Finite Group Databasebeta

← #13410356

Subgroups — #13410356

Order 1792.

1442 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#546C211yesmaximal, characteristic
2C2×#89827C211yesmaximal, characteristic, complemented
3#11739829C211yesmaximal, characteristic, complemented
4C2×#89779C211yesmaximal, characteristic, complemented
5#11729889C212yesmaximal, complemented
6#11729882C212yesmaximal, complemented
7#11729475C211yesmaximal, characteristic, complemented
8#11729449C212yesmaximal, complemented
9#11729433C212yesmaximal, complemented
10#11729369C211yesmaximal, characteristic, complemented
11#11729363C211yesmaximal, characteristic, complemented
12C22×C7×#41C2211yescharacteristic
13C23×C7×D4C2211yescharacteristic
14C7×#546—24no—
15C7×#546—24no—
16C2×C7×#124C2211yescharacteristic
17C2×C7×#123—21no—
18C2×C7×C22⋊Q8C2211yescharacteristic
19C2×C7×#121—21no—
20C2×C7×#121C2211yescharacteristic
21C2×C7×#120—21no—
22C2×C4×C7×D4C2211yescharacteristic
23C2×C4×C7×D4C2211yescharacteristic
24C22×C7×C22⋊C4—21no—
25C2×#6432C2211yescharacteristic, complemented
26C2×#6409C2212yescomplemented
27C2×#6409C2212yescomplemented
28#89993—21no—
29C2×#6406C2212yes—
30C2×#6406C2212yes—
31C2×#6406C2212yes—
32C2×#6406C2212yes—
33#89984C2212yes—
34#89984C2212yescomplemented
35C2×C22⋊(C7⋊C8)C2212yes—
36C2×C22⋊(C7⋊C8)C2212yescomplemented
37#89827—24no—
38#89779—24no—
39#89749—21no—
40C2×#6381C2212yescomplemented
41#89719—21no—
42#89719—21no—
43C4×C4⋊Dic7C2211yescharacteristic, complemented
44C2×C4⋊(C7⋊C8)C2211yescharacteristic, complemented
45C2×#6377C2211yescharacteristic, complemented
46#89324C2212yescomplemented
47#25897—71nomaximal, Sylow
48C25×C7—21no—
49C2×C7×#41—22no—
50C2×C7×#41—22no—
51C2×C7×#41—24no—
52C2×C7×#41—42no—
53C22×C7×Q8C2311yescharacteristic
54C22×C7×D4—21no—
55C22×C7×D4—21no—
56C22×C7×D4—22no—
57C22×C7×D4—22no—
58C22×C7×D4—24no—
59C22×C7×D4—21no—
60C22×C7×D4—42no—
61C22×C7×D4D411yescharacteristic
62C22×C7×D4C2311yescharacteristic
63C22×C7×D4C2311yescharacteristic
64C23×C4×C7—21no—
65C23×C4×C7—21no—
66C23×C4×C7C2311yescharacteristic
67C23×C4×C7C2311yescharacteristic
68C22×Dic14—21no—
69C7×#124—22no—
70C7×#124D414yes—
71C7×#123—28no—
72C7×C22⋊Q8—22no—
73C7×C22⋊Q8—22no—
74C7×#121—28no—
75C7×#121—22no—
76C7×#121—22no—
77C7×#120—28no—
78C4×C7×D4—22no—
79C4×C7×D4—22no—
80C4×C7×D4—22no—
81C4×C7×D4D414yes—
82C2×C7×C4⋊C4—21no—
83C2×C7×C4⋊C4D411yescharacteristic
84C2×C7×C4⋊C4C2311yescharacteristic
85C2×C7×C22⋊C4—28no—
86C2×C7×C22⋊C4—21no—
87C2×C7×C22⋊C4—21no—
88C2×C7×C22⋊C4—21no—
89C2×C7×C22⋊C4—21no—
90C2×C7×C22⋊C4—21no—
91C2×C7×C22⋊C4D411yescharacteristic
92C2×C7×C22⋊C4D411yescharacteristic
93C2×C42×C7C2311yescharacteristic
