Finite Group Databasebeta

← #16929915

Subgroups — #16929915

Order 4352.

674 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762866C211yesmaximal, characteristic, complemented
2C2×#11762543C212yesmaximal
3#14527011C211yesmaximal, characteristic, complemented
4#14526662C212yesmaximal, complemented
5C17×#2854C211yesmaximal, characteristic
6#11762866—24no—
7C2×C4⋊#3622—21no—
8C2×C22⋊#3622C2212yes—
9C2×#10587474—21no—
10C2×#10587502—21no—
11C2×#10587502—21no—
12C2×#10587502C2212yes—
13#11762543—24no—
14C2×C17×#123—21no—
15C2×C17×#121C2211yescharacteristic
16C22×C42×C17C2211yescharacteristic
17C17×#390C411yescharacteristic, complemented
18C17×#390C411yescharacteristic, complemented
19C17×#389C411yescharacteristic, complemented
20C17×#389C411yescharacteristic, complemented
21C17×#388—21no—
22C17×#388—21no—
23C17×#386—21no—
24C17×#380C2211yescharacteristic
25C22×D4×C17—21no—
26C22×D4×C17C2×C411yescharacteristic
27C23×C4×C17D411yescharacteristic
28C23×C4×C17D411yescharacteristic
29C23×C4×C17C2311yescharacteristic
30C23×Dic17—22no—
31C17×#123—44no—
32C17×#121—22no—
33C17×#121—22no—
34C2×C4⋊C4×C17—21no—
35C2×C4⋊C4×C17—21no—
36C2×C4⋊C4×C17C2×C411yescharacteristic
37C2×C4⋊C4×C17C2×C411yescharacteristic
38C2×C4⋊C4×C17C2×C411yescharacteristic
39C2×C22⋊C4×C17—21no—
40C2×C22⋊C4×C17—21no—
41C2×C22⋊C4×C17—21no—
42C2×C22⋊C4×C17—21no—
43C2×C22⋊C4×C17—21no—
44C2×C22⋊C4×C17—21no—
45C2×C42×C17—22no—
46C2×C42×C17—22no—
47C2×C42×C17C2×C411yescharacteristic
48C2×C42×C17C2×C411yescharacteristic
49C2×C42×C17D411yescharacteristic
50C2×C42×C17D411yescharacteristic
51C22×#3622—22no—
52C17×#95—21no—
53C17×#95—21no—
54C17×#95—21no—
55#10587502—24no—
56#10587502—22no—
57#10587502—22no—
58#10587502—42no—
59C22⋊#3622—22no—
60C22⋊#3622D414yes—
61C4⋊#3622—24no—
62#10587474—24no—
63C24×C17—21no—
64C24×C17C2×D411yescharacteristic
65C2×D4×C17—22no—
66C2×D4×C17—41no—
67C2×D4×C17—41no—
68C2×D4×C17—42no—
69C2×D4×C17—42no—
70C2×D4×C17—41no—
71C2×D4×C17—41no—
72C22×C4×C17—22no—
73C22×C4×C17—22no—
74C22×C4×C17—22no—
75C22×C4×C17—21no—
76C22×C4×C17—21no—
77C22×C4×C17—22no—
78C22×C4×C17—21no—
79C22×C4×C17—21no—
80C22×C4×C17—21no—
81C22×C4×C17—41no—
82C22×C4×C17—41no—
83C22×C4×C17—41no—
84C22×C4×C17C22⋊C411yescharacteristic
85C22×C4×C17C22⋊C411yescharacteristic
86C22×C4×C17C22⋊C411yescharacteristic
87C22×C4×C17C22⋊C411yescharacteristic
88C22×C4×C17C22×C411yescharacteristic
89C22×C4×C17C2×D412yes—
90C22×C4×C17C2×D411yescharacteristic
91C22×C4×C17#4112yes—
92C22×Dic17—24no—
93C22×Dic17—42no—
94C22×Dic17—42no—
95C22×Dic17—42no—
96C22×Dic17—42no—
97C22×Dic17—42no—
98C22×Dic17—42no—
99C4⋊C4×C17—22no—
100C4⋊C4×C17—41no—
101C4⋊C4×C17—41no—
102C4⋊C4×C17—42no—
103C4⋊C4×C17—42no—
104C22⋊C4×C17—42no—
105C22⋊C4×C17—42no—
106C22⋊C4×C17—42no—
107C22⋊C4×C17—42no—
108C22⋊C4×C17—42no—
109C22⋊C4×C17—41no—
110C22⋊C4×C17—41no—
111C42×C17—22no—
112C42×C17—22no—
113C42×C17—21no—
114C42×C17—21no—
115C42×C17—21no—
116C42×C17—21no—
117C2×#3622—24no—
118C2×#3622—42no—
119C2×#3622—42no—
120#9118—171nomaximal, Sylow
121C23×C17—21no—
122C23×C17—21no—
123C23×C17—21no—
124C23×C17—21no—
125C23×C17—41no—
126C23×C17—41no—
127C23×C17—41no—
128C23×C17—41no—
129C23×C17—41no—
