Finite Group Databasebeta

← #17197801

Subgroups — #17197801

Order 4352.

2286 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763380C211yesmaximal, characteristic
2#14531963C212yesmaximal, complemented
3#14531927C212yesmaximal, complemented
4#14531917C212yesmaximal, complemented
5#14531872C212yesmaximal, complemented
6#14531811C212yesmaximal, complemented
7#14531789C212yesmaximal, complemented
8#14531679C212yesmaximal, complemented
9#14531616C212yesmaximal, complemented
10#14531593C214yesmaximal, complemented
11#14531571C214yesmaximal, complemented
12#14531462C212yesmaximal, complemented
13#14531312C212yesmaximal, complemented
14#14531275C211yesmaximal, characteristic
15C17×#2311C211yesmaximal, characteristic
16C23×Dic34C2211yescharacteristic
17#11763380C4116yescomplemented
18C2×#10587630C2212yes—
19C2×#10587622C2212yes—
20C2×#10587621C2212yes—
21C2×#10587621C2212yes—
22C2×#10587610C2212yes—
23C2×#10587610C2212yes—
24C2×#10587573C2212yes—
25C2×#10587573C2212yes—
26C2×#10587612C2211yescharacteristic
27C2×#10587612C2211yescharacteristic
28C22×C4×Dic17C2211yescharacteristic
29#11762716C2211yescharacteristic
30#11762716C2211yescharacteristic
31#11762714C2212yes—
32#11762714C2212yes—
33#11762713C2214yes—
34#11762713C2214yes—
35#11762712C2211yescharacteristic
36#11762712C2211yescharacteristic
37#11762706C2214yes—
38#11762705C2212yes—
39#11762705C2212yes—
40#11762704C2214yes—
41#11762703C2214yes—
42#11762703C2214yes—
43#11762697C2214yes—
44#11762697C2214yes—
45#11762697C2214yes—
46#11762697C2214yes—
47#11762696C2214yes—
48#11762693C2214yes—
49#11762691C2214yes—
50#11762690C2214yes—
51#11762689C2212yes—
52#11762689C2212yes—
53#11762682C2214yes—
54#11762681C2214yes—
55#11762676C2214yes—
56#11762667C2214yes—
57C4⋊C4×Dic17C2212yes—
58C4⋊C4×Dic17C2212yes—
59C4×#63225C2214yes—
60#11762660C2214yes—
61#11762660C2214yes—
62C22⋊C4×Dic17C2214yes—
63C22⋊C4×Dic17C2214yes—
64C4×C22⋊Dic17C2214yes—
65C4×C22⋊Dic17C2212yes—
66C4×C22⋊Dic17C2212yes—
67C22×#41×C17C2211yescharacteristic
68C2×C17×#117C2211yescharacteristic
69C22×C4⋊C4×C17C2211yescharacteristic
70C17×#391C2212yes—
71C17×#390C2212yes—
72C17×#386C2212yes—
73C17×#386C2212yes—
74C17×#380C2212yes—
75C17×#380C2212yes—
76C2×#41×C17—42no—
77C2×#41×C17—42no—
78C2×#41×C17C2×C418yes—
79C22×Q8×C17C2311yescharacteristic
80C22×D4×C17C2311yescharacteristic
81C22×D4×C17C2311yescharacteristic
82C22×D4×C17C2311yescharacteristic
83C23×C4×C17C2311yescharacteristic
84C23×C4×C17C2311yescharacteristic
85C23×C4×C17C2311yescharacteristic
