Finite Group Databasebeta

← #17978106

Subgroups — #17978106

Order 4352.

606 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14544886C211yesmaximal, characteristic, complemented
2#14544886C211yesmaximal, characteristic, complemented
3#14544882C211yesmaximal, characteristic, complemented
4#14544857C211yesmaximal, characteristic
5#14544857C211yesmaximal, characteristic
6#14544853C211yesmaximal, characteristic
7#14544844C211yesmaximal, characteristic, complemented
8#14544841C211yesmaximal, characteristic, complemented
9#14544837C211yesmaximal, characteristic, complemented
10#14544731C211yesmaximal, characteristic
11#14544731C211yesmaximal, characteristic
12#14544731C211yesmaximal, characteristic
13#14544718C211yesmaximal, characteristic, complemented
14#14544713C211yesmaximal, characteristic, complemented
15#14532086C211yesmaximal, characteristic
16C22×#63247C2211yescharacteristic
17C2×#10587544C2211yescharacteristic
18C2×#10587544C2211yescharacteristic
19C2×#10587544C2211yescharacteristic
20C2×#10587544C2211yescharacteristic
21C2×#10587544C2211yescharacteristic
22C2×C4⋊#3624C2211yescharacteristic
23C2×C4⋊#3624C2211yescharacteristic
24C2×C4⋊#3624C2211yescharacteristic
25C2×C4⋊#3624C2211yescharacteristic
26C2×C22⋊#3624C2211yescharacteristic
27C2×C22⋊#3624C2211yescharacteristic
28C2×C22⋊#3624C2211yescharacteristic
29C2×#10587542C2211yescharacteristic
30C2×#10587542C2211yescharacteristic
31C2×C4×#3624C2211yescharacteristic
32#11763675C2211yescharacteristic
33#11763675C2211yescharacteristic
34#11763675C2211yescharacteristic
35#11763675C2211yescharacteristic
36#11763675C2211yescharacteristic
37#11763675C2211yescharacteristic
38#11763675C2211yescharacteristic
39#11763675C2211yescharacteristic
40#11763675C2211yescharacteristic
41#11763675C2211yescharacteristic
42#11763675C2211yescharacteristic
43#11763675C2211yescharacteristic
44C17⋊#401C411yescharacteristic
45#11762748C411yescharacteristic
46#11762748C411yescharacteristic
47#11762748C411yescharacteristic
48#11762734C411yescharacteristic
49#11762734C411yescharacteristic
50#11762734C411yescharacteristic
51C4⋊C4×Dic17C2211yescharacteristic
52C4⋊C4×Dic17C2211yescharacteristic
53C4⋊C4×Dic17C2211yescharacteristic
54C22⋊C4×Dic17C2211yescharacteristic
55C22⋊C4×Dic17C2211yescharacteristic
56C22⋊C4×Dic17C2211yescharacteristic
57#11762654C2211yescharacteristic
58C17×#396C411yescharacteristic
59C23×Dic17C2311yescharacteristic
60C2×#63247—41no—
61C2×#63247—41no—
62C2×#63247—41no—
63C22×#3624C2311yescharacteristic
64C22×#3624C2311yescharacteristic
65C22×#3624C2311yescharacteristic
66C22×#3624C2311yescharacteristic
67C22×#3624C2311yescharacteristic
68C22×#3624C2311yescharacteristic
69C22×#3624C2311yescharacteristic
70C22×#3624C2311yescharacteristic
