Finite Group Databasebeta

← #13355186

Subgroups — #13355186

Order 1792.

2046 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6495C211yesmaximal, characteristic
2C2×C4×#1157C211yesmaximal, characteristic, complemented
3C2×C7×#390C211yesmaximal, characteristic, complemented
4C2×#89987C211yesmaximal, characteristic, complemented
5C2×#89738C211yesmaximal, characteristic, complemented
6#11729312C211yesmaximal, characteristic, complemented
7#11729281C211yesmaximal, characteristic, complemented
8#11727414C214yesmaximal, complemented
9#11727411C214yesmaximal, complemented
10C23×C7×D4C2211yescharacteristic
11C22×#1157C2211yescharacteristic
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C7×C22⋊C4—22no—
14C22×C42×C7C2211yescharacteristic
15C22×#1142C2212yes—
16C2×#6495—24no—
17C2×#6495—24no—
18C2×#6495C418yescomplemented
19C22×#1141C2212yes—
20C2×C4×C22⋊D7—22no—
21C2×#6489C2211yescharacteristic
22C22×#1136C2211yescharacteristic
23C22×#1136C2211yescharacteristic
24C4×#1157—24no—
25C4×#1157—24no—
26C2×C4×D28C2212yescomplemented
27C2×#6435—21no—
28C2×C42×D7—21no—
29C2×C4×Dic14C2212yescomplemented
30C7×#390—21no—
31C7×#390—21no—
32C7×#390C2214yes—
33#89987—24no—
34C2×#6406C2214yes—
35C2×#6384C2212yescomplemented
36#89738—24no—
37C2×#6383C2212yescomplemented
38#89324C2214yescomplemented
39#89319C2214yescomplemented
40#20928—71nomaximal, Sylow
41C25×C7—21no—
42C22×C7×D4—22no—
43C22×C7×D4—22no—
44C22×C7×D4—22no—
45C22×C7×D4—21no—
46C22×C7×D4—22no—
47C22×C7×D4—22no—
48C22×C7×D4C2×C414yescomplemented
49C22×C7×D4C2312yes—
50C23×C4×C7—21no—
51C23×C4×C7D412yes—
52C23×C4×C7C2311yescharacteristic
53C22×C22⋊D7—21no—
54C2×#1157—22no—
55C2×#1157—22no—
56C2×#1157—22no—
57C2×#1157C2×C414yes—
58C2×#1157D414yes—
59C2×#1157D414yes—
60C22×D28C2311yescharacteristic
61C22×C4×D7—21no—
62C22×C4×D7—22no—
63C22×Dic14C2311yescharacteristic
64C2×C7×C4⋊C4—22no—
65C2×C7×C4⋊C4—22no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4C2312yes—
68C2×C7×C22⋊C4—24no—
69C2×C7×C22⋊C4—24no—
70C2×C7×C22⋊C4—48no—
71C2×C42×C7—22no—
72C2×C42×C7—22no—
73C2×C42×C7D411yescharacteristic
74C2×C42×C7D411yescharacteristic
75C2×C42×C7C2312yes—
76C2×C22⋊Dic7—21no—
77C2×#1142—24no—
78C2×#1142—24no—
79C2×#1142C2×C418yescomplemented
80#6495—232no—
81C2×#1141—24no—
82C2×#1141—24no—
83C2×#1141C2×C418yescomplemented
84C4×C22⋊D7—216no—
85C2×(C2×D7)⋊C4—22no—
86#6489—22no—
87#6489D414yes—
88C2×C4⋊Dic7C2312yes—
89C2×Dic7⋊C4—22no—
90C2×C4×Dic7—22no—
91C2×C4×Dic7—21no—
92C2×#1136—22no—
93C2×#1136—22no—
94C2×#1136—22no—
95C2×#1136—22no—
96C2×#1136—22no—
97C2×#1136C2×C414yescomplemented
98C22×C7⋊C8C2312yes—
99C22×C7⋊C8C2312yes—
