Finite Group Databasebeta

← #13410367

Subgroups — #13410367

Order 1792.

1462 conjugacy classes of subgroups.

The subgroup lattice has 650 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#11739827C211yesmaximal, characteristic, complemented
4C2×#89780C211yesmaximal, characteristic, complemented
5#11729889C212yesmaximal, complemented
6#11729882C212yesmaximal, complemented
7#11729475C211yesmaximal, characteristic, complemented
8#11729448C212yesmaximal, complemented
9#11729434C212yesmaximal, 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×#6499—21no—
26C2×#6432C2211yescharacteristic, complemented
27C2×#6409C2212yescomplemented
28C2×#6409C2212yescomplemented
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#89780—24no—
39#89746—21no—
40C2×#6382C2212yescomplemented
41#89725—21no—
42#89725—21no—
43C4×C4⋊Dic7C2211yescharacteristic, complemented
44C2×C4⋊(C7⋊C8)C2211yescharacteristic, complemented
45C2×#6377C2211yescharacteristic, complemented
46#89324C2212yescomplemented
47#25898—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—
96#6499—44no—
97C2×C4⋊Dic7—21no—
98C2×C4⋊Dic7C2312yes—
99C2×C4⋊Dic7C2312yes—
100C2×C4×Dic7—21no—
101C22×C7⋊C8C2312yes—
102C22×C7⋊C8C2312yes—
103#6432—22no—
104#6432—22no—
105#6409—24no—
106#6409—42no—
107#6406—24no—
108#6406—24no—
109#6406—24no—
110#6406—42no—
111C7⋊#95—21no—
112C7⋊#95—22no—
113C7⋊#95—22no—
114C22⋊(C7⋊C8)—42no—
115C22⋊(C7⋊C8)—42no—
116#6382—24no—
117C4⋊(C7⋊C8)—22no—
118#6377—22no—
119C2×#546D711yescharacteristic
120C2×#489—71no—
121C2×#476—71no—
122#2547—71no—
123#2035—72no—
124#2034—72no—
125#1983—71no—
126#1955—71no—
127#1788—72no—
128#1786—72no—
129#1762—71no—
130C24×C7—21no—
131C24×C7—24no—
132C24×C7—21no—
133C24×C7—44no—
134C24×C7—42no—
135C24×C7—42no—
136C24×C7—82no—
137C24×C7C2×D411yescharacteristic
138C24×C7C2×D411yescharacteristic
139C7×#41—48no—
140C7×#41—44no—
141C7×#41—44no—
142C2×C7×Q8—21no—
143C2×C7×Q8—22no—
144C2×C7×Q8—22no—
145C2×C7×Q8C2×D412yes—
146C2×C7×D4—24no—
147C2×C7×D4—24no—
148C2×C7×D4—22no—
149C2×C7×D4—24no—
150C2×C7×D4—22no—
151C2×C7×D4—21no—
152C2×C7×D4—21no—
153C2×C7×D4—22no—
154C2×C7×D4—22no—
155C2×C7×D4—22no—
156C2×C7×D4—41no—
157C2×C7×D4—44no—
158C2×C7×D4—48no—
159C2×C7×D4—44no—
160C2×C7×D4—44no—
161C2×C7×D4—44no—
162C2×C7×D4—82no—
163C2×C7×D4C2×D412yes—
164C2×C7×D4C2×D412yes—
165C2×C7×D4#4114yes—
166C22×C4×C7—24no—
167C22×C4×C7—21no—
168C22×C4×C7—21no—
169C22×C4×C7—21no—
170C22×C4×C7—21no—
171C22×C4×C7—21no—
172C22×C4×C7—21no—
173C22×C4×C7—22no—
174C22×C4×C7—22no—
175C22×C4×C7—22no—
176C22×C4×C7—22no—
177C22×C4×C7—24no—
178C22×C4×C7—44no—
179C22×C4×C7—82no—
180C22×C4×C7C2×D411yescharacteristic
181C22×C4×C7C2×D411yescharacteristic
182C22×C4×C7C2×D411yescharacteristic
183C22×C4×C7C2×D411yescharacteristic
184C22×C4×C7C2411yescharacteristic
185C22×Dic7—22no—
186C22×Dic7—22no—
187C2×Dic14—42no—
188C2×Dic14—42no—
189C2×Dic14—42no—
190C7×C4⋊C4—24no—
191C7×C4⋊C4—22no—
192C7×C4⋊C4—21no—
193C7×C4⋊C4C2×D412yes—
