Finite Group Databasebeta

← #16955654

Subgroups — #16955654

Order 4352.

1710 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762703C212yesmaximal
2C2×C4×C22⋊Dic17C211yesmaximal, characteristic
3#14527021C212yesmaximal
4#14527015C212yesmaximal, complemented
5#14527012C211yesmaximal, characteristic, complemented
6#14527010C211yesmaximal, characteristic, complemented
7#14527008C212yesmaximal, complemented
8#14527007C211yesmaximal, characteristic
9#95×Dic17C211yesmaximal, characteristic, complemented
10#95×Dic17C211yesmaximal, characteristic, complemented
11C17×C22⋊#95C211yesmaximal, characteristic
12C22×#63225C2212yes—
13C22×C22⋊Dic17C2211yescharacteristic
14C22×C22⋊Dic17C2212yes—
15C22×C22⋊Dic17C2211yescharacteristic
16C22×C4×Dic17C2212yes—
17C22×C4×Dic17C2211yescharacteristic
18C22×C4×Dic17C2211yescharacteristic
19C22×C4×Dic17C2212yes—
20#11762703C4116yescomplemented
21C4×C22⋊Dic17C418yescomplemented
22C2×#10587502C2211yescharacteristic
23C2×#10587502C2211yescharacteristic
24C2×#10587502C2212yes—
25C2×#10587502C2212yes—
26C2×#10587502C2212yes—
27C2×#10587502C2212yes—
28C2×#10587502C2211yescharacteristic
29C2×#10587502C2211yescharacteristic
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C24×C4×C17C2211yescharacteristic
33C22×C22⋊C4×C17C2212yes—
34C2×C17×#95C2211yescharacteristic
35C2×C17×#95C2211yescharacteristic
36C2×C17×#95C2212yes—
37C25×C17C2311yescharacteristic
38C23×C4×C17—22no—
39C23×C4×C17—22no—
40C23×C4×C17—42no—
41C23×C4×C17—42no—
42C23×C4×C17C2×C414yes—
43C23×C4×C17C2312yes—
44C23×C4×C17C2312yes—
45C23×C4×C17C2311yescharacteristic
46C23×C4×C17C2311yescharacteristic
47C23×Dic17C2312yes—
48C23×Dic17C2312yes—
49C23×Dic17C2311yescharacteristic
50C23×Dic17C2311yescharacteristic
51C23×Dic17C2311yescharacteristic
52C23×Dic17C2311yescharacteristic
53C2×C22⋊C4×C17—42no—
54C2×C22⋊C4×C17—42no—
55C2×C22⋊C4×C17C2×C418yes—
56C2×C22⋊Dic17—24no—
57C2×C22⋊Dic17—42no—
58C2×C22⋊Dic17—42no—
59C2×C22⋊Dic17C2×C414yes—
60C2×C22⋊Dic17C2×C418yes—
61C2×C22⋊Dic17C2×C418yes—
62C2×C22⋊Dic17C2×C414yes—
63C2×#63225—42no—
64C2×#63225—42no—
65C2×#63225C2×C418yescomplemented
66C2×C4×Dic17—24no—
67C2×C4×Dic17—42no—
68C2×C4×Dic17—42no—
69C2×C4×Dic17—42no—
70C2×C4×Dic17—42no—
71C2×C4×Dic17—42no—
72C2×C4×Dic17—42no—
73C2×C4×Dic17C2×C414yes—
74C2×C4×Dic17C2×C414yes—
75C2×C4×Dic17C2×C418yescomplemented
76C17×#95—22no—
77C17×#95—22no—
78C17×#95—42no—
79#10587502—22no—
80#10587502—22no—
81#10587502—24no—
82#10587502—42no—
83#10587502—42no—
84#10587502—42no—
85#10587502C2×C414yes—
86#10587502C2×C414yes—
87#10587502C2×C418yescomplemented
88#10587502C2×C418yescomplemented
89C24×C17—21no—
90C24×C17—21no—
91C24×C17—22no—
92C24×C17—22no—
93C24×C17—22no—
94C24×C17—21no—
95C24×C17—21no—
