Finite Group Databasebeta

← #16955899

Subgroups — #16955899

Order 4352.

1778 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762704C212yesmaximal
2C2×C4×#63225C212yesmaximal
3C2×C4×#63225C211yesmaximal, characteristic
4C2×#11762660C211yesmaximal, characteristic
5#14527017C212yesmaximal
6#14527016C212yesmaximal
7#14527012C211yesmaximal, characteristic
8#14527011C211yesmaximal, characteristic
9#14527008C211yesmaximal, characteristic
10#95×Dic17C211yesmaximal, characteristic
11C2×C4×C4⋊C4×C17C211yesmaximal, characteristic
12C22×#63225C2212yes—
13C22×#63225C2212yes—
14C22×#63225C2211yescharacteristic
15C22×#63225C2211yescharacteristic
16C22×C4×Dic17C2212yes—
17C22×C4×Dic17C2211yescharacteristic
18C22×C4×Dic17C2212yes—
19C22×C4×Dic17C2211yescharacteristic
20#11762704C4116yescomplemented
21C4×#63225—28no—
22C4×#63225C418yescomplemented
23#11762660—24no—
24C2×#10587502C2211yescharacteristic
25C2×#10587502C2212yes—
26C2×#10587502C2212yes—
27C2×#10587502C2211yescharacteristic
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2211yescharacteristic
33C2×#10587502C2211yescharacteristic
34C22×C4⋊C4×C17C2212yes—
35C22×C42×C17C2212yes—
36C22×C42×C17C2211yescharacteristic
37C4×C4⋊C4×C17—24no—
38C2×C17×#95C2211yescharacteristic
39C2×C17×#95C2211yescharacteristic
40C23×C4×C17C2312yes—
41C23×C4×C17C2311yescharacteristic
42C23×C4×C17C2311yescharacteristic
43C23×C4×C17C2312yes—
44C23×C4×C17C2311yescharacteristic
45C23×Dic17C2312yes—
46C23×Dic17C2312yes—
47C23×Dic17C2311yescharacteristic
48C23×Dic17C2311yescharacteristic
49C23×Dic17C2311yescharacteristic
50C23×Dic17C2311yescharacteristic
51C2×C4⋊C4×C17—24no—
52C2×C4⋊C4×C17—24no—
53C2×C4⋊C4×C17C2×C418yes—
54C2×C42×C17—22no—
55C2×C42×C17—22no—
56C2×C42×C17—24no—
57C2×C42×C17—24no—
58C2×C42×C17—42no—
59C2×C42×C17C2×C414yes—
60C2×#63225—24no—
61C2×#63225—24no—
62C2×#63225—24no—
63C2×#63225—24no—
64C2×#63225C2×C418yes—
65C2×#63225C2×C418yes—
66C2×#63225C2×C418yes—
67C2×#63225C2×C414yes—
68C2×#63225C2×C414yes—
69C2×#63225D414yes—
70C2×#63225Q814yes—
71C2×C4×Dic17—22no—
72C2×C4×Dic17—24no—
73C2×C4×Dic17—22no—
74C2×C4×Dic17—24no—
75C2×C4×Dic17—24no—
76C2×C4×Dic17—24no—
77C2×C4×Dic17—24no—
78C2×C4×Dic17—42no—
79C2×C4×Dic17C2×C418yes—
80C2×C4×Dic17C2×C414yes—
81C2×C4×Dic17C2×C414yes—
82C17×#95—22no—
83C17×#95D414yes—
84#10587502—22no—
85#10587502—24no—
86#10587502—22no—
87#10587502—24no—
88#10587502—24no—
89#10587502—42no—
90#10587502C2×C418yes—
91#10587502C2×C418yes—
92#10587502C2×C414yes—
93#10587502C2×C414yes—
94C24×C17C2411yescharacteristic
95C22×C4×C17—22no—
