Finite Group Databasebeta

← #17199516

Subgroups — #17199516

Order 4352.

2132 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763516C211yesmaximal, characteristic
2#14531944C214yesmaximal, complemented
3#14531876C212yesmaximal, complemented
4#14531873C211yesmaximal, characteristic, complemented
5#14531811C212yesmaximal, complemented
6#14531732C212yesmaximal
7#14531696C211yesmaximal, characteristic, complemented
8#14531655C212yesmaximal, complemented
9#14531521C212yesmaximal, complemented
10#14531462C212yesmaximal, complemented
11#14531459C214yesmaximal, complemented
12#14531421C212yesmaximal
13#14531400C212yesmaximal, complemented
14#14531351C211yesmaximal, characteristic, complemented
15#14531258C211yesmaximal, characteristic, complemented
16#14531249C211yesmaximal, characteristic
17C17×#2354C211yesmaximal, characteristic
18#11763516C4116yescomplemented
19C2×#10587555C2211yescharacteristic
20C2×#10587576C2212yes—
21C2×#10587610C2212yes—
22C2×#10587564C2214yes—
23C2×C4×Dic34C2211yescharacteristic
24C2×#10587573C2212yes—
25C2×#10587573C2212yes—
26C2×D4×Dic17C2212yes—
27C2×#10587612C2212yes—
28C2×#10587563C2214yes—
29C22×C4⋊Dic17C2212yes—
30#11762714C2212yes—
31#11762714C2211yescharacteristic
32#11762714C2211yescharacteristic
33#11762713C2214yes—
34#11762712C2211yescharacteristic
35#11762712C2211yescharacteristic
36#11762708C2211yescharacteristic
37#11762708C2211yescharacteristic
38#11762707C2212yes—
39#11762707C2214yes—
40#11762707C2214yes—
41#11762706C2214yes—
42#11762705C2212yes—
43#11762705C2212yes—
44#11762705C2212yes—
45#11762705C2211yescharacteristic
46#11762705C2212yes—
47#11762705C2211yescharacteristic
48#11762704C2214yes—
49#11762703C2214yes—
50#11762703C2214yes—
51#11762702C2212yes—
52#11762702C2212yes—
53#11762697C2214yes—
54#11762697C2214yes—
55#11762696C2214yes—
56#11762693C2214yes—
57#11762692C2214yes—
58#11762692C2214yes—
59#11762691C2212yes—
60#11762691C2212yes—
61#11762678C2212yes—
62#11762670C2214yes—
63#11762667C2214yes—
64#11762667C2214yes—
65C4⋊C4×Dic17C2212yes—
66C4⋊C4×Dic17C2212yes—
67C4⋊C4×Dic17C2212yes—
68C4⋊C4×Dic17C2212yes—
69C4×#63225C2214yes—
70C4×#63225C2212yes—
71C4×C4⋊Dic17C2211yescharacteristic
72C4×C4⋊Dic17C2212yes—
73C4×C22⋊Dic17C2212yes—
74#11762655C2212yes—
75C42×Dic17C2211yescharacteristic
76C2×C4×D4×C17C2211yescharacteristic
77C2×C17×#117C2212yes—
78C17×#390C2212yes—
79C17×#388C2214yes—
80C17×#384C2211yescharacteristic
81C17×#380C2212yes—
82C4×C4⋊C4×C17C2211yescharacteristic
83C4×C4⋊C4×C17C2212yes—
84C22×D4×C17C2311yescharacteristic
85C23×C4×C17C2312yes—