96C24×C17—21no—
97C24×C17—21no—
98C24×C17C22×C412yes—
99C24×C17C22×C414yes—
100C24×C17C2411yescharacteristic
101C22×C4×C17—22no—
102C22×C4×C17—22no—
103C22×C4×C17—21no—
104C22×C4×C17—21no—
105C22×C4×C17—22no—
106C22×C4×C17—22no—
107C22×C4×C17—22no—
108C22×C4×C17—22no—
109C22×C4×C17—22no—
110C22×C4×C17—22no—
111C22×C4×C17—22no—
112C22×C4×C17—22no—
113C22×C4×C17—22no—
114C22×C4×C17—22no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—22no—
118C22×C4×C17—22no—
119C22×C4×C17—28no—
120C22×C4×C17—44no—
121C22×C4×C17—44no—
122C22×C4×C17—48no—
123C22×C4×C17—48no—
124C22×C4×C17C22×C412yes—
125C22×C4×C17C22×C414yes—
126C22×C4×C17C22×C414yes—
127C22×C4×C17C22×C412yes—
128C22×C4×C17#4112yes—
129C22×C4×C17#4112yes—
130C22×Dic17—22no—
131C22×Dic17—22no—
132C22×Dic17—21no—
133C22×Dic17—21no—
134C22×Dic17—22no—
135C22×Dic17—21no—
136C22×Dic17—21no—
137C22×Dic17—21no—
138C22×Dic17—21no—
139C22×Dic17—22no—
140C22×Dic17—22no—
141C22×Dic17—22no—
142C22×Dic17—22no—
143C22×Dic17—22no—
144C22×Dic17—22no—
145C22×Dic17—21no—
146C22×Dic17—22no—
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—22no—
151C22×Dic17—22no—
152C22×Dic17—22no—
153C22×Dic17—22no—
154C22×Dic17—21no—
155C22×Dic17C22×C412yes—
156C22×Dic17C22×C414yes—
157C22×Dic17C22×C414yes—
158C22×Dic17C22×C412yes—
159C22×Dic17C22×C412yes—
160C22×Dic17C22×C414yes—
161C22×Dic17C22×C414yes—
162C22×Dic17C22×C412yes—
163C22×Dic17#4114yes—
164C22⋊C4×C17—48no—
165C22⋊Dic17—28no—
166C22⋊Dic17—48no—
167C22⋊Dic17C42116yescomplemented
168#63225—48no—
169C4×Dic17—44no—
170C4×Dic17—44no—
171C4×Dic17—48no—
172C4×Dic17—48no—
173#13788—171nomaximal, Sylow
174C23×C17—22no—
175C23×C17—22no—
176C23×C17—24no—
177C23×C17—24no—
178C23×C17—24no—
179C23×C17—24no—
180C23×C17—22no—
181C23×C17—22no—
182C23×C17—24no—
183C23×C17—22no—
184C23×C17—22no—
185C23×C17—24no—
186C23×C17—42no—
187C23×C17—42no—
188C23×C17—42no—
189C23×C17—42no—
190C23×C17—42no—
191C23×C17—41no—
192C23×C17—41no—
193C23×C17—41no—
194C23×C17—42no—
195C23×C17—41no—
196C23×C17C2×C4214yes—
197C23×C17C23×C411yescharacteristic
198C23×C17C23×C412yes—
199C23×C17C2×#4111yescharacteristic
200C23×C17C2×#4111yescharacteristic
201C23×C17C2×#4112yes—
202C23×C17H(5,𝔽2)12yes—
203C23×C17H(5,𝔽2)11yescharacteristic
204C23×C17H(5,𝔽2)12yes—
205C23×C17H(5,𝔽2)11yescharacteristic
206C23×C172−1+411yescharacteristic
207C23×C172−1+411yescharacteristic
208C2×C4×C17—22no—
209C2×C4×C17—22no—
210C2×C4×C17—24no—
211C2×C4×C17—24no—
212C2×C4×C17—22no—
213C2×C4×C17—24no—
214C2×C4×C17—24no—
215C2×C4×C17—24no—
216C2×C4×C17—24no—
217C2×C4×C17—22no—
218C2×C4×C17—22no—
219C2×C4×C17—24no—
220C2×C4×C17—24no—
221C2×C4×C17—416no—
222C2×C4×C17—42no—
223C2×C4×C17—42no—
224C2×C4×C17—42no—
225C2×C4×C17—42no—
