Finite Group Databasebeta

← #17052670

Subgroups — #17052670

Order 4352.

1010 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762909C211yesmaximal, characteristic
2C2×#11762909C211yesmaximal, characteristic
3C2×C4×#63229C211yesmaximal, characteristic
4#14531704C211yesmaximal, characteristic, complemented
5#14529519C211yesmaximal, characteristic, complemented
6#14529514C211yesmaximal, characteristic, complemented
7#14529051C211yesmaximal, characteristic, complemented
8#14529045C211yesmaximal, characteristic
9#14529037C211yesmaximal, characteristic, complemented
10#14529031C211yesmaximal, characteristic
11#14528950C211yesmaximal, characteristic, complemented
12#14528794C211yesmaximal, characteristic
13#14528786C211yesmaximal, characteristic, complemented
14#14528659C211yesmaximal, characteristic, complemented
15D4×C4⋊C4×C17C211yesmaximal, characteristic
16C22×#63229C2211yescharacteristic
17C2×C4×Dic34C2211yescharacteristic
18C2×C4×Dic34C2211yescharacteristic
19C2×C4×Dic34C2211yescharacteristic
20#11762909—24no—
21#11762909C418yescomplemented
22C4×#63229—24no—
23C2×#10587478C2211yescharacteristic
24C2×#10587478C2211yescharacteristic
25C2×#10587478C2211yescharacteristic
26C2×#10587479C2211yescharacteristic
27C2×#10587479C2211yescharacteristic
28C2×C4⋊#3622C2211yescharacteristic
29C2×#10587481C2211yescharacteristic
30C2×#10587481C2211yescharacteristic
31C2×#10587481C2211yescharacteristic
32C2×#10587503C2211yescharacteristic
33C2×#10587503C2211yescharacteristic
34C2×#10587503C2211yescharacteristic
35C2×#10587474C2211yescharacteristic
36C2×#10587474C2211yescharacteristic
37C2×C4×#3622C2211yescharacteristic
38#11762707C2211yescharacteristic
39#11762707C2211yescharacteristic
40#11762704—21no—
41#11762704—21no—
42#11762675C2211yescharacteristic
43#11762673—21no—
44#11762660—21no—
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762564C2211yescharacteristic
50#11762564C2211yescharacteristic
51C2×C4×D4×C17C2211yescharacteristic
52C2×C4×D4×C17C2211yescharacteristic
53C2×C4×D4×C17C2211yescharacteristic
54C22×C4⋊C4×C17—21no—
55C17×#389C2211yescharacteristic
56C17×#389C2211yescharacteristic
57C17×#385—21no—
58C17×#385—21no—
59C17×#384C2211yescharacteristic
60C17×#380—21no—
61C4×C4⋊C4×C17C2211yescharacteristic
62C22×D4×C17C2311yescharacteristic
63C23×C4×C17—21no—
64C23×C4×C17—21no—
65C23×C4×C17—21no—
66C22×Dic34C2311yescharacteristic
67C4×D4×C17—22no—
68C4×D4×C17—22no—
69C4×D4×C17—22no—
70C4×D4×C17—41no—
71C4×D4×C17—41no—
72C4×D4×C17C2×C414yes—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17—21no—
76C2×C4⋊C4×C17—21no—
77C2×C4⋊C4×C17—44no—
78C2×C4⋊C4×C17D411yescharacteristic
79C2×C4⋊C4×C17D411yescharacteristic
80C2×C4⋊C4×C17C2311yescharacteristic
81C2×C4⋊C4×C17C2311yescharacteristic
82C2×C4⋊C4×C17C2311yescharacteristic
83C2×C22⋊C4×C17—21no—
84C2×C22⋊C4×C17—21no—
85C2×C22⋊C4×C17—21no—
86C2×C42×C17C2311yescharacteristic
87C2×C42×C17C2311yescharacteristic