71C2×C4⋊C4×C17C2×C411yescharacteristic
72C2×C4⋊C4×C17C2×C411yescharacteristic
73C2×C4⋊C4×C17C2×C411yescharacteristic
74C2×C22⋊C4×C17C2×C411yescharacteristic
75C2×C22⋊C4×C17C2×C411yescharacteristic
76C2×C22⋊C4×C17C2×C411yescharacteristic
77C2×C22⋊Dic17C2×C411yescharacteristic
78C2×C22⋊Dic17C2×C411yescharacteristic
79C2×C22⋊Dic17C2×C411yescharacteristic
80C2×C4⋊Dic17C2×C411yescharacteristic
81C2×C4⋊Dic17C2×C411yescharacteristic
82C2×C4⋊Dic17C2×C411yescharacteristic
83C2×C4×Dic17C2311yescharacteristic
84C2×C4×Dic17C2311yescharacteristic
85C2×C4×Dic17C2311yescharacteristic
86C2×C4×Dic17C2311yescharacteristic
87C2×C4×Dic17C2311yescharacteristic
88C2×C4×Dic17C2311yescharacteristic
89C22⋊#3624—41no—
90C22⋊#3624—41no—
91C22⋊#3624—41no—
92#10587544—41no—
93#10587544—41no—
94#10587544—41no—
95#10587544—41no—
96#10587544—41no—
97#10587542—41no—
98#10587542—41no—
99C4⋊#3624—41no—
100C4⋊#3624—41no—
101C4⋊#3624—41no—
102C4⋊#3624—41no—
103C4×#3624—41no—
104C17×#95C2×C411yescharacteristic
105#10587502C2×C411yescharacteristic
106#10587502C2×C411yescharacteristic
107#10587502C2×C411yescharacteristic
108#10587502C2×C411yescharacteristic
109#10587502C2×C411yescharacteristic
110#10587502C2×C411yescharacteristic
111#10587502C2×C411yescharacteristic
112#10587502C2×C411yescharacteristic
113#10587502C2×C411yescharacteristic
114#10587502C2×C411yescharacteristic
115#10587502C2×C411yescharacteristic
116#10587502C2×C411yescharacteristic
117#10587502C2×C411yescharacteristic
118#10587502C2×C411yescharacteristic
119#10587502C2×C411yescharacteristic
120C24×C17C22×C411yescharacteristic
121C22×C4×C17C22×C411yescharacteristic
122C22×C4×C17C22×C411yescharacteristic
123C22×C4×C17C22×C411yescharacteristic
124C22×C4×C17C22×C411yescharacteristic
125C22×C4×C17C22×C411yescharacteristic
126C22×C4×C17C22×C411yescharacteristic
127C22×Dic17—21no—
128C22×Dic17—21no—
129C22×Dic17—21no—
130C22×Dic17—21no—
131C22×Dic17—21no—
132C22×Dic17—21no—
133C22×Dic17C22×C411yescharacteristic
134C22×Dic17C22×C411yescharacteristic
135C22×Dic17C22×C411yescharacteristic
136C22×Dic17C22×C411yescharacteristic
137C22×Dic17C22×C411yescharacteristic
138C22×Dic17C22×C411yescharacteristic
139C22×Dic17C22×C411yescharacteristic
140C22×Dic17C2411yescharacteristic
141#63247—44no—
142C2×#3624—21no—
143C2×#3624—21no—
144C2×#3624—21no—
145C2×#3624—21no—
146C2×#3624—21no—
147C2×#3624—21no—
148C2×#3624—21no—
149C2×#3624—21no—
150C2×#3624—21no—
151C2×#3624—21no—
152C2×#3624—21no—
153C2×#3624—21no—
154C2×#3624—21no—
155C2×#3624—21no—
156C2×#3624—21no—
157C2×#3624—21no—
158C2×#3624—21no—
159C2×#3624—21no—
160C2×#3624—21no—
161C2×#3624—21no—
162C2×#3624—21no—
163C2×#3624—21no—
164C2×#3624—21no—
165C2×#3624—21no—
166C4⋊C4×C17—22no—
167C4⋊C4×C17—41no—