130C23×C17—41no—
131C23×C17—41no—
132C23×C17C2×C22⋊C411yescharacteristic
133C23×C17#12112yes—
134C23×C17AGL(1,ℤ/8)12yes—
135C23×C17AGL(1,ℤ/8)11yescharacteristic
136C23×C17#13711yescharacteristic
137D4×C17—82no—
138D4×C17—82no—
139D4×C17—82no—
140C2×C4×C17—21no—
141C2×C4×C17—21no—
142C2×C4×C17—21no—
143C2×C4×C17—21no—
144C2×C4×C17—21no—
145C2×C4×C17—21no—
146C2×C4×C17—21no—
147C2×C4×C17—21no—
148C2×C4×C17—21no—
149C2×C4×C17—21no—
150C2×C4×C17—21no—
151C2×C4×C17—21no—
152C2×C4×C17—21no—
153C2×C4×C17—21no—
154C2×C4×C17—21no—
155C2×C4×C17—42no—
156C2×C4×C17—42no—
157C2×C4×C17—44no—
158C2×C4×C17—42no—
159C2×C4×C17—42no—
160C2×C4×C17—41no—
161C2×C4×C17—41no—
162C2×C4×C17—41no—
163C2×C4×C17—41no—
164C2×C4×C17—41no—
165C2×C4×C17—41no—
166C2×C4×C17—41no—
167C2×C4×C17—41no—
168C2×C4×C17—41no—
169C2×C4×C17#10412yes—
170C2×C4×C17C4×D412yes—
171C2×C4×C17#12011yescharacteristic
172C2×C4×C17#12311yescharacteristic
173C2×Dic17—44no—
174C2×Dic17—44no—
175C2×Dic17—44no—
176C2×Dic17—82no—
177C2×Dic17—82no—
178C2×Dic17—82no—
179C2×Dic17—82no—
180#3622—44no—
181C2×#432—171no—
182#2854D1711yescharacteristic
183C2×#327—172no—
184#1434—171no—
185#1294—172no—
186C22×C17—22no—
187C22×C17—22no—
188C22×C17—22no—
189C22×C17—21no—
190C22×C17—21no—
191C22×C17—21no—
192C22×C17—21no—
193C22×C17—22no—
194C22×C17—81no—
195C22×C17—81no—
196C22×C17—81no—
197C22×C17—81no—
198C22×C17—81no—
199C22×C17—81no—
200C22×C17—81no—
201C22×C17#38611yescharacteristic
202C22×C17#41011yescharacteristic
203C22×C17#41011yescharacteristic
204C22×C17#41711yescharacteristic
205C22×C17#41811yescharacteristic
206C22×C17C2×#10411yescharacteristic
207C22×C17#42111yescharacteristic
208C22×C17#46812yes—
209C22×C17#46912yes—
210C4×C17—22no—
211C4×C17—21no—
212C4×C17—22no—
213C4×C17—21no—
214C4×C17—41no—
215C4×C17—41no—
216C4×C17—41no—
217C4×C17—81no—
218C4×C17—81no—
219C4×C17—81no—
220Dic17—84no—
221C2×#123—21no—
222C2×#121C2×D1711yescharacteristic
223C22×C42C2×D1711yescharacteristic
224#432—344no—
225C2×C4⋊C8—341no—
226C2×C22⋊C8—172no—
227C2×#97—341no—
228#390Dic1711yescharacteristic, complemented
229#390Dic1711yescharacteristic, complemented
230#389Dic1711yescharacteristic, complemented
231#389Dic1711yescharacteristic, complemented
232#388—21no—
233#388—21no—
234#386—21no—
235#380C2×D1711yescharacteristic
236C2×#95—172no—
237C2×#95—341no—
238C2×#95—341no—
239#327—344no—
240C2×C17—22no—
241C2×C17—22no—
242C2×C17—161no—
243C2×C17#150511yescharacteristic
244C2×C17#150911yescharacteristic
245C2×C17#151211yescharacteristic
246C2×C17#151411yescharacteristic
247C2×C17#187111yescharacteristic
248C2×C17#189511yescharacteristic
249C2×C17#189911yescharacteristic
250C22×D4—21no—
251C22×D4C2×Dic1711yescharacteristic
252C23×C4—342no—
253C23×C4C22⋊D1711yescharacteristic
254C23×C4C22⋊D1711yescharacteristic
255C23×C4C22×D1711yescharacteristic
256C22×C8—342no—
257#123—44no—
258#121—22no—
259#121—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4C2×Dic1711yescharacteristic
263C2×C4⋊C4C2×Dic1711yescharacteristic
264C2×C4⋊C4C2×Dic1711yescharacteristic
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42C2×Dic1711yescharacteristic
274C2×C42C2×Dic1711yescharacteristic