88C2×C42×C17C2311yescharacteristic
89C2×#63229C2×C414yes—
90C2×#63229D414yes—
91C2×#63229Q814yes—
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95C2×#63225—21no—
96C2×#63225—21no—
97C2×#63225—21no—
98C2×#63225—21no—
99C2×#63225—21no—
100C2×C4×Dic17—21no—
101C2×C4×Dic17—21no—
102C2×C4×Dic17—21no—
103C22×#3622C2311yescharacteristic
104C22×#3622C2311yescharacteristic
105C22×#3622C2311yescharacteristic
106C22×#3622C2311yescharacteristic
107C4×Dic34—22no—
108C4×Dic34—22no—
109C4×Dic34—22no—
110C4×Dic34—41no—
111C4×Dic34—41no—
112C4×Dic34C2×C414yescomplemented
113C17×#95—21no—
114C17×#95—21no—
115C17×#95D411yescharacteristic
116C17×#95D411yescharacteristic
117#10587503—22no—
118#10587503—22no—
119#10587503C2×C414yes—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587481—22no—
125#10587481—22no—
126#10587481C2×C414yescomplemented
127#10587479—22no—
128#10587479—41no—
129#10587478—22no—
130#10587478—41no—
131#10587478C2×C414yescomplemented
132C4⋊#3622—22no—
133#10587474—22no—
134#10587474C2×C414yescomplemented
135C4×#3622—22no—
136C24×C17—21no—
137C2×D4×C17—21no—
138C2×D4×C17—21no—
139C2×D4×C17—21no—
140C2×D4×C17C22×C412yes—
141C2×D4×C17C2×D412yes—
142C2×D4×C17C2×Q812yes—
143C22×C4×C17—21no—
144C22×C4×C17—24no—
145C22×C4×C17—21no—
146C22×C4×C17—21no—
147C22×C4×C17—42no—
148C22×C4×C17—42no—
149C22×C4×C17—42no—
150C22×C4×C17—42no—
151C22×C4×C17—42no—
152C22×C4×C17—42no—
153C22×C4×C17—81no—
154C22×C4×C17—81no—
155C22×C4×C17C2×D411yescharacteristic
156C22×C4×C17C2×D411yescharacteristic
157C22×C4×C17C2×D411yescharacteristic
158C22×C4×C17C2×D411yescharacteristic
159C22×C4×C17C2×D411yescharacteristic
160C22×C4×C17C2×D411yescharacteristic
161C22×C4×C17C2411yescharacteristic
162C22×Dic17—21no—
163C22×Dic17—21no—
164C22×Dic17—21no—
165C22×Dic17—21no—
166C2×Dic34—21no—
167C2×Dic34—21no—
168C2×Dic34—21no—
169C2×Dic34C22×C412yes—
170C2×Dic34C2×D412yes—
171C2×Dic34C2×Q812yes—
172C4⋊C4×C17—22no—
173C4⋊C4×C17—21no—
174C4⋊C4×C17—21no—
175C4⋊C4×C17—21no—
176C4⋊C4×C17—21no—
177C4⋊C4×C17—21no—
178C4⋊C4×C17—41no—
179C4⋊C4×C17—42no—
180C4⋊C4×C17—42no—
181C4⋊C4×C17—42no—
182C4⋊C4×C17—81no—
183C4⋊C4×C17C22×C412yes—
184C22⋊C4×C17—24no—
185C22⋊C4×C17—42no—
186C22⋊C4×C17—42no—
187C42×C17—21no—
188C42×C17—21no—
189C42×C17—21no—
190C42×C17—21no—
191C42×C17C22×C412yes—
192C42×C17#4112yes—
193#63229C4⋊C4116yescomplemented
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17C22×C412yes—
200#63225—24no—
201#63225—42no—
202#63225—42no—
203#63225—42no—
204#63225—42no—
205C4×Dic17—24no—
206C4×Dic17—42no—
207C4×Dic17—42no—
208C2×#3622—21no—
209C2×#3622—21no—
210C2×#3622—21no—
211C2×#3622—21no—
212C2×#3622—21no—
213C2×#3622—21no—
214C2×#3622—21no—
215C2×#3622—21no—
216C2×#3622C22×C412yes—
217C2×#3622C22×C412yes—
218C2×#3622C2×D412yes—
219C2×#3622C2×Q812yes—
220#22677—171nomaximal, Sylow
221C23×C17—22no—
222C23×C17—22no—
223C23×C17—22no—
224C23×C17—41no—
