Finite Group Databasebeta

← #17199528

Subgroups — #17199528

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#14531951C214yesmaximal, complemented
3#14531885C212yesmaximal, complemented
4#14531873C211yesmaximal, characteristic, complemented
5#14531809C212yesmaximal, complemented
6#14531733C211yesmaximal, characteristic
7#14531732C212yesmaximal
8C4⋊C4×Dic34C211yesmaximal, characteristic, complemented
9#14531651C212yesmaximal, complemented
10#14531524C212yesmaximal, complemented
11#14531466C212yesmaximal, complemented
12#14531463C214yesmaximal, complemented
13#14531421C212yesmaximal
14#14531400C212yesmaximal, complemented
15C4×#10587555C211yesmaximal, characteristic, complemented
16#14531261C211yesmaximal, characteristic, complemented
17C17×#2293C211yesmaximal, characteristic
18#11763516C4116yescomplemented
19C2×#10587555C2211yescharacteristic
20C2×#10587576C2212yes—
21C2×#10587610C2212yes—
22C2×#10587564C2214yes—
23C2×C4×Dic34C2212yes—
24C2×C4×Dic34C2212yes—
25C2×C4×Dic34C2211yescharacteristic
26C2×D4×Dic17C2212yes—
27C2×#10587612C2212yes—
28C2×#10587563C2214yes—
29C22×C4⋊Dic17C2212yes—
30#11762715C2212yes—
31#11762715C2211yescharacteristic
32#11762715C2211yescharacteristic
33#11762713C2214yes—
34#11762712C2212yes—
35#11762707C2212yes—
36#11762707C2212yes—
37#11762707C2212yes—
38#11762707C2211yescharacteristic
39#11762707C2212yes—
40#11762707C2211yescharacteristic
41#11762706C2214yes—
42#11762705C2212yes—
43#11762705C2214yes—
44#11762705C2214yes—
45#11762704C2214yes—
46#11762703C2214yes—
47#11762703C2214yes—
48#11762702C2214yes—
49#11762697C2214yes—
50#11762697C2214yes—
51#11762694C2214yes—
52#11762694C2214yes—
53#11762693C2212yes—
54#11762693C2212yes—
55#11762691C2214yes—
56#11762690C2214yes—
57#11762679C2211yescharacteristic
58#11762678C2212yes—
59#11762670C2214yes—
60#11762669C2212yes—
61#11762667C2214yes—
62#11762667C2214yes—
63C4⋊C4×Dic17C2212yes—
64C4⋊C4×Dic17C2212yes—
65C4⋊C4×Dic17C2212yes—
66C4×#63225C2212yes—
67C4×C4⋊Dic17C2212yes—
68C4×C4⋊Dic17C2212yes—
69C4×C4⋊Dic17C2212yes—
70C4×C4⋊Dic17C2211yescharacteristic
71#11762660C2214yes—
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×Dic34C2311yescharacteristic
88C22×Dic34C2312yes—
89C4×D4×C17C2×C418yes—
90C17×#117—44no—
91C2×C4⋊C4×C17C2311yescharacteristic
92C2×C4⋊C4×C17C2311yescharacteristic
93C2×C4⋊C4×C17C2312yes—
94C2×C4⋊C4×C17C2312yes—
95C2×C4⋊C4×C17C2311yescharacteristic
96C2×C22⋊C4×C17C2312yes—
97C2×C22⋊C4×C17C2312yes—
98C2×C22⋊C4×C17C2312yes—
99C2×C22⋊C4×C17C2312yes—
100C2×C22⋊C4×C17C2312yes—
101C2×C42×C17C2311yescharacteristic
102C2×C42×C17C2311yescharacteristic
103C2×C42×C17C2312yes—
104C2×C42×C17C2311yescharacteristic
105C2×C42×C17C2311yescharacteristic
106C2×C42×C17C2311yescharacteristic
107C2×C42×C17C2311yescharacteristic
108C2×C42×C17C2311yescharacteristic
109C2×C22⋊Dic17C2314yes—
110C2×C22⋊Dic17C2314yes—
111C2×C22⋊Dic17C2314yes—
112D4×Dic17C2×C4116yes—
113#10587612—44no—
114C2×C4⋊Dic17C2×C4116yes—
115C2×C4⋊Dic17C2311yescharacteristic
116C2×C4⋊Dic17C2312yes—
117C2×C4⋊Dic17C2312yes—
118C2×C4⋊Dic17C2311yescharacteristic
119C2×C4⋊Dic17C2312yes—
120C2×C4⋊Dic17C2314yes—
121C2×C4⋊Dic17C2312yes—
122C2×C4⋊Dic17C2311yescharacteristic