86C23×Dic17C2314yes—
87C22×Dic34C2311yescharacteristic
88C22×Dic34C2312yes—
89C4×D4×C17C2×C418yes—
90C17×#117—44no—
91C2×C4⋊C4×C17C2311yescharacteristic
92C2×C4⋊C4×C17C2312yes—
93C2×C4⋊C4×C17C2311yescharacteristic
94C2×C4⋊C4×C17C2312yes—
95C2×C4⋊C4×C17C2312yes—
96C2×C4⋊C4×C17C2311yescharacteristic
97C2×C22⋊C4×C17C2314yes—
98C2×C22⋊C4×C17C2314yes—
99C2×C22⋊C4×C17C2312yes—
100C2×C42×C17C2312yes—
101C2×C42×C17C2314yes—
102C2×C42×C17C2311yescharacteristic
103C2×C22⋊Dic17C2312yes—
104C2×C22⋊Dic17C2312yes—
105C2×C22⋊Dic17C2312yes—
106C2×C22⋊Dic17C2312yes—
107C2×C22⋊Dic17C2314yes—
108D4×Dic17C2×C4116yes—
109#10587612—44no—
110C2×C4⋊Dic17C2×C4116yes—
111C2×C4⋊Dic17C2311yescharacteristic
112C2×C4⋊Dic17C2311yescharacteristic
113C2×C4⋊Dic17C2312yes—
114C2×C4⋊Dic17C2312yes—
115C2×C4⋊Dic17C2314yes—
116C2×C4⋊Dic17C2311yescharacteristic
117C2×C4⋊Dic17C2312yes—
118C2×C4⋊Dic17C2312yes—
119#10587610C2×C4116yes—
120C2×#63225C2312yes—
121C2×#63225C2312yes—
122C2×#63225C2314yes—
123C2×#63225C2314yes—
124C2×#63225C2314yes—
125C2×#63225C2314yes—
126C2×#63225C2312yes—
127C2×#63225C2314yes—
128C2×C4×Dic17C2311yescharacteristic
129C2×C4×Dic17C2314yes—
130C2×C4×Dic17C2311yescharacteristic
131C2×C4×Dic17C2311yescharacteristic
132C2×C4×Dic17C2312yes—
133C2×C4×Dic17C2311yescharacteristic
134C2×C4×Dic17C2311yescharacteristic
135C2×C4×Dic17C2311yescharacteristic
136C2×C4×Dic17C2314yes—
137C2×C4×Dic17C2314yes—
138C2×C4×Dic17C2312yes—
139#10587576C2×C4116yes—
140#10587573—44no—
141#10587573—44no—
142#10587564C2×C4132yes—
143#10587563—48no—
144#10587555C2×C418yes—
145C4×Dic34C2×C418yes—
146C17×#95C2312yes—
147C17×#95C2314yes—
148#10587502C2312yes—
149#10587502C2312yes—
150#10587502C2312yes—
151#10587502C2312yes—
152#10587502C2312yes—
153#10587502C2312yes—
154#10587502C2312yes—
155#10587502C2312yes—
156#10587502C2314yes—
157#10587502C2314yes—
158#10587502C2314yes—
159#10587502C2314yes—
160#10587502C2314yes—
161#10587502C2312yes—
162C24×C17C2412yes—
163C2×D4×C17—22no—
164C2×D4×C17—22no—
165C2×D4×C17C22×C414yes—
166C22×C4×C17—24no—
167C22×C4×C17—24no—
168C22×C4×C17C22×C418yes—
169C22×C4×C17C2412yes—
170C22×C4×C17C2412yes—
171C22×C4×C17C2414yes—
172C22×C4×C17C2411yescharacteristic
173C22×C4×C17C2412yes—
174C22×C4×C17C2411yescharacteristic
175C22×C4×C17C2411yescharacteristic
176C22×Dic17—44no—
177C22×Dic17—44no—
178C22×Dic17C22×C4116yes—
179C22×Dic17C2411yescharacteristic
180C22×Dic17C2411yescharacteristic
181C22×Dic17C2412yes—
182C22×Dic17C2412yes—
183C22×Dic17C2411yescharacteristic
184C22×Dic17C2414yes—
185C22×Dic17C2412yes—