194C7×C22⋊C4—24no—
195C7×C22⋊C4—24no—
196C7×C22⋊C4—22no—
197C7×C22⋊C4—24no—
198C7×C22⋊C4—24no—
199C7×C22⋊C4—24no—
200C7×C22⋊C4—22no—
201C42×C7C2×D412yes—
202C42×C7C2×D412yes—
203C22⋊Dic7—44no—
204C22⋊Dic7—44no—
205C4⋊Dic7—22no—
206C4⋊Dic7—22no—
207C4⋊Dic7—22no—
208C4⋊Dic7—22no—
209C4⋊Dic7—42no—
210C4×Dic7—42no—
211C2×C7⋊C8—22no—
212C2×C7⋊C8—22no—
213C2×C7⋊C8—22no—
214C2×C7⋊C8—22no—
215C2×C7⋊C8—22no—
216C2×C7⋊C8—22no—
217C22×#41C2×D711yescharacteristic
218C23×D4C2×D711yescharacteristic
219#546—24no—
220#546—24no—
221C2×C4⋊Q8—71no—
222C2×#124—141no—
223C2×#124C2×D711yescharacteristic
224C2×#123—21no—
225C2×C22⋊Q8C2×D711yescharacteristic
226C2×#121—21no—
227C2×#121C2×D711yescharacteristic
228C2×#120—21no—
229C2×C4×D4C2×D711yescharacteristic
230C2×C4×D4C2×D711yescharacteristic
231C22×C22⋊C4—21no—
232#489—144no—
233#476—144no—
234C2×#106—72no—
235C2×C4⋊C8—71no—
236C2×#103—72no—
237C2×#103—72no—
238C2×#102—72no—
239C2×#102—72no—
240C2×#102—72no—
241C2×#102—72no—
242C2×C22⋊C8—72no—
243C2×C22⋊C8—72no—
244C2×#97—71no—
245#397—141no—
246#397—141no—
247#391—141no—
248#380—72no—
249#380—72no—
250C4×C4⋊C4—71no—
251#340—72no—
252C23×C7—21no—
253C23×C7—22no—
254C23×C7—22no—
255C23×C7—24no—
256C23×C7—21no—
257C23×C7—22no—
258C23×C7—22no—
259C23×C7—21no—
260C23×C7—22no—
261C23×C7—21no—
262C23×C7—21no—
263C23×C7—22no—
264C23×C7—48no—
265C23×C7—48no—
266C23×C7—41no—
267C23×C7—41no—
268C23×C7—42no—
269C23×C7—42no—
270C23×C7—41no—
271C23×C7—41no—
272C23×C7—42no—
273C23×C7—88no—
274C23×C7—84no—
275C23×C7—88no—
276C23×C7AGL(1,ℤ/8)12yes—
277C23×C7C22×D411yescharacteristic
278C7×Q8—24no—
279C7×Q8—41no—
280C7×Q8—41no—
281C7×D4—22no—
282C7×D4—24no—
283C7×D4—24no—
284C7×D4—44no—
285C7×D4—48no—
286C7×D4—48no—
287C7×D4—41no—
288C7×D4—41no—
289C7×D4—84no—
290C7×D4#12414yescomplemented
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—22no—
294C2×C4×C7—22no—
295C2×C4×C7—21no—
296C2×C4×C7—22no—
297C2×C4×C7—21no—
298C2×C4×C7—22no—
299C2×C4×C7—22no—
300C2×C4×C7—48no—
301C2×C4×C7—42no—
302C2×C4×C7—42no—
303C2×C4×C7—41no—
304C2×C4×C7—41no—
305C2×C4×C7—41no—
306C2×C4×C7—41no—
307C2×C4×C7—42no—
308C2×C4×C7—42no—
309C2×C4×C7—84no—
310C2×C4×C7#12012yes—
311C2×C4×C7#12012yes—
312C2×C4×C7C22×D411yescharacteristic
313C2×C4×C7C22×D411yescharacteristic
314C2×C4×C7C2×#4112yes—
315C2×C4×C7H(5,𝔽2)12yes—
316C2×Dic7—42no—
317C2×Dic7—42no—
318C2×Dic7—42no—
319C2×Dic7—42no—
320C2×Dic7—42no—
321C2×Dic7—42no—
322Dic14—84no—
323C7⋊C8—42no—
324C7⋊C8—42no—
325C25—21no—
326C2×#41—22no—
327C2×#41—22no—
328C2×#41—24no—
329C2×#41—42no—
330C22×Q8—141no—
331C22×Q8C22×D711yescharacteristic
332C22×D4—21no—
333C22×D4—21no—
334C22×D4—22no—
335C22×D4—22no—
336C22×D4—24no—
337C22×D4—21no—
338C22×D4—42no—
339C22×D4C22⋊D711yescharacteristic
340C22×D4C22×D711yescharacteristic
341C22×D4C22×D711yescharacteristic
342C23×C4—21no—
343C23×C4—21no—
344C23×C4C22×D711yescharacteristic
345C23×C4C22×D711yescharacteristic
346C22×C8—72no—
347C22×C8—72no—