86C23×C4×C17C2312yes—
87C23×Dic17C2312yes—
88C23×Dic17C2312yes—
89C22×Dic34—42no—
90C22×Dic34—42no—
91C22×Dic34C2×C418yescomplemented
92C22×Dic34C2312yes—
93C22×Dic34C2312yes—
94C17×#117—42no—
95C2×C4⋊C4×C17—42no—
96C2×C4⋊C4×C17C2312yes—
97C2×C4⋊C4×C17C2312yes—
98C2×C4⋊C4×C17C2312yes—
99C2×C22⋊C4×C17C2312yes—
100C2×C22⋊C4×C17C2314yes—
101C2×C22⋊C4×C17C2314yes—
102C2×C42×C17C2312yes—
103C2×C22⋊Dic17C2312yes—
104C2×C22⋊Dic17C2312yes—
105C2×C22⋊Dic17C2312yes—
106C2×C22⋊Dic17C2312yes—
107C2×C22⋊Dic17C2312yes—
108C2×C22⋊Dic17C2312yes—
109C2×C22⋊Dic17C2314yes—
110C2×C22⋊Dic17C2314yes—
111#10587630C2×C4116yescomplemented
112#10587622C2×C4116yes—
113#10587621C2×C4116yescomplemented
114#10587621C2×C4116yescomplemented
115#10587612—42no—
116#10587612C2×C418yescomplemented
117C2×C4⋊Dic17C2311yescharacteristic
118C2×C4⋊Dic17C2311yescharacteristic
119C2×C4⋊Dic17C2312yes—
120#10587610C2×C4116yescomplemented
121#10587610C2×C4116yescomplemented
122C2×#63225C2314yes—
123C2×#63225C2314yes—
124C2×#63225C2314yes—
125C2×#63225C2314yes—
126C2×#63225C2314yes—
127C2×#63225C2314yes—
128C2×C4×Dic17—42no—
129C2×C4×Dic17C2312yes—
130C2×C4×Dic17C2312yes—
131C2×C4×Dic17C2314yes—
132C2×C4×Dic17C2311yescharacteristic
133C2×C4×Dic17C2314yes—
134C2×C4×Dic17C2314yes—
135C2×C4×Dic17C2314yes—
136C2×C4×Dic17C2311yescharacteristic
137C2×C4×Dic17C2312yes—
138#10587573—44no—
139#10587573—44no—
140C17×#95C2314yes—
141C17×#95C2314yes—
142#10587502C2312yes—
143#10587502C2312yes—
144#10587502C2312yes—
145#10587502C2312yes—
146#10587502C2314yes—
147#10587502C2314yes—
148#10587502C2314yes—
149#10587502C2314yes—
150#10587502C2314yes—
151#10587502C2314yes—
152#10587502C2314yes—
153#10587502C2314yes—
154C24×C17C2411yescharacteristic
155C24×C17C2411yescharacteristic
156C24×C17C2411yescharacteristic
157#41×C17—416no—
158C2×Q8×C17—41no—
159C2×Q8×C17—41no—
160C2×Q8×C17C22×C414yes—
161C2×D4×C17—41no—
162C2×D4×C17—41no—
163C2×D4×C17—41no—
164C2×D4×C17—41no—
165C2×D4×C17—41no—
166C2×D4×C17—41no—
167C2×D4×C17C22×C414yes—
168C2×D4×C17C22×C414yes—
169C2×D4×C17C22×C414yes—
170C22×C4×C17—22no—
171C22×C4×C17—22no—
172C22×C4×C17—22no—
173C22×C4×C17—22no—
174C22×C4×C17—22no—
175C22×C4×C17—22no—
176C22×C4×C17—44no—
177C22×C4×C17—42no—
178C22×C4×C17C22×C414yes—
179C22×C4×C17C22×C414yes—
180C22×C4×C17C22×C414yes—
181C22×C4×C17C2414yes—
182C22×C4×C17C2414yes—
183C22×C4×C17C2411yescharacteristic
184C22×C4×C17C2411yescharacteristic
185C22×C4×C17C2411yescharacteristic
186C22×C4×C17C2411yescharacteristic
187C22×Dic17—24no—
