Finite Group Databasebeta

← #16934603

Subgroups — #16934603

Order 4352.

734 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 113 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×#11763137C211yesmaximal, characteristic, complemented
2#14534591C211yesmaximal, characteristic
3C2×#11762736C211yesmaximal, characteristic
4#14528871C211yesmaximal, characteristic, complemented
5C2×#11762532C211yesmaximal, characteristic
6#14526663C211yesmaximal, characteristic, complemented
7C2×C17×#328C211yesmaximal, characteristic, complemented
8C22×C4⋊Dic17C2211yescharacteristic
9C22×#63227—21no—
10C22×#63227C2211yescharacteristic
11#11763137—22no—
12#11763137—22no—
13C2×#10587479—21no—
14C2×#10587506—21no—
15C2×C22⋊#3622C2211yescharacteristic
16#11762744—21no—
17#11762736—22no—
18#11762736—22no—
19#11762731—22no—
20#11762720—21no—
21#11762681—21no—
22#11762669C2211yescharacteristic
23#11762664—21no—
24C2×#10587502—21no—
25#11762572—22no—
26#11762532—22no—
27C2×C17×C22⋊Q8C2211yescharacteristic
28C2×C17×C22⋊C8C2211yescharacteristic
29C2×C17×#95C2211yescharacteristic
30C17×#328—22no—
31C24×D17D411yescharacteristic
32C22×Q8×C17D411yescharacteristic
33C23×C4×C17—21no—
34C23×C4×C17C2311yescharacteristic
35C23×Dic17—21no—
36C23×Dic17D411yescharacteristic
37C22×C8×C17—21no—
38C17×C22⋊Q8—21no—
39C17×C22⋊Q8—21no—
40C17×C22⋊Q8—21no—
41C17×C22⋊Q8—21no—
42C2×C4⋊C4×C17—21no—
43C2×C4⋊C4×C17D411yescharacteristic
44C2×C22⋊C4×C17—21no—
45C2×C22⋊Dic17—21no—
46C2×#63227—24no—
47C2×#63227—22no—
48C2×#63227—22no—
49C2×#63227—21no—
50C2×#63227—21no—
51C2×#63227—21no—
52C2×#63227—21no—
53C2×#63227—21no—
54C2×#63227—21no—
55C2×#63227—42no—
56C2×#63227—42no—
57C2×C4⋊Dic17—21no—
58C2×C4⋊Dic17—21no—
59C2×C4⋊Dic17—21no—
60C2×C4⋊Dic17—21no—
61C2×C4⋊Dic17—21no—
62C2×C4⋊Dic17D412yes—
63C2×C4⋊Dic17D412yes—
64C2×#63225—22no—
65C22×#3622—21no—
66C17×C22⋊C8—21no—
67C17×C22⋊C8—21no—
68C17×#95—21no—
69C17×#95D412yes—
70#10587506—24no—
71#10587502—22no—
72#10587502—24no—
73#10587502—21no—
74C22⋊#3622—21no—
75C22⋊#3622D412yes—
76#10587479—24no—
77C24×C17C2×D411yescharacteristic
78C23×D17—21no—
79C23×D17—21no—
80C23×D17—21no—
81C23×D17—21no—
82C23×D17—21no—
83C23×D17—21no—
84C23×D17—42no—
85C23×D17—42no—
86C23×D17SD1612yes—
87C2×Q8×C17—21no—
88C2×Q8×C17—21no—
89C2×Q8×C17—42no—
90C22×C4×C17—22no—
91C22×C4×C17—21no—
92C22×C4×C17—21no—
93C22×C4×C17—42no—
94C22×C4×C17—42no—
95C22×C4×C17—42no—
96C22×C4×C17—41no—
97C22×C4×C17—41no—
98C22×C4×C17C2×D411yescharacteristic
99C22×C4×C17C2×D411yescharacteristic
100C22×C4×C17C2×D411yescharacteristic
101C22×C4×C17C2×D411yescharacteristic
102C22×C4×C17C2×D411yescharacteristic
103C22×C4×C17#4111yescharacteristic