275C2×C42C22⋊D1711yescharacteristic
276C2×C42C22⋊D1711yescharacteristic
277C4⋊C8—344no—
278C22⋊C8—174no—
279C22⋊C8—342no—
280#97—344no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—344no—
285#95—342no—
286#95—342no—
287#95—682no—
288C17#911811yescharacteristic, complemented, Sylow
289C24—21no—
290C24C2×C22⋊D1711yescharacteristic
291C2×D4—22no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—41no—
297C2×D4—41no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—344no—
311C22×C4—682no—
312C22×C4—682no—
313C22×C4—682no—
314C22×C4—682no—
315C22×C4—682no—
316C22×C4—682no—
317C22×C4C22⋊Dic1711yescharacteristic
318C22×C4C22⋊Dic1711yescharacteristic
319C22×C4C22⋊Dic1711yescharacteristic
320C22×C4C22⋊Dic1711yescharacteristic
321C22×C4D4×D1712yes—
322C22×C4#6325412yes—
323C22×C4C22×Dic1711yescharacteristic
324C22×C4C2×C22⋊D1711yescharacteristic
325C2×C8—344no—
326C2×C8—682no—
327C2×C8—682no—
328C4⋊C4—22no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C42—22no—
341C42—22no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23#1058761812yes—
358C23#1058762512yes—
359C23#1058763511yescharacteristic
360C23#1058763711yescharacteristic
361C23C2×C22⋊Dic1711yescharacteristic
362D4—82no—
363D4—82no—
364D4—82no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—684no—
395C2×C4—684no—
396C2×C4—684no—
397C2×C4—1362no—
398C2×C4—1362no—
399C2×C4—1362no—
400C2×C4—1362no—
401C2×C4#1058750812yes—
402C2×C4D4×Dic1712yes—
403C2×C4#1058762111yescharacteristic
404C2×C4#1058762311yescharacteristic
405C8—684no—
406C22—22no—
407C22—22no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—22no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22#1176268911yescharacteristic
422C22#1176278311yescharacteristic
423C22#1176278311yescharacteristic
424C22#1176281011yescharacteristic
425C22#1176281411yescharacteristic
426C22C2×#1058750811yescharacteristic
427C22#1176282611yescharacteristic
428C22#1176305312yes—
429C22#1176306012yes—
430C4—22no—
431C4—21no—
432C4—22no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—81no—
438C4—81no—
439C4—81no—
440C4—1364no—
441C2—22no—
442C2—22no—
443C2—161no—
444C2#1452729211yescharacteristic
445C2#1452730911yescharacteristic
446C2#1452732211yescharacteristic
447C2#1452733111yescharacteristic
448C2#1452908711yescharacteristic
449C2#1452922311yescharacteristic
450C2#1452924311yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic17
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#2854C2
socleC23×C17C2×C22⋊C4
prime radicalC2×C17×#121C22
cosocleC23×C4×C17C23
hypercenter#2854D17
hypocenterC17#9118
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2854

Complements

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

NH ≅ G/Ntype
1. C2×#11762866C2semidirect
3. #14527011C2semidirect
4. #14526662C2semidirect
17. C17×#390C4semidirect
18. C17×#390C4semidirect
19. C17×#389C4semidirect
20. C17×#389C4semidirect
228. #390Dic17semidirect
229. #390Dic17semidirect
230. #389Dic17semidirect
231. #389Dic17semidirect
288. C17#9118semidirect

Maximal subgroups

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

subgroupindex
C2×#117628662
C2×#117625432
#145270112
#145266622
C17×#28542
#911817