225C23×C17—41no—
226C23×C17—41no—
227C23×C17—41no—
228C23×C17C22×D411yescharacteristic
229D4×C17—22no—
230D4×C17—22no—
231D4×C17—22no—
232D4×C17C2×C4⋊C414yes—
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—22no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
244C2×C4×C17—21no—
245C2×C4×C17—21no—
246C2×C4×C17—21no—
247C2×C4×C17—21no—
248C2×C4×C17—21no—
249C2×C4×C17—48no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17—41no—
253C2×C4×C17—41no—
254C2×C4×C17—41no—
255C2×C4×C17—41no—
256C2×C4×C17—41no—
257C2×C4×C17—41no—
258C2×C4×C17—84no—
259C2×C4×C17—84no—
260C2×C4×C17C4×D412yes—
261C2×C4×C17C4×D412yes—
262C2×C4×C17AGL(1,ℤ/8)12yes—
263C2×C4×C17C23×C411yescharacteristic
264C2×C4×C17C22×D411yescharacteristic
265C2×C4×C17C22×Q811yescharacteristic
266C2×C4×C17C2×#4111yescharacteristic
267C2×C4×C17H(5,𝔽2)11yescharacteristic
268C2×C4×C172−1+411yescharacteristic
269C2×Dic17—22no—
270C2×Dic17—22no—
271C2×Dic17—22no—
272C2×Dic17—22no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281Dic34—22no—
282Dic34—22no—
283Dic34—22no—
284Dic34C2×C4⋊C414yescomplemented
285#3622—22no—
286#3622—41no—
287#3622—41no—
288#3622C2×C4⋊C414yescomplemented
289C2×#440—171no—
290C2×#440—171no—
291C2×C4×SD16—171no—
292#2351—171no—
293D4×C4⋊C4D1711yescharacteristic
294#1943—171no—
295#1942—171no—
296#1865—171no—
297#1864—171no—
298#1863—171no—
299#1862—171no—
300#1847—171no—
301#1804—171no—
302#1803—171no—
303#1771—171no—
304C22×C17—24no—
305C22×C17—42no—
306C22×C17—42no—
307C22×C17—42no—
308C22×C17—42no—
309C22×C17—42no—
310C22×C17—42no—
311C22×C17C2×C4×D411yescharacteristic
312C22×C17D4211yescharacteristic
313C22×C17D4×Q811yescharacteristic
314C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
315C22×C17#57511yescharacteristic
316C22×C17#57711yescharacteristic
317C22×C17#57811yescharacteristic
318C4×C17—22no—
319C4×C17—22no—
320C4×C17—22no—
321C4×C17—42no—
322C4×C17—41no—
323C4×C17—41no—
324C4×C17—41no—
325C4×C17—81no—
326C4×C17—81no—
327C4×C17C22×C4⋊C411yescharacteristic
328C4×C17#52011yescharacteristic
329C4×C17#54711yescharacteristic
330C4×C17#55411yescharacteristic
331Dic17—24no—
332Dic17—42no—
333Dic17—81no—
334Dic17—81no—
335C22×SD16—171no—
336C2×C4×Q8—171no—
337C2×C4×Q8—171no—
338C2×C4×Q8—171no—
339C2×C4×D4C2×D1711yescharacteristic
340C2×C4×D4C2×D1711yescharacteristic
341C2×C4×D4C2×D1711yescharacteristic
342C22×C4⋊C4—21no—
343#440—178no—
344#440—344no—
345C4×SD16—344no—
346C2×#107—171no—
347C2×#107—171no—
348C2×#107—171no—
349C2×#106—171no—
350C2×#106—171no—
351C2×C4⋊C8—171no—
352C2×#103—171no—
353C2×#103—171no—
354C2×#103—171no—
355C2×#102—171no—
356C2×#102—171no—
357C2×#102—171no—
358C2×#97—171no—
359C2×#97—171no—
360C2×C4×C8—171no—
361#389—171no—
362#389—171no—
363#389—341no—
364#389—341no—
365#389C2×D1711yescharacteristic
366#389C2×D1711yescharacteristic
367#385—21no—
368#385—21no—
369#384C2×D1711yescharacteristic
370#382—171no—
