Finite Group Databasebeta

← #16958041

Subgroups — #16958041

Order 4352.

2006 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762684C211yesmaximal, characteristic
2C2×#11762684C211yesmaximal, characteristic
3C2×C22⋊C4×Dic17C211yesmaximal, characteristic
4C22×#10587502C211yesmaximal, characteristic
5#14527041C212yesmaximal, complemented
6#14527037C212yesmaximal, complemented
7#14527018C212yesmaximal, complemented
8#14527013C212yesmaximal, complemented
9#14527007C211yesmaximal, characteristic
10#14527006C211yesmaximal, characteristic
11C2×C17×#385C211yesmaximal, characteristic
12C24×Dic17C2211yescharacteristic
13C24×Dic17C2211yescharacteristic
14C22×C4⋊Dic17C2211yescharacteristic
15C22×C4⋊Dic17C2212yes—
16C22×C4⋊Dic17C2211yescharacteristic
17C22×C22⋊Dic17C2211yescharacteristic
18C22×C22⋊Dic17C2211yescharacteristic
19C22×C4×Dic17C2212yes—
20#11762684—24no—
21#11762684—24no—
22C22⋊C4×Dic17—24no—
23C2×#10587502—24no—
24C2×#10587502C2212yes—
25C2×#10587502C2212yes—
26C2×#10587502C2212yes—
27C2×#10587502C2211yescharacteristic
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2211yescharacteristic
31C2×#10587502C2212yes—
32C2×#10587502C2212yes—
33C2×#10587502C2212yes—
34C24×C4×C17C2211yescharacteristic
35C22×C4⋊C4×C17C2212yes—
36C22×C22⋊C4×C17C2211yescharacteristic
37C22×C22⋊C4×C17C2211yescharacteristic
38C17×#385C418yescomplemented
39C2×C17×#95C2212yes—
40C25×C17C2311yescharacteristic
41C23×C4×C17—22no—
42C23×C4×C17—22no—
43C23×C4×C17—42no—
44C23×C4×C17—42no—
45C23×C4×C17C2×C414yes—
46C23×C4×C17C2311yescharacteristic
47C23×C4×C17C2311yescharacteristic
48C23×C4×C17C2312yes—
49C23×C4×C17C2312yes—
50C23×Dic17—22no—
51C23×Dic17—22no—
52C23×Dic17—22no—
53C23×Dic17—24no—
54C23×Dic17—24no—
55C23×Dic17—22no—
56C23×Dic17—22no—
57C23×Dic17—42no—
58C23×Dic17—42no—
59C23×Dic17D414yes—
60C23×Dic17C2311yescharacteristic
61C23×Dic17C2311yescharacteristic
62C23×Dic17C2311yescharacteristic
63C23×Dic17C2312yes—
64C23×Dic17C2312yes—
65C23×Dic17C2311yescharacteristic
66C2×C4⋊C4×C17—42no—
67C2×C4⋊C4×C17—42no—
68C2×C4⋊C4×C17C2×C418yescomplemented
69C2×C22⋊C4×C17—22no—
70C2×C22⋊C4×C17—22no—
71C2×C22⋊C4×C17C2×C414yes—
72C2×C22⋊C4×C17C2×C414yes—
73C2×C22⋊C4×C17D414yes—
74C2×C22⋊C4×C17Q814yes—
75C2×C22⋊Dic17—22no—
76C2×C22⋊Dic17—22no—
77C2×C22⋊Dic17—22no—
78C2×C22⋊Dic17—22no—
79C2×C22⋊Dic17—22no—
80C2×C22⋊Dic17Q814yes—
81C2×C4⋊Dic17—24no—
82C2×C4⋊Dic17—22no—
83C2×C4⋊Dic17—22no—
84C2×C4⋊Dic17—24no—
85C2×C4⋊Dic17—24no—
86C2×C4⋊Dic17—42no—
87C2×C4⋊Dic17—42no—
88C2×C4×Dic17—24no—
89C2×C4×Dic17—24no—
90C2×C4×Dic17—42no—
91C17×#95C2×C418yescomplemented
92#10587502—24no—
93#10587502—24no—
94#10587502—24no—
95#10587502—24no—
96#10587502—22no—
97#10587502—22no—
98#10587502—24no—
99#10587502—42no—
100#10587502—42no—
101#10587502—42no—
102C24×C17—22no—
103C24×C17—22no—
104C24×C17—22no—
105C24×C17—22no—
106C24×C17C22×C412yes—
107C24×C17C2×D412yes—
108C24×C17C2×D412yes—
109C24×C17C2×Q812yes—
110C24×C17C2×Q812yes—