104C22×Dic17—22no—
105C22×Dic17—21no—
106C22×Dic17—21no—
107C22×Dic17—22no—
108C22×Dic17—21no—
109C22×Dic17—42no—
110C22×Dic17—41no—
111C22×Dic17—41no—
112C22×Dic17—41no—
113C22×Dic17—41no—
114C22×Dic17—41no—
115C22×Dic17—41no—
116C2×C8×C17—42no—
117C2×C8×C17—41no—
118C4⋊C4×C17—21no—
119C4⋊C4×C17—21no—
120C4⋊C4×C17—41no—
121C4⋊C4×C17—41no—
122C22⋊C4×C17—41no—
123C22⋊C4×C17—41no—
124C22⋊Dic17—41no—
125C22⋊Dic17—41no—
126#63227—48no—
127#63227—41no—
128#63227—41no—
129#63227—41no—
130#63227—41no—
131#63227—82no—
132#63227—82no—
133C4⋊Dic17—22no—
134C4⋊Dic17—22no—
135C4⋊Dic17—41no—
136C4⋊Dic17—41no—
137C4⋊Dic17—42no—
138#63225—82no—
139C2×#3622—22no—
140C2×#3622—42no—
141#9926—171nomaximal, Sylow
142C23×C17—22no—
143C23×C17—22no—
144C23×C17#12011yescharacteristic
145C23×C17#12011yescharacteristic
146C23×C17#12111yescharacteristic
147C23×C17C2×SD1611yescharacteristic
148C23×C17#13511yescharacteristic
149C23×C17AGL(1,ℤ/8)11yescharacteristic
150C23×C17#13711yescharacteristic
151C22×D17—22no—
152C22×D17—22no—
153C22×D17—22no—
154C22×D17—44no—
155C22×D17—44no—
156C22×D17—42no—
157C22×D17—42no—
158C22×D17—41no—
159C22×D17—41no—
160C22×D17—41no—
161C22×D17—42no—
162C22×D17—42no—
163C22×D17—41no—
164C22×D17—82no—
165C22×D17—82no—
166C22×D17—81no—
167C22×D17—81no—
168Q8×C17—42no—
169Q8×C17—81no—
170C2×C4×C17—22no—
171C2×C4×C17—22no—
172C2×C4×C17—41no—
173C2×C4×C17—41no—
174C2×C4×C17—41no—
175C2×C4×C17—41no—
176C2×C4×C17—42no—
177C2×C4×C17—42no—
178C2×C4×C17—41no—
179C2×C4×C17—41no—
180C2×C4×C17—42no—
181C2×C4×C17—42no—
182C2×C4×C17—42no—
183C2×C4×C17—82no—
184C2×C4×C17—82no—
185C2×C4×C17#12011yescharacteristic
186C2×C4×C17#12111yescharacteristic
187C2×C4×C17#12111yescharacteristic
188C2×C4×C17#12711yescharacteristic
189C2×Dic17—42no—
190C2×Dic17—41no—
191C2×Dic17—42no—
192C2×Dic17—42no—
193C2×Dic17—42no—
194C2×Dic17—42no—
195C2×Dic17—42no—
196C2×Dic17—42no—
197C2×Dic17—42no—
198C2×Dic17—81no—
199C2×Dic17—81no—
200C2×Dic17—81no—
201C2×Dic17—81no—
202C2×Dic17—81no—
203C8×C17—81no—
204#3622—42no—
205C2×#483—171no—
206#2626—171no—
207C2×#397—171no—
208#1823—171no—
209C2×#328D1711yescharacteristic, complemented
210C2×#323—171no—
211#1294—171no—
212C22×C17—22no—
213C22×C17—22no—
214C22×C17—22no—
215C22×C17—22no—
216C22×C17—21no—
217C22×C17—21no—
218C22×C17—21no—
219C22×C17—21no—
220C22×C17#39211yescharacteristic
221C22×C17#44811yescharacteristic
222C22×C17#44911yescharacteristic
223C22×C17#45011yescharacteristic
224C22×C17#45211yescharacteristic
225C22×C17#45512yes—
226C22×C17#45812yes—
227C22×C17#46511yescharacteristic
228C22×C17#46811yescharacteristic
229C2×D17—44no—
230C2×D17—44no—
231C2×D17—42no—
