Finite Group Databasebeta

← #17062897

Subgroups — #17062897

Order 4352.

1002 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762857C211yesmaximal, characteristic, complemented
2#14537348C211yesmaximal, characteristic, complemented
3#14537301C211yesmaximal, characteristic
4C2×#11762808C211yesmaximal, characteristic
5C2×#11762791C211yesmaximal, characteristic
6#14530148C214yesmaximal, complemented
7#14530141C214yesmaximal, complemented
8C2×#11762578C211yesmaximal, characteristic, complemented
9C2×C17×#428C211yesmaximal, characteristic, complemented
10C22×#63252C2211yescharacteristic
11C22×#63223C2211yescharacteristic
12C22×#63223C2211yescharacteristic
13C22×#63223C2211yescharacteristic
14#11762857—24no—
15C2×#10587493C2214yes—
16#11762836—22no—
17#11762836C2214yes—
18#11762833—22no—
19#11762817C2212yes—
20#11762813C2212yes—
21#11762808—21no—
22#11762808—21no—
23#11762808—22no—
24#11762808—22no—
25#11762803—22no—
26C2×#10587481C2212yes—
27C2×#10587480C2212yes—
28#11762791—24no—
29C2×#10587495C2212yes—
30C2×#10587494C2212yes—
31#11762578C2214yes—
32C22×M16×C17C2211yescharacteristic
33C2×C17×#117C2211yescharacteristic
34C22×C4⋊C4×C17C2211yescharacteristic
35C17×#428—22no—
36C17×#428C2214yes—
37C2×C17×#107—21no—
38C2×C17×#106—21no—
39C23×C4×C17—21no—
40C23×C4×C17C2311yescharacteristic
41C22×C22⋊D17—21no—
42C2×#63252—22no—
43C2×#63252—21no—
44C2×#63252—21no—
45C2×#63252—21no—
46C2×#63252—21no—
47C2×#63252—22no—
48C2×#63252—22no—
49C2×#63252Q814yes—
50C22×D68C2311yescharacteristic
51C22×C4×D17—21no—
52C22×Dic34C2311yescharacteristic
53C2×M16×C17—21no—
54C2×M16×C17—22no—
55C2×M16×C17D414yes—
56C2×M16×C17C2312yes—
57C22×C8×C17—21no—
58C17×#117—21no—
59C17×#117—21no—
60C17×#117D412yes—
61C17×#117C2312yes—
62C2×C4⋊C4×C17—21no—
63C2×C4⋊C4×C17—21no—
64C2×C4⋊C4×C17—21no—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17—21no—
67C2×C4⋊C4×C17D411yescharacteristic
68C2×C4⋊C4×C17D411yescharacteristic
69C2×C4⋊C4×C17C2312yes—
70C2×C22⋊C4×C17—21no—
71C2×C42×C17—21no—
72C2×#63223—22no—
73C2×#63223—21no—
74C2×#63223—21no—
75C2×#63223—22no—
76C2×#63223—22no—
77C2×#63223—22no—
78C2×#63223—22no—
79C2×#63223—22no—
80C2×#63223D412yes—
81C2×#63223D412yes—
82C2×#63223C2312yes—
83C2×#63223C2312yes—
84C22×#3622—21no—
85C22×#3622—21no—
86C22×#3622C2312yes—
87C17×#107—24no—
88C17×#106—24no—
89#10587495—24no—
90#10587494—24no—
91#10587493—28no—
92#10587481—24no—
93#10587481—22no—
94#10587481—22no—
95#10587480—24no—
96#10587480—22no—
97#10587480—22no—
98C4⋊#3622—28no—
99C24×C17#4111yescharacteristic
100C23×D17—21no—
101C22×C4×C17—21no—
102C22×C4×C17—21no—
103C22×C4×C17—22no—
104C22×C4×C17—41no—
105C22×C4×C17—41no—
106C22×C4×C17—82no—
107C22×C4×C17C2×D412yes—
108C22×C4×C17C2×D411yescharacteristic
109C22×C4×C17C2×D411yescharacteristic
110C22×C4×C17C2×D411yescharacteristic
111C22×C4×C17C2×D411yescharacteristic
112C22×C4×C17C2×Q812yes—
113C22×C4×C17#4112yes—
114C22×C4×C17#4111yescharacteristic
115C22×C4×C17#4112yes—
