Finite Group Databasebeta

← #17199589

Subgroups — #17199589

Order 4352.

2244 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic34C211yesmaximal, characteristic
2#14531944C214yesmaximal, complemented
3#14531870C212yesmaximal, complemented
4#14531840C211yesmaximal, characteristic, complemented
5#14531809C212yesmaximal, complemented
6#14531756C212yesmaximal
7#14531739C211yesmaximal, characteristic, complemented
8#14531733C211yesmaximal, characteristic
9C22⋊C4×Dic34C212yesmaximal, complemented
10#14531541C214yesmaximal, complemented
11#14531531C212yesmaximal, complemented
12#14531466C212yesmaximal, complemented
13#14531421C212yesmaximal
14#14531368C212yesmaximal, complemented
15C4×#10587555C211yesmaximal, characteristic, complemented
16#14531261C211yesmaximal, characteristic, complemented
17C17×#2293C211yesmaximal, characteristic
18C23×Dic34C2212yes—
19D4×Dic34C4116yescomplemented
20C2×#10587555C2211yescharacteristic
21C2×#10587574C2212yes—
22C2×#10587610C2212yes—
23C2×#10587564C2214yes—
24C2×C4×Dic34C2212yes—
25C2×C4×Dic34C2212yes—
26C2×C4×Dic34C2211yescharacteristic
27C2×D4×Dic17C2212yes—
28C2×#10587612C2212yes—
29#11762715C2212yes—
30#11762715C2212yes—
31#11762715C2211yescharacteristic
32#11762715C2212yes—
33#11762715C2211yescharacteristic
34#11762714C2214yes—
35#11762714C2214yes—
36#11762713C2214yes—
37#11762713C2214yes—
38#11762712C2212yes—
39#11762707C2212yes—
40#11762707C2212yes—
41#11762707C2211yescharacteristic
42#11762707C2211yescharacteristic
43#11762705C2212yes—
44#11762704C2214yes—
45#11762703C2214yes—
46#11762703C2214yes—
47#11762702C2214yes—
48#11762697C2214yes—
49#11762697C2214yes—
50#11762696C2214yes—
51#11762693C2212yes—
52#11762693C2212yes—
53#11762692C2214yes—
54#11762691C2214yes—
55#11762690C2214yes—
56#11762679C2211yescharacteristic
57#11762674C2212yes—
58#11762670C2214yes—
59#11762670C2214yes—
60#11762669C2212yes—
61#11762667C2214yes—
62C4⋊C4×Dic17C2212yes—
63C4⋊C4×Dic17C2212yes—
64C4⋊C4×Dic17C2212yes—
65C4×#63225C2212yes—
66C4×C4⋊Dic17C2211yescharacteristic
67C4×C4⋊Dic17C2212yes—
68C4×C4⋊Dic17C2212yes—
69C4×C4⋊Dic17C2212yes—
70#11762660C2214yes—
71C22⋊C4×Dic17C2214yes—
72#11762655C2212yes—
73C2×C4×D4×C17C2211yescharacteristic
74C2×C17×#117C2212yes—
75C17×#390C2211yescharacteristic
76C17×#390C2211yescharacteristic
77C17×#389C2211yescharacteristic
78C17×#389C2211yescharacteristic
79C17×#388C2212yes—
80C17×#388C2212yes—
81C4×C4⋊C4×C17C2211yescharacteristic
82C4×C22⋊C4×C17C2212yes—
83C43×C17C2211yescharacteristic
84C22×D4×C17C2311yescharacteristic
85C23×C4×C17C2312yes—
86C23×Dic17C2314yes—