94C2×C22⋊Dic7—22no—
95C2×C22⋊Dic7—22no—
96C2×C4⋊Dic7—21no—
97C2×C4⋊Dic7C2312yes—
98C2×C4⋊Dic7C2312yes—
99C2×C4×Dic7—21no—
100C22×C7⋊C8C2312yes—
101C22×C7⋊C8C2312yes—
102#6432—22no—
103#6432—22no—
104#6409—24no—
105#6409—42no—
106#6406—24no—
107#6406—24no—
108#6406—24no—
109#6406—42no—
110C7⋊#95—21no—
111C7⋊#95—22no—
112C7⋊#95—22no—
113C22⋊(C7⋊C8)—42no—
114C22⋊(C7⋊C8)—42no—
115#6381—24no—
116C4⋊(C7⋊C8)—22no—
117#6377—22no—
118C2×#546D711yescharacteristic
119C2×#489—71no—
120C2×#474—71no—
121#2563—71no—
122#2036—72no—
123#2033—72no—
124#1983—71no—
125#1955—71no—
126#1788—72no—
127#1786—72no—
128#1762—71no—
129C24×C7—21no—
130C24×C7—24no—
131C24×C7—21no—
132C24×C7—44no—
133C24×C7—42no—
134C24×C7—42no—
135C24×C7—82no—
136C24×C7C2×D411yescharacteristic
137C24×C7C2×D411yescharacteristic
138C7×#41—48no—
139C7×#41—44no—
140C7×#41—44no—
141C2×C7×Q8—21no—
142C2×C7×Q8—22no—
143C2×C7×Q8—22no—
144C2×C7×Q8C2×D412yes—
145C2×C7×D4—24no—
146C2×C7×D4—24no—
147C2×C7×D4—22no—
148C2×C7×D4—24no—
149C2×C7×D4—22no—
150C2×C7×D4—21no—
151C2×C7×D4—21no—
152C2×C7×D4—22no—
153C2×C7×D4—22no—
154C2×C7×D4—22no—
155C2×C7×D4—41no—
156C2×C7×D4—44no—
157C2×C7×D4—48no—
158C2×C7×D4—44no—
159C2×C7×D4—44no—
160C2×C7×D4—44no—
161C2×C7×D4—82no—
162C2×C7×D4C2×D412yes—
163C2×C7×D4C2×D412yes—
164C2×C7×D4#4114yes—
165C22×C4×C7—24no—
166C22×C4×C7—21no—
167C22×C4×C7—21no—
168C22×C4×C7—21no—
169C22×C4×C7—21no—
170C22×C4×C7—21no—
171C22×C4×C7—21no—
172C22×C4×C7—22no—
173C22×C4×C7—22no—
174C22×C4×C7—22no—
175C22×C4×C7—22no—
176C22×C4×C7—24no—
177C22×C4×C7—44no—
178C22×C4×C7—82no—
179C22×C4×C7C2×D411yescharacteristic
180C22×C4×C7C2×D411yescharacteristic
181C22×C4×C7C2×D411yescharacteristic
182C22×C4×C7C2×D411yescharacteristic
183C22×C4×C7C2411yescharacteristic
184C22×Dic7—22no—
185C22×Dic7—22no—
186C2×Dic14—42no—
187C2×Dic14—82no—
188C7×C4⋊C4—24no—
189C7×C4⋊C4—22no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4C2×D412yes—
192C7×C22⋊C4—24no—
193C7×C22⋊C4—24no—
194C7×C22⋊C4—22no—
195C7×C22⋊C4—24no—
196C7×C22⋊C4—24no—
197C7×C22⋊C4—24no—
198C7×C22⋊C4—22no—
199C42×C7C2×D412yes—
200C42×C7C2×D412yes—
201C22⋊Dic7—82no—
202C22⋊Dic7—82no—
203C4⋊Dic7—22no—
204C4⋊Dic7—22no—
205C4⋊Dic7—22no—
206C4⋊Dic7—22no—
207C4⋊Dic7—42no—
208C4×Dic7—42no—
209C2×C7⋊C8—22no—
210C2×C7⋊C8—22no—
211C2×C7⋊C8—22no—
212C2×C7⋊C8—22no—
213C2×C7⋊C8—22no—
214C2×C7⋊C8—22no—
215C22×#41C2×D711yescharacteristic
216C23×D4C2×D711yescharacteristic
217#546—24no—
218#546—24no—
219C2×C4⋊Q8—71no—
220C2×#124C2×D711yescharacteristic
221C2×#123—21no—
222C2×C22⋊Q8C2×D711yescharacteristic
223C2×#121—21no—
224C2×#121C2×D711yescharacteristic
225C2×#120—21no—
226C2×C4×D4C2×D711yescharacteristic
227C2×C4×D4C2×D711yescharacteristic
