Finite Group Databasebeta

← #16944790

Subgroups — #16944790

Order 4352.

586 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C2×#11762576C211yesmaximal, characteristic
2#14527111C211yesmaximal, characteristic
3#14526808C212yesmaximal, complemented
4#14526784C212yesmaximal, complemented
5C17×#2482C211yesmaximal, characteristic
6C22×C22⋊Dic17—21no—
7#11762773—21no—
8#11762576—21no—
9#11762576C2212yes—
10#11762533—21no—
11#11762533C2212yes—
12C2×C17×#117C411yescharacteristic, complemented
13C22×C22⋊C4×C17C2211yescharacteristic
14C17×#399C411yescharacteristic, complemented
15C17×#399C2211yescharacteristic
16C17×#397C411yescharacteristic, complemented
17C17×#397C2211yescharacteristic
18C17×#388—21no—
19C17×#388—21no—
20C17×#385—21no—
21C17×#381C411yescharacteristic, complemented
22C17×#379—21no—
23C25×C17D411yescharacteristic
24C23×C4×C17C2×C411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C23×Dic17—41no—
27C17×#117—41no—
28C17×#117—41no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17C2×C411yescharacteristic, complemented
32C2×C4⋊C4×C17C2×C411yescharacteristic, complemented
33C2×C22⋊C4×C17—21no—
34C2×C22⋊C4×C17—21no—
35C2×C22⋊C4×C17—21no—
36C2×C22⋊C4×C17—21no—
37C2×C22⋊C4×C17—21no—
38C2×C22⋊C4×C17—21no—
39C2×C22⋊C4×C17—41no—
40C2×C22⋊C4×C17—41no—
41C2×C22⋊C4×C17C2×C411yescharacteristic
42C2×C22⋊C4×C17D411yescharacteristic
43C2×C22⋊C4×C17D411yescharacteristic
44C2×C22⋊C4×C17C2311yescharacteristic
45C2×C42×C17—21no—
46C2×C22⋊Dic17—24no—
47C2×C22⋊Dic17—22no—
48C2×C22⋊Dic17—41no—
49C2×C22⋊Dic17—42no—
50C2×#63223—41no—
51C17×#95—21no—
52C17×#95—21no—
53C17×#95—21no—
54C17×#95C2×C411yescharacteristic, complemented
55C17×#95C2×C411yescharacteristic, complemented
56C22⋊#3622—22no—
57C22⋊#3622—41no—
58C24×C17—21no—
59C24×C17—21no—
60C24×C17—22no—
61C24×C17—21no—
62C24×C17—21no—
63C24×C17—42no—
64C24×C17—81no—
65C24×C17C22⋊C411yescharacteristic
66C24×C17C22⋊C411yescharacteristic
67C24×C17C2×D411yescharacteristic
68C22×C4×C17—21no—
69C22×C4×C17—21no—
70C22×C4×C17—21no—
71C22×C4×C17—21no—
72C22×C4×C17—21no—
73C22×C4×C17—21no—
74C22×C4×C17—21no—
75C22×C4×C17—21no—
76C22×C4×C17—42no—
77C22×C4×C17—81no—
78C22×C4×C17C22⋊C411yescharacteristic
79C22×C4×C17C22⋊C411yescharacteristic
80C22×C4×C17C22×C411yescharacteristic
81C22×C4×C17C2×D411yescharacteristic
82C22×Dic17—44no—
83C22×Dic17—42no—
84C22×Dic17—82no—
85C22×Dic17—82no—
86C4⋊C4×C17—41no—
87C4⋊C4×C17—41no—
88C4⋊C4×C17—81no—
89C4⋊C4×C17—81no—
90C22⋊C4×C17—41no—
91C22⋊C4×C17—41no—
92C22⋊C4×C17—41no—
93C22⋊C4×C17—41no—
94C22⋊C4×C17—42no—
95C22⋊C4×C17—41no—
96C22⋊C4×C17—41no—
97C22⋊C4×C17—42no—
98C22⋊C4×C17—41no—
99C22⋊C4×C17—41no—
100C22⋊C4×C17—81no—
101C22⋊C4×C17—81no—
102C42×C17—42no—
103C22⋊Dic17—24no—
104C22⋊Dic17—44no—
105C22⋊Dic17—42no—
106#63223—82no—
107C2×#3622—42no—
108#11755—171nomaximal, Sylow
109C23×C17—22no—
110C23×C17—22no—
111C23×C17—22no—
112C23×C17—22no—
113C23×C17—22no—
114C23×C17—21no—
115C23×C17—21no—
116C23×C17—21no—
117C23×C17—21no—
118C23×C17—21no—
119C23×C17—21no—
120C23×C17—41no—
121C23×C17—42no—
122C23×C17—41no—
123C23×C17—41no—
124C23×C17—41no—
125C23×C17—41no—
