Finite Group Databasebeta

← #13380652

Subgroups — #13380652

Order 1792.

2206 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#537C211yesmaximal, characteristic, complemented
2C2×#90521C211yesmaximal, characteristic, complemented
3C22×#6512C211yesmaximal, characteristic, complemented
4C2×#90006C211yesmaximal, characteristic, complemented
5C2×#90005C211yesmaximal, characteristic, complemented
6C22×#6383C211yesmaximal, characteristic, complemented
7#11729453C214yesmaximal, complemented
8#11729438C214yesmaximal, complemented
9#11729312C211yesmaximal, characteristic, complemented
10C22×C7×#41C2211yescharacteristic
11C22×C7×#41C2211yescharacteristic
12C23×D28C2211yescharacteristic
13C7×#537—24no—
14C7×#537—24no—
15C2×C7×C22⋊Q8—21no—
16C2×C7×C22⋊Q8C2212yescomplemented
17C2×C7×#121—21no—
18C2×C7×#121—22no—
19C2×C7×#121C2212yescomplemented
20C22×C7×C4⋊C4C2211yescharacteristic
21#90521—24no—
22#90521—24no—
23C2×#6512—24no—
24C2×#6512—24no—
25C2×#6512—24no—
26C2×#6510—21no—
27C22×#1143C2212yes—
28C2×#6503—21no—
29C2×#6500—22no—
30C22×#1141C2212yes—
31C2×#6493C2214yes—
32C2×#6489C2211yescharacteristic
33C22×#1136C2211yescharacteristic
34C23×C7⋊C8C2211yescharacteristic
35#90006—24no—
36#90005—24no—
37C2×#6409C2212yescomplemented
38C2×#6406C2212yescomplemented
39C2×C22⋊(C7⋊C8)C2214yescomplemented
40C2×#6383—24no—
41C2×#6383C2214yes—
42#89324C2214yescomplemented
43#23393—71nomaximal, Sylow
44C24×D7—21no—
45C2×C7×#41—22no—
46C2×C7×#41—22no—
47C2×C7×#41—22no—
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C2×C7×#41—22no—
51C2×C7×#41—22no—
52C2×C7×#41—22no—
53C2×C7×#41—22no—
54C2×C7×#41—22no—
55C2×C7×#41—22no—
56C2×C7×#41D414yescomplemented
57C22×C7×Q8D411yescharacteristic
58C22×C7×Q8C2311yescharacteristic
59C22×C7×D4—21no—
60C22×C7×D4—22no—
61C22×C7×D4D412yes—
62C22×C7×D4D411yescharacteristic
63C22×C7×D4C2311yescharacteristic
64C23×C4×C7—21no—
65C23×C4×C7—21no—
66C23×C4×C7C2312yes—
67C23×C4×C7C2311yescharacteristic
68C22×C22⋊D7—24no—
69C22×D28—22no—
70C22×D28—22no—
71C22×D28—22no—
72C22×D28—22no—
73C22×D28—21no—
74C22×D28—22no—
75C22×D28D414yescomplemented
76C22×D28C2312yes—
77C7×C22⋊Q8—28no—
78C7×C22⋊Q8—24no—
79C7×C22⋊Q8—24no—
80C7×#121—216no—
81C7×#121—28no—
82C7×#121—24no—
83C7×#121—24no—
84C2×C7×C4⋊C4—21no—
85C2×C7×C4⋊C4—22no—
86C2×C7×C4⋊C4—22no—
87C2×C7×C4⋊C4D414yescomplemented
88C2×C7×C4⋊C4C2312yes—
89C2×C7×C22⋊C4—22no—
90C2×C7×C22⋊C4—22no—
91C2×C22⋊Dic7—21no—
92#6512—232no—
93#6510—28no—
94C2×#1143—24no—
95C2×#1143—24no—
96C2×#1143—24no—
97#6503—28no—
98#6500—216no—
99C2×#1141—24no—
100C2×#1141—24no—
101C2×#1141—24no—
102#6493—28no—
103#6493—28no—
104C2×(C2×D7)⋊C4—24no—
105#6489—22no—
106#6489D414yescomplemented
107C2×C4⋊Dic7C2312yes—
108C2×C4×Dic7—21no—
109C2×#1136—22no—
110C2×#1136—22no—
111C2×#1136D414yescomplemented
112C22×C7⋊C8—22no—
113C22×C7⋊C8—22no—