96C22×C4×C17—21no—
97C22×C4×C17—21no—
98C22×C4×C17—22no—
99C22×C4×C17—22no—
100C22×C4×C17—21no—
101C22×C4×C17—21no—
102C22×C4×C17—22no—
103C22×C4×C17—22no—
104C22×C4×C17—21no—
105C22×C4×C17—21no—
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×C17C22×C414yes—
115C22×C4×C17C22×C412yes—
116C22×C4×C17C22×C414yes—
117C22×C4×C17C22×C412yes—
118C22×C4×C17C22×C412yes—
119C22×C4×C17C22×C414yes—
120C22×C4×C17C2×D412yes—
121C22×C4×C17C2×D412yes—
122C22×C4×C17C2×D414yes—
123C22×C4×C17C2×Q812yes—
124C22×C4×C17#4112yes—
125C22×C4×C17#4114yes—
126C22×Dic17—21no—
127C22×Dic17—21no—
128C22×Dic17—21no—
129C22×Dic17—21no—
130C22×Dic17—22no—
131C22×Dic17—21no—
132C22×Dic17—21no—
133C22×Dic17—22no—
134C22×Dic17—22no—
135C22×Dic17—22no—
136C22×Dic17—22no—
137C22×Dic17—22no—
138C22×Dic17—22no—
139C22×Dic17—22no—
140C22×Dic17—22no—
141C22×Dic17—22no—
142C22×Dic17—22no—
143C22×Dic17—21no—
144C22×Dic17—22no—
145C22×Dic17—22no—
146C22×Dic17—21no—
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17C22×C412yes—
150C22×Dic17C22×C414yes—
151C22×Dic17C22×C414yes—
152C22×Dic17C22×C412yes—
153C22×Dic17C22×C414yes—
154C22×Dic17C22×C412yes—
155C22×Dic17C22×C412yes—
156C22×Dic17C22×C414yes—
157C22×Dic17C2×D412yes—
158C22×Dic17C2×D412yes—
159C22×Dic17C2×Q812yes—
160C22×Dic17C2×Q812yes—
161C22×Dic17#4114yes—
162C4⋊C4×C17—216no—
163C42×C17—28no—
164C42×C17—48no—
165#63225—216no—
166#63225—216no—
167#63225C42116yescomplemented
168#63225C4⋊C4116yescomplemented
169C4×Dic17—28no—
170C4×Dic17—28no—
171C4×Dic17—216no—
172C4×Dic17—48no—
173#13821—171nomaximal, Sylow
174C23×C17C23×C411yescharacteristic
175C23×C17C23×C412yes—
176C23×C17C22×D411yescharacteristic
177C23×C17C22×D411yescharacteristic
178C23×C17C22×Q811yescharacteristic
179C23×C17C2×#4111yescharacteristic
180C23×C17C2×#4112yes—
181C23×C17C2×#4112yes—
182C23×C17H(5,𝔽2)11yescharacteristic
183C23×C172−1+411yescharacteristic
184C23×C172−1+412yes—
185C2×C4×C17—22no—
186C2×C4×C17—22no—
187C2×C4×C17—22no—
188C2×C4×C17—24no—
189C2×C4×C17—24no—
190C2×C4×C17—22no—
191C2×C4×C17—24no—
192C2×C4×C17—22no—
193C2×C4×C17—24no—
194C2×C4×C17—22no—
195C2×C4×C17—22no—
196C2×C4×C17—24no—
197C2×C4×C17—24no—
198C2×C4×C17—24no—
199C2×C4×C17—22no—
200C2×C4×C17—24no—
201C2×C4×C17—22no—
202C2×C4×C17—24no—
203C2×C4×C17—24no—
204C2×C4×C17—22no—
205C2×C4×C17—24no—
206C2×C4×C17—24no—
207C2×C4×C17—24no—
208C2×C4×C17—42no—
209C2×C4×C17—42no—
210C2×C4×C17—42no—
211C2×C4×C17—42no—
212C2×C4×C17C2×C4214yes—
213C2×C4×C17C2×C4⋊C414yes—
214C2×C4×C17C4×D414yes—
215C2×C4×C17C4×D418yes—
216C2×Dic17—22no—