111C24×C17#4112yes—
112C24×C17#4112yes—
113C24×C17C2411yescharacteristic
114C22×C4×C17—22no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—22no—
118C22×C4×C17—28no—
119C22×C4×C17—22no—
120C22×C4×C17—21no—
121C22×C4×C17—21no—
122C22×C4×C17—22no—
123C22×C4×C17—22no—
124C22×C4×C17—22no—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—22no—
128C22×C4×C17—22no—
129C22×C4×C17—22no—
130C22×C4×C17—22no—
131C22×C4×C17—22no—
132C22×C4×C17—22no—
133C22×C4×C17—48no—
134C22×C4×C17—44no—
135C22×C4×C17—44no—
136C22×C4×C17—48no—
137C22×C4×C17C22×C414yes—
138C22×C4×C17C22×C412yes—
139C22×C4×C17C22×C412yes—
140C22×C4×C17C22×C414yes—
141C22×C4×C17C2×D414yes—
142C22×C4×C17C2×Q814yes—
143C22×Dic17—21no—
144C22×Dic17—21no—
145C22×Dic17—22no—
146C22×Dic17—28no—
147C22×Dic17—28no—
148C22×Dic17—21no—
149C22×Dic17—22no—
150C22×Dic17—22no—
151C22×Dic17—22no—
152C22×Dic17—21no—
153C22×Dic17—22no—
154C22×Dic17—22no—
155C22×Dic17—22no—
156C22×Dic17—22no—
157C22×Dic17—22no—
158C22×Dic17—22no—
159C22×Dic17—22no—
160C22×Dic17—22no—
161C22×Dic17—22no—
162C22×Dic17—22no—
163C22×Dic17—22no—
164C22×Dic17—28no—
165C22×Dic17—21no—
166C22×Dic17—21no—
167C22×Dic17—21no—
168C22×Dic17—21no—
169C22×Dic17—22no—
170C22×Dic17—22no—
171C22×Dic17—21no—
172C22×Dic17—21no—
173C22×Dic17—22no—
174C22×Dic17—22no—
175C22×Dic17—44no—
176C22×Dic17—48no—
177C22×Dic17—48no—
178C22×Dic17—48no—
179C22×Dic17—48no—
180C22×Dic17—44no—
181C22×Dic17—44no—
182C22×Dic17C2×D412yes—
183C22×Dic17C2×D412yes—
184C22×Dic17C2×Q814yes—
185C22×Dic17#4114yes—
186C22×Dic17#4114yes—
187C4⋊C4×C17—48no—
188C22⋊C4×C17—28no—
189C22⋊C4×C17C4⋊C4116yescomplemented
190C22⋊Dic17—28no—
191C22⋊Dic17—44no—
192C4⋊Dic17—48no—
193C4⋊Dic17—44no—
194C4⋊Dic17—44no—
195C4×Dic17—48no—
196#14044—171nomaximal, Sylow
197C23×C17—24no—
198C23×C17—24no—
199C23×C17—22no—
200C23×C17—22no—
201C23×C17—24no—
202C23×C17—22no—
203C23×C17—22no—
204C23×C17—24no—
205C23×C17—24no—
206C23×C17—24no—
207C23×C17—24no—
208C23×C17—24no—
209C23×C17—24no—
210C23×C17—24no—
211C23×C17—24no—
212C23×C17—24no—
213C23×C17C2×C4⋊C414yes—
214C23×C17C4×D414yes—
215C23×C17C4×Q814yes—
216C23×C17#12114yes—
217C23×C17C22⋊Q814yes—
218C23×C17#12514yes—
219C23×C17C4⋊Q814yes—
220C23×C17C23×C411yescharacteristic
221C23×C17C22×D411yescharacteristic
222C23×C17C22×D411yescharacteristic
223C23×C17C22×Q811yescharacteristic
224C23×C17C22×Q811yescharacteristic
225C23×C17C2×#4111yescharacteristic
226C23×C17C2×#4111yescharacteristic
227C23×C17C2×#4112yes—
228C23×C17C2×#4112yes—
229C23×C17H(5,𝔽2)12yes—
230C23×C17H(5,𝔽2)12yes—
231C2×C4×C17—24no—
232C2×C4×C17—24no—
233C2×C4×C17—24no—
234C2×C4×C17—24no—
235C2×C4×C17—24no—
236C2×C4×C17—24no—
237C2×C4×C17—22no—
238C2×C4×C17—22no—
239C2×C4×C17—24no—
240C2×C4×C17—24no—
241C2×C4×C17—24no—
242C2×C4×C17—24no—
243C2×C4×C17—24no—
244C2×C4×C17—416no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—42no—
248C2×C4×C17—42no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17—42no—
252C2×C4×C17—42no—