87C22×Dic34—44no—
88C22×Dic34—44no—
89C22×Dic34C2×C4116yes—
90C22×Dic34C2311yescharacteristic
91C22×Dic34C2312yes—
92C22×Dic34C2314yes—
93C4×D4×C17C2×C418yes—
94C17×#117—44no—
95C2×C4⋊C4×C17C2311yescharacteristic
96C2×C4⋊C4×C17C2311yescharacteristic
97C2×C4⋊C4×C17C2312yes—
98C2×C4⋊C4×C17C2312yes—
99C2×C4⋊C4×C17C2311yescharacteristic
100C2×C22⋊C4×C17C2312yes—
101C2×C22⋊C4×C17C2312yes—
102C2×C22⋊C4×C17C2312yes—
103C2×C22⋊C4×C17C2312yes—
104C2×C22⋊C4×C17C2312yes—
105C2×C42×C17C2311yescharacteristic
106C2×C42×C17C2311yescharacteristic
107C2×C42×C17C2311yescharacteristic
108C2×C42×C17C2311yescharacteristic
109C2×C42×C17C2312yes—
110C2×C42×C17C2311yescharacteristic
111C2×C42×C17C2311yescharacteristic
112C2×C42×C17C2311yescharacteristic
113C2×C22⋊Dic17C2314yes—
114C2×C22⋊Dic17C2314yes—
115C2×C22⋊Dic17C2314yes—
116D4×Dic17C2×C4116yes—
117#10587612—44no—
118C2×C4⋊Dic17C2312yes—
119C2×C4⋊Dic17C2311yescharacteristic
120C2×C4⋊Dic17C2311yescharacteristic
121C2×C4⋊Dic17C2312yes—
122C2×C4⋊Dic17C2312yes—
123C2×C4⋊Dic17C2312yes—
124C2×C4⋊Dic17C2312yes—
125C2×C4⋊Dic17C2311yescharacteristic
126#10587610C2×C4116yes—
127C2×#63225C2312yes—
128C2×#63225C2314yes—
129C2×#63225C2314yes—
130C2×#63225C2312yes—
131C2×#63225C2314yes—
132C2×#63225C2312yes—
133C2×C4×Dic17C2312yes—
134C2×C4×Dic17C2314yes—
135C2×C4×Dic17C2314yes—
136C2×C4×Dic17C2314yes—
137C2×C4×Dic17C2314yes—
138C2×C4×Dic17C2312yes—
139#10587574C2×C4116yes—
140#10587564C2×C4132yes—
141#10587555C2×C418yes—
142C4×Dic34—44no—
143C4×Dic34—44no—
144C4×Dic34C2×C418yes—
145C17×#95C2312yes—
146C17×#95C2312yes—
147C17×#95C2312yes—
148#10587502C2312yes—
149#10587502C2314yes—
150#10587502C2314yes—
151#10587502C2314yes—
152#10587502C2312yes—
153#10587502C2314yes—
154#10587502C2312yes—
155#10587502C2314yes—
156#10587502C2312yes—
157#10587502C2312yes—
158#10587502C2312yes—
159#10587502C2312yes—
160#10587502C2314yes—
161#10587502C2314yes—
162#10587502C2314yes—
163C24×C17C2412yes—
164C2×D4×C17—22no—
165C2×D4×C17—22no—
166C2×D4×C17C22×C414yes—
167C22×C4×C17—24no—
168C22×C4×C17—24no—
169C22×C4×C17C22×C418yes—
170C22×C4×C17C2411yescharacteristic
171C22×C4×C17C2412yes—
172C22×C4×C17C2411yescharacteristic
173C22×C4×C17C2412yes—
174C22×C4×C17C2411yescharacteristic
175C22×C4×C17C2412yes—
176C22×C4×C17C2411yescharacteristic
177C22×C4×C17C2411yescharacteristic
178C22×C4×C17C2411yescharacteristic
179C22×C4×C17C2411yescharacteristic
180C22×Dic17—44no—
181C22×Dic17—44no—
182C22×Dic17C22×C4116yes—
183C22×Dic17C2412yes—
184C22×Dic17C2412yes—
185C22×Dic17C2414yes—
186C22×Dic17C2412yes—
187C22×Dic17C2414yes—