348C4⋊Q8—142no—
349C4⋊Q8—142no—
350#124—22no—
351#124—284no—
352#124C22⋊D714yes—
353#123—28no—
354C22⋊Q8—22no—
355C22⋊Q8—22no—
356#121—28no—
357#121—22no—
358#121—22no—
359#120—28no—
360C4×D4—22no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4C22⋊D714yes—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—72no—
366C2×C4⋊C4—72no—
367C2×C4⋊C4—141no—
368C2×C4⋊C4C22⋊D711yescharacteristic
369C2×C4⋊C4C22×D711yescharacteristic
370C2×C22⋊C4—28no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—142no—
377C2×C22⋊C4—142no—
378C2×C22⋊C4C22⋊D711yescharacteristic
379C2×C22⋊C4C22⋊D711yescharacteristic
380C2×C42—141no—
381C2×C42C22×D711yescharacteristic
382#106—144no—
383C4⋊C8—142no—
384#103—144no—
385#103—282no—
386#102—144no—
387#102—144no—
388#102—144no—
389#102—282no—
390C22⋊C8—282no—
391C22⋊C8—282no—
392#97—142no—
393#95—141no—
394#95—142no—
395#95—142no—
396C22×C7—24no—
397C22×C7—22no—
398C22×C7—24no—
399C22×C7—41no—
400C22×C7—42no—
401C22×C7—42no—
402C22×C7—41no—
403C22×C7—44no—
404C22×C7—42no—
405C22×C7—44no—
406C22×C7—41no—
407C22×C7—42no—
408C22×C7—41no—
409C22×C7—42no—
410C22×C7—41no—
411C22×C7—42no—
412C22×C7—42no—
413C22×C7—42no—
414C22×C7—816no—
415C22×C7—82no—
416C22×C7—82no—
417C22×C7C2×#12011yescharacteristic
418C22×C7#54612yes—
419C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
420C22×C7#57511yescharacteristic
421C22×C7#57712yes—
422C4×C7—22no—
423C4×C7—22no—
424C4×C7—41no—
425C4×C7—41no—
426C4×C7—42no—
427C4×C7—42no—
428C4×C7C2×#12411yescharacteristic
429C4×C7#53511yescharacteristic
430C4×C7#54612yes—
431Dic7—82no—
432Dic7—82no—
433C24—21no—
434C24—24no—
435C24—21no—
436C24—44no—
437C24—42no—
438C24—42no—
439C24—82no—
440C24C2×C22⋊D711yescharacteristic
441C24C2×C22⋊D711yescharacteristic
442#41—48no—
443#41—44no—
444#41—44no—
445C2×Q8—21no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—282no—
449C2×Q8—282no—
450C2×Q8—282no—
451C2×Q8C2×C22⋊D712yes—
452C2×D4—24no—
453C2×D4—24no—
454C2×D4—22no—
455C2×D4—24no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—41no—
463C2×D4—44no—
464C2×D4—48no—
465C2×D4—44no—
466C2×D4—44no—
467C2×D4—44no—
468C2×D4—82no—
469C2×D4#115914yes—
470C2×D4C2×C22⋊D712yes—
471C2×D4C2×C22⋊D712yes—
472C22×C4—24no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—24no—
484C22×C4—44no—
485C22×C4—82no—
486C22×C4—142no—
487C22×C4—142no—
488C22×C4C2×C22⋊D711yescharacteristic
489C22×C4C2×C22⋊D711yescharacteristic
490C22×C4C2×C22⋊D711yescharacteristic
491C22×C4C2×C22⋊D711yescharacteristic
492C22×C4C23×D711yescharacteristic
493C2×C8—142no—
494C2×C8—142no—
495C2×C8—142no—
496C2×C8—142no—
497C2×C8—142no—
498C2×C8—142no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—142no—
503C4⋊C4—142no—
504C4⋊C4—142no—
505C4⋊C4—142no—
506C4⋊C4—282no—
507C4⋊C4C2×C22⋊D712yes—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—22no—
511C22⋊C4—24no—
512C22⋊C4—24no—
513C22⋊C4—24no—
514C22⋊C4—22no—
515C22⋊C4—284no—
516C22⋊C4—284no—
517C42—282no—