168C4⋊C4×C17—41no—
169C22⋊C4×C17—22no—
170C22⋊C4×C17—41no—
171C22⋊C4×C17—41no—
172C22⋊Dic17—41no—
173C22⋊Dic17—41no—
174C22⋊Dic17—41no—
175C4⋊Dic17—22no—
176C4⋊Dic17—41no—
177C4⋊Dic17—41no—
178C4×Dic17—21no—
179C4×Dic17—21no—
180C4×Dic17—21no—
181C4×Dic17—21no—
182C4×Dic17—21no—
183C4×Dic17—21no—
184C4×Dic17—21no—
185C4×Dic17—21no—
186C4×Dic17—21no—
187C4×Dic17—21no—
188C4×Dic17—21no—
189C4×Dic17—21no—
190#23382—171nomaximal, Sylow
191C23×C17—21no—
192C23×C17—21no—
193C23×C17—21no—
194C23×C17—21no—
195C23×C17—21no—
196C23×C17—21no—
197C23×C17—21no—
198C23×C17C23×C411yescharacteristic
199C2×C4×C17—21no—
200C2×C4×C17—21no—
201C2×C4×C17—21no—
202C2×C4×C17—21no—
203C2×C4×C17—21no—
204C2×C4×C17—21no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—21no—
208C2×C4×C17—21no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17#13112yes—
217C2×Dic17—21no—
218C2×Dic17—21no—
219C2×Dic17—21no—
220C2×Dic17—21no—
221C2×Dic17—21no—
222C2×Dic17—21no—
223C2×Dic17—21no—
224C2×Dic17—21no—
225C2×Dic17—21no—
226C2×Dic17—21no—
227C2×Dic17—21no—
228C2×Dic17—21no—
229C2×Dic17—21no—
230C2×Dic17—21no—
231C2×Dic17—21no—
232C2×Dic17—21no—
233C2×Dic17—21no—
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—41no—
239C2×Dic17—41no—
240C2×Dic17—41no—
241C2×Dic17—41no—
242C2×Dic17H(5,𝔽2)11yescharacteristic
243C2×Dic17H(5,𝔽2)11yescharacteristic
244C2×Dic17H(5,𝔽2)11yescharacteristic
245C2×Dic172−1+411yescharacteristic
246C2×Dic172−1+411yescharacteristic
247C2×Dic172−1+411yescharacteristic
248#3624—41no—
249#3624—41no—
250#3624—41no—
251#3624—41no—
252#3624—41no—
253#3624—41no—
254#3624—41no—
255#3624—41no—
256#2401—171no—
257#1940—171no—
258#1940—171no—
259#1939—171no—
260#1924—171no—
261#1924—171no—
262#1923—171no—
263#1917—171no—
264#1917—171no—
265#1916—171no—
266#1851—171no—
267#1851—171no—
268#1851—171no—
269#1843—171no—
270#1840—171no—
271C22×C17—41no—
272C22×C17—41no—
273C22×C17—41no—
274C22×C17—41no—
275C22×C17—41no—
276C22×C17—41no—
277C22×C17—41no—
278C22×C17#52011yescharacteristic
279C22×C17#52011yescharacteristic
280C22×C17#52011yescharacteristic
281C22×C17C2×#13111yescharacteristic
282C22×C17#56811yescharacteristic
283C22×C17#56811yescharacteristic
284C22×C17#56811yescharacteristic
285C4×C17—22no—
286C4×C17—41no—
287C4×C17—41no—
288C4×C17—41no—
289C4×C17—41no—
290C4×C17—41no—
291Dic17—41no—
292Dic17—41no—
293Dic17—41no—
294Dic17—41no—
295Dic17—41no—
296Dic17—41no—
297Dic17—41no—
298Dic17#56011yescharacteristic
299Dic17#56311yescharacteristic
300Dic17#56311yescharacteristic
301Dic17#56311yescharacteristic
302C22×M16—171no—
303C2×C4⋊C8—171no—
304C2×C4⋊C8—171no—
305C2×C4⋊C8—171no—
306C2×C4⋊C8—171no—
307C2×C4⋊C8—171no—
308C2×C4⋊C8—171no—
309C2×C4⋊C8—171no—
310C2×C4⋊C8—171no—
