Finite Group Databasebeta

← #16929171

Subgroups — #16929171

Order 4352.

540 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#63223C211yesmaximal, characteristic
2#14529729C211yesmaximal, characteristic, complemented
3C4×#10587503C211yesmaximal, characteristic
4#14527239C211yesmaximal, characteristic, complemented
5#14526686C211yesmaximal, characteristic, complemented
6#14526620C211yesmaximal, characteristic, complemented
7C17×#3101C211yesmaximal, characteristic
8C22×#63223—21no—
9#11762887—21no—
10#11762881—22no—
11D4×#3622—21no—
12#11762833C2211yescharacteristic
13C2×#10587503—21no—
14#11762773—21no—
15C4×#63223C2211yescharacteristic
16C2×C4×#3622C2211yescharacteristic
17C4×C4⋊Dic17C2211yescharacteristic
18#11762571—21no—
19#11762568—21no—
20#11762568—21no—
21Q8⋊#3622—22no—
22C17×#533C2211yescharacteristic
23C2×C4×D4×C17C2211yescharacteristic
24C17×#483—21no—
25C17×#476C411yescharacteristic, complemented
26C17×#476C411yescharacteristic, complemented
27C17×#461C411yescharacteristic, complemented
28C17×#461C411yescharacteristic, complemented
29C17×#450—21no—
30C4×SD16×C17—21no—
31C4×SD16×C17—21no—
32C17×#431C2211yescharacteristic
33C22×D4×C17D411yescharacteristic
34C23×C4×C17—21no—
35C2×SD16×C17—21no—
36C2×SD16×C17—21no—
37C17×C4⋊Q8C2×C411yescharacteristic
38C17×C4⋊Q8C2×C411yescharacteristic
39C17×#125—21no—
40C17×C22⋊Q8—21no—
41C17×C22⋊Q8—21no—
42C4×Q8×C17—21no—
43C4×Q8×C17—21no—
44C4×D4×C17—21no—
45C4×D4×C17—21no—
46C4×D4×C17C2×C411yescharacteristic
47C4×D4×C17C2×C411yescharacteristic
48C4×D4×C17D411yescharacteristic
49C4×D4×C17D411yescharacteristic
50C17×#117—21no—
51C2×C4⋊C4×C17D411yescharacteristic
52C2×C22⋊C4×C17—21no—
53C2×C42×C17C2311yescharacteristic
54C2×C4⋊Dic17—21no—
55C2×C4×Dic17—21no—
56C2×#63223—22no—
57C2×#63223—21no—
58C2×#63223—42no—
59C2×#63223—42no—
60C2×#63223D411yescharacteristic
61C2×#63223D411yescharacteristic
62C22×#3622—22no—
63C22×#3622—21no—
64C17×#106—21no—
65C17×#106—21no—
66C17×C4⋊C8C2×C411yescharacteristic
67C17×C4⋊C8C2×C411yescharacteristic
68C17×#103—21no—
69C17×#103—21no—
70C17×#102—21no—
71C17×#102—21no—
72C17×C22⋊C8—21no—
73C4×C8×C17—21no—
74#10587503—24no—
75#10587502—21no—
76C22⋊#3622—22no—
77C22⋊#3622—22no—
78C4⋊#3622—21no—
79C4⋊#3622D412yes—
80#10587474—21no—
81C4×#3622—21no—
82C4×#3622—21no—
83C4×#3622—21no—
84C24×C17—21no—
85C2×Q8×C17—21no—
86C2×Q8×C17—21no—
87C2×D4×C17—22no—
88C2×D4×C17—21no—
89C2×D4×C17—41no—
90C2×D4×C17C22⋊C411yescharacteristic
91C2×D4×C17C22⋊C411yescharacteristic
92C22×C4×C17—21no—
93C22×C4×C17—21no—
94C22×C4×C17—21no—
95C22×C4×C17—21no—
96C22×C4×C17—21no—
97C22×C4×C17—41no—
98C22×C4×C17—41no—
99C22×C4×C17—41no—
100C22×C4×C17—41no—
101C22×C4×C17C2×D411yescharacteristic
102C22×C4×C17C2×D411yescharacteristic
103C22×C4×C17#4111yescharacteristic
104C22×Dic17—41no—
105SD16×C17—42no—
106SD16×C17—42no—
107C2×C8×C17—21no—
108C2×C8×C17—21no—
109C4⋊C4×C17—21no—
110C4⋊C4×C17—21no—
111C4⋊C4×C17—21no—
112C4⋊C4×C17—21no—
113C4⋊C4×C17—21no—
114C4⋊C4×C17—41no—
115C4⋊C4×C17—41no—
116C4⋊C4×C17C22⋊C411yescharacteristic
117C4⋊C4×C17C22⋊C411yescharacteristic
118C22⋊C4×C17—21no—
119C22⋊C4×C17—21no—
120C22⋊C4×C17—21no—
121C22⋊C4×C17—21no—
122C22⋊C4×C17—41no—
123C42×C17—41no—
124C42×C17C22×C411yescharacteristic
125C42×C17C2×D411yescharacteristic
126C42×C17C2×D411yescharacteristic
127C42×C17#4111yescharacteristic
128C4⋊Dic17—22no—
129C4⋊Dic17—41no—
130C4×Dic17—41no—
131C4×Dic17—41no—
132#63223—21no—
133#63223—21no—
134#63223—21no—
135#63223—44no—
136#63223—41no—
137#63223—41no—
138#63223SD1612yescomplemented
