Finite Group Databasebeta

← #17060785

Subgroups — #17060785

Order 4352.

622 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14529725C211yesmaximal, characteristic, complemented
2#14529721C211yesmaximal, characteristic, complemented
3#14529508C211yesmaximal, characteristic, complemented
4#14529508C211yesmaximal, characteristic, complemented
5#14529399C211yesmaximal, characteristic
6#14529399C211yesmaximal, characteristic
7#14529399C211yesmaximal, characteristic
8#14529392C211yesmaximal, characteristic
9#14529350C211yesmaximal, characteristic, complemented
10#14529350C211yesmaximal, characteristic, complemented
11#14529350C211yesmaximal, characteristic, complemented
12#14529343C211yesmaximal, characteristic, complemented
13#14528855C211yesmaximal, characteristic
14#14528768C211yesmaximal, characteristic
15C17×#2396C211yesmaximal, characteristic
16C2×C4⋊#3622C2211yescharacteristic
17C2×C4⋊#3622C2211yescharacteristic
18C2×C4⋊#3622C2211yescharacteristic
19C2×C4⋊#3622C2211yescharacteristic
20C2×C4⋊#3622C2211yescharacteristic
21C2×C4⋊#3622C2211yescharacteristic
22C2×C22⋊#3622C2211yescharacteristic
23C2×C22⋊#3622C2211yescharacteristic
24C2×C22⋊#3622C2211yescharacteristic
25C2×C22⋊#3622C2211yescharacteristic
26C2×#10587474C2211yescharacteristic
27C2×#10587474C2211yescharacteristic
28C2×#10587474C2211yescharacteristic
29C2×C4×#3622C2211yescharacteristic
30#11762564C2211yescharacteristic
31#11762564C2211yescharacteristic
32#11762564C2211yescharacteristic
33#11762564C2211yescharacteristic
34#11762564C2211yescharacteristic
35#11762564C2211yescharacteristic
36#11762564C2211yescharacteristic
37#11762564C2211yescharacteristic
38#11762564C2211yescharacteristic
39#11762564C2211yescharacteristic
40#11762564C2211yescharacteristic
41#11762564C2211yescharacteristic
42#11762564C2211yescharacteristic
43#11762564C2211yescharacteristic
44C2×C4×Q8×C17C2211yescharacteristic
45C2×C17×#117C2211yescharacteristic
46C17×#400C411yescharacteristic
47C17×#399C411yescharacteristic
48C17×#398C411yescharacteristic
49C17×#398C411yescharacteristic
50C17×#396C411yescharacteristic
51C17×#396C411yescharacteristic
52C17×#395C411yescharacteristic
53C17×#393C411yescharacteristic
54C4×C4⋊C4×C17C2211yescharacteristic
55C4×C4⋊C4×C17C2211yescharacteristic
56C4×C4⋊C4×C17C2211yescharacteristic
57C4×C22⋊C4×C17C2211yescharacteristic
58C4×C22⋊C4×C17C2211yescharacteristic
59C22×Q8×C17C2×C411yescharacteristic
60C23×C4×C17C2311yescharacteristic
61C4×Q8×C17—41no—
62C4×Q8×C17—41no—
63C17×#117—41no—
64C17×#117—41no—
65C2×C4⋊C4×C17C2×C411yescharacteristic
66C2×C4⋊C4×C17C2×C411yescharacteristic
67C2×C4⋊C4×C17C2×C411yescharacteristic
68C2×C4⋊C4×C17C2×C411yescharacteristic
69C2×C4⋊C4×C17C2×C411yescharacteristic
70C2×C4⋊C4×C17C2×C411yescharacteristic
71C2×C4⋊C4×C17C2×C411yescharacteristic
72C2×C4⋊C4×C17C2×C411yescharacteristic
73C2×C4⋊C4×C17C2×C411yescharacteristic
74C2×C4⋊C4×C17C2×C411yescharacteristic
75C2×C4⋊C4×C17C2×C411yescharacteristic
76C2×C22⋊C4×C17C2×C411yescharacteristic
77C2×C22⋊C4×C17C2×C411yescharacteristic
78C2×C22⋊C4×C17C2×C411yescharacteristic
79C2×C22⋊C4×C17C2×C411yescharacteristic
80C2×C22⋊C4×C17C2×C411yescharacteristic
81C2×C22⋊C4×C17C2×C411yescharacteristic
82C2×C42×C17C2311yescharacteristic
83C2×C42×C17C2311yescharacteristic
84C2×C42×C17C2311yescharacteristic
85C2×C42×C17C2311yescharacteristic
86C2×C42×C17C2311yescharacteristic
87C2×C42×C17C2311yescharacteristic