217C2×Dic17—22no—
218C2×Dic17—24no—
219C2×Dic17—22no—
220C2×Dic17—24no—
221C2×Dic17—22no—
222C2×Dic17—22no—
223C2×Dic17—22no—
224C2×Dic17—22no—
225C2×Dic17—22no—
226C2×Dic17—22no—
227C2×Dic17—24no—
228C2×Dic17—24no—
229C2×Dic17—24no—
230C2×Dic17—22no—
231C2×Dic17—24no—
232C2×Dic17—24no—
233C2×Dic17—24no—
234C2×Dic17—22no—
235C2×Dic17—42no—
236C2×Dic17—42no—
237C2×Dic17—42no—
238C2×Dic17—42no—
239C2×Dic17—42no—
240C2×Dic17—42no—
241C2×Dic17—42no—
242C2×Dic17—42no—
243C2×Dic17—42no—
244C2×Dic17—42no—
245C2×Dic17C2×C4214yes—
246C2×Dic17C2×C4214yes—
247C2×Dic17C2×C4⋊C414yes—
248C2×Dic17C2×C4⋊C414yes—
249C2×Dic17#11714yes—
250C2×Dic17C4×D418yes—
251C2×Dic17C4×Q818yes—
252C2×#389—172no—
253C2×C4×C4⋊C4—171no—
254C2×C4×C4⋊C4—172no—
255C2×C4×C4⋊C4—171no—
256C2×C4×C4⋊C4D1711yescharacteristic
257#1436—172no—
258#1436—172no—
259C4⋊#95—171no—
260#1434—171no—
261#1434—171no—
262C4×#95—171no—
263C22×C17C22×C4211yescharacteristic
264C22×C17C22×C4⋊C411yescharacteristic
265C22×C17C2×#11711yescharacteristic
266C22×C17C2×C4×D412yes—
267C22×C17C2×C4×D411yescharacteristic
268C22×C17C2×C4×D412yes—
269C22×C17C2×C4×Q812yes—
270C22×C17C4×#4111yescharacteristic
271C22×C17C4×#4112yes—
272C22×C17#51912yes—
273C22×C17#51911yescharacteristic
274C22×C17#52012yes—
275C22×C17#52011yescharacteristic
276C22×C17D4211yescharacteristic
277C22×C17#54712yes—
278C22×C17#54711yescharacteristic
279C22×C17#54812yes—
280C22×C17D4×Q811yescharacteristic
281C22×C17#55212yes—
282C22×C17#55312yes—
283C22×C17#55411yescharacteristic
284C22×C17#55612yes—
285C22×C17#55712yes—
286C4×C17—24no—
287C4×C17—24no—
288C4×C17—28no—
289C4×C17—24no—
290C4×C17—44no—
291Dic17—24no—
292Dic17—24no—
293Dic17—42no—
294Dic17—44no—
295Dic17—44no—
296Dic17C4×C4⋊C418yescomplemented
297C22×C4⋊C4—171no—
298C22×C4⋊C4—171no—
299C22×C4⋊C4—172no—
300C22×C4⋊C4—172no—
301C22×C4⋊C4C2×D1712yes—
302C22×C42—171no—
303C22×C42—172no—
304C22×C42—171no—
305C22×C42—172no—
306C22×C42C2×D1712yes—
307C22×C42C2×D1711yescharacteristic
308#389—1716no—
309C4×C4⋊C4—24no—
310C4×C4⋊C4—178no—
311C4×C4⋊C4—348no—
312C4×C4⋊C4—344no—
313C2×#95—172no—
314C2×#95—172no—
315C2×#95—171no—
316C2×#95—171no—
317C2×#95—171no—
318C2×#95—172no—
319C2×#95—171no—
320C2×#95—172no—
321C2×#95—172no—
322C2×#95—172no—
323C2×#95C2×D1711yescharacteristic
324C2×#95C2×D1711yescharacteristic
325C2×C17C2×C4×C4⋊C411yescharacteristic
326C2×C17C42×D411yescharacteristic
327C2×C17#227611yescharacteristic
328C2×C17#234411yescharacteristic
329C2×C17#234711yescharacteristic
330C2×C17D4×C4⋊C411yescharacteristic