100C4×D28—24no—
101C4×D28—24no—
102#6435—28no—
103C42×D7—28no—
104C4×Dic14—24no—
105C4×Dic14—24no—
106#6406—28no—
107#6384—24no—
108#6383—24no—
109C22×AGL(1,ℤ/8)—71no—
110C2×C4×#41—71no—
111C2×#417—71no—
112C2×#418—71no—
113C2×#390D711yescharacteristic, complemented
114#1880—74no—
115#1879—74no—
116#1736—71no—
117#1728—71no—
118C24×C7—24no—
119C24×C7—22no—
120C24×C7—44no—
121C24×C7—44no—
122C24×C7—42no—
123C24×C7—42no—
124C24×C7C2×D411yescharacteristic
125C23×D7—21no—
126C2×C7×D4—28no—
127C2×C7×D4—28no—
128C2×C7×D4—24no—
129C2×C7×D4—22no—
130C2×C7×D4—22no—
131C2×C7×D4—22no—
132C2×C7×D4—22no—
133C2×C7×D4—22no—
134C2×C7×D4—42no—
135C2×C7×D4—44no—
136C2×C7×D4—42no—
137C2×C7×D4—44no—
138C2×C7×D4C22×C414yes—
139C22×C4×C7—24no—
140C22×C4×C7—24no—
141C22×C4×C7—22no—
142C22×C4×C7—42no—
143C22×C4×C7—44no—
144C22×C4×C7—42no—
145C22×C4×C7C22×C412yes—
146C22×C4×C7C2×D412yes—
147C22×C4×C7C2×D412yes—
148C22×C4×C7C2×D412yes—
149C22×C4×C7C2×D412yes—
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D412yes—
152C22×C4×C7#4112yes—
153C22×C4×C7#4112yes—
154C22×C4×C7C2411yescharacteristic
155C2×C22⋊D7—24no—
156C2×C22⋊D7—24no—
157C2×C22⋊D7—24no—
158C22×Dic7—21no—
159C22×Dic7—22no—
160#1157—28no—
161#1157—28no—
162#1157—28no—
163#1157C22⋊C4116yescomplemented
164C2×D28—21no—
165C2×D28—21no—
166C2×D28—21no—
167C2×D28C22×C412yes—
168C2×D28C2×D412yes—
169C2×D28C2×D412yes—
170C2×C4×D7—24no—
171C2×C4×D7—24no—
172C2×C4×D7—24no—
173C2×C4×D7—28no—
174C2×C4×D7—44no—
175C2×C4×D7—44no—
176C2×Dic14—21no—
177C2×Dic14—21no—
178C2×Dic14—21no—
179C2×Dic14C22×C412yes—
180C2×Dic14C2×D412yes—
181C2×Dic14C2×D412yes—
182C7×C4⋊C4—22no—
183C7×C4⋊C4—22no—
184C7×C4⋊C4—42no—
185C7×C22⋊C4—48no—
186C7×C22⋊C4—48no—
187C42×C7—22no—
188C42×C7—22no—
189C42×C7—22no—
190C42×C7—22no—
191C22⋊Dic7—24no—
192#1142—216no—
193#1141—216no—
194(C2×D7)⋊C4—28no—
195C4⋊Dic7—22no—
196C4⋊Dic7C2×D414yes—
197Dic7⋊C4—28no—
198C4×Dic7—28no—
199C4×Dic7—24no—
200#1136—28no—
201#1136—44no—
202C2×C7⋊C8—22no—
203C2×C7⋊C8—22no—
204C2×C7⋊C8—22no—
205C2×C7⋊C8—22no—
206C2×C7⋊C8—22no—
207C2×C7⋊C8C22×C414yes—
208C22×#41—71no—
209C23×D4C2×D711yescharacteristic
210C2×AGL(1,ℤ/8)—78no—
211C2×AGL(1,ℤ/8)—144no—
212C2×AGL(1,ℤ/8)—144no—
213C22×SD16—72no—
214C22×D8—72no—
215C22×M16—71no—
216C22×M16—71no—
217C4×#41—144no—
218C4×#41—144no—
219C2×C4×Q8—72no—
220C2×C4×D4—72no—
221C2×C4×D4—142no—
222C2×#117—71no—
223C2×#117—141no—
224C22×C4⋊C4C2×D711yescharacteristic
225C22×C22⋊C4—22no—
226C22×C42—141no—
227C22×C42C2×D711yescharacteristic
228#418—144no—
229#417—144no—
230C2×#103—72no—
231C2×#102—74no—
232C2×#102—72no—
233#390—21no—
234#390—21no—
235#390C2×D714yes—
236#340—74no—