114C22×C7⋊C8—22no—
115C22×C7⋊C8C2312yes—
116C22×C7⋊C8C2312yes—
117#6409—24no—
118#6406—24no—
119C22⋊(C7⋊C8)—28no—
120#6383—28no—
121C22×AGL(1,ℤ/8)—71no—
122C2×#537D711yescharacteristic, complemented
123C2×#535—71no—
124C2×#416—71no—
125C22×#102—71no—
126C2×#408—71no—
127#2000—74no—
128#1999—74no—
129#1736—71no—
130C24×C7—21no—
131C24×C7C2×D412yes—
132C24×C7C2×D411yescharacteristic
133C23×D7—22no—
134C23×D7—24no—
135C23×D7—42no—
136C23×D7—42no—
137C23×D7—82no—
138C23×D7—82no—
139C7×#41—28no—
140C7×#41—28no—
141C7×#41—44no—
142C7×#41—44no—
143C7×#41—44no—
144C7×#41—44no—
145C7×#41—44no—
146C7×#41—44no—
147C2×C7×Q8—22no—
148C2×C7×Q8—22no—
149C2×C7×Q8—22no—
150C2×C7×Q8—21no—
151C2×C7×Q8—21no—
152C2×C7×Q8—21no—
153C2×C7×Q8—21no—
154C2×C7×Q8—21no—
155C2×C7×Q8C2×D412yes—
156C2×C7×D4—24no—
157C2×C7×D4—24no—
158C2×C7×D4—24no—
159C2×C7×D4—24no—
160C2×C7×D4—28no—
161C2×C7×D4—22no—
162C2×C7×D4—22no—
163C2×C7×D4—22no—
164C2×C7×D4—21no—
165C2×C7×D4—21no—
166C2×C7×D4—21no—
167C2×C7×D4—21no—
168C2×C7×D4—21no—
169C2×C7×D4—42no—
170C2×C7×D4—42no—
171C2×C7×D4—84no—
172C2×C7×D4C2×D412yes—
173C22×C4×C7—22no—
174C22×C4×C7—21no—
175C22×C4×C7—24no—
176C22×C4×C7—24no—
177C22×C4×C7—22no—
178C22×C4×C7—22no—
179C22×C4×C7—22no—
180C22×C4×C7—22no—
181C22×C4×C7—22no—
182C22×C4×C7—22no—
183C22×C4×C7—42no—
184C22×C4×C7—42no—
185C22×C4×C7—82no—
186C22×C4×C7C2×D412yes—
187C22×C4×C7C2×D411yescharacteristic
188C22×C4×C7C2×D412yes—
189C22×C4×C7C2×D412yes—
190C22×C4×C7C2×D412yes—
191C22×C4×C7C2×D412yes—
192C22×C4×C7C2×D412yes—
193C22×C4×C7#4112yes—
194C22×C4×C7C2411yescharacteristic
195C2×C22⋊D7—216no—
196C2×C22⋊D7—84no—
197C2×C22⋊D7—84no—
198C22×Dic7—22no—
199C2×D28—28no—
200C2×D28—28no—
201C2×D28—22no—
202C2×D28—22no—
203C2×D28—22no—
204C2×D28—22no—
205C2×D28—24no—
206C2×D28—22no—
207C2×D28—42no—
208C2×D28—44no—
209C2×D28—42no—
210C2×D28—44no—
211C2×D28C2×D414yes—
212C7×C4⋊C4—28no—
213C7×C4⋊C4—24no—
214C7×C4⋊C4—22no—
215C7×C4⋊C4C2×D414yes—
216C7×C22⋊C4—28no—
217C7×C22⋊C4—28no—
218C22⋊Dic7—24no—
219#1143—216no—
220#1141—216no—
221(C2×D7)⋊C4—216no—
222C4⋊Dic7—22no—
223C4⋊Dic7C2×D414yes—
224C4×Dic7—24no—
225#1136—28no—
226C2×C7⋊C8—22no—
227C2×C7⋊C8—22no—
228C2×C7⋊C8—22no—
229C2×C7⋊C8—22no—
230C2×C7⋊C8—22no—
231C2×C7⋊C8—44no—
232C2×C7⋊C8C2×D414yes—
233C22×#41C2×D711yescharacteristic
234C22×#41C2×D711yescharacteristic
235C23×D4—71no—
236C2×AGL(1,ℤ/8)—144no—
237C2×AGL(1,ℤ/8)—144no—
238C2×AGL(1,ℤ/8)—144no—
239C22×SD16—72no—
240C22×D8—72no—
241C22×M16—71no—
242C23×C8—71no—
243#537—24no—
244#537—24no—
245#535—144no—
246#535—144no—
247C2×#127—141no—
248C2×#124—141no—
249C2×C22⋊Q8—21no—
250C2×C22⋊Q8C2×D712yescomplemented
251C2×#121—21no—
252C2×#121—22no—
253C2×#121—74no—
254C2×#121C2×D712yescomplemented
255C2×#120—142no—
256C2×#117—71no—
257C22×C4⋊C4C2×D711yescharacteristic
258#416—144no—
259C2×#103—72no—