331C2×C17#235111yescharacteristic
332C2×C17#236112yes—
333C2×C17#236212yes—
334C2×C17#236512yes—
335C2×C17#237012yes—
336C23×C4—171no—
337C23×C4—171no—
338C23×C4—171no—
339C23×C4—172no—
340C23×C4—172no—
341C23×C4—171no—
342C23×C4C22×D1712yes—
343C23×C4C22×D1711yescharacteristic
344C23×C4C22×D1712yes—
345C23×C4C22×D1711yescharacteristic
346C23×C4C22×D1711yescharacteristic
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—174no—
350C2×C4⋊C4—174no—
351C2×C4⋊C4—178no—
352C2×C4⋊C4—174no—
353C2×C4⋊C4—174no—
354C2×C4⋊C4—178no—
355C2×C4⋊C4—178no—
356C2×C4⋊C4—344no—
357C2×C4⋊C4—344no—
358C2×C4⋊C4—344no—
359C2×C4⋊C4—344no—
360C2×C4⋊C4C4×D1718yes—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42—42no—
366C2×C42—174no—
367C2×C42—174no—
368C2×C42—178no—
369C2×C42—344no—
370C2×C42—344no—
371C2×C42—344no—
372C2×C42—342no—
373C2×C42—342no—
374C2×C42—344no—
375C2×C42—344no—
376C2×C42—682no—
377C2×C42C4×D1714yes—
378#95—22no—
379#95—174no—
380#95—174no—
381#95—178no—
382#95—178no—
383#95—344no—
384#95—342no—
385#95—344no—
386#95—344no—
387#95—342no—
388#95—682no—
389#95C22⋊D1714yes—
390C17#1382111yescharacteristic, complemented, Sylow
391C24C23×D1711yescharacteristic
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—172no—
412C22×C4—174no—
413C22×C4—172no—
414C22×C4—172no—
415C22×C4—174no—
416C22×C4—172no—
417C22×C4—172no—
418C22×C4—174no—
419C22×C4—172no—
420C22×C4—172no—
421C22×C4—172no—
422C22×C4—174no—
423C22×C4—174no—
424C22×C4—341no—
425C22×C4—341no—
426C22×C4—342no—
427C22×C4—341no—
428C22×C4—341no—
429C22×C4—341no—
430C22×C4—341no—
431C22×C4—342no—
432C22×C4—342no—
433C22×C4—342no—
434C22×C4—342no—
435C22×C4—342no—
436C22×C4—342no—
437C22×C4—341no—
438C22×C4—342no—
439C22×C4—342no—
440C22×C4—341no—
441C22×C4—342no—
442C22×C4—342no—
443C22×C4—342no—
444C22×C4—342no—
445C22×C4—342no—
446C22×C4—342no—
447C22×C4C2×C4×D1714yes—
448C22×C4C2×C4×D1712yes—
449C22×C4C2×C4×D1714yes—
450C22×C4C2×C4×D1712yes—
451C22×C4C2×C4×D1712yes—
452C22×C4C2×C4×D1714yes—
453C22×C4#6325212yes—
454C22×C4#6325214yes—
455C22×C4D4×D1712yes—
456C22×C4Q8×D1712yes—
457C22×C4C2×C22⋊D1712yes—
458C22×C4C2×C22⋊D1714yes—
459C4⋊C4—216no—
460C4⋊C4—1716no—
461C4⋊C4—1716no—
462C4⋊C4—3416no—
463C4⋊C4—3416no—
464C42—28no—
465C42—48no—
466C42—348no—
467C42—3416no—
468C42—348no—
469C42—688no—
470C23C22×C4×D1711yescharacteristic
471C23C22×C4×D1712yes—
472C23C2×#6325211yescharacteristic
473C23C2×#6325212yes—
474C23C2×D4×D1711yescharacteristic
475C23#1058769111yescharacteristic
476C23C2×Q8×D1711yescharacteristic
477C23#1058769411yescharacteristic
