Finite Group Databasebeta

← #17084518

Subgroups — #17084518

Order 4352.

818 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763117C211yesmaximal, characteristic, complemented
2#14533950C211yesmaximal, characteristic
3#14533693C211yesmaximal, characteristic
4#14530217C211yesmaximal, characteristic, complemented
5#14530212C211yesmaximal, characteristic, complemented
6#14529988C211yesmaximal, characteristic, complemented
7#14529981C211yesmaximal, characteristic, complemented
8#14529749C211yesmaximal, characteristic, complemented
9#14529554C211yesmaximal, characteristic, complemented
10#14529217C211yesmaximal, characteristic, complemented
11#14529217C211yesmaximal, characteristic, complemented
12#14528817C211yesmaximal, characteristic, complemented
13#14528817C211yesmaximal, characteristic, complemented
14C4×#10587487C211yesmaximal, characteristic, complemented
15C2×C17×#436C211yesmaximal, characteristic
16C2×#10587556C2211yescharacteristic
17C2×#10587610C2211yescharacteristic
18C2×#10587610C2211yescharacteristic
19C2×C4×Dic34C2211yescharacteristic
20#11763117—24no—
21C2×#10587489C2211yescharacteristic
22C2×#10587489C2211yescharacteristic
23C2×#10587488C2211yescharacteristic
24C2×#10587488C2211yescharacteristic
25C2×#10587487C2211yescharacteristic
26C2×#10587487C2211yescharacteristic
27C2×#10587487C2211yescharacteristic
28C2×#10587487C2211yescharacteristic
29#11762747—21no—
30#11762740—21no—
31#11762723—21no—
32#11762723—21no—
33#11762703—21no—
34#11762703—21no—
35#11762697—21no—
36#11762697—21no—
37#11762675C2211yescharacteristic
38#11762669C2211yescharacteristic
39#11762669C2211yescharacteristic
40#11762669C2211yescharacteristic
41#11762669C2211yescharacteristic
42#11762669C2211yescharacteristic
43#11762669C2211yescharacteristic
44C4×C4⋊Dic17C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762572C2211yescharacteristic
51#11762572C2211yescharacteristic
52#11762572C2211yescharacteristic
53C22×M16×C17C2211yescharacteristic
54C22×M16×C17C2211yescharacteristic
55C2×C4×D4×C17C2211yescharacteristic
56C17×#436—24no—
57C2×C17×C4⋊C8C2211yescharacteristic
58C2×C17×C22⋊C8C2211yescharacteristic
59C2×C17×C22⋊C8C2211yescharacteristic
60C2×C4×C8×C17C2211yescharacteristic
61C22×D4×C17D411yescharacteristic
62C23×C4×C17C2311yescharacteristic
63C23×C4×C17C2311yescharacteristic
64C22×Dic34C2311yescharacteristic
65C2×M16×C17—22no—
66C2×M16×C17—22no—
67C2×M16×C17—22no—
68C2×M16×C17—22no—
69C2×M16×C17—41no—
70C2×M16×C17—41no—
71C22×C8×C17C2311yescharacteristic
72C22×C8×C17C2311yescharacteristic
73C22×C8×C17C2311yescharacteristic
74C22×C8×C17C2311yescharacteristic
75C4×D4×C17—22no—
76C4×D4×C17—22no—
77C2×C4⋊C4×C17D411yescharacteristic
78C2×C22⋊C4×C17D411yescharacteristic
79C2×C22⋊C4×C17D411yescharacteristic