188C22×Dic17C2412yes—
189C2×Dic34—22no—
190C2×Dic34—22no—
191C2×Dic34—24no—
192C2×Dic34—24no—
193C2×Dic34—432no—
194C2×Dic34—44no—
195C2×Dic34—44no—
196C2×Dic34C22×C414yes—
197C2×Dic34C22×C418yes—
198C2×Dic34C22×C4116yes—
199C4⋊C4×C17—22no—
200C4⋊C4×C17—22no—
201C4⋊C4×C17—42no—
202C4⋊C4×C17—42no—
203C4⋊C4×C17C22×C414yes—
204C22⋊C4×C17—24no—
205C22⋊C4×C17—42no—
206C22⋊C4×C17—42no—
207C22⋊C4×C17—42no—
208C22⋊C4×C17C22×C418yes—
209C42×C17—22no—
210C42×C17—24no—
211C42×C17—22no—
212C42×C17—22no—
213C42×C17—22no—
214C42×C17—22no—
215C42×C17—41no—
216C42×C17C22×C414yes—
217C22⋊Dic17—44no—
218C22⋊Dic17—44no—
219C22⋊Dic17C22×C4116yes—
220C4⋊Dic17—22no—
221C4⋊Dic17—24no—
222C4⋊Dic17—22no—
223C4⋊Dic17—24no—
224C4⋊Dic17—42no—
225C4⋊Dic17C22×C414yes—
226C4⋊Dic17C22×C418yes—
227C4⋊Dic17C22×C418yes—
228#63225—24no—
229#63225—44no—
230#63225—44no—
231#63225—42no—
232#63225C22×C418yes—
233#63225C22×C4116yes—
234C4×Dic17—24no—
235C4×Dic17—44no—
236C4×Dic17—44no—
237C4×Dic17—44no—
238C4×Dic17—44no—
239C4×Dic17C22×C418yes—
240#34487—171nomaximal, Sylow
241C23×C17—22no—
242C23×C17—22no—
243C23×C17—22no—
244C23×C17—22no—
245C23×C17—22no—
246C23×C17—22no—
247C23×C17C23×C414yes—
248C23×C17C2511yescharacteristic
249D4×C17—28no—
250C2×C4×C17—21no—
251C2×C4×C17—22no—
252C2×C4×C17—22no—
253C2×C4×C17—21no—
254C2×C4×C17—21no—
255C2×C4×C17—21no—
256C2×C4×C17—216no—
257C2×C4×C17—21no—
258C2×C4×C17—21no—
259C2×C4×C17—22no—
260C2×C4×C17—22no—
261C2×C4×C17—21no—
262C2×C4×C17—21no—
263C2×C4×C17—21no—
264C2×C4×C17—21no—
265C2×C4×C17—21no—
266C2×C4×C17—22no—
267C2×C4×C17—22no—
268C2×C4×C17—22no—
269C2×C4×C17—21no—
270C2×C4×C17—21no—
271C2×C4×C17—22no—
272C2×C4×C17C23×C412yes—
273C2×C4×C17C23×C414yes—
274C2×C4×C17C23×C412yes—
275C2×C4×C17C23×C412yes—
276C2×C4×C17H(5,𝔽2)12yes—
277C2×C4×C17H(5,𝔽2)12yes—
278C2×C4×C172−1+412yes—
279C2×C4×C172−1+412yes—
280C2×C4×C172−1+412yes—
281C2×Dic17—22no—
282C2×Dic17—22no—
283C2×Dic17—24no—
284C2×Dic17—24no—
285C2×Dic17—22no—
286C2×Dic17—22no—
287C2×Dic17—22no—
288C2×Dic17—24no—
289C2×Dic17—22no—
290C2×Dic17—22no—
291C2×Dic17—22no—
292C2×Dic17—22no—
293C2×Dic17—24no—
294C2×Dic17—22no—
295C2×Dic17—24no—
296C2×Dic17—416no—
297C2×Dic17C23×C414yes—
298C2×Dic17C23×C418yes—
299C2×Dic17C23×C414yes—
300Dic34—28no—
301Dic34—216no—
302Dic34—416no—
303C2×D4×Q8—171no—
304#2378—174no—
305#2369—172no—
306#2365—171no—
307#2362—172no—
308#2357—172no—
309#2355—171no—
310#2354—171no—
311Q8×C22⋊C4—172no—
312#2332—174no—
313#2331—172no—
314#2323—172no—
315#2319—172no—
316#2310—172no—
317C4×C4⋊Q8—171no—
318#2295—171no—
319#2293D1711yescharacteristic
