Finite Group Databasebeta

← #17008444

Subgroups — #17008444

Order 4352.

1172 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1172 conjugacy classes of subgroups — too many to draw. Only its 183 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
1#14539586C212yesmaximal, complemented
2#14533715C211yesmaximal, characteristic
3#14527809C214yesmaximal, complemented
4#14527707C214yesmaximal, complemented
5#14527304C212yesmaximal, complemented
6#14527111C211yesmaximal, characteristic, complemented
7C2×C17×#543C211yesmaximal, characteristic, complemented
8C2×#10587615—21no—
9C2×#10587616C2211yescharacteristic
10C2×#10587639—21no—
11#11763142C2214yescomplemented
12#11763122—22no—
13#11763057—24no—
14#11762959C2214yescomplemented
15#11762948—22no—
16#11762773C2212yes—
17#11762745C2212yes—
18#11762718C2212yes—
19#11762709—22no—
20C4×C22⋊Dic17C2211yescharacteristic
21#11762546C2218yescomplemented
22#11762533C2214yes—
23C17×#543—22no—
24C17×#543—24no—
25C17×#543C2214yes—
26C2×C17×C22⋊Q8—22no—
27C2×C17×C22⋊Q8—22no—
28C2×C17×C22⋊Q8—21no—
29C2×C17×C22⋊Q8C2212yes—
30C22×C22⋊C4×C17—21no—
31C22×C22⋊C4×C17C2211yescharacteristic
32C25×C17D411yescharacteristic
33C22×Q8×C17—21no—
34C22×Q8×C17D412yescomplemented
35C23×C4×C17—21no—
36C23×C4×C17—21no—
37C23×C4×C17D411yescharacteristic
38C23×C4×C17C2311yescharacteristic
39C22×C22⋊D17—21no—
40C22×C22⋊D17—22no—
41C22×D68C2311yescharacteristic
42C17×C22⋊Q8—28no—
43C17×C22⋊Q8—28no—
44C17×C22⋊Q8—24no—
45C17×C22⋊Q8—28no—
46C17×C22⋊Q8—24no—
47C17×C22⋊Q8—22no—
48C17×C22⋊Q8—22no—
49C17×C22⋊Q8—28no—
50C17×C22⋊Q8D414yescomplemented
51C17×C22⋊Q8C2314yes—
52C2×C4⋊C4×C17—22no—
53C2×C4⋊C4×C17—21no—
54C2×C4⋊C4×C17—22no—
55C2×C4⋊C4×C17D412yescomplemented
56C2×C22⋊C4×C17—22no—
57C2×C22⋊C4×C17—24no—
58C2×C22⋊C4×C17—22no—
59C2×C22⋊C4×C17—24no—
60C2×C22⋊C4×C17—22no—
61C2×C22⋊C4×C17—22no—
62C2×C22⋊C4×C17D412yes—
63C2×C22⋊C4×C17C2312yes—
64#10587639—44no—
65C2×C22⋊Dic17—21no—
66C2×C22⋊Dic17—22no—
67C2×#63230—24no—
68#10587616—22no—
69#10587616—22no—
70#10587615—44no—
71C2×#63227—21no—
72C2×#63227—22no—
73C2×C4⋊Dic17C2311yescharacteristic
74C2×#63225—22no—
75C2×C4×Dic17—21no—
76C2×#63223—22no—
77#10587506—24no—
78#10587502C2312yes—
79C22⋊#3622—22no—
80C22⋊#3622C2314yes—
81#10587480—24no—
82#10587478—24no—
83C24×C17—21no—
84C24×C17—21no—
85C24×C17—24no—
86C24×C17—42no—
87C24×C17—81no—
88C24×C17C2×D411yescharacteristic
89C24×C17C2×D412yes—
90C23×D17—21no—
91C2×Q8×C17—24no—
92C2×Q8×C17—22no—
93C2×Q8×C17—82no—
94C2×Q8×C17—82no—
95C2×Q8×C17C2×D414yes—
96C22×C4×C17—22no—
97C22×C4×C17—22no—
98C22×C4×C17—21no—
99C22×C4×C17—24no—
100C22×C4×C17—24no—
101C22×C4×C17—21no—
102C22×C4×C17—42no—
103C22×C4×C17—81no—
104C22×C4×C17—82no—
105C22×C4×C17—82no—
106C22×C4×C17C2×D412yes—
107C22×C4×C17C2×D411yescharacteristic
108C22×C4×C17C2×D412yes—
109C22×C4×C17C2×D412yes—
110C22×C4×C17C2411yescharacteristic
111C2×C22⋊D17—44no—
112C2×C22⋊D17—42no—
113C2×C22⋊D17—42no—
114C2×C22⋊D17—41no—
115C2×C22⋊D17—41no—
116C2×C22⋊D17—44no—
117C2×C22⋊D17—82no—
118C22×Dic17—22no—
119C22×Dic17—21no—
120C2×D68—22no—
121C2×D68—81no—
122C4⋊C4×C17—24no—
123C4⋊C4×C17—28no—
124C4⋊C4×C17—22no—
125C4⋊C4×C17—24no—
126C4⋊C4×C17—24no—
127C4⋊C4×C17C2×D414yes—
128C22⋊C4×C17—28no—
129C22⋊C4×C17—24no—
130C22⋊C4×C17—24no—
131C22⋊C4×C17—24no—
132C22⋊C4×C17—24no—
133C22⋊Dic17—24no—
