Finite Group Databasebeta

← #16961366

Subgroups — #16961366

Order 4352.

1650 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762723C211yesmaximal, characteristic
2C2×#11762693C211yesmaximal, characteristic, complemented
3#14527043C211yesmaximal, characteristic, complemented
4#14527034C211yesmaximal, characteristic, complemented
5#14527031C212yesmaximal, complemented
6#14527027C212yesmaximal, complemented
7#14527025C211yesmaximal, characteristic
8#14527021C212yesmaximal, complemented
9#14527021C212yesmaximal, complemented
10C4×#10587502C211yesmaximal, characteristic, complemented
11C2×C17×#390C211yesmaximal, characteristic
12C22×#63225C2211yescharacteristic
13C22×#63225C2211yescharacteristic
14C22×C22⋊Dic17C2212yes—
15C22×C22⋊Dic17C2212yes—
16C22×C22⋊Dic17C2212yes—
17C22×C22⋊Dic17C2212yes—
18C22×C4×Dic17C2212yes—
19#11762723C418yescomplemented
20#11762693—24no—
21C2×#10587502C2211yescharacteristic
22C2×#10587502C2212yes—
23C2×#10587502C2211yescharacteristic
24C2×#10587502C2211yescharacteristic
25C2×#10587502C2211yescharacteristic
26C2×#10587502C2211yescharacteristic
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2211yescharacteristic
32C23×D4×C17C2211yescharacteristic
33C22×C4⋊C4×C17C2211yescharacteristic
34C22×C22⋊C4×C17C2212yes—
35C22×C22⋊C4×C17C2212yes—
36C22×C42×C17C2211yescharacteristic
37C17×#390—24no—
38C25×C17C2312yes—
39C22×D4×C17—22no—
40C22×D4×C17—22no—
41C22×D4×C17—42no—
42C22×D4×C17—42no—
43C22×D4×C17C2×C414yes—
44C23×C4×C17C2312yes—
45C23×C4×C17C2311yescharacteristic
46C23×C4×C17C2311yescharacteristic
47C23×C4×C17C2311yescharacteristic
48C23×Dic17C2311yescharacteristic
49C23×Dic17C2311yescharacteristic
50C23×Dic17C2312yes—
51C23×Dic17C2311yescharacteristic
52C23×Dic17C2311yescharacteristic
53C23×Dic17C2312yes—
54C2×C4⋊C4×C17—22no—
55C2×C4⋊C4×C17—42no—
56C2×C22⋊C4×C17—24no—
57C2×C22⋊C4×C17—24no—
58C2×C22⋊C4×C17—24no—
59C2×C22⋊C4×C17—24no—
60C2×C22⋊C4×C17—42no—
61C2×C22⋊C4×C17—42no—
62C2×C42×C17—42no—
63C2×C42×C17D414yes—
64C2×C22⋊Dic17—24no—
65C2×C22⋊Dic17—24no—
66C2×C22⋊Dic17—42no—
67C2×C22⋊Dic17—42no—
68C2×C22⋊Dic17—42no—
69C2×C22⋊Dic17—42no—
70C2×C22⋊Dic17—42no—
71C2×C22⋊Dic17—42no—
72C2×C22⋊Dic17—42no—
73C2×C22⋊Dic17—42no—
74C2×C22⋊Dic17C2×C418yes—
75C2×C22⋊Dic17C2×C418yes—
76C2×#63225—22no—
77C2×#63225—22no—
78C2×#63225—22no—
79C2×#63225—42no—
80C2×#63225C2×C414yes—
81C2×C4×Dic17—24no—
82C2×C4×Dic17—42no—
83C2×C4×Dic17—42no—
84#10587502—22no—
85#10587502—22no—
86#10587502—22no—
87#10587502—22no—
88#10587502—22no—
89#10587502—24no—
90#10587502—42no—
91#10587502—42no—
92#10587502—42no—
93#10587502—42no—
94#10587502C2×C414yes—
95C24×C17—22no—
96C24×C17—22no—
97C24×C17—22no—
98C24×C17—22no—
99C24×C17—22no—
100C24×C17—22no—
101C24×C17—22no—
102C24×C17—22no—
103C24×C17—22no—