232C2×D17—82no—
233C2×D17—82no—
234C2×D17—82no—
235C2×D17—81no—
236C2×D17—81no—
237C2×D17—82no—
238C4×C17—21no—
239C4×C17—21no—
240C4×C17—21no—
241C4×C17—21no—
242C4×C17—82no—
243C4×C17—81no—
244C4×C17—81no—
245Dic17—82no—
246Dic17—82no—
247Dic17—81no—
248C2×C22⋊Q8C2×D1711yescharacteristic
249C22×C4⋊C4—171no—
250C22×C22⋊C4—171no—
251C22×C22⋊C4—341no—
252#483—342no—
253#483—342no—
254C2×#106—341no—
255C2×#103—341no—
256C2×C22⋊C8—171no—
257C2×C22⋊C8C2×D1711yescharacteristic
258#399—341no—
259#397—342no—
260#397—342no—
261#396—342no—
262#393—341no—
263#385—341no—
264#380—171no—
265#379—341no—
266C2×#95—341no—
267C2×#95C2×D1711yescharacteristic
268#340—342no—
269#328—22no—
270#323—342no—
271C2×C17—21no—
272C2×C17—21no—
273C2×C17—21no—
274C2×C17—21no—
275C2×C17#160812yes—
276C2×C17#161212yes—
277C2×C17#200011yescharacteristic
278C2×C17#200411yescharacteristic
279C2×C17#202211yescharacteristic
280D17—82no—
281C25—171no—
282C22×Q8D6811yescharacteristic
283C23×C4—21no—
284C23×C4—171no—
285C23×C4—341no—
286C23×C4C22×D1711yescharacteristic
287C22×C8—21no—
288C22×C8—341no—
289C22⋊Q8—21no—
290C22⋊Q8—21no—
291C22⋊Q8—21no—
292C22⋊Q8—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—172no—
295C2×C4⋊C4—172no—
296C2×C4⋊C4—341no—
297C2×C4⋊C4—341no—
298C2×C4⋊C4—341no—
299C2×C4⋊C4—341no—
300C2×C4⋊C4—341no—
301C2×C4⋊C4—342no—
302C2×C4⋊C4D6811yescharacteristic
303C2×C22⋊C4—21no—
304C2×C22⋊C4—344no—
305C2×C22⋊C4—341no—
306C2×C22⋊C4—341no—
307C2×C22⋊C4—341no—
308C2×C22⋊C4—341no—
309C2×C22⋊C4—341no—
310C2×C22⋊C4—341no—
311C2×C22⋊C4—341no—
312C2×C22⋊C4—342no—
313C2×C22⋊C4—342no—
314C2×C22⋊C4—682no—
315C2×C22⋊C4—682no—
316#106—344no—
317#103—344no—
318C22⋊C8—21no—
319C22⋊C8—21no—
320C22⋊C8—172no—
321C22⋊C8—341no—
322#95—21no—
323#95—344no—
324#95—342no—
325#95—341no—
326#95C22⋊D1712yes—
327C17#992611yescharacteristic, complemented, Sylow
328C24—172no—
329C24—341no—
330C24—341no—
331C24—341no—
332C24—341no—
333C24—341no—
334C24—341no—
335C24—682no—
336C24—682no—
337C24D4×D1711yescharacteristic
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—42no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—342no—
350C22×C4—341no—
351C22×C4—341no—
352C22×C4—342no—
353C22×C4—341no—
354C22×C4—682no—
355C22×C4—681no—
356C22×C4—681no—
357C22×C4—681no—
358C22×C4—681no—
359C22×C4—681no—
360C22×C4—681no—
361C22×C4C2×D6811yescharacteristic
362C22×C4#6325211yescharacteristic
363C22×C4D4×D1711yescharacteristic
364C22×C4D4×D1711yescharacteristic
365C22×C4D4×D1711yescharacteristic
366C22×C4C2×C22⋊D1711yescharacteristic
367C2×C8—42no—
368C2×C8—41no—
369C2×C8—342no—
370C2×C8—682no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—342no—