237#336—74no—
238C23×C7—22no—
239C23×C7—22no—
240C23×C7—24no—
241C23×C7—416no—
242C23×C7—42no—
243C23×C7—42no—
244C23×C7—44no—
245C23×C7—44no—
246C23×C7—48no—
247C23×C7—88no—
248C23×C7—84no—
249C23×C7—84no—
250C23×C7C4×D412yes—
251C23×C7#12012yes—
252C23×C7#12112yes—
253C23×C7C22×D411yescharacteristic
254C22×D7—22no—
255C22×D7—22no—
256C22×D7—22no—
257C22×D7—42no—
258C22×D7—42no—
259C7×D4—24no—
260C7×D4—24no—
261C7×D4—42no—
262C7×D4—42no—
263C7×D4—48no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—48no—
270C2×C4×C7—42no—
271C2×C4×C7—44no—
272C2×C4×C7—84no—
273C2×C4×C7C2×C22⋊C414yes—
274C2×C4×C7C4×D414yes—
275C2×C4×C7C4×D412yes—
276C2×C4×C7#12012yes—
277C2×C4×C7#12114yes—
278C2×C4×C7#12112yes—
279C2×C4×C7#12314yes—
280C2×C4×C7AGL(1,ℤ/8)14yescomplemented
281C2×C4×C7C23×C411yescharacteristic
282C2×C4×C7C22×D411yescharacteristic
283C2×C4×C7C22×D411yescharacteristic
284C2×C4×C7C22×D411yescharacteristic
285C2×C4×C7C2×#4111yescharacteristic
286C2×C4×C7C2×#4111yescharacteristic
287C22⋊D7—216no—
288C2×Dic7—22no—
289C2×Dic7—22no—
290C2×Dic7—22no—
291C2×Dic7—24no—
292C2×Dic7—42no—
293C2×Dic7—42no—
294D28—22no—
295D28—22no—
296D28—22no—
297D28C2×C22⋊C414yescomplemented
298C4×D7—216no—
299C4×D7—416no—
300Dic14—22no—
301Dic14—22no—
302Dic14—22no—
303Dic14C2×C22⋊C414yescomplemented
304C7⋊C8—24no—
305C7⋊C8—42no—
306C25—21no—
307C2×#41—74no—
308C2×#41—74no—
309C2×#41—74no—
310C2×#41—142no—
311C2×#41—142no—
312C2×#41—142no—
313C22×Q8—71no—
314C22×D4—22no—
315C22×D4—22no—
316C22×D4—22no—
317C22×D4—22no—
318C22×D4—22no—
319C22×D4—21no—
320C22×D4—71no—
321C22×D4—141no—
322C22×D4C4×D714yescomplemented
323C22×D4C22×D712yes—
324C23×C4—21no—
325C23×C4—141no—
326C23×C4—142no—
327C23×C4C22⋊D712yes—
328C23×C4C22×D711yescharacteristic
329AGL(1,ℤ/8)—1432no—
330C2×SD16—78no—
331C2×SD16—144no—
332C2×SD16—144no—
333C2×D8—78no—
334C2×D8—144no—
335C2×D8—144no—
336C2×M16—74no—
337C2×M16—142no—
338C2×M16—142no—
339C2×M16—142no—
340C2×M16—142no—
341C2×M16—142no—
342C22×C8—72no—
343C22×C8—72no—
344C4×Q8—144no—
345C4×Q8—144no—
346C4×D4—144no—
347C4×D4—144no—
348C4×D4—1416no—
349#117—74no—
350#117—148no—
351#117—142no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—72no—
356C2×C4⋊C4—142no—
357C2×C4⋊C4C22×D712yes—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—48no—
361C2×C22⋊C4—141no—
362C2×C22⋊C4—142no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—148no—
366C2×C42—141no—
367C2×C42—142no—
368C2×C42C22⋊D711yescharacteristic
369C2×C42C22⋊D711yescharacteristic
370C2×C42C22×D712yes—
371#103—144no—
372#102—144no—
373#102—148no—
374C22×C7—24no—
375C22×C7—24no—
376C22×C7—22no—
377C22×C7—22no—