228C22×C22⋊C4—21no—
229#489—144no—
230#474—144no—
231C2×#107—72no—
232C2×C4⋊C8—71no—
233C2×#103—72no—
234C2×#103—72no—
235C2×#102—72no—
236C2×#102—72no—
237C2×#102—72no—
238C2×#102—72no—
239C2×C22⋊C8—72no—
240C2×C22⋊C8—72no—
241C2×#97—71no—
242#393—141no—
243#393—141no—
244#390—141no—
245#382—141no—
246#380—72no—
247#380—72no—
248C4×C4⋊C4—71no—
249#340—72no—
250C23×C7—21no—
251C23×C7—22no—
252C23×C7—22no—
253C23×C7—24no—
254C23×C7—21no—
255C23×C7—22no—
256C23×C7—22no—
257C23×C7—21no—
258C23×C7—22no—
259C23×C7—21no—
260C23×C7—21no—
261C23×C7—22no—
262C23×C7—48no—
263C23×C7—48no—
264C23×C7—41no—
265C23×C7—41no—
266C23×C7—42no—
267C23×C7—42no—
268C23×C7—41no—
269C23×C7—41no—
270C23×C7—42no—
271C23×C7—88no—
272C23×C7—84no—
273C23×C7—88no—
274C23×C7#13712yes—
275C23×C7C22×D411yescharacteristic
276C7×Q8—24no—
277C7×Q8—41no—
278C7×Q8—41no—
279C7×D4—22no—
280C7×D4—24no—
281C7×D4—24no—
282C7×D4—44no—
283C7×D4—48no—
284C7×D4—48no—
285C7×D4—41no—
286C7×D4—41no—
287C7×D4—84no—
288C7×D4#12414yes—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7—22no—
292C2×C4×C7—22no—
293C2×C4×C7—21no—
294C2×C4×C7—22no—
295C2×C4×C7—21no—
296C2×C4×C7—22no—
297C2×C4×C7—22no—
298C2×C4×C7—48no—
299C2×C4×C7—42no—
300C2×C4×C7—42no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7—41no—
305C2×C4×C7—42no—
306C2×C4×C7—42no—
307C2×C4×C7—84no—
308C2×C4×C7#12012yes—
309C2×C4×C7#12012yes—
310C2×C4×C7C22×D411yescharacteristic
311C2×C4×C7C22×D411yescharacteristic
312C2×C4×C7C2×#4112yes—
313C2×C4×C7H(5,𝔽2)12yes—
314C2×Dic7—42no—
315C2×Dic7—42no—
316C2×Dic7—42no—
317C2×Dic7—42no—
318C2×Dic7—42no—
319C2×Dic7—42no—
320Dic14—84no—
321C7⋊C8—42no—
322C7⋊C8—42no—
323C25—21no—
324C2×#41—22no—
325C2×#41—22no—
326C2×#41—24no—
327C2×#41—42no—
328C22×Q8—141no—
329C22×Q8C22×D711yescharacteristic
330C22×D4—21no—
331C22×D4—21no—
332C22×D4—22no—
333C22×D4—22no—
334C22×D4—24no—
335C22×D4—21no—
336C22×D4—42no—
337C22×D4C22⋊D711yescharacteristic
338C22×D4C22×D711yescharacteristic
339C22×D4C22×D711yescharacteristic
340C23×C4—21no—
341C23×C4—21no—
342C23×C4C22×D711yescharacteristic
343C23×C4C22×D711yescharacteristic
344C22×C8—72no—
345C22×C8—72no—
346C4⋊Q8—142no—
347C4⋊Q8—142no—
348#124—22no—
349#124C22⋊D714yes—
350#123—28no—
351C22⋊Q8—22no—
352C22⋊Q8—22no—
353#121—28no—
354#121—22no—
355#121—22no—
356#120—28no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4C22⋊D714yes—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—72no—
363C2×C4⋊C4—72no—
364C2×C4⋊C4—141no—
365C2×C4⋊C4C22⋊D711yescharacteristic
366C2×C4⋊C4C22×D711yescharacteristic
367C2×C22⋊C4—28no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—142no—
374C2×C22⋊C4—142no—
375C2×C22⋊C4C22⋊D711yescharacteristic