226C2×C4×C17—42no—
227C2×C4×C17—42no—
228C2×C4×C17—42no—
229C2×C4×C17—42no—
230C2×C4×C17—42no—
231C2×C4×C17—42no—
232C2×C4×C17#11714yes—
233C2×Dic17—24no—
234C2×Dic17—22no—
235C2×Dic17—22no—
236C2×Dic17—22no—
237C2×Dic17—24no—
238C2×Dic17—24no—
239C2×Dic17—22no—
240C2×Dic17—22no—
241C2×Dic17—22no—
242C2×Dic17—24no—
243C2×Dic17—24no—
244C2×Dic17—24no—
245C2×Dic17—22no—
246C2×Dic17—24no—
247C2×Dic17—24no—
248C2×Dic17—24no—
249C2×Dic17—24no—
250C2×Dic17—22no—
251C2×Dic17—24no—
252C2×Dic17—22no—
253C2×Dic17—24no—
254C2×Dic17—42no—
255C2×Dic17—42no—
256C2×Dic17—42no—
257C2×Dic17—42no—
258C2×Dic17—42no—
259C2×Dic17—42no—
260C2×Dic17—42no—
261C2×Dic17—42no—
262C2×Dic17—42no—
263C2×Dic17—42no—
264C2×Dic17C2×C4214yes—
265C2×Dic17C2×C4214yes—
266C2×Dic17#11714yes—
267C2×#388—172no—
268C2×C4×C22⋊C4—171no—
269#1437—172no—
270#1436—172no—
271C4⋊#95—171no—
272#1434—172no—
273#1434—171no—
274C22⋊#95—171no—
275C22⋊#95D1711yescharacteristic
276C4×#95—171no—
277C4×#95—171no—
278C22×C17—24no—
279C22×C17—24no—
280C22×C17—24no—
281C22×C17—42no—
282C22×C17—42no—
283C22×C17—42no—
284C22×C17—42no—
285C22×C17—44no—
286C22×C17—44no—
287C22×C17—44no—
288C22×C17—44no—
289C22×C17C22×C4211yescharacteristic
290C22×C17C2×#11711yescharacteristic
291C22×C17C2×#11711yescharacteristic
292C22×C17C4×#4112yes—
293C22×C17C4×#4112yes—
294C22×C17#51812yes—
295C22×C17#51811yescharacteristic
296C22×C17#51811yescharacteristic
297C22×C17#51812yes—
298C22×C17#52012yes—
299C22×C17#52011yescharacteristic
300C22×C17#52011yescharacteristic
301C22×C17#52012yes—
302C22×C17#53812yes—
303C22×C17#53812yes—
304C22×C17#53812yes—
305C22×C17#53912yes—
306C22×C17#54012yes—
307C22×C17#54011yescharacteristic
308C22×C17#54111yescharacteristic
309C22×C17#54211yescharacteristic
310C22×C17#54212yes—
311C22×C17#54211yescharacteristic
312C4×C17—24no—
313C4×C17—42no—
314C4×C17—44no—
315C4×C17—44no—
316Dic17—24no—
317Dic17—24no—
318Dic17—24no—
319Dic17—42no—
320Dic17—44no—
321Dic17—44no—
322C24×C4C2×D1711yescharacteristic
323C22×C4⋊C4—172no—
324C22×C22⋊C4—171no—
325C22×C22⋊C4—171no—
326C22×C22⋊C4—172no—
327C22×C22⋊C4C2×D1712yes—
328C22×C42—171no—
329C22×C42—172no—
330C22×C42—171no—
331C22×C42—172no—
332#388—1716no—
333C4×C22⋊C4—178no—
334C2×#95—172no—
335C2×#95—171no—
336C2×#95—172no—
337C2×#95—171no—
338C2×#95—171no—
339C2×#95—171no—
340C2×#95—172no—
341C2×#95—172no—
342C2×#95—172no—
343C2×#95—172no—
344C2×#95C2×D1711yescharacteristic
345C2×#95C2×D1711yescharacteristic
346C2×#95C2×D1712yes—
347C2×C17—44no—
348C2×C17C4×#11711yescharacteristic
349C2×C17#227511yescharacteristic
350C2×C17#227511yescharacteristic
351C2×C17#231211yescharacteristic
352C2×C17#231411yescharacteristic
353C2×C17#231811yescharacteristic
354C2×C17#231911yescharacteristic