320C22×C17—24no—
321C22×C17—24no—
322C22×C17—24no—
323C22×C17—42no—
324C22×C17—42no—
325C22×C17—42no—
326C22×C17—42no—
327C22×C17C24×C411yescharacteristic
328C22×C17C2×H(5,𝔽2)11yescharacteristic
329C22×C17C2×H(5,𝔽2)11yescharacteristic
330C22×C17C2×2−1+411yescharacteristic
331C22×C17C2×2−1+411yescharacteristic
332C22×C17C2×2−1+411yescharacteristic
333C22×C17#58511yescharacteristic
334C4×C17—24no—
335C4×C17—22no—
336C4×C17—22no—
337C4×C17—41no—
338C4×C17—42no—
339C4×C17—41no—
340C4×C17—42no—
341C4×C17—41no—
342C4×C17#51814yes—
343C4×C17#51914yes—
344Dic17—24no—
345Dic17—28no—
346Dic17—24no—
347Dic17—42no—
348Dic17—42no—
349Dic17—44no—
350C23×Q8—172no—
351D4×Q8—1716no—
352C2×C4⋊Q8—172no—
353C2×C4⋊Q8—171no—
354C2×C22⋊Q8—172no—
355C2×C22⋊Q8—174no—
356C2×C4×Q8—171no—
357C2×C4×Q8—172no—
358C2×C4×Q8—172no—
359C2×C4×D4—172no—
360C2×C4×D4C2×D1711yescharacteristic
361C2×#117—172no—
362C2×#117C2×D1712yes—
363#391—174no—
364#391—174no—
365#391—172no—
366#391—171no—
367#391—172no—
368#391—172no—
369#391—171no—
370#391—174no—
371#391—174no—
372#390—172no—
373#390C2×D1711yescharacteristic
374#390C2×D1711yescharacteristic
375#389—172no—
376#389—174no—
377#389—172no—
378#389—172no—
379#389—171no—
380#389—171no—
381#389C2×D1711yescharacteristic
382#389C2×D1711yescharacteristic
383#388—174no—
384#388—174no—
385#388—174no—
386#388—174no—
387#388—174no—
388#388C2×D1712yes—
389#388C2×D1712yes—
390#387—174no—
391#387—172no—
392#387—172no—
393#387—174no—
394#387—174no—
395#387—174no—
396#384—171no—
397#382—172no—
398#381—174no—
399#381—174no—
400#380—172no—
401#380—174no—
402C4×C4⋊C4—172no—
403C4×C4⋊C4—172no—
404C4×C4⋊C4—172no—
405C4×C4⋊C4—172no—
406C4×C4⋊C4—172no—
407C4×C4⋊C4—172no—
408C4×C4⋊C4—171no—
409C4×C4⋊C4—172no—
410C4×C4⋊C4—174no—
411C4×C4⋊C4C2×D1711yescharacteristic
412C4×C22⋊C4—174no—
413C4×C22⋊C4C2×D1712yes—
414#376—172no—
415C43C2×D1711yescharacteristic
416C2×C17—44no—
417C2×C17C2×#51811yescharacteristic
418C2×C17C2×#51911yescharacteristic
419C2×C17C4×2−1+411yescharacteristic
420C2×C17#354711yescharacteristic
421C2×C17#355011yescharacteristic
422C2×C17#355211yescharacteristic
423C2×C17#355411yescharacteristic
424C22×Q8—1716no—
425C22×Q8—172no—
426C22×Q8—171no—
427C22×Q8—174no—
428C22×Q8—684no—
429C22×Q8—684no—
430C22×D4C22×D1711yescharacteristic
431C23×C4—174no—
432C23×C4C22×D1712yes—
433C4⋊Q8—1716no—
434C4⋊Q8—178no—
435C22⋊Q8—1716no—
436C22⋊Q8—1732no—
437C4×Q8—178no—
438C4×Q8—684no—
439C4×Q8—684no—
440C4×D4—1716no—
441C4×D4C4×D1718yes—
442#117—44no—
443#117—684no—
444C2×C4⋊C4—172no—
445C2×C4⋊C4—171no—
446C2×C4⋊C4—172no—
