Finite Group Databasebeta

← #16951839

Subgroups — #16951839

Order 4352.

820 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 820 conjugacy classes of subgroups — too many to draw. Only its 111 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×#11762783C211yesmaximal, characteristic, complemented
2C2×#11762596C212yesmaximal, complemented
3#14528712C211yesmaximal, characteristic, complemented
4#14526902C212yesmaximal, complemented
5C17×#2404C211yesmaximal, characteristic
6C2×#10587612—21no—
7C2×#10587507—21no—
8C2×#10587504C2212yes—
9#11762783—24no—
10C2×#10587505—21no—
11C2×#10587505C2212yes—
12#11762773—21no—
13#11762596—24no—
14C22×#41×C17C2211yescharacteristic
15C2×C17×#120C411yescharacteristic, complemented
16C2×C17×#120C2211yescharacteristic
17C22×C22⋊C4×C17C411yescharacteristic, complemented
18C17×#393—21no—
19C17×#392—21no—
20C17×#386—21no—
21C17×#386—21no—
22C17×#386C411yescharacteristic, complemented
23C17×#386C2211yescharacteristic
24C17×#381C411yescharacteristic, complemented
25C25×C17C2×C411yescharacteristic
26C2×#41×C17—22no—
27C2×#41×C17—22no—
28C2×#41×C17—41no—
29C2×#41×C17—41no—
30C2×#41×C17—42no—
31C22×Q8×C17D411yescharacteristic
32C22×D4×C17—21no—
33C22×D4×C17—21no—
34C22×D4×C17C2×C411yescharacteristic
35C22×D4×C17D411yescharacteristic
36C22×D4×C17C2311yescharacteristic
37C23×C4×C17C2×C411yescharacteristic
38C23×C4×C17C2×C411yescharacteristic
39C23×C4×C17D411yescharacteristic
40C23×C4×C17D411yescharacteristic
41C17×#120—22no—
42C17×#120—22no—
43C17×#120—42no—
44C2×C22⋊C4×C17—21no—
45C2×C22⋊C4×C17—21no—
46C2×C22⋊C4×C17—21no—
47C2×C22⋊C4×C17—21no—
48C2×C22⋊C4×C17—21no—
49C2×C22⋊C4×C17—21no—
50C2×C22⋊C4×C17—41no—
51C2×C22⋊C4×C17—41no—
52C2×C22⋊C4×C17C2×C411yescharacteristic
53C2×C22⋊C4×C17C2×C411yescharacteristic
54C2×C22⋊Dic17—22no—
55#10587612—24no—
56#10587612—42no—
57C2×C4⋊Dic17—41no—
58C2×C4×Dic17—41no—
59C2×#63223—22no—
60C17×#95—21no—
61C17×#95—21no—
62C17×#95—21no—
63C17×#95—21no—
64#10587507—42no—
65#10587505—24no—
66#10587505—22no—
67#10587505D414yescomplemented
68#10587504—22no—
69#10587504—22no—
70C22⋊#3622—41no—
71C22⋊#3622—41no—
72C24×C17—21no—
73C24×C17—22no—
74C24×C17—42no—
75C24×C17—41no—
76C24×C17—41no—
77C24×C17—81no—
78C24×C17C22⋊C411yescharacteristic
79C24×C17C22×C411yescharacteristic
80C24×C17C2×D411yescharacteristic
81#41×C17—44no—
82#41×C17—44no—
83#41×C17—44no—
84#41×C17—44no—
85C2×Q8×C17—22no—
86C2×Q8×C17—22no—
87C2×Q8×C17—41no—
88C2×D4×C17—22no—
89C2×D4×C17—21no—
90C2×D4×C17—21no—
91C2×D4×C17—22no—
92C2×D4×C17—41no—
93C2×D4×C17—41no—
94C2×D4×C17—41no—
95C2×D4×C17—41no—
96C2×D4×C17—41no—
97C2×D4×C17—42no—
98C2×D4×C17—42no—
99C2×D4×C17—42no—
100C2×D4×C17—41no—
101C2×D4×C17—41no—
102C2×D4×C17—81no—
103C2×D4×C17—81no—
104C2×D4×C17C2×D412yes—
105C2×D4×C17#4112yes—
106C22×C4×C17—21no—
107C22×C4×C17—21no—
108C22×C4×C17—21no—
109C22×C4×C17—21no—
110C22×C4×C17—22no—
111C22×C4×C17—22no—
112C22×C4×C17—21no—
113C22×C4×C17—21no—
114C22×C4×C17—21no—
115C22×C4×C17—21no—
116C22×C4×C17—22no—
117C22×C4×C17—21no—
118C22×C4×C17—21no—
119C22×C4×C17—21no—
120C22×C4×C17—41no—
121C22×C4×C17—41no—
122C22×C4×C17—41no—
123C22×C4×C17—81no—
124C22×C4×C17C22⋊C411yescharacteristic
125C22×C4×C17C22⋊C411yescharacteristic
126C22×C4×C17C22⋊C411yescharacteristic
127C22×C4×C17C2×D411yescharacteristic
128C22×Dic17—42no—
129C22⋊C4×C17—22no—
130C22⋊C4×C17—42no—
131C22⋊C4×C17—41no—
132C22⋊C4×C17—41no—
133C22⋊C4×C17—42no—
134C22⋊C4×C17—42no—
135C22⋊C4×C17—41no—
136C22⋊C4×C17—81no—
137C22⋊C4×C17—81no—
138C22⋊Dic17—24no—
139C22⋊Dic17—42no—
140C4⋊Dic17—44no—
141C4×Dic17—44no—
142#63223—42no—
