Finite Group Databasebeta

← #16930384

Subgroups — #16930384

Order 4352.

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

normal subgroups only

This group has 762 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
1#14534520C211yesmaximal, characteristic
2#14531397C211yesmaximal, characteristic, complemented
3#14527352C211yesmaximal, characteristic, complemented
4C2×#11762532C211yesmaximal, characteristic, complemented
5#14526652C211yesmaximal, characteristic, complemented
6#14526651C211yesmaximal, characteristic, complemented
7C17×#1519C211yesmaximal, characteristic, complemented
8C2×#10587623—21no—
9C22×#63227C2211yescharacteristic
10C2×C22⋊C4×D17—21no—
11C2×C22⋊#3622C2211yescharacteristic
12#11762735C412yescomplemented
13#11762725—22no—
14#11762717C412yescomplemented
15#11762689—21no—
16#11762683—22no—
17#11762680—22no—
18#11762669C2211yescharacteristic
19#11762669C2211yescharacteristic
20#11762667—21no—
21#11762543—22no—
22#11762532—22no—
23C2×C17×#121C2211yescharacteristic
24C2×C17×C22⋊C8C2211yescharacteristic
25C17×#380C2211yescharacteristic
26C17×#327—22no—
27C24×D17D411yescharacteristic
28C22×D4×C17—21no—
29C22×D4×C17C2×C411yescharacteristic, complemented
30C23×C4×C17C2311yescharacteristic
31C22×C22⋊D17—22no—
32C23×Dic17D411yescharacteristic
33C22×C4×D17—22no—
34C22×C8×C17—21no—
35C17×#121—21no—
36C17×#121—21no—
37C17×#121—21no—
38C17×#121—21no—
39C2×C4⋊C4×C17—21no—
40C2×C4⋊C4×C17C2×C411yescharacteristic, complemented
41C2×C22⋊C4×C17—21no—
42C2×C22⋊C4×C17—21no—
43C2×C22⋊Dic17—21no—
44C2×C22⋊Dic17—21no—
45C2×C22⋊Dic17—22no—
46#10587623—42no—
47#10587623—42no—
48C2×#63227—22no—
49C2×#63227—22no—
50C2×#63227—21no—
51C2×#63227—21no—
52C2×#63227—21no—
53C2×#63227—21no—
54C2×#63227C2×C412yes—
55C2×#63227D412yes—
56C2×C4⋊Dic17—21no—
57C2×C4⋊Dic17—21no—
58C2×#63225—22no—
59C22×#3622—21no—
60C22⋊C4×D17—42no—
61C22⋊C4×D17—42no—
62C17×C22⋊C8—21no—
63C17×C22⋊C8—21no—
64C17×#95D412yescomplemented
65#10587502—22no—
66#10587502—22no—
67#10587502C2×C412yescomplemented
68#10587502D412yescomplemented
69C22⋊#3622—21no—
70C22⋊#3622—21no—
71C24×C17—21no—
72C24×C17C2×D411yescharacteristic
73C23×D17—21no—
74C23×D17—21no—
75C23×D17—21no—
76C23×D17—21no—
77C23×D17—81no—
78C23×D17—81no—
79C23×D17C22⋊C412yes—
80C23×D17D812yes—
81C23×D17SD1612yes—
82C2×D4×C17—21no—
83C2×D4×C17—21no—
84C2×D4×C17—41no—
85C2×D4×C17—41no—
86C2×D4×C17—41no—
87C2×D4×C17—41no—
88C2×D4×C17—41no—
89C2×D4×C17—41no—
90C2×D4×C17—81no—
91C22×C4×C17—21no—
92C22×C4×C17—22no—
93C22×C4×C17—81no—
94C22×C4×C17C22×C411yescharacteristic
95C22×C4×C17C2×D411yescharacteristic
96C22×C4×C17C2×D411yescharacteristic
97C22×C4×C17C2×D411yescharacteristic
98C22×C4×C17#4111yescharacteristic
99C22×C4×C17#4111yescharacteristic
100C2×C22⋊D17—44no—
101C2×C22⋊D17—82no—
102C2×C22⋊D17—82no—
103C22×Dic17—21no—
104C22×Dic17—21no—