139C2×#3622—21no—
140C2×#3622—22no—
141C2×#3622—21no—
142C2×#3622—22no—
143C2×#3622—44no—
144C2×#3622—41no—
145C2×#3622—41no—
146#8989—171nomaximal, Sylow
147C23×C17—21no—
148C23×C17—21no—
149C23×C17—42no—
150C23×C17—41no—
151C23×C17—41no—
152C23×C17—81no—
153C23×C17#12111yescharacteristic
154Q8×C17—42no—
155Q8×C17—41no—
156Q8×C17—41no—
157D4×C17—21no—
158D4×C17—21no—
159D4×C17—21no—
160D4×C17—42no—
161D4×C17#10412yes—
162C2×C4×C17—21no—
163C2×C4×C17—21no—
164C2×C4×C17—21no—
165C2×C4×C17—21no—
166C2×C4×C17—41no—
167C2×C4×C17—41no—
168C2×C4×C17—41no—
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×C17C2×C22⋊C411yescharacteristic
180C2×C4×C17C4×D411yescharacteristic
181C2×C4×C17C4×D411yescharacteristic
182C2×C4×C17#12011yescharacteristic
183C2×C4×C17#12111yescharacteristic
184C2×C4×C17#12311yescharacteristic
185C2×C4×C17C2×SD1611yescharacteristic
186C2×C4×C17#13511yescharacteristic
187C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
188C2×C4×C17#13711yescharacteristic
189C2×Dic17—42no—
190C2×Dic17—81no—
191C2×Dic17—81no—
192C8×C17—41no—
193C8×C17—41no—
194#3622—22no—
195#3622—42no—
196#3622—42no—
197#3622—41no—
198#3101D1711yescharacteristic
199D4×M16—171no—
200#1986—171no—
201C4×#102—171no—
202#1493—171no—
203#1303—171no—
204#1278—171no—
205C22×C17—21no—
206C22×C17—21no—
207C22×C17—22no—
208C22×C17—41no—
209C22×C17—41no—
210C22×C17—41no—
211C22×C17—41no—
212C22×C17—41no—
213C22×C17—82no—
214C22×C17—82no—
215C22×C17#38611yescharacteristic
216C22×C17#46111yescharacteristic
217C22×C17#46311yescharacteristic
218C4×C17—21no—
219C4×C17—21no—
220C4×C17—21no—
221C4×C17—41no—
222C4×C17—41no—
223C4×C17—81no—
224C4×C17—81no—
225C4×C17C2×#10411yescharacteristic
226C4×C17#42111yescharacteristic
227C4×C17C4×SD1611yescharacteristic
228C4×C17#44011yescharacteristic
229C4×C17#45011yescharacteristic
230C4×C17#45211yescharacteristic
231Dic17—82no—
232C22×M16—341no—
233#533C2×D1711yescharacteristic
234C2×C4×D4C2×D1711yescharacteristic
235#483—21no—
236#476Dic1711yescharacteristic, complemented
237#476Dic1711yescharacteristic, complemented
238#461Dic1711yescharacteristic, complemented
239#461Dic1711yescharacteristic, complemented
240#450—21no—
241C4×SD16—21no—
242C4×SD16—21no—
243#436—341no—
244#435—342no—
245C8×D4—341no—
246#431C2×D1711yescharacteristic
247#423—171no—
248C2×#102—341no—
249#407—341no—
250C4×M16—171no—
251C2×C4×C8—171no—
252C4×C4⋊C4—171no—
253#340—341no—
254#339—341no—
255#339—341no—
256Q8⋊C8—342no—
257C2×C17—21no—
258C2×C17—21no—
259C2×C17—42no—
260C2×C17—81no—
261C2×C17#187111yescharacteristic
262C2×C17#187711yescharacteristic
263C2×C17#188111yescharacteristic
264C22×D4C22⋊D1711yescharacteristic
265C23×C4—21no—
266C2×SD16—21no—
267C2×SD16—21no—
268C2×M16—171no—
269C2×M16—171no—
270C2×M16—342no—
271C2×M16—341no—
272C2×M16—682no—
273C2×M16—682no—
274C22×C8—342no—
275C22×C8—341no—
276C4⋊Q8C2×Dic1711yescharacteristic
277C4⋊Q8C2×Dic1711yescharacteristic
278#125—21no—
279C22⋊Q8—21no—
280C22⋊Q8—21no—
281C4×Q8—21no—
282C4×Q8—21no—
283C4×D4—21no—
284C4×D4—21no—
285C4×D4C2×Dic1711yescharacteristic
286C4×D4C2×Dic1711yescharacteristic
287C4×D4C22⋊D1711yescharacteristic
288C4×D4C22⋊D1711yescharacteristic
289#117—21no—
290C2×C4⋊C4—341no—
291C2×C4⋊C4C22⋊D1711yescharacteristic
292C2×C22⋊C4—21no—
293C2×C42—341no—
294C2×C42C22×D1711yescharacteristic
295#106—21no—
296#106—21no—
297C4⋊C8—172no—
298C4⋊C8—341no—
299C4⋊C8C2×Dic1711yescharacteristic
300C4⋊C8C2×Dic1711yescharacteristic
301#103—21no—
302#103—21no—
303#102—21no—
304#102—21no—
305#102—344no—
306C22⋊C8—21no—
