Finite Group Databasebeta

← #16929367

Subgroups — #16929367

Order 4352.

740 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762746C211yesmaximal, characteristic
2#14531381C211yesmaximal, characteristic
3#14528875C211yesmaximal, characteristic, complemented
4#14528872C211yesmaximal, characteristic, complemented
5C2×#11762532C211yesmaximal, characteristic
6#14526663C211yesmaximal, characteristic, complemented
7C17×#1294C211yesmaximal, characteristic, complemented
8C22×C4⋊Dic17—21no—
9C22×C4⋊Dic17C2211yescharacteristic
10C22×#63227—21no—
11C22×#63227C2211yescharacteristic
12C2×C22⋊#3622C2211yescharacteristic
13#11762746—22no—
14#11762746C414yescomplemented
15#11762701—22no—
16#11762699—22no—
17#11762669C2211yescharacteristic
18#11762664—21no—
19C22⋊C4×Dic17—21no—
20#11762572—22no—
21#11762571—22no—
22#11762532—22no—
23C2×C17×C22⋊C8C2211yescharacteristic
24C17×#380C2211yescharacteristic
25C2×C17×#95C2211yescharacteristic
26C24×D17D411yescharacteristic
27C23×C4×C17—21no—
28C23×C4×C17C2311yescharacteristic
29C23×Dic17—21no—
30C23×Dic17D411yescharacteristic
31C22×C8×C17—21no—
32C2×C4⋊C4×C17D412yes—
33C2×C22⋊C4×C17—21no—
34C2×C22⋊Dic17—21no—
35C2×#63227—22no—
36C2×#63227—22no—
37C2×#63227—24no—
38C2×#63227—21no—
39C2×#63227—21no—
40C2×#63227—21no—
41C2×#63227—21no—
42C2×#63227—44no—
43C2×#63227C2×C412yes—
44C2×#63227D412yes—
45C2×C4⋊Dic17—21no—
46C2×C4⋊Dic17—24no—
47C2×C4⋊Dic17—21no—
48C2×C4⋊Dic17—21no—
49C2×C4⋊Dic17—21no—
50C2×C4⋊Dic17—21no—
51C2×C4⋊Dic17—44no—
52C2×C4⋊Dic17C2×C412yes—
53C2×C4⋊Dic17D412yes—
54C2×C4×Dic17—22no—
55C22×#3622—21no—
56C17×C22⋊C8—21no—
57C17×C22⋊C8—21no—
58C17×#95—21no—
59C17×#95—21no—
60C17×#95C2×C412yes—
61#10587502—22no—
62#10587502—21no—
63C22⋊#3622—21no—
64C22⋊#3622—21no—
65C24×C17C2×D411yescharacteristic
66C23×D17—21no—
67C23×D17—21no—
68C23×D17—21no—
69C23×D17—21no—
70C23×D17—21no—
71C23×D17—21no—
72C23×D17—42no—
73C23×D17—42no—
74C23×D17C22⋊C412yes—
75C22×C4×C17—24no—
76C22×C4×C17—22no—
77C22×C4×C17—22no—
78C22×C4×C17—42no—
79C22×C4×C17—42no—
80C22×C4×C17—41no—
81C22×C4×C17—41no—
82C22×C4×C17C22×C411yescharacteristic
83C22×C4×C17C2×D411yescharacteristic
84C22×C4×C17C2×D411yescharacteristic
85C22×C4×C17C2×D411yescharacteristic
86C22×C4×C17#4111yescharacteristic
87C22×C4×C17#4111yescharacteristic
88C22×Dic17—21no—
89C22×Dic17—21no—
90C22×Dic17—22no—
91C22×Dic17—22no—
92C22×Dic17—42no—
93C22×Dic17—42no—
94C22×Dic17—42no—
95C22×Dic17—41no—
96C22×Dic17—41no—
97C22×Dic17C22⋊C412yes—
98C2×C8×C17—42no—
99C2×C8×C17—41no—
100C4⋊C4×C17—42no—
101C22⋊C4×C17—41no—
102C22⋊C4×C17—41no—
103C22⋊Dic17—41no—
104C22⋊Dic17—41no—
105#63227—22no—
106#63227—22no—
107#63227—22no—
108#63227—22no—
109#63227—48no—
110#63227—82no—
111#63227—82no—
112C4⋊Dic17—22no—
113C4⋊Dic17—22no—
114C4⋊Dic17—48no—
115C4⋊Dic17—41no—
116C4⋊Dic17—41no—
117C4⋊Dic17—42no—
118C4×Dic17—82no—
119C2×#3622—42no—
120C2×#3622—41no—
121#9024—171nomaximal, Sylow
122C23×C17—22no—
123C23×C17—22no—
124C23×C17C2×C22⋊C411yescharacteristic
125C23×C17#12011yescharacteristic
126C23×C17#12311yescharacteristic
127C23×C17#13511yescharacteristic
128C23×C17#13511yescharacteristic
129C23×C17#13711yescharacteristic
130C23×C17#13711yescharacteristic
131C22×D17—22no—
132C22×D17—22no—