105C22×Dic17—22no—
106C22×Dic17—22no—
107C22×Dic17—81no—
108C22×Dic17C22⋊C412yes—
109C2×C4×D17—44no—
110C2×C4×D17—82no—
111C2×C4×D17—82no—
112C2×C8×C17—41no—
113C2×C8×C17—81no—
114C4⋊C4×C17—21no—
115C4⋊C4×C17—21no—
116C4⋊C4×C17—81no—
117C22⋊C4×C17—41no—
118C22⋊C4×C17—41no—
119C22⋊C4×C17—41no—
120C22⋊C4×C17—41no—
121C22⋊Dic17—41no—
122C22⋊Dic17—41no—
123C22⋊Dic17—41no—
124C22⋊Dic17—41no—
125C22⋊Dic17—82no—
126#63227—22no—
127#63227—22no—
128#63227—22no—
129#63227—22no—
130#63227—44no—
131#63227—44no—
132C4⋊Dic17—81no—
133C4⋊Dic17—81no—
134#63225—82no—
135C2×#3622—41no—
136C2×#3622—81no—
137#9203—171nomaximal, Sylow
138C23×C17—41no—
139C23×C17—41no—
140C23×C17—42no—
141C23×C17—42no—
142C23×C17—41no—
143C23×C17—41no—
144C23×C17—41no—
145C23×C17C2×C22⋊C411yescharacteristic
146C23×C17#12011yescharacteristic
147C23×C17#12311yescharacteristic
148C23×C17C2×D811yescharacteristic
149C23×C17C2×SD1611yescharacteristic
150C23×C17AGL(1,ℤ/8)11yescharacteristic
151C23×C17#13711yescharacteristic
152C22×D17—22no—
153C22×D17—22no—
154C22×D17—22no—
155C22×D17—22no—
156C22×D17—22no—
157C22×D17—22no—
158C22×D17—42no—
159C22×D17—42no—
160C22×D17—42no—
161C22×D17—42no—
162C22×D17—82no—
163C22×D17—82no—
164C22×D17—81no—
165C22×D17—81no—
166C22×D17—82no—
167C22×D17—82no—
168C22×D17#10214yes—
169D4×C17—81no—
170D4×C17—81no—
171D4×C17—81no—
172D4×C17—81no—
173D4×C17—81no—
174D4×C17—81no—
175C2×C4×C17—41no—
176C2×C4×C17—41no—
177C2×C4×C17—41no—
178C2×C4×C17—41no—
179C2×C4×C17—41no—
180C2×C4×C17—41no—
181C2×C4×C17—42no—
182C2×C4×C17—42no—
183C2×C4×C17—41no—
184C2×C4×C17—41no—
185C2×C4×C17—41no—
186C2×C4×C17—42no—
187C2×C4×C17C4×D411yescharacteristic
188C2×C4×C17C4×D411yescharacteristic
189C2×C4×C17#12111yescharacteristic
190C2×C4×C17#12111yescharacteristic
191C22⋊D17—88no—
192C2×Dic17—22no—
193C2×Dic17—42no—
194C2×Dic17—42no—
195C2×Dic17—42no—
196C2×Dic17—42no—
197C2×Dic17—42no—
198C2×Dic17—42no—
199C2×Dic17—42no—
200C2×Dic17—42no—
201C2×Dic17—81no—
202C4×D17—88no—
203C8×C17—81no—
204#3622—81no—
205#2617—171no—
206#2315—171no—
207#1519—171no—
208#1519D1711yescharacteristic, complemented
209C2×#323—171no—
210#1289—171no—
211#1289—171no—
212C22×C17—21no—
213C22×C17—21no—
214C22×C17—21no—
215C22×C17—21no—
216C22×C17—21no—
217C22×C17—21no—
218C22×C17—41no—
219C22×C17—41no—
220C22×C17—41no—
221C22×C17—41no—
222C22×C17—82no—
223C22×C17—82no—
224C22×C17—81no—
225C22×C17—81no—
226C22×C17—81no—
227C22×C17#38611yescharacteristic
228C22×C17C2×#10211yescharacteristic
229C22×C17#41711yescharacteristic
230C22×C17#44711yescharacteristic
231C22×C17#44811yescharacteristic
232C22×C17#48011yescharacteristic
233C22×C17#48111yescharacteristic
234C2×D17—24no—
235C2×D17—44no—
236C2×D17—44no—
237C2×D17—42no—
238C2×D17—42no—
239C2×D17—44no—
240C2×D17—82no—
241C2×D17—82no—
242C2×D17—82no—
243C4×C17—21no—
244C4×C17—21no—