355C2×C17#233312yes—
356C2×C17#233412yes—
357C2×C17#233512yes—
358C2×C17#233712yes—
359C25C22×D1711yescharacteristic
360C23×C4—22no—
361C23×C4—22no—
362C23×C4—42no—
363C23×C4—42no—
364C23×C4—171no—
365C23×C4—171no—
366C23×C4—171no—
367C23×C4—172no—
368C23×C4—172no—
369C23×C4—171no—
370C23×C4C4×D1714yes—
371C23×C4C22×D1712yes—
372C23×C4C22×D1712yes—
373C23×C4C22×D1711yescharacteristic
374C23×C4C22×D1711yescharacteristic
375C2×C4⋊C4—178no—
376C2×C4⋊C4—682no—
377C2×C4⋊C4—682no—
378C2×C22⋊C4—42no—
379C2×C22⋊C4—42no—
380C2×C22⋊C4—174no—
381C2×C22⋊C4—178no—
382C2×C22⋊C4—174no—
383C2×C22⋊C4—178no—
384C2×C22⋊C4—344no—
385C2×C22⋊C4—682no—
386C2×C22⋊C4—682no—
387C2×C22⋊C4C4×D1718yes—
388C2×C42—174no—
389C2×C42—174no—
390C2×C42—178no—
391C2×C42—344no—
392C2×C42—682no—
393C2×C42—682no—
394C2×C42—682no—
395C2×C42—682no—
396C2×C42—682no—
397C2×C42—682no—
398#95—22no—
399#95—22no—
400#95—42no—
401#95—174no—
402#95—174no—
403#95—178no—
404#95—178no—
405#95—344no—
406#95—342no—
407#95—342no—
408#95—682no—
409#95—682no—
410#95—682no—
411C17#1378811yescharacteristic, complemented, Sylow
412C24—21no—
413C24—21no—
414C24—21no—
415C24—22no—
416C24—22no—
417C24—21no—
418C24—22no—
419C24—21no—
420C24—21no—
421C24C2×C4×D1714yes—
422C24C2×C4×D1712yes—
423C24C23×D1711yescharacteristic
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—28no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—174no—
448C22×C4—172no—
449C22×C4—172no—
450C22×C4—172no—
451C22×C4—174no—
452C22×C4—174no—
453C22×C4—172no—
454C22×C4—174no—
455C22×C4—174no—
456C22×C4—342no—
457C22×C4—342no—
458C22×C4—341no—
459C22×C4—342no—
460C22×C4—341no—
461C22×C4—341no—
462C22×C4—341no—
463C22×C4—341no—
464C22×C4—341no—
465C22×C4—342no—
466C22×C4—342no—
467C22×C4—342no—
468C22×C4—342no—
469C22×C4—342no—
470C22×C4—342no—
471C22×C4—342no—
472C22×C4—342no—
473C22×C4—342no—
474C22×C4—342no—
475C22×C4—342no—
476C22×C4—342no—
477C22×C4—341no—
478C22×C4—342no—
479C22×C4—342no—
480C22×C4—341no—
481C22×C4C2×C4×D1714yes—
482C22×C4C2×C4×D1712yes—
483C22×C4C2×C4×D1714yes—
484C22×C4C2×C4×D1712yes—
485C22×C4#6325412yes—
486C22×C4#6325412yes—
487C4⋊C4—688no—
488C22⋊C4—48no—
489C22⋊C4—1716no—
490C22⋊C4—348no—
491C22⋊C4—688no—
492C42—684no—
493C42—684no—
494C42—688no—
495C42—688no—
496C23—22no—
497C23—22no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—41no—
518C23C42×D1714yes—
519C23C22×C4×D1711yescharacteristic
520C23C22×C4×D1712yes—
521C23C2×#6325411yescharacteristic
522C23C2×#6325411yescharacteristic
523C23#1058769112yes—
524C23#1058769111yescharacteristic
525C23#1058769112yes—
526C23#1058769411yescharacteristic
527C23#41×D1712yes—
528C23#1058769611yescharacteristic
529C23#1058769711yescharacteristic