104C24×C17—22no—
105C24×C17—22no—
106C24×C17—22no—
107C24×C17—22no—
108C24×C17—22no—
109C24×C17C22×C414yes—
110C24×C17C2411yescharacteristic
111C2×D4×C17—28no—
112C2×D4×C17—48no—
113C2×D4×C17—44no—
114C2×D4×C17—44no—
115C2×D4×C17—48no—
116C22×C4×C17—22no—
117C22×C4×C17—22no—
118C22×C4×C17—21no—
119C22×C4×C17—22no—
120C22×C4×C17—21no—
121C22×C4×C17—22no—
122C22×C4×C17—22no—
123C22×C4×C17—21no—
124C22×C4×C17—21no—
125C22×C4×C17—22no—
126C22×C4×C17—22no—
127C22×C4×C17—22no—
128C22×C4×C17—22no—
129C22×C4×C17—22no—
130C22×C4×C17—22no—
131C22×C4×C17—22no—
132C22×C4×C17C22×C412yes—
133C22×C4×C17C2×D412yes—
134C22×C4×C17C2×D412yes—
135C22×C4×C17C2×D412yes—
136C22×C4×C17#4112yes—
137C22×C4×C17#4114yes—
138C22×Dic17—21no—
139C22×Dic17—21no—
140C22×Dic17—22no—
141C22×Dic17—22no—
142C22×Dic17—22no—
143C22×Dic17—22no—
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—22no—
151C22×Dic17—22no—
152C22×Dic17—22no—
153C22×Dic17—22no—
154C22×Dic17—22no—
155C22×Dic17—22no—
156C22×Dic17—22no—
157C22×Dic17—21no—
158C22×Dic17—21no—
159C22×Dic17—21no—
160C22×Dic17—21no—
161C22×Dic17—21no—
162C22×Dic17—22no—
163C22×Dic17—22no—
164C22×Dic17—21no—
165C22×Dic17—22no—
166C22×Dic17—22no—
167C22×Dic17—21no—
168C22×Dic17—21no—
169C22×Dic17—21no—
170C22×Dic17C22×C414yes—
171C22×Dic17C22×C412yes—
172C22×Dic17C22×C412yes—
173C22×Dic17#4114yes—
174C4⋊C4×C17—44no—
175C22⋊C4×C17—48no—
176C22⋊C4×C17—48no—
177C42×C17—44no—
178C22⋊Dic17—48no—
179C22⋊Dic17—48no—
180C22⋊Dic17—48no—
181C22⋊Dic17—48no—
182#63225—28no—
183#63225—44no—
184C4×Dic17—48no—
185#14325—171nomaximal, Sylow
186C23×C17—24no—
187C23×C17—24no—
188C23×C17—24no—
189C23×C17—24no—
190C23×C17—24no—
191C23×C17—24no—
192C23×C17—24no—
193C23×C17—42no—
194C23×C17—42no—
195C23×C17—42no—
196C23×C17—42no—
197C23×C17—42no—
198C23×C17—42no—
199C23×C17—42no—
200C23×C17—42no—
201C23×C17—42no—
202C23×C17—42no—
203C23×C17—42no—
204C23×C17—42no—
205C23×C17—42no—
206C23×C17—42no—
207C23×C17—42no—
208C23×C17—42no—
209C23×C17—42no—
210C23×C17—42no—
211C23×C17—42no—
212C23×C17—42no—
213C23×C17—42no—
214C23×C17—42no—
215C23×C17—42no—
216C23×C17—42no—
217C23×C17—42no—
218C23×C17—42no—
219C23×C17—42no—
220C23×C17—42no—
221C23×C17C23×C411yescharacteristic
222C23×C17C22×D411yescharacteristic
223C23×C17C2×#4111yescharacteristic
224C23×C17C2×#4112yes—
225C23×C17C2×#4112yes—
226C23×C17H(5,𝔽2)12yes—
227C23×C17H(5,𝔽2)12yes—
228C23×C17H(5,𝔽2)11yescharacteristic
229C23×C17H(5,𝔽2)11yescharacteristic
230C23×C17H(5,𝔽2)11yescharacteristic
231C23×C172−1+411yescharacteristic
232D4×C17—416no—
233C2×C4×C17—24no—
234C2×C4×C17—22no—
235C2×C4×C17—22no—
236C2×C4×C17—22no—
237C2×C4×C17—22no—
238C2×C4×C17—24no—
239C2×C4×C17—24no—
240C2×C4×C17—24no—
241C2×C4×C17—22no—
242C2×C4×C17—22no—
243C2×C4×C17—22no—
244C2×C4×C17—42no—