371#382—341no—
372#380—21no—
373C4×C4⋊C4—341no—
374C4×C4⋊C4C2×D1711yescharacteristic
375#340—171no—
376#340—171no—
377#340—171no—
378#340—171no—
379#336—171no—
380#336—171no—
381C2×C17—44no—
382C2×C17D4×C4⋊C411yescharacteristic
383C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
384C2×C17#294711yescharacteristic
385C2×C17#326511yescharacteristic
386C2×C17#326911yescharacteristic
387C2×C17#338611yescharacteristic
388C2×C17#338811yescharacteristic
389C22×Q8—171no—
390C22×D4C22×D1711yescharacteristic
391C23×C4—21no—
392C23×C4—21no—
393C23×C4—21no—
394C2×SD16—174no—
395C2×SD16—174no—
396C2×SD16—174no—
397C22×C8—171no—
398C22×C8—171no—
399C22×C8—171no—
400C22×C8—171no—
401C4×Q8—174no—
402C4×Q8—342no—
403C4×Q8—342no—
404C4×Q8—342no—
405C4×Q8—681no—
406C4×Q8—681no—
407C4×D4—22no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—41no—
411C4×D4—41no—
412C4×D4C4×D1714yes—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—44no—
418C2×C4⋊C4—171no—
419C2×C4⋊C4—171no—
420C2×C4⋊C4—171no—
421C2×C4⋊C4—341no—
422C2×C4⋊C4—341no—
423C2×C4⋊C4—341no—
424C2×C4⋊C4—341no—
425C2×C4⋊C4—341no—
426C2×C4⋊C4C22⋊D1711yescharacteristic
427C2×C4⋊C4C22⋊D1711yescharacteristic
428C2×C4⋊C4C22×D1711yescharacteristic
429C2×C4⋊C4C22×D1711yescharacteristic
430C2×C4⋊C4C22×D1711yescharacteristic
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C42—341no—
435C2×C42—341no—
436C2×C42—341no—
437C2×C42C22×D1711yescharacteristic
438C2×C42C22×D1711yescharacteristic
439C2×C42C22×D1711yescharacteristic
440#107—174no—
441#107—342no—
442#107—681no—
443#106—342no—
444#106—681no—
445C4⋊C8—342no—
446#103—174no—
447#103—342no—
448#103—342no—
449#102—174no—
450#102—342no—
451#102—342no—
452#97—174no—
453#97—342no—
454C4×C8—342no—
455#95—21no—
456#95—21no—
457#95—341no—
458#95—341no—
459#95—341no—
460#95—341no—
461#95C22⋊D1711yescharacteristic
462#95C22⋊D1711yescharacteristic
463C17#2267711yescharacteristic, complemented, Sylow
464C24—21no—
465C2×Q8—172no—
466C2×Q8—172no—
467C2×Q8—172no—
468C2×Q8—341no—
469C2×Q8—341no—
470C2×Q8—341no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4C2×C4×D1712yes—
475C2×D4D4×D1712yes—
476C2×D4Q8×D1712yes—
477C22×C4—21no—
478C22×C4—24no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—81no—
488C22×C4—81no—
489C22×C4—341no—
490C22×C4—341no—
491C22×C4—341no—
492C22×C4—341no—
493C22×C4C2×C22⋊D1711yescharacteristic
494C22×C4C2×C22⋊D1711yescharacteristic
495C22×C4C2×C22⋊D1711yescharacteristic
496C22×C4C2×C22⋊D1711yescharacteristic
497C22×C4C2×C22⋊D1711yescharacteristic
498C22×C4C2×C22⋊D1711yescharacteristic
499C22×C4C23×D1711yescharacteristic
500SD16—1716no—
501C2×C8—172no—
502C2×C8—172no—
503C2×C8—172no—
504C2×C8—172no—
505C2×C8—341no—
506C2×C8—341no—
507C2×C8—341no—
508C2×C8—341no—
509C2×C8—341no—
510C2×C8—341no—
511C2×C8—341no—
512C2×C8—341no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—41no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—81no—