133C22×D17—22no—
134C22×D17—42no—
135C22×D17—42no—
136C22×D17—44no—
137C22×D17—44no—
138C22×D17—41no—
139C22×D17—41no—
140C22×D17—42no—
141C22×D17—42no—
142C22×D17—41no—
143C22×D17—41no—
144C22×D17—81no—
145C22×D17—81no—
146C22×D17—82no—
147C22×D17—82no—
148C2×C4×C17—22no—
149C2×C4×C17—22no—
150C2×C4×C17—24no—
151C2×C4×C17—41no—
152C2×C4×C17—41no—
153C2×C4×C17—41no—
154C2×C4×C17—41no—
155C2×C4×C17—44no—
156C2×C4×C17—44no—
157C2×C4×C17—42no—
158C2×C4×C17—42no—
159C2×C4×C17—42no—
160C2×C4×C17—42no—
161C2×C4×C17—42no—
162C2×C4×C17C4×D411yescharacteristic
163C2×C4×C17C4×D411yescharacteristic
164C2×C4×C17#12111yescharacteristic
165C2×C4×C17#12111yescharacteristic
166C2×Dic17—22no—
167C2×Dic17—42no—
168C2×Dic17—42no—
169C2×Dic17—42no—
170C2×Dic17—42no—
171C2×Dic17—42no—
172C2×Dic17—42no—
173C2×Dic17—42no—
174C2×Dic17—42no—
175C2×Dic17—81no—
176C2×Dic17—82no—
177C2×Dic17—82no—
178C8×C17—81no—
179#3622—81no—
180C2×#399—171no—
181#2312—171no—
182#1824—171no—
183#1824—171no—
184C2×#323—171no—
185#1294—171no—
186#1294D1711yescharacteristic, complemented
187C22×C17—22no—
188C22×C17—22no—
189C22×C17—22no—
190C22×C17—22no—
191C22×C17—21no—
192C22×C17—21no—
193C22×C17—21no—
194C22×C17—21no—
195C22×C17#38611yescharacteristic
196C22×C17#41611yescharacteristic
197C22×C17#41911yescharacteristic
198C22×C17#45211yescharacteristic
199C22×C17#45211yescharacteristic
200C22×C17#45312yes—
201C22×C17#45812yes—
202C22×C17#48511yescharacteristic
203C22×C17#48511yescharacteristic
204C2×D17—44no—
205C2×D17—42no—
206C2×D17—42no—
207C2×D17—42no—
208C2×D17—82no—
209C2×D17—82no—
210C2×D17—82no—
211C2×D17—82no—
212C2×D17—82no—
213C4×C17—21no—
214C4×C17—21no—
215C4×C17—21no—
216C4×C17—21no—
217C4×C17—44no—
218C4×C17—82no—
219Dic17—42no—
220Dic17—82no—
221Dic17—82no—
222C22×C4⋊C4—171no—
223C22×C4⋊C4—341no—
224C22×C22⋊C4—171no—
225C22×C22⋊C4—341no—
226C2×C22⋊C8—171no—
227C2×C22⋊C8C2×D1711yescharacteristic
228#399—174no—
229#399—342no—
230#388—342no—
231#388—342no—
232#380—171no—
233#380C2×D1711yescharacteristic
234#379—341no—
235C4×C22⋊C4—341no—
236C2×#95C2×D1711yescharacteristic
237#340—342no—
238#340—342no—
239#323—342no—
240C2×C17—21no—
241C2×C17—21no—
242C2×C17—21no—
243C2×C17—21no—
244C2×C17#161812yes—
245C2×C17#161812yes—
246C2×C17#187311yescharacteristic
247C2×C17#188711yescharacteristic
248C2×C17#189011yescharacteristic
249D17—82no—
250C25—171no—
251C23×C4—21no—
252C23×C4—171no—
253C23×C4—341no—
254C23×C4C22×D1711yescharacteristic
255C22×C8—21no—
256C22×C8—341no—
257C2×C4⋊C4—172no—
258C2×C4⋊C4—172no—
259C2×C4⋊C4—341no—
260C2×C4⋊C4—341no—
261C2×C4⋊C4—341no—
262C2×C4⋊C4—341no—
263C2×C4⋊C4—344no—
264C2×C4⋊C4—341no—
265C2×C4⋊C4—684no—
266C2×C4⋊C4D6812yes—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—172no—
269C2×C22⋊C4—172no—
270C2×C22⋊C4—341no—
271C2×C22⋊C4—341no—
272C2×C22⋊C4—341no—
273C2×C22⋊C4—341no—
274C2×C22⋊C4—344no—
275C2×C22⋊C4—341no—
276C2×C22⋊C4—342no—
277C2×C22⋊C4—342no—
278C2×C22⋊C4—684no—
279C2×C42—342no—
280C22⋊C8—21no—
281C22⋊C8—21no—
282C22⋊C8—341no—
283C22⋊C8—341no—
284#95—21no—
285#95—21no—
286#95—342no—
287#95—341no—
288#95C4×D1712yes—
289C17#902411yescharacteristic, complemented, Sylow
290C24—172no—
291C24—341no—
292C24—341no—
293C24—341no—
294C24—341no—
295C24—341no—
