Finite Group Databasebeta

← #17898007

Subgroups — #17898007

Order 4352.

2694 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763574C211yesmaximal, characteristic
2#14541984C211yesmaximal, characteristic
3D17×#520C211yesmaximal, characteristic
4#14537383C213yesmaximal, complemented
5#14537382C211yesmaximal, characteristic, complemented
6#14537380C213yesmaximal, complemented
7#14537379C211yesmaximal, characteristic, complemented
8#14537378C213yesmaximal, complemented
9#14537377C211yesmaximal, characteristic, complemented
10#14537376C213yesmaximal, complemented
11#14537372C213yesmaximal, complemented
12#14537371C213yesmaximal, complemented
13C2×#11762850C213yesmaximal, complemented
14D17×#428C213yesmaximal, complemented
15C17×#2913C211yesmaximal, characteristic, complemented
16C2×#41×D17C2211yescharacteristic
17#11763574C4116yescomplemented
18C2×#10587593C2213yes—
19C2×#10587634C2211yescharacteristic
20C2×#10587587C2213yes—
21C22×#63218C2213yes—
22C2×M16×D17C2213yes—
23#11763292C2211yescharacteristic
24#11763292C2211yescharacteristic
25#11763291—21no—
26#11763290C2213yes—
27#11763290C2213yes—
28#11763289C2213yes—
29#11763289C2213yes—
30#11763288—23no—
31#11763285—21no—
32#11763281—23no—
33C4×#63256—21no—
34C4×#63254—23no—
35C4×Q8×D17—21no—
36C4×D4×D17—23no—
37C2×#10587578C2213yes—
38#117×D17C2213yes—
39#117×D17C2213yes—
40C2×C4⋊C4×D17C2213yes—
41#11762852C2213yescomplemented
42#11762852C2213yescomplemented
43#11762851C2213yescomplemented
44#11762851C2213yescomplemented
45#11762850C2216yes—
46#11762850C2213yes—
47#11762850C2211yescharacteristic
48#11762850C2216yescomplemented
49#11762850C2216yescomplemented
50#11762850C2213yescomplemented
51#11762850C2213yescomplemented
52#11762849C2213yescomplemented
53#11762849C2213yescomplemented
54#11762848C2213yes—
55#11762848C2213yes—
56#11762848C2213yescomplemented
57#11762848C2213yescomplemented
58#11762847C2213yescomplemented
59#11762846C2216yescomplemented
60#11762845C2213yescomplemented
61#11762845C2213yescomplemented
62#11762844C2216yescomplemented
63C2×#10587478C2213yescomplemented
64#107×D17C2216yescomplemented
65C2×#10587488C2213yescomplemented
66C2×#10587479C2213yescomplemented
67#106×D17C2216yescomplemented
68C2×C17×#131C2211yescharacteristic
69C17×#520C2211yescharacteristic
70C17×#520C2211yescharacteristic
71C17×#428C2213yescomplemented
72C17×#428C2213yescomplemented
73C17×#427C2213yescomplemented
74C2×C17×#107C2213yescomplemented
75C2×#41×C17C2311yescharacteristic
76#41×D17C2×C418yes—
77#41×D17Q818yes—
78C2×#63256D411yescharacteristic
79C2×Q8×D17D411yescharacteristic
80C2×#63254D413yes—
81C2×D4×D17D413yes—
82C2×#63252C2313yes—
83C22×C4×D17—23no—
84C22×C4×D17C2313yes—
85C17×#131C2×C418yescomplemented