378C22×C7—48no—
379C22×C7—44no—
380C22×C7—816no—
381C22×C7—84no—
382C22×C7—82no—
383C22×C7—82no—
384C22×C7#38614yes—
385C22×C7C2×C4×D411yescharacteristic
386C22×C7C2×#12011yescharacteristic
387C22×C7C2×#12111yescharacteristic
388C22×C7C2×AGL(1,ℤ/8)12yes—
389C22×C7#57512yes—
390C2×D7—24no—
391C2×D7—44no—
392C2×D7—44no—
393C2×D7—42no—
394C2×D7—42no—
395C4×C7—24no—
396C4×C7—42no—
397C4×C7—82no—
398C4×C7#38614yes—
399C4×C7C22×C22⋊C411yescharacteristic
400C4×C7C2×C4×D411yescharacteristic
401C4×C7C2×#12111yescharacteristic
402C4×C7C2×#12311yescharacteristic
403Dic7—24no—
404Dic7—44no—
405C24—24no—
406C24—22no—
407C24—44no—
408C24—44no—
409C24—42no—
410C24—42no—
411C24—141no—
412C24C2×C22⋊D711yescharacteristic
413#41—716no—
414#41—148no—
415#41—148no—
416#41—148no—
417C2×Q8—72no—
418C2×Q8—72no—
419C2×Q8—72no—
420C2×Q8—141no—
421C2×Q8—141no—
422C2×Q8—141no—
423C2×D4—28no—
424C2×D4—28no—
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—72no—
436C2×D4—72no—
437C2×D4—72no—
438C2×D4—144no—
439C2×D4—144no—
440C2×D4—144no—
441C2×D4—141no—
442C2×D4—141no—
443C2×D4—141no—
444C2×D4C2×C4×D714yes—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—148no—
452C22×C4—141no—
453C22×C4—144no—
454C22×C4—144no—
455C22×C4—144no—
456C22×C4—142no—
457C22×C4—284no—
458C22×C4—284no—
459C22×C4C2×C4×D712yes—
460C22×C4#115712yes—
461C22×C4D4×D712yes—
462C22×C4D4×D712yes—
463C22×C4D4×D712yes—
464C22×C4#115912yes—
465C22×C4C2×C22⋊D712yes—
466C22×C4C2×C22⋊D712yes—
467C22×C4C2×C22⋊D711yescharacteristic
468C22×C4C23×D711yescharacteristic
469SD16—1416no—
470D8—1416no—
471M16—148no—
472M16—284no—
473C2×C8—74no—
474C2×C8—142no—
475C2×C8—142no—
476C2×C8—142no—
477C2×C8—142no—
478C2×C8—142no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—42no—
482C4⋊C4—74no—
483C4⋊C4—148no—
484C4⋊C4—142no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—148no—
488C22⋊C4—144no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—148no—
494C42—144no—
495C2×C7—24no—
496C2×C7—84no—
497C2×C7C2×#38611yescharacteristic
498C2×C7C4×AGL(1,ℤ/8)12yes—
499C2×C7#300612yes—
500C2×C7#303811yescharacteristic
501C2×C7#304011yescharacteristic
502D7—44no—
503C23—22no—
504C23—22no—
505C23—24no—
506C23—416no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23—48no—
511C23—44no—
512C23—88no—
513C23—84no—
514C23—84no—
515C23—142no—
516C23—142no—
517C23—142no—
518C23—282no—
519C23—282no—
520C23C4×C22⋊D712yes—
521C23#650012yes—
522C23#650112yes—
523C23C22×C22⋊D711yescharacteristic
524Q8—74no—
525Q8—142no—
526Q8—142no—
527Q8—142no—
528D4—24no—
529D4—24no—
530D4—42no—
531D4—42no—
532D4—48no—
533D4—74no—
534D4—1416no—
535D4—142no—
536D4—142no—