376C4⋊C4—342no—
377C4⋊C4—682no—
378C4⋊C4—681no—
379C4⋊C4—681no—
380C4⋊C4—1362no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—688no—
384C22⋊C4—681no—
385C22⋊C4—681no—
386C22⋊C4—681no—
387C22⋊C4—681no—
388C22⋊C4—681no—
389C22⋊C4—681no—
390C22⋊C4—1362no—
391C22⋊C4—1362no—
392C23—22no—
393C23—22no—
394C23—342no—
395C23—342no—
396C23—342no—
397C23—684no—
398C23—682no—
399C23—684no—
400C23—682no—
401C23—681no—
402C23—681no—
403C23—681no—
404C23—682no—
405C23—682no—
406C23—681no—
407C23—1362no—
408C23—1361no—
409C23—1362no—
410C23—1361no—
411C23#1058756811yescharacteristic
412C23#1058757011yescharacteristic
413C23SD16×D1711yescharacteristic
414C23#1058760011yescharacteristic
415C23#1058760411yescharacteristic
416C23#1058760511yescharacteristic
417C23#1058762311yescharacteristic
418Q8—42no—
419Q8—81no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—682no—
436C2×C4—682no—
437C2×C4—682no—
438C2×C4—682no—
439C2×C4—681no—
440C2×C4—682no—
441C2×C4—682no—
442C2×C4—682no—
443C2×C4—682no—
444C2×C4—1361no—
445C2×C4—1361no—
446C2×C4—1361no—
447C2×C4—1361no—
448C2×C4—1361no—
449C2×C4#1058756811yescharacteristic
450C2×C4#1058757011yescharacteristic
451C2×C4#1058761611yescharacteristic
452C2×C4#1058762611yescharacteristic
453C8—81no—
454C8—682no—
455C22—21no—
456C22—21no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—21no—
462C22—21no—
463C22—684no—
464C22—684no—
465C22—682no—
466C22—1362no—
467C22—1362no—
468C22—1362no—
469C22—1361no—
470C22—1362no—
471C22—1361no—
472C22#1176271811yescharacteristic
473C22#1176294911yescharacteristic
474C22#1176295711yescharacteristic
475C22#1176295811yescharacteristic
476C22#1176297011yescharacteristic
477C22#1176298812yes—
478C22#1176300012yes—
479C22#1176304111yescharacteristic
480C22#1176305811yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—82no—
486C4—81no—
487C4—81no—
488C4—1361no—
489C4—1362no—
490C4—1362no—
491C2—21no—
492C2—21no—
493C2—21no—
494C2—21no—
495C2—1362no—
496C2#1452777612yes—
497C2#1452780112yes—
498C2#1452978611yescharacteristic
499C2#1452981211yescharacteristic
500C2#1452991411yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC2×C17×#328C2
socleC23×C17#120
prime radicalC24×D17D4
hypercenterC2×#328D17
hypocenterC17#9926
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
1. C2×#11763137C2semidirect
4. #14528871C2semidirect
6. #14526663C2semidirect
7. C2×C17×#328C2semidirect
209. C2×#328D17semidirect
327. C17#9926semidirect

Maximal subgroups

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

subgroupindex
C2×#117631372
#145345912
C2×#117627362
#145288712
C2×#117625322
#145266632
C2×C17×#3282
#992617