518C42C2×C22⋊D712yes—
519C42C2×C22⋊D712yes—
520C2×C7—42no—
521C2×C7—44no—
522C2×C7—81no—
523C2×C7—82no—
524C2×C7#243511yescharacteristic
525C2×C7#300611yescharacteristic
526C2×C7#300911yescharacteristic
527C2×C7#330612yes—
528C2×C7#331112yes—
529C23—22no—
530C23—24no—
531C23—21no—
532C23—22no—
533C23—21no—
534C23—22no—
535C23—22no—
536C23—21no—
537C23—21no—
538C23—22no—
539C23—22no—
540C23—21no—
541C23—48no—
542C23—48no—
543C23—41no—
544C23—41no—
545C23—42no—
546C23—42no—
547C23—41no—
548C23—41no—
549C23—42no—
550C23—88no—
551C23—84no—
552C23—88no—
553C23#651212yes—
554C23C22×C22⋊D711yescharacteristic
555Q8—24no—
556Q8—41no—
557Q8—41no—
558Q8—564no—
559D4—22no—
560D4—24no—
561D4—24no—
562D4—44no—
563D4—48no—
564D4—48no—
565D4—41no—
566D4—41no—
567D4—84no—
568D4#649914yescomplemented
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—48no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—84no—
588C2×C4—282no—
589C2×C4—282no—
590C2×C4—282no—
591C2×C4—282no—
592C2×C4—282no—
593C2×C4—282no—
594C2×C4#651612yes—
595C2×C4#651612yes—
596C2×C4C2×#115912yes—
597C2×C4#654812yes—
598C2×C4C22×C22⋊D711yescharacteristic
599C2×C4C22×C22⋊D711yescharacteristic
600C8—282no—
601C8—282no—
602C7#2589811yescharacteristic, complemented, Sylow
603C22—24no—
604C22—24no—
605C22—22no—
606C22—44no—
607C22—42no—
608C22—44no—
609C22—41no—
610C22—42no—
611C22—42no—
612C22—41no—
613C22—42no—
614C22—42no—
615C22—41no—
616C22—42no—
617C22—41no—
618C22—42no—
619C22—41no—
620C22—42no—
621C22—816no—
622C22—82no—
623C22—82no—
624C22#9049712yes—
625C22C2×#651211yescharacteristic
626C22#9051311yescharacteristic
627C22#9052712yes—
628C22C2×#651611yescharacteristic
629C4—22no—
630C4—22no—
631C4—41no—
632C4—41no—
633C4—42no—
634C4—42no—
635C4—562no—
636C4—562no—
637C4C2×#649911yescharacteristic
638C4#9049611yescharacteristic
639C4#9049712yes—
640C2—42no—
641C2—44no—
642C2—81no—
643C2—82no—
644C2#1174094112yes—
645C2#1174094212yes—
646C2#1174149511yescharacteristic
647C2#1174152711yescharacteristic
648C2#1174155011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
Fitting, prime radicalC2×C7×#546C2
socleC23×C7C22×D4
hypercenterC2×#546D7
hypocenterC7#25898
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. #11739827C2semidirect
4. C2×#89780C2semidirect
5. #11729889C2semidirect
6. #11729882C2semidirect
7. #11729475C2semidirect
8. #11729448C2semidirect
9. #11729434C2semidirect
10. #11729369C2semidirect
11. #11729363C2semidirect
26. C2×#6432C22semidirect
27. C2×#6409C22semidirect
28. C2×#6409C22semidirect
34. #89984C22semidirect
36. C2×C22⋊(C7⋊C8)C22semidirect
40. C2×#6382C22semidirect
43. C4×C4⋊Dic7C22semidirect
44. C2×C4⋊(C7⋊C8)C22semidirect
45. C2×#6377C22semidirect
46. #89324C22semidirect
290. C7×D4#124semidirect
568. D4#6499semidirect
602. C7#25898semidirect

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
#117398272
C2×#897802
#117298892
#117298822
#117294752
#117294482
#117294342
#117293692
#117293632
#258987