307C22⋊C8—342no—
308C22⋊C8—342no—
309#97—341no—
310C4×C8—21no—
311C4×C8—341no—
312C4×C8—341no—
313C4×C8—341no—
314#95—341no—
315C17#898911yescharacteristic, complemented, Sylow
316C24—21no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×D4—22no—
320C2×D4—21no—
321C2×D4—41no—
322C2×D4C22⋊Dic1711yescharacteristic
323C2×D4C22⋊Dic1711yescharacteristic
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—681no—
334C22×C4D4×D1711yescharacteristic
335C22×C4#6325411yescharacteristic
336C22×C4C2×C22⋊D1711yescharacteristic
337SD16—42no—
338SD16—42no—
339M16—172no—
340M16—341no—
341M16—341no—
342M16—341no—
343M16—684no—
344M16—681no—
345M16—681no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—341no—
349C2×C8—342no—
350C2×C8—341no—
351C2×C8—342no—
352C2×C8—684no—
353C2×C8—681no—
354C2×C8—681no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—342no—
363C4⋊C4—681no—
364C4⋊C4C22⋊Dic1711yescharacteristic
365C4⋊C4C22⋊Dic1711yescharacteristic
366C22⋊C4—21no—
367C22⋊C4—21no—
368C22⋊C4—21no—
369C22⋊C4—21no—
370C22⋊C4—41no—
371C42—41no—
372C42—681no—
373C42—681no—
374C42D4×D1711yescharacteristic
375C42#6325411yescharacteristic
376C42C22×Dic1711yescharacteristic
377C42C2×C22⋊D1711yescharacteristic
378C23—21no—
379C23—21no—
380C23—42no—
381C23—41no—
382C23—41no—
383C23—81no—
384C23#1058762511yescharacteristic
385Q8—42no—
386Q8—41no—
387Q8—41no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—42no—
392D4#1058750812yes—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—682no—
411C2×C4—1361no—
412C2×C4—1361no—
413C2×C4SD16×D1711yescharacteristic
414C2×C4#1058760011yescharacteristic
415C2×C4#1058760111yescharacteristic
416C2×C4#1058760211yescharacteristic
417C2×C4D4×Dic1711yescharacteristic
418C2×C4D4×Dic1711yescharacteristic
419C2×C4#1058762111yescharacteristic
420C2×C4#1058762311yescharacteristic
421C2×C4#1058762511yescharacteristic
422C2×C4C2×C22⋊Dic1711yescharacteristic
423C8—41no—
424C8—41no—
425C8—342no—
426C8—682no—
427C8—682no—
428C8—681no—
429C22—21no—
430C22—21no—
431C22—22no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—82no—
438C22—82no—
439C22#1176268911yescharacteristic
440C22#1176301211yescharacteristic
441C22#1176302611yescharacteristic
442C4—21no—
443C4—21no—
444C4—21no—
445C4—41no—
446C4—41no—
447C4—81no—
448C4—81no—
449C4—1362no—
450C4C2×#1058750811yescharacteristic
451C4#1176282611yescharacteristic
452C4SD16×Dic1711yescharacteristic
453C4#1176291011yescharacteristic
454C4#1176295811yescharacteristic
455C4#1176296811yescharacteristic
456C2—21no—
457C2—21no—
458C2—42no—
459C2—81no—
460C2#1452908711yescharacteristic
461C2#1452912011yescharacteristic
462C2#1452914811yescharacteristic
subgroupG/H
centerC2×C4#10587623
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#3101C2
socleC22×C17#386
prime radicalC22×D4×C17D4
cosocleC2×C42×C17C23
hypercenter#3101D17
hypocenterC17#8989
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3101

Complements

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

NH ≅ G/Ntype
2. #14529729C2semidirect
4. #14527239C2semidirect
5. #14526686C2semidirect
6. #14526620C2semidirect
25. C17×#476C4semidirect
26. C17×#476C4semidirect
27. C17×#461C4semidirect
28. C17×#461C4semidirect
138. #63223SD16semidirect
236. #476Dic17semidirect
237. #476Dic17semidirect
238. #461Dic17semidirect
239. #461Dic17semidirect
315. C17#8989semidirect

Maximal subgroups

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

subgroupindex
D4×#632232
#145297292
C4×#105875032
#145272392
#145266862
#145266202
C17×#31012
#898917