Finite Group Databasebeta

← #17034019

Subgroups — #17034019

Order 4352.

1330 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763575C211yesmaximal, characteristic
2C2×#11762858C211yesmaximal, characteristic, complemented
3#14537301C211yesmaximal, characteristic
4C2×#11762808C211yesmaximal, characteristic
5#14528849C214yesmaximal, complemented
6#14528844C214yesmaximal, complemented
7#14528523C211yesmaximal, characteristic, complemented
8C2×#11762578C211yesmaximal, characteristic, complemented
9C2×C17×#428C211yesmaximal, characteristic, complemented
10C22×#63252C2211yescharacteristic
11#11763575—24no—
12#11763575C418yescomplemented
13C2×#10587593—22no—
14C2×#10587587C2214yes—
15C22×#63223C2211yescharacteristic
16C22×#63223C2211yescharacteristic
17C22×#63223C2211yescharacteristic
18C2×M16×D17—22no—
19#11762858—24no—
20C2×#10587490C2214yescomplemented
21#11762817—21no—
22#11762813—21no—
23#11762808—22no—
24#11762808C2212yes—
25#11762808C2214yes—
26#11762808C2212yes—
27#11762803—22no—
28C2×#10587481—21no—
29C2×#10587480—21no—
30#11762578C2214yescomplemented
31#11762578C2212yescomplemented
32#11762578C2212yescomplemented
33C22×M16×C17C2211yescharacteristic
34C2×C17×#117C2211yescharacteristic
35C22×C4⋊C4×C17C2211yescharacteristic
36C17×#428—22no—
37C17×#428—22no—
38C2×C17×#107C2212yescomplemented
39C2×C17×#106C2212yescomplemented
40C23×C4×C17—21no—
41C23×C4×C17C2311yescharacteristic
42C22×C22⋊D17—21no—
43C2×#63252—21no—
44C2×#63252—22no—
45C2×#63252—21no—
46C2×#63252—22no—
47C2×#63252C2×C414yes—
48C2×#63252D412yes—
49C2×#63252Q814yes—
50C2×#63252C2312yes—
51C22×D68D411yescharacteristic
52C22×C4×D17—21no—
53C22×Dic34D411yescharacteristic
54C2×M16×C17C2×C414yes—
55C2×M16×C17D414yes—
56C2×M16×C17D412yes—
57C2×M16×C17D412yes—
58C22×C8×C17C2312yes—
59C17×#117—21no—
60C17×#117—21no—
61C17×#117—21no—
62C17×#117—21no—
63C2×C4⋊C4×C17—21no—
64C2×C4⋊C4×C17—21no—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17—21no—
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17C2312yes—
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C22⋊C4×C17—21no—
72C2×C42×C17—21no—
73C2×#63223—21no—
74C2×#63223—22no—
75C2×#63223—21no—
76C2×#63223—21no—
77C2×#63223—22no—
78C2×#63223—22no—
79C2×#63223—22no—
80C2×#63223—22no—
81C2×#63223C2×C414yes—
82C2×#63223C2312yes—
83C2×#63223C2312yes—
84C2×#63223C2312yes—
85C22×#3622—21no—
86C22×#3622—21no—
87C22×#3622—21no—
88#10587593—216no—
89M16×D17—216no—
90#10587587—28no—
91#10587587C2×C4116yescomplemented
92C2×#63218—24no—
93C2×C8×D17—24no—
94C17×#107—24no—
95C17×#106—24no—
96#10587490—28no—
97#10587481—24no—
98#10587481—22no—
99#10587481—22no—
100#10587480—24no—
101#10587480—22no—
102#10587480—22no—
103C24×C17#4111yescharacteristic
104C23×D17—21no—
105C22×C4×C17—22no—
106C22×C4×C17—21no—
107C22×C4×C17—21no—
108C22×C4×C17—41no—
109C22×C4×C17—41no—
110C22×C4×C17—82no—
111C22×C4×C17C22×C412yes—
112C22×C4×C17C2×D412yes—
113C22×C4×C17C2×D411yescharacteristic
114C22×C4×C17C2×D411yescharacteristic
115C22×C4×C17C2×D411yescharacteristic
116C22×C4×C17C2×D411yescharacteristic
117C22×C4×C17C2×Q812yes—