116C22×C4×C17C2411yescharacteristic
117C2×C22⋊D17—24no—
118C2×C22⋊D17—41no—
119C2×C22⋊D17—41no—
120C2×C22⋊D17—42no—
121C22×Dic17—21no—
122#63252—24no—
123#63252—24no—
124#63252—24no—
125#63252—24no—
126#63252—42no—
127#63252—42no—
128#63252—42no—
129#63252—42no—
130C2×D68—21no—
131C2×D68—21no—
132C2×D68—21no—
133C2×D68—21no—
134C2×D68—21no—
135C2×D68C2×Q812yes—
136C2×C4×D17—24no—
137C2×C4×D17—41no—
138C2×C4×D17—41no—
139C2×C4×D17—42no—
140C2×Dic34—21no—
141C2×Dic34—21no—
142C2×Dic34—21no—
143C2×Dic34—21no—
144C2×Dic34—21no—
145C2×Dic34C2×Q812yes—
146M16×C17—24no—
147M16×C17—22no—
148M16×C17C2×D414yes—
149C2×C8×C17—24no—
150C2×C8×C17—22no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17—22no—
154C4⋊C4×C17—22no—
155C4⋊C4×C17—41no—
156C4⋊C4×C17—41no—
157C4⋊C4×C17C2×D412yes—
158C4⋊C4×C17C2×D412yes—
159C22⋊C4×C17—22no—
160C22⋊C4×C17—22no—
161C42×C17—22no—
162C42×C17—22no—
163#63223—24no—
164#63223—24no—
165#63223—22no—
166#63223—22no—
167#63223—22no—
168#63223—22no—
169#63223—42no—
170#63223—42no—
171C2×#3622—22no—
172C2×#3622—22no—
173C2×#3622—22no—
174C2×#3622—22no—
175C2×#3622—42no—
176C2×#3622—42no—
177C2×#3622C2×D414yes—
178#23604—171nomaximal, Sylow
179C23×C17—41no—
180C23×C17—41no—
181C23×C17#12111yescharacteristic
182C23×C17#12111yescharacteristic
183C23×C17C22⋊Q812yes—
184C23×C17#12612yes—
185C23×C17C2×#4111yescharacteristic
186C22×D17—22no—
187C22×D17—41no—
188C22×D17—41no—
189C22×D17—81no—
190C22×D17—81no—
191C2×C4×C17—22no—
192C2×C4×C17—22no—
193C2×C4×C17—42no—
194C2×C4×C17—81no—
195C2×C4×C17—81no—
196C2×C4×C17—82no—
197C2×C4×C17—82no—
198C2×C4×C17—82no—
199C2×C4×C17#12112yes—
200C2×C4×C17#12112yes—
201C2×C4×C17#12112yes—
202C2×C4×C17#12112yes—
203C2×C4×C17#12111yescharacteristic
204C2×C4×C17#12111yescharacteristic
205C2×C4×C17C22⋊Q812yes—
206C2×C4×C17C22⋊Q812yes—
207C2×C4×C17C22⋊Q812yes—
208C2×C4×C17C22⋊Q812yes—
209C2×C4×C17C22⋊Q812yes—
210C2×C4×C17#12612yes—
211C2×C4×C17C22×D411yescharacteristic
212C2×C4×C17C22×D411yescharacteristic
213C2×C4×C17C22×D411yescharacteristic
214C2×C4×C17C22×Q811yescharacteristic
215C2×C4×C17C2×#4111yescharacteristic
216C2×C4×C17C2×#4111yescharacteristic
217C22⋊D17—44no—
218C22⋊D17—44no—
219C2×Dic17—22no—
220C2×Dic17—41no—
221C2×Dic17—41no—
222D68—22no—
223D68—22no—
224D68—41no—
225D68—41no—
226C4×D17—44no—
227C4×D17—44no—
228Dic34—22no—
229Dic34—22no—
230Dic34—41no—
231Dic34—41no—
232C8×C17—24no—
233#3622—24no—
234#3622—42no—
235#3622—42no—
236C2×#430—171no—
237C2×#428D1711yescharacteristic, complemented
238#2905—171no—
239#2896—171no—
240C2×#417—171no—
241C2×#413—171no—
242#2064—174no—
243#2063—174no—
244C2×#343—171no—
245C22×C17—21no—
246C22×C17—21no—
247C22×C17—21no—
248C22×C17—21no—
249C22×C17—21no—
250C22×C17—21no—
251C22×C17—42no—
252C22×C17—42no—
253C22×C17#39612yes—
254C22×C17#39612yes—
255C22×C17C2×#12111yescharacteristic
256C22×C17C2×C22⋊Q811yescharacteristic
257C22×C17C2×#12611yescharacteristic
258C2×D17—42no—
259C2×D17—82no—
260C2×D17—81no—
261C2×D17—81no—