86C2×M16×C17C2313yes—
87C22×C8×C17C2313yes—
88C4×Q8×C17—21no—
89C4×Q8×C17—21no—
90C4×D4×C17—23no—
91C4×D4×C17—23no—
92C17×#117C2313yes—
93C17×#117C2313yes—
94C2×C4⋊C4×C17C2313yes—
95C2×C4⋊C4×C17C2313yes—
96#10587634C2×C418yescomplemented
97Q8×Dic17—21no—
98Q8×Dic17—21no—
99D4×Dic17—23no—
100D4×Dic17—23no—
101C4×C22⋊D17—26no—
102C4×C22⋊D17—26no—
103C2×#63227—23no—
104#10587612C2313yes—
105#10587612C2313yes—
106C2×C4⋊Dic17C2313yes—
107C2×C4⋊Dic17C2313yes—
108C2×#63225—23no—
109C2×C4×Dic17—23no—
110C2×#63223C2313yes—
111C22×#3622C2313yes—
112#10587593C2×C4124yescomplemented
113M16×D17C2×C4124yescomplemented
114#10587587C2×C4124yescomplemented
115C2×#63218C2×C4124yescomplemented
116C2×#63218C2313yes—
117C2×#63218C2311yescharacteristic
118C2×#63218C2313yes—
119C2×#63218C2313yes—
120C2×C8×D17C2316yes—
121#10587579—23no—
122#10587579—23no—
123#10587578—23no—
124#10587578—23no—
125#10587578C2311yescharacteristic
126#10587578C2313yes—
127#10587578C2313yes—
128#10587578C2313yes—
129#10587578C2316yes—
130C4⋊C4×D17—23no—
131C4⋊C4×D17—23no—
132C4⋊C4×D17C2313yes—
133C4⋊C4×D17C2316yes—
134C4⋊C4×D17C2311yescharacteristic
135C4⋊C4×D17C2313yes—
136C4⋊C4×D17C2313yes—
137#10587573—23no—
138#10587573—23no—
139#10587567—26no—
140#10587567—26no—
141C22⋊C4×D17—23no—
142C22⋊C4×D17—23no—
143#10587563—23no—
144#10587563—23no—
145C4×D68—23no—
146C4×D68—23no—
147#10587558—23no—
148#10587558—23no—
149C42×D17—23no—
150C42×D17—23no—
151C4×Dic34—23no—
152C4×Dic34—23no—
153C17×#107C2316yes—
154C17×#107C2313yes—
155C17×#107C2311yescharacteristic
156C17×#106C2313yes—
157C17×#106C2313yes—
158#10587489C2313yes—
159#10587489C2313yes—
160#10587488C2316yes—
161#10587488C2313yes—
162#10587488C2311yescharacteristic
163#10587479C2313yes—
164#10587479C2316yes—
165#10587479C2313yes—
166#10587479C2311yescharacteristic
167#10587479C2313yes—
168#10587478C2313yes—
169#10587478C2316yes—
170#10587478C2313yes—
171#10587478C2313yes—
172#10587478C2311yescharacteristic
173C23×D17C2×D413yes—
174#41×C17C22×C414yes—
175#41×C17C2×Q814yes—
176C2×Q8×C17C2×D411yescharacteristic
177C2×D4×C17C2×D413yes—
178C22×C4×C17—23no—
179C22×C4×C17—23no—
180C22×C4×C17C2413yes—
181C2×C22⋊D17C2×D416yes—
182C22×Dic17—23no—
183C22×Dic17—23no—
184C22×Dic17C2×D413yes—
185#63256C4⋊C418yes—
186Q8×D17C4⋊C418yes—
187#63254C4⋊C4124yes—
188D4×D17C4⋊C4124yes—
189#63252C22×C4112yes—
190#63252C2×Q8112yes—
191C2×D68C2×D413yes—
192C2×C4×D17—23no—
193C2×C4×D17—21no—
194C2×C4×D17—26no—
195C2×C4×D17—23no—
196C2×C4×D17—23no—
197C2×C4×D17—86no—
198C2×C4×D17C22×C4112yes—
199C2×C4×D17C2×D413yes—
200C2×C4×D17C2×D413yes—
201C2×C4×D17C2×Q8112yes—
202C2×C4×D17C2411yescharacteristic
203C2×C4×D17C2413yes—
204C2×Dic34C2×D413yes—
205M16×C17C22×C4112yescomplemented