188C22×Dic17—42no—
189C22×Dic17—42no—
190C22×Dic17—42no—
191C22×Dic17—42no—
192C22×Dic17C22×C418yes—
193C22×Dic17C2412yes—
194C22×Dic17C2412yes—
195C22×Dic17C2412yes—
196C22×Dic17C2414yes—
197C22×Dic17C2414yes—
198C22×Dic17C2412yes—
199C2×Dic34—24no—
200C2×Dic34—24no—
201C2×Dic34—416no—
202C2×Dic34—42no—
203C2×Dic34—42no—
204C2×Dic34C22×C418yes—
205C2×Dic34C22×C418yes—
206C4⋊C4×C17—24no—
207C4⋊C4×C17—24no—
208C4⋊C4×C17—42no—
209C22⋊C4×C17—44no—
210C22⋊C4×C17—44no—
211C22⋊C4×C17—42no—
212C42×C17—42no—
213C22⋊Dic17—24no—
214C22⋊Dic17—24no—
215C22⋊Dic17—42no—
216C22⋊Dic17—42no—
217C22⋊Dic17—42no—
218C22⋊Dic17C22×C4116yes—
219C22⋊Dic17C22×C4116yes—
220C22⋊Dic17C22×C418yes—
221C4⋊Dic17—41no—
222C4⋊Dic17—41no—
223C4⋊Dic17C22×C418yes—
224#63225—28no—
225#63225—44no—
226#63225—44no—
227#63225—44no—
228#63225C22×C4116yes—
229#63225C22×C4116yes—
230C4×Dic17—24no—
231C4×Dic17—24no—
232C4×Dic17—44no—
233C4×Dic17—44no—
234C4×Dic17—41no—
235C4×Dic17—41no—
236C4×Dic17—44no—
237C4×Dic17—44no—
238C4×Dic17C22×C418yes—
239#34413—171nomaximal, Sylow
240C23×C17—21no—
241C23×C17—21no—
242C23×C17—21no—
243C23×C17—21no—
244C23×C17—21no—
245C23×C17—21no—
246C23×C17—22no—
247C23×C17—22no—
248C23×C17—22no—
249C23×C17—22no—
250C23×C17—22no—
251C23×C17—22no—
252C23×C17C23×C412yes—
253C23×C17C23×C412yes—
254C23×C17C23×C412yes—
255C23×C17C2511yescharacteristic
256Q8×C17—44no—
257D4×C17—44no—
258D4×C17—44no—
259D4×C17—44no—
260C2×C4×C17—21no—
261C2×C4×C17—21no—
262C2×C4×C17—21no—
263C2×C4×C17—21no—
264C2×C4×C17—21no—
265C2×C4×C17—21no—
266C2×C4×C17—28no—
267C2×C4×C17—28no—
268C2×C4×C17—28no—
269C2×C4×C17—24no—
270C2×C4×C17—24no—
271C2×C4×C17—24no—
272C2×C4×C17—24no—
273C2×C4×C17—24no—
274C2×C4×C17—24no—
275C2×C4×C17—84no—
276C2×C4×C17C23×C412yes—
277C2×C4×C17C23×C412yes—
278C2×C4×C17C23×C412yes—
279C2×C4×C17C23×C412yes—
280C2×C4×C17H(5,𝔽2)12yes—
281C2×C4×C17H(5,𝔽2)12yes—
282C2×Dic17—24no—
283C2×Dic17—24no—
284C2×Dic17—22no—
285C2×Dic17—24no—
286C2×Dic17—22no—
287C2×Dic17—24no—
288C2×Dic17—22no—
289C2×Dic17—22no—
290C2×Dic17—22no—
291C2×Dic17—22no—
292C2×Dic17—22no—
293C2×Dic17—22no—
294C2×Dic17—22no—
295C2×Dic17—22no—
296C2×Dic17—22no—
297C2×Dic17—48no—
298C2×Dic17—48no—
299C2×Dic17C23×C418yes—
300C2×Dic17C23×C418yes—
301C2×Dic172−1+414yes—
302Dic34—216no—
303Dic34—48no—
304C2×#536—171no—
305#2381—172no—
306#2376—172no—
307#2374—172no—
308#2369—172no—
309#2362—172no—
310#2360—172no—
311#2347—172no—
312Q8×C22⋊C4—172no—
313#2337—174no—
314#2335—174no—
315#2323—172no—
316#2311D1711yescharacteristic