478C23#41×D1712yes—
479C23#1058769712yes—
480C23C22×C22⋊D1711yescharacteristic
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—174no—
509C2×C4—174no—
510C2×C4—174no—
511C2×C4—178no—
512C2×C4—178no—
513C2×C4—174no—
514C2×C4—174no—
515C2×C4—342no—
516C2×C4—342no—
517C2×C4—342no—
518C2×C4—342no—
519C2×C4—342no—
520C2×C4—344no—
521C2×C4—344no—
522C2×C4—342no—
523C2×C4—342no—
524C2×C4—342no—
525C2×C4—342no—
526C2×C4—344no—
527C2×C4—344no—
528C2×C4—344no—
529C2×C4—342no—
530C2×C4—344no—
531C2×C4—344no—
532C2×C4—344no—
533C2×C4—342no—
534C2×C4—682no—
535C2×C4—682no—
536C2×C4—682no—
537C2×C4—682no—
538C2×C4—682no—
539C2×C4—682no—
540C2×C4—682no—
541C2×C4—682no—
542C2×C4—682no—
543C2×C4—682no—
544C2×C4C42×D1714yes—
545C2×C4C4⋊C4×D1714yes—
546C2×C4C4×C22⋊D1714yes—
547C2×C4C4×C22⋊D1718yes—
548C22C2×C42×D1711yescharacteristic
549C22C2×C4⋊C4×D1711yescharacteristic
550C22#117×D1711yescharacteristic
551C22C4×D4×D1712yes—
552C22C2×C4×C22⋊D1711yescharacteristic
553C22C2×C4×C22⋊D1712yes—
554C22C4×Q8×D1712yes—
555C22C4×#6325212yes—
556C22C4×#6325211yescharacteristic
557C22#1176328512yes—
558C22#1176328611yescharacteristic
559C22#1176328812yes—
560C22#1176329011yescharacteristic
561C22D4×C22⋊D1711yescharacteristic
562C22#1176345812yes—
563C22#1176346011yescharacteristic
564C22#1176346812yes—
565C22Q8×C22⋊D1711yescharacteristic
566C22#1176349212yes—
567C22#1176350812yes—
568C22#1176351411yescharacteristic
569C22#1176353112yes—
570C22#1176353712yes—
571C4—24no—
572C4—24no—
573C4—24no—
574C4—28no—
575C4—44no—
576C4—178no—
577C4—344no—
578C4—344no—
579C4—682no—
580C4—684no—
581C4—684no—
582C2C4×C4⋊C4×D1711yescharacteristic
583C2C42×C22⋊D1711yescharacteristic
584C2#1453114811yescharacteristic
585C2#1453165411yescharacteristic
586C2#1453167811yescharacteristic
587C2C4⋊C4×C22⋊D1711yescharacteristic
588C2#1453170411yescharacteristic
589C2#1453180112yes—
590C2#1453181212yes—
591C2#1453183812yes—
592C2#1453188112yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC22×C17C22×C42
FittingC2×C4×C4⋊C4×C17C2
socle, prime radical, cosocleC24×C17C24
hypercenterC2×C4×C4⋊C4D17
hypocenterC17#13821
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
20. #11762704C4semidirect
22. C4×#63225C4semidirect
167. #63225C42semidirect
168. #63225C4⋊C4semidirect
296. Dic17C4×C4⋊C4semidirect
390. C17#13821semidirect

Maximal subgroups

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

subgroupindex
C2×#117627042
C2×C4×#632252
C2×#117626602
#145270172
#145270162
#145270122
#145270112
#145270082
#95×Dic172
C2×C4×C4⋊C4×C172
#1382117