
410C4—81no—
411C4—81no—
412C4—81no—
413C2—22no—
414C2—22no—
415C2—41no—
416C2C17×#200511yescharacteristic
417C2C17×#201011yescharacteristic
418C2C17×#202211yescharacteristic
subgroupG/H
center, socleC22×C17#392
commutator, Frattini, cosocleC23×C4C23×C17
prime radicalC24×C17C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C17 ◁ C23×C4×C17 ◁ G

Complements

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

NH ≅ G/Ntype
135. #9945C17direct (N × H)
291. C17#9945direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#30952
C17×#25832
C17×#24842
C17×#18262
C17×#15112
C17×#14632
C17×#12842
#994517