186C22×Dic17C2412yes—
187C22×Dic17C2411yescharacteristic
188C2×Dic34—22no—
189C2×Dic34—22no—
190C2×Dic34—42no—
191C2×Dic34—42no—
192C2×Dic34C22×C414yes—
193C2×Dic34C22×C418yes—
194C4⋊C4×C17—24no—
195C4⋊C4×C17—22no—
196C4⋊C4×C17—24no—
197C4⋊C4×C17—22no—
198C4⋊C4×C17—42no—
199C4⋊C4×C17C22×C414yes—
200C22⋊C4×C17—44no—
201C22⋊C4×C17—44no—
202C22⋊C4×C17C22×C418yes—
203C42×C17—24no—
204C42×C17—44no—
205C42×C17C22×C414yes—
206C22⋊Dic17—24no—
207C22⋊Dic17—42no—
208C22⋊Dic17—42no—
209C22⋊Dic17—42no—
210C22⋊Dic17C22×C4116yes—
211C4⋊Dic17—22no—
212C4⋊Dic17—22no—
213C4⋊Dic17—42no—
214C4⋊Dic17—42no—
215C4⋊Dic17C22×C414yes—
216C4⋊Dic17C22×C418yes—
217C4⋊Dic17C22×C418yes—
218C4⋊Dic17C22×C4116yes—
219#63225—24no—
220#63225—44no—
221#63225—44no—
222#63225—44no—
223#63225—44no—
224#63225—42no—
225#63225C22×C418yes—
226#63225C22×C4116yes—
227C4×Dic17—24no—
228C4×Dic17—22no—
229C4×Dic17—22no—
230C4×Dic17—22no—
231C4×Dic17—22no—
232C4×Dic17—28no—
233C4×Dic17—22no—
234C4×Dic17—44no—
235C4×Dic17—44no—
236C4×Dic17—41no—
237C4×Dic17C22×C418yes—
238#34484—171nomaximal, Sylow
239C23×C17—22no—
240C23×C17—22no—
241C23×C17—22no—
242C23×C17—22no—
243C23×C17—22no—
244C23×C17—22no—
245C23×C17C23×C414yes—
246C23×C17C2511yescharacteristic
247D4×C17—28no—
248C2×C4×C17—22no—
249C2×C4×C17—21no—
250C2×C4×C17—22no—
251C2×C4×C17—21no—
252C2×C4×C17—216no—
253C2×C4×C17—22no—
254C2×C4×C17—24no—
255C2×C4×C17—22no—
256C2×C4×C17—24no—
257C2×C4×C17—22no—
258C2×C4×C17—22no—
259C2×C4×C17—22no—
260C2×C4×C17—24no—
261C2×C4×C17—22no—
262C2×C4×C17C23×C412yes—
263C2×C4×C17C23×C414yes—
264C2×C4×C17C23×C412yes—
265C2×C4×C17C23×C412yes—
266C2×C4×C17H(5,𝔽2)12yes—
267C2×C4×C172−1+412yes—
268C2×C4×C172−1+412yes—
269C2×C4×C172−1+412yes—
270C2×Dic17—22no—
271C2×Dic17—24no—
272C2×Dic17—22no—
273C2×Dic17—22no—
274C2×Dic17—21no—
275C2×Dic17—21no—
276C2×Dic17—24no—
277C2×Dic17—22no—
278C2×Dic17—22no—
279C2×Dic17—21no—
280C2×Dic17—21no—
281C2×Dic17—22no—
282C2×Dic17—22no—
283C2×Dic17—21no—
284C2×Dic17—21no—
285C2×Dic17—21no—
286C2×Dic17—21no—
287C2×Dic17—21no—
288C2×Dic17—22no—
289C2×Dic17—22no—
290C2×Dic17—21no—
291C2×Dic17—22no—
292C2×Dic17—21no—
293C2×Dic17—416no—
294C2×Dic17C23×C414yes—
295C2×Dic17C23×C418yes—
296C2×Dic17C23×C414yes—
297C2×Dic17H(5,𝔽2)12yes—
298Dic34—28no—
299Dic34—48no—
300C2×#554—171no—
301#2378—174no—
302#2370—172no—
303#2369—171no—
304#2362—172no—
305#2354—172no—
306#2354D1711yescharacteristic
307Q8×C4⋊C4—171no—
308#2344—172no—
309#2330—172no—
310#2323—174no—
311#2323—172no—
312#2319—172no—
313#2316—172no—
314C4×C4⋊Q8—171no—
315#2295—171no—