262C2×D17—82no—
263C4×C17—42no—
264C4×C17—42no—
265C4×C17—82no—
266C4×C17#39612yes—
267C4×C17#39612yes—
268C4×C17C2×#12111yescharacteristic
269C4×C17C2×#12111yescharacteristic
270C4×C17C2×C22⋊Q811yescharacteristic
271C4×C17C2×C22⋊Q811yescharacteristic
272Dic17—42no—
273C22×#41—171no—
274C22×M16—171no—
275C22×M16—171no—
276C22×M16—171no—
277C22×M16C2×D1711yescharacteristic
278C2×#117C2×D1711yescharacteristic
279C22×C4⋊C4C2×D1711yescharacteristic
280#430—344no—
281C2×#108—174no—
282#428—22no—
283#428C2×D1714yes—
284C2×#107—21no—
285C2×#106—21no—
286#424—174no—
287#424—342no—
288#423—342no—
289#419—172no—
290#418—172no—
291#417—342no—
292#417—342no—
293#417—341no—
294#417—341no—
295#416—342no—
296C2×#103—172no—
297C2×#102—172no—
298#413—344no—
299C2×#101—172no—
300C2×#100—172no—
301#343—174no—
302C2×C17—21no—
303C2×C17—21no—
304C2×C17—22no—
305C2×C17—22no—
306C2×C17C2×#39611yescharacteristic
307C2×C17#306811yescharacteristic
308C2×C17#306911yescharacteristic
309D17—82no—
310C2×#41—174no—
311C2×#41—342no—
312C2×#41—342no—
313C2×#41—341no—
314C2×#41—341no—
315C2×#41—342no—
316C2×#41—341no—
317C2×#41—341no—
318C22×Q8—171no—
319C22×D4—171no—
320C22×D4—341no—
321C23×C4—21no—
322C23×C4—341no—
323C23×C4C22×D1711yescharacteristic
324C2×M16—22no—
325C2×M16—21no—
326C2×M16—172no—
327C2×M16—172no—
328C2×M16—172no—
329C2×M16—172no—
330C2×M16—342no—
331C2×M16—342no—
332C2×M16—341no—
333C2×M16—341no—
334C2×M16—342no—
335C2×M16—342no—
336C2×M16—342no—
337C2×M16—342no—
338C2×M16D6814yes—
339C2×M16C22×D1712yes—
340C22×C8—21no—
341C22×C8—172no—
342C22×C8—341no—
343C22×C8—341no—
344#117—21no—
345#117—21no—
346#117C22⋊D1712yes—
347#117C22×D1712yes—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4C22⋊D1711yescharacteristic
354C2×C4⋊C4C22⋊D1711yescharacteristic
355C2×C4⋊C4C22×D1712yes—
356C2×C22⋊C4—21no—
357C2×C42—21no—
358#108—348no—
359#107—24no—
360#106—24no—
361C4⋊C8—348no—
362#103—344no—
363#103—342no—
364#103—342no—
365#102—344no—
366#102—342no—
367#102—342no—
368#101—344no—
369#100—344no—
370C17#2360411yescharacteristic, complemented, Sylow
371C24—341no—
372C24#6325411yescharacteristic
373#41—344no—
374#41—344no—
375#41—344no—
376#41—344no—
377#41—682no—
378#41—682no—
379#41—682no—
380#41—682no—
381C2×Q8—172no—
382C2×Q8—341no—
383C2×Q8—341no—
384C2×Q8—341no—
385C2×Q8—341no—
386C2×Q8—341no—
387C2×D4—172no—
388C2×D4—344no—
389C2×D4—341no—
390C2×D4—341no—
391C2×D4—341no—
392C2×D4—341no—
393C2×D4—341no—
394C2×D4—682no—
395C2×D4—681no—
396C2×D4—681no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—82no—
403C22×C4—341no—
404C22×C4—344no—
405C22×C4—682no—
406C22×C4—681no—
407C22×C4—681no—
408C22×C4C2×D6812yes—
409C22×C4#6325212yes—
410C22×C4D4×D1711yescharacteristic
411C22×C4D4×D1711yescharacteristic
412C22×C4#6325411yescharacteristic
413C22×C4Q8×D1712yes—
414C22×C4#6325612yes—
415C22×C4C2×C22⋊D1711yescharacteristic
416C22×C4C2×C22⋊D1711yescharacteristic
417C22×C4C23×D1711yescharacteristic
418M16—22no—
419M16—24no—