126C23×C17—42no—
127C23×C17—81no—
128C23×C17—81no—
129C23×C17—84no—
130C23×C17—82no—
131C23×C17—82no—
132C23×C17#9911yescharacteristic
133C23×C17#9911yescharacteristic
134C23×C17#10412yes—
135C23×C17C2×C22⋊C411yescharacteristic
136C23×C17AGL(1,ℤ/8)11yescharacteristic
137C23×C17#13711yescharacteristic
138C2×C4×C17—21no—
139C2×C4×C17—21no—
140C2×C4×C17—21no—
141C2×C4×C17—21no—
142C2×C4×C17—42no—
143C2×C4×C17—42no—
144C2×C4×C17—42no—
145C2×C4×C17—41no—
146C2×C4×C17—41no—
147C2×C4×C17—41no—
148C2×C4×C17—41no—
149C2×C4×C17—41no—
150C2×C4×C17—41no—
151C2×C4×C17—41no—
152C2×C4×C17—41no—
153C2×C4×C17—41no—
154C2×C4×C17—41no—
155C2×C4×C17—41no—
156C2×C4×C17—41no—
157C2×C4×C17—41no—
158C2×C4×C17—41no—
159C2×C4×C17—41no—
160C2×C4×C17—41no—
161C2×C4×C17—81no—
162C2×Dic17—44no—
163C2×Dic17—88no—
164C2×Dic17—82no—
165C2×Dic17—82no—
166#3622—82no—
167#2482D1711yescharacteristic
168C2×#342—171no—
169#1462—171no—
170#1352—172no—
171#1344—172no—
172C22×C17—21no—
173C22×C17—21no—
174C22×C17—21no—
175C22×C17—21no—
176C22×C17—21no—
177C22×C17—21no—
178C22×C17—21no—
179C22×C17—21no—
180C22×C17—22no—
181C22×C17—21no—
182C22×C17—42no—
183C22×C17—42no—
184C22×C17—42no—
185C22×C17—41no—
186C22×C17—41no—
187C22×C17—44no—
188C22×C17—44no—
189C22×C17—41no—
190C22×C17—41no—
191C22×C17—41no—
192C22×C17—41no—
193C22×C17—81no—
194C22×C17—81no—
195C22×C17—82no—
196C22×C17—82no—
197C22×C17#35212yes—
198C22×C17C2×#9911yescharacteristic
199C22×C17#41711yescharacteristic
200C22×C17C2×#10411yescharacteristic
201C4×C17—41no—
202C4×C17—41no—
203C4×C17—41no—
204C4×C17—41no—
205C4×C17—81no—
206C4×C17—81no—
207C4×C17—81no—
208C4×C17—81no—
209Dic17—84no—
210C2×#117Dic1711yescharacteristic, complemented
211C22×C22⋊C4—341no—
212C22×C22⋊C4C2×D1711yescharacteristic
213#407—341no—
214#399Dic1711yescharacteristic, complemented
215#399C2×D1711yescharacteristic
216#397Dic1711yescharacteristic, complemented
217#397C2×D1711yescharacteristic
218#388—21no—
219#388—21no—
220#385—21no—
221#381Dic1711yescharacteristic, complemented
222#379—21no—
223#342—172no—
224#342—341no—
225#323—172no—
226#323—341no—
227C2×C17—21no—
228C2×C17—21no—
229C2×C17—22no—
230C2×C17—41no—
231C2×C17—41no—
232C2×C17—42no—
233C2×C17—41no—
234C2×C17#150411yescharacteristic
235C2×C17C2×#35211yescharacteristic
236C2×C17#212711yescharacteristic
237C25C22⋊D1711yescharacteristic
238C23×C4—681no—
239C23×C4C2×Dic1711yescharacteristic
240C23×C4C22⋊D1711yescharacteristic
241C2×M16—681no—
242#117—41no—
243#117—41no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4C2×Dic1711yescharacteristic, complemented
247C2×C4⋊C4C2×Dic1711yescharacteristic, complemented
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—344no—
257C2×C22⋊C4—342no—
258C2×C22⋊C4—681no—
259C2×C22⋊C4—682no—
260C2×C22⋊C4C2×Dic1711yescharacteristic
261C2×C22⋊C4C22⋊D1711yescharacteristic
262C2×C22⋊C4C22⋊D1711yescharacteristic
263C2×C22⋊C4C22×D1711yescharacteristic
264C2×C42—21no—
265C22⋊C8—342no—
266C22⋊C8—681no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95C2×Dic1711yescharacteristic, complemented
271#95C2×Dic1711yescharacteristic, complemented