376C2×C22⋊C4C22⋊D711yescharacteristic
377C2×C42—141no—
378C2×C42C22×D711yescharacteristic
379#107—144no—
380C4⋊C8—142no—
381#103—144no—
382#103—282no—
383#102—144no—
384#102—144no—
385#102—144no—
386#102—282no—
387C22⋊C8—282no—
388C22⋊C8—282no—
389#97—142no—
390#95—141no—
391#95—142no—
392#95—142no—
393C22×C7—24no—
394C22×C7—22no—
395C22×C7—24no—
396C22×C7—41no—
397C22×C7—42no—
398C22×C7—42no—
399C22×C7—41no—
400C22×C7—44no—
401C22×C7—42no—
402C22×C7—44no—
403C22×C7—41no—
404C22×C7—42no—
405C22×C7—41no—
406C22×C7—42no—
407C22×C7—41no—
408C22×C7—42no—
409C22×C7—42no—
410C22×C7—42no—
411C22×C7—816no—
412C22×C7—82no—
413C22×C7—82no—
414C22×C7C2×#12011yescharacteristic
415C22×C7#54612yes—
416C22×C7C2×#13711yescharacteristic
417C22×C7#57511yescharacteristic
418C22×C7#57612yes—
419C4×C7—22no—
420C4×C7—22no—
421C4×C7—41no—
422C4×C7—41no—
423C4×C7—42no—
424C4×C7—42no—
425C4×C7C2×#12411yescharacteristic
426C4×C7#53511yescharacteristic
427C4×C7#54612yes—
428Dic7—82no—
429Dic7—82no—
430C24—21no—
431C24—24no—
432C24—21no—
433C24—44no—
434C24—42no—
435C24—42no—
436C24—82no—
437C24C2×C22⋊D711yescharacteristic
438C24C2×C22⋊D711yescharacteristic
439#41—48no—
440#41—44no—
441#41—44no—
442C2×Q8—21no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—282no—
446C2×Q8—562no—
447C2×Q8C2×C22⋊D712yes—
448C2×D4—24no—
449C2×D4—24no—
450C2×D4—22no—
451C2×D4—24no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—41no—
459C2×D4—44no—
460C2×D4—48no—
461C2×D4—44no—
462C2×D4—44no—
463C2×D4—44no—
464C2×D4—82no—
465C2×D4#115914yes—
466C2×D4C2×C22⋊D712yes—
467C2×D4C2×C22⋊D712yes—
468C22×C4—24no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—24no—
480C22×C4—44no—
481C22×C4—82no—
482C22×C4—142no—
483C22×C4—142no—
484C22×C4C2×C22⋊D711yescharacteristic
485C22×C4C2×C22⋊D711yescharacteristic
486C22×C4C2×C22⋊D711yescharacteristic
487C22×C4C2×C22⋊D711yescharacteristic
488C22×C4C23×D711yescharacteristic
489C2×C8—142no—
490C2×C8—142no—
491C2×C8—142no—
492C2×C8—142no—
493C2×C8—142no—
494C2×C8—142no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—142no—
499C4⋊C4—142no—
500C4⋊C4—142no—
501C4⋊C4—142no—
502C4⋊C4—282no—
503C4⋊C4C2×C22⋊D712yes—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—22no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—22no—
511C22⋊C4—562no—
512C22⋊C4—562no—
513C42—282no—
514C42C2×C22⋊D712yes—
515C42C2×C22⋊D712yes—
516C2×C7—42no—
517C2×C7—44no—
518C2×C7—81no—
519C2×C7—82no—
520C2×C7#243511yescharacteristic
521C2×C7#300711yescharacteristic
522C2×C7#300811yescharacteristic
523C2×C7#331412yes—
524C2×C7#331912yes—
525C23—22no—
526C23—24no—
527C23—21no—
528C23—22no—
529C23—21no—
530C23—22no—
531C23—22no—