88C22×#3622C2311yescharacteristic
89C22×#3622C2311yescharacteristic
90C22×#3622C2311yescharacteristic
91C22×#3622C2311yescharacteristic
92C22×#3622C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C17×#95C2×C411yescharacteristic
97C17×#95C2×C411yescharacteristic
98C17×#95C2×C411yescharacteristic
99C17×#95C2×C411yescharacteristic
100C17×#95C2×C411yescharacteristic
101C17×#95C2×C411yescharacteristic
102C17×#95C2×C411yescharacteristic
103C17×#95C2×C411yescharacteristic
104C17×#95C2×C411yescharacteristic
105C17×#95C2×C411yescharacteristic
106C22⋊#3622—41no—
107C22⋊#3622—41no—
108C22⋊#3622—41no—
109C22⋊#3622—41no—
110C4⋊#3622—41no—
111C4⋊#3622—41no—
112C4⋊#3622—41no—
113C4⋊#3622—41no—
114C4⋊#3622—41no—
115C4⋊#3622—41no—
116#10587474—41no—
117#10587474—41no—
118#10587474—41no—
119C4×#3622—41no—
120C24×C17C22×C411yescharacteristic
121C2×Q8×C17—22no—
122C2×Q8×C17—41no—
123C2×Q8×C17—41no—
124C22×C4×C17—21no—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17C22×C411yescharacteristic
131C22×C4×C17C22×C411yescharacteristic
132C22×C4×C17C22×C411yescharacteristic
133C22×C4×C17C22×C411yescharacteristic
134C22×C4×C17C22×C411yescharacteristic
135C22×C4×C17C22×C411yescharacteristic
136C22×C4×C17C22×C411yescharacteristic
137C22×C4×C17C22×C411yescharacteristic
138C22×C4×C17C22×C411yescharacteristic
139C22×C4×C17C22×C411yescharacteristic
140C22×C4×C17C22×C411yescharacteristic
141C22×C4×C17C22×C411yescharacteristic
142C22×C4×C17C22×C411yescharacteristic
143C22×C4×C17C2411yescharacteristic
144C4⋊C4×C17—22no—
145C4⋊C4×C17—41no—
146C4⋊C4×C17—41no—
147C4⋊C4×C17—41no—
148C4⋊C4×C17—41no—
149C4⋊C4×C17—41no—
150C4⋊C4×C17—41no—
151C4⋊C4×C17—41no—
152C4⋊C4×C17—41no—
153C4⋊C4×C17—41no—
154C4⋊C4×C17—41no—
155C22⋊C4×C17—22no—
156C22⋊C4×C17—41no—
157C22⋊C4×C17—41no—
158C22⋊C4×C17—41no—
159C22⋊C4×C17—41no—
160C22⋊C4×C17—41no—
161C42×C17—21no—
162C42×C17—21no—
163C42×C17—21no—
164C42×C17—21no—
165C42×C17—21no—
166C42×C17—21no—
167C42×C17—21no—
168C42×C17—21no—
169C42×C17—21no—
170C42×C17—21no—
171C42×C17—21no—
172C42×C17—21no—
173C2×#3622—21no—
174C2×#3622—21no—
175C2×#3622—21no—
176C2×#3622—21no—
177C2×#3622—21no—
178C2×#3622—21no—
179C2×#3622—21no—
180C2×#3622—21no—
181C2×#3622—21no—
182C2×#3622—21no—
183C2×#3622—21no—
184C2×#3622—21no—
185C2×#3622—21no—
186C2×#3622—21no—
187C2×#3622—21no—
188C2×#3622—21no—
189C2×#3622—21no—
190C2×#3622—21no—
191C2×#3622—21no—
192C2×#3622—21no—
193C2×#3622—21no—
194C2×#3622—21no—
195C2×#3622—21no—
196C2×#3622—21no—
197#23345—171nomaximal, Sylow
198C23×C17—21no—
199C23×C17—21no—
200C23×C17—21no—
201C23×C17—21no—
202C23×C17—21no—
203C23×C17—21no—
204C23×C17—21no—
205C23×C17C23×C411yescharacteristic
206Q8×C17—44no—
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—21no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
244C2×C4×C17—21no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17—41no—
249C2×C4×C17#13112yes—
250C2×C4×C17H(5,𝔽2)11yescharacteristic
251C2×C4×C17H(5,𝔽2)11yescharacteristic
252C2×C4×C17H(5,𝔽2)11yescharacteristic
253C2×C4×C172−1+411yescharacteristic
254C2×C4×C172−1+411yescharacteristic
255C2×C4×C172−1+411yescharacteristic
256#3622—41no—
257#3622—41no—
258#3622—41no—
259#3622—41no—
260#3622—41no—
261#3622—41no—
262#3622—41no—
263#3622—41no—
264#2396D1711yescharacteristic
265#1985—171no—
266#1984—171no—
267#1940—171no—
268#1940—171no—
269#1924—171no—
270#1924—171no—
271#1924—171no—