206C2×C8×C17C22×C4112yescomplemented
207C2×C8×C17C2411yescharacteristic
208C2×C8×C17C2413yes—
209C4⋊C4×C17—23no—
210C4⋊C4×C17—23no—
211C4⋊C4×C17C2411yescharacteristic
212C4⋊C4×C17C2413yes—
213C4⋊C4×C17C2411yescharacteristic
214C4⋊C4×C17C2413yes—
215C22⋊C4×C17—23no—
216C22⋊C4×C17—23no—
217C42×C17—23no—
218C42×C17—23no—
219C22⋊Dic17—23no—
220C22⋊Dic17—23no—
221#63227—21no—
222#63227—23no—
223#63227—23no—
224#63227—23no—
225#63227—26no—
226C4⋊Dic17—23no—
227C4⋊Dic17—23no—
228C4⋊Dic17C2413yes—
229C4⋊Dic17C2411yescharacteristic
230C4⋊Dic17C2413yes—
231C4⋊Dic17C2411yescharacteristic
232#63225—23no—
233#63225—26no—
234#63225—21no—
235#63225—23no—
236#63225—23no—
237C4×Dic17—23no—
238C4×Dic17—23no—
239C4×Dic17—23no—
240C4×Dic17—26no—
241C4×Dic17—21no—
242C4×Dic17—23no—
243C4×Dic17—23no—
244#63223C22×C4112yescomplemented
245C2×#3622C22×C4112yescomplemented
246C2×#3622C2413yes—
247C2×#3622C2411yescharacteristic
248#63218C22×C414yes—
249#63218C22×C4112yes—
250#63218C22×C4112yescomplemented
251#63218C22×C4112yescomplemented
252C8×D17C22×C4124yescomplemented
253#60109—171nomaximal, Sylow
254C23×C17C22×D413yes—
255C22×D17—43no—
256C22×D17—43no—
257C22×D17C2×C4⋊C4112yes—
258C22×D17C22×D411yescharacteristic
259C22×D17C22×D413yes—
260Q8×C17C2×C4⋊C414yes—
261D4×C17C2×C4⋊C4112yes—
262C2×C4×C17—21no—
263C2×C4×C17—21no—
264C2×C4×C17—23no—
265C2×C4×C17—23no—
266C2×C4×C17—83no—
267C2×C4×C17—83no—
268C2×C4×C17C23×C416yes—
269C2×C4×C17C22×D413yes—
270C2×C4×C17C22×Q816yes—
271C2×C4×C17C2511yescharacteristic
272C22⋊D17C2×C4⋊C4124yes—
273C2×Dic17—21no—
274C2×Dic17—21no—
275C2×Dic17—23no—
276C2×Dic17—23no—
277C2×Dic17—83no—
278C2×Dic17—83no—
279C2×Dic17C2×C4⋊C4112yes—
280C2×Dic17C22×D413yes—
281C2×Dic17C22×D411yescharacteristic
282D68C2×C4⋊C4112yes—
283C4×D17—42no—
284C4×D17—46no—
285C4×D17—83no—
286C4×D17—83no—
287C4×D17—86no—
288C4×D17C2×C4⋊C4112yes—
289C4×D17C2×C4⋊C4112yes—
290C4×D17C23×C412yes—
291C4×D17C23×C416yes—
292C4×D17C22×Q812yes—
293C4×D17C22×Q816yes—
294Dic34C2×C4⋊C4112yes—
295C8×C17C23×C412yes—
296C8×C17C23×C416yes—
297#3622C23×C416yes—
298#3622C23×C412yes—
299C2×#568—171no—
300#3428—171no—
301C2×#520—171no—
302#2913—173no—
303#2913—173no—
304#2913—171no—
305#2913D1711yescharacteristic, complemented
306#2912—173no—
307#2912—173no—
308#2912—171no—
309#2912—171no—
310#2911—173no—
311#2911—173no—
312C2×#428—173no—
313C2×#428—173no—
314C22×C17—23no—
315C22×C17—23no—
316C22×C17C22×C4⋊C416yes—
317C22×C17C23×D411yescharacteristic
318C2×D17—21no—
319C2×D17—23no—
320C2×D17—21no—
321C2×D17—23no—
322C2×D17—412no—
323C2×D17C22×C4⋊C412yes—
324C2×D17C22×C4⋊C416yes—
325C4×C17—43no—
326C4×C17—43no—
327C4×C17—41no—
328C4×C17—41no—
329C4×C17C22×C4⋊C416yes—
330C4×C17C24×C411yescharacteristic
331C4×C17C23×Q811yescharacteristic