317C4×C22⋊Q8—172no—
318#2297—171no—
319C22×C17—22no—
320C22×C17—22no—
321C22×C17—22no—
322C22×C17—24no—
323C22×C17—24no—
324C22×C17—24no—
325C22×C17—42no—
326C22×C17—42no—
327C22×C17—42no—
328C22×C17—41no—
329C22×C17—41no—
330C22×C17—41no—
331C22×C17—41no—
332C22×C17—41no—
333C22×C17—41no—
334C22×C17C24×C411yescharacteristic
335C22×C17C2×H(5,𝔽2)11yescharacteristic
336C22×C17C2×H(5,𝔽2)11yescharacteristic
337C22×C17C2×2−1+412yes—
338C22×C17#58512yes—
339C4×C17—22no—
340C4×C17—22no—
341C4×C17—22no—
342C4×C17—44no—
343C4×C17—44no—
344C4×C17#51814yes—
345Dic17—28no—
346Dic17—28no—
347Dic17—24no—
348Dic17—42no—
349Dic17—42no—
350Dic17—42no—
351C22×#41C2×D1711yescharacteristic
352C23×Q8—171no—
353#536—1716no—
354C2×C4⋊Q8—172no—
355C2×#124—172no—
356C2×#123—172no—
357C2×#123—172no—
358C2×C22⋊Q8—172no—
359C2×C22⋊Q8—172no—
360C2×C4×Q8—172no—
361C2×C4×Q8—172no—
362C2×#117—171no—
363C2×#117—171no—
364C2×#117C2×D1711yescharacteristic
365C22×C4⋊C4C2×D1711yescharacteristic
366C22×C42—171no—
367#391—171no—
368#391—171no—
369#391—172no—
370#391—172no—
371#391—174no—
372#391—174no—
373#391C2×D1712yes—
374#390—171no—
375#390—171no—
376#390C2×D1712yes—
377#389—174no—
378#389—172no—
379#389—172no—
380#389—174no—
381#388—174no—
382#388—174no—
383#388—174no—
384#388—174no—
385#388—174no—
386#388—174no—
387#387—174no—
388#387—174no—
389#387—174no—
390#387—174no—
391#386—172no—
392#386—172no—
393#386C2×D1712yes—
394#386C2×D1712yes—
395#385—174no—
396#385—174no—
397#383—174no—
398#380—174no—
399#380C2×D1712yes—
400#380C2×D1712yes—
401C4×C4⋊C4—172no—
402C4×C4⋊C4—172no—
403C4×C4⋊C4—174no—
404C4×C4⋊C4—174no—
405C4×C4⋊C4—174no—
406C4×C22⋊C4—174no—
407C4×C22⋊C4—172no—
408C4×C22⋊C4—172no—
409C4×C22⋊C4—174no—
410C4×C22⋊C4—174no—
411C2×C17—42no—
412C2×C17—42no—
413C2×C17—42no—
414C2×C17C2×#51811yescharacteristic
415C2×C17C4×2−1+412yes—
416C2×C17#354112yes—
417C2×C17#355011yescharacteristic
418C2×C17#355111yescharacteristic
419C2×#41—42no—
420C2×#41—42no—
421C2×#41C4×D1718yes—
422C22×Q8—178no—
423C22×Q8—172no—
424C22×Q8—172no—
425C22×Q8—682no—
426C22×Q8—682no—
427C22×Q8C22×D1711yescharacteristic
428C22×D4C22×D1711yescharacteristic
429C22×D4C22×D1711yescharacteristic
430C22×D4C22×D1711yescharacteristic
431C23×C4—172no—
432C23×C4—172no—
433C23×C4C22×D1711yescharacteristic
434C23×C4C22×D1711yescharacteristic
435C23×C4C22×D1711yescharacteristic
436C23×C4C22×D1712yes—
437C4⋊Q8—1716no—
438#124—1716no—
439#123—1716no—
440#123—1716no—
441C22⋊Q8—1716no—
442C22⋊Q8—1716no—
443C4×Q8—684no—
444C4×Q8—684no—