272#1923—171no—
273#1917—171no—
274#1917—171no—
275#1917—171no—
276#1916—171no—
277#1818—171no—
278#1798—171no—
279C22×C17—41no—
280C22×C17—41no—
281C22×C17—41no—
282C22×C17—41no—
283C22×C17—41no—
284C22×C17—41no—
285C22×C17—41no—
286C22×C17#51811yescharacteristic
287C22×C17#51911yescharacteristic
288C22×C17#52011yescharacteristic
289C22×C17C2×#13111yescharacteristic
290C22×C17#56811yescharacteristic
291C22×C17#56811yescharacteristic
292C22×C17#56811yescharacteristic
293C4×C17—22no—
294C4×C17—41no—
295C4×C17—41no—
296C4×C17—41no—
297C4×C17—41no—
298C4×C17—41no—
299C4×C17—41no—
300C4×C17—41no—
301C4×C17—41no—
302C4×C17—41no—
303C4×C17—41no—
304C4×C17—41no—
305C4×C17—41no—
306C4×C17#56211yescharacteristic
307C4×C17#56211yescharacteristic
308C4×C17#56311yescharacteristic
309C4×C17#56311yescharacteristic
310C2×C4×Q8C2×D1711yescharacteristic
311C2×#117C2×D1711yescharacteristic
312C2×C4⋊C8—171no—
313C2×C4⋊C8—171no—
314C2×C4⋊C8—171no—
315C2×C4⋊C8—171no—
316C2×C4⋊C8—171no—
317C2×C4⋊C8—171no—
318C2×C22⋊C8—171no—
319C2×C22⋊C8—171no—
320C2×C22⋊C8—171no—
321C2×C22⋊C8—171no—
322C2×#97—171no—
323C2×#97—171no—
324C2×#97—171no—
325C2×C4×C8—171no—
326#400Dic1711yescharacteristic
327#399Dic1711yescharacteristic
328#398Dic1711yescharacteristic
329#398Dic1711yescharacteristic
330#396Dic1711yescharacteristic
331#396Dic1711yescharacteristic
332#395Dic1711yescharacteristic
333#393Dic1711yescharacteristic
334C4×C4⋊C4C2×D1711yescharacteristic
335C4×C4⋊C4C2×D1711yescharacteristic
336C4×C4⋊C4C2×D1711yescharacteristic
337C4×C22⋊C4C2×D1711yescharacteristic
338C4×C22⋊C4C2×D1711yescharacteristic
339#336—171no—
340#336—171no—
341#336—171no—
342#336—171no—
343#336—171no—
344#336—171no—
345#336—171no—
346#336—171no—
347#336—171no—
348#336—171no—
349#336—171no—
350#336—171no—
351#336—171no—
352#336—171no—
353C2×C17—81no—
354C2×C17#237811yescharacteristic
355C2×C17#297611yescharacteristic
356C2×C17#298011yescharacteristic
357C2×C17#298511yescharacteristic
358C2×C17#298711yescharacteristic
359C2×C17#299311yescharacteristic
360C2×C17#299511yescharacteristic
361C22×Q8C2×Dic1711yescharacteristic
362C23×C4C22×D1711yescharacteristic
363C22×C8—171no—
364C22×C8—171no—
365C22×C8—171no—
366C22×C8—171no—
367C22×C8—171no—
368C22×C8—171no—
369C22×C8—171no—
370C22×C8—171no—
371C4×Q8—41no—
372C4×Q8—41no—
373#117—41no—
374#117—41no—
375C2×C4⋊C4C2×Dic1711yescharacteristic
376C2×C4⋊C4C2×Dic1711yescharacteristic
377C2×C4⋊C4C2×Dic1711yescharacteristic
378C2×C4⋊C4C2×Dic1711yescharacteristic
379C2×C4⋊C4C2×Dic1711yescharacteristic
380C2×C4⋊C4C2×Dic1711yescharacteristic
381C2×C4⋊C4C2×Dic1711yescharacteristic
382C2×C4⋊C4C2×Dic1711yescharacteristic
383C2×C4⋊C4C2×Dic1711yescharacteristic
384C2×C4⋊C4C2×Dic1711yescharacteristic
385C2×C4⋊C4C2×Dic1711yescharacteristic
386C2×C22⋊C4C2×Dic1711yescharacteristic
387C2×C22⋊C4C2×Dic1711yescharacteristic
388C2×C22⋊C4C2×Dic1711yescharacteristic
389C2×C22⋊C4C2×Dic1711yescharacteristic
390C2×C22⋊C4C2×Dic1711yescharacteristic
391C2×C22⋊C4C2×Dic1711yescharacteristic
392C2×C42C22×D1711yescharacteristic
393C2×C42C22×D1711yescharacteristic
394C2×C42C22×D1711yescharacteristic
395C2×C42C22×D1711yescharacteristic
396C2×C42C22×D1711yescharacteristic
397C2×C42C22×D1711yescharacteristic
398C4⋊C8—681no—
399C4⋊C8—681no—
400C4⋊C8—681no—
401C4⋊C8—681no—
402C4⋊C8—681no—
403C4⋊C8—681no—
404C22⋊C8—681no—
405C22⋊C8—681no—
406C22⋊C8—681no—
407C22⋊C8—681no—
408#97—681no—