118C22×C4×C17#4111yescharacteristic
119C22×C4×C17#4112yes—
120C22×C4×C17C2411yescharacteristic
121C2×C22⋊D17—24no—
122C2×C22⋊D17—22no—
123C2×C22⋊D17—24no—
124C2×C22⋊D17—22no—
125C22×Dic17—21no—
126#63252—24no—
127#63252—24no—
128#63252—24no—
129#63252—24no—
130#63252—24no—
131#63252—22no—
132#63252—22no—
133#63252C4⋊C418yes—
134#63252C22×C414yes—
135#63252C2×Q814yes—
136C2×D68—22no—
137C2×D68—21no—
138C2×D68C4⋊C414yes—
139C2×D68C2×D412yes—
140C2×C4×D17—24no—
141C2×C4×D17—22no—
142C2×C4×D17—24no—
143C2×C4×D17—22no—
144C2×Dic34—22no—
145C2×Dic34—21no—
146C2×Dic34C4⋊C414yes—
147C2×Dic34C2×D412yes—
148M16×C17C22⋊C418yes—
149M16×C17C22⋊C418yes—
150C2×C8×C17C22×C414yes—
151C2×C8×C17C2×D414yes—
152C2×C8×C17C2×D414yes—
153C4⋊C4×C17—22no—
154C4⋊C4×C17—21no—
155C4⋊C4×C17—21no—
156C4⋊C4×C17—21no—
157C4⋊C4×C17—22no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—41no—
160C4⋊C4×C17—41no—
161C22⋊C4×C17—41no—
162C22⋊C4×C17—41no—
163C42×C17—41no—
164C42×C17—41no—
165#63223—24no—
166#63223—22no—
167#63223—22no—
168#63223—22no—
169#63223—22no—
170#63223—22no—
171#63223—42no—
172#63223—42no—
173#63223C22×C414yes—
174C2×#3622—24no—
175C2×#3622—22no—
176C2×#3622—22no—
177C2×#3622—22no—
178C2×#3622—42no—
179C2×#3622—42no—
180#63218—216no—
181C8×D17—216no—
182#20679—171nomaximal, Sylow
183C23×C17—41no—
184C23×C17—41no—
185C23×C17#11712yes—
186C23×C17#12111yescharacteristic
187C23×C17#12111yescharacteristic
188C23×C17C22⋊Q812yes—
189C23×C17C2×#4111yescharacteristic
190C22×D17—22no—
191C22×D17—24no—
192C22×D17—81no—
193C22×D17—81no—
194C2×C4×C17—41no—
195C2×C4×C17—41no—
196C2×C4×C17—42no—
197C2×C4×C17—82no—
198C2×C4×C17—82no—
199C2×C4×C17—81no—
200C2×C4×C17—81no—
201C2×C4×C17—82no—
202C2×C4×C17C2×C22⋊C412yes—
203C2×C4×C17C2×C22⋊C412yes—
204C2×C4×C17C2×C4⋊C412yes—
205C2×C4×C17C2×C4⋊C412yes—
206C2×C4×C17#11712yes—
207C2×C4×C17#12112yes—
208C2×C4×C17#12112yes—
209C2×C4×C17#12111yescharacteristic
210C2×C4×C17#12111yescharacteristic
211C2×C4×C17C22⋊Q812yes—
212C2×C4×C17C22⋊Q812yes—
213C2×C4×C17C22⋊Q812yes—
214C2×C4×C17C23×C411yescharacteristic
215C2×C4×C17C22×D411yescharacteristic
216C2×C4×C17C22×D411yescharacteristic
217C2×C4×C17C22×D411yescharacteristic
218C2×C4×C17C22×Q811yescharacteristic
219C2×C4×C17C2×#4111yescharacteristic
220C22⋊D17—24no—
221C22⋊D17—24no—
222C22⋊D17—28no—
223C2×Dic17—22no—
224C2×Dic17—24no—
225D68—22no—
226D68—24no—
227D68C2×C4⋊C414yes—
228C4×D17—24no—
229C4×D17—24no—
230C4×D17—28no—
231Dic34—22no—
232Dic34—24no—
233Dic34C2×C4⋊C414yes—
234C8×C17C2×C22⋊C418yes—
235#3622—24no—
236#3622—42no—
237#3622—42no—
238C2×#568—171no—
239C2×#430—171no—
240C2×#428D1711yescharacteristic, complemented
241#2896—171no—
242C2×#417—171no—
243#1816—174no—
244#1815—174no—
245#1742—171no—
246C2×#343—171no—
247C22×C17—21no—
248C22×C17—21no—
249C22×C17—21no—
250C22×C17—21no—
251C22×C17—21no—
252C22×C17—21no—
253C22×C17—42no—
254C22×C17—42no—
255C22×C17#38012yes—
256C22×C17#38012yes—
257C22×C17C2×#11711yescharacteristic
258C22×C17C2×#12111yescharacteristic
259C22×C17C2×C22⋊Q811yescharacteristic
260C2×D17—24no—
261C2×D17—42no—
262C2×D17—42no—