316#2293—171no—
317C22×C17—24no—
318C22×C17—24no—
319C22×C17—24no—
320C22×C17—42no—
321C22×C17—42no—
322C22×C17—42no—
323C22×C17—42no—
324C22×C17C24×C411yescharacteristic
325C22×C17C2×H(5,𝔽2)11yescharacteristic
326C22×C17C2×H(5,𝔽2)11yescharacteristic
327C22×C17C2×2−1+411yescharacteristic
328C22×C17C2×2−1+411yescharacteristic
329C22×C17C2×2−1+411yescharacteristic
330C22×C17#58511yescharacteristic
331C4×C17—24no—
332C4×C17—22no—
333C4×C17—42no—
334C4×C17—42no—
335C4×C17—44no—
336C4×C17#51914yes—
337C4×C17#52014yes—
338Dic17—24no—
339Dic17—28no—
340Dic17—24no—
341Dic17—22no—
342Dic17—41no—
343Dic17—42no—
344Dic17—42no—
345Dic17—41no—
346Dic17—41no—
347#554—1716no—
348C2×C4⋊Q8—171no—
349C2×#125—172no—
350C2×C22⋊Q8—172no—
351C2×C22⋊Q8—174no—
352C2×C4×Q8—172no—
353C2×C4×Q8—172no—
354C2×C4×Q8—171no—
355C2×C4×D4—172no—
356C2×C4×D4C2×D1711yescharacteristic
357C2×#117—172no—
358C2×#117—174no—
359C2×#117C2×D1712yes—
360C22×C4⋊C4—172no—
361#391—172no—
362#391—171no—
363#391—171no—
364#391—174no—
365#390—171no—
366#390—171no—
367#390C2×D1712yes—
368#389—172no—
369#389—172no—
370#389—171no—
371#389—172no—
372#389—172no—
373#389—174no—
374#389—171no—
375#389—171no—
376#389—171no—
377#389—174no—
378#389—174no—
379#389—172no—
380#389—174no—
381#388—174no—
382#388—174no—
383#388—172no—
384#388—172no—
385#388—174no—
386#388—174no—
387#388C2×D1714yes—
388#387—174no—
389#387—172no—
390#387—172no—
391#387—174no—
392#387—174no—
393#387—174no—
394#384—172no—
395#384C2×D1711yescharacteristic
396#381—174no—
397#380—174no—
398#380—174no—
399#380C2×D1712yes—
400C4×C4⋊C4—172no—
401C4×C4⋊C4—172no—
402C4×C4⋊C4—172no—
403C4×C4⋊C4—172no—
404C4×C4⋊C4—174no—
405C4×C4⋊C4—172no—
406C4×C4⋊C4—171no—
407C4×C4⋊C4—172no—
408C4×C4⋊C4C2×D1712yes—
409C4×C4⋊C4C2×D1711yescharacteristic
410C4×C22⋊C4—172no—
411#376—172no—
412C43—171no—
413C2×C17—44no—
414C2×C17C2×#51911yescharacteristic
415C2×C17C2×#52011yescharacteristic
416C2×C17C4×H(5,𝔽2)11yescharacteristic
417C2×C17#354411yescharacteristic
418C2×C17#354711yescharacteristic
419C2×C17#354811yescharacteristic
420C2×C17#355511yescharacteristic
421C22×Q8—171no—
422C22×Q8—172no—
423C22×D4C22×D1711yescharacteristic
424C23×C4—174no—
425C23×C4C22×D1712yes—
426C4⋊Q8—178no—
427#125—1716no—
428C22⋊Q8—1716no—
429C22⋊Q8—1732no—
430C4×Q8—178no—
431C4×Q8—684no—
432C4×Q8—684no—
433C4×D4—1716no—
434C4×D4C4×D1718yes—
435#117—44no—
436#117—684no—
437#117—688no—
438C2×C4⋊C4—1716no—
439C2×C4⋊C4—171no—
440C2×C4⋊C4—174no—
441C2×C4⋊C4—172no—
442C2×C4⋊C4—172no—
443C2×C4⋊C4—172no—
444C2×C4⋊C4—174no—
445C2×C4⋊C4—172no—