332Dic17—43no—
333Dic17—41no—
334Dic17—41no—
335Dic17—43no—
336Dic17C22×C4⋊C416yes—
337Dic17C22×C4⋊C412yes—
338C22×#41—171no—
339#568—1716no—
340C2×#131—171no—
341C2×#131—173no—
342C2×#131—173no—
343C2×#131C2×D1711yescharacteristic
344C22×M16—173no—
345C22×M16—173no—
346#520—171no—
347#520—171no—
348#520—173no—
349#520—173no—
350#520—173no—
351#520—173no—
352#520—343no—
353#520—341no—
354#520C2×D1711yescharacteristic
355#520C2×D1711yescharacteristic
356#519—341no—
357#518—343no—
358C4×#41—343no—
359C4×#41—341no—
360C2×C4×Q8—341no—
361C2×C4×D4—343no—
362C2×#117—173no—
363C2×#117—173no—
364C2×#117—173no—
365C22×C4⋊C4—173no—
366#428—173no—
367#428—173no—
368#428—173no—
369#428—173no—
370#428—173no—
371#428—173no—
372#428—176no—
373#428—173no—
374#428—176no—
375#428—176no—
376#428—171no—
377#428—173no—
378#428—173no—
379#428—173no—
380#428—173no—
381#428—173no—
382#428—173no—
383#428C2×D1713yescomplemented
384#428C2×D1713yescomplemented
385#427—173no—
386#427—173no—
387#427—173no—
388#427—176no—
389#427—176no—
390#427C2×D1713yescomplemented
391C2×#107—173no—
392C2×#107—176no—
393C2×#107C2×D1713yescomplemented
394C2×#106—173no—
395C2×#106—173no—
396C2×#106—176no—
397C2×C17—26no—
398C2×C17C23×C4⋊C411yescharacteristic
399C2×C17#358511yescharacteristic
400C2×C17#358611yescharacteristic
401D17—22no—
402D17—26no—
403C2×#41—178no—
404C2×#41—178no—
405C2×#41—171no—
406C2×#41—173no—
407C2×#41—173no—
408C2×#41C22×D1711yescharacteristic
409C22×Q8—171no—
410C22×D4—173no—
411C23×C4—173no—
412C23×C4—343no—
413#131—178no—
414#131—1724no—
415#131—1724no—
416#131C4×D1718yescomplemented
417C2×M16—173no—
418C2×M16—1724no—
419C2×M16—1724no—
420C2×M16—171no—
421C2×M16—173no—
422C2×M16—173no—
423C2×M16—173no—
424C2×M16C22×D1713yes—
425C22×C8—173no—
426C22×C8—176no—
427C22×C8C22×D1713yes—
428C4×Q8—21no—
429C4×Q8—21no—
430C4×Q8—343no—
431C4×Q8—343no—
432C4×Q8—343no—
433C4×Q8—343no—
434C4×Q8—341no—
435C4×Q8—341no—
436C4×D4—23no—
437C4×D4—23no—
438C4×D4—343no—
439C4×D4—343no—
440C4×D4—346no—
441C4×D4—346no—
442C4×D4—343no—
443C4×D4—343no—
444C4×D4—346no—
445C4×D4—346no—
446C4×D4—343no—
447C4×D4—343no—
448#117—173no—
449#117—173no—
450#117—173no—
451#117—171no—
452#117—176no—
453#117—173no—
454#117—173no—
455#117—343no—
456#117—343no—
457#117—343no—
458#117—343no—
459#117—343no—
460#117—343no—
461#117C22×D1713yes—
462#117C22×D1713yes—
463C2×C4⋊C4—173no—
464C2×C4⋊C4—173no—
465C2×C4⋊C4—173no—
466C2×C4⋊C4—176no—
467C2×C4⋊C4—171no—
468C2×C4⋊C4—173no—
469C2×C4⋊C4—173no—
470C2×C4⋊C4—343no—
471C2×C4⋊C4—343no—
472C2×C4⋊C4—343no—
473C2×C4⋊C4C22×D1713yes—
474C2×C4⋊C4C22×D1713yes—
475C2×C22⋊C4—343no—
476C2×C22⋊C4—343no—
477C2×C22⋊C4—343no—
478C2×C42—343no—
479C2×C42—343no—
480C2×C42—343no—
481#107—173no—