253C2×C4×C17—42no—
254C2×C4×C17—42no—
255C2×C4×C17C2×C4⋊C418yes—
256C2×Dic17—22no—
257C2×Dic17—22no—
258C2×Dic17—24no—
259C2×Dic17—24no—
260C2×Dic17—22no—
261C2×Dic17—24no—
262C2×Dic17—22no—
263C2×Dic17—24no—
264C2×Dic17—22no—
265C2×Dic17—24no—
266C2×Dic17—24no—
267C2×Dic17—24no—
268C2×Dic17—24no—
269C2×Dic17—416no—
270C2×Dic17—416no—
271C2×Dic17—42no—
272C2×Dic17—42no—
273C2×Dic17—42no—
274C2×Dic17—42no—
275C2×Dic17—42no—
276C2×Dic17—42no—
277C2×Dic17—42no—
278C2×Dic17—42no—
279C2×Dic17—42no—
280C2×Dic17—42no—
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×Dic17#12314yes—
292C2×#385—171no—
293C2×#385—171no—
294C2×#385D1711yescharacteristic
295C2×C4×C22⋊C4—171no—
296C22×#95—171no—
297#1442—172no—
298#1441—172no—
299#1436—172no—
300C4⋊#95—172no—
301C22⋊#95—171no—
302C22⋊#95—171no—
303C22×C17—24no—
304C22×C17—24no—
305C22×C17—28no—
306C22×C17—28no—
307C22×C17—24no—
308C22×C17—24no—
309C22×C17—28no—
310C22×C17—24no—
311C22×C17—28no—
312C22×C17—24no—
313C22×C17#38918yes—
314C22×C17C22×C4⋊C411yescharacteristic
315C22×C17C2×C4×D411yescharacteristic
316C22×C17C2×C4×Q811yescharacteristic
317C22×C17C4×#4112yes—
318C22×C17#51812yes—
319C22×C17C2×#12111yescharacteristic
320C22×C17C2×C22⋊Q811yescharacteristic
321C22×C17C2×#12311yescharacteristic
322C22×C17#52511yescharacteristic
323C22×C17C2×#12511yescharacteristic
324C22×C17#52912yes—
325C22×C17C2×C4⋊Q811yescharacteristic
326C22×C17#53411yescharacteristic
327C22×C17#53511yescharacteristic
328C22×C17#53812yes—
329C22×C17#54612yes—
330C22×C17#54612yes—
331C22×C17#55112yes—
332C22×C17#55112yes—
333C22×C17#55412yes—
334C22×C17#55412yes—
335C22×C17#55412yes—
336C22×C17#55412yes—
337C4×C17—28no—
338C4×C17—42no—
339C4×C17—44no—
340C4×C17—42no—
341Dic17—24no—
342Dic17—44no—
343Dic17—42no—
344Dic17—44no—
345Dic17—81no—
346Dic17—81no—
347C24×C4—171no—
348C24×C4—171no—
349C24×C4C2×D1711yescharacteristic
350C22×C4⋊C4—171no—
351C22×C4⋊C4—172no—
352C22×C4⋊C4—171no—
353C22×C4⋊C4C2×D1712yes—
354C22×C22⋊C4—171no—
355C22×C22⋊C4—171no—
356C22×C22⋊C4C2×D1711yescharacteristic
357C22×C22⋊C4C2×D1711yescharacteristic
358C22×C42—172no—
359#385—344no—
360#385—344no—
361#385Dic1718yescomplemented
362C4×C22⋊C4—344no—
363C2×#95—172no—
364C2×#95—171no—
365C2×#95—172no—
366C2×#95—172no—
367C2×#95—172no—
368C2×#95—172no—
369C2×#95—171no—
370C2×#95—172no—
371C2×#95—172no—
372C2×#95—172no—
373C2×#95—344no—
374C2×#95C2×D1712yes—
375C2×C17—28no—
376C2×C17C2×#38911yescharacteristic
377C2×C17C4×#12311yescharacteristic
378C2×C17#232111yescharacteristic
379C2×C17#234912yes—
380C2×C17#236812yes—
381C2×C17#241111yescharacteristic
382C2×C17#241211yescharacteristic
383C2×C17#245012yes—
384C2×C17#250211yescharacteristic
385C2×C17#250512yes—
386C2×C17#258311yescharacteristic
387C25C22×D1711yescharacteristic
388C23×C4—22no—
389C23×C4—22no—
390C23×C4—42no—
391C23×C4—42no—
392C23×C4—171no—
393C23×C4—172no—
394C23×C4—171no—
395C23×C4—172no—
396C23×C4—174no—
397C23×C4—171no—