311C2×C4⋊C8—171no—
312C2×C22⋊C8—171no—
313C2×C22⋊C8—171no—
314C2×C22⋊C8—171no—
315C2×#97—171no—
316C2×#97—171no—
317C2×C4×C8—171no—
318#401—171no—
319#400—171no—
320#400—171no—
321#400—171no—
322#397—171no—
323#397—171no—
324#397—171no—
325#396#59511yescharacteristic
326C4×C4⋊C4—171no—
327C4×C4⋊C4—171no—
328C4×C4⋊C4—171no—
329C4×C22⋊C4—171no—
330C4×C22⋊C4—171no—
331C4×C22⋊C4—171no—
332#376—171no—
333#336—171no—
334#336—171no—
335#336—171no—
336#336—171no—
337#336—171no—
338#336—171no—
339#336—171no—
340#336—171no—
341#336—171no—
342#336—171no—
343#336—171no—
344#336—171no—
345C2×C17—81no—
346C2×C17#238311yescharacteristic
347C2×C17#299211yescharacteristic
348C2×C17#299311yescharacteristic
349C2×C17#299311yescharacteristic
350C2×C17#299511yescharacteristic
351C2×C17#299511yescharacteristic
352C2×C17#299511yescharacteristic
353C23×C4—171no—
354C2×M16—681no—
355C2×M16—681no—
356C2×M16—681no—
357C22×C8—171no—
358C22×C8—171no—
359C22×C8—171no—
360C22×C8—171no—
361C22×C8—171no—
362C22×C8—171no—
363C22×C8—171no—
364C22×C8—171no—
365C2×C4⋊C4—171no—
366C2×C4⋊C4—171no—
367C2×C4⋊C4—171no—
368C2×C4⋊C4C2×#59511yescharacteristic
369C2×C4⋊C4C2×#59511yescharacteristic
370C2×C4⋊C4C2×#59511yescharacteristic
371C2×C22⋊C4—171no—
372C2×C22⋊C4—171no—
373C2×C22⋊C4—171no—
374C2×C22⋊C4C2×#59511yescharacteristic
375C2×C22⋊C4C2×#59511yescharacteristic
376C2×C22⋊C4C2×#59511yescharacteristic
377C2×C42—171no—
378C2×C42—171no—
379C2×C42—171no—
380C2×C42—171no—
381C2×C42—171no—
382C2×C42—171no—
383C4⋊C8—681no—
384C4⋊C8—681no—
385C4⋊C8—681no—
386C4⋊C8—681no—
387C4⋊C8—681no—
388C4⋊C8—681no—
389C4⋊C8—681no—
390C4⋊C8—681no—
391C4⋊C8—681no—
392C22⋊C8—681no—
393C22⋊C8—681no—
394C22⋊C8—681no—
395#97—681no—
396#97—681no—
397C4×C8—681no—
398#95—171no—
399#95—171no—
400#95—171no—
401#95—171no—
402#95—171no—
403#95—171no—
404#95—171no—
405#95—171no—
406#95—171no—
407#95—171no—
408#95—171no—
409#95—171no—
410#95—171no—
411#95—171no—
412#95—171no—
413#95C2×#59511yescharacteristic
414C17#2338211yescharacteristic, complemented, Sylow
415C24C22×#59511yescharacteristic
416C22×C4—171no—
417C22×C4—171no—
418C22×C4—171no—
419C22×C4—171no—
420C22×C4—171no—
421C22×C4—171no—
422C22×C4—171no—
423C22×C4—171no—
424C22×C4—341no—
425C22×C4—341no—
426C22×C4—341no—
427C22×C4—341no—
428C22×C4—341no—
429C22×C4—341no—
430C22×C4C22×#59511yescharacteristic
431C22×C4C22×#59511yescharacteristic
432C22×C4C22×#59511yescharacteristic
433C22×C4C22×#59511yescharacteristic
434C22×C4C22×#59511yescharacteristic
435C22×C4C22×#59511yescharacteristic
436M16—684no—
437C2×C8—341no—
438C2×C8—341no—
439C2×C8—341no—
440C2×C8—341no—
441C2×C8—341no—
442C2×C8—341no—
443C2×C8—341no—
444C2×C8—341no—
445C2×C8—341no—
446C2×C8—341no—
447C2×C8—341no—
448C2×C8—341no—