80C2×C42×C17C2311yescharacteristic
81C2×C22⋊Dic17—21no—
82C2×C22⋊Dic17—21no—
83C2×C22⋊Dic17—21no—
84C2×C22⋊Dic17—21no—
85C2×C22⋊Dic17—21no—
86C2×C22⋊Dic17—21no—
87C2×C22⋊Dic17—21no—
88C2×C22⋊Dic17—21no—
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96#10587610—22no—
97#10587610—22no—
98#10587610—41no—
99#10587610—41no—
100C2×#63225—21no—
101C2×#63225—21no—
102C2×#63225—21no—
103C2×#63225—21no—
104C2×#63225—21no—
105C2×C4×Dic17—21no—
106C2×C4×Dic17—21no—
107#10587556—22no—
108#10587556—22no—
109C4×Dic34—22no—
110C4×Dic34—22no—
111C17×C4⋊C8D414yes—
112C17×C22⋊C8—22no—
113C17×C22⋊C8—22no—
114C4×C8×C17—22no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587489—22no—
125#10587489—41no—
126#10587488—22no—
127#10587488—41no—
128#10587487—22no—
129#10587487—22no—
130#10587487—22no—
131#10587487—22no—
132C24×C17C2×D411yescharacteristic
133C24×C17C2×D411yescharacteristic
134C2×D4×C17—22no—
135C2×D4×C17—41no—
136C2×D4×C17—41no—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17—21no—
143C22×C4×C17—21no—
144C22×C4×C17—21no—
145C22×C4×C17—21no—
146C22×C4×C17—21no—
147C22×C4×C17—21no—
148C22×C4×C17—21no—
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2×D411yescharacteristic
151C22×C4×C17C2×D411yescharacteristic
152C22×C4×C17C2×D411yescharacteristic
153C22×C4×C17C2411yescharacteristic
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17—21no—
157C22×Dic17—21no—
158C22×Dic17—21no—
159C22×Dic17—21no—
160C22×Dic17—21no—
161C22×Dic17—21no—
162C2×Dic34—21no—
163C2×Dic34—21no—
164C2×Dic34—21no—
165C2×Dic34—21no—
166C2×Dic34—21no—
167C2×Dic34#4112yes—
168M16×C17—44no—
169M16×C17—44no—
170C2×C8×C17—21no—
171C2×C8×C17—21no—
172C2×C8×C17—21no—
173C2×C8×C17—21no—
174C2×C8×C17—21no—
175C2×C8×C17—21no—
176C2×C8×C17—21no—
177C2×C8×C17—21no—
178C2×C8×C17—21no—
179C2×C8×C17C2×D412yes—
180C2×C8×C17C2×D412yes—
181C2×C8×C17#4112yes—
182C4⋊C4×C17—22no—
183C22⋊C4×C17—22no—
184C22⋊C4×C17—22no—
185C42×C17C2×D412yes—
186C42×C17#4112yes—
187C22⋊Dic17—42no—
188C22⋊Dic17—42no—
189C22⋊Dic17—42no—
190C22⋊Dic17—42no—
191C22⋊Dic17—42no—
192C22⋊Dic17—42no—
193C22⋊Dic17—81no—
194C22⋊Dic17—81no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17—21no—
200C4⋊Dic17—21no—
201C4⋊Dic17—21no—
202C4⋊Dic17—21no—
203C4⋊Dic17—21no—
204C4⋊Dic17—21no—
205C4⋊Dic17—21no—
206C4⋊Dic17—21no—
207C4⋊Dic17—21no—
208C4⋊Dic17—21no—
209#63225—42no—
210#63225—42no—
211#63225—42no—
212#63225—42no—
213#63225—42no—
214C4×Dic17—42no—
215C4×Dic17—81no—
216#25345—171nomaximal, Sylow
217C23×C17—21no—
218C23×C17—21no—
219C23×C17—21no—
220C23×C17—21no—
221C23×C17—21no—
222C23×C17—21no—
223C23×C17—21no—
224C23×C17—21no—
225C23×C17—21no—
226C23×C17—21no—
227C23×C17—21no—
228C23×C17—21no—
229C23×C17—21no—
230C23×C17—21no—
231C23×C17C22×D411yescharacteristic