272C17#1175511yescharacteristic, complemented, Sylow
273C24—21no—
274C24—21no—
275C24—22no—
276C24—21no—
277C24—21no—
278C24—42no—
279C24—81no—
280C24C22⋊Dic1711yescharacteristic
281C24C22⋊Dic1711yescharacteristic
282C24C2×C22⋊D1711yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—42no—
292C22×C4—81no—
293C22×C4—684no—
294C22×C4—682no—
295C22×C4—1362no—
296C22×C4—1362no—
297C22×C4C22⋊Dic1711yescharacteristic
298C22×C4C22⋊Dic1711yescharacteristic
299C22×C4C22×Dic1711yescharacteristic
300C22×C4C2×C22⋊D1711yescharacteristic
301M16—1362no—
302C2×C8—682no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—81no—
306C4⋊C4—81no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—42no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—42no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—81no—
318C22⋊C4—81no—
319C22⋊C4—344no—
320C22⋊C4—684no—
321C22⋊C4—682no—
322C42—42no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—22no—
332C23—21no—
333C23—21no—
334C23—41no—
335C23—42no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—42no—
341C23—81no—
342C23—81no—
343C23—84no—
344C23—82no—
345C23—82no—
346C23#1058750511yescharacteristic
347C23#1058750511yescharacteristic
348C23#1058750812yes—
349C23#1058763511yescharacteristic
350C23#1058763711yescharacteristic
351C23C2×C22⋊Dic1711yescharacteristic
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—81no—
376C2×C4—684no—
377C2×C4—1368no—
378C2×C4—1362no—
379C2×C4—1362no—
380C8—1362no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—22no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—41no—
395C22—41no—
396C22—44no—
397C22—44no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—81no—
403C22—81no—
404C22—82no—
405C22—82no—
406C22#1176259912yes—
407C22C2×#1058750511yescharacteristic
408C22#1176281011yescharacteristic
409C22C2×#1058750811yescharacteristic
410C4—41no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—81no—
415C4—81no—
416C4—81no—
417C4—81no—
418C4—1364no—
419C2—21no—
420C2—21no—
421C2—22no—
422C2—41no—
423C2—41no—
424C2—42no—
425C2—41no—
426C2#1452728811yescharacteristic
427C2C2×#1176259911yescharacteristic
428C2#1453045311yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#2482C2
socleC22×C17C2×#99
prime radicalC25×C17D4
cosocleC2×C22⋊C4×C17C23
hypercenter#2482D17
hypocenterC17#11755
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2482

Complements

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

NH ≅ G/Ntype
3. #14526808C2semidirect
4. #14526784C2semidirect
12. C2×C17×#117C4semidirect
14. C17×#399C4semidirect
16. C17×#397C4semidirect
21. C17×#381C4semidirect
31. C2×C4⋊C4×C17C2×C4semidirect
32. C2×C4⋊C4×C17C2×C4semidirect
54. C17×#95C2×C4semidirect
55. C17×#95C2×C4semidirect
210. C2×#117Dic17semidirect
214. #399Dic17semidirect
216. #397Dic17semidirect
221. #381Dic17semidirect
246. C2×C4⋊C4C2×Dic17semidirect
247. C2×C4⋊C4C2×Dic17semidirect
270. #95C2×Dic17semidirect
271. #95C2×Dic17semidirect
272. C17#11755semidirect

Maximal subgroups

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

subgroupindex
C2×#117625762
#145271112
#145268082
#145267842
C17×#24822
#1175517