245C2×C4×C17—42no—
246C2×C4×C17—42no—
247C2×C4×C17—42no—
248C2×C4×C17—42no—
249C2×C4×C17—42no—
250C2×C4×C17—42no—
251C2×C4×C17—42no—
252C2×C4×C17—42no—
253C2×C4×C17C4×D414yes—
254C2×C4×C17#12414yes—
255C2×Dic17—22no—
256C2×Dic17—22no—
257C2×Dic17—24no—
258C2×Dic17—22no—
259C2×Dic17—24no—
260C2×Dic17—24no—
261C2×Dic17—24no—
262C2×Dic17—22no—
263C2×Dic17—24no—
264C2×Dic17—24no—
265C2×Dic17—24no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17—41no—
269C2×Dic17—41no—
270C2×Dic17—42no—
271C2×Dic17—42no—
272C2×Dic17—42no—
273C2×Dic17—42no—
274C2×Dic17—42no—
275C2×Dic17—42no—
276C2×Dic17—41no—
277C2×Dic17—42no—
278C2×Dic17—42no—
279C2×Dic17—42no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283C2×Dic17—41no—
284C2×Dic17—41no—
285C2×Dic17—41no—
286C2×Dic17—41no—
287C2×Dic17—42no—
288C2×Dic17—42no—
289C2×Dic17—42no—
290C2×Dic17—42no—
291C2×#394—171no—
292C2×#390D1711yescharacteristic
293C2×#387—171no—
294#1442—171no—
295#1441—171no—
296#1440—172no—
297#1439—172no—
298#1438—171no—
299#1437—172no—
300#1437—172no—
301C4×#95—171no—
302C22×C17—44no—
303C22×C17—44no—
304C22×C17—44no—
305C22×C17—44no—
306C22×C17—44no—
307C22×C17—44no—
308C22×C17—44no—
309C22×C17—82no—
310C22×C17—82no—
311C22×C17—82no—
312C22×C17—82no—
313C22×C17—82no—
314C22×C17—82no—
315C22×C17—82no—
316C22×C17—82no—
317C22×C17C2×C4×D411yescharacteristic
318C22×C17C4×#4112yes—
319C22×C17#51811yescharacteristic
320C22×C17#51812yes—
321C22×C17#52011yescharacteristic
322C22×C17C2×#12411yescharacteristic
323C22×C17#52912yes—
324C22×C17#53211yescharacteristic
325C22×C17#53411yescharacteristic
326C22×C17#53511yescharacteristic
327C22×C17#53511yescharacteristic
328C22×C17#53711yescharacteristic
329C22×C17#53812yes—
330C22×C17#53812yes—
331C22×C17#53812yes—
332C22×C17#53812yes—
333C22×C17#53912yes—
334C22×C17#54012yes—
335C22×C17#54011yescharacteristic
336C22×C17#54012yes—
337C22×C17#54012yes—
338C22×C17#54212yes—
339C22×C17#54211yescharacteristic
340C4×C17—24no—
341C4×C17—24no—
342C4×C17—42no—
343C4×C17—82no—
344Dic17—42no—
345Dic17—42no—
346Dic17—44no—
347Dic17—44no—
348Dic17—81no—
349Dic17—81no—
350C23×D4C2×D1711yescharacteristic
351C22×C4⋊C4—171no—
352C22×C4⋊C4—171no—
353C22×C4⋊C4C2×D1711yescharacteristic
354C22×C22⋊C4—172no—
355C22×C22⋊C4—172no—
356C22×C22⋊C4—172no—
357C22×C22⋊C4—172no—
358C22×C22⋊C4C2×D1712yes—
359C22×C22⋊C4C2×D1712yes—
360C22×C42—172no—
361C22×C42C2×D1711yescharacteristic
362#394—178no—
363#390—24no—
364#387—344no—
365C2×#95—171no—
366C2×#95—171no—
367C2×#95—172no—
368C2×#95—171no—
369C2×#95—171no—
370C2×#95—171no—
371C2×#95—171no—
372C2×#95—172no—
373C2×#95—172no—
374C2×#95—172no—
375C2×#95—172no—
376C2×C17—84no—
377C2×C17C4×#12411yescharacteristic
378C2×C17#232111yescharacteristic
379C2×C17#232411yescharacteristic
380C2×C17#233312yes—