532C23—21no—
533C23—21no—
534C23—22no—
535C23—22no—
536C23—21no—
537C23—48no—
538C23—48no—
539C23—41no—
540C23—41no—
541C23—42no—
542C23—42no—
543C23—41no—
544C23—41no—
545C23—42no—
546C23—88no—
547C23—84no—
548C23—88no—
549C23#651412yes—
550C23C22×C22⋊D711yescharacteristic
551Q8—24no—
552Q8—41no—
553Q8—41no—
554Q8—564no—
555D4—22no—
556D4—24no—
557D4—24no—
558D4—44no—
559D4—48no—
560D4—48no—
561D4—41no—
562D4—41no—
563D4—84no—
564D4#649914yes—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—48no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—84no—
584C2×C4—282no—
585C2×C4—282no—
586C2×C4—282no—
587C2×C4—282no—
588C2×C4—282no—
589C2×C4—282no—
590C2×C4#651612yes—
591C2×C4#651612yes—
592C2×C4C2×#115912yes—
593C2×C4#654812yes—
594C2×C4C22×C22⋊D711yescharacteristic
595C2×C4C22×C22⋊D711yescharacteristic
596C8—282no—
597C8—282no—
598C7#2589711yescharacteristic, complemented, Sylow
599C22—24no—
600C22—24no—
601C22—22no—
602C22—44no—
603C22—42no—
604C22—44no—
605C22—41no—
606C22—42no—
607C22—42no—
608C22—41no—
609C22—42no—
610C22—42no—
611C22—41no—
612C22—42no—
613C22—41no—
614C22—42no—
615C22—41no—
616C22—42no—
617C22—816no—
618C22—82no—
619C22—82no—
620C22#9049712yes—
621C22#9051311yescharacteristic
622C22C2×#651411yescharacteristic
623C22#9052612yes—
624C22C2×#651611yescharacteristic
625C4—22no—
626C4—22no—
627C4—41no—
628C4—41no—
629C4—42no—
630C4—42no—
631C4—562no—
632C4—562no—
633C4C2×#649911yescharacteristic
634C4#9049611yescharacteristic
635C4#9049712yes—
636C2—42no—
637C2—44no—
638C2—81no—
639C2—82no—
640C2#1174094312yes—
641C2#1174094412yes—
642C2#1174149511yescharacteristic
643C2#1174152911yescharacteristic
644C2#1174154611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
Fitting, prime radicalC2×C7×#546C2
socleC23×C7C22×D4
hypercenterC2×#546D7
hypocenterC7#25897
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#546

Complements

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

NH ≅ G/Ntype
2. C2×#89827C2semidirect
3. #11739829C2semidirect
4. C2×#89779C2semidirect
5. #11729889C2semidirect
6. #11729882C2semidirect
7. #11729475C2semidirect
8. #11729449C2semidirect
9. #11729433C2semidirect
10. #11729369C2semidirect
11. #11729363C2semidirect
25. C2×#6432C22semidirect
26. C2×#6409C22semidirect
27. C2×#6409C22semidirect
34. #89984C22semidirect
36. C2×C22⋊(C7⋊C8)C22semidirect
40. C2×#6381C22semidirect
43. C4×C4⋊Dic7C22semidirect
44. C2×C4⋊(C7⋊C8)C22semidirect
45. C2×#6377C22semidirect
46. #89324C22semidirect
598. C7#25897semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5462
C2×#898272
#117398292
C2×#897792
#117298892
#117298822
#117294752
#117294492
#117294332
#117293692
#117293632
#258977