263C2×D17—84no—
264C4×C17—81no—
265C4×C17—82no—
266C4×C17—81no—
267C4×C17#38012yes—
268C4×C17#38012yes—
269C4×C17C22×C22⋊C411yescharacteristic
270C4×C17C22×C4⋊C411yescharacteristic
271C4×C17C2×#12111yescharacteristic
272C4×C17C2×C22⋊Q811yescharacteristic
273Dic17—24no—
274C22×#41—171no—
275#568—178no—
276#568—344no—
277C2×#131—174no—
278C2×#131—342no—
279C22×M16—171no—
280C22×M16—171no—
281C22×M16—171no—
282C22×M16—342no—
283C22×M16C2×D1711yescharacteristic
284C2×#117C2×D1711yescharacteristic
285C22×C4⋊C4C2×D1711yescharacteristic
286#430—344no—
287C2×#108—174no—
288#428—22no—
289#428—22no—
290C2×#107C2×D1712yescomplemented
291C2×#106C2×D1712yescomplemented
292#419—341no—
293#418—341no—
294#417—172no—
295#417—174no—
296#417—172no—
297#417—342no—
298#416—342no—
299C2×#103—341no—
300C2×#102—341no—
301#343—172no—
302#343—174no—
303#343—172no—
304C2×C17—22no—
305C2×C17—21no—
306C2×C17—21no—
307C2×C17—22no—
308C2×C17C2×#38011yescharacteristic
309C2×C17#306811yescharacteristic
310C2×C17#306911yescharacteristic
311D17—44no—
312C2×#41—174no—
313C2×#41—172no—
314C2×#41—174no—
315C2×#41—172no—
316C2×#41—342no—
317C2×#41—341no—
318C2×#41—342no—
319C2×#41—341no—
320C22×Q8—171no—
321C22×D4—171no—
322C22×D4—341no—
323C23×C4—21no—
324C23×C4—341no—
325C23×C4C22×D1711yescharacteristic
326#131—1716no—
327#131—3416no—
328#131—348no—
329C2×M16—174no—
330C2×M16—172no—
331C2×M16—172no—
332C2×M16—172no—
333C2×M16—342no—
334C2×M16—341no—
335C2×M16—3416no—
336C2×M16—341no—
337C2×M16—341no—
338C2×M16—344no—
339C2×M16—342no—
340C2×M16—342no—
341C2×M16—342no—
342C2×M16—342no—
343C2×M16C4×D1714yes—
344C2×M16D6814yes—
345C2×M16C22⋊D1712yes—
346C2×M16C22⋊D1712yes—
347C22×C8—341no—
348C22×C8—344no—
349C22×C8—341no—
350C22×C8—341no—
351C22×C8C22×D1712yes—
352#117—21no—
353#117—21no—
354#117—21no—
355#117—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4C22×D1712yes—
362C2×C4⋊C4C22×D1711yescharacteristic
363C2×C4⋊C4C22×D1711yescharacteristic
364C2×C22⋊C4—21no—
365C2×C42—21no—
366#108—348no—
367#107—24no—
368#106—24no—
369#103—344no—
370#103—342no—
371#103—342no—
372#102—344no—
373#102—342no—
374#102—342no—
375C17#2067911yescharacteristic, complemented, Sylow
376C24—341no—
377C24#6325411yescharacteristic
378#41—174no—
379#41—174no—
380#41—178no—
381#41—342no—
382#41—342no—
383#41—344no—
384#41—344no—
385#41—344no—
386#41—344no—
387#41—344no—
388C2×Q8—172no—
389C2×Q8—174no—
390C2×Q8—342no—
391C2×Q8—341no—
392C2×D4—172no—
393C2×D4—174no—
394C2×D4—342no—
395C2×D4—344no—
396C2×D4—344no—
397C2×D4—342no—
398C2×D4—342no—
399C2×D4—341no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—82no—
406C22×C4—341no—
407C22×C4—342no—
408C22×C4—344no—
409C22×C4—342no—
410C22×C4—344no—
411C22×C4C2×C4×D1712yes—
412C22×C4C2×D6812yes—
413C22×C4D4×D1711yescharacteristic
414C22×C4D4×D1711yescharacteristic
415C22×C4#6325411yescharacteristic
416C22×C4Q8×D1712yes—
417C22×C4#6325612yes—
418C22×C4C2×C22⋊D1711yescharacteristic
419C22×C4C2×C22⋊D1711yescharacteristic
420C22×C4C23×D1711yescharacteristic
421M16—174no—
422M16—342no—
423M16—344no—
424M16—3416no—
425M16—342no—