260C2×#102—72no—
261C2×#102—74no—
262C2×#102—144no—
263#408—144no—
264C2×C22⋊C8—74no—
265#340—74no—
266C23×C7—22no—
267C23×C7—22no—
268C23×C7—22no—
269C23×C7—22no—
270C23×C7—22no—
271C23×C7—22no—
272C23×C7—24no—
273C23×C7—24no—
274C23×C7—41no—
275C23×C7—41no—
276C23×C7—42no—
277C23×C7—42no—
278C23×C7#12012yes—
279C23×C7#12012yes—
280C23×C7#12112yes—
281C23×C7C22×D411yescharacteristic
282C22×D7—24no—
283C22×D7—42no—
284C22×D7—42no—
285C22×D7—44no—
286C22×D7—44no—
287C22×D7—48no—
288C22×D7—88no—
289C22×D7—84no—
290C22×D7—84no—
291C22×D7—88no—
292C7×Q8—22no—
293C7×Q8—22no—
294C7×Q8—41no—
295C7×Q8—41no—
296C7×Q8—44no—
297C7×D4—22no—
298C7×D4—22no—
299C7×D4—48no—
300C7×D4—48no—
301C7×D4—41no—
302C7×D4—41no—
303C7×D4—44no—
304C7×D4—88no—
305C2×C4×C7—22no—
306C2×C4×C7—24no—
307C2×C4×C7—22no—
308C2×C4×C7—22no—
309C2×C4×C7—22no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
312C2×C4×C7—44no—
313C2×C4×C7—42no—
314C2×C4×C7—42no—
315C2×C4×C7—42no—
316C2×C4×C7—42no—
317C2×C4×C7—48no—
318C2×C4×C7—84no—
319C2×C4×C7#12014yes—
320C2×C4×C7#12012yes—
321C2×C4×C7#12012yes—
322C2×C4×C7#12114yes—
323C2×C4×C7#12114yes—
324C2×C4×C7#12112yes—
325C2×C4×C7#12714yes—
326C2×C4×C7C22×D411yescharacteristic
327C2×C4×C7C22×D411yescharacteristic
328C2×C4×C7C22×D411yescharacteristic
329C2×C4×C7C22×D411yescharacteristic
330C2×C4×C7C22×D411yescharacteristic
331C2×C4×C7C2×#4111yescharacteristic
332C22⋊D7—816no—
333C2×Dic7—24no—
334C2×Dic7—42no—
335C2×Dic7—42no—
336D28—24no—
337D28—24no—
338D28—42no—
339D28—42no—
340D28—48no—
341C7⋊C8—24no—
342C7⋊C8—42no—
343C25—141no—
344C2×#41—22no—
345C2×#41—22no—
346C2×#41—22no—
347C2×#41—22no—
348C2×#41—22no—
349C2×#41—22no—
350C2×#41—22no—
351C2×#41—22no—
352C2×#41—22no—
353C2×#41—22no—
354C2×#41—22no—
355C2×#41C22⋊D714yescomplemented
356C22×Q8C22⋊D711yescharacteristic
357C22×Q8C22×D711yescharacteristic
358C22×D4—21no—
359C22×D4—22no—
360C22×D4—74no—
361C22×D4—72no—
362C22×D4—144no—
363C22×D4—142no—
364C22×D4—142no—
365C22×D4—142no—
366C22×D4—142no—
367C22×D4—142no—
368C22×D4—141no—
369C22×D4C22⋊D712yes—
370C22×D4C22⋊D711yescharacteristic
371C22×D4C22×D711yescharacteristic
372C23×C4—21no—
373C23×C4—21no—
374C23×C4C22×D712yes—
375C23×C4C22×D711yescharacteristic
376AGL(1,ℤ/8)—1432no—
377C2×SD16—144no—
378C2×SD16—144no—
379C2×SD16—144no—
380C2×D8—144no—
381C2×D8—144no—
382C2×D8—144no—
383C2×M16—74no—
384C2×M16—142no—
385C2×M16—142no—
386C22×C8—72no—
387C22×C8—72no—
388C22×C8—142no—
389C22×C8—142no—
390C22×C8—142no—
391#127—148no—
392#124—148no—
393C22⋊Q8—28no—
394C22⋊Q8—24no—
395C22⋊Q8—24no—
396#121—216no—
397#121—28no—
398#121—24no—
399#121—24no—
400#121—148no—
401#121—148no—
402#120—1416no—
403#117—74no—
404#117—142no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—22no—
407C2×C4⋊C4—22no—
408C2×C4⋊C4—72no—
409C2×C4⋊C4C22⋊D714yescomplemented
410C2×C4⋊C4C22×D712yes—