245C4×C17—21no—
246C4×C17—21no—
247C4×C17—82no—
248C4×C17—81no—
249Dic17—42no—
250Dic17—82no—
251Dic17—82no—
252C2×#121C2×D1711yescharacteristic
253C2×#120—341no—
254C22×C22⋊C4—171no—
255C22×C22⋊C4—341no—
256C2×C22⋊C8—171no—
257C2×C22⋊C8C2×D1711yescharacteristic
258#397—172no—
259#394—342no—
260#392—172no—
261#386—341no—
262#385—342no—
263#385—342no—
264#380—171no—
265#380—171no—
266#380—341no—
267#380C2×D1711yescharacteristic
268#327—22no—
269#327—342no—
270#323—342no—
271C2×C17—21no—
272C2×C17—21no—
273C2×C17—21no—
274C2×C17—21no—
275C2×C17—21no—
276C2×C17—21no—
277C2×C17—161no—
278C2×C17#187411yescharacteristic
279C2×C17#191111yescharacteristic
280C2×C17#191311yescharacteristic
281D17—44no—
282C25—171no—
283C22×D4—21no—
284C22×D4—342no—
285C22×D4C4×D1711yescharacteristic, complemented
286C23×C4—171no—
287C23×C4—342no—
288C23×C4C22×D1711yescharacteristic
289C22×C8—21no—
290C22×C8—341no—
291#121—21no—
292#121—21no—
293#121—21no—
294#121—21no—
295#120—682no—
296#120—682no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—342no—
299C2×C4⋊C4—341no—
300C2×C4⋊C4—341no—
301C2×C4⋊C4C4×D1711yescharacteristic, complemented
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—172no—
305C2×C22⋊C4—172no—
306C2×C22⋊C4—341no—
307C2×C22⋊C4—341no—
308C2×C22⋊C4—341no—
309C2×C22⋊C4—341no—
310C2×C22⋊C4—342no—
311C2×C22⋊C4—342no—
312C2×C22⋊C4—341no—
313C2×C22⋊C4—341no—
314C2×C22⋊C4—342no—
315C2×C22⋊C4—682no—
316C2×C22⋊C4—682no—
317C22⋊C8—21no—
318C22⋊C8—21no—
319C22⋊C8—341no—
320C22⋊C8—341no—
321#95—172no—
322#95—172no—
323#95—342no—
324#95—342no—
325#95C22⋊D1712yescomplemented
326C17#920311yescharacteristic, complemented, Sylow
327C24—21no—
328C24—172no—
329C24—172no—
330C24—172no—
331C24—341no—
332C24—341no—
333C24—341no—
334C24—341no—
335C24—1361no—
336C24—1361no—
337C24D4×D1711yescharacteristic
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—81no—
347C2×D4—684no—
348C2×D4—1362no—
349C2×D4—1362no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—81no—
353C22×C4—172no—
354C22×C4—341no—
355C22×C4—341no—
356C22×C4—342no—
357C22×C4—342no—
358C22×C4—684no—
359C22×C4—1362no—
360C22×C4—1362no—
361C22×C4—1361no—
362C22×C4C2×C4×D1711yescharacteristic
363C22×C4#6325211yescharacteristic
364C22×C4D4×D1711yescharacteristic
365C22×C4D4×D1711yescharacteristic
366C22×C4#6325411yescharacteristic
367C22×C4C2×C22⋊D1711yescharacteristic
368C2×C8—41no—
369C2×C8—81no—
370C2×C8—681no—
371C2×C8—1361no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—81no—
375C4⋊C4—1362no—
376C4⋊C4—1361no—
377C4⋊C4—1361no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—342no—
383C22⋊C4—342no—
384C22⋊C4—342no—
385C22⋊C4—342no—
386C22⋊C4—684no—
387C22⋊C4—684no—
388C22⋊C4—681no—
389C22⋊C4—681no—
390C22⋊C4—681no—
391C22⋊C4—681no—
392C22⋊C4—1362no—
393C23—41no—
394C23—41no—
395C23—42no—