409#97—681no—
410#97—681no—
411C4×C8—681no—
412#95C2×Dic1711yescharacteristic
413#95C2×Dic1711yescharacteristic
414#95C2×Dic1711yescharacteristic
415#95C2×Dic1711yescharacteristic
416#95C2×Dic1711yescharacteristic
417#95C2×Dic1711yescharacteristic
418#95C2×Dic1711yescharacteristic
419#95C2×Dic1711yescharacteristic
420#95C2×Dic1711yescharacteristic
421#95C2×Dic1711yescharacteristic
422C17#2334511yescharacteristic, complemented, Sylow
423C24C22×Dic1711yescharacteristic
424C2×Q8—22no—
425C2×Q8—41no—
426C2×Q8—41no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4C22×Dic1711yescharacteristic
434C22×C4C22×Dic1711yescharacteristic
435C22×C4C22×Dic1711yescharacteristic
436C22×C4C22×Dic1711yescharacteristic
437C22×C4C22×Dic1711yescharacteristic
438C22×C4C22×Dic1711yescharacteristic
439C22×C4C22×Dic1711yescharacteristic
440C22×C4C22×Dic1711yescharacteristic
441C22×C4C22×Dic1711yescharacteristic
442C22×C4C22×Dic1711yescharacteristic
443C22×C4C22×Dic1711yescharacteristic
444C22×C4C22×Dic1711yescharacteristic
445C22×C4C22×Dic1711yescharacteristic
446C22×C4C23×D1711yescharacteristic
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—
461C2×C8—341no—
462C2×C8—341no—
463C2×C8—341no—
464C2×C8—341no—
465C2×C8—341no—
466C2×C8—341no—
467C2×C8—341no—
468C2×C8—341no—
469C2×C8—341no—
470C2×C8—341no—
471C4⋊C4—22no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C22⋊C4—22no—
483C22⋊C4—41no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23C23×Dic1711yescharacteristic
508Q8—44no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4#1058763412yes—
552C2×C4#1058769111yescharacteristic
553C2×C4#1058769111yescharacteristic
554C2×C4#1058769411yescharacteristic
555C2×C4#1058769411yescharacteristic
556C2×C4#1058769611yescharacteristic
557C2×C4#1058769711yescharacteristic
558C8—681no—
559C8—681no—
560C8—681no—
561C8—681no—
562C8—681no—
563C8—681no—
564C8—681no—
565C8—681no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22#1176328411yescharacteristic
574C22#1176328711yescharacteristic
575C22#1176329211yescharacteristic
576C22C2×#1058763411yescharacteristic
577C22#1176357611yescharacteristic
578C22#1176357611yescharacteristic
579C22#1176357611yescharacteristic
580C4—22no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4#1176355511yescharacteristic
594C4#1176355511yescharacteristic
595C4#1176356011yescharacteristic
596C4#1176356011yescharacteristic
597C2—81no—
598C2#1453195011yescharacteristic
599C2#1453783211yescharacteristic
600C2#1453785511yescharacteristic
601C2#1453787911yescharacteristic
602C2#1453789011yescharacteristic
603C2#1453793311yescharacteristic
604C2#1453795411yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D17
commutator, socleC23×C17C23×C4
FittingC17×#2396C2
prime radicalC24×C17C22×C4
cosocleC22×C4×C17C24
hypercenter#2396D17
hypocenterC17#23345
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ #2396

Complements

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

NH ≅ G/Ntype
1. #14529725C2semidirect
2. #14529721C2semidirect
3. #14529508C2semidirect
4. #14529508C2semidirect
9. #14529350C2semidirect
10. #14529350C2semidirect
11. #14529350C2semidirect
12. #14529343C2semidirect
422. C17#23345semidirect

Maximal subgroups

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

subgroupindex
#145297252
#145297212
#145295082
#145293992
#145293922
#145293502
#145293432
#145288552
#145287682
C17×#23962
#2334517