426M16—342no—
427M16—342no—
428M16—342no—
429M16—682no—
430M16—682no—
431M16#6322718yes—
432M16#6322718yes—
433C2×C8—342no—
434C2×C8—344no—
435C2×C8—3416no—
436C2×C8—342no—
437C2×C8—342no—
438C2×C8—682no—
439C2×C8—682no—
440C2×C8C2×C4×D1714yes—
441C2×C8C2×D6814yes—
442C2×C8C2×C22⋊D1714yes—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C42—41no—
454C42—41no—
455C23—41no—
456C23—41no—
457C23—342no—
458C23—344no—
459C23—1361no—
460C23—1361no—
461C23#1058757812yes—
462C23#1058758312yes—
463C23#1058762411yescharacteristic
464C23#1058762411yescharacteristic
465C23C2×#6325411yescharacteristic
466Q8—174no—
467Q8—342no—
468Q8—344no—
469D4—174no—
470D4—344no—
471D4—344no—
472D4—348no—
473D4—342no—
474D4—344no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—342no—
484C2×C4—344no—
485C2×C4—344no—
486C2×C4—344no—
487C2×C4—348no—
488C2×C4C4⋊C4×D1712yes—
489C2×C4C4⋊C4×D1712yes—
490C2×C4#1058757812yes—
491C2×C4#1058758112yes—
492C2×C4#1058758112yes—
493C2×C4#1058758312yes—
494C2×C4C2×#6322712yes—
495C2×C4C2×#6322712yes—
496C2×C4#1058762411yescharacteristic
497C2×C4#1058762411yescharacteristic
498C2×C4#1058763212yes—
499C2×C4#1058763212yes—
500C2×C4C22×C4×D1711yescharacteristic
501C2×C4C22×D6811yescharacteristic
502C2×C4C2×D4×D1711yescharacteristic
503C2×C4C2×Q8×D1711yescharacteristic
504C2×C4C2×#6325611yescharacteristic
505C2×C4C22×C22⋊D1711yescharacteristic
506C8—344no—
507C8—682no—
508C8—682no—
509C8C2×#6322718yes—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—42no—
517C22—42no—
518C22—344no—
519C22—682no—
520C22—682no—
521C22—1364no—
522C22#1176266612yes—
523C22#1176266612yes—
524C22C2×#1058757811yescharacteristic
525C22C2×#1058762411yescharacteristic
526C22C2×#1058758311yescharacteristic
527C4—82no—
528C4—81no—
529C4—81no—
530C4—344no—
531C4#1176266612yes—
532C4#1176266612yes—
533C4C22×#6322711yescharacteristic
534C4C2×C4⋊C4×D1711yescharacteristic
535C4C2×#1058758111yescharacteristic
536C4C2×#1058763211yescharacteristic
537C2—22no—
538C2—21no—
539C2—21no—
540C2—22no—
541C2—684no—
542C2C2×#1176266611yescharacteristic
543C2#1453850811yescharacteristic
544C2#1453851711yescharacteristic
subgroupG/H
centerC22C2×#10587624
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#428C2
socleC22×C17C2×#121
prime radicalC22×#63252C22
cosocleC22×C4×C17C24
hypercenterC2×#428D17
hypocenterC17#20679
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×#11762858C2semidirect
5. #14528849C2semidirect
6. #14528844C2semidirect
7. #14528523C2semidirect
8. C2×#11762578C2semidirect
9. C2×C17×#428C2semidirect
12. #11763575C4semidirect
20. C2×#10587490C22semidirect
30. #11762578C22semidirect
31. #11762578C22semidirect
32. #11762578C22semidirect
38. C2×C17×#107C22semidirect
39. C2×C17×#106C22semidirect
91. #10587587C2×C4semidirect
240. C2×#428D17semidirect
290. C2×#107C2×D17semidirect
291. C2×#106C2×D17semidirect
375. C17#20679semidirect

Maximal subgroups

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

subgroupindex
C2×#117635752
C2×#117628582
#145373012
C2×#117628082
#145288492
#145288442
#145285232
C2×#117625782
C2×C17×#4282
#2067917