123C2×C4⋊Dic17C2312yes—
124#10587610C2×C4116yes—
125C2×#63225C2312yes—
126C2×#63225C2314yes—
127C2×#63225C2312yes—
128C2×#63225C2314yes—
129C2×#63225C2314yes—
130C2×#63225C2314yes—
131C2×#63225C2314yes—
132C2×#63225C2312yes—
133C2×C4×Dic17C2314yes—
134C2×C4×Dic17C2312yes—
135C2×C4×Dic17C2314yes—
136C2×C4×Dic17C2314yes—
137C2×C4×Dic17C2314yes—
138C2×C4×Dic17C2312yes—
139#10587576C2×C4116yes—
140#10587564C2×C4132yes—
141#10587563—48no—
142#10587555C2×C418yes—
143C4×Dic34—44no—
144C4×Dic34—44no—
145C4×Dic34C2×C418yes—
146C17×#95C2312yes—
147C17×#95C2312yes—
148C17×#95C2312yes—
149#10587502C2312yes—
150#10587502C2314yes—
151#10587502C2314yes—
152#10587502C2314yes—
153#10587502C2312yes—
154#10587502C2314yes—
155#10587502C2314yes—
156#10587502C2312yes—
157#10587502C2312yes—
158#10587502C2312yes—
159#10587502C2312yes—
160#10587502C2312yes—
161#10587502C2314yes—
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×C17C2411yescharacteristic
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×Dic17—44no—
180C22×Dic17—44no—
181C22×Dic17C22×C4116yes—
182C22×Dic17C2412yes—
183C22×Dic17C2412yes—
184C22×Dic17C2414yes—
185C22×Dic17C2414yes—
186C22×Dic17C2412yes—
187C22×Dic17C2412yes—
188C2×Dic34—22no—
189C2×Dic34—22no—
190C2×Dic34—42no—
191C2×Dic34—42no—
192C2×Dic34C22×C414yes—
193C2×Dic34C22×C418yes—
194C4⋊C4×C17—22no—
195C4⋊C4×C17—22no—
196C4⋊C4×C17—42no—
197C4⋊C4×C17—42no—
198C4⋊C4×C17C22×C414yes—
199C22⋊C4×C17—24no—
200C22⋊C4×C17—42no—
201C22⋊C4×C17—42no—
202C22⋊C4×C17—42no—
203C22⋊C4×C17C22×C418yes—
204C42×C17—22no—
205C42×C17—24no—
206C42×C17—22no—
207C42×C17—22no—
208C42×C17—22no—
209C42×C17—22no—
210C42×C17—41no—
211C42×C17C22×C414yes—
212C22⋊Dic17—44no—
213C22⋊Dic17—44no—
214C22⋊Dic17C22×C4116yes—
215C4⋊Dic17—24no—
216C4⋊Dic17—22no—
217C4⋊Dic17—24no—
218C4⋊Dic17—22no—
219C4⋊Dic17—42no—
220C4⋊Dic17C22×C414yes—
221C4⋊Dic17C22×C418yes—
222C4⋊Dic17C22×C418yes—
223C4⋊Dic17C22×C4116yes—
224#63225—24no—
225#63225—44no—
226#63225—44no—
227#63225—44no—
228#63225—44no—
229#63225—42no—
230#63225C22×C418yes—
231#63225C22×C4116yes—
232C4×Dic17—28no—
233C4×Dic17—24no—
234C4×Dic17—44no—
235C4×Dic17—44no—
236C4×Dic17—44no—
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—21no—
249C2×C4×C17—22no—
250C2×C4×C17—21no—
251C2×C4×C17—21no—
252C2×C4×C17—22no—
253C2×C4×C17—21no—
254C2×C4×C17—216no—
255C2×C4×C17—21no—
256C2×C4×C17—21no—
257C2×C4×C17—22no—
258C2×C4×C17—22no—
259C2×C4×C17—21no—
260C2×C4×C17—22no—
261C2×C4×C17—21no—
262C2×C4×C17—21no—
263C2×C4×C17—21no—
264C2×C4×C17—21no—
265C2×C4×C17—22no—
266C2×C4×C17—22no—
267C2×C4×C17—21no—
268C2×C4×C17—21no—
269C2×C4×C17—22no—
270C2×C4×C17C23×C412yes—
271C2×C4×C17C23×C414yes—
272C2×C4×C17C23×C412yes—
273C2×C4×C17C23×C412yes—
274C2×C4×C17H(5,𝔽2)12yes—
275C2×C4×C17H(5,𝔽2)12yes—
276C2×C4×C172−1+412yes—
277C2×C4×C172−1+412yes—
278C2×C4×C172−1+412yes—
279C2×Dic17—22no—
280C2×Dic17—24no—
281C2×Dic17—22no—
282C2×Dic17—22no—
283C2×Dic17—24no—
284C2×Dic17—22no—
285C2×Dic17—22no—
286C2×Dic17—22no—
287C2×Dic17—24no—
288C2×Dic17—24no—
289C2×Dic17—22no—
290C2×Dic17—22no—
291C2×Dic17—22no—
292C2×Dic17—24no—
293C2×Dic17—22no—
294C2×Dic17—416no—
295C2×Dic17C23×C414yes—