398C23×C4—171no—
399C23×C4—344no—
400C23×C4—344no—
401C23×C4—342no—
402C23×C4—342no—
403C23×C4—342no—
404C23×C4—342no—
405C23×C4—342no—
406C23×C4—682no—
407C23×C4—682no—
408C23×C4C2×Dic1714yes—
409C23×C4C22×D1711yescharacteristic
410C23×C4C22×D1711yescharacteristic
411C23×C4C22×D1712yes—
412C23×C4C22×D1712yes—
413C2×C4⋊C4—42no—
414C2×C4⋊C4—42no—
415C2×C4⋊C4—342no—
416C2×C4⋊C4—342no—
417C2×C4⋊C4—344no—
418C2×C4⋊C4—344no—
419C2×C4⋊C4—344no—
420C2×C4⋊C4—682no—
421C2×C4⋊C4—682no—
422C2×C4⋊C4C2×Dic1718yescomplemented
423C2×C22⋊C4—22no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—174no—
426C2×C22⋊C4—342no—
427C2×C22⋊C4—342no—
428C2×C22⋊C4—342no—
429C2×C22⋊C4—342no—
430C2×C22⋊C4—342no—
431C2×C22⋊C4Dic3414yes—
432C2×C22⋊C4D6814yes—
433C2×C22⋊C4C2×Dic1714yes—
434C2×C22⋊C4C2×Dic1714yes—
435C2×C42—344no—
436C2×C42—344no—
437C2×C42—682no—
438#95—344no—
439#95—344no—
440#95—344no—
441#95—342no—
442#95—342no—
443#95—344no—
444#95—344no—
445#95—682no—
446#95—682no—
447#95—682no—
448#95C2×Dic1718yescomplemented
449C17#1404411yescharacteristic, complemented, Sylow
450C24—22no—
451C24—22no—
452C24—22no—
453C24—22no—
454C24C2×Dic3412yes—
455C24C2×D6812yes—
456C24D4×D1712yes—
457C24#6325412yes—
458C24Q8×D1712yes—
459C24#6325612yes—
460C24C22×Dic1712yes—
461C24C23×D1711yescharacteristic
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—28no—
481C22×C4—48no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—48no—
485C22×C4—172no—
486C22×C4—174no—
487C22×C4—172no—
488C22×C4—174no—
489C22×C4—174no—
490C22×C4—348no—
491C22×C4—341no—
492C22×C4—342no—
493C22×C4—341no—
494C22×C4—348no—
495C22×C4—341no—
496C22×C4—342no—
497C22×C4—342no—
498C22×C4—342no—
499C22×C4—342no—
500C22×C4—342no—
501C22×C4—342no—
502C22×C4—342no—
503C22×C4—342no—
504C22×C4—342no—
505C22×C4—342no—
506C22×C4—341no—
507C22×C4—342no—
508C22×C4—342no—
509C22×C4—342no—
510C22×C4—342no—
511C22×C4—348no—
512C22×C4—341no—
513C22×C4—341no—
514C22×C4—341no—
515C22×C4—341no—
516C22×C4—342no—
517C22×C4—342no—
518C22×C4—341no—
519C22×C4—341no—
520C22×C4—342no—
521C22×C4—342no—
522C22×C4—684no—
523C22×C4—688no—
524C22×C4—688no—
525C22×C4—688no—
526C22×C4—688no—
527C22×C4—684no—
528C22×C4—684no—
529C22×C4C2×Dic3414yes—
530C22×C4C2×D6814yes—
531C22×C4C22×Dic1712yes—
532C22×C4C22×Dic1714yes—
533C22×C4C22×Dic1712yes—
534C22×C4C22×Dic1714yes—
535C4⋊C4—48no—
536C4⋊C4—688no—
537C4⋊C4—684no—
538C4⋊C4—684no—
539C22⋊C4—28no—
540C22⋊C4—348no—
541C22⋊C4—684no—
542C22⋊C4C4⋊Dic17116yescomplemented
543C42—688no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—22no—
553C23—22no—
554C23—24no—
555C23—22no—
556C23—22no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23#1058757414yes—
561C23#1058757614yes—
562C23#1058758114yes—
563C23#1058758314yes—
564C23C2×C4⋊Dic1714yes—
565C23D4×Dic1714yes—
566C23Q8×Dic1714yes—
567C23C22×Dic3411yescharacteristic
568C23C22×D6811yescharacteristic
569C23C2×D4×D1711yescharacteristic
570C23C2×#6325411yescharacteristic
571C23#1058769112yes—