232D4×C17—44no—
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—41no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17#12112yes—
252C2×C4×C17#12112yes—
253C2×C4×C17C22×D411yescharacteristic
254C2×C4×C17C2×#4111yescharacteristic
255C2×C4×C17C2×#4111yescharacteristic
256C2×C4×C17C2×#4111yescharacteristic
257C2×C4×C17H(5,𝔽2)11yescharacteristic
258C2×C4×C172−1+411yescharacteristic
259C2×Dic17—22no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17—41no—
269C2×Dic17—41no—
270C2×Dic17—41no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283Dic34—22no—
284Dic34—22no—
285Dic34—41no—
286Dic34—41no—
287C8×C17—22no—
288C8×C17—22no—
289C8×C17—22no—
290C8×C17—41no—
291C2×#479—171no—
292C2×#436D1711yescharacteristic
293#2562—171no—
294#2536—171no—
295#2077—171no—
296#2076—171no—
297#2036—171no—
298#2035—171no—
299#1991—171no—
300#1949—171no—
301#1894—171no—
302#1894—171no—
303#1809—171no—
304#1809—171no—
305C4×#103—171no—
306C22×C17—41no—
307C22×C17—41no—
308C22×C17—41no—
309C22×C17—41no—
310C22×C17—41no—
311C22×C17—41no—
312C22×C17—41no—
313C22×C17—41no—
314C22×C17—41no—
315C22×C17—41no—
316C22×C17—41no—
317C22×C17—41no—
318C22×C17—41no—
319C22×C17—41no—
320C22×C17C2×#12111yescharacteristic
321C22×C17#54611yescharacteristic
322C22×C17#54711yescharacteristic
323C22×C17#57511yescharacteristic
324C22×C17#57511yescharacteristic
325C22×C17#57711yescharacteristic
326C22×C17#57811yescharacteristic
327C4×C17—22no—
328C4×C17—22no—
329C4×C17—41no—
330C4×C17—41no—
331C4×C17#52911yescharacteristic
332C4×C17#54211yescharacteristic
333C4×C17#54611yescharacteristic
334C4×C17#54811yescharacteristic
335Dic17—42no—
336Dic17—81no—
337Dic17—81no—
338Dic17—81no—
339Dic17—81no—
340Dic17—81no—
341Dic17—81no—
342Dic17—81no—
343C22×M16C2×D1711yescharacteristic
344C22×M16C2×D1711yescharacteristic
345C2×#125—171no—
346C2×C22⋊Q8—171no—
347C2×C22⋊Q8—171no—
348C2×C4×Q8—171no—
349C2×C4×D4C2×D1711yescharacteristic
350#479—344no—
351#436—24no—
352C2×#107—171no—
353C2×#107—171no—
354C2×#106—171no—
355C2×#106—171no—
356C2×C4⋊C8C2×D1711yescharacteristic
357C2×#103—171no—
358C2×#103—171no—
359C2×#103—171no—
360C2×#103—171no—
361C2×C22⋊C8C2×D1711yescharacteristic
362C2×C22⋊C8C2×D1711yescharacteristic
363C2×C4×C8C2×D1711yescharacteristic
364#400—341no—
365#398—341no—
366#394—341no—
367#394—341no—
368#388—341no—
369#388—341no—
370#388—341no—
371#388—341no—
372#382—171no—
373#380—171no—
374#380—171no—
375#380—171no—
376#380—171no—
377#380—171no—
378#380—171no—
379C4×C4⋊C4—171no—
380#340—171no—
381#340—171no—
382#340—171no—
383#340—171no—
384#340—171no—
385#340—171no—
386#340—171no—
387#340—171no—
388C2×C17—81no—
389C2×C17—81no—
390C2×C17#242711yescharacteristic
391C2×C17#305411yescharacteristic
392C2×C17#306311yescharacteristic