134C22⋊Dic17—44no—
135#63230—84no—
136#63227—44no—
137#63227—42no—
138C4⋊Dic17—22no—
139#63225—44no—
140C4×Dic17—81no—
141#63223—82no—
142C2×#3622—24no—
143#18318—171nomaximal, Sylow
144C23×C17—24no—
145C23×C17—24no—
146C23×C17—22no—
147C23×C17—22no—
148C23×C17—22no—
149C23×C17—22no—
150C23×C17—41no—
151C23×C17—42no—
152C23×C17—42no—
153C23×C17—42no—
154C23×C17—42no—
155C23×C17—81no—
156C23×C17—81no—
157C23×C17—88no—
158C23×C17#12011yescharacteristic
159C23×C17#12011yescharacteristic
160C23×C17#13714yes—
161C23×C17C22×D411yescharacteristic
162C22×D17—42no—
163C22×D17—41no—
164C22×D17—41no—
165C22×D17—41no—
166C22×D17—42no—
167Q8×C17—44no—
168Q8×C17—82no—
169Q8×C17—84no—
170C2×C4×C17—24no—
171C2×C4×C17—22no—
172C2×C4×C17—24no—
173C2×C4×C17—41no—
174C2×C4×C17—42no—
175C2×C4×C17—42no—
176C2×C4×C17—41no—
177C2×C4×C17—44no—
178C2×C4×C17—42no—
179C2×C4×C17—44no—
180C2×C4×C17—42no—
181C2×C4×C17—81no—
182C2×C4×C17—84no—
183C2×C4×C17—84no—
184C2×C4×C17#12012yes—
185C2×C4×C17C22×D412yes—
186C22⋊D17—88no—
187C22⋊D17—82no—
188C22⋊D17—82no—
189C2×Dic17—24no—
190C2×Dic17—42no—
191C2×Dic17—42no—
192C2×Dic17—41no—
193C2×Dic17—41no—
194C2×Dic17—41no—
195D68—44no—
196#3622—44no—
197C2×#543D1711yescharacteristic, complemented
198#3193—172no—
199#2539—171no—
200#1614—174no—
201#1596—174no—
202#1508—172no—
203#1462—171no—
204C22×C17—21no—
205C22×C17—21no—
206C22×C17—21no—
207C22×C17—21no—
208C22×C17—21no—
209C22×C17—24no—
210C22×C17—21no—
211C22×C17—21no—
212C22×C17—44no—
213C22×C17—44no—
214C22×C17—48no—
215C22×C17—42no—
216C22×C17—42no—
217C22×C17—82no—
218C22×C17—84no—
219C22×C17#45312yes—
220C22×C17C2×#12011yescharacteristic
221C22×C17C2×#13712yes—
222C2×D17—82no—
223C2×D17—82no—
224C2×D17—81no—
225C2×D17—81no—
226C2×D17—81no—
227C4×C17—22no—
228C4×C17—22no—
229C4×C17—42no—
230C4×C17—44no—
231C4×C17—44no—
232Dic17—44no—
233Dic17—81no—
234#543—24no—
235#543—22no—
236#543C2×D1714yes—
237C2×#123—341no—
238C2×C22⋊Q8—22no—
239C2×C22⋊Q8—21no—
240C2×C22⋊Q8—22no—
241C2×C22⋊Q8C2×D1712yes—
242C2×#121—171no—
243C2×#120—341no—
244C22×C22⋊C4—21no—
245C22×C22⋊C4C2×D1711yescharacteristic
246#484—174no—
247#480—342no—
248#468—344no—
249#450—174no—
250#448—342no—
251#407—172no—
252#399—172no—
253#392—172no—
254#390—342no—
255C4×C22⋊C4—171no—
256#328—178no—
257#323—174no—
258C2×C17—22no—
259C2×C17—22no—
260C2×C17—41no—
261C2×C17—44no—
262C2×C17#300411yescharacteristic
263C2×C17C2×#45311yescharacteristic
264C2×C17#302611yescharacteristic
265D17—161no—
266C25C22⋊D1711yescharacteristic
267C22×Q8—21no—
268C22×Q8C22⋊D1712yescomplemented
269C22×D4—171no—
270C22×D4—341no—
271C22×D4—342no—
272C23×C4—21no—
273C23×C4—21no—
274C23×C4C22⋊D1711yescharacteristic
275C23×C4C22×D1711yescharacteristic
276C2×SD16—344no—
277C2×M16—342no—
278#123—684no—
279C22⋊Q8—28no—
280C22⋊Q8—28no—
281C22⋊Q8—24no—
282C22⋊Q8—28no—
283C22⋊Q8—24no—
284C22⋊Q8—22no—
285C22⋊Q8—22no—
286C22⋊Q8—28no—
287C22⋊Q8C22⋊D1714yescomplemented
288C22⋊Q8C22×D1714yes—
289#121—342no—
290#121—342no—
291#120—684no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—171no—
296C2×C4⋊C4—342no—
297C2×C4⋊C4C22⋊D1712yescomplemented
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—341no—
305C2×C22⋊C4—342no—
306C2×C22⋊C4—341no—
307C2×C22⋊C4—342no—
308C2×C22⋊C4C22⋊D1712yes—
309C2×C22⋊C4C22×D1712yes—
310C2×C42—341no—
311#107—344no—