482#107—173no—
483#107—176no—
484#107—173no—
485#107—173no—
486#107—173no—
487#107—171no—
488#107C22×D1716yes—
489#107C22×D1713yes—
490#107C22×D1711yescharacteristic
491#106—173no—
492#106—176no—
493#106—171no—
494#106—176no—
495#106—173no—
496#106—173no—
497#106—173no—
498#106—171no—
499#106C22×D1713yes—
500#106C22×D1713yes—
501C17#6010911yescharacteristic, complemented, Sylow
502C24—173no—
503#41—178no—
504#41—1724no—
505#41—1712no—
506#41—1712no—
507#41C2×C4×D1714yes—
508#41Q8×D1714yes—
509C2×Q8—178no—
510C2×Q8—173no—
511C2×Q8D4×D1711yescharacteristic
512C2×D4—176no—
513C2×D4—1724no—
514C2×D4—173no—
515C2×D4D4×D1713yes—
516C22×C4—23no—
517C22×C4—23no—
518C22×C4—173no—
519C22×C4—1712no—
520C22×C4—173no—
521C22×C4—173no—
522C22×C4—1712no—
523C22×C4—173no—
524C22×C4—171no—
525C22×C4—341no—
526C22×C4—343no—
527C22×C4—343no—
528C22×C4—343no—
529C22×C4—346no—
530C22×C4—343no—
531C22×C4—343no—
532C22×C4—1366no—
533C22×C4C23×D1713yes—
534M16—1712no—
535M16—1712no—
536M16—174no—
537M16—1712no—
538M16—1712no—
539M16C2×C4×D17112yescomplemented
540C2×C8—1712no—
541C2×C8—171no—
542C2×C8—173no—
543C2×C8—1724no—
544C2×C8C2×C4×D17112yescomplemented
545C2×C8C23×D1711yescharacteristic
546C2×C8C23×D1713yes—
547C4⋊C4—23no—
548C4⋊C4—23no—
549C4⋊C4—173no—
550C4⋊C4—173no—
551C4⋊C4—171no—
552C4⋊C4—171no—
553C4⋊C4—343no—
554C4⋊C4—346no—
555C4⋊C4—341no—
556C4⋊C4—343no—
557C4⋊C4—343no—
558C4⋊C4—343no—
559C4⋊C4—343no—
560C4⋊C4C23×D1711yescharacteristic
561C4⋊C4C23×D1713yes—
562C4⋊C4C23×D1711yescharacteristic
563C4⋊C4C23×D1713yes—
564C22⋊C4—23no—
565C22⋊C4—23no—
566C22⋊C4—341no—
567C22⋊C4—343no—
568C22⋊C4—343no—
569C22⋊C4—346no—
570C22⋊C4—343no—
571C22⋊C4—343no—
572C22⋊C4—343no—
573C42—23no—
574C42—23no—
575C42—341no—
576C42—343no—
577C42—343no—
578C42—346no—
579C42—343no—
580C42—343no—
581C42—343no—
582C23—171no—
583C23—173no—
584C23—1712no—
585C23—683no—
586C23—683no—
587C23C2×D4×D1713yes—
588Q8—1712no—
589Q8C4⋊C4×D1714yes—
590D4—1724no—
591D4—1712no—
592D4C4⋊C4×D17112yes—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—23no—
596C2×C4—23no—
597C2×C4—83no—
598C2×C4—83no—
599C2×C4—1712no—
600C2×C4—171no—
601C2×C4—173no—
602C2×C4—1712no—
603C2×C4—1712no—
604C2×C4—176no—
605C2×C4—172no—
606C2×C4—172no—
607C2×C4—176no—
608C2×C4—341no—
609C2×C4—341no—
610C2×C4—343no—
611C2×C4—343no—
612C2×C4—686no—
613C2×C4—682no—
614C2×C4—1366no—
615C2×C4—1363no—
616C2×C4—1363no—
617C2×C4—1363no—
618C2×C4—1363no—
619C2×C4C22×C4×D1716yes—
620C2×C4C2×D4×D1713yes—
621C2×C4C2×Q8×D1716yes—
622C2×C4C24×D1711yescharacteristic
623C8—172no—
624C8—176no—
625C8C22×C4×D1716yes—
626C8C22×C4×D1712yes—
627C22—23no—
628C22—23no—
629C22—172no—
630C22—176no—
631C22—341no—
632C22—343no—
633C22—341no—
634C22—343no—