143#63223—42no—
144C2×#3622—42no—
145#13087—171nomaximal, Sylow
146C23×C17—21no—
147C23×C17—21no—
148C23×C17—21no—
149C23×C17—21no—
150C23×C17—22no—
151C23×C17—21no—
152C23×C17—21no—
153C23×C17—21no—
154C23×C17—41no—
155C23×C17—41no—
156C23×C17—41no—
157C23×C17—41no—
158C23×C17—41no—
159C23×C17—41no—
160C23×C17—41no—
161C23×C17—41no—
162C23×C17—41no—
163C23×C17—42no—
164C23×C17—42no—
165C23×C17—41no—
166C23×C17—41no—
167C23×C17—41no—
168C23×C17—82no—
169C23×C17—84no—
170C23×C17—84no—
171C23×C17—82no—
172C23×C17C2×C22⋊C411yescharacteristic
173C23×C17C4×D412yes—
174C23×C17#12011yescharacteristic
175C23×C17#12311yescharacteristic
176Q8×C17—44no—
177D4×C17—42no—
178D4×C17—42no—
179D4×C17—44no—
180D4×C17—42no—
181D4×C17—42no—
182D4×C17—82no—
183D4×C17—82no—
184D4×C17—84no—
185C2×C4×C17—22no—
186C2×C4×C17—22no—
187C2×C4×C17—22no—
188C2×C4×C17—22no—
189C2×C4×C17—21no—
190C2×C4×C17—21no—
191C2×C4×C17—21no—
192C2×C4×C17—21no—
193C2×C4×C17—21no—
194C2×C4×C17—21no—
195C2×C4×C17—21no—
196C2×C4×C17—21no—
197C2×C4×C17—21no—
198C2×C4×C17—42no—
199C2×C4×C17—42no—
200C2×C4×C17—42no—
201C2×C4×C17—42no—
202C2×C4×C17—41no—
203C2×C4×C17—41no—
204C2×C4×C17—41no—
205C2×C4×C17—41no—
206C2×C4×C17—41no—
207C2×C4×C17—41no—
208C2×C4×C17—41no—
209C2×C4×C17—41no—
210C2×C4×C17—41no—
211C2×C4×C17—82no—
212C2×C4×C17#9911yescharacteristic
213C2×C4×C17#9911yescharacteristic
214C2×C4×C17#9912yes—
215C2×C4×C17#12112yes—
216C2×Dic17—44no—
217C2×Dic17—82no—
218C2×Dic17—82no—
219#3622—82no—
220C2×#410—171no—
221#2404D1711yescharacteristic
222C2×AΣL(1,𝔽16)—172no—
223#1785—171no—
224#1390—172no—
225C22×C17—21no—
226C22×C17—21no—
227C22×C17—21no—
228C22×C17—21no—
229C22×C17—22no—
230C22×C17—41no—
231C22×C17—41no—
232C22×C17—42no—
233C22×C17—42no—
234C22×C17—42no—
235C22×C17—42no—
236C22×C17—42no—
237C22×C17—41no—
238C22×C17—41no—
239C22×C17—41no—
240C22×C17—41no—
241C22×C17—41no—
242C22×C17—42no—
243C22×C17—41no—
244C22×C17—41no—
245C22×C17—81no—
246C22×C17—81no—
247C22×C17—81no—
248C22×C17—81no—
249C22×C17—84no—
250C22×C17—84no—
251C22×C17—81no—
252C22×C17—81no—
253C22×C17#38611yescharacteristic
254C22×C17C2×#9911yescharacteristic
255C22×C17C2×#9911yescharacteristic
256C4×C17—22no—
257C4×C17—21no—
258C4×C17—21no—
259C4×C17—22no—
260C4×C17—41no—
261C4×C17—81no—
262C4×C17—81no—
263C4×C17—81no—
264Dic17—84no—
265C22×#41C2×D1711yescharacteristic
266C2×#120Dic1711yescharacteristic, complemented
267C2×#120C2×D1711yescharacteristic
268C2×#117—341no—
269C22×C22⋊C4Dic1711yescharacteristic, complemented
270C2×#101—341no—
271C2×#100—172no—
272#410—344no—
273C2×#99—172no—
274C2×#99—341no—
275#407—341no—
276#393—21no—
277#392—21no—
278#386—21no—
279#386—21no—
280#386Dic1711yescharacteristic, complemented
281#386C2×D1711yescharacteristic
282#381Dic1711yescharacteristic, complemented
283AΣL(1,𝔽16)—344no—
284C2×C17—21no—
285C2×C17—21no—
286C2×C17—41no—
287C2×C17—42no—
288C2×C17—41no—
289C2×C17—81no—
290C2×C17—82no—
291C2×C17#189811yescharacteristic
292C2×C17#212011yescharacteristic
293C2×C17#212211yescharacteristic
294C25C2×Dic1711yescharacteristic
295C2×#41—22no—
296C2×#41—22no—
297C2×#41—41no—
298C2×#41—41no—
299C2×#41—42no—
300C22×Q8C22⋊D1711yescharacteristic
301C22×D4—21no—
302C22×D4—21no—
303C22×D4C2×Dic1711yescharacteristic
304C22×D4C22⋊D1711yescharacteristic
305C22×D4C22×D1711yescharacteristic
306C23×C4C2×Dic1711yescharacteristic
307C23×C4C2×Dic1711yescharacteristic
308C23×C4C22⋊D1711yescharacteristic
309C23×C4C22⋊D1711yescharacteristic
310C2×M16—342no—
311#120—22no—
312#120—22no—
313#120—42no—
314#117—344no—
315#117—682no—