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—416no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—174no—
555C2×C4—174no—
556C2×C4—174no—
557C2×C4—344no—
558C2×C4—342no—
559C2×C4—342no—
560C2×C4—342no—
561C2×C4—344no—
562C2×C4—344no—
563C2×C4—342no—
564C2×C4—342no—
565C2×C4—344no—
566C2×C4—342no—
567C2×C4—344no—
568C2×C4—344no—
569C2×C4—342no—
570C2×C4—344no—
571C2×C4—344no—
572C2×C4—344no—
573C2×C4—344no—
574C2×C4—342no—
575C2×C4—344no—
576C2×C4—342no—
577C2×C4—344no—
578C2×C4—682no—
579C2×C4—682no—
580C2×C4—682no—
581C2×C4—682no—
582C2×C4—682no—
583C2×C4—682no—
584C2×C4—682no—
585C2×C4—682no—
586C2×C4—682no—
587C2×C4—682no—
588C2×C4#1058756314yes—
589C22—24no—
590C22—24no—
591C22—24no—
592C22—44no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22C2×C42×D1711yescharacteristic
601C22#117×D1711yescharacteristic
602C22C2×#1058756311yescharacteristic
603C22C4×#6325412yes—
604C22C4×#6325412yes—
605C22#1176328112yes—
606C22#1176328212yes—
607C22#1176328211yescharacteristic
608C22#1176328211yescharacteristic
609C22#1176328812yes—
610C22#1176328911yescharacteristic
611C22#1176329011yescharacteristic
612C22#1176329012yes—
613C22#1176339112yes—
614C22#1176339412yes—
615C22#1176339412yes—
616C22#1176340512yes—
617C22#1176341112yes—
618C22#1176341211yescharacteristic
619C22#1176341911yescharacteristic
620C22#1176342611yescharacteristic
621C22#1176342712yes—
622C22#1176343211yescharacteristic
623C4—24no—
624C4—44no—
625C4—42no—
626C4—44no—
627C4—344no—
628C4—344no—
629C4—344no—
630C4—682no—
631C4—684no—
632C4—684no—
633C2—44no—
634C2C4×#1058756311yescharacteristic
635C2#1453114211yescharacteristic
636C2#1453114311yescharacteristic
637C2#1453138011yescharacteristic
638C2#1453139111yescharacteristic
639C2#1453141911yescharacteristic
640C2#1453142411yescharacteristic
641C2#1453155012yes—
642C2#1453156012yes—
643C2#1453156912yes—
644C2#1453159312yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC22×C17C22×C42
FittingC17×C22⋊#95C2
socle, cosocleC24×C17C24
prime radicalC25×C17C23
hypercenterC22⋊#95D17
hypocenterC17#13788
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
4. #14527015C2semidirect
5. #14527012C2semidirect
6. #14527010C2semidirect
7. #14527008C2semidirect
9. #95×Dic17C2semidirect
10. #95×Dic17C2semidirect
20. #11762703C4semidirect
21. C4×C22⋊Dic17C4semidirect
65. C2×#63225C2×C4semidirect
75. C2×C4×Dic17C2×C4semidirect
87. #10587502C2×C4semidirect
88. #10587502C2×C4semidirect
167. C22⋊Dic17C42semidirect
411. C17#13788semidirect

Maximal subgroups

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

subgroupindex
C2×#117627032
C2×C4×C22⋊Dic172
#145270212
#145270152
#145270122
#145270102
#145270082
#145270072
#95×Dic172
C17×C22⋊#952
#1378817