396C23—42no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—174no—
401C23—342no—
402C23—342no—
403C23—342no—
404C23—342no—
405C23—342no—
406C23—342no—
407C23—682no—
408C23—682no—
409C23—682no—
410C23—682no—
411C23—1361no—
412C23—1361no—
413C23—1362no—
414C23—1362no—
415C23—1362no—
416C23—1362no—
417C23C22⋊C4×D1711yescharacteristic
418C23#1058756911yescharacteristic
419C23D8×D1711yescharacteristic
420C23#1058759711yescharacteristic
421C23SD16×D1711yescharacteristic
422C23#1058760111yescharacteristic
423C23#1058762311yescharacteristic
424D4—81no—
425D4—81no—
426D4—81no—
427D4—81no—
428D4—81no—
429D4—81no—
430D4—1368no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—342no—
444C2×C4—682no—
445C2×C4—682no—
446C2×C4—682no—
447C2×C4—682no—
448C2×C4—682no—
449C2×C4—682no—
450C2×C4—682no—
451C2×C4—682no—
452C2×C4—1368no—
453C2×C4—1361no—
454C2×C4#1058756711yescharacteristic
455C2×C4#1058757011yescharacteristic
456C2×C4C4×C22⋊D1711yescharacteristic
457C2×C4#1058762411yescharacteristic
458C8—81no—
459C8—1361no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—82no—
471C22—82no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22—344no—
476C22—684no—
477C22—684no—
478C22—684no—
479C22—682no—
480C22—682no—
481C22—1362no—
482C22—1362no—
483C22—1362no—
484C22#1176268511yescharacteristic
485C22#102×D1711yescharacteristic
486C22#1176280711yescharacteristic
487C22#1176294111yescharacteristic
488C22#1176294611yescharacteristic
489C22#1176312111yescharacteristic
490C22#1176312611yescharacteristic
491C4—21no—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—82no—
496C4—81no—
497C4—682no—
498C4—1362no—
499C4—1362no—
500C2—21no—
501C2—21no—
502C2—21no—
503C2—21no—
504C2—21no—
505C2—21no—
506C2—161no—
507C2—684no—
508C2#1452909811yescharacteristic
509C2#1452931311yescharacteristic
510C2#1452932311yescharacteristic
subgroupG/H
centerC22#11762685
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1519C2
socleC22×C17#386
prime radical#14534520C2
cosocleC23×C4×C17C23
hypercenter#1519D17
hypocenterC17#9203
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1519

Complements

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

NH ≅ G/Ntype
2. #14531397C2semidirect
3. #14527352C2semidirect
4. C2×#11762532C2semidirect
5. #14526652C2semidirect
6. #14526651C2semidirect
7. C17×#1519C2semidirect
12. #11762735C4semidirect
14. #11762717C4semidirect
29. C22×D4×C17C2×C4semidirect
40. C2×C4⋊C4×C17C2×C4semidirect
64. C17×#95D4semidirect
67. #10587502C2×C4semidirect
68. #10587502D4semidirect
208. #1519D17semidirect
285. C22×D4C4×D17semidirect
301. C2×C4⋊C4C4×D17semidirect
325. #95C22⋊D17semidirect
326. C17#9203semidirect

Maximal subgroups

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

subgroupindex
#145345202
#145313972
#145273522
C2×#117625322
#145266522
#145266512
C17×#15192
#920317