445#117—42no—
446#117—178no—
447#117—682no—
448C2×C4⋊C4—42no—
449C2×C4⋊C4—172no—
450C2×C4⋊C4—174no—
451C2×C4⋊C4—174no—
452C2×C4⋊C4—174no—
453C2×C4⋊C4—171no—
454C2×C4⋊C4—171no—
455C2×C4⋊C4—174no—
456C2×C4⋊C4—174no—
457C2×C4⋊C4—174no—
458C2×C4⋊C4C22×D1712yes—
459C2×C4⋊C4C22×D1712yes—
460C2×C4⋊C4C22×D1712yes—
461C2×C22⋊C4—172no—
462C2×C22⋊C4—174no—
463C2×C22⋊C4—174no—
464C2×C22⋊C4—172no—
465C2×C22⋊C4—172no—
466C2×C22⋊C4—172no—
467C2×C22⋊C4—172no—
468C2×C22⋊C4—172no—
469C2×C22⋊C4C22×D1712yes—
470C2×C22⋊C4C22×D1714yes—
471C2×C22⋊C4C22×D1714yes—
472C2×C42—172no—
473C2×C42—172no—
474C2×C42—172no—
475C2×C42—174no—
476C2×C42—174no—
477C2×C42—174no—
478C2×C42—171no—
479C2×C42—174no—
480C2×C42—171no—
481C2×C42—682no—
482C2×C42C22×D1712yes—
483#95—174no—
484#95—172no—
485#95—172no—
486#95—174no—
487#95—172no—
488#95—174no—
489#95—174no—
490#95—172no—
491#95—174no—
492#95—174no—
493#95—174no—
494#95—174no—
495#95C22×D1714yes—
496#95C22×D1714yes—
497C17#3441311yescharacteristic, complemented, Sylow
498C24C23×D1711yescharacteristic
499C24C23×D1711yescharacteristic
500C24C23×D1711yescharacteristic
501#41—416no—
502C2×Q8—41no—
503C2×Q8—41no—
504C2×Q8—178no—
505C2×Q8—178no—
506C2×Q8—344no—
507C2×Q8—344no—
508C2×Q8—682no—
509C2×Q8—682no—
510C2×Q8—6816no—
511C2×Q8C2×C4×D1714yes—
512C2×D4—41no—
513C2×D4—41no—
514C2×D4—41no—
515C2×D4—41no—
516C2×D4—41no—
517C2×D4—41no—
518C2×D4C2×C4×D1714yes—
519C2×D4C2×C4×D1714yes—
520C2×D4C2×C4×D1714yes—
521C22×C4—22no—
522C22×C4—22no—
523C22×C4—22no—
524C22×C4—22no—
525C22×C4—22no—
526C22×C4—22no—
527C22×C4—42no—
528C22×C4—44no—
529C22×C4—172no—
530C22×C4—174no—
531C22×C4—174no—
532C22×C4—178no—
533C22×C4—172no—
534C22×C4—172no—
535C22×C4—172no—
536C22×C4—344no—
537C22×C4—682no—
538C22×C4—682no—
539C22×C4—682no—
540C22×C4—682no—
541C22×C4C2×C4×D1714yes—
542C22×C4C2×C4×D1714yes—
543C22×C4C2×C4×D1714yes—
544C22×C4C23×D1714yes—
545C22×C4C23×D1714yes—
546C22×C4C23×D1711yescharacteristic
547C22×C4C23×D1711yescharacteristic
548C22×C4C23×D1711yescharacteristic
549C22×C4C23×D1711yescharacteristic
550C4⋊C4—24no—
551C4⋊C4—24no—
552C4⋊C4—42no—
553C4⋊C4—178no—
554C4⋊C4—1716no—
555C4⋊C4—1716no—
556C4⋊C4—348no—
557C4⋊C4—681no—
558C4⋊C4—681no—
559C4⋊C4—684no—
560C4⋊C4—684no—
561C4⋊C4—684no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—42no—
565C22⋊C4—178no—
566C22⋊C4—1716no—
567C22⋊C4—1716no—
568C22⋊C4—344no—
569C22⋊C4—344no—
570C22⋊C4—682no—
571C22⋊C4—682no—
572C22⋊C4—682no—
573C42—42no—
574C42—178no—
575C42—344no—
576C42—344no—
577C42—681no—