420M16—344no—
421M16—344no—
422M16—342no—
423M16—342no—
424M16—342no—
425M16—342no—
426M16—682no—
427M16—682no—
428M16C2×D6814yes—
429C2×C8—22no—
430C2×C8—24no—
431C2×C8—174no—
432C2×C8—342no—
433C2×C8—342no—
434C2×C8—342no—
435C2×C8—342no—
436C2×C8—682no—
437C2×C8—682no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4C2×C22⋊D1712yes—
445C4⋊C4C2×C22⋊D1712yes—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C42—22no—
449C42—22no—
450C23—41no—
451C23—41no—
452C23—342no—
453C23—681no—
454C23—681no—
455C23—1361no—
456C23—1361no—
457C23#1058758312yes—
458C23#1058758412yes—
459C23#1058762511yescharacteristic
460C23#1058762511yescharacteristic
461C23C2×#6325411yescharacteristic
462Q8—342no—
463Q8—342no—
464Q8—681no—
465Q8—681no—
466D4—342no—
467D4—342no—
468D4—684no—
469D4—684no—
470D4—681no—
471D4—681no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—42no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—342no—
481C2×C4—681no—
482C2×C4—681no—
483C2×C4—684no—
484C2×C4—684no—
485C2×C4#1058758112yes—
486C2×C4#1058758112yes—
487C2×C4#1058758212yes—
488C2×C4#1058758212yes—
489C2×C4#1058758312yes—
490C2×C4#1058758412yes—
491C2×C4#1058761612yes—
492C2×C4#1058761612yes—
493C2×C4#1058762511yescharacteristic
494C2×C4#1058762511yescharacteristic
495C2×C4#1058763212yes—
496C2×C4#1058763212yes—
497C2×C4C22×D6811yescharacteristic
498C2×C4C2×#6325211yescharacteristic
499C2×C4C2×D4×D1711yescharacteristic
500C2×C4C2×Q8×D1711yescharacteristic
501C2×C4C2×#6325611yescharacteristic
502C2×C4C22×C22⋊D1711yescharacteristic
503C8—24no—
504C8—344no—
505C8—682no—
506C8—682no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—42no—
514C22—42no—
515C22—682no—
516C22—1362no—
517C22—1362no—
518C22—1361no—
519C22—1361no—
520C22#1176273112yes—
521C22#1176273112yes—
522C22C2×#1058762511yescharacteristic
523C22C2×#1058758311yescharacteristic
524C22C2×#1058758411yescharacteristic
525C4—42no—
526C4—42no—
527C4—82no—
528C4—682no—
529C4#1176273112yes—
530C4#1176273112yes—
531C4C2×#1058761611yescharacteristic
532C4C2×#1058758111yescharacteristic
533C4C2×#1058763211yescharacteristic
534C4C2×#1058758211yescharacteristic
535C2—21no—
536C2—21no—
537C2—22no—
538C2—22no—
539C2—1362no—
540C2C2×#1176273111yescharacteristic
541C2#1453850911yescharacteristic
542C2#1453851811yescharacteristic
subgroupG/H
centerC22C2×#10587625
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#428C2
socleC22×C17C2×#121
prime radicalC22×#63252C22
hypercenterC2×#428D17
hypocenterC17#23604
LCSG ▷ C22×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
1. C2×#11762857C2semidirect
2. #14537348C2semidirect
6. #14530148C2semidirect
7. #14530141C2semidirect
8. C2×#11762578C2semidirect
9. C2×C17×#428C2semidirect
237. C2×#428D17semidirect
370. C17#23604semidirect

Maximal subgroups

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

subgroupindex
C2×#117628572
#145373482
#145373012
C2×#117628082
C2×#117627912
#145301482
#145301412
C2×#117625782
C2×C17×#4282
#2360417