524C4⋊C4—172no—
525C4⋊C4—341no—
526C4⋊C4—344no—
527C4⋊C4—341no—
528C4⋊C4—341no—
529C4⋊C4—341no—
530C4⋊C4—341no—
531C4⋊C4—682no—
532C4⋊C4—682no—
533C4⋊C4—682no—
534C4⋊C4—682no—
535C4⋊C4C2×C4×D1712yes—
536C22⋊C4—24no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—344no—
544C42—682no—
545C42—682no—
546C42C2×C4×D1712yes—
547C42#6325212yes—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23C22×C22⋊D1711yescharacteristic
556Q8—174no—
557Q8—342no—
558Q8—342no—
559Q8—342no—
560D4—22no—
561D4—22no—
562D4—22no—
563D4C4⋊C4×D1714yes—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—48no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—342no—
592C2×C4—342no—
593C2×C4—342no—
594C2×C4—342no—
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×C4C4×C22⋊D1712yes—
604C2×C4C4×C22⋊D1712yes—
605C2×C4#1058761812yes—
606C2×C4C22×C4×D1711yescharacteristic
607C2×C4C2×#6325211yescharacteristic
608C2×C4C2×D4×D1711yescharacteristic
609C2×C4#1058769111yescharacteristic
610C2×C4C2×Q8×D1711yescharacteristic
611C2×C4#1058769411yescharacteristic
612C8—174no—
613C8—342no—
614C8—681no—
615C8—681no—
616C22—24no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22C2×C4×C22⋊D1711yescharacteristic
624C22D4×C22⋊D1711yescharacteristic
625C22Q8×C22⋊D1711yescharacteristic
626C22C2×#1058761811yescharacteristic
627C22#1176360711yescharacteristic
628C22#1176362011yescharacteristic
629C22#1176362411yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—41no—
634C4—41no—
635C4—42no—
636C4—41no—
637C4—81no—
638C4—81no—
639C4—344no—
640C4—682no—
641C4—1361no—
642C4—1361no—
643C4C2×C4⋊C4×D1711yescharacteristic
644C4#1176329011yescharacteristic
645C4#1176346011yescharacteristic
646C4#1176351411yescharacteristic
647C2—44no—
648C2C4⋊C4×C22⋊D1711yescharacteristic
649C2C4×#1058761811yescharacteristic
650C2#1453762311yescharacteristic
651C2#1454027511yescharacteristic
652C2#1454033511yescharacteristic
653C2#1454170511yescharacteristic
654C2#1454172911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingD4×C4⋊C4×C17C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenterD4×C4⋊C4D17
hypocenterC17#22677
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #14531704C2semidirect
5. #14529519C2semidirect
6. #14529514C2semidirect
7. #14529051C2semidirect
9. #14529037C2semidirect
11. #14528950C2semidirect
13. #14528786C2semidirect
14. #14528659C2semidirect
21. #11762909C4semidirect
112. C4×Dic34C2×C4semidirect
126. #10587481C2×C4semidirect
131. #10587478C2×C4semidirect
134. #10587474C2×C4semidirect
193. #63229C4⋊C4semidirect
284. Dic34C2×C4⋊C4semidirect
288. #3622C2×C4⋊C4semidirect
463. C17#22677semidirect

Maximal subgroups

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

subgroupindex
C2×#117629092
C2×C4×#632292
#145317042
#145295192
#145295142
#145290512
#145290452
#145290372
#145290312
#145289502
#145287942
#145287862
#145286592
D4×C4⋊C4×C172
#2267717