578C42—681no—
579C42—684no—
580C42—684no—
581C42—684no—
582C42—684no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—22no—
590C23—22no—
591C23—22no—
592C23—22no—
593C23—22no—
594C23—22no—
595C23C22×C4×D1712yes—
596C23C22×C4×D1712yes—
597C23C22×C4×D1712yes—
598C23C24×D1711yescharacteristic
599Q8—44no—
600Q8—3416no—
601Q8—688no—
602D4—44no—
603D4—44no—
604D4—44no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—84no—
621C2×C4—178no—
622C2×C4—178no—
623C2×C4—174no—
624C2×C4—342no—
625C2×C4—344no—
626C2×C4—344no—
627C2×C4—344no—
628C2×C4—344no—
629C2×C4—342no—
630C2×C4—342no—
631C2×C4—342no—
632C2×C4—342no—
633C2×C4—342no—
634C2×C4—342no—
635C2×C4—342no—
636C2×C4—342no—
637C2×C4—342no—
638C2×C4—342no—
639C2×C4—688no—
640C2×C4—688no—
641C2×C4C22×C4×D1712yes—
642C2×C4C22×C4×D1712yes—
643C2×C4C22×C4×D1712yes—
644C2×C4C22×C4×D1712yes—
645C2×C4#1058769112yes—
646C2×C4#1058769112yes—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22C23×C4×D1711yescharacteristic
663C22C2×#1058769111yescharacteristic
664C22C2×#1058769111yescharacteristic
665C222−1+4×D1712yes—
666C22#1176364412yes—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—44no—
671C4—44no—
672C4—348no—
673C4—348no—
674C4—344no—
675C4—682no—
676C4—682no—
677C4—682no—
678C4#1176328214yes—
679C2—42no—
680C2—42no—
681C2—42no—
682C2C2×#1176328211yescharacteristic
683C2#1454199512yes—
684C2#1454306512yes—
685C2#1454319311yescharacteristic
686C2#1454320811yescharacteristic
subgroupG/H
center, FrattiniC23C24×D17
commutatorC22×C17C24×C4
FittingC17×#2311C2
socle, cosocleC23×C17C25
prime radicalC22×#41×C17C22
hypercenter#2311D17
hypocenterC17#34413
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. #14531963C2semidirect
3. #14531927C2semidirect
4. #14531917C2semidirect
5. #14531872C2semidirect
6. #14531811C2semidirect
7. #14531789C2semidirect
8. #14531679C2semidirect
9. #14531616C2semidirect
10. #14531593C2semidirect
11. #14531571C2semidirect
12. #14531462C2semidirect
13. #14531312C2semidirect
17. #11763380C4semidirect
91. C22×Dic34C2×C4semidirect
111. #10587630C2×C4semidirect
113. #10587621C2×C4semidirect
114. #10587621C2×C4semidirect
116. #10587612C2×C4semidirect
120. #10587610C2×C4semidirect
121. #10587610C2×C4semidirect
497. C17#34413semidirect

Maximal subgroups

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

subgroupindex
C2×#117633802
#145319632
#145319272
#145319172
#145318722
#145318112
#145317892
#145316792
#145316162
#145315932
#145315712
#145314622
#145313122
#145312752
C17×#23112
#3441317