446C2×C4⋊C4—171no—
447C2×C4⋊C4—171no—
448C2×C4⋊C4—174no—
449C2×C4⋊C4—172no—
450C2×C4⋊C4—172no—
451C2×C4⋊C4—174no—
452C2×C4⋊C4—174no—
453C2×C4⋊C4—172no—
454C2×C4⋊C4—174no—
455C2×C4⋊C4C22×D1711yescharacteristic
456C2×C4⋊C4C22×D1712yes—
457C2×C4⋊C4C22×D1711yescharacteristic
458C2×C4⋊C4C22×D1712yes—
459C2×C4⋊C4C22×D1712yes—
460C2×C4⋊C4C22×D1711yescharacteristic
461C2×C22⋊C4—174no—
462C2×C22⋊C4—172no—
463C2×C22⋊C4—172no—
464C2×C22⋊C4—172no—
465C2×C22⋊C4—172no—
466C2×C22⋊C4C22×D1714yes—
467C2×C22⋊C4C22×D1714yes—
468C2×C22⋊C4C22×D1712yes—
469C2×C42—172no—
470C2×C42—171no—
471C2×C42—172no—
472C2×C42—174no—
473C2×C42—171no—
474C2×C42—171no—
475C2×C42—171no—
476C2×C42—171no—
477C2×C42—174no—
478C2×C42—174no—
479C2×C42—171no—
480C2×C42C22×D1712yes—
481C2×C42C22×D1714yes—
482C2×C42C22×D1711yescharacteristic
483#95—172no—
484#95—172no—
485#95—172no—
486#95—172no—
487#95—172no—
488#95—172no—
489#95—172no—
490#95—174no—
491#95—174no—
492#95—174no—
493#95—174no—
494#95—174no—
495#95—172no—
496#95—172no—
497#95C22×D1714yes—
498#95C22×D1712yes—
499C17#3448411yescharacteristic, complemented, Sylow
500C24C23×D1712yes—
501C2×Q8—174no—
502C2×Q8—178no—
503C2×Q8—342no—
504C2×Q8—342no—
505C2×Q8—682no—
506C2×Q8—682no—
507C2×D4—22no—
508C2×D4—22no—
509C2×D4C2×C4×D1714yes—
510C22×C4—24no—
511C22×C4—24no—
512C22×C4—171no—
513C22×C4—172no—
514C22×C4—174no—
515C22×C4—172no—
516C22×C4—1716no—
517C22×C4—171no—
518C22×C4—172no—
519C22×C4—171no—
520C22×C4—171no—
521C22×C4—172no—
522C22×C4—684no—
523C22×C4—684no—
524C22×C4C2×C4×D1718yes—
525C22×C4C23×D1712yes—
526C22×C4C23×D1712yes—
527C22×C4C23×D1714yes—
528C22×C4C23×D1711yescharacteristic
529C22×C4C23×D1712yes—
530C22×C4C23×D1711yescharacteristic
531C22×C4C23×D1711yescharacteristic
532C4⋊C4—22no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—22no—
536C4⋊C4—42no—
537C4⋊C4—174no—
538C4⋊C4—1716no—
539C4⋊C4—178no—
540C4⋊C4—178no—
541C4⋊C4—178no—
542C4⋊C4—1716no—
543C4⋊C4—344no—
544C4⋊C4—342no—
545C4⋊C4—342no—
546C4⋊C4—684no—
547C4⋊C4—682no—
548C4⋊C4—682no—
549C4⋊C4—684no—
550C4⋊C4—684no—
551C4⋊C4—682no—
552C4⋊C4—684no—
553C4⋊C4C2×C4×D1714yes—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—1716no—
557C22⋊C4—344no—
558C22⋊C4—682no—
559C22⋊C4—682no—
560C22⋊C4—682no—
561C22⋊C4C2×C4×D1718yes—
562C42—24no—
563C42—44no—
564C42—178no—
565C42—348no—
566C42—342no—
567C42—342no—
568C42—344no—
569C42—342no—
570C42—342no—
571C42—342no—
572C42—684no—
573C42—684no—
574C42—681no—
575C42C2×C4×D1714yes—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23C22×C4×D1714yes—
583C23C24×D1711yescharacteristic