393C2×C17#332611yescharacteristic
394C2×C17#332711yescharacteristic
395C2×C17#335111yescharacteristic
396C2×C17#335211yescharacteristic
397C22×Q8—171no—
398C22×D4D6811yescharacteristic
399C23×C4C22×D1711yescharacteristic
400C23×C4C22×D1711yescharacteristic
401C2×M16—22no—
402C2×M16—22no—
403C2×M16—22no—
404C2×M16—22no—
405C2×M16—41no—
406C2×M16—41no—
407C22×C8C22×D1711yescharacteristic
408C22×C8C22×D1711yescharacteristic
409C22×C8C22×D1711yescharacteristic
410C22×C8C22×D1711yescharacteristic
411#125—342no—
412#125—342no—
413C22⋊Q8—342no—
414C22⋊Q8—342no—
415C22⋊Q8—681no—
416C22⋊Q8—681no—
417C4×Q8—342no—
418C4×Q8—342no—
419C4×D4—22no—
420C4×D4—22no—
421C2×C4⋊C4—171no—
422C2×C4⋊C4—171no—
423C2×C4⋊C4—171no—
424C2×C4⋊C4—171no—
425C2×C4⋊C4—171no—
426C2×C4⋊C4—171no—
427C2×C4⋊C4—171no—
428C2×C4⋊C4—341no—
429C2×C4⋊C4—341no—
430C2×C4⋊C4—341no—
431C2×C4⋊C4—341no—
432C2×C4⋊C4—341no—
433C2×C4⋊C4D6811yescharacteristic
434C2×C22⋊C4—341no—
435C2×C22⋊C4—341no—
436C2×C22⋊C4—341no—
437C2×C22⋊C4—341no—
438C2×C22⋊C4—341no—
439C2×C22⋊C4—341no—
440C2×C22⋊C4—341no—
441C2×C22⋊C4—341no—
442C2×C22⋊C4D6811yescharacteristic
443C2×C22⋊C4D6811yescharacteristic
444C2×C42—341no—
445C2×C42—341no—
446C2×C42C22×D1711yescharacteristic
447#107—342no—
448#107—681no—
449#106—342no—
450#106—681no—
451C4⋊C8C22⋊D1714yes—
452#103—342no—
453#103—342no—
454#103—342no—
455#103—342no—
456C22⋊C8—22no—
457C22⋊C8—22no—
458C4×C8—22no—
459#95—341no—
460#95—341no—
461#95—341no—
462#95—341no—
463#95—341no—
464#95—341no—
465#95—341no—
466#95—341no—
467#95—341no—
468C17#2534511yescharacteristic, complemented, Sylow
469C24C2×D6811yescharacteristic
470C24C2×D6811yescharacteristic
471C2×Q8—172no—
472C2×Q8—341no—
473C2×Q8—341no—
474C2×Q8—341no—
475C2×Q8—341no—
476C2×Q8—341no—
477C2×D4—22no—
478C2×D4—41no—
479C2×D4—41no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—341no—
493C22×C4—341no—
494C22×C4—341no—
495C22×C4—341no—
496C22×C4—341no—
497C22×C4—341no—
498C22×C4—341no—
499C22×C4—341no—
500C22×C4C2×D6811yescharacteristic
501C22×C4C2×D6811yescharacteristic
502C22×C4C2×D6811yescharacteristic
503C22×C4C2×D6811yescharacteristic
504C22×C4C23×D1711yescharacteristic
505M16—44no—
506M16—44no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8#6325412yes—
517C2×C8C2×C22⋊D1712yes—
518C2×C8C2×C22⋊D1712yes—
519C4⋊C4—22no—
520C4⋊C4—341no—
521C4⋊C4—341no—
522C4⋊C4—341no—
523C4⋊C4—341no—
524C4⋊C4—341no—
525C4⋊C4—341no—
526C4⋊C4—341no—
527C4⋊C4—341no—
528C4⋊C4—341no—
529C4⋊C4—341no—
530C4⋊C4—341no—
531C4⋊C4—341no—
532C4⋊C4—341no—
533C4⋊C4—341no—
534C4⋊C4—682no—
535C4⋊C4—682no—
536C4⋊C4—682no—
537C4⋊C4—682no—
538C4⋊C4—682no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—682no—
542C22⋊C4—682no—
543C22⋊C4—682no—
544C22⋊C4—682no—
545C22⋊C4—682no—
546C22⋊C4—682no—