296C2×Dic17C23×C418yes—
297C2×Dic17C23×C414yes—
298Dic34—28no—
299Dic34—48no—
300C2×#554—171no—
301#2378—174no—
302#2370—172no—
303#2369—171no—
304#2362—172no—
305#2354—172no—
306#2354—171no—
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#2293D1711yescharacteristic
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—22no—
334C4×C17—41no—
335C4×C17—42no—
336C4×C17—41no—
337C4×C17—42no—
338C4×C17—41no—
339C4×C17#51914yes—
340C4×C17#52014yes—
341Dic17—24no—
342Dic17—28no—
343Dic17—24no—
344Dic17—42no—
345Dic17—42no—
346Dic17—44no—
347#554—1716no—
348C2×C4⋊Q8—171no—
349C2×#125—172no—
350C2×C22⋊Q8—172no—
351C2×C22⋊Q8—174no—
352C2×C4×Q8—171no—
353C2×C4×Q8—172no—
354C2×C4×Q8—172no—
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—172no—
366#390C2×D1711yescharacteristic
367#390C2×D1711yescharacteristic
368#389—174no—
369#389—174no—
370#389—172no—
371#389—174no—
372#389—172no—
373#389—172no—
374#389—171no—
375#389—172no—
376#389—172no—
377#389—171no—
378#389—174no—
379#389C2×D1711yescharacteristic
380#389C2×D1711yescharacteristic
381#388—174no—
382#388—174no—
383#388—174no—
384#388—174no—
385#388—174no—
386#388C2×D1712yes—
387#388C2×D1712yes—
388#387—174no—
389#387—172no—
390#387—172no—
391#387—174no—
392#387—174no—
393#387—174no—
394#384—172no—
395#384—171no—
396#381—174no—
397#380—172no—
398#380—174no—
399#380—174no—
400C4×C4⋊C4—172no—
401C4×C4⋊C4—172no—
402C4×C4⋊C4—172no—
403C4×C4⋊C4—172no—
404C4×C4⋊C4—172no—
405C4×C4⋊C4—172no—
406C4×C4⋊C4—171no—
407C4×C4⋊C4—172no—
408C4×C4⋊C4—174no—
409C4×C4⋊C4C2×D1711yescharacteristic
410C4×C22⋊C4C2×D1712yes—
411#376—172no—
412C43C2×D1711yescharacteristic
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—688no—
437#117—684no—
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—172no—
448C2×C4⋊C4—171no—
449C2×C4⋊C4—172no—
450C2×C4⋊C4—174no—
451C2×C4⋊C4—174no—
452C2×C4⋊C4—174no—
453C2×C4⋊C4—172no—
454C2×C4⋊C4—172no—
455C2×C4⋊C4—174no—
456C2×C4⋊C4C22×D1711yescharacteristic
457C2×C4⋊C4C22×D1711yescharacteristic
458C2×C4⋊C4C22×D1712yes—
459C2×C4⋊C4C22×D1712yes—
460C2×C4⋊C4C22×D1711yescharacteristic
461C2×C22⋊C4—174no—
462C2×C22⋊C4—174no—
463C2×C22⋊C4—174no—
464C2×C22⋊C4C22×D1712yes—
465C2×C22⋊C4C22×D1712yes—
466C2×C22⋊C4C22×D1712yes—
467C2×C22⋊C4C22×D1712yes—
468C2×C22⋊C4C22×D1712yes—
469C2×C42—172no—
470C2×C42—174no—
471C2×C42—172no—
472C2×C42—174no—
473C2×C42—174no—
474C2×C42—174no—
475C2×C42C22×D1711yescharacteristic
476C2×C42C22×D1712yes—
477C2×C42C22×D1711yescharacteristic
478C2×C42C22×D1711yescharacteristic
479C2×C42C22×D1711yescharacteristic
480C2×C42C22×D1711yescharacteristic
481C2×C42C22×D1711yescharacteristic
482C2×C42C22×D1711yescharacteristic
483#95—172no—
484#95—172no—
485#95—172no—
486#95—172no—
487#95—172no—
488#95—174no—
489#95—174no—
490#95—174no—
491#95—174no—
492#95—174no—
493#95—174no—
494#95—172no—
495#95—172no—
496#95C22×D1712yes—
497#95C22×D1712yes—
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—172no—
513C22×C4—174no—
514C22×C4—172no—
515C22×C4—1716no—
516C22×C4—172no—
517C22×C4—172no—
518C22×C4—174no—
519C22×C4—684no—
520C22×C4—684no—
521C22×C4C2×C4×D1718yes—
522C22×C4C23×D1711yescharacteristic
523C22×C4C23×D1711yescharacteristic