449C2×C8—341no—
450C2×C8—341no—
451C2×C8—341no—
452C2×C8—341no—
453C2×C8—341no—
454C2×C8—341no—
455C2×C8—341no—
456C2×C8—341no—
457C2×C8—341no—
458C2×C8—341no—
459C2×C8—341no—
460C2×C8—341no—
461C4⋊C4—22no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—342no—
465C4⋊C4—681no—
466C4⋊C4—681no—
467C22⋊C4—22no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—681no—
471C22⋊C4—681no—
472C22⋊C4—681no—
473C42—341no—
474C42—341no—
475C42—341no—
476C42—341no—
477C42—341no—
478C42—341no—
479C42—341no—
480C42—341no—
481C42—341no—
482C42—341no—
483C42—341no—
484C42—341no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23C23×#59511yescharacteristic
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—171no—
511C2×C4—171no—
512C2×C4—171no—
513C2×C4—171no—
514C2×C4—171no—
515C2×C4—171no—
516C2×C4—341no—
517C2×C4—341no—
518C2×C4—341no—
519C2×C4—341no—
520C2×C4—341no—
521C2×C4—341no—
522C2×C4—341no—
523C2×C4—341no—
524C2×C4—341no—
525C2×C4—341no—
526C2×C4—341no—
527C2×C4—341no—
528C2×C4—341no—
529C2×C4—341no—
530C2×C4—341no—
531C2×C4—341no—
532C2×C4—341no—
533C2×C4—341no—
534C2×C4—341no—
535C2×C4—341no—
536C2×C4—341no—
537C2×C4—681no—
538C2×C4—681no—
539C2×C4—681no—
540C2×C4—681no—
541C2×C4#1058767812yes—
542C8—681no—
543C8—681no—
544C8—681no—
545C8—681no—
546C8—681no—
547C8—681no—
548C8—681no—
549C8—681no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22#1176387811yescharacteristic
558C22#1176387811yescharacteristic
559C22#1176387811yescharacteristic
560C22C2×#1058767811yescharacteristic
561C22#1176388711yescharacteristic
562C22#1176388711yescharacteristic
563C22#1176388911yescharacteristic
564C4—22no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—171no—
571C4—171no—
572C4—171no—
573C4—171no—
574C4—681no—
575C4—681no—
576C4—681no—
577C4—681no—
578C4—681no—
579C4—681no—
580C4—681no—
581C2—81no—
582C2#1454533111yescharacteristic
583C2#1454572011yescharacteristic
584C2#1454572611yescharacteristic
585C2#1454572911yescharacteristic
586C2#1454573711yescharacteristic
587C2#1454574111yescharacteristic
588C2#1454574111yescharacteristic
subgroupG/H
center, FrattiniC23C23×#595
commutator, socleC23×C17C23×C4
FittingC17×#396C4
prime radicalC24×C17C22×C4
cosocleC22×Dic17C24
hypercenter#396#595
hypocenterC17#23382
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. #14544886C2semidirect
2. #14544886C2semidirect
3. #14544882C2semidirect
7. #14544844C2semidirect
8. #14544841C2semidirect
9. #14544837C2semidirect
13. #14544718C2semidirect
14. #14544713C2semidirect
414. C17#23382semidirect

Maximal subgroups

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

subgroupindex
#145448862
#145448822
#145448572
#145448532
#145448442
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