537D4—142no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—48no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—84no—
547C2×C4—142no—
548C2×C4—142no—
549C2×C4—142no—
550C2×C4—1416no—
551C2×C4—144no—
552C2×C4—2816no—
553C2×C4—282no—
554C2×C4—282no—
555C2×C4D7×C22⋊C414yes—
556C2×C4(C22⋊D7)⋊C414yes—
557C2×C4#644614yes—
558C2×C4#644714yes—
559C2×C4C4×C22⋊D712yes—
560C2×C4#649514yescomplemented
561C2×C4#650012yes—
562C2×C4#650112yes—
563C2×C4C22×C4×D711yescharacteristic
564C2×C4C2×#115711yescharacteristic
565C2×C4C2×D4×D711yescharacteristic
566C2×C4C2×D4×D711yescharacteristic
567C2×C4C2×D4×D711yescharacteristic
568C2×C4C2×#115911yescharacteristic
569C8—144no—
570C8—282no—
571C7#2092811yescharacteristic, complemented, Sylow
572C22—24no—
573C22—24no—
574C22—22no—
575C22—22no—
576C22—48no—
577C22—44no—
578C22—816no—
579C22—84no—
580C22—82no—
581C22—82no—
582C22—144no—
583C22—284no—
584C22—284no—
585C22—282no—
586C22—282no—
587C22#8972814yes—
588C22C2×C4×C22⋊D711yescharacteristic
589C22C2×#649512yes—
590C22C2×#650011yescharacteristic
591C22C2×#650111yescharacteristic
592C22#9051312yes—
593C4—24no—
594C4—42no—
595C4—82no—
596C4—144no—
597C4—284no—
598C4#8972814yes—
599C4C2×D7×C22⋊C411yescharacteristic
600C4C2×(C22⋊D7)⋊C411yescharacteristic
601C4C2×#644611yescharacteristic
602C4C2×#644711yescharacteristic
603C2—24no—
604C2—84no—
605C2—284no—
606C2C2×#8972811yescharacteristic
607C2C4×#649512yes—
608C2#1174006912yes—
609C2#1174039011yescharacteristic
610C2#1174061011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#390C2
socleC23×C7C22×D4
prime radicalC22×#6495C2
cosocleC22×C4×C7C24
hypercenterC2×#390D7
hypocenterC7#20928
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C2×C4×#1157C2semidirect
3. C2×C7×#390C2semidirect
4. C2×#89987C2semidirect
5. C2×#89738C2semidirect
6. #11729312C2semidirect
7. #11729281C2semidirect
8. #11727414C2semidirect
9. #11727411C2semidirect
18. C2×#6495C4semidirect
26. C2×C4×D28C22semidirect
29. C2×C4×Dic14C22semidirect
35. C2×#6384C22semidirect
37. C2×#6383C22semidirect
38. #89324C22semidirect
39. #89319C22semidirect
48. C22×C7×D4C2×C4semidirect
79. C2×#1142C2×C4semidirect
83. C2×#1141C2×C4semidirect
97. C2×#1136C2×C4semidirect
113. C2×#390D7semidirect
163. #1157C22⋊C4semidirect
280. C2×C4×C7AGL(1,ℤ/8)semidirect
297. D28C2×C22⋊C4semidirect
303. Dic14C2×C22⋊C4semidirect
322. C22×D4C4×D7semidirect
560. C2×C4#6495semidirect
571. C7#20928semidirect

Maximal subgroups

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

subgroupindex
C22×#64952
C2×C4×#11572
C2×C7×#3902
C2×#899872
C2×#897382
#117293122
#117292812
#117274142
#117274112
#209287