572C23#1058769112yes—
573C23C2×Q8×D1711yescharacteristic
574C23C2×#6325611yescharacteristic
575C23#41×D1712yes—
576C23#41×D1712yes—
577C23C23×Dic1711yescharacteristic
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—416no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—174no—
603C2×C4—344no—
604C2×C4—342no—
605C2×C4—342no—
606C2×C4—344no—
607C2×C4—344no—
608C2×C4—344no—
609C2×C4—344no—
610C2×C4—342no—
611C2×C4—344no—
612C2×C4—342no—
613C2×C4—344no—
614C2×C4—342no—
615C2×C4—344no—
616C2×C4—6816no—
617C2×C4—6816no—
618C2×C4—682no—
619C2×C4—682no—
620C2×C4—682no—
621C2×C4—682no—
622C2×C4—682no—
623C2×C4—682no—
624C2×C4—682no—
625C2×C4—682no—
626C2×C4—682no—
627C2×C4—682no—
628C2×C4—682no—
629C2×C4—682no—
630C2×C4—681no—
631C2×C4—681no—
632C2×C4—681no—
633C2×C4—681no—
634C2×C4—681no—
635C2×C4—681no—
636C2×C4—682no—
637C2×C4—682no—
638C2×C4C2×C4⋊Dic1718yes—
639C22—24no—
640C22—28no—
641C22—24no—
642C22—24no—
643C22—28no—
644C22—24no—
645C22—24no—
646C22—28no—
647C22—24no—
648C22—28no—
649C22#1176270818yes—
650C22C22×C4⋊Dic1711yescharacteristic
651C22C2×D4×Dic1711yescharacteristic
652C22C2×Q8×Dic1711yescharacteristic
653C22#41×Dic1712yes—
654C22#1176328412yes—
655C22C2×#1058758111yescharacteristic
656C22C2×#1058758311yescharacteristic
657C22#123×D1711yescharacteristic
658C22#1176331711yescharacteristic
659C22C2×#1058757611yescharacteristic
660C22#1176333612yes—
661C22C2×#1058757411yescharacteristic
662C22#1176336911yescharacteristic
663C22#1176337411yescharacteristic
664C22#1176339412yes—
665C22#1176345412yes—
666C22#1176345412yes—
667C22#1176348612yes—
668C22#1176348612yes—
669C22#1176351312yes—
670C22#1176351312yes—
671C22#1176351812yes—
672C22#1176351812yes—
673C4—28no—
674C4—44no—
675C4—42no—
676C4—42no—
677C4—344no—
678C4—684no—
679C4—682no—
680C4—684no—
681C4—1361no—
682C4—1361no—
683C2—28no—
684C2C2×#1176270811yescharacteristic
685C2#123×Dic1711yescharacteristic
686C2#1453144511yescharacteristic
687C2#1453169412yes—
688C2#1453186912yes—
689C2#1453216711yescharacteristic
690C2#1453217811yescharacteristic
691C2#1453260612yes—
692C2#1453326011yescharacteristic
693C2#1453329112yes—
694C2#1453416311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC2×C17×#385C2
socle, cosocleC24×C17C24
prime radicalC25×C17C23
hypercenterC2×#385D17
hypocenterC17#14044
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
5. #14527041C2semidirect
6. #14527037C2semidirect
7. #14527018C2semidirect
8. #14527013C2semidirect
38. C17×#385C4semidirect
68. C2×C4⋊C4×C17C2×C4semidirect
91. C17×#95C2×C4semidirect
189. C22⋊C4×C17C4⋊C4semidirect
361. #385Dic17semidirect
422. C2×C4⋊C4C2×Dic17semidirect
448. #95C2×Dic17semidirect
449. C17#14044semidirect
542. C22⋊C4C4⋊Dic17semidirect

Maximal subgroups

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

subgroupindex
C2×#117626842
C2×C22⋊C4×Dic172
C22×#105875022
#145270412
#145270372
#145270182
#145270132
#145270072
#145270062
C2×C17×#3852
#1404417