312#103—344no—
313#102—344no—
314C22⋊C8—174no—
315C22⋊C8—342no—
316#95—172no—
317C17#1831811yescharacteristic, complemented, Sylow
318C24—21no—
319C24—24no—
320C24—21no—
321C24—42no—
322C24—81no—
323C24—341no—
324C24C2×C22⋊D1711yescharacteristic
325C24C2×C22⋊D1712yes—
326C2×Q8—24no—
327C2×Q8—22no—
328C2×Q8—82no—
329C2×Q8—82no—
330C2×Q8C2×C22⋊D1714yes—
331C2×D4—342no—
332C2×D4—684no—
333C2×D4—682no—
334C2×D4—682no—
335C2×D4—684no—
336C2×D4—681no—
337C2×D4—681no—
338C2×D4—1361no—
339C2×D4—1362no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—21no—
346C22×C4—42no—
347C22×C4—81no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4—342no—
351C22×C4—341no—
352C22×C4C2×C22⋊D1712yes—
353C22×C4C2×C22⋊D1712yes—
354C22×C4C2×C22⋊D1712yes—
355C22×C4C2×C22⋊D1711yescharacteristic
356C22×C4C23×D1711yescharacteristic
357SD16—1364no—
358M16—1362no—
359C2×C8—344no—
360C4⋊C4—24no—
361C4⋊C4—28no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—342no—
366C4⋊C4—684no—
367C4⋊C4C2×C22⋊D1714yes—
368C22⋊C4—28no—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—24no—
373C22⋊C4—344no—
374C22⋊C4—684no—
375C22⋊C4—684no—
376C22⋊C4—682no—
377C42—1361no—
378C23—24no—
379C23—24no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—41no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—81no—
390C23—81no—
391C23—88no—
392C23—681no—
393C23—681no—
394C23—682no—
395C23—681no—
396C23—682no—
397C23#1058762814yes—
398C23#1058763911yescharacteristic
399C23#1058763911yescharacteristic
400C23C22×C22⋊D1711yescharacteristic
401Q8—44no—
402Q8—82no—
403Q8—84no—
404D4—684no—
405D4—1368no—
406D4—1362no—
407D4—1362no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—81no—
420C2×C4—84no—
421C2×C4—84no—
422C2×C4—344no—
423C2×C4—682no—
424C2×C4—682no—
425C2×C4—681no—
426C2×C4—681no—
427C2×C4—681no—
428C2×C4#1058763912yes—
429C2×C4C22×C22⋊D1712yes—
430C8—684no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—24no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—44no—
440C22—44no—
441C22—48no—
442C22—42no—
443C22—42no—
444C22—82no—
445C22—84no—
446C22—1362no—
447C22—1362no—
448C22—1361no—
449C22—1361no—
450C22—1361no—
451C22#1176297712yes—
452C22C2×#1058763911yescharacteristic
453C22C2×#1058762812yes—
454C4—22no—
455C4—22no—
456C4—42no—
457C4—44no—
458C4—44no—
459C4—684no—
460C4—1361no—
461C2—22no—
462C2—22no—
463C2—41no—
464C2—44no—
465C2—2721no—
466C2#1453801811yescharacteristic
467C2C2×#1176297711yescharacteristic
468C2#1453818111yescharacteristic
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#543C2
socleC22×C17C2×#120
prime radical#14533715C2
hypercenterC2×#543D17
hypocenterC17#18318
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
1. #14539586C2semidirect
3. #14527809C2semidirect
4. #14527707C2semidirect
5. #14527304C2semidirect
6. #14527111C2semidirect
7. C2×C17×#543C2semidirect
11. #11763142C22semidirect
14. #11762959C22semidirect
21. #11762546C22semidirect
34. C22×Q8×C17D4semidirect
50. C17×C22⋊Q8D4semidirect
55. C2×C4⋊C4×C17D4semidirect
197. C2×#543D17semidirect
268. C22×Q8C22⋊D17semidirect
287. C22⋊Q8C22⋊D17semidirect
297. C2×C4⋊C4C22⋊D17semidirect
317. C17#18318semidirect

Maximal subgroups

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

subgroupindex
#145395862
#145337152
#145278092
#145277072
#145273042
#145271112
C2×C17×#5432
#1831817