447C2×C4⋊C4—174no—
448C2×C4⋊C4—172no—
449C2×C4⋊C4—171no—
450C2×C4⋊C4—172no—
451C2×C4⋊C4—171no—
452C2×C4⋊C4—172no—
453C2×C4⋊C4—172no—
454C2×C4⋊C4—174no—
455C2×C4⋊C4—174no—
456C2×C4⋊C4—172no—
457C2×C4⋊C4—172no—
458C2×C4⋊C4C22×D1711yescharacteristic
459C2×C4⋊C4C22×D1711yescharacteristic
460C2×C4⋊C4C22×D1712yes—
461C2×C4⋊C4C22×D1712yes—
462C2×C4⋊C4C22×D1711yescharacteristic
463C2×C22⋊C4—174no—
464C2×C22⋊C4—174no—
465C2×C22⋊C4—174no—
466C2×C22⋊C4C22×D1712yes—
467C2×C22⋊C4C22×D1712yes—
468C2×C22⋊C4C22×D1712yes—
469C2×C22⋊C4C22×D1712yes—
470C2×C22⋊C4C22×D1712yes—
471C2×C42—172no—
472C2×C42—172no—
473C2×C42—174no—
474C2×C42—174no—
475C2×C42—174no—
476C2×C42—174no—
477C2×C42C22×D1711yescharacteristic
478C2×C42C22×D1712yes—
479C2×C42C22×D1711yescharacteristic
480C2×C42C22×D1711yescharacteristic
481C2×C42C22×D1711yescharacteristic
482C2×C42C22×D1711yescharacteristic
483C2×C42C22×D1711yescharacteristic
484C2×C42C22×D1711yescharacteristic
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—172no—
494#95—174no—
495#95—174no—
496#95—174no—
497#95—174no—
498#95—174no—
499#95—172no—
500#95C22×D1712yes—
501#95C22×D1712yes—
502#95C22×D1712yes—
503C17#3448711yescharacteristic, complemented, Sylow
504C24C23×D1712yes—
505C2×Q8—174no—
506C2×Q8—178no—
507C2×Q8—1716no—
508C2×Q8—344no—
509C2×Q8—342no—
510C2×Q8—344no—
511C2×Q8—342no—
512C2×Q8—684no—
513C2×Q8—684no—
514C2×Q8—6832no—
515C2×D4—22no—
516C2×D4—22no—
517C2×D4C2×C4×D1714yes—
518C22×C4—24no—
519C22×C4—24no—
520C22×C4—172no—
521C22×C4—172no—
522C22×C4—174no—
523C22×C4—1716no—
524C22×C4—172no—
525C22×C4—172no—
526C22×C4—174no—
527C22×C4—684no—
528C22×C4—684no—
529C22×C4C2×C4×D1718yes—
530C22×C4C23×D1711yescharacteristic
531C22×C4C23×D1712yes—
532C22×C4C23×D1711yescharacteristic
533C22×C4C23×D1712yes—
534C22×C4C23×D1711yescharacteristic
535C22×C4C23×D1712yes—
536C22×C4C23×D1711yescharacteristic
537C22×C4C23×D1711yescharacteristic
538C22×C4C23×D1711yescharacteristic
539C22×C4C23×D1711yescharacteristic
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—178no—
545C4⋊C4—178no—
546C4⋊C4—174no—
547C4⋊C4—1716no—
548C4⋊C4—178no—
549C4⋊C4—342no—
550C4⋊C4—342no—
551C4⋊C4—344no—
552C4⋊C4—344no—
553C4⋊C4—344no—
554C4⋊C4—684no—
555C4⋊C4—684no—
556C4⋊C4—682no—
557C4⋊C4—682no—
558C4⋊C4C2×C4×D1714yes—
559C22⋊C4—24no—
560C22⋊C4—42no—
561C22⋊C4—42no—
562C22⋊C4—42no—
563C22⋊C4—1716no—
564C22⋊C4—684no—
565C22⋊C4—684no—
566C22⋊C4C2×C4×D1718yes—
567C42—22no—
568C42—24no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—41no—
574C42—178no—
575C42—344no—
576C42—684no—
577C42—684no—
578C42—684no—