411C2×C22⋊C4—22no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—141no—
414C2×C22⋊C4—144no—
415C2×C42—141no—
416#103—144no—
417#102—144no—
418#102—148no—
419C22⋊C8—148no—
420C22×C7—22no—
421C22×C7—22no—
422C22×C7—24no—
423C22×C7—24no—
424C22×C7—42no—
425C22×C7—42no—
426C22×C7—42no—
427C22×C7—42no—
428C22×C7—42no—
429C22×C7—42no—
430C22×C7—44no—
431C22×C7—44no—
432C22×C7—82no—
433C22×C7—82no—
434C22×C7#39214yes—
435C22×C7C2×#12011yescharacteristic
436C22×C7C2×#12011yescharacteristic
437C22×C7C2×#12111yescharacteristic
438C22×C7#57612yes—
439C22×C7#57712yes—
440C2×D7—44no—
441C2×D7—48no—
442C2×D7—816no—
443C2×D7—82no—
444C2×D7—82no—
445C2×D7—84no—
446C4×C7—42no—
447C4×C7—42no—
448C4×C7—44no—
449C4×C7#39214yes—
450C4×C7C2×#12011yescharacteristic
451C4×C7C2×#12111yescharacteristic
452C4×C7C2×#12111yescharacteristic
453C4×C7C2×#12711yescharacteristic
454Dic7—44no—
455C24—21no—
456C24—142no—
457C24—144no—
458C24—282no—
459C24—282no—
460C24—562no—
461C24—562no—
462C24C2×C22⋊D712yes—
463C24C2×C22⋊D711yescharacteristic
464#41—28no—
465#41—28no—
466#41—44no—
467#41—44no—
468#41—44no—
469#41—44no—
470#41—44no—
471#41—44no—
472C2×Q8—22no—
473C2×Q8—22no—
474C2×Q8—22no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8C2×C22⋊D712yes—
481C2×D4—24no—
482C2×D4—24no—
483C2×D4—24no—
484C2×D4—24no—
485C2×D4—28no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—42no—
495C2×D4—42no—
496C2×D4—74no—
497C2×D4—84no—
498C2×D4—1416no—
499C2×D4—148no—
500C2×D4—144no—
501C2×D4—148no—
502C2×D4—142no—
503C2×D4—142no—
504C2×D4—142no—
505C2×D4—142no—
506C2×D4—142no—
507C2×D4—284no—
508C2×D4—282no—
509C2×D4—284no—
510C2×D4—282no—
511C2×D4—564no—
512C2×D4—564no—
513C2×D4C2×C22⋊D712yes—
514C22×C4—22no—
515C22×C4—21no—
516C22×C4—24no—
517C22×C4—24no—
518C22×C4—22no—
519C22×C4—22no—
520C22×C4—22no—
521C22×C4—22no—
522C22×C4—22no—
523C22×C4—22no—
524C22×C4—42no—
525C22×C4—42no—
526C22×C4—82no—
527C22×C4—142no—
528C22×C4D4×D712yes—
529C22×C4D4×D712yes—
530C22×C4D4×D712yes—
531C22×C4#115912yes—
532C22×C4C2×C22⋊D712yes—
533C22×C4C2×C22⋊D711yescharacteristic
534C22×C4C2×C22⋊D712yes—
535C22×C4C2×C22⋊D712yes—
536C22×C4C23×D711yescharacteristic
537SD16—1416no—
538D8—1416no—
539M16—148no—
540C2×C8—74no—
541C2×C8—142no—
542C2×C8—142no—
543C2×C8—142no—
544C2×C8—142no—
545C2×C8—142no—
546C2×C8—284no—
547C4⋊C4—28no—
548C4⋊C4—24no—
549C4⋊C4—22no—
550C4⋊C4—74no—
551C4⋊C4—142no—
552C4⋊C4C2×C22⋊D714yes—
553C22⋊C4—28no—
554C22⋊C4—28no—
555C22⋊C4—1416no—
556C22⋊C4—144no—
557C42—144no—
558C2×C7—24no—
559C2×C7—82no—
560C2×C7—82no—
561C2×C7C2×#39211yescharacteristic
562C2×C7#300811yescharacteristic
563C2×C7#301011yescharacteristic
564C2×C7#301212yes—
565C2×C7#306212yes—
566D7—84no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—24no—
572C23—24no—
573C23—22no—
574C23—22no—
575C23—41no—
576C23—41no—