381C2×C17#233512yes—
382C2×C17#263911yescharacteristic
383C2×C17#264111yescharacteristic
384C2×C17#264812yes—
385C2×C17#265112yes—
386C2×C17#266111yescharacteristic
387C2×C17#266211yescharacteristic
388C25C22×D1712yes—
389C22×D4—22no—
390C22×D4—22no—
391C22×D4—42no—
392C22×D4—42no—
393C22×D4C4×D1714yes—
394C23×C4—171no—
395C23×C4—171no—
396C23×C4—172no—
397C23×C4—172no—
398C23×C4—171no—
399C23×C4—171no—
400C23×C4C22×D1712yes—
401C23×C4C22×D1711yescharacteristic
402C23×C4C22×D1711yescharacteristic
403C23×C4C22×D1711yescharacteristic
404C2×C4⋊C4—22no—
405C2×C4⋊C4—42no—
406C2×C4⋊C4—174no—
407C2×C4⋊C4—342no—
408C2×C4⋊C4—342no—
409C2×C4⋊C4—342no—
410C2×C4⋊C4—682no—
411C2×C22⋊C4—24no—
412C2×C22⋊C4—24no—
413C2×C22⋊C4—24no—
414C2×C22⋊C4—24no—
415C2×C22⋊C4—42no—
416C2×C22⋊C4—42no—
417C2×C22⋊C4—178no—
418C2×C22⋊C4—178no—
419C2×C22⋊C4—344no—
420C2×C22⋊C4—344no—
421C2×C22⋊C4—682no—
422C2×C22⋊C4—682no—
423C2×C22⋊C4—682no—
424C2×C22⋊C4—682no—
425C2×C22⋊C4—682no—
426C2×C22⋊C4—682no—
427C2×C22⋊C4—682no—
428C2×C22⋊C4—682no—
429C2×C42—42no—
430C2×C42—344no—
431C2×C42—682no—
432C2×C42—682no—
433C2×C42C22⋊D1714yes—
434#95—174no—
435#95—344no—
436#95—342no—
437#95—342no—
438#95—342no—
439#95—342no—
440#95—342no—
441#95—682no—
442#95—682no—
443#95—682no—
444#95—682no—
445C17#1432511yescharacteristic, complemented, Sylow
446C24—22no—
447C24—22no—
448C24—22no—
449C24—22no—
450C24—22no—
451C24—22no—
452C24—22no—
453C24—22no—
454C24—22no—
455C24—22no—
456C24—22no—
457C24—22no—
458C24—22no—
459C24—22no—
460C24C2×C4×D1714yes—
461C24C23×D1711yescharacteristic
462C2×D4—28no—
463C2×D4—48no—
464C2×D4—44no—
465C2×D4—44no—
466C2×D4—48no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—174no—
484C22×C4—172no—
485C22×C4—172no—
486C22×C4—174no—
487C22×C4—342no—
488C22×C4—341no—
489C22×C4—341no—
490C22×C4—342no—
491C22×C4—342no—
492C22×C4—341no—
493C22×C4—342no—
494C22×C4—342no—
495C22×C4—342no—
496C22×C4—342no—
497C22×C4—342no—
498C22×C4—342no—
499C22×C4—342no—
500C22×C4—341no—
501C22×C4—342no—
502C22×C4—342no—
503C22×C4—342no—
504C22×C4—342no—
505C22×C4—341no—
506C22×C4—341no—
507C22×C4—341no—
508C22×C4—341no—
509C22×C4—341no—
510C22×C4—342no—
511C22×C4—341no—
512C22×C4—342no—
513C22×C4—342no—
514C22×C4—342no—
515C22×C4—341no—
516C22×C4—341no—
517C22×C4—341no—
518C22×C4—341no—
519C22×C4C2×C4×D1712yes—
520C22×C4#6325212yes—
521C22×C4#6325414yes—
522C22×C4C2×C22⋊D1712yes—
523C22×C4C2×C22⋊D1712yes—
524C22×C4C2×C22⋊D1712yes—
525C4⋊C4—44no—
526C4⋊C4—348no—
527C4⋊C4—684no—
528C22⋊C4—48no—
529C22⋊C4—48no—
530C22⋊C4—688no—
531C22⋊C4—688no—
532C22⋊C4—688no—
533C22⋊C4—688no—
534C42—44no—
535C42—688no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23C22×C4×D1711yescharacteristic
572C23C2×#6325211yescharacteristic
573C23C2×#6325412yes—