547C22⋊C4—1361no—
548C22⋊C4—1361no—
549C42—682no—
550C42—1361no—
551C42#6325212yes—
552C42C2×C22⋊D1712yes—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23C22×D6811yescharacteristic
568Q8—342no—
569Q8—342no—
570Q8—681no—
571Q8—681no—
572D4—44no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—342no—
592C2×C4—681no—
593C2×C4—681no—
594C2×C4—681no—
595C2×C4—681no—
596C2×C4—681no—
597C2×C4—681no—
598C2×C4—681no—
599C2×C4—681no—
600C2×C4—681no—
601C2×C4—681no—
602C2×C4—681no—
603C2×C4—681no—
604C2×C4—681no—
605C2×C4—681no—
606C2×C4—681no—
607C2×C4—681no—
608C2×C4—681no—
609C2×C4—681no—
610C2×C4—681no—
611C2×C4—681no—
612C2×C4—681no—
613C2×C4—681no—
614C2×C4—681no—
615C2×C4#1058761612yes—
616C2×C4#1058761612yes—
617C2×C4C2×#6325211yescharacteristic
618C2×C4C2×#6325411yescharacteristic
619C2×C4#1058769111yescharacteristic
620C2×C4#41×D1711yescharacteristic
621C2×C4#1058769711yescharacteristic
622C2×C4C22×C22⋊D1711yescharacteristic
623C8—22no—
624C8—22no—
625C8—22no—
626C8—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22C2×#1058761611yescharacteristic
642C22#1176345411yescharacteristic
643C22#1176346311yescharacteristic
644C22#1176360911yescharacteristic
645C22#1176360911yescharacteristic
646C22#1176362111yescharacteristic
647C22#1176362611yescharacteristic
648C4—22no—
649C4—22no—
650C4—41no—
651C4—41no—
652C4—682no—
653C4—1361no—
654C4—1361no—
655C4—1361no—
656C4—1361no—
657C4—1361no—
658C4—1361no—
659C4—1361no—
660C4#1176334111yescharacteristic
661C4#1176342411yescharacteristic
662C4#1176345311yescharacteristic
663C4#1176346611yescharacteristic
664C2—81no—
665C2—81no—
666C2#1453235511yescharacteristic
667C2#1453839311yescharacteristic
668C2#1453847511yescharacteristic
669C2#1454103911yescharacteristic
670C2#1454105411yescharacteristic
671C2#1454134611yescharacteristic
672C2#1454135811yescharacteristic
subgroupG/H
centerC23C22×D68
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#436C2
socleC23×C17C22×D4
prime radicalC22×D4×C17D4
hypercenterC2×#436D17
hypocenterC17#25345
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#436

Complements

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

NH ≅ G/Ntype
1. C2×#11763117C2semidirect
4. #14530217C2semidirect
5. #14530212C2semidirect
6. #14529988C2semidirect
7. #14529981C2semidirect
8. #14529749C2semidirect
9. #14529554C2semidirect
10. #14529217C2semidirect
11. #14529217C2semidirect
12. #14528817C2semidirect
13. #14528817C2semidirect
14. C4×#10587487C2semidirect
468. C17#25345semidirect

Maximal subgroups

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

subgroupindex
C2×#117631172
#145339502
#145336932
#145302172
#145302122
#145299882
#145299812
#145297492
#145295542
#145292172
#145288172
C4×#105874872
C2×C17×#4362
#2534517