577C23—42no—
578C23—42no—
579C23—144no—
580C23—282no—
581C23—288no—
582C23—284no—
583C23—284no—
584C23—282no—
585C23—568no—
586C23—564no—
587C23—564no—
588C23—568no—
589C23#650012yes—
590C23#650212yes—
591C23#651612yes—
592C23C22×C22⋊D711yescharacteristic
593Q8—22no—
594Q8—22no—
595Q8—41no—
596Q8—41no—
597Q8—44no—
598D4—22no—
599D4—22no—
600D4—48no—
601D4—48no—
602D4—41no—
603D4—41no—
604D4—44no—
605D4—88no—
606D4—144no—
607D4—144no—
608D4—282no—
609D4—282no—
610D4—288no—
611D4—5616no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—48no—
625C2×C4—84no—
626C2×C4—144no—
627C2×C4—282no—
628C2×C4—282no—
629C2×C4#650014yes—
630C2×C4#650012yes—
631C2×C4#650114yes—
632C2×C4#650214yes—
633C2×C4#650212yes—
634C2×C4#650314yes—
635C2×C4#651612yes—
636C2×C4C2×D4×D711yescharacteristic
637C2×C4C2×D4×D711yescharacteristic
638C2×C4C2×D4×D711yescharacteristic
639C2×C4C2×#115911yescharacteristic
640C2×C4C22×C22⋊D711yescharacteristic
641C2×C4C22×C22⋊D711yescharacteristic
642C8—144no—
643C8—282no—
644C7#2339311yescharacteristic, complemented, Sylow
645C22—22no—
646C22—22no—
647C22—24no—
648C22—24no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—44no—
656C22—44no—
657C22—82no—
658C22—82no—
659C22—284no—
660C22—288no—
661C22—5616no—
662C22—562no—
663C22—562no—
664C22—564no—
665C22#8999514yes—
666C22C2×#650011yescharacteristic
667C22C2×#650211yescharacteristic
668C22#9052612yes—
669C22#9052812yes—
670C22C2×#651611yescharacteristic
671C4—42no—
672C4—42no—
673C4—44no—
674C4—284no—
675C4#8999514yes—
676C4C2×#650011yescharacteristic
677C4C2×#650111yescharacteristic
678C4C2×#650211yescharacteristic
679C4C2×#650311yescharacteristic
680C2—24no—
681C2—82no—
682C2—82no—
683C2—564no—
684C2#1174086712yes—
685C2#1174086912yes—
686C2C2×#8999511yescharacteristic
687C2#1174154611yescharacteristic
688C2#1174155811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#537C2
socleC23×C7C22×D4
hypercenterC2×#537D7
hypocenterC7#23393
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. C2×C7×#537C2semidirect
2. C2×#90521C2semidirect
3. C22×#6512C2semidirect
4. C2×#90006C2semidirect
5. C2×#90005C2semidirect
6. C22×#6383C2semidirect
7. #11729453C2semidirect
8. #11729438C2semidirect
9. #11729312C2semidirect
16. C2×C7×C22⋊Q8C22semidirect
19. C2×C7×#121C22semidirect
37. C2×#6409C22semidirect
38. C2×#6406C22semidirect
39. C2×C22⋊(C7⋊C8)C22semidirect
42. #89324C22semidirect
56. C2×C7×#41D4semidirect
75. C22×D28D4semidirect
87. C2×C7×C4⋊C4D4semidirect
106. #6489D4semidirect
111. C2×#1136D4semidirect
122. C2×#537D7semidirect
250. C2×C22⋊Q8C2×D7semidirect
254. C2×#121C2×D7semidirect
355. C2×#41C22⋊D7semidirect
409. C2×C4⋊C4C22⋊D7semidirect
644. C7#23393semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5372
C2×#905212
C22×#65122
C2×#900062
C2×#900052
C22×#63832
#117294532
#117294382
#117293122
#233937