574C23#1058769111yescharacteristic
575C23#1058769112yes—
576C23#1058769112yes—
577C23#1058769111yescharacteristic
578C23#41×D1712yes—
579C23#1058769611yescharacteristic
580C23#1058769711yescharacteristic
581C23C22×C22⋊D1711yescharacteristic
582D4—416no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—342no—
604C2×C4—342no—
605C2×C4—344no—
606C2×C4—342no—
607C2×C4—344no—
608C2×C4—344no—
609C2×C4—344no—
610C2×C4—342no—
611C2×C4—344no—
612C2×C4—344no—
613C2×C4—344no—
614C2×C4—681no—
615C2×C4—681no—
616C2×C4—681no—
617C2×C4—681no—
618C2×C4—682no—
619C2×C4—682no—
620C2×C4—682no—
621C2×C4—682no—
622C2×C4—682no—
623C2×C4—682no—
624C2×C4—682no—
625C2×C4—681no—
626C2×C4—682no—
627C2×C4—682no—
628C2×C4—681no—
629C2×C4—681no—
630C2×C4—681no—
631C2×C4—681no—
632C2×C4—681no—
633C2×C4—681no—
634C2×C4—681no—
635C2×C4—682no—
636C2×C4—682no—
637C2×C4—682no—
638C2×C4—682no—
639C2×C4C4×C22⋊D1714yes—
640C2×C4#1058762214yes—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22C2×C4×C22⋊D1711yescharacteristic
657C22C4×#6325412yes—
658C22#1176328111yescharacteristic
659C22#1176328212yes—
660C22#1176328811yescharacteristic
661C22C2×#1058762211yescharacteristic
662C22#1176334112yes—
663C22#1176335511yescharacteristic
664C22#1176336811yescharacteristic
665C22#1176337611yescharacteristic
666C22#1176337611yescharacteristic
667C22#1176338711yescharacteristic
668C22#1176339112yes—
669C22#1176339312yes—
670C22#1176339612yes—
671C22#1176339712yes—
672C22#1176340412yes—
673C22#1176340912yes—
674C22#1176341011yescharacteristic
675C22#1176341312yes—
676C22#1176341312yes—
677C22#1176342312yes—
678C22#1176342411yescharacteristic
679C4—24no—
680C4—24no—
681C4—42no—
682C4—82no—
683C4—682no—
684C4—684no—
685C4—682no—
686C4—684no—
687C4—1361no—
688C4—1361no—
689C2—84no—
690C2C4×#1058762211yescharacteristic
691C2#1453144211yescharacteristic
692C2#1453147111yescharacteristic
693C2#1453154312yes—
694C2#1453157112yes—
695C2#1453471711yescharacteristic
696C2#1453472911yescharacteristic
697C2#1453477612yes—
698C2#1453482112yes—
699C2#1453491311yescharacteristic
700C2#1453492311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC2×C17×#390C2
socle, cosocleC24×C17C24
prime radicalC23×D4×C17C22
hypercenterC2×#390D17
hypocenterC17#14325
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C2×#11762693C2semidirect
3. #14527043C2semidirect
4. #14527034C2semidirect
5. #14527031C2semidirect
6. #14527027C2semidirect
8. #14527021C2semidirect
9. #14527021C2semidirect
10. C4×#10587502C2semidirect
19. #11762723C4semidirect
445. C17#14325semidirect

Maximal subgroups

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

subgroupindex
C2×#117627232
C2×#117626932
#145270432
#145270342
#145270312
#145270272
#145270252
#145270212
C4×#105875022
C2×C17×#3902
#1432517