316C2×C4⋊C4—681no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—41no—
324C2×C22⋊C4—41no—
325C2×C22⋊C4—342no—
326C2×C22⋊C4C2×Dic1711yescharacteristic
327C2×C22⋊C4C2×Dic1711yescharacteristic
328C2×C42—681no—
329#101—682no—
330#100—342no—
331#100—342no—
332#99—174no—
333#99—344no—
334#99—342no—
335C22⋊C8—681no—
336C22⋊C8—681no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341C17#1308711yescharacteristic, complemented, Sylow
342C24—21no—
343C24—22no—
344C24—41no—
345C24—42no—
346C24—41no—
347C24—81no—
348C24C22⋊Dic1711yescharacteristic
349C24C22×Dic1711yescharacteristic
350C24C2×C22⋊D1711yescharacteristic
351#41—44no—
352#41—44no—
353#41—44no—
354#41—44no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—41no—
358C2×D4—22no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—22no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—81no—
373C2×D4—81no—
374C2×D4D4×D1712yes—
375C2×D4#6325412yes—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—81no—
394C22×C4—682no—
395C22×C4C22⋊Dic1711yescharacteristic
396C22×C4C22⋊Dic1711yescharacteristic
397C22×C4C22⋊Dic1711yescharacteristic
398C22×C4C2×C22⋊D1711yescharacteristic
399M16—682no—
400M16—682no—
401C2×C8—682no—
402C4⋊C4—684no—
403C22⋊C4—22no—
404C22⋊C4—42no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—41no—
410C22⋊C4—81no—
411C22⋊C4—81no—
412C22⋊C4—344no—
413C22⋊C4—682no—
414C42—684no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—22no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—42no—
433C23—42no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—82no—
438C23—84no—
439C23—84no—
440C23—82no—
441C23D4×Dic1712yes—
442C23#1058762111yescharacteristic
443C23#1058762311yescharacteristic
444C23C2×C22⋊Dic1711yescharacteristic
445Q8—44no—
446D4—42no—
447D4—42no—
448D4—44no—
449D4—42no—
450D4—42no—
451D4—82no—
452D4—82no—
453D4—84no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—82no—
481C2×C4—684no—
482C2×C4—1362no—
483C2×C4—1362no—
484C2×C4#1058750512yes—
485C2×C4#1058750511yescharacteristic
486C2×C4#1058750511yescharacteristic
487C2×C4#1058762512yes—
488C8—1362no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—22no—
494C22—41no—
495C22—41no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—42no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—84no—
514C22—84no—
515C22—81no—
516C22—81no—
517C22#1176268911yescharacteristic
518C22C2×#1058750511yescharacteristic
519C22C2×#1058750511yescharacteristic
520C4—22no—
521C4—21no—
522C4—21no—
523C4—22no—
524C4—41no—
525C4—81no—
526C4—81no—
527C4—81no—
528C4—1364no—
529C2—21no—
530C2—21no—
531C2—41no—
532C2—42no—
533C2—41no—
534C2—81no—
535C2—82no—
536C2#1452923811yescharacteristic
537C2#1453042511yescharacteristic
538C2#1453043411yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC24×C17C22×C4
FrattiniC22×D4C22×D17
Fitting, prime radicalC17×#2404C2
socleC22×C17C2×#99
cosocleC22×D4×C17C23
hypercenter#2404D17
hypocenterC17#13087
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2404

Complements

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

NH ≅ G/Ntype
1. C2×#11762783C2semidirect
2. C2×#11762596C2semidirect
3. #14528712C2semidirect
4. #14526902C2semidirect
15. C2×C17×#120C4semidirect
17. C22×C22⋊C4×C17C4semidirect
22. C17×#386C4semidirect
24. C17×#381C4semidirect
67. #10587505D4semidirect
266. C2×#120Dic17semidirect
269. C22×C22⋊C4Dic17semidirect
280. #386Dic17semidirect
282. #381Dic17semidirect
341. C17#13087semidirect

Maximal subgroups

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

subgroupindex
C2×#117627832
C2×#117625962
#145287122
#145269022
C17×#24042
#1308717