296C24—341no—
297C24—682no—
298C24—682no—
299C24D4×D1711yescharacteristic
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—172no—
308C22×C4—341no—
309C22×C4—341no—
310C22×C4—342no—
311C22×C4—342no—
312C22×C4—682no—
313C22×C4—682no—
314C22×C4—682no—
315C22×C4—681no—
316C22×C4—681no—
317C22×C4C2×C4×D1711yescharacteristic
318C22×C4C2×D6811yescharacteristic
319C22×C4#6325211yescharacteristic
320C22×C4D4×D1711yescharacteristic
321C22×C4D4×D1711yescharacteristic
322C22×C4#6325611yescharacteristic
323C2×C8—42no—
324C2×C8—41no—
325C2×C8—682no—
326C2×C8—681no—
327C4⋊C4—42no—
328C4⋊C4—342no—
329C4⋊C4—342no—
330C4⋊C4—682no—
331C4⋊C4—688no—
332C4⋊C4—681no—
333C4⋊C4—681no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—342no—
337C22⋊C4—342no—
338C22⋊C4—342no—
339C22⋊C4—342no—
340C22⋊C4—688no—
341C22⋊C4—681no—
342C22⋊C4—681no—
343C22⋊C4—1362no—
344C22⋊C4—1362no—
345C42—1362no—
346C23—22no—
347C23—22no—
348C23—342no—
349C23—342no—
350C23—342no—
351C23—682no—
352C23—682no—
353C23—684no—
354C23—684no—
355C23—681no—
356C23—681no—
357C23—682no—
358C23—682no—
359C23—681no—
360C23—681no—
361C23—1361no—
362C23—1361no—
363C23—1362no—
364C23—1362no—
365C23C22⋊C4×D1711yescharacteristic
366C23#1058756811yescharacteristic
367C23#1058758011yescharacteristic
368C23#1058759811yescharacteristic
369C23#1058760111yescharacteristic
370C23#1058760211yescharacteristic
371C23#1058760411yescharacteristic
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—342no—
387C2×C4—682no—
388C2×C4—682no—
389C2×C4—682no—
390C2×C4—682no—
391C2×C4—682no—
392C2×C4—682no—
393C2×C4—682no—
394C2×C4—682no—
395C2×C4—1361no—
396C2×C4—1362no—
397C2×C4—1362no—
398C2×C4C4×D6811yescharacteristic
399C2×C4#1058757011yescharacteristic
400C2×C4#1058757911yescharacteristic
401C2×C4#1058758111yescharacteristic
402C8—81no—
403C8—1361no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—684no—
413C22—682no—
414C22—682no—
415C22—682no—
416C22—1362no—
417C22—1362no—
418C22—1362no—
419C22—1362no—
420C22—1362no—
421C22#1176268711yescharacteristic
422C22#1176280211yescharacteristic
423C22#1176281611yescharacteristic
424C22#1176297211yescharacteristic
425C22#1176297411yescharacteristic
426C22#1176297812yes—
427C22#1176300012yes—
428C22#1176315211yescharacteristic
429C22#1176315311yescharacteristic
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—44no—
435C4—82no—
436C4—682no—
437C4—1362no—
438C4—1362no—
439C2—21no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—1362no—
444C2#1452783312yes—
445C2#1452783512yes—
446C2#1452909511yescharacteristic
447C2#1452918111yescharacteristic
448C2#1452919611yescharacteristic
subgroupG/H
centerC23#10587568
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1294C2
socleC23×C17#120
prime radicalC24×D17D4
cosocleC23×C4×C17C23
hypercenter#1294D17
hypocenterC17#9024
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
3. #14528875C2semidirect
4. #14528872C2semidirect
6. #14526663C2semidirect
7. C17×#1294C2semidirect
14. #11762746C4semidirect
186. #1294D17semidirect
289. C17#9024semidirect

Maximal subgroups

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

subgroupindex
C2×#117627462
#145313812
#145288752
#145288722
C2×#117625322
#145266632
C17×#12942
#902417