524C22×C4C23×D1712yes—
525C22×C4C23×D1711yescharacteristic
526C22×C4C23×D1712yes—
527C22×C4C23×D1712yes—
528C22×C4C23×D1711yescharacteristic
529C22×C4C23×D1711yescharacteristic
530C22×C4C23×D1711yescharacteristic
531C22×C4C23×D1711yescharacteristic
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—174no—
537C4⋊C4—1716no—
538C4⋊C4—178no—
539C4⋊C4—178no—
540C4⋊C4—178no—
541C4⋊C4—1716no—
542C4⋊C4—342no—
543C4⋊C4—344no—
544C4⋊C4—344no—
545C4⋊C4—344no—
546C4⋊C4—342no—
547C4⋊C4—684no—
548C4⋊C4—684no—
549C4⋊C4—684no—
550C4⋊C4—682no—
551C4⋊C4—682no—
552C4⋊C4—684no—
553C4⋊C4C2×C4×D1714yes—
554C22⋊C4—24no—
555C22⋊C4—42no—
556C22⋊C4—42no—
557C22⋊C4—42no—
558C22⋊C4—1716no—
559C22⋊C4—684no—
560C22⋊C4—684no—
561C22⋊C4C2×C4×D1718yes—
562C42—22no—
563C42—24no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—41no—
569C42—178no—
570C42—348no—
571C42—344no—
572C42—684no—
573C42—684no—
574C42—684no—
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—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—216no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—174no—
610C2×C4—174no—
611C2×C4—178no—
612C2×C4—342no—
613C2×C4—344no—
614C2×C4—342no—
615C2×C4—342no—
616C2×C4—342no—
617C2×C4—344no—
618C2×C4—342no—
619C2×C4—344no—
620C2×C4—344no—
621C2×C4—342no—
622C2×C4—342no—
623C2×C4—342no—
624C2×C4—342no—
625C2×C4—344no—
626C2×C4—342no—
627C2×C4—6816no—
628C2×C4C22×C4×D1712yes—
629C2×C4C22×C4×D1714yes—
630C2×C4C22×C4×D1712yes—
631C2×C4C22×C4×D1712yes—
632C2×C4#1058769112yes—
633C2×C4#1058769612yes—
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
645C22C2×#1058769111yescharacteristic
646C22C2×#1058769611yescharacteristic
647C22C2×#1058769711yescharacteristic
648C22C2×#1058769711yescharacteristic
649C22C2×#1058769711yescharacteristic
650C22#1176364511yescharacteristic
651C4—22no—
652C4—24no—
653C4—22no—
654C4—42no—
655C4—41no—
656C4—41no—
657C4—42no—
658C4—41no—
659C4—344no—
660C4—344no—
661C4—348no—
662C4—682no—
663C4—682no—
664C4—684no—
665C4#1176328614yes—
666C4#1176328814yes—
667C2—44no—
668C2C2×#1176328611yescharacteristic
669C2C2×#1176328811yescharacteristic
670C2C4×#1058769611yescharacteristic
671C2#1454312411yescharacteristic
672C2#1454317311yescharacteristic
673C2#1454318211yescharacteristic
674C2#1454327411yescharacteristic
subgroupG/H
center, FrattiniC23C24×D17
commutatorC22×C17C24×C4
FittingC17×#2293C2
socle, cosocleC23×C17C25
prime radicalC22×D4×C17C23
hypercenter#2293D17
hypocenterC17#34484
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ #2293

Complements

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

NH ≅ G/Ntype
2. #14531951C2semidirect
3. #14531885C2semidirect
4. #14531873C2semidirect
5. #14531809C2semidirect
8. C4⋊C4×Dic34C2semidirect
9. #14531651C2semidirect
10. #14531524C2semidirect
11. #14531466C2semidirect
12. #14531463C2semidirect
14. #14531400C2semidirect
15. C4×#10587555C2semidirect
16. #14531261C2semidirect
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
#145319512
#145318852
#145318732
#145318092
#145317332
#145317322
C4⋊C4×Dic342
#145316512
#145315242
#145314662
#145314632
#145314212
#145314002
C4×#105875552
#145312612
C17×#22932
#3448417