635C22—6812no—
636C22C2×C4⋊C4×D1716yes—
637C22C22×D4×D1711yescharacteristic
638C4—43no—
639C4—43no—
640C4—41no—
641C4—41no—
642C4—172no—
643C4—176no—
644C4—683no—
645C4—681no—
646C4—681no—
647C4—683no—
648C4C2×C4⋊C4×D1716yes—
649C4C23×C4×D1711yescharacteristic
650C4C22×Q8×D1711yescharacteristic
651C2—26no—
652C2—342no—
653C2—346no—
654C2C22×C4⋊C4×D1711yescharacteristic
655C2#1454355911yescharacteristic
656C2#1454356911yescharacteristic
subgroupG/H
centerC22C22×D4×D17
commutatorC4×C17C24×C4
FrattiniC2×C4C24×D17
FittingC17×#2913C2
socleC22×C17C23×D4
prime radicalC2×#41×D17C22
cosocleC2×C4×C17C25
hypercenter#2913D17
hypocenterC17#60109
LCSG ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×#41 ◁ #2913

Complements

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

NH ≅ G/Ntype
4. #14537383C2semidirect
5. #14537382C2semidirect
6. #14537380C2semidirect
7. #14537379C2semidirect
8. #14537378C2semidirect
9. #14537377C2semidirect
10. #14537376C2semidirect
11. #14537372C2semidirect
12. #14537371C2semidirect
13. C2×#11762850C2semidirect
14. D17×#428C2semidirect
15. C17×#2913C2semidirect
17. #11763574C4semidirect
41. #11762852C22semidirect
42. #11762852C22semidirect
43. #11762851C22semidirect
44. #11762851C22semidirect
48. #11762850C22semidirect
49. #11762850C22semidirect
50. #11762850C22semidirect
51. #11762850C22semidirect
52. #11762849C22semidirect
53. #11762849C22semidirect
56. #11762848C22semidirect
57. #11762848C22semidirect
58. #11762847C22semidirect
59. #11762846C22semidirect
60. #11762845C22semidirect
61. #11762845C22semidirect
62. #11762844C22semidirect
63. C2×#10587478C22semidirect
64. #107×D17C22semidirect
65. C2×#10587488C22semidirect
66. C2×#10587479C22semidirect
67. #106×D17C22semidirect
71. C17×#428C22semidirect
72. C17×#428C22semidirect
73. C17×#427C22semidirect
74. C2×C17×#107C22semidirect
85. C17×#131C2×C4semidirect
96. #10587634C2×C4semidirect
112. #10587593C2×C4semidirect
113. M16×D17C2×C4semidirect
114. #10587587C2×C4semidirect
115. C2×#63218C2×C4semidirect
205. M16×C17C22×C4semidirect
206. C2×C8×C17C22×C4semidirect
244. #63223C22×C4semidirect
245. C2×#3622C22×C4semidirect
250. #63218C22×C4semidirect
251. #63218C22×C4semidirect
252. C8×D17C22×C4semidirect
305. #2913D17semidirect
383. #428C2×D17semidirect
384. #428C2×D17semidirect
390. #427C2×D17semidirect
393. C2×#107C2×D17semidirect
416. #131C4×D17semidirect
501. C17#60109semidirect
539. M16C2×C4×D17semidirect
544. C2×C8C2×C4×D17semidirect

Maximal subgroups

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

subgroupindex
C2×#117635742
#145419842
D17×#5202
#145373832
#145373822
#145373802
#145373792
#145373782
#145373772
#145373762
#145373722
#145373712
C2×#117628502
D17×#4282
C17×#29132
#6010917