579C42—684no—
580C42C2×C4×D1714yes—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—22no—
585C23—22no—
586C23—22no—
587C23C22×C4×D1714yes—
588C23C24×D1711yescharacteristic
589Q8—348no—
590Q8—3416no—
591Q8—6816no—
592D4—28no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—216no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—174no—
616C2×C4—174no—
617C2×C4—178no—
618C2×C4—344no—
619C2×C4—342no—
620C2×C4—344no—
621C2×C4—342no—
622C2×C4—342no—
623C2×C4—342no—
624C2×C4—342no—
625C2×C4—342no—
626C2×C4—344no—
627C2×C4—342no—
628C2×C4—342no—
629C2×C4—342no—
630C2×C4—344no—
631C2×C4—344no—
632C2×C4—342no—
633C2×C4—6816no—
634C2×C4C22×C4×D1712yes—
635C2×C4C22×C4×D1714yes—
636C2×C4C22×C4×D1712yes—
637C2×C4C22×C4×D1712yes—
638C2×C4#1058769112yes—
639C2×C4#1058769612yes—
640C2×C4#1058769712yes—
641C2×C4#1058769712yes—
642C2×C4#1058769712yes—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22C23×C4×D1711yescharacteristic
651C22C2×#1058769111yescharacteristic
652C22C2×#1058769611yescharacteristic
653C22C2×#1058769711yescharacteristic
654C22C2×#1058769711yescharacteristic
655C22C2×#1058769711yescharacteristic
656C22#1176364511yescharacteristic
657C4—22no—
658C4—24no—
659C4—22no—
660C4—41no—
661C4—42no—
662C4—42no—
663C4—41no—
664C4—41no—
665C4—344no—
666C4—348no—
667C4—344no—
668C4—682no—
669C4—682no—
670C4—684no—
671C4#1176328114yes—
672C4#1176328614yes—
673C2—44no—
674C2C2×#1176328111yescharacteristic
675C2C2×#1176328611yescharacteristic
676C2C4×#1058769711yescharacteristic
677C2#1454317311yescharacteristic
678C2#1454319811yescharacteristic
679C2#1454322911yescharacteristic
680C2#1454325711yescharacteristic
subgroupG/H
center, FrattiniC23C24×D17
commutatorC22×C17C24×C4
FittingC17×#2293C2
socle, cosocleC23×C17C25
prime radicalC22×D4×C17C23
hypercenter#2293D17
hypocenterC17#34487
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ #2293

Complements

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

NH ≅ G/Ntype
2. #14531944C2semidirect
3. #14531870C2semidirect
4. #14531840C2semidirect
5. #14531809C2semidirect
7. #14531739C2semidirect
9. C22⋊C4×Dic34C2semidirect
10. #14531541C2semidirect
11. #14531531C2semidirect
12. #14531466C2semidirect
14. #14531368C2semidirect
15. C4×#10587555C2semidirect
16. #14531261C2semidirect
19. D4×Dic34C4semidirect
503. C17#34487semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic342
#145319442
#145318702
#145318402
#145318092
#145317562
#145317392
#145317332
C22⋊C4×Dic342
#145315412
#145315312
#145314662
#145314212
#145313682
C4×#105875552
#145312612
C17×#22932
#3448717