584Q8—348no—
585Q8—688no—
586D4—28no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—216no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—174no—
602C2×C4—174no—
603C2×C4—178no—
604C2×C4—172no—
605C2×C4—344no—
606C2×C4—342no—
607C2×C4—342no—
608C2×C4—341no—
609C2×C4—341no—
610C2×C4—342no—
611C2×C4—342no—
612C2×C4—344no—
613C2×C4—341no—
614C2×C4—342no—
615C2×C4—342no—
616C2×C4—341no—
617C2×C4—341no—
618C2×C4—342no—
619C2×C4—341no—
620C2×C4—341no—
621C2×C4—341no—
622C2×C4—341no—
623C2×C4—342no—
624C2×C4—342no—
625C2×C4—341no—
626C2×C4—341no—
627C2×C4—342no—
628C2×C4—6816no—
629C2×C4C22×C4×D1712yes—
630C2×C4C22×C4×D1714yes—
631C2×C4C22×C4×D1712yes—
632C2×C4C22×C4×D1712yes—
633C2×C4#1058769112yes—
634C2×C4#1058769712yes—
635C2×C4#1058769712yes—
636C2×C4#1058769712yes—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22C23×C4×D1711yescharacteristic
645C22H(5,𝔽2)×D1711yescharacteristic
646C22C2×#1058769111yescharacteristic
647C22C2×#1058769711yescharacteristic
648C22C2×#1058769711yescharacteristic
649C22C2×#1058769711yescharacteristic
650C22#1176364611yescharacteristic
651C4—24no—
652C4—22no—
653C4—42no—
654C4—42no—
655C4—44no—
656C4—342no—
657C4—344no—
658C4—344no—
659C4—348no—
660C4—682no—
661C4—681no—
662C4—681no—
663C4—682no—
664C4—681no—
665C4#1176328614yes—
666C4#1176328814yes—
667C2—44no—
668C2C2×#1176328611yescharacteristic
669C2C2×#1176328811yescharacteristic
670C2#1454199011yescharacteristic
671C2#1454311911yescharacteristic
672C2#1454316111yescharacteristic
673C2#1454317711yescharacteristic
674C2#1454326011yescharacteristic
subgroupG/H
center, FrattiniC23C24×D17
commutatorC22×C17C24×C4
FittingC17×#2354C2
socle, cosocleC23×C17C25
prime radicalC22×D4×C17C23
hypercenter#2354D17
hypocenterC17#34484
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ #2354

Complements

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

NH ≅ G/Ntype
2. #14531944C2semidirect
3. #14531876C2semidirect
4. #14531873C2semidirect
5. #14531811C2semidirect
7. #14531696C2semidirect
8. #14531655C2semidirect
9. #14531521C2semidirect
10. #14531462C2semidirect
11. #14531459C2semidirect
13. #14531400C2semidirect
14. #14531351C2semidirect
15. #14531258C2semidirect
18. #11763516C4semidirect
499. C17#34484semidirect

Maximal subgroups

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

subgroupindex
C2×#117635162
#145319442
#145318762
#145318732
#145318112
#145317322
#145316962
#145316552
#145315212
#145314622
#145314592
#145314212
#145314002
#